ADVANCES IN
APPLIED MECHANICS
VOLUME 7
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ADVANCES IN
APPLIED MECHANICS
Editors
TH. VON K ~ R M ~ N
H. L. DRYDEN
Managing Editor
G. KUERTI
Case Institute of Technology, Cleveland, Ohio
Associate Editors
F. H. VAN DEN DUNCEN
L. HOWARTH
VOLUME 7
1962
ACADEMIC PRESS
NEW YORK AND LONDON
COPYRIGHT
0 1982, B Y ACADEMIC P R E S S INC.
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ACADEMIC PRESS INC.
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LTD.
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S Q U A R E HOUSE,LONDON
W. 1
Library of Congress Catalog Card Number: 48-8503
PRINTED
IN THE UNITEDSTATES
OF AMERICA
CONTRIBUTORS
TO VOLUME
7
G. I. BARENBLATT,
Institute of Geology and Development of Combustible
Minerals of the U.S.S.R. Academy of Sciences, Moscow, U.S.S.R.I
RAYMOND
HIDE,Physics Department, King’s College (University of Durham),
Newcastle-upon-T yne, England2
HAROLDMIRELS, Lewis Research Center, National Aeronautics and Space
Administration, Cleveland, Ohios
W. OLSZAK,Institute of Fundamental Technical Problems, Polish Academy
of Sciences, Warsaw, Poland
PAULH. ROBERTS,Physics Department, King’s College (University of
Durham), Newcastle-upon-Tyne, England4
J. RYCHLEWSKI,
Institute of Fundamental Technical Problems, Polish Academy of Sciences, Warsaw, Poland
W. URBANOWSKI,
Institute of Fwdamental Technical Problems, Polish Academy of Sciences, Warsaw, Poland
l Present
address: Institute of Mechanics, Moscow State University, Moscow, USSR.
Department of Geology and Geophysics, Massachusetts Institute
of Technology, Cambridge, Massachusetts.
a Present address : Aerospace Corporation, El Segundo, California.
Present address: Yerkes Observatory (University of Chicago), Williams Bay,
Wisconsin.
a Present address:
V
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Preface
The seventh volume of Advances in Applied Mechanics includes two
extensive reviews of topics in solid mechanics and an account of recent
analytical results obtained in the field of hypersonic obstacle flow.
A detailed presentation of the basic physical principles and problems of
phenomenological magneto-hydrodynamics concludes this volume ; it may
serve as an introduction into this comparatively new branch of hydrodynamics.
THE EDITORS
July, 1962
Vii
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Contents
CONTRIBUTORS
TO VOLUME
7
.......................
v
vii
PREFACE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Hypersonic Flow o ~ e rSlender Bodies Associated with Power-Law Shocks
.
BY HAROLD
MIRELS.Lewis Research Center National Aeronautics and Space Administration. Cleveland. Ohio
I . Introduction . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 2
I1 Hypersonic Slender-Body Theory . . . . . . . . . . . . . . . . . . 4
I11. Flows Associated with Power-Law Shocks . . . . . . . . . . . . . . 8
IV . Flows Associated with Slightly Perturbed Power-Law Shocks . . . . . 26
V Integral Methods . . . . . . . . . . . . . . . . . . . . .
. . . . . 36
VI . Validity of Self-similar Solutions . . . . . . . . . . . . . . . . . . 43
VII Further Discussion of Integral Methods . . . . . . . . . . . . . . . 41
VIII Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . 49
. . . . . 51
Notation . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 52
References . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 311
Addendum . . . . . . . . . . . . . . . . . . . . . . . . . .
.
.
.
.
The Mathematical Theory of Equllibrium Cracks In Brittle Fracture
.
BY G. I BARENBLATT.
Institute of Geology and Development of Combustible Minerals of
the U.S.S.R. Academy of Sciences. Moscow. U.S.S.R.
.
.
.
.
I Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
I1 The Development of the Equilibrium Crack Theory . . . . . . . . . . 62
I11 The Structure of the Edge of an Equilibrium Crack in a Brittle Body
69
IV Basic Hypotheses and General Statement of the Problem of Equilibrium
Cracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
V . Special Problems in the Theory of Equilibrium Cracks . . . . . . . . 90
VI Wedging; Dynamic Problems in the Theory of Cracks . . . . . . . . . 114
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
125
.
.
Plasticity Under Non-Homogeneous Conditions
.
.
BY W OLSZAK.J RYCHLEWSKI
A N D W . URBANOWSKI. Institute Of Fundamentat
Technical Problems. Polish Academy of Sciences. Warsaw
.
.
.
.
I Physical Foundations . . . . . . . . . . . . . . . . . . . . . . . .
132
11 Plane Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . .
151
I11 Particular Solutions . . . . . . . . . . . . . . . . . . . . . . . . .
183
IV Elastic-plastic Non-homogeneous Plates . . . . . . . . . . . . . . . . 190
ix
CONTENTS
X
V . Limit Analysis and Limit Design . . . . . . . . . . . . . . . . . . 191
VI Propagation of Elastic-plastic Waves in a Non-homogeneous Medium . . 201
V I I . Other Problems . . . . . . . . . . . . . . . . . . . . . . . . . .
203
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
206
.
Some Elementary Problems in Magneto-hydrudynrtmics
BY RAYMOND
HIDE A N D PAULH . ROBERTS.Physics Department. King’s College
(University of Durham) Newcastle.upon.Tyne.
1. England
.
I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
216
11. Basic Equations of Magneto-hydrodynamics . . . . . . . . . . . . . . 219
111. Electromagnetic and. Mechanical Effects ; Dimensionless Parameters . . . 224
I V . Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . .
233
V . Plane Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . .
244
VI . Alfvbn Waves in Systems of Finite Extent . . . . . . . . . . . . . . 261
VII . Gravity Waves : Rayleigh-Taylor Instability . . . . . . . . . . . . . 267
V I I I . Gravitational Instability: Jeans’ Criterion . . . . . . . . . . . . . . 270
IX . Steady Flow between Parallel Planes . . . . . . . . . . . . . . . . 274
X . Flow due t o an Oscillating- Plane: Rayleigh’s Problem . . . . . . . . . 286
XI . Steady Two-dimensional Inertial Flow in the Presence of a Magnetic Field . 300
Appendix A : The Hydromagnetic Energy Equation . . . . . . . . . . . . 305
Appendix B : Relativistic Magneto-Hydrodynamics . . . . . . . . . . . . . 311
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
313
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
313
AUTHORINDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
321
SUBJECTINDEX. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
327
Hypersonic Flow over Slender Bodies
Associated with Power-Law Shocks
BY HAROLD MIRELS
Lewis Research Center+
National Aeronautics and Space Administration
Cleveland. Ohio
Page
I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
I1. Hypersonic Slender-Body Theory
. . . . . . . . . . . . . . . . . .
4
I11. Flows Associated with Power-Law Shocks . . . . . . . . . . . . . .
1. Basic Equations . . . . . . . . . . . . . . . . . . . . . . . . .
2. Alternative Formulations . . . . . . . . . . . . . . . . . . . . .
Stream-function formulation . . . . . . . . . . . . . . . . . .
Lagrangian formulation . . . . . . . . . . . . . . . . . . . .
Sedov formulation . . . . . . . . . . . . . . . . . . . . . .
3. Analytic Solutions . . . . . . . . . . . . . . . . . . . . . . . .
Blast wave . . . . . . . . . . . . . . . . . . . . . . . . . .
Newtonian theory . . . . . . . . . . . . . . . . . . . . . . .
“Sharp-blow’’ solution . . . . . . . . . . . . . . . . . . . . .
Approximate solutions . . . . . . . . . . . . . . . . . . . . .
4 Nature of the Flow . . . . . . . . . . . . . . . . . . . . . . .
8
8
13
13
14
16
16
16
18
19
21
23
.
.
IV Flows Associated with Slightly Perturbed Power-Law Shocks . . . . .
1 . Basic Equations . . . . . . . . . . . . . . . . . . . . . . . . .
2. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . .
Boundary-layer effects . . . . . . . . . . . . . . . . . . . . .
Angle-of-attack effects . . . . . . . . . . . . . . . . . . . . .
Effect of blunting the nose of very slender wedges and cones . .
3. Effect of
#0 . . . . . . . . . . . . . . . . . . . . . .
V. Integral Methods . . . . . . . . . . . . . . . . . . .
1. Continuity Integral . . . . . . . . . . . . . . . .
General case . . . . . . . . . . . . . . . . . .
Hypersonic slender body approximations . . . . .
Slender blunt-nosed bodies a t infinite Mach number
2. Momentum Integral . . . . . . . . . . . . . . . .
General case . . . . . . . . . . . . . . . . . .
Hypersonic slender body approximation . . . . . .
26
26
29
29
31
33
34
. . . . . . 36
. . . . . . 37
. . . . . .
37
. . . . . . 38
. . . . . . 39
. . . . . . 40
. . . . . . 40
. . . . . . 41
VI . Validity of Self-similar Solutions . . . . . . . . . . . . . . . . . . 43
1. Infinite Mach Number . . . . . . . . . . . . . . . . . . . . . .
43
2 . Finite Mach Number . . . . . . . . . . . . . . . . . . . . . . .
46
* Present Address: Aerospace Corporation. El Segundo. California
1
2
HAROLD MIRELS
. . . . . . . . . . . . . . .
V I I I . Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . .
Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Addendum.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
VII. Further Discussion of Integral Methods
References.
49
51
317
I. INTRODUCTION
The steady-state equations of motion for hypersonic flow over slender
bodies can be reduced to simpler form by incorporating the “hypersonicslender-body approximations” (e.g., Hayes [l] and Van Dyke [?]). The
reduced equations are valid provided S 2 < 1 and ( M 6 ) - 2 is not near one,
where M is the free.stream Mach number and S is a characteristic shock
slope. If the streamwise coordinate is considered as time, these reduced
equations are identical with the full (exact) equations for a corresponding
unsteady flow in one space variable less. Forebody drag on a hypersonic
slender body is equivalent to the net energy perturbation (from the undisturbed state) in the corresponding unsteady flow.
Taylor [3, 41 has treated the unsteady constant-energy flow field behind
the spherical “blast” wave which is generated when a finite amount of
energy is released instantaneously at a point. The analysis assumes a very
strong wave and is valid (for a perfect gas) until the decay of shock strength
is sufficient to violate the strong shock assumptions. The problem of planar,
cylindrical, and spherical blast waves was treated in a unified manner by
Sakurai [5,6] and the flow-field modifications associated with more moderate
shock strengths were found by a perturbation analysis. The solution for the
cylindrical blast wave was obtained, independently, by Lin [7]. References
[7] to [O] have pointed out that, within the framework of hypersonic slenderbody theory (in the limit (M6)-2--+0),the hypersonic flow over a bluntnosed flat plate, or circular cylinder, may be considered as the steady-state
analog of the constant-energy planar, or cylindrical, blast-wave problem,
respectively. The nose drag in the steady problem is equivalent to the
finite energy which is instantaneously released in the blast-wave problem.
The steady-flow solution is not correct near the nose (where the hypersonic
slender-body approximation S2 << 1 is violated) and far downstream of the
nose (where the strong-shock assumption, ( M 6 ) - 2-* 0, is violated). However,
useful results are expected for the intermediate region.
The blast-wave problems all exhibit flow similarity. That is, the flow
fields at different times are similar, except for a scale factor on both the
dependent and independent variables. Lees and Kubota [9,10] have observed
that such similarity exists for hypersonic flows whenever the shock shape
follows a power-law variation with the streamwise distance, provided the
HYPERSONIC FLOW OVER SLENDER BODIES
3
hypersonic slender-body equations are considered in the limit as (Mb)-2-p 0.
This led to numerical solutions of the hypersonic flow over slender powerlaw bodes. The effect of nonvanishing values of (MS)-2 was also found by
Kubota [lo] using a numerical perturbation analysis similar to that of
Sakurai [5, 61.
As noted by Hayes and Probstein [ll], all these authors were apparently
unaware of the extensive early work of various Russian authors. Unsteady
self-similar motions were studied in great generality by Sedov [12],
Stanyukovich [ 131, and others. The application to hypersonic flows was
developed by Grodzovskii [14], Chernyi [15, 161, and co-workers. A historical account of these developments is given in [ll].
More recently, Mirels [17, 181 has obtained additional numerical results
and approximate analytical solutions for hypersonic flow over slender
power-law bodies. In [18], the flow fields and bodies associated with slightly
perturbed power-law shocks were investigated. Studies of the singular
points of the equations of motion have been presented in [12, 19-21]. These
indicate the conditions under which streamlines through a power-law shock
terminate a t a limit line. Integral methods for finding the hypersonic flow
past arbitrary blunt nosed slender bodies have been developed by Chernyi
[22] and Cheng et al. [23]. The validity of the use of hypersonic slender
body theory for slender bodies with blunted noses has been discussed in
papers by Cheng [24] and Sychev 1251. The ionization of the flow field
associated with hypersonic blunt-nosed bodies has also been investigated
by Feldman 126, 271 in connection with communication with and detection
of bodies entering the earth’s atmosphere.
The purpose of the present paper is to give a unified discussion of the
application of hypersonic slender-body theory, in the limit (M8)-2 << 1,
for finding the self-similar flows associated with slender power-law shocks
and related bodies. (“Related bodies” refers to any body associated with a
power-law shock or a slightly perturbed power-law shock.) Integral methods
for finding the non-self-similar hypersonic flow over more general body
shapes are also discussed. The specific topics considered are as follows.
Chapter I1 outlines hypersonic slender-body theory which is valid for 6 8 << 1
and (MS)-2not near one. The resulting equations are applied in Chapter 111,
in the limit ( M S ) - 2 - 0,to find the self-similar flows associated with powerlaw shocks. Several alternative formulations of the equations of motion
are noted, some analytical solutions are given and the general features of
the flow field are discussed. In Chapter IV, the effects of slightly perturbing
a power-law shock are considered. Numerical results are presented which
permit the determination of the effects of thin boundary layers, small angles
of attack, nose blunting and small but nonvanishing values of (MS)-2.
Chapter V develops continuity and momentum-integral methods for finding
the hypersonic flow over arbitrarily shaped slender bodies. The continuity
4
HAROLD MIRELS
integral is applied in Chapter VI to examine the validity of the self-similar
solutions of Chapter 111. In particular, the violation of the hypersonic
slender body assumption of small shock slope d2 << 1 at the nose of a powerlaw shock is investigated. Criteria are developed which define the downstream influence of nose bluntness. Chapter VII simplifies the integral
methods for the case where y , the ratio of specific heats, is near 1 and most
of the drag occurs a t the nose of the body. In Chapter 1’111, the validity
of the blast wave solutions of Chapter I11 for describing the flow over bluntnosed cylinders is examined in more detail by comparison with exact solutions
obtained by the method of characteristics (e.g. [26, 281). The importance
of nose shape, which is neglected in blast wave theory, is demonstrated.
The present discussion is an elaboration and extension of [as]. Nondimensional variables are used in Chapters 11 to IV. Here dimensional
variables have a superscript bar. However, in Chapters V to VIII, dimensional variables are used exclusively, and the superscript bar is omitted.
The notation is summarized after Chapter VIII.
11. HYPERSONIC
SLENDER-BODY
THEORY
The equations of motion for hypersonic flow over slender bodies (e.g. [a])
are summarized herein. Two dimensional and axisymmetric flows are
considered with (Z,?)and (G,d) being the
streamwise and transverse coordinates and
velocities, respectively. The superscript
bar indicates a dimensional quantity.
The equations of motion for an inviscid,
non-conducting fluid are :
continuity:
apil
FIG.1. Flow across oblique shock.
--
(2.la)
ati
(2.lb)
x-momentum: 6 -
(d.lc)
r-momentum :
(2.ld)
az
energy : (c
a0
G-
ax
+ d- aa
+ 1 apaa = 0
a?
ag
1 ap
+da? + ar = 0,
Ty
a + c P)(k)
= 0,
ar
py
--
a p
+7
+ o re = 0,
HYPERSONIC FLOW OVER SLENDER BODIES
5
where a = 0 , 1 for two dimensional and axisymmetric flows, respectively.
Equation (2.ld) assumes a perfect gas and is equivalent to stating that
entropy is constant along a streamline (except for possible jumps across
shock waves). Boundary conditions across an oblique shock (Fig. 1) may
be expressed as
(2.2b)
(2.2c)
(2.2d)
where subscripts 00 and s represent conditions upstream and downstream
of the shock, respectively, M is the free-stream Mach number, and t is the
angle which the shock forms with the free-stream direction.
No assumptions regarding 6 and M are made in (2.1)and (2.2). Attention
is now restricted to flows for which d2 << 1 (where 6 is a characteristic shock
slope) and M 2 >> 1. The prod-L r,v
uct 6M can be interpreted as
the ratio of shock slope to Mach
angle, 1/M, and is always
greater than 1. Consider (Md)
not very close to one (i.e., exGPD
clude bodies which are sIender
0
L
relatil’e t o the Mach angle,
FIG.3. Physical quantities for study of hyperMdb << 1, and thus can be
sonic flow over slender bodies.
treated by linearized supersonic flow techniques). Eqs. (2.2) then suggest the following non-dimensional
dependent variables for studying hypersonic flow over slender bodies :
(2.3a)
p = fi/prnZimw,
p =p / p m ,
u = (a - G m ) / d m P ,
21 = 77/27,6.
These quantities remain of order 1 as a2+ 0 and M + 03 provided ( M 6 ) 2
is not close to one. The independent variables may be non-dimensionalized
according to
(2.3b)
x = %/L,
r =qs1,
where L is a characteristic streamwise length (Fig. 2).
6
HAROLD MIRELS
The equations of motion can now be written, after substituting (2.3)
into (2.1) and neglecting terms of order S2 compared with one,
(2.4a)
continuity:
(2.4b)
r-momentum:
(2.4~)
energy:
aPv
aP
Z+F+
=o,
a!?
r
1 ap
--=O,
p ar
av
av
-+
v-+
ax
a7
($+ v "(2)
ar
0.
py
=
Equations (2.4) can be solved independently of the x-momentum equation
which is therefore omitted. After solving (2.4) the value of u can be found
from Bernoulli's equation, which to the present order, becomes
If rb(x) and R ( x ) denote the non-dimensional body and shock ordinates,
respectively, the boundary conditions associated with (2.4) are :
(2.6a)
a t body surface:
vb = rb',
(2.6b)
upstream of shock: urn= vm = 0,
p m = 1,
p m = l/yM2d2,
on the downstream side of shock:
(2.6~)
(2.6d)
ps =
(2.6e)
21,
[
2
7
+
1+
YY-
=-
2
Y + l
I'
(M6R')-2
,
R'[1 - ( M S R ' ) - 2 ] .
Equations (2.3) t o (2.6) constitute the hypersonic small disturbance equations
as originally described by Van Dyke [2].
Hayes [ 13 has shown that the hypersonic small-disturbance equations
correspond to the full equations (no restrictions regarding 6 or M) for
unsteady flow in the transverse plane. This follows from the fact that the
streamwise velocity ti, remains unchanged, to order d2, in hypersonic
slender-body theory. The flow in the plane of an observer moving with
then appears as an unsteady transverse flow. Formally, this
velocity iim
corresponds to replacing ti by ti, and 2 by zZ,l in (2.la), (2.lc), and (2.ld)
HYPERSONIC FLOW OVER SLENDER BODIES
7
or to replacing x by t in the non-dimensional equations. The quantity 6M
can be eliminated from ( 2 . 6 ~to
) (2.6e) by noting a, G 1/M6. The quantity
R'la, is then the shock Mach number in the unsteady case. Equivalent
steady and unsteady non-dimensionalized flows are illustrated in Figs. 3
and 4. The unsteady flow in Fig. 4 is due to the expansion of a plane (a = 0)
or cylindrical (u = 1) piston bounded by two parallel walls.
v,=u,
'q p,= I
=o
FIG.3. Nondimensional hypersonic
flow over slender body.
FIG.4. Unsteady (piston driven) flow which
is equivalent t o steady flow of Fig. 3.
(a) Flow at t = 1,. (b) Piston and shock
displacement versus time.
The forebody drag on a slender body can be determined by a pressure
integration along the body surface or by considering the momentum flux
across an appropriate control surface. These give, respectively (employing
the hypersonic slender-body approximations)
x
=
where D ( x ) D ( f ) / p , ~ 7 ~ ~ 6 ~ ( L is
6 )the
" + ~forebody drag up to station x ,
D(0)is the finite drag addition (if any) a t x = 0 and p b is the surface pressure.
For u = 0, (2.7) gives the drag of a symmetric two-dimensional body (rather
than half this value). The integration in (2.7b) is conducted at constant x .
Forebody drag in a steady hypersonic flow corresponds to energy addition
in the equivalent unsteady flow. Thus, the integral in (2.7a) may be interpreted as the energy addition by a piston (note, t'b = d r b / d x ) and (2.7b)
may be interpreted as the net energy perturbation up till time t = x for
unsteady flows of the type indicated in Fig. 4. The derivation of (2.7) is
discussed in more detail in Section V.2.
8
HAROLD MIRELS
A single solution of the non-dimensional equations of motion (2.4)-(2.6)
for a given value of M S and of y and a given body shape corresponds to an
infinite number of physical flows. The physical flows are those having the
same values of MS and y and body shapes which are reduced by (2.3b) to
the non-dimensional body whose solution is known. This law of similitude
has been applied to the case of real gas flows (in particular, flows in local
thermodynamic equilibrium) by Cheng [24]. In [24], Cheng shows that
similitude exists (in terms of the non-dimensional quantities defined by (2.3))
for flows in local thermal equilibrium provided the free-stream atmosphere
(composition, pressure and density) is the same. The latter requirement is
replaced by the condition that y be constant in the case of ideal gas flow.
Additional discussion of the law of similitude for hypersonic flow over
slender bodies is given in [ll]. Except for the discussion of Chapter VIII,
the present paper is restricted to ideal gas flows with constant y .
111. FLOWS
ASSOCIATEDWITH POWER-LAW
SHOCKS
The hypersonic slender body equations of Chapter I1 are now used to
investigate the hypersonic flow field associated with slender power-law
shocks. The limiting case ( M c ? ) - ~ -0 (i.e., M - a)is considered. It is
shown that the motion is self-similar. Several formulations of the equations
of motion are given, some exact and approximate solutions are indicated,
and the general nature of the flow field is discussed.
1 . Basic Equations
In physical coordinates, the equation for a power-law shock is
(3.1)
R ( f )= C f " ,
where C and m are constants. To nondimensionalize, let 1 be the streamwise
length of interest and R ( L ) be the corresponding shock ordinate. The characteristic shock slope is then (Fig. 2)
(3.2)
L
Note that 6 and L are not independent and that the characteristic geometric
bL(l-"). Thus S and L always appear in the
parameters are m and C
combination SL('-") in the treatment of power-law shocks. I t is assumed
that S2 << 1 so that hypersonic slender body theory is applicable.*
* The assumption P<< 1 or (M6)-2 + 0 is violated at the nose of power law bodies
when vn # 1. The effect of this violation is discussed in Chapter VI.
HYPERSONIC FLOW OVER SLENDER BODIES
9
In nondimensional variables the shock shape is
R = xm.
(3.3)
New independent variables are introduced, namely
(3.4)
7 =rlxm,
f = x,
so that
-.a
(3.5)
av
The variable y is the ratio of Y to the local shock ordinate and equals 1 a t
the shock. Equations ( 2 . 6 ~ to
) (2.6e), with ( M 6 ) - 2 = 0, suggest that the
dependent variables be expressed as
(3.6)
p = ,2re(m - 1) F,
p=$,
v=rnfm-ly,
where F, I), and y are functions of 7 . Substitution into (2.4) then yields
(3.7a)
(a, -
(3.7b)
(3.7c)
(9-y)
;(
--y-
);
-(a+l)B=O.
The boundary conditions a t the shock, y = 1, (from (2.6) and (3.6)) are
~ ( 1=F(l)
)
=
2
~
y+l’
*(1) = y+l.
Y-1
The parameter p is here a function of m, namely
(3.9)
p E-a-+2 l [RR”/(R’)2]
= ___ ( 1 - 1).
a+l m
Equations (3.7) and (3.8) completely define the flow associated with a
power-law shock. Any streamline through the shock may be considered as
defining a body, and the flow between this streamline and the shock can be
found from a numerical integration of (3.7) and (3.8). Unless otherwise
stated, the body will be taken to be that corresponding to the streamline
through the origin (Fig. 5 ) . This body shape, for E > 0, is given by y = r / b or
(3.10)
Yb = qbEm,
10
HAROLD MIRELS
where 'qb is the value of q at which
fP=v
(3.11)
(so that (2.6a) is satisfied). Comparing (3.3) and (3.10) shows that the body
is similar to the shock and the ratio of their ordinates is q b . Both follow a
power-law variation with 6,
FIG.5. Power-law shock with body corresponding to streamline through origin
(0< p < 1).
The pressure distribution on the body can be found from (3.0) and (3.3) as
(3.12)
p b = (dR/dE)'Fb = (drb/dt)'Fbl'qb',
where Fb = F ( q b ) . The drag up to station 6 may be expressed (from (2.7a)
and (2.7b), respectively) as
where
1
(3.13~)
The quantity Ib = I(qb)is a function of IJ, y and p. Equations (3.13a) and
(3.13b) give the drag on the body defined by the streamline through the
origin. The limitation on in (3.13b) is due to the appearance of limit lines
for u = 1 and p > 1 as discussed in Section 111.4. For u = 1, (3.13b) applies
for a limited range of p greater than one (i.e. 1 < p PSB) as can be deduced
from the discussion in 111.4. The drag on the upper surface of a streamline,
up to the point (E,q), is given by
-=
(3.13d)
TABLE1. QUANTITIES
DEFINING
BODYSHAPEAND SURFACE
PRESSURE ASSOCIATEDWITH A POWER-LAW
SHOCK
(ZERO ORDERPROBLEM)
Y
P
Numerical solution
(Eqs. (3.7))
Approximate solution (Eqs. (3.41))
a=o
a=O
u=1
Approximate solution
(Eqs. (5.16))
a = l
a=O
a = l
_
a=O,l
_
Fb
rlb
%'b
Fb
0.947
0.774
0.682
0.608
0.531
0.453
0.410
0.930
0.886
0.843
0.789
0.698
0.514
0
0.964
0.941
0.918
0.888
0.835
0.717
0
0.930
0.764
0.680
0.618
0.555
0.493
0.430
~
qb
F b
rlb
F b
co
Do
0
1/3
1/2
5/8
3/4
7/8
L
0.930
0.891
0.852
0.803
0.716
0.535
O
0.930
0.761
0.675
0.611
0.546
0.481
0.41.5
0.965
0.945
0.924
0.898
0.845
0.735
0
0.948
0.775
0.688
0.621
0.553
0.484
0.411
0
2.93
5.18
7.16
8.52
6.16
0.465
1.255
1.535
1.800
2.13
2.68
3.80
7.67
0.930 0.930
0.891 0.766
0.852 0.672
0.801 0.598
0.710 0.520
0.513 0.442
0
0.412
-2.53
2.05
0.965
+1.636 1.317 0.945
3.64
1.642 0.924
5.53
1.981 0.897
7.11
2.53
0.846
0.724
5.68
3.62
0.465 7.13
0
1.4
0
1/3
1/2
5/8
3/4
7/8
1
0.833 0.833 0.915
0.759 0.666 0.875
0.695 0.581 0.839
0.623 0.518 0.796
0.513 0.454 0.725
0.333 0.390 0,589
0
0.325 0
0.875 0
0.696 0.962
0.607 1.240
0.538 1.296
0.467 1.167
0.392 0.817
0.311 0.417
1.556
1.678
1.779
1.888
2.04
2.30
3.50
0.833 0.833
0.760 0.679
0.695 0.584
0.619 0.504
0.499 0.415
0.284 0.320
0
0.316
-1.019
$0.383
0.757
0.930
0.949
0.757
0.417
2.11
1.228
1.485
1.631
1.798
2.03
2.92
0.915 0.872
0.875 0.704
0.839 0.611
0.795 0.529
0.719 0.438
0.561 0.337
0
0.302
0.833
0.755
0.688
0.613
0.501
0.322
0
0.913
0.869
0.829
0.783
0.708
0.567
0
0.833
0.667
0.583
0.521
0.458
0.396
0.333
1.67
0
1/3
1/2
5/8
314
0.749 0.749 0.870
0.658 0.587 0.819
0.585 0.507 0.776
0.509 0.446 0.727
0.404 0.386 0.652
0.248 0.326 0.518
0
0.264 0
0.811 0
0.634 0.535
0.544 0.644
0.474 0.655
0.403 0.608
0.326 0.504
0.241 0.375
1.788 0.749 0.749
1.788 0.660 0.605
1.770 0.586 0.512
1.762 0.505 0.432
1.752 0.385 0.340
1.738 0.186 0.239
2.49
0
0.250
-0.625
+0.170
0.365
0.452
0.483
0.459
0.375
2.16
1.161
1.405
1.450
1.457
1.436
1.867
0.870
0.819
0.776
0.726
0.644
0.480
0
0.749
0.660
0.588
0.513
0.408
0.252
0
0.865
0.812
0.767
0.717
0.639
0.502
0
0.749
0.582
0.499
0.437
0.314
0.312
0.249
1.15
7/8
1
Reference
~ 7 1
POI
%'b
~1
F b
co
Da
Tlb
~ 7 1
0.805
0.645
0.554
0.469
0.370
0.257
0.255
_
m
X
bz
I
0
12
HAROLD MIRELS
Equation (3.13d) gives the drag of a symmetrical profile when a = 0. The
integral I ( q ) is a measure of the energy of the transverse flow between (6,~)
and the shock. Equations (3.13a) and (3.13b) show that I b and Fb are
related by
(3.14)
Equations (3.7) were integrated numerically for various values of P and y
by Kubota [lo] and Mirels [17], for G = 1,0, respectively. The numerical
results for ?]b and F b are given in Table 1. These define the body shape and
surface-pressure distribution corresponding to a given power law shock.
Plots showing the variation of pl, F and #, with q, for u = 1, y = 1.4 and
various values of ,f?, are also presented in [lo] and 1171.
TABLE2. EVALUATION
OF I b FOR BLASTWAVE, p = 1
Y
(0.
+
- 1)Ib
Numerical integration
(Eqs. (3.7, 3.13))
1.0
1.15
1.2
1.a
1.4
1.667
1.67
Approximate solution
(Eq. ( 5 . 1 5 ~ ) )
o=o
o = l
Ref.
o = 0.1
1.0
1.088
1.109
1.140
1.164
1.213
1.206
1.0
1.098
1.134
1.170
1.203
1.249
1.278
~ 7 1
[51
[51
15, 171
[51
~ 7 1
1.0
1.064
1.082
1.111
1.133
1.167
1.167
When P = 0, the body is a wedge (u = 0) or cone (a = 1). Values of P
in the range O < ,4l < 1 correspond to power-law bodies with infinite slope
at [ = 0. For p = 1, it is found that qb = 0 and all the drag is added a t
6 = 0 (3.13b). This is the "blast wave" or "constant energy" case which
can be used to estimate the shock shape and pressure distribution associated
with a flat plate of semi thickness TN or a blunt-nosed circular cylinder of
radius TN. If D(0)is the drag addition a t 6 = 0, the shock shape corresponding
to P = 1 is (from (3.13b) and noting C = Sl('-"))
(3.15a)
_ -_ '?N
2
HYPERSONIC FLOW OVER SLENDER BODIES
13
where CD, is the nose drag coefficient
(3.15b)
CD, 3 D(O)/[7fpm6m2v,~+1/20].
Numerical values of I b for use in (3.15a)are given in Table 2. The surface
pressure distribution is found from (3.12). An exact analytical solution
for the 3/ = 1 case was found by Sedov [la]and, independently, by Latter
[30]. The Sedov solution is discussed in Section 111.3.
The nature of the flow field, particularly for /I> 1, will be discussed
further in Section 111.4. When no confusion results, the variables 6 and x
will be used interchangeably.
2. Alternative Formulations
The differential equations describing hypersonic flow through power-law
shocks (3.7)and (3.8),are now reformulated from other viewpoints. The
advantages of these new approaches are noted.
Stream-function formulation. The continuity equation (2.44 is satisfied
by a stream function $ such that
(3.16)
Considering a power-law shock, a suitable form for $ is
(3.17)
where 0 is a function of 7 and e(1) = 1.
The quantity p/p' is constant along a streamline, downstream of the
shock, and may be written as
(3.18)
where w is a function of
PlpY = w ,
6. Considering conditions at 7 = 1 yields
(3.19)
Eliminating 6 in (3.19)gives o as a function of $, and (3.18)becomes
(3.20)
14
HAROLD MIRELS
Equations (3.16), (3.17), and (3.20) permit p , v , and p to be expressed as
functions of 6 and 8. Thus
(3.21a)
(3.21b)
(3.21c)
and the momentum equation (2.4b) becomes
with the boundary conditions
(3.22b)
e(1) = 1,
eyl) = (a
+ l ) ( y+ l)/(v - 1).
The streamline through the origin corresponds to 8 = 0.
The streamlines downstream of the shock are defined, parametrically
(with q as parameter), by
(3.23)
where (xi,Ri) are the coordinates of a streamline a t its intersection with the
shock (Fig. 5 ) . If x / x , , r / R j and I(q) are tabulated as functions of q, in the
course of integrating (3.22), each streamline and its drag are defined.
Equations (3.21) and (3.22) are somewhat simpler than (3.7) and (3.8)
in that the problem has been reduced to finding a single dependent variable 0
in terms of q. Equations (3.21) and (3.22) were used by Mirels [17] as a
guide to find approximate analytical solutions. These are discussed in
Section 111.3.
Lagrangian formulation. In the present section 0 and 6 are the independent
variables. Equation (3.23) shows
(3.24)
e =(R~/R)#+~
so that 0 can be considered as a Lagrangian variable. Note that for [ fixed,
the local value of 0 is the ratio of the mass flow between the local streamline
HYPERSONIC FLOW OVER SLENDER BODIES
15
and the streamline 8 = 0 to the mass flow between the shock and the
streamline 8 = 0. The present formulation may also be considered in effect
to be a Mises transformation in that a stream function is used as an independent variable.
The dependent variable considered herein is
(3.25)
f &!LeF
2
so that f = 1 at 0 = 1. If is fixed, f is the ratio of the local pressure to the
value directly downstream of the shock. Equations (3.21c), (3.21b), and
(3.2la) show, respectively,
(3.26a)
(3.26b)
(3.26~)
Noting from (3.21~)that
(3.27a)
it follows that
(3.27b)
Equation (3.22a) can now be written
16
HAROLD MIRELS
Substitution of (3.26a) into (3.28a) reduces it to an integro-differential
are found from
equation for f in terms of 0. The quantities y, !~,t, and
from (3.26).
The main advantage of (3.28) is that the role of y is explicitly indicated.
In particular, the nature of the flow for y near 1 can be readily deduced.
The variation of f with 0 is nearly linear for < 1 and y near 1 so that a
Taylor-series expansion of f about 0 = 1, which can be obtained directly
from (3.28), represents an accurate solution. Reference [21] discusses the
variation of f with B in great detail, for 0 = 0, and presents the Taylor
expansion for f up to and including terms of order (0 - l)4.
Sedov formulation.
Sedov [12] introduced the variables V E rnqlq,
z = ym2F/q2t,h and reduced the equations of motion to the form
V - m dz
(3.29a) ~2
dV
- [2(V - 1)
+ a(y - l ) V ] (V - m)2+ ( y - 1)(1- m)V(V - m ) - [2(V - 1)+ ~ ( -y 1))
V(V - 1)(V - m ) - [(l+ a ) V K]Z
(3.29b)
(3.2%)
dlny z - (V - m)2
V ( V - 1 ) ( V - m ) - [(l a ) V - 4 2 ’
dV
+
- (a + 1)V
dl n (V - m)gL d In y
V-m
+
where = 2(1 - m)/y. The shock location corresponds to V = 2m/(y
1)
and z
2y(y - l ) m 2 / ( y 1)2.The streamline through the origin is defined
m for ,8< 1 and V = m l y for ,8 = 1.
by V
Equations (3.29) are particularly useful for studying the singularities
and the general properties of the self-similar flows associated with-power law
shocks. For details, the reader is referred to Sedov [12]. General properties
of self-similar flows are discussed in 111.4.
+
3. Analytic Solutions
Blast wave. Sedov obtained an analytic solution of (3.29a) for the case
of a blast wave (/I
= 1 or m = 2/(a
3)). His solution is
+
(3.30)
z = ( y - 1)V2(m - V)/2(V - m/y).
Equation (3.30) was deduced by noting that the energy between a similarity
line (i.e., q = constant) and the shock is constant when fi = 1 (see (3.13d)).
Thus the transfer of energy across a similarity line is zero. This can be
17
HYPERSONIC FLOW OVER SLENDER BODIES
+
+
expressed in the form [ F / ( y- 1) ($p2/2)](p- q) Fp, = 0, which
leads to (3.30).
Following.the notation of [ l l ] , the blast wave solution may be expressed as
(3.31~)
0 =m [ y
+ 1 + u(y - l)],
@ =m
2y - 0
The solution is tabulated, for various y , in [12]. Analytic solutions to the
blast wave problem were obtained independently by Latter [30] for (T = 2
and Kubota [lo] who generalized Latter's solution for u = 0,1, 2.
Equations (3.31) can be considerably simplified for small values of
E
(y - l)/(y 1). First, note that F , varies between 0 and 1. Neglecting
terms of order E compared with 1 it can then be shown that F , = F, = 1,
F, = 2 - F , and that equations (3.31) become
+
(3.32a)
F3
em---2 - F3
[ ye]
2e( o+l)
(3.32b)
(3.32~)
(3.32d)
q~m q M F3W(u+1)
~
p/(a+l),
18
HAROLD MIRELS
A similar expansion is given by Freeman [31] and Brocher [32]. Equation
(3.32b) shows that q essentially equals 1 for all values of 0 except 0 very
nearly zero so that most of the disturbed mass is located at the shock wave.
The pressure on the body (0 = 0) is half of that directly behind the shock.
Newtonian theory. Hypersonic flow in the limit of large M and small E
is termed Newtonian flow (cf. e.g., [ l l ] ) . The Newtonian approximation for
hypersonic flow associated with slender power-law shocks is now discussed
by considering the solution of (3.26) and (3.28) for E = 0 and for E + 0.
When E = 0, (3.26) and (3.28) show
(3.33)
f = 1 - p ( i - ey2,
= 7 = 1,
*/*(i)= eq.
These equations are valid for all p and constitute an exact solution. Since
q = 1 for all 8, the disturbed flow has zero thickness and the shock wave
coincides with the body (or effective body) surface (i.e., q b = 1). There is
a pressure variation across this infinitesimal layer due to centrifugal force.
Now consider E - + 0 and p < 1. It can be shown that q = 1 O(E)and
d / / d e = (/3/2)[1 O ( E ) ]so that (3.26) gives
+
+
(3.34a)
(3.34c)
(3.34d)
The parameter q b is found by setting 0 = 0 in (3.34b). Equations (3.34)
converge, uniformly over the flow field, to (3.33) as a limit when E - 0 .
Equations (3.34) also represent the solution for ,8 = 1, in the limit E -+ 0,
as may be seen by comparison with the exact solution (3.32). In particular,
integration of (3.34b) yields
(3.35)
in agreement with (3.3213). Comparison of (3.35) and (3.33)shows that, for
p = 1, the solution for E + 0 does not converge uniformly over the flow field
HYPERSONIC FLOW OVER SLENDER BODIES
19
to the solution for E = 0. While both solutions indicate that most of the
mass is concentrated at the shock wave, (3.35) gives l;lb = 0, while (3.33)
gives l;lb = 1. This nonuniform convergence was discussed in [31], by a
similar comparison of blast-wave and Newtonian theory. The nonuniform
convergence, at 8 = 0, also occurs for fi > 1 as may be deduced from the
discussion in Section 111.4.
For small E and
into (3.13~))
p < 1, (3.32) and (3.34) give (after substituting
(3.36)
Interpreting I b as a measure of the sum of the net internal and kinetic
energy perturbation a t a fixed x (i.e., unsteady viewpoint), it is found that
the ratio of internal to kinetic energy is y / ( y - p), to the present order.
The ratio is 1 for p = 0 and approaches infinity for fi = 1 and E + 0.
“Sharp-blow” solutiovt. A closed-form solution of (3.26) and (3.28) can
be obtained for the case a = 0, y = 715, ,8 = 413 (i.e., m = 3/5). This
solution was obtained in [33-351 (as noted in [21]). It is an example of a
flow which Russian authors refer to (according to the translation of [21])
as the “sharp-blow” problem.
The solution is
(3.37a)
The streamline through the shock point (xi,Ri) is (from (3.23) and (3.37a))
(3.37b)
where R, = x : ‘ ~ . This streamline is indicated in Fig. 6. For x/xi large,
) . streamline
the streamline approaches the straight line r = - X / ( ~ X ; / ~The
through the origin coincides with the negative Y-axis. The pressure, along
each streamline, decreases as x increases and approaches zero as x + 00.
The drag on the upper surface of a streamline, D(x)/2, is (from (3.13d)
and (3.37))
20
HAROLD MIRELS
-
(3.38b)
3 1
32 xi1/‘
for
x+w.
The drag is finite except for the streamline going through xi = 0.
8-
3FIG.6. Streamline through (xi, Ri) for “sharp blow” problem (Eq. 3.37b).
The flow is illustrated in Fig. 7(a). An infinite drag is imposed a t (0,O)
to flow in the upper half plane (Y
0) and a lateral expansion (into a vacuum
occupying Y < 0) is permitted for x > 0. I t is qualitatively similar to
>
FIG. 7. Flow associated with “sharp blow” problem (u = 0). (a) Steady flow described by (3.37). (b) Qualitatively similar steady flow. ( c ) Equivalent unsteady flow.
unseparated flow about the edge of a plate which is normal to a hypersonic
stream (Fig. 7(b)).
The corresponding unsteady flow is illustrated in Fig. 7(c) and was
referred to as the “sharp-blow’’ problem in [21]. This unsteady flow may be
described as follows (for u = 0 and all 7). Consider a massless piston a t
Y = 0 to separate a semi-infinite column of gas (r > 0) from a semi-infinite
21
HYPERSONIC FLOW OVER SLENDER BODIES
vacuum (r < 0). The piston is unrestrained, a t time t = 0, and an infinite
amount of energy is added to the gas column at the piston face. A shock is
generated, which propagates into the gas and the piston is accelerated,
instantaneously, to r = - 00.
The resulting shock-shape parameter,
designated PsB, varies from 2 for y = 1, to 1.117 for y- w . Corresponding
values of PsB and y , from [21] and [34], are given in Table 3.
TABLE3. SHOCK-SHAPE
PARAMETER
FOR SHARP-BLOW
PROBLEM
~~
Y
1
1.1
1.4
5/3
2.8
00
mSB
PSB
0.5
2
Ref.
-
0.56888
1.5157
[34]
0.6
4/3
[33, 34, 351
0.61073
1.2748
[34]
0.626704
1.1913
[341
0.6416
1.117
[211
From physical considerations, it is expected that the sharp-blow flow
has no net momentum in the r-direction. This can be readily demonstrated.
Equating surface pressure to rate of change of v-momentum gives, in general
(for u = 0)
1
(3.39)
For the sharp-blow problem, F6 is zero, and therefore the integral (a measure
of the net v-momentum at any section) must be zero. Substitution of (3.37)
into the integral verifies this property for y = 7/5. Thus, there must be no
net v-momentum downstream of x if the flow in Fig. 7(b) is to be qualitatively
similar to the sharp-blow problem.
The “free layer” flow discussed in [ l l ] (for u = 0 , l ; y = 1 ; M = 00,
6 = unrestricted) approaches, with increasing x, a power-law shock with
p = 2 and is thus a special case (for u = 0) of the sharp-blow problem.
(However, for the u = 0 free layer flow, there is finite v-momentum addition
at x = 0 and the v-momentum remains constant for x > 0.)
Approximate solutions. An approximate analytical solution of the blast
wave problem ,8 = 1 was obtained by Taylor [3] for u = 3 and by Sakurai [5]
for u = 1, 2. These have been generalized by Mirels [17] for all /3
1 using
the following approach.
<
The asymptotic form for 8 , near 8 = 0, is (from (3.22a))
22
HAROLD MIRELS
where
This suggests the following approximate expression for 8
where C,, Do,and qb are, as yet, unknown constants. The corresponding
values of F , # and 9 are (from (3.21))
(3.41b)
Equation (3.41a) satisfies the boundary conditions 8 = 0 a t q = q b and
8 = 1 at q = 1 and has the proper form near 8 = 0.
The constants C,, Do and qb are determined by satisfying the boundary
conditions on O', 8" and 8"' a t q = 1 as found from (3.22). For p = 1,
it is known that qb = 0 so that C, and Docan be found by satisfying only
8' and 8" a t q = 1. The latter procedure gives, for P = 1,
which agrees with [ 5 ] .
Numerical results for C,, Do, r j b and Fb are presented in Table 1. Comparison of these values of qb and Fb with those obtained from a numerical integration of the equations of motion shows that the approximate analytical
solution is accurate when the shock is relatively close to the body (P near 0
HYPERSONIC FLOW OVER SLENDER BODIES
23
and y near 1). Sufficient results are presented to evaluate the accuracy of
the approximate method for various values of 3f, and y . The constants
C,, Do and q b presented in Table 1 can be used in (3.41) to obtain the variation of 8, F , t,h, and q~ with q.
4. Nature of the Flow
The physical nature of flow field associated with a power-law shock
is now considered in more detail. First flows with a = 0, y = 1.4, and
FIG. 8. Schematic representation of flow fields and surface pressures for power law
shocks (a = 0, y = 1.4). (a) Flow fields.
various P-values are discussed. These are generalized to a = 0 , y > 1.
Finally, flows with a = 1 are considered. The discussion is based primarily
on the results of Adamski and Popov [21].
The flow fields for a = 0, y = 1.4 are indicated in Fig. 8(a). The corresponding pressure variation, as a function of 8, is given in Fig! 8(b)
24
HAROLD MIRELS
and represents the integration of (3.28a). Referring to Fig. 8, case (1)
has convex shock and body shape, with zero slope a t x = 0; case (2)
corresponds to a wedge; and case (3) is a slender blunt-nose power-law
body. Case (4) is the blast-wave problem. Finite drag is added a t (0,O).
Note, Y b = 0. Case (5) represents a value of p between the blast-wave
case, p = 1, and the sharp-blow case, pss = 413. Here q b is a finite
negative number. The streamlines of the disturbed flow are concave
t'
1
'\
.2
-4
OSAWLE POINT
OCRlTlCAL POINT
.6
I
I
.8
1.0
1
b
FIG. 8 concluded. Schematic representation of flow fields and surface pressures for
power law shocks (a = 0, y = 1.4). (b) Surface pressure.
downward near the shock and concave upward near the body. Thus, at
fixed x , the pressure first decreases and then increases in going from R
to rb (Fig. 8(b)). When p = pSs = 413, two flows are possible as noted
in [21]. These are designated cases (6) and (7) in Fig. 8. The two possibilities arise because (3.28) has a singular point (in particular, a saddle
point) at O S p = (1/7)3/2. The corresponding value of q is qsp = - 112.
In both cases the solution for O s p
6
1 is f = 6 as indicated in Fig. 8(b).
However, when continuing the solution for 0 6
O S p , two paths are
possible. For one, case (6), f increases with decreasing 6 to the value
f = 0.1101 at 0 = 0 [21]. The other possibility is case (7), for which f
decreases with decreasing 8 according to f = 0. Case (7) then corresponds
to the sharp-blow problem. The locus of the saddle point, in the physical
plane (Fig. €!(a)) is indicated by the dashed curve r S p = qspxm. Cases (8)
and (9) correspond to p slightly greater and much greater than Pse, respec-
< <
< <
25
HYPERSONIC F L O W OVER SLENDER BODIES
tively. For p > pss, the solution for f (starting at 0 = 1) first decreases
with decreasing 8 but then reaches a critical point (dfld8 = co) beyond
which f becomes a multivalued fundtion of 8 (Fig. 8(b)). This multivaluedness
is not physically admissible, and the solution starting at 8 = 1, f = 1 cannot
be continued past the critical point, as is indicated by the dashed curves
in Fig. 8(b). If the value of q at the critical point is designated by qc, the
curve rC = qcxmis a limit line in the physical flow. The solution for flow
through a power-law shock cannot be continued past this line.
A physical interpretation of the limit line can be readily given. I t is
a line of constant q which is also a Mach line. This is shown as follows. The
slope a t any point along a line of constant q is
drldx = mqxm- l .
(3.43)
The slope of the Mach line a t any point is (only the upper family need be
considered)
(3.44)
dr _
- ~ + a a m x ~ - ~
dx
where a is the local speed of sound. Equating (3.43) and (3.44), a line of
constant q will be a Mach line when
(3.45)
q c = plc
+ l/yFcI*c.
The subscript c is used since (3.45) defines the critical point of the flow,
as may be seen by writing (3.45) in the form (using (3.26b) and (3.26~))
(3.46)
and noting that the denominator of (3.28a) is zero at this point. Thus,
defining a power-law shock uniquely determines the flow up to but not
beyond the limit line. The flow can be continued past this line only when it
corresponds to a singular point (as in cases (6) and ( 7 ) of Fig. 8).
Figure 8 is representative of the flow field for u = 0 and y > 1. For
1 < p < pss, q b is finite and negative, while for p > pss there occur limit
lines. Note that the shock shape is relatively insensitive to the body shape
when
is negative. For example, when y = 715, m varies only over the
small range 213 to 3/5 as q b goes from 0 to - 0 0 .
For (T = 1 and y > 1, the flow associated with a power-law shock is
confined to q 2 0 and a limit line occurs in the flow field for all > 1
(e.g., [19]). Cases (1) to (4), represent the flows for CT = 1, -. 1 < p
1,
while case (9) represents the u = 1, p > 1 case. There is no flow to compare
with the cr = 0 “sharp-blow’’ problem (except when y = 1).
<
26
HAROLD MIRELS
Studies of the integral curves and singular points of the differential
equations considered herein have been presented in [lo, 12, 19, 20, 21, 34, 361
and also by other investigators. The discussion in [12] is particularly
detailed.
IV. FLOWS
ASSOCIATED
WITH SLIGHTLY
PERTURBED
POWER-LAW
SHOCKS
The effect of a power-law perturbation of a power-law shock is now
considered. The assumption (MIJ)-~-+
0 is retained except in Section IV.3
wherein the effect of small but nonvanishing ( M 6 ) - a is evaluated. The
material is based primarily on [17, 181.
1. Basic Equations
The flow associated with a power-law shock, found in Chapter 111, is
considered to be a zero-order solution and is denoted by subscript zero. Thus
the zero-order shock and body shapes are denoted by Ro = xm, Yb,O = rb,0xrn
where q b , O is tabulated (as q,) in
N
R=xm(l+r2a2x 1
Table 1, etc. The effect of a small
Ro8 x
powerlaw shock perturbation is denoted
by a subscript 2 in agreement with
b#a9b$m
the notation of [18]. The shock shape
may
then be expressed as (Fig. 9)
'0
Fp:xm
R/HM
.
R4
X
(4.1)
+
R = zm(l E ~ u ~ x ~ ) ,
FIG. 9. Power law perturbation of
power-law shock and corresponding
body shape (0 < p < I ) .
where E2 iS Small.* The problem iS to
find the flow field associated with this
shock. The body streamline (corresponding to 0 = 0) can be shown to have the form
(4.2a)
(4.2b)
rb = Xm(qb,O f E 2 X N )
= p ( & z X N ) l ! ( o + 1)
(p < I),
(p = 1).
The quantity e2 is a measure of the magnitude of the body perturbation. The
ratio of the shock perturbation to the body perturbation is u2 except for the
3/ = u = 1 case where the ratio is u,(E,$'"''~.
The independent variables ([,q), as defined in (3.4), are again used.
Substitution of (4.1)into (2.6) with (MS)-2 = 0 suggests that the dependent
variables be expressed as
* However, E~ should be larger than 6* since terms of order 6*have been neglected
in obtaining the hypersonic slender body equations used herein.
27
HYPERSONIC FLOW OVER SLENDER BODIES
(4.3a)
(4.3b)
(4.3c)
where po,p2,F,, etc. are functions of 7. The quantities v0,Fo, and t,bo are
the same as v, F, and JI of Chapter 111. Substituting (4.3) into (2.4) and
collecting terms of order cZo and eZ1 yield, respectively, the zero-order
equations (3.7) and the perturbation equations. The latter are [18]
U
(4.4a)
(4.4b)
P2’
~-
v - To
- .$2’ +
V2’
~-
71 - Po
$0
1
-
($+17
- Po +Pi
0)
P2
V-Po
I
v
+ +1h
(0
*2
- 0,
v - Po
*O
F
_o F_2 ’ +
(17 - Po)2$0 Fo
v -‘Po
71 - Po
+
where p = N / [ m ( u l)]. It is convenient to satisfy the shock boundary
conditions a t the zero-order shock location, v = 1. The boundary conditions are
Equations (4.4) can be integrated numerically, starting a t 17 = 1, if p2/a,,
F2/a2,and $Ja2 are considered as the dependent variables. The value of
a2 is then determined by satisfying the tangent-flow boundary condition
a t the body surface. This condition shows that a t r]b,O (from [18])
28
HAROLD MIRELS
TABLE4.RESULTS
OF NUMERICAL
INTEGRATION
OF PERTURBATION
EQUATIONS
(EQ.(4.4))
(Data from ref. [la]
Y
P
o=o
Case
1.15
0
1/3
3/4
1
1.4
0
1/3
3/4
1
1.67
0
1/3
3/4
1
a Effect
p
o=l
P
aa
Fb, 2
0.798
2.00
1.18
1.71
2.09
1.39
1.58
2.03
1.53
1.45
1.56
0.506
0.483
0.869
- 1/4
0
0
1.05
1.04
1.06
0.863
1.96
1.64
0
1/8
1.08
1.02
1.49
1.99
0
5/16
1.18
0.734
1.31
1.96
1
0.0518
0.503
- 1/4
1.13
0.843
1.20
1.77
1.32
1.14
1.86
1.44
1.02
1.95
1.44
0.660
0.443
0.217
2.00
1.39
1.76
2.03
1.64
1.67
1.90
1.77
1.53
1.36
0.611
0.655
0
0
1.09
1.14
1.91
1.59
0
1/8
1.19
1.06
1.45
1.87
0
5/16
1.38
0.655
1.29
1.65
1
0.0389
0.643
1.74
1.33
2.28
1.52
1.25
2.40
1.71
1.10
2.48
1.74
0.701
0.546
0.244
0.934
2.00
1.55
1.78
1.99
1.82
1.73
1.84
1.91
1.59
1.31
0.670
0.605
- 1/4
0
0
1.20
1.15
1.22
0.823
1.86
1.55
0
1/a
1.29
1.11
1.40
1.80
0
5/16
1.53
0.645
1.24
1.53
1
0.0294
0.828
@a
Fb,2
1.16
1.07
1.29
1.12
1.04
1.34
1.17
0.974
1.40
1.15
0.693
0.325
0.188
1.44
of boundary-layer development.
* Effect of small angles of attack.
Effect of wedge and cone nose blunting,
Case
HYPERSONIC
FLOW OVER SLENDER BODIES
29
(4.6a)
(4.6b)
Analytic procedures are required near l;)b,o, since this is a critical point of
the differential equations. Numerical results for a2 and Fb,2 are given in
Table'4 for a variety of physical problems which are discussed later.
If a2 is known, the shock shape corresponding to a given body perturbation, and vice versa, is found from (4.1) and (4.2). The pressure distribution
on the perturbed body is
(4.7)
+
p b = (dRo/dx)2[Fb,0
E2Fb,2xN].
Expressions for drag are given in [ 1 8 ] .
For P = 1 , (4.2b) shows e2xN = ( Y ~ / R ~ )so" +that
~ the perturbation
analysis is valid for (rb/RO)'+l << 1 . The perturbed shock shape can be
written
(4.8)
R/Ro= 1
+ a2(rb/Ro)"+1.
Eq. (4.8) indicates that for 0 = 1 a body perturbation rb/Ro causes a
shock perturbation of order ( Y ~ / RThus
~ ) ~ the
. shock is relatively insensitive to body changes for the u = P = 1 case.
2. Applications
Boundary-layer effects. If the boundary layer on a power-law body induces
only small perturbations in the zero-order flow field, the magnitude of these
perturbations can be found using the equations of the previous section.
Expressions applicable for finding the boundary-layer displacement
thickness 6* on a slender power-law body at hypersonic speeds are derived
in [lo, 18, 23, 371 and other papers. Following the method of ClS],
the non-dimensional boundary-layer displacement thickness S* is (assuming
a linear variation of viscosity with temperature)
where
(4.9b)
30
HAROLD MIRELS
TABLE5. BOUNDARY-LAYER
PARAMETERS
(FROM
[18] BASEDO N NUMERICALDATAO F [38, 391.)
gw
J2
JI + J s
Js
0
-0.14
0
0.60
2.00
0.495
0.470
0.4235
0.383
0.134
0
- 0.257
- 0.538
0.2
-0.14
0
0.50
1.60
0.504
0.470
0.409
0.366
0.692
0.619
0.199
- 0.083
0.6
- 0.20
2.00
0.554
0.470
0.380
0.294
2.034
1.556
1.185
0.759
1
0
0.05
0.10
0.20
0.40
0.60
0.80
0.470
0.452
0.435
0.408
0.370
0.336
0.312
2.69
2.53
2.48
2.41
2.33
2.27
2.24
2
-0.10
0
0.30
0.50
1.00
0.543
0.470
0.333
0.274
0.1765
5.67
5.19
5.49
6.01
0
0.50
7.85
2(1 - m )
y
2m(o+ 1) - 1 '
Here R,L = fiwL/Pw,
o = TP,,iie/Tep,=constant, g, is the ratio of the
fluid enthalpy at the wall (assumed constant) to the free-stream stagnation
parameters which
enthalpy, and J 1 , J z , and J 3 are boundary-layer profile are tabulated in Table 6 . For m = 1 or y = 1, v 2 ( J , Jz J 3 ) =
0.664 1.73gw. Equation (4.9) was derived for Prandtl number 1 and is
valid for /3 < 1, and for p = 1, ~7 = 0. The case p = ~7 = 1 is excluded.
For y + 1, (4.9b) simplifies to
+
+ +
HYPERSONIC FLOW OVER S L E N D ER BODIES
31
If 7b.O is a power-law body and d* is the boundary-layer displacement
d*. Using (4.0a) for 6*, this
thickness, the effective body is Yb = Yb,O
body corresponds to (4.2) with
+
(4.11a)
p = - p3 4
2(a
+ 1)
The perturbation procedure is valid provided E~ is small.* (It is small when
the boundary layer is thin compared with the body radius, for p < 1, and
with the shock radius, for p = 1.) Note that F~ contains the ratio of the
hypersonic boundary-layer interaction parameter M3vo)/Re,i to the
hypersonic similarity parameter (Md)2and may also be interpreted as the
ratio of the boundary-layer induced pressure to the zero-order shock-induced
pressure.
Numerical results for u2 and Fb,2are given in Table 4 for p defined by
(4.11a) and various values of y and /I.These determine the shock and surfacepressure perturbations induced by boundary-layer development on a powerlaw body. For m = 314, the perturbed shock and body shape are similar.
I t is then possible to solve for the flow field without assuming small perturbations. The power-law exponent m = 314 corresponds to the strong interaction case for hypersonic flow over a flat plate [ l l ] .
Angle-of-attack effects. The equations of Section IV.l can be used
directly to find the effect of very small angles of attack on two-dimensional
power-law bodies. If the body is a t angle of attack a the equation of
the upper surface becomes
(4.12a)
7 b = yb,O
f EzX,
where
(4.12b)
E2 = - u p .
Assuming E~ << 1, the resulting flow (in the upper half plane) is found from
the equations with N = 1 - m, or
(4.13)
p =pi2
* The boundary layer effect is of interest provided it is larger than order 6a and
larger than the entropy layer effect discussed in Section VI.l.
32
HAROLD MIRELS
The flow in the lower half plane is found by symmetry (F2(q)= - F 2 ( - q ) ,
etc.). Numerical solutions for u2 and F b , 2 have been obtained and are also
noted in Table 4.
The lift problem for a = 1 and p < 1 is formulated in [18] but no
numerical results are given. A cylindrical coordinate system (x,r,8) is used
where x is in the stream direction and (r,8) are polar coordinates in the
transverse plane. The body and shock shapes are given by
+
sin
~
8),
(4.14a)
r b = x"(qb,O
(4.14b)
R = xm(l+ ~ ~ u ~ ~ ~ 6-) ." s i n
E
~ -X
The dependent variables are expressed in the form indicated by (4.3)
but with the perturbation terms multiplied by sin 8. In addition, the
velocity in the 8 direction is
w = ma,R, cos 8
(4.15)
where R, is a function of q. The differential equations defining the motion
are the same as (4.4) except that a term Q2/q(po-- q ) must be added to
the left hand side of (4.4a) and the &momentum equation
(4.16)
must be considered. The boundary conditions a t the shock are given by
(4.5) with the additional condition
(4.17)
+ 1))
Qz(l)/% = - 2/(r
and the boundary condition at the body surface is given by (4.6~3). The
latter equations must be solved numerically to find the effect of small angles
of attack on axially symmetric power-law bodies. The case corresponding
t o p = 1 has not been formulated.
The surface pressure distribution for both a = 0 and u = 1 may be
expressed as
(4.18)
f i b = m2z+
- l) [ F b , O
+
&2Fb,2X1 -
sin 81,
where 8 = n/2 and 3x12 for the upper and lower surfaces, respectively,
when a = 0. If L ( x ) is the net lift up to station x , (4.18) shows
(4.19a)
(4.19b)
HYPERSONIC FLOW OVER SLENDER BODIES
33
The location of the center of pressure depends only on u and m, for given x .
The lift of slender axisymmetric bodies (not necessarily power-law bodies)
has been treated by Cole [40] using Newtonian theory. Gonor [41] has
considered the special case of a lifting cone. The effect of power-law perturbations of the centerline of power-law bodies is also discussed in [18].
Effect of blunting the nose of very slender wedges and cones. The effect
of blunting the nose of very slender wedges and cones is now considered.
It is assumed that the wedge or cone is sufficiently slender so that the major
FIG. 10. Hypersonic flow past blunted slender wedge or cone. (a) Actual flow.
(b) Hypersonic slender body approximation.
contribution to the drag is a t the nose. The zero-order flow is then a constantenergy (,d = 1) flow. The divergence of the body downstream of the nose
induces a small perturbation in this zero-body flow, which can be found
from the equations of Section IV.l.
Fig. 10 illustrates the flow under consideration. The body shape, in
dimensional variables, is
(4.20)
fb = fN
+ 663,
where Sb is the vertex semi-angle of the wedge or cone. An equivalent body
which has zero area a t 3 = 0 but has the same area divergence downstream
of 3 as does (4.20) is given by f { + l = (6b%)'+1 2ufNdbR. In nondimensional variables, the latter becomes
+
(4.21b)
-6bx
6
for
0 =
o or u = 1, Y N d I X d b < < 1.
Equation (4.21b) is the hypersonic slender body approximation of a blunted
slender cone or wedge. Both the physical flow and the approximation are
illustrated in Fig. 10. The approximation is poor near x = 0.
34
HAROLD MIRELS
Comparison of (4.21b) with (4.2b) shows
(4.22)
&2 = (6b/6)a+1,
= ((T
+ 1)/2.
Numerical solutions have been obtained for (T = 0, 1 and y = 1.15, 1.4,
and 1.67. The results for a, and Fb,2 are tabulated in Table 4. Surface
pressure and shock shape can then be found from (4.7) and (4.8).
Chernyi (15, 161 has studied the hypersonic flow past blunted wedges
and cones using an integral method which is described in V.2. The integral
method is not restricted to flows wherein the major drag contribution is
a t x = 0.
3. Effect of (M6)-2 # 0
Up till now the limiting case (MS)-2-* 0, which corresponds to infinite
free-stream Mach number, has been considered. In this limit, the flow past
a slender power-law body gives rise to a power-law shock. If the Mach
number is decreased, the shock will depart from its power-law shape and the
pressure distribution on the power-law body will be modified. This effect
is studied herein. In particular, small but non-vanishing values of the
parameter
s1 G (MS)-2
(4.23)
will be assumed and (2.4) and (2.6) will be solved by a perturbation procedure.
The body shape is given by yb = r&rm and the zero-order solution is the same
as discussed in Chapter 111. This problem is sometimes referred to as the
effect of "counter pressure" in Russian literature and as "second order"
blast wave theory (in the case of a blast wave).
Let subscript 1 represent perturbation quantities (as in [17]). The shock
boundary conditions (2.6) suggest that the perturbed shock has the form
(4.24)
R = tm[ l + ela1E2(l- "'1,
and that the dependent variables can be expressed as
(4.25b)
= mtm- [po + ~ ~2 0 -p"'I, ~ t
p = m 2 [ 2 ( m - 1) [F, + E1F1t2(1- m, I,
(4.25~)
p = $bo
(4.25a)
+
E1$b1t2(m - 1 ) .
Equations (4.25) are in the form of (4.3) with N = 2(1 - m),or equiyalently,
(4.26)
p =P.
Thus, substitution of (4.25) into (2.4) yields perturbation equations which
are identical to (4.4) with p = /? and with the subscript 1 replacing the
HYPERSONIC FLOW OVER SLENDER BODIES
35
subscript 2 therein. The shock boundary conditions to be satisfied at 7 = 1
are [lo, 171
-2
v
1
=
m
+ q 2 - ( U + v + 2
y f l
4uy
Y+l
]
a,,
(4.27a)
(4.27b)
(4.27~)
The flow at the body surface must satisfy
(4.28)
P)l(Vb) = O*
since the body is unperturbed from its zero order power-law shape.
For a given body, a, is determined such that (4.4) (with p = P ) and (4.27)
yield a solution satisfying (4.28). To avoid trial-and-error choices for a,
in a numerical integration of (4.4) and (4.27) it is advisable to decompose
each dependent variable and its boundary condition into two parts, one
independent of a, and the other proportional to a,
(4.29)
( )1 = ( 11.1
+(
)1,2a,.
+
For example, v, = pl,,
v1,2a,, etc. The solution for ( )1,1 and ( )1,2 can
then be found. The final solution is given by (4.29) with a, determined from
(4.30)
= - 9l31(~b)/~l,2(~b).
Numerical results for a, and Fb,l are given in Table 6. Similar results are
presented in [lo] for u = 1. (The latter appear to be somewhat in error,
particularly for a,, as noted in [17].) The surface pressure and shock shape
are found from
The pressure perturbation is of the same order as the free-stream pressure,
p a = E J ~ ,so that the latter must be included when computing local depar-
36
HAROLD MIRELS
tures from free-stream values. Equations (4.30) and Table 6 indicate that
the surface pressure and the shock ordinate are increased by non zero values
of E ~ . The surface pressure perturbation is independent of x .
TABLE6. NUMERICAL
RESULTS
FOR EFFECTOF ( M 6 ) - a # 0.
Y
1.15
1.4
1.67
P
0
1/3
1/2
FROM
[17]
a = l
a=O
a1
Fb.1
a1
Fb,l
1.000
1.34
1.43
1.800
2.79
3.16
2.77
0.910
0.489
0.885
1.08
1.17
1.18
1.10
1.23
1.10
2.50
3.37
3.84
3.79
2.93
1.32
0.476
0.807
0.932
0.976
0.976
0.964
0.992
0.918
1.97
2.49
2.67
2.51
1.95
1.14
0.465
0.783
1.63
2.00
2.11
1.98
1.57
0.931
5/8
-
314
718
1
1.31
1.03
0
1/3
1/2
1.000
1.21
1.21
518
-
3/4
718
1
1.07
0.965
1.548
2.17
2.26
1.78
0.799
0
1/3
112
5/8
3/4
7/8
1
1.000
1.14
1.11
0.991
1.348
1.77
1.78
1.39
0.930
0.707
-
-
0.762
0.863
0.900
0.911
0.922
0.969
V. INTEGRAL
METHODS
Hypersonic slender body theory, which assumes d2 << 1 and (Md)-2 not
near one, was used in Chapters I11 and IV to find the flow associated with
power law and perturbed power law shocks and bodies. In order that the
equations of motion yield similarity solutions it was necessary to further
) - ~ either zero or very small. The treatment of more
assume that ( M C ~ was
general bodies and Mach numbers requires either a numerical integration
of the full equations of motion or the use of integral methods. Some integral
37
HYPERSONIC FLOW OVER SLENDER BODIES
methods are outlined herein. The object is to provide a means for finding
the flow over more general body shapes than considered in Chapters I11
and IV. Also, the integral formulation will be used in Chapter VI to estimate
the error introduced by disregarding the violation of the hypersonic slender
body assumptions at the nose of power-law shocks.
The development in the remainder of this article is in terms of dimensional
quantities and the superscript bar is omitted.
1. Continuity Integral
Conservation of mass is expressed in integral form for supersonic and
hypersonic flow over an arbitrary body. The resulting expressions are then
simplified by introducing approximations.
General case. Consider the streamtube which crosses the shock at (xi,Ri)
and has a lateral width dRi (Fig. 11). At some downstream station x the
streamtube has an ordinate r
and a width dr. Conservation
of mass yields (in physical variables)
bum
p
(2nRJ"dR; = pw (2nr)"dr.
X
(5.1)
XI
Integrating (5.1) at a fixed value
of x and noting d8
(a 1)
R t d R j / R U f gives
1
= +
X
FIG.11. Notation for continuity
integral.
Equation (5.2) relates r and 8. For 8 = 1, r = R, (5.2) becomes
If p / p , and u / u , could be expressed as functions of 8, (5.3) would relate
the shock and body shape in a given problem. This might be done as follows.
The flow along a streamline is isentropic downstream of the shock so that
PIP' = PiIpiY or
(5.4a)
38
HAROLD MIRELS
where subscripts i and s represent conditions directly behind the shock at
xi and x , respectively, and f E p / p , is the pressure at ( x , ~ divided
)
by the
pressure behind the shock at ( x , R ) (Fig. 11). Similarly, Bernoulli's equation gives
(5.4b)
(5)=I(+)
2 -,"@if-s].
2
a
Y-
m
Ps
P
1
PWUW
All the variables on the right hand sides of (5.4) can be expressed in
terms of shock shape except v and f . If a reasonable variation of v and f
with 8 is assumed, (5.3) can be solved to find the body shape corresponding
to a given shock, and vice versa. No other assumptions are involved. For
example, it may be possible to assume that at each x the variation of f with
0 depends only on the local value of p and is given by the zero-order similarity solutions. In problems of hypersonic flow over slender bodies the
term involving z' is negligible.
Hypersonic slender body approximations. If the hypersonic slender-body
approximations are introduced, (5.4b) becomes u / u m= 1 and (5.4a) becomes
+
where E = ( y - l ) / ( y 1) is not necessarily small. Equations (5.5)and (5.3)
yield (3.26a) if it is noted that for power law shocks, Ri' = 19-p/~R',
and if
the limit of infinite Mach number is taken.
The Newtonian theory of Cole [42] assumes E<< 1, (MRi')-2/(y- 1)=0(1),
E ( M R ' ) -<<
~ 1, E ( M R ~ ' )<<- ~1. U'ith these assumptions (5.5) and (5.3)
become
2
0
+
To the same order f = 1 - p(1 - 0)/2, where ,d = - 2 [ R R " / ( R ' ) 2 ] / ( o 1).
Equation (5.6) gives the body shape corresponding to a given shock shape.
The shock and body ordinates, R and rb, are nearly equal. Thus if r b is
given and R is t o be found, the derivatives of R and Ri can be replaced by
derivatives of rb and Yb,,, respectively, in (5.6) and in the expression for f .
The latter form is the one used by Cole [42] who in addition takes the
exponent l / y in (5.6) to be 1.
HYPERSONIC FLOW OVER SLENDER BODIES
39
If (5.6) is used to study blunt-nosed slender bodies with finite drag
addition a t the nose (i.e., p = 1 a t x = 0 ) , the derivatives of R,Ri cannot
be replaced by the derivatives of rb and rb,). This is because the shock
separation from the body is large. In addition it is necessary to retain the
exponent as l/y.* Mirels [29] derived (5.6) as a generalization of Cole’s
Newtonian solution. Although its validity for describing the flow over
blunted (p = 1) slender bodies may be questioned (due to the violation at
the nose of the hypersonic slender-body assumptions) it is certainly valid
for the equivalent unsteady flow of a perfect gas.
Slender blunt-nosed bodies at infinite Mach number. In the present section
the assumption of body slenderness is not applied a t the nose. The effects
of nose bluntness are therefore retained and can be evaluated. The limiting case M - , 00 is considered and the development follows that of
Sychev !25].
Recall that t is the local shock angle relative to the free stream
(Fig. 1). No assumption regarding the magnitude of t is made. Note that
tan t = R’, sin2t = (R’)2/(1 R12),etc. The boundary conditions a t the
shock, for M + 00, then become [cf. (2.41
+
(R’)2
2
Ps
-pmum2
5
1
1
_
ps -_,
+ ( ~ ’ 1 ~ ’ Pm
E
(5.7)
% _
-I-um
2
R12
y + 1 1+(R’)2’
vs
um
-
2
R’
y + l 1+(R’)2’
and Eqs. (5.4) become
[(
_
p - -1 _
R’)’ 1 + (R,’)2f]”’
-
pm
E
Ri‘
1
+ (R‘)2
Equations (5.8) can be further simplified if it is assumed that the bluntness
of the body is confined to the nose region and that x is sufficiently far
) ~ 1, and
(5.8) and
downstream so that (R’)2<< 1. Then ( Y / U ~ <(
(5.2) yield
* If the exponent l/y is replaced by 1 in (5.6) the integral diverges for p = 1 and
therefore does not exhibit the correct limiting behavior for y near 1 . Replacing l/y
by 1 is equivalent to assuming isothermal rather than isentropic flow along a streamline
as was discussed b y Freeman [31].
40
HAROLD MIRELS
(5.9)
-l+!+(
($+l
The term involving (R')2(y-1)/v is retained in (5.9) so that the equation is
correct to order R12. If r is replaced by R and the limit f3 is replaced by 1,
(5.9)relates shock and body shape. A reasonable estimate for f in the integral
of (5.9) is to use its variation with 0 and P as given by the zero-order similarity
solutions as is discussed later.
Sychev 1251 used (5.9) to find the body shapes associated with a powerlaw blast wave and compared the result with the zero order similarity solution. He also investigated the validity of the similarity solutions for various
degrees of nose bluntness. This work is described in Chapter VI.
2. M o m e n t u m Integral
General case. The net forebody drag up to station x can be found by
considering the flux of momentum across the upstream and downstream
faces of the control surface indicated
in Fig. 12. The result gives
axI(#
(5.10)
D(x)
2~
- (Pm
rb(x12
0
Ra+1
+ pm~m2)
-
* -
FIG.12. Control surface for momenturn integral.
O
f
1
+ pu2)Pdr,
'b
where u = 0 corresponds to a symmetric two-dimensional body. Conservation of energy across these surfaces yields
R
'b
R
(5.11)
HYPERSONIC FLOW OVER SLENDER BODIES
41
Replacing the integral in (5.10) by the appropriate expression from (5.11)
gives the final result
(5.12)
Rafl
y ( y - 1)M2 fJ 1 '
1
-
+
Equation (5.12) gives the net drag up to station x in terms of an integration
with respect to 7 at that station.
Hypersonic slender body approximation. For hypersonic slender bodies
(5.12) becomes
where terms of order d2 have been dropped and D ( x ) has been replaced by
the sum of the finite drag addition at x = 0 (if any) and the surface pressure
drag downstream of x = 0. Equation (5.13) is the dimensional form of (2.7).
This equation was used by Chernyi to study non-self-similar hypersonic
flows over blunt nosed bodies [15, 161. Chernyi's method, described in [22],
is now outlined.
Chernyi assumes that the pressure and velocity may be expressed in the
form
(6.14a)
V
Hm
-
2
R'[1 - (MR')-2]
y+l
+ O(&Rf),
(5.14b)
Equations (5.14) are correct in the limit E << 1. However, they are also
valid directly behind the shock for all E . Chernyi lets p be constant, a t
42
HAROLD MIRELS
its wall value (0 = 0 ) , when evaluating the integral on the right hand side
of (5.13). This procedure is valid for ,B small since the term containing 0
is then negligible in (5.14b). I t is also valid for P = 1 and E << 1 since most
of the fluid is concentrated at the shock and the region from the base of the
shock to the wall is essentially at the wall pressure. W'ith fi = fib, the integral
becomes (for M
do)
-
(5.15a)
i
[. . . ] Y " ~=
YR"+'RI2Ib,
*h
where
Here qb = rb/R and I b is defined by (3.13~).Substitution of (5.14) and (5.15)
into (5.13) yields an integral equation relating rb and R. The surface pressure
is found from (5.14b).
The accuracy of this method can be estimated by solving for the flow
associated with a power-law shock, in the limit M - t 00, and comparing
with the zero-order solutions obtained in 111. The result for qb and
is [22]
(5.16b)
F b = 2 / ( y $- 1) - P/2.
Numerical values for qb and F b are listed in Table 1 for various values of
u, y , and P. These agree within about 4% with the exact numerical solutions of (3.7). Hence (5.16) provides simple and relatively accurate estimates
of 156 and Fb for the entire range of u, y , and P of interest. Values of I b
obtained from (5.15~)are listed in Table 2. These are correct within 4% for
u = 0 and 10% for u = 1. Since z b is raised to the l/(u 3) power in (3.15a)
an error of less than 3O/, results from the use of (5.15~)therein.
Chernyi numerically integrated (5.13), with (5.14) and M - do, to find
the hypersonic flow over blunt-nosed slender wedges and cones [15, 161.
His solutions are not limited to small departures from a blast wave flow
as are the more exact perturbation solutions of Section IV.2. The parameters
evolved in [lB, 161 have been used in [43] to correlate numerical data
obtained from characteristic solutions of the hypersonic flow over blunt
nosed plates at angle of attack.
+
43
HYPERSONIC FLOW OVER SLENDER BODIES
VI. VALIDITYOF SELF-SIMILAR
SOLUTIONS
The hypersonic slender body equations of motion were integrated in
Chapter 111 in the limit (Md)-z+ 0, to find the self-similar flows associated
1). I t was
with slender blunt-nosed power-law shocks and bodies (0 < p
recognized that the solutions are not valid near the nose, where the assumption of slenderness is violated, but it was expected that the solutions become
valid downstream of the nose. The validity of the similar solutions, for
various degrees of nose bluntness, has been studied by Cheng [24], for
u = 0, and by Sychev [25]. Their results are discussed herein.
<
1. Infinite Mach Number
The indirect problem is considered first. Given a power-law shock R = Cxm,
in the limit M - 00, what is the corresponding body shape and surfacepressure distribution? The shock is illustrated in Fig. 13. The point (x,,R,)
is the point on the shock at which
the local shock slope is unity. The
(x,, R ; & ~ ~ ~
streamlines crossing the shock above
R, can be treated by hypersonic
LAYER
slender-body theory. The flow below
(xi'Ri)-X
0
X
Re crosses a nearly normal shock and
is not correctly described by that
FIG. 13. Entropy layer, Re' = 1.
theory, The latter flow undergoes a
relatively large entropy increase when crossing the shock and is termed the
entropy layer. Station x , in Fig. 13, is a downstream station at which
<< 1. The streamline through (xe,Rc),bounds the entropy layer and
has the ordinate re a t x . Hypersonic slender-body theory neglects terms
of order
compared with one. The present problem is to determine
the range of y and p for which the entropy layer at x creates effects which
are larger than terms of order (R') and therefore must be correctly evaluated.
From the assumption of a power law shock and R,'
that at x
- -
= 1 it
follows
-
Equation (6.1) gives the fraction of the perturbed mass flow which is in the
entropy layer. Using the estimates zd urn, v
vbR'urn, pu(rc - rb)
prnu,R8,1qba, and PbNprnu,2R'2 in ( 2 . 1 ~ )shows that the pressure differential across the entropy layer at x is of order
(6.2)
(pe -pb)/pb
(R')2'P.
44
HAROLD MIRELS
< <
for 0 p 1 and
The pressure differential is less than or equal to order
therefore can be ignored. Thus, the pressure is uniform across the entropy
layer and equals the wall surface value given by the zero order similarity
solution corresponding to the power-law shock. The zero-order solution
(which is valid outside the entropy layer) gives the correct variation of
pressure with 8 throughout the entire flow.
The thickness of the entropy layer at x can be estimated from (5.2)
and (5.9). Considering u u,, f
1, and (1 Ri'2)/Ri'2 1 (for
0
8 8,) gives
- -
< <
-
+
where E is not necessarily small. Equation (6.3) is a measure of the crosssectional area of the entropy layer relative to the area enclosed by the shock.
This area ratio is
O(R')2for fi in the range
<
(6.4)
O<P<y/(y+
1).
In this range of ,8 the body shape associated with the power-law shock is
given correctly, to order R12, by the zero-order similar solution and is a
power-law body. However, for ,8 in the range y / ( y
1) < p
1, the
calculation of body shape must take the entropy layer properly into account
The latter range of p decreases as y
if it is to be correct to order
increases. I t includes blast-wave and nearly blast-wave flows. The body
shape can be found from a numerical integration of (5.9) or by approximate
methods based on that equation.
Equation (6.3) may be compared with the corresponding entropy-layer
thickness obtained from the zero-order similar solution (3.26a)
+
<
The ratio (6.5) t o (6.3) is
The entropy-layer thickness indicated by the similar solution is considerably
larger than the correct value when and y are near 1. This is due to the
fact that similarity theory considerably underestimates the fluid density
near the wall. Equations (6.2) and (6.3) were obtained in [24] (for cr = 0),
and the remaining equations where obtained in [25] (for u = 0,l).
45
HYPERSONIC FLOW OVER SLENDER BODIES
Sychev [25] has found the body shape associated with an axisymmetric
blast wave by numerically integrating (5.9). He chose R = 2/g,y = 1.4,
and used Sedov’s exact solution [12] to relate f and 8. The resulting body
shape is given in Fig. 14(a), and the ratio Y b / R is given, as a function of x ,
in Fig. 14(b). The latter may be contrasted with the zero-order similarity
result rblR = 0. The actual body shape associated with a blast wave has
considerable thickness when compared with the similar solution. When
blast-wave theory is used to estimate hypersonic flow over a blunt-nosed
circularcylinder (with axis parallel to the stream direction) the shock shape
R = VZx corresponds to a cylinder radius rN = l . 6 / 2 / G 1.6 for y = 1.4
(3.15a). The difference between the body shape indicated in Fig. 14(a) and
-
rl
12
-
0
8
16
24
32
a
-
b
FIG.14. Body associated with blast wave R = v 2 x , u = 1, y = 1.4, M -+ m (from
ref. [25]). (a) Shock and corresponding body. (b) Ratio of body to shock ordinates.
-
rN 1.6 may not be very significant with regard to surface pressure
and shock shape since blast-wave theory indicates very little mass flow in
the region near the body and is relatively insensitive to small changes in
body shape (4.8).
The direct problem is now considered. Given a power-law body, in the
limit M-+ bo, what is the corresponding shock shape and surface-pressure
distribution ? When L,? y / ( y l), the zero order similar solutions are
applicable. But when p > y / ( y I), these solutions are in error. One method
for finding the shock shape and pressure distribution is to perturb the zeroorder similar solution to account for the excessive size of the entropy layer.
This may be done by solving the perturbation equations of IV.l, using an
“equivalent body” which is somewhat smaller than the given power-law
body. The equivalent body to be used with the equations of IV.l should
equal the cross-sectional area of the actual body minus the difference between
the area of the entropy layer as given by the zero-order similar solution (6.5)
and the actual entropy-layer thickness (6.6). This gives
Yb =
<
-
+
+
where E~
(- 1)[ ( y - l ) / ( y - ,L?)]d2(y-B)’YB,
Yb.eq is the equivalent body, Ro
is the zero-order shock and q b , O is the zero-order body to shock radius ratio.
46
HAROLD MIRELS
Equation (6.7) can be reduced to the same form as (4.2) and the procedure
of IV.l can be used to find the shock shape and surface pressure corresponding
to the original power law body. (Note that R,’/6 in (6.7) corresponds to
mx*’l in (4.2).) The shock and surface-pressure perturbations are of the
form given in (4.1) and (4.7) with N = 2(y - P)(m - l)/yP. These perturbations cannot be evaluated exactly since E~ is known only to within an order
of magnitude. Inasmuch as E~ is negative, the perturbed shock is closer to
the body and the surface pressure is lower than indicated by the zero-order
solution.
The above discussion regarding power-law shocks and bodies in the limit
When 0 p
y / ( y 1) the
similarity solutions describe the flow downstream of the nose to order (R’)2.
The shock and body shapes are similar. When y / ( y 1) < p
1, the
body associated with a given power-law shock (indirect problem) is thicker
than indicated by similarity theory, due to the excessively low fluid density
near the wall in the similarity solution, but the surface pressure is correctly
given. For the direct problem and y / ( y 1) < p
1 the shock associated
with a given power-law body is closer to the body* and the surface pressures
are somewhat lower than indicated by the similarity solution. For all p,
the flow asymptotically approaches the similarity solution as x + bo (R’+ 0).
< <
+
M + bo may be summarized as follows.
+
+
<
<
2. Finite Mach Nwmber
In order that the zero-order similarity solutions be valid for describing
hypersonic flow associated with a slender power-law shock or body it is also
- ~1. This condition assures a strong shock and was
necessary that ( M c ~ )<<
automatically satisfied in the previous section because of the assumption of
M-, 00. However, for M large but finite, the similarity solutions are not
valid far downstream of the nose due to violation of the condition (M6)-2<<1.
(The first departures from similarity theory have been described in Section IV.3.) Hence, for M # bo, the flow associated with a power law shock
or body does not asymptotically approach the similarity solution as x-+ bo.
It may be concluded that for 6 small and M large, the similarity solutions
are valid (subject to the limitations discussed in the previous section) in the
intermediate region between the nose and far downstream locations.
* In an actual flow-problem, the nose of the shock is upstream of the body.
In regions relatively near the nose this effect tends to increase the shock ordinate
relative to the similarity value.
HYPERSONIC FLOW OVER SLENDER BODIES
47
VII. FURTHER
DISCUSSION
OF INTEGRAL METHODS
The direct problem of determining the hypersonic flow associated with
an arbitrary blunt nosed body by integral methods requires further discussion. The validity of the integral methods, in the presence of entropy layers,
is discussed herein. Some properties of the entropy layer are noted. Finally,
a simplification of the integral methods is introduced which is valid when y
is near 1 and when most of the drag addition occurs a t the nose.
The hypersonic slender-body form of the momentum integral (5.13) is
obtained from the general form (5.12)by assuming u = u, in the integration
with respect to r. The assumption u = u, is correct to order ( R ’ ) 2outside
the entropy layer but requires
-
inside the entropy layer. Eq. (7.1) is found from ( 5 . 8 ) by assuming
(Ri’)2/[1 (Ri’)2] 1 and (R’)2<< 1 . If the local value of R‘ is sufficiently
small so that (7.1) is satisfied, the hypersonic slender-body form of the
momentum integral is valid regardless of the degree of nose blunting. If (7.1)
is satisfied, the continuity integral can be written
+
Outside the entropy layer, R i t 2 is negligible compared to unity but inside
the entropy layer it must be retained.
An interesting property of the entropy layer can be deduced from ( 7 . 2 ) .
For r = Y , and f3 = OL, (7.2) shows
Re
where use is made of f3 F (Ri/R)“+’and f = f b (in the entropy layer). The
right-hand side of (7.3) is a constant which depends on the upstream shape
of the shock. Equation (7.3) is a direct consequence of the fact that the
streamwise velocity is constant to the present approximation (7.1). The
cross-sectional area of the entropy layer a t any section is then inversely
proportional to the average density in the entropy layer which, in.turn, is
proportional to the local pressure to the l / y power.
48
HAROLD MIRELS
Cheng [23] has simplified the procedure for finding the flow over arbitrarily shaped blunt-nosed bodies. He assumes (7.1) is satisfied, M-+w
and E << 1. In addition, he assumes that the cross-sectional area of the
entropy layer is large compared with the cross-sectional area of the disturbed
flow between the entropy layer and the shock (Fig. 15). The latter assumption may be expressed, from (7.2)
The numerator is obtained by assuming (Ri')2<< 1 and (R'/Ri')2~w 1 in the
integral of (7.2). The inequality noted
in (7.4) is satisfied for R' small
provided 8, is not too small (i.e., the
ROPY LAYER
mass flow in the entropy layer is not
X
small relative to the total disturbed
mass flow).
FIG.16. Entropy layer which occupies
With y = 1, R = re, and f b =
major portion of disturbed flow.
1 - p/2 (7.3) becomes
Equation (7.6) provides a relatively simple relation between R and yb.
However, the constant on the right-hand side can not be estimated since
the upstream shape of the shock is not known.
In order to evaluate the constant in ( 7 4 , Cheng resorts to the momentum
integral (5.13) which can be written for M + 00 and E <( 1
The left-hand side assumes that most of the drag is added at the nose. The
right-hand states that most of the energy of the transverse flow is in the
form of internal energy corresponding to the pressure Pa. (The first error
term on the right hand side is the error due to the pressure departing from
pb at points outside the entropy layer and the second error term is the ratio
of the net transverse kinetic energy to the net internal energy, which is
small for E << 1.) Neglecting the error terms, (7.6) is the same as (7.5) except
that the constant is evaluated, viz
49
HYPERSONIC FLOW OVER SLENDER BODIES
Equation (7.7) is relatively simple and has been solved in [23] for a variety
of problems involving blunted flat plates. Studies of shock-boundary layer
interactions were also made using this equation.
In (7.7) the lateral displacement between the shock and the body is due
entirely to the entropy layer. The cross-sectional area of the entropy layer
is inversely proportional to the local pressure in order to conserve mass
(continuity approach) and to conserve the internal energy (momentum
integral approach). Rut the local pressure depends on the shock shape
parameter B. Hence a consistent solution relating shock shape, pressure, and
cross-sectional area is obtained from (7.7). This is similar to the strong
interaction between shock waves and boundary layers wherein a consistent
solution is also required ( e g , 11111). The effect of the downstream body
shape rb is to laterally displace the entropy layer and thus effect the final
self-consistent solution. Since (7.7) is derived directly from the momentum
integral equation, recourse to the continuity integral was unnecessary.
However, the continuity integral provided additional insight into the flow
process. Equating (7.7) and (7.3) also provides an estimate of Re, namely
(RI/YN)u+lM CD,.
I t should be noted that the continuity integral is more sensitive to the
details of the upstream shock shape than is the momentum integral. In
order to integrate (7.2) across the entropy layer, it is necessary to estimate
both the value of Re and the variation of R,' in the range 0
R, Re.
The latter estimates define the variation of density across the entropy layer
and thereby its thickness. However, in the momentum-integral method,
the internal energy in the entropy layer depends only on the pressure f i b
and on the cross-sectional area. The details of the density distribution are
not required. Hence, the momentum integral is relatively insensitive to
the upstream shock shape and provides more information (assuming the nose
drag is known) concerning the downstream flow than does the continuity
integral.
< <
VIII. CONCLUDINGREMARKS
Several investigators [26, 28, 43, 441 have computed the inviscid hypersonic flow over blunt-nosed plates and circular cylinders using the method of
characteristics and have compared the results with blast-wave theory. Since
the method of characteristics provides an exact solution (within the accuracy
of the computational procedure) it can be used to gauge the accuracy of
blast-wave theory. The results of Vaglio-Laurin and Trella [28] are discussed
herein. Reference [28] determined the axisymmetric flow about three
configurations with long cylindrical afterbodies namely a sphere-cylinder
60
HAROLD MIRELS
(Fig. 16(a)) and two sphere-cone-cylinder combinations (Figs. 16(b), 16(c)).
Several different flight conditions were assumed. The fluid was taken to be
either a perfect gas (with y = 1.4) or air in equilibrium dissociation.
The following was found with regard to the pressure distribution on the
cylindrical afterbody. For the spherical nose configuration, f i b / + , - 0.40
was correlated versus M 2 C Z i rN/x, as indicated by "second" order blast
wave theory (4.30 a) with y = 1.4, provided the origin for x was determined
such that the pressure at the upstream
end of the cylindrical afterbody
matched the pressure obtained from a
consideration of the hypersonic flow
over a hemisphere. Good pressure correlation also resulted if the origin of the
shock was arbitrarily located at a distance 2rN upstream of the body, regardless of Mach number and gas behavior (i.e., y = 1.4 or thermodynamic
equilibrium). However, in order to cor&*
,..I
relate the afterbody surface pressures
associated with the less blunt spherecone-cylinder combinations, the value
of rN used in M2Cg;rN/x had to be
reduced. A procedure for obtaining
the effective value of y N was suggested
in [28] which resulted in good pressure
correlation.
I
t
FIG.16. Nose configurations studied
in [28].
The blast wave predictions for
shock shape and for the flow between
the body and the shock were less
satisfactory. Only the spherical nose configuration resulted in the
parabolic shock indicated by the zero order blast wave theory. Moreover,
the latter correlation required that the x coordinate be measured from
the apex of the nose, that the y coordinate be displaced outward from
the axis by (2/5)rN and that the coefficients in the blast wave prediction be evaluated for gas conditions appropriate to the shock point at which
the streamline is deflected approximately 20°. This is the streamline which
[28] considers as bounding the entropy layer. The shock shape associated with
each sphere-cone-cylinder combination could be correlated for different flight
/ ~ xo)"'
( X where
conditions by an equation of the form y - yo = ( C O , / I ~ ) ~ xo,yo and m depend on nose shape and Ib, which expresses the effect of gas
behavior, is selected consistent with conditions at the shock point where the
streamline is deflected 20°.
HYPERSONIC F L O W OVER SLENDER BODIES
51
The disturbed flow between the 20° streamline and the shock satisfies
the requirements of hypersonic slender body theory and was correlated for
different flight conditions by non-dimensionalized flow variables such as
those noted in (2.3). The correlation existed for the thermal-equilibrium
flows as well as the flows with constant y in accordance with the hypersonic
similitude discussion of Cheng [24]. The flow in the entropy layer has gone
through a strong shock and does not satisfy the hypersonic slender body
requirements. As a consequence, the entropy layer flow did not exhibit
similitude.
It may be concluded that blast-wave theory provides a good guide for
estimating the inviscid hypersonic flow over the afterbodies of blunt nosed
bodies particularly for distances far downstream of the nose. However,
refinements are necessary for a more accurate determination of the flow a t
stations which are not far downstream of the nose. The nose shape plays
a greater role than accounted for in blast wave theory. The influence of
the nose shape is most strongly felt in the entropy layer. In future developments it is expected that this influence will be taken into account to a
greater extent and that reliable analytic estimates of the complete flow
field about blunt nosed bodies will be possible.
Experimental studies have been made of the hypersonic flow over slender
blunt-nosed bodies [e.g., 10, 23, 45-48]. Viscous boundary layer effects
complicate the problem of relating the experimental shock shapes and surface
pressure to the predictions of blast wave theory. A full discussion is beyond
the scope of the present study. Reference [23] is recommended for the reader
who is interested in the experimental correlations of blast wave theory.
NOTATION
In Chapters I1 to IV, barred quantities are dimensional and unbarred quantities
are non-dimensionalized according to (2.3). (2.7),etc. However, in Chapters V t o V I I I
only non-dimensional quantities are used and the superscript bar is omitted.
a
Speed of sound
a,, a,
Constants, (4.1). (4124)
C
Constant, (3.1)
CD,
Nose drag coefficient, (3.15b)
D(n)
F(7)
Drag up to station n, (2.7)
Pressure similarity variable, (3.6)
Ratio of local pressure to value behind shock
Energy integral, function of y , u, p , 7, ( 3 . 1 3 ~ )
Characteristic streamwise length
Free-stream Mach number
Power-law exponent, (3.1)
Perturbation power-law exponent, (4.1)
Pressure
Shock ordinate
52
HAROLD MIRELS
Body ordinate
Semi thickness or radius (a) a t x = 0 or (b) of plate or circular cylinder
T
Temperature
1
Time
Streamwise and lateral velocities, respectively
(u,v)
Streamwise and lateral coordinates, respectively
(%,I)
(%$,Re) Shock point where slope is 1
(xi,Ri) Coordinates of streamline a t intersection with shock
- 2 RR”
___
- 1 1 for power-law shock
P
.+ 1 R’a ’
Y
Ratio of specific heats
Characteristic shock slope, equals R ( L ) / Lfor power-law shock, (3.2)
6
Vertex-semi angle of wedge or cone
ab
E
(Y - l ) / ( Y
1)
rb(%)
VN
[ ]
81
a+l
[
+
(M8)-Z
Body perturbation parameter, (4.2)
Similarity variable, r / x m
Ratio of body to shock ordinate, Yb/R
(R,/R)u+*,function of 11 for power-law shock
( j =
P
N / [ W 1)1
E
x
P
Density
0,l for two-dimensional or axisymmetric bodies, respectively
a
Similarity variable for 3, (3.6)
v
Similarity variable for p, (3.6)
Subscripts :
b
Evaluated a t body surface
c
Critical point, dflde = w
e
Associated with streamline bounding entropy layer
S
Downstream side of shock
SB
Sharp blow
SP
Singular point (saddle point)
0
Zero order solution (slender power-law shock)
Perturbation due to el # 0
1
Perturbation due t o 8% # 0
2
rn
Free-stream conditions
Prime indicates differentiation with respect to independent variable
( )’
(usually q or n )
Pa
7
76
+
*
References
1. HAYES,W. D., On hypersonic similitude, Quart. Jour. Appl. Math. 5, 105-106 (1947).
2. VANDYKE,M. D., A study of hypersonic small-disturbance theory. N A C A R e p .
1194 (1954). (Supersedes N A C A T N 3173.)
3. TAYLOR,
G. I., The formation of a blast wave by a very intense explosion, Pt. I,
Theoretical discussion. Proc. Roy. Soc. ( L o n d o n ) , ser. A , 201, 159-174 (1950).
4. TAYLOR,
G. I., The formation of a blast wave by a very intense explosion, Pt. 11.
The atomic explosion of 1945. Proc. Roy. Soc. ( L o n d o n ) , ser. A , 201, 175-188
(1950).
HYPERSONIC FLOW OVER SLENDER BODIES
53
5. SAKURAI,
AKIRA,On the propagation and structure of a blast wave, I, Jour. Phys.
Soc. (Japan) 8, 662-669 (1953).
6. SAKURAI,
AKIRA, On the propagation and structure of a blast wave, 11, Jour.
Phys. Soc. ( J a p a n ) 9, 2 5 6 2 6 6 (1954).
7. LIN, S. C., Cylindrical shock waves produced by instantaneous energy release,
JOUY.APPl. Phys. 26, 54-57 (1954).
8. CHENG,H. K., and PALLONE,
A. J., Inviscid leading-edge effect in hypersonic
flow, Jouv. Aero. Sci. 23, 700-702 (1956).
9. LEES, L., and KUBOTA,T., Inviscid hypersonic flow over blunt-nosed slender
bodies, Jour. Aevo. Sci. 24, 195-202 (1957).
10. KUBOTA,T., Investigation of flow around simple bodies in hypersonic flow,
Mem. 40, Guggenheim Aero. Lab., C.I.T., June 25, 1957. (Contract DA04-495-ORD-19.)
11. HAYES,W. D., and PROBSTEIN,
R. F.. “Hypersonic Flow Theory”. Academic
Press, 1969.
12. SEDOV,L. I., “Similarity and Dimensional Methods in Mechanics” (English Translation). Academic Press, 1959.
13. STANYUKOVICH,
K. P., “Unsteady Motion of Continuous Media”. (English Translation). Pergamon Press, 1960.
14. GRODZOVSKII,
G. L., Certain peculiarities of the flow around bodies a t high supersonic
velocities, Izv. Akad. N a u k , S S S R , Otd. Tekhn. 1957, no. 6, 86-92. (Translated
by Morris D. Friedman, Inc.)
15. CHERNYI,G. G., The effect of slight blunting of the leading edge of a profile on
flow a t high supersonic speeds, Appl. Sci. Sec., Academy Sci. U S S R . 114, 197-200
(May-June 1957).
16. CHERNYI,
G. G., Flow around a thin blunt cone a t high supersonic speeds, Appl.
Sci. Sec., Academy Sci. U S S R 116, 263-265 (July-Aug. 1957).
17. MIRELS,H., Approximate analytical solutions for hypersonic flow over slender
power-law bodies, N A S A T R R-15, 1959.
P. R., Effect of body perturbations on hypersonic
18. MIRELS,H., and THORNTON,
flow over slender power-law bodies. N A S A T R R-45, 1959.
S. S., Cauchy’s problem and the problem of a piston for one-dimen19. GRIGORIAN,
sional, nonsteady motions of a gas (automodel motion). P M M Jour. Appl.
Math. and Mech. 22, 244 (1958).
N. N., and MEL’NIKOVA,N. S., On the steady motion of gas driven
20. KOCHINA,
outward by a piston, neglecting the counter pressure, P M M Jour. Appl. Math.
and Mech. 22, 622 (1958).
21. ADAMSKI,
V. B., and POPOV,
N. A., The motion of a gas under the action of a pressure
on a piston, varying according to a power law, P M M . Jour. Appl. Math. and
Mech. 28, 3 (1959).
22. CHERNYI,G. G . , Application of integral relationships in problems of propagation of strong shock waves, P M M J O U Y . A p p l . Math. and Mech. 24, 1 (1960).
23. CHENG,H. K., HALL,J. G., GOLIAN,
T. C., and HERTZBERG,
A., Boundary-layer
displacement and leading-edge bluntness effects in high temperature hypersonic
flow, Jour. Aevo. Space Sci. 28, 5 (1961).
24. CHENG,H. K., Similitude of hypersonic real-gas flows over slender bodies with
blunted noses. Jour. Aevo./Space Sci. 26, 9 (1959).
25. SYCHEV,
V. V., On the theory of hypersonic gas flow with a power-law shock wave,
P M M Jour. Appl. Math. and Mech. 24, No. 3 (1960).
26. FELDMAN,
S., Numerical comparison between exact and approximate theories of
hypersonic inviscid flow past slender blunt-nosed bodies, A R S Journal 80,
No. 5 (1960).
64
HAROLD MIRELS
27. FELDMAN,
S., On trails of axisymmetric hypersonic blunt bodies flying through
the atmosphere, Jouv. Aero. Space Sci. 28, No. 6 (1961).
R., and TRELLA,M., A study of flow fields about some typical
28. VAGLIO-LAURIN,
blunt-nosed slender bodies, Polytechnic Institute of Brooklyn, PIBAL REP O RT
No. 623, AFOSR 2, Dec. 1960.
29. MIRELS,H., Similarity solutions for inviscid hypersonic flow over power-law and
related bodies, Amer. Rocket SOC.,Preprint 1111-60, 1960.
30. LATTER,R., Similarity solution for a spherical shock wave. Jour. AppZ. P h y s .
26, 954-961 (1955).
31. FREEMAN,
N. C., A note on the explosion theory of Sedov with application t o the
Newtonian theory of unsteady hypersonic flow, Jour. AerolSpace Scz. 27, 7 7 (1960).
32. BROCHER,E. F., Comments on the behavior of Sedov's blast wave solution as
y -+ 1, Jour. AerolSpace Sci. 27, 12 (1960).
33. VON HOERNER,S., Losungen der hydrodynamischen Gleichungen mit linearem
Verlauf der Geschwindigkeit, Z s . Natuvfor. 10a. 687 (1955).
34. HAFELE, W., Zur analytischen Behandlung ebener, starker, instationarer StoDwellen, 2s. Naturfor. 10a, 1006 (1955).
35. ZHUKOV.A. I., and KASHDAN,IA. M., Dvizhenie gaza pod deistviem kratkovremennogo impul'sa (The Motion of a Gas Under the Action of a Short Lived
Impulse), Akust. Z h . 2 , 352 (1956).
36. GUDERLEY,G., Starke kugelige und zylindrische VerdichtungsstbDe in der Nahe
des Kugelmittelpunktes bzw. der Zylinderachse, Luftfahrtforschung 19, 302 (1942).
37. BERTRAM,M. H., and FELLER,W . V., A simple method for determining heat
transfer, skin friction, and boundary-layer thickness for hypersonic laminar
boundary-layer flows in a pressure gradient, N A S A M E M O 5-24-59L, 1959.
E., Similar solutions for the compressible laminar
38. COHEN,C. B., and RESHOTKO,
boundary layer with heat transfer and pressure gradient, N A C A T R 1293, 1956.
39. SMITH,A. M. O., Rapid laminar boundary-layer calculations by piecewise application of similar solutions, Jour. Aero. Sci. 23, 901-912 (1956).
40. COLE, J. D., Lift of slender nose shapes according to Newtonian theory, Rep.
P-1270, Rand Corp., Feb. 1958.
41. GONOR,A. L., Hypersonic flow around a cone at a n angle of attack, Zzv. A N U S S R ,
O . T . N . , No. 7 , 102 (1958). (Translated b y Morris D. Friedman, Inc.)
42. COLE, J . D., Newtonian flow theory for slender bodies, Jour. Aero. Sci. 24, 448
(1957).
D. L., and BERTRAM,
M. H.. The blunt plate in hypersonic flow. N A S A
43. BARADELL,
T N D-408 (1960).
44. CASACCIO,
A., Theoretical pressure distribution on a hemisphere cylinder combination, Jour. Aero/Space Sci. 28, No. 1 (1959).
A., JR., and JOHNSTON,PATRICK
J., Fluid-dynamic properties of
45. HENDERSON,
some simple sharp- and blunt-nosed shapes a t Mach numbers from 16 to 24 in
helium flow, N A S A M E M O 5-8-59L, 1959.
46. CREAGER,
M. O., The effect of leading-edge sweep and surface inclination on the
hypersonic flow field over a blunt flat plate, N A S A M E M O 12-26-58A. 1959.
M. H., and HENDERSON,
A., JR., Effects of boundary-layer displacement
47. BERTRAM,
and leading edge bluntness on pressure distribution, skin friction, and heat
transfer of bodies at hypersonic speeds. N A C A T N 4301, 1958.
48. VAS, I. E., and BOGDONOFF,
S. M., Mach and Reynolds number effects on the
flows over blunt flat plates a t hypersonic speeds, Princeton University Rept. 529,
ARL Tech. Note 60-164, 1960.
The Mathematical Thsory of Equilibrium Cracks in Brittle Fracture
BY G . I . BARENBLATT
Institute of Geology and Development of Combustible Minerals of the U.S.S.R. Academy
of Sciences. Moscow. U.S.S.R.'
Page
.
............................
56
I1. The Development of the Equilibrium Crack Theory . . . . . . . . . . 62
I11. The Structure of the Edge of an Equilibrium Crack in a Brittle Body . . 89
I Introduction
1. Stresses and Strains Near the Edge of an Arbitrary Surface of Discontinuity of Normal Displacement . . . . . . . . . . . . . . . . . 69
2 . Stresses and Strains Near the Edge of an Equilibrium Crack . . . . . 73
3 . Determination of the Boundaries of Equilibrium Cracks . . . . . . . 74
.
IV Basic Hypotheses and General Statement of the Problem of Equilibrium
Cracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1. Forces of Cohesion; Inner and Edge Regions; Basic Hypotheses . . .
2 . Modulus of Cohesion . . . . . . . . . . . . . . . . . . . . . . .
3 . The Boundary Condition a t the Contour of an Equilibrium Crack . .
4 . Basic Problems in the Theory of Equilibrium Cracks . . . . . . . .
5 . Derivation of the Boundary Condition a t the Contour of an Equilibrium
Crack by Energy Considerations . . . . . . . . . . . . . . . . . .
8. Experimental Confirmation of the Theory of Brittle Fracture; Quasi-Brittle
Fracture
7 . Cracks in thin Plates . . . . . . . . . . . . . . . . . . . . . . .
...........................
76
76
80
81
82
84
85
89
V. Special Problems in the Theory of Equilibrium Cracks . . . . . . . . . 90
1. Isolated Straight Cracks . . . . . . . . . . . . . . . . . . . . . .
90
2 . Plane Axisymmetrical Cracks . . . . . . . . . . . . . . . . . . . 98
3 . The Extension of Isolated Cracks Under Proportional Loading; Stability
97
of Isolated Cracks . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 103
4 . Cracks Extending to the Surface of the Body
5. Cracks Near Boundaries of a Body; Systems of Cracks . . . . . . . 107
8 Cracks in Rocks . . . . . . . . . . . . . . . . . . . . . . . . .
112
.
VI . Wedging; Dynamic Problems in the Theory of Cracks . . . . . . . . . 114
1 Wedging of an Infinite Body . . . . . . . . . . . . . . . . . . . 114
2 . Wedging of a Strip . . . . . . . . . . . . . . . . . . . . . . . .
119
3 . Dynamic Problems in the Theory of Cracks . . . . . . . . . . . . 121
.
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
* Present address: Institute
Moscow. USSR .
of
Mechanics.
56
Moscow
State
125
University.
56
G . I . BARENBLATT
I. INTRODUCTION
In recent years the interest in the problem of brittle fracture and, in
particular, in the theory of cracks has grown appreciably in connection with
various technical applications. Numerous investigations have been carried
out, enlarging in essential points the classical concepts of cracks and the
methods of analysis. The qualitative features of the problems of cracks,
associated with their peculiar non-linearity as revealed in these investigations, makes the theory of cracks stand out distinctly from the whole range
of problems in the present theory of elasticity. The purpose of the present
paper is to present a unified view of how the basic problems in the theory
of equilibrium cracks are formulated, and to discuss the results obtained.
FIG.1.
FIG.2.
The object of the theory of equilibrium cracks is the study of the equilibrium of solids in the presence of cracks. Consider a solid having cracks
(Fig. 1) which are in equilibrium under the action of a system of loads. The
body, able to sustain any finite stresses, is assumed to be perfectly brittle,
i.e. to retain the property of linear elasticity up to fracture. The possibility
of applying the model of a perfectly brittle body to real materials will be
discussed later.
The opening of a crack (the distance between the opposite faces) is always
much smaller than its longitudinal dimensions; therefore cracks can be
considered as surfaces of discontinuity of the material, i.e. of the displacement
vector. Henceforth, unless the contrary is stated, plane cracks of normal
discontinuity are considered, i.e. cracks are pieces of a plane bounded by
closed curves (crack colztozlrs), at which only the normal component of the
displacement vector has a discontinuity. The case when the tangential
component of the displacement vector is discontinuous a t the discontinuity
surface can be treated in the same manner.
One might think that the investigation of the equilibrium of elastic
bodies with cracks can be carried out by the usual methods of the theory
of elasticity in the same way as it is done for bodies with cavities (Fig. 2).
MATHEMATICAL THEORY OF EQUILIBRIUM CRACKS
57
However, there exists a fundamental distinction between these two problems,
The form of a cavity undergoes only slight changes even under a considerable
variation in the load acting upon the body, whereas cracks, whose surface
also constitutes a part of the body boundary, can expand a good deal even
with small increase of the load, to which the body is subjected. (In Figs. 1
and 2, dotted lines indicate additional loads and the corresponding positions
of the body boundaries.)
Thus, one of the basic assumptions of the classical linear theory of
elasticity is not satisfied in problems of the theory of cracks, namely the
assumption about the smallness of changes in the boundaries of a body under
loading, which permits one to satisfy the boundary conditions at the surface
of the unstrained body. This fact makes the problem of the equilibrium of
a body with cracks, unlike traditional problems of the theory of elasticity,
essentially non-linear. In the theory of cracks one must determine from the
condition of equilibrium not only the distribution of stresses and strains
but also the boundary of the region, in which the solution of the equilibrium
equations is constructed.
Non-linear problems of this type (“problems with unknown boundaries”)
have long been known in various fields of continuum physics. Suffice it to
mention the theory of jets and the theory of finite-amplitude waves in
hydrodynamics, the theory of flow past bodies in the presence of shock waves
in gas dynamics, Stefan’s problem of freezing in the theory of heat conduction, etc. The main difficulty in all these problems lies in the determination
of the boundary of the region in which the solution is sought. Likewise, the
basic problem in the theory of equilibrium cracks is the determination of the
surfaces of cracks when a given load is applied.
The differential equations of equilibrium and the usual boundary conditions of the theory of elasticity cannot in principle give the solution of this
problem without the introduction of some additional considerations. This
may be seen from the fact that we can construct a formal solution of the
equations satisfying the usual boundary conditions no matter how we prescribe
crack surfaces. The analysis of these solutions shows that in general the
tensile stress u normal to the surface of a crack is infinite a t the crack contour.
More exactly, near an arbitrary point of the crack contour
N
u =-
VS
+
finite quantity,
where s is the distance of a point of the body lying in the plane of a crack
from the crack contour, N is the stress intensity factor, a quantity dependent
on the applied loads, the form of the crack contour, and the coordinates of
the point considered, but independent of s. The form of a normal section
of the deformed crack surface near the contour appears in such cases
unnaturally rounded (as in Fig. 3 or somewhat different; see details below).
58
G . I. BARENBLATT
Generally speaking, however, there exist such exceptional contours of
cracks for which stresses at the edges are finite ( N = 0) under a given load;
at the same time the opposite faces of cracks close smoothly at the edges.
The form of a section of the crack surface near the edge appears then as a
cusp, cf. Fig. 4. It can be shown that for such contours, and only for them,
the energy released by a small change in the contour of a crack is equal to
zero. It follows that only such contours can bound equilibrium cracks.
FIG.3.
FIG.4.
Thus, when all loads acting upon a body are given, the problem of the
theory of equilibrium cracks may be formulated as follows: for a given
position of initial cracks and a given system of forces acting upon the body
one requires the determination of the stresses, the strains, and the contours
of cracks so as to satisfy the differential equations of equilibrium and the
boundary conditions, and to insure finiteness of stresses (or, which is the
same, a smooth closing of the opposing faces at the crack edges). If the
position of the initial cracks is not given, then, since according to our model
the body can sustain any finite stress, the solution of the problem formulated
above is not unique. This is only natural because at one and the same load
in one and the same body there need not be any cracks, or there may be
one crack, or two, and so on.
In the general case of curved cracks, the shape is determined not only
by the load existing at a given moment but also by the whole history of
loading. If however, the symmetry of the body and the applied monotonically
increasing loads assure the development of plane cracks, then the contours
of cracks are determined by the current load alone. All the results at
present available in the theory of cracks correspond to particular cases of
this simplified formulation of the problem.
A given system of forces acting upon the body should in general include
not only the loads applied to the body. The following example illustrates
what is meant. Let us attempt to determine the contour of an equilibrium
crack in the case of the loads represented in Fig. 1. If, in accordance with
the usual approach in the theory of elasticity (as in the case of the cavity
shown in Fig. 2), the surface of the crack is considered to be free of stresses,
the result will be paradoxical: whatever contour of the crack we would
MATHEMATICAL THEORY OF EQUILIBRIUM CRACKS
59
take, the tensile stresses at its edge are always infinite. Consequently, there
cannot exist an equilibrium crack; however small the force of extension
may be, the body that has a crack breaks in two!
Such an obvious lack of agreement with reality can be easily explained:
simply using the model of an elastic body, we have not taken into consideration all forces acting upon the body. It appears that - and this is also one
of the main distinctions between the problems of the theory of cracks and
the traditional problems of the theory of elasticity - for developing an
adequate theory of cracks it is necessary to consider molecular forces of
cohesion acting near the edge of a crack, where the distance between the
opposite faces of the crack is small and the mutual attraction strong.
Although consideration of forces of cohesion settles the matter in principle,
it complicates a great deal the analysis. The difficulty is that neither the
distribution of forces of cohesion over the crack surface nor the dependence
of the intensity of these forces on the distance between the opposite faces
are known. Moreover, the distribution of forces of cohesion in general
depends on the applied loads. However, if cracks are not too small, there is
a way out of the difficulty: with increasing distance between the opposite
faces the intensity of forces of cohesion reaches very quickly a large maximum,
which approaches Young’s modulus and then diminishes rapidly.
Therefore two simplifying assumptions can be made. The first is that the
area of the part of the crack surface acted upon by the forces of cohesion
can be considered as negligibly small compared to the entire area of the
crack surface. According to the second assumption the form of the crack
surface (and, consequently, the local distribution of forces of cohesion)
near the adges, a t which the forces of cohesion have the maximum intensity,
does not depend on the applied load.*
The intensity of the forces of cohesion has the highest possible value for
a given material under given conditions. This happens for instance a t all
edge points of a crack formed at the initial rupture of the material as the load
increases. For most real materials cracks are irreversible under ordinary
conditions. If an irreversible crack is produced by an artificial cut without
subsequent expansion or is obtained from a crack that existed under a greater
load by diminishing the load, then the intensity of forces of cohesion a t the
crack contour will be lower than the maximum possible one. The forces of
cohesion that act a t the surface of a crack compensate the applied extensional
loads and secure finiteness of stresses and smooth closing of the crack faces.
With an increase in extensional loads the forces of cohesion grow, thus adjusting themselves to the increasing tensile stresses, and the crack does not
* Sh. A . Sergaziev very neatly compared cracks for which these assumptions are
satisfied with “zippers”.
60
G . I. BARENBLATT
expand further until the highest possible intensity of forces of cohesion is
reached. The crack starts expanding+ only upon reaching the highest
possible intensity of forces of cohesion at the edge.
Successive expansion of the crack edge under increasing extensional
load is represented schematically in Fig. 5.
FIG. 5.
1.2. The intensity of forces of cohesion is less than the maximum.
3,4. The intensity of forces of cohesion is equal to the maximum.
If use is made of the first of the above assumptions, molecular forces of
cohesion will enter in the conditions that determine the position of crack
edges only in the form of the integral
d
where G ( t ) is the intensity of the forces of cohesion acting near the crack
edge, t is the distance along the crack surface taken along the normal to the
crack edge, and d is the width of the region subject to the forces of cohesion.
For those contour points, at which the second assumption applies, this
integral represents a constant of the given material under given conditions
(temperature, composition, and pressure of the surrounding atmosphere, etc.),
which determines its resistance to the formation of cracks. It can be shown
that the quantity K is related to the surface tension of the material To,
the modulus of elasticity E , and Poisson's ratio v by means of the simple
equation
Furthermore, for all points of the crack edge at which the intensity of
forces of cohesion is a maximum, the stress-intensity factor N , entering in
+ Quite a similar situation arises when a body moves over a rough horizontal
surface under the action of a horizontal force. The motion of the body begins only after
the force exceeds the highest possible value of the friction for the given body and the
given surface.
MATHEMATICAL T H E O R Y OF EQUILIBRIUM CRACKS
61
(1.1) and calculated without taking into account the forces of cohesion
should be equal to Kln. For all points of the edge at which the intensity
of forces of cohesion is below the maximum, the stress intensity factor
calculated without considering forces of cohesion is smaller than Kln.
The foregoing considerations elucidate sufficiently the nature of the
forces of cohesion involved in this problem, and it is now possible to formulate
the fundamental problem of the theory of equilibrium cracks.* When the
symmetry of the body, of the initial cracks, and of the monotonously increasing forces insures development of a system of plane cracks, this problem
can be stated as follows.
Let a system of contours of initial cracks be given in a body. I t is required
to find the stress and displacement fields corresponding to a given load as
well as a system of contours of plane cracks surrounding the contours of the
initial cracks (and perhaps coinciding with them partly).
Mathematically the problem consists in constructing such a system of
contours that the factor of intensity of the tensile stress, calculated without
taking into account the forces of cohesion, should be equal to K l n at all
edge points, not lying on the contours of the initial cracks, and should not
exceed K l n a t all points of contours, lying on the contours of the initial
cracks.
The foregoing formulation of the problem eliminates from our direct
consideration the molecular forces of cohesion (they enter only through
the constant K ) . Therefore stress and strain fields furnished by the solution
of this problem will not be realistic in a small neighbourhood of the crack
edges.
When cracks are reversible, or when the applied load is great enough to
cause the contours of all the cracks to expand beyond the contours of the
initial cracks, the form of the latter evidently is no longer of any importance.
The equilibrium state corresponding to the highest possible intensity
of forces of cohesion a t least at one point of the crack contour can be stable
or unstable. Accordingly, further extension of the crack with increasing
load proceeds in essentially different ways. In the case of stable equilibrium,
a slow quasi-static transition of the crack from one equilibrium state to
another takes place, when the load is increased gradually. If the equilibrium
is unstable, the slightest excess over the equilibrium load is accompanied by
a rapid crack extension that has a dynamic character. In some cases, when
there exist no neighbouring stable states of equilibrium, this leads to the
complete rupture of the body. The theory of cracks developed in such a
way that problems of this latter type were mainly treated until recently.
* Such general formulations of problems seem advisable to us despite the fact
that their general solution in effective form exceeds by far the possibilities of present
mathematics. General statements of problems are a help in realizing the meaning of
specific solutions and difficulties arising in developing the theory.
62
G. I. BARENBLATT
Sometimes the condition for the onset of crack extension is therefore
identified with the condition for complete fracture of the body. It should
be clearly understood, however, that this is only true in special cases, the
practical significance of which must not be exaggerated.
Below, following a brief outline of the development of the mathematical
theory of cracks, the fundamentals of the theory of equilibrium cracks are
given as well as the results for the most typical special problems treated
hitherto. A t the end of this review dynamic problems in the theory of cracks
are discussed briefly.
When writing this article the author endeavoured to avoid the repetition
of available presentations of some aspects of brittle fracture. Thus the
review deals with the theory of cracks proper, i.e. with the mathematical
theory of brittle fracture. The numerous available experimental investigations are referred to only inasmuch as they are necessary for confirming the
theory presented and establishing the limits of its applicability. Experimental investigations of brittle fracture, unlike the mathematical theory, were
discussed more than once in reviews and monographs. At the same time,
questions concerning exclusively mathematical techniques of solving the
problems of elasticity theory are discussed only briefly, if a t all. Also the
question of the formation of the initial cracks will not be touched.
Trying to preserve a unified point of view in discussing certain results
of other investigators, the author permitted himself sometimes a deviation
from the original treatment.
11. THE DEVELOPMENT
OF THE EQUILIBRIUM
CRACKTHEORY
Investigations in the field of the theory of cracks were started by
C. E. Inglis [ l ] about fifty years ago. His paper presents the solution of a
problem within the classical theory of elasticity concerning the equilibrium
of an infinite body with an isolated elliptical cavity (in particular, with a
straight-line cut) in a uniform stress field. N. I. Muskhelishvili [2] - also
within the classical theory of elasticity - obtained in a simpler and more
effective form the solution of a problem concerning the equilibrium of an
infinite body having an elliptical cavity in an arbitrary stress field.
However, in spite of their outstanding significance for subsequent
investigations, papers [l, 21 did not prepare the foundations for the theory
of cracks proper. The fact is that the solutions obtained in these papers
possess two properties which were difficult to explain. First, the length of
a crack was found to be indefinite at a given load so that it was possible to
construct a solution with an arbitrary value of this parameter. Everyday
experience suggests nevertheless that the dimensions of cracks existing in
a body should be connected somehow with the extensional loads applied to
MATHEMATICAL THEORY OF EQUILIBRIUM CRACKS
03
the body. As the load increases, cracks existing in the body do not expand
at first when the load is small; upon reaching a certain load they begin to
expand, the expansion depending on the manner in which the load is applied.
In some cases cracks expand rapidly up to complete rupture of the body with
the load maintained constant, in other cases they expand slowly, stopping
as soon as the increase of the load is suspended. Since the opening of a crack
is usually small compared to its longitudinal dimensions, it is natural to
represent a crack as a cut; but then the tensile stresses at the crack edges
in Inglis’ problem are infinite, and in general the same thing happens in the
problem treated by Muskhelishvili. Clearly solutions with infinite tensile
stresses at the edges of a crack are unacceptable in a physically correct
model of a brittle body. Thus, direct application of the classical scheme of
the theory of elasticity to the problem of cracks leads to a problem which
is incomplete and yields physically unacceptable solutions.
A. A. Griffith’s papers [3, 41 are rightly considered fundamental for the
theory of cracks of brittle fracture. The important idea, first advanced in
these papers is that an adequate theory of cracks requires the improvement
of the model accepted for a brittle body by the consideration of molecular
forces of cohesion acting near the edge of a crack.
Griffith treated the following problem : An infinite brittle body stretched
by a uniform stress Po at infinity has a straight crack of a certain size 21.
I t is required to determine the critical value of Po at which the crack
begins to expand. The molecular forces of cohesion were considered as
forces of surface tension being internal forces for the given body; their
effect on the stress and strain field was neglected.
Under this condition the change dF of free energy (“total potential
energy” in Griffith’s terminology) of a brittle body with a crack, compared
to the same body under the same loads but without a crack, is equal to the
difference between the surface energy of the crack U and the decrease in
strain energy of the body due to formation of the crack W . For the crack
to expand, the change in free energy of the body must not grow with an
increase in the size 21 of the crack. Thus, the parameters of the critical
equilibrium state are obtained from the condition
a(u- w) = 0.
ai
But the surface energy of the crack U is equal to the product of the
surface area of the crack and the energy To required to form the unit surface
of the crack. Under certain sufficiently general assumptions, the quantity To,
the surface tension, can be considered constant for a given material under
given conditions. Therefore, according to Griffith, the determination of
the critical load reduces to the determination of the quantity aWla1, “the
elastic energy release rate”. Analysing the simplest case, Griffith calculated
64
G . I . BARENBLATT
this quantity by using Inglis’ results [l] and obtained relations determining
the critical values of tensile stress in the forms
for plane strain and plain stress, respectively.
The theoretical part of Griffith’s paper contains also the results of the
investigation of the structure of a crack near its ends. This is carried out
on the basis of the classical solution of elasticity theory, constructed without
considering forces of cohesion, hence with infinite tensile stresses at the ends
of the crack, if it has the shape of a cut. Griffith made an attempt to improve
this description of a crack by considering it as an elliptical cavity with a
finite radius of curvature p at the end (Fig. 3). However, according to his
own estimate the magnitude of the radius of curvature at the end of the
crack was of the order of the intermolecular distance, which clearly indicates
the incorrectness of the approach: in any investigation based on the concept
of a continuous medium distances of intermolecular order of magnitude
cannot be considered as finite.
This part of Griffith’s work is inadequate for the following reason. In
determining the equilibrium size of a crack, the effect of molecular forces
of cohesion on the stress and strain fields can be neglected, but this cannot
be done in analysing the structure of a crack near its ends. The distance at
which the effect of forces of cohesion is appreciable is comparable to the
distance over which the form of a crack varies essentially. Therefore, to
a considerable part, Griffith’s analysis of the structure of crack edges cannot
be accepted as correct, and in particular his conclusion concerning the
rounded form of cracks near the ends is wrong, as will be shown in detail
later. This aspect of the matter, obviously of prime importance, remained
unclarified until recently and led in a number of cases to misinterpretations
of Griffith’s results [B].
In addition t o the basic shortcoming pointed out here, there were some
errors in calculations in the theoretical part of the paper [3]. Shortly after
it had appeared, A. Smekal [6] published a detailed comment on it, containing
also quite an interesting general discussion of the problem of brittle fracture
and correcting the errors.
In a subsequent paper by K. Wolf [7] a more precise and simpler account
of Griffith’s results was given, and similar calculations were made for
somewhat different (but also uniform) states of stress. In [7] the relation of
Griffith’s theory of fracture to previously proposed theories of strength
was also discussed.
In connection with his experiments on the splitting of mica I. V. Obreimov
investigated [8] the tearing-off of a thin shaving from a body by a splitting
wedge that slides over its surface and has a single point of contact with the
MATHEMATICAL T H E O R Y OF EQUILIBRIUM CRACKS
65
shaving. Using the approximate methods of thin-beam theory, Obreimov
established the relation between the form parameters of a crack and the
surface tension by means of an energy method similar to that used in
Griffith’s paper. The method of paper [8] was continued later by many
investigators [9-121.
The determination of the elastic energy release rate awlal for tensile
stress fields more complex than a uniform one, as well as for other configurations of cracks encountered considerable mathematical difficulties. The
investigations of H. M. Westergaard [13], I. N. Sneddon [14, 151, I. N.
Sneddon and H. A. Elliot [16], M. L. Williams [17] clarified the distribution
of stresses and strains near the discontinuity surfaces of the displacement.
Together with the classical papers by Muskhelishvili [ 2 , 18, 191 the investigations of Westergaard and Sneddon constitute the mathematical basis of
subsequent works on the theory of cracks. However, the conditions of
equilibrium for new particular cases and, still less, for a somewhat more
general case of loading were not obtained in these papers.
In the papers by R. A. Sack [20], T. J . Willmore [21], and 0. L. Bowie
[22] the conditions of equilibrium were obtained for some new special cases
of loading and position of cracks. The energy method was applied directly
in these papers, and thus considerable difficulties in the calculations had
to be overcome. In view of the fact that the equilibrium states in the problems
treated in [20-221 are unstable and unique, the conditions of equilibrium
are identical with those for complete fracture of the body.
The papers by G. R. Irwin [23] and E. 0. Orowan [24], in which the
concept of quasi-brittle fracture was developed, represent an important
stage in the theory of cracks. Irwin and Orowan noticed that a number
of materials, which behave as highly ductile in standard tensile tests, fracture
by a quasi-brittle mechanism when cracks are forming. This means that the
arising plastic deformations are concentrated in a very narrow layer near
the surface of a crack. As was shown by Irwin and Orowan, it is possible
in such cases to employ Griffith’s theory of brittle fracture, introducing
instead of surface tension the effective density of surface energy. This
quantity, in addition to the specific work required to produce rupture of
internal bonds (= surface tension), includes the specific work required to
produce plastic deformations in the surface layer of a crack; it is sometimes
several orders of magnitude larger than the surface tension.
The idea of quasi-brittle fracture extended considerably the range of
applicability of the theory of brittle fracture and was undoubtedly one of
the main reasons for reviving interest in this problem. Irwin, Orowan and
other authors published a series of papers [23-321 devoted to the development of the generalized theory of brittle fracture, to the investigation
of the limits of its applicability, and to the analysis of experimental data
from the view point of this theory. Special notice deserves the paper by
66
G . I. BARENBLATT
H. F. Bueckner [33] in which a quite general energy analysis of brittle and
quasi-brittle fracture was carried out on the basis of the Griffith-IrwinOrowan scheme.
In all the foregoing papers the question of the structure of a crack near
its edge remained without clarification. In a very interesting paper [34]
devoted to the physico-chemical analysis of deformation processes,
P. A. Rebinder first expressed the thought about the wedge-like form of a
crack at its ends and about the necessity of a corresponding development
of Griffith’s theory. H. A . Elliot [35], N. F. Mott [36], and Ya. I. Frenkel [5],
in analysing the form of a crack, proceeded from the idea of a crack of
infinite length between two solid blocks of the material, which were at
normal intermolecular distance from each other before formation of the
crack.
In [35] the blocks were considered to be semiinfinite. Starting from
the classical solution for a straight-line crack [l] and a disk-shaped crack
[20] having a diameter 2c in a uniform tensile stress field fi, the distributions
of normal stresses cry and lateral displacements v were determined in [35]
for points of the planes distant half the normal intermolecular distance from
the crack plane. The function ay(2v)containing fi and c as parameters was
identified with the relation between molecular forces of cohesion and the
distance ; by integrating this function, the surface tension was determined,
which thus was found to be connected with p and c. The author identified
this relation with the condition of fracture, which of course differed from
Griffith’s condition. The distribution of the lateral displacements so obtained
was identified with the form of the crack.
Such an approach is inadequate for the following reasons. The formal
application of the apparatus of classical elasticity for the determination of
stresses and deformations near the edge of a crack is unjustifiable, since in
applying this apparatus all distances (even those which are considered small)
must be large compared to the intermolecular distance. Moreover, forces
of cohesion act not only inside the body but also on a part of the crack surface.
If this fact is taken into account, the edges of a crack have a pointed rather
than a rounded shape, and there is no infinite stress concentration at the ends.
This will be shown below in detail. Thus, stress and displacement distributions near the edge of the crack surface differ essentially from the corresponding distributions obtained according to the solutions of Inglis [11
and Sack [20], in which the surface of cracks was supposed to be free of
stress. Note also that the decrease of a,(2v) with increasing v is very slow
in paper [35], much slower indeed than the natural velocity of diminution
of the intensity of forces of cohesion,
Ya. I. Frenkel [5] treated the problem of a crack of infinite length cutting
through a thin strip in longitudinal direction. The use of the approximate
theory of thin beams, which is unsuitable for analysing the form of a crack
MATHEMATICAL THEORY OF EQUILIBRIUM CRACKS
67
near its ends, did not permit him to obtain an adequate result. Incidentally,
the comments on Griffith’s theory contained in this paper cannot be accepted
as well justified either. Frenkel criticizes Griffith because of the instability
of the equilibrium in the case of a straight crack in a uniform tensile stress
field (considered by Griffith) and he ascribes this instability to Griffith’s
wrong idea about the form of the crack ends. This is not true. The conclusion
about stability or instability of equilibrium of a crack does not depend on
considerations concerning the structure of the crack ends. As will be shown
later, instability of a crack in a uniform field occurs even when allowance
is made for smooth closing of cracks a t the ends; it is a part of the problem
itself rather than a consequence of the peculiar crack shape assumed.
Frenkel’s conclusion concerning the existence of a stable state of equilibrium
in addition to the unstable one is due to his incorrect replacement of the
uniform state of stress by another one.*
In a paper by A. R. Rzhanitsyn [37] an attempt was made to solve the
problem of a circular crack in a body subjected to a uniform tensile stress
under consideration of the molecular forces of cohesion distributed over
the crack surface and with smooth closing of the crack. Unfortunately the
application of inadequate methods (averaging stresses and strains) did not
allow the author to obtain the correct conditions of equilibrium.
An idea first suggested by S. A. Khristianovitch [38] was of great importance for the proper understanding of the structure of cracks near their ends.
Khristianovitch considered, in connection with the theory of the so-called
hydraulic fracture of an oil-bearing stratum, an isolated crack in an infinite
body under a constant all-round compressive stress a t infinity, maintained
by a uniformly distributed pressure of a fluid contained inside the crack.
The problem was treated in the quasi-static formulation. In solving it,
Khristianovitch hit upon the indefiniteness of the crack length. He noticed,
however, the following circumstance. Under the assumption that the fluid
fills the crack completely, tensile stresses a t the end of the crack are always
infinite, whatever the size of the crack. But if the fluid fills the crack only
partially, so that there is a free portion of the crack surface which is not
wetted by the fluid, then at one exceptional value of the crack length tensile
stresses a t the ends of the crack are finite. I t turned out that for this value
of the crack length (and only for this one) the opposite faces of the crack
close smoothly at its edges. Khristianovitch advanced a hypothesis of
finiteness of stresses or, which is the same, of smooth closing of the opposite
faces of a crack a t its edges as a fundamental condition determining the
size of a crack. The use of this hypothesis made it possible to solve a number
of problems concerning formation and expansion of cracks in rocks [38--431.
* Besides these basic shortcomings there are some errors in calculations in [5]
indicated in [37].
68
G. I. BARENBLATT
In all these papers, however, molecular forces of cohesion were not taken
into account directly. Now in dealing with cracks in rock massifs it is quite
permissible to neglect forces of cohesion. The estimates show that the effect
of rock pressure is far greater here than the action of forces of molecular
cohesion, particularly if the natural fissuring of rocks is taken into consideration. Under other conditions (in particular, in many cases when massifs
are simulated in laboratories) forces of cohesion play an important part and
their consideration is of great significance in analysing the conditions of
equilibrium and expansion of cracks.
A very interesting early work by H. M. Westergaard [44] should be
mentioned in connection with these investigations (see also [13]). On the
basis of the analogy with the contact problem noted by the author, it is
stated that there is no stress concentration at the end of a crack in such
brittle materials as concrete. The same paper gives formulas which describe
correctly stresses and strains near the ends of equilibrium cracks of brittle
fracture in the absence of forces of cohesion. However, Westergaard did
not connect the condition of finiteness of stress with the determination of
the longitudinal dimension of a crack, which he assumed to be given.
I n papers [45, 461 by G. R. Irwin (see also [47, 48, 49, 331) an important
formula was established that correlates the strain-energy release rate
with the stress intensity factor near the ends of a crack in a problem of the
classical theory of elasticity. On the basis of this formula the strain-energy
release rate was determined, and the conditions of fracture were obtained
for several new cases of loading and position of cracks [47, 50, 32, 51, 521.
Beginning with the work of Griffith, in most of the theoretical investigations problems of a similar type were treated: the equilibrium state,
.,in which the intensity of forces of cohesion at the contour is a maximum,
turns out to be unstable, and the condition for the onset of expansion of a
crack coincides with the condition for the beginning of complete fracture
of the body. Thus the condition for onset of the expansion is identified in
some papers with the onset of rapid crack propagation and fracture for all
cracks. In general, that is not true. Cracks actually may be stable so that
the beginning of crack development is not necessarily connected with the
fracture of a body; and one should not imagine that stable cracks are rare,
that they are not encountered in practice and are difficult to produce
experimentally. As the experimental investigations carried out by numerous
authors beginning from I. V. Obreimov [8] show, the extension of cracks
is stable in many cases throughout the greater part of the process of fracture.
A. A. Wells [30] obtained stable cracks over a certain range of extensional
forces in steel plates under combined external tensile stresses and internal
stresses due to welded seams. F. C. Roesler [53] and J. J. Benbow [54]
investigated stable conical cracks in glass and silica. The same authors [ Q ]
obtained stable cracks in wedging a strip of organic glass. Recently
MATHEMATICAL THEORY OF EQUILIBRIUM CRACKS
69
J. P. Romualdi and P. H. Sanders [52] obtained stable cracks within certain
limits of loads for a tensile plate stiffened by riveted ribs. References to
other investigations in which stable cracks were obtained and analysed can
be found in a monograph by B. A. Drozdovsky and Ya. B . Fridman [65].
All these papers confirm strongly the possibility of using the concept of
brittle and quasi-brittle fracture for stable cracks.
Consideration of stable cracks greatly extends the problems that can be
formulated in the theory of equilibrium cracks. Indeed, in the case of
unstable cracks, only the determination of the load a t which a crack begins
to expand is of interest, since the process becomes dynamic upon reaching
this load. In the case of stable cracks, however, one has to investigate the
quasi-static expansion of cracks with change in loads.
In papers [56-611 the formulation of problems in the theory of equilibrium cracks of brittle fracture was improved and supplemented in accordance
with the foregoing considerations. In these papers a new approach to
problems of the theory of cracks was proposed, which is based on the general
formulation of problems concerning elastic equilibrium of bodies in the
presence of cracks, as it was given in [40]. The further discussions in this
review are based on this approach, which is presented in the following
chapter. A number of new problems of the theory of cracks were formulated
and solved on that basis.
111. THE STRUCTURE
OF THE EDGEOF AN EQUILIBRIUM
CRACK I N A
BRITTLE
BODY
1. Stresses and Strains Near the Edge of an Arbitrary Surface of Discontinuity
of Normal Displacement
As has already been pointed out, one can construct a formal solution of
the differential equations of the theory of elasticity, which satisfies the
boundary conditions corresponding to the applied load, if one prescribes
arbitrarily a surface of discontinuity of the displacement. In the present
section the behavior of the solutions of the equations of elasticity near the
edge of a surface of discontinuity of displacement is investigated. For
simplicity we shall restrict ourselves here to surfaces of discontinuity of
normal displacement, appearing as plane faces bounded by closed curves
(contowrs).
Near an arbitrary point 0 a t the contour of such a surface, let us take
a vicinity whose characteristic dimension is small compared to the radius
of curvature of the contour at the point 0. Deformation in this vicinity
can be considered as plane and corresponding to a straight infinite cut in
70
G. I. BARENBLATT
an infinite body subjected to a system of symmetrical loads (see Fig. 8;
the plane of deformation is a plane normal to the contour of the discontinuity
surface at the point 0; the trace of the cut in the drawing is the intersection
of that plane with the discontinuity surface). Loads can be applied a t the
surface of the cut and inside the body; the loads a t the surface can be
assumed to be normal without losing the generality of the further analysis.
Consider now this configuration in more detail.
FIG.8.
The stress and displacement fields can be presented as the sum of two
fields (Fig. 6), the first of which corresponds to a continuous body under
loads applied inside the body; the second belongs to a body with a cut,
symmetrical loads being applied at the surface of the cut only. The shape
of the deformed surface of the cut is determined by the second state of
stress, since normal displacements at the place of the cut for the first state
of stress are equal to zero by symmetry.*
The analysis of the first state of stress can be carried out by the usual
methods of the theory of elasticity and is of no special interest; we shall
consider this state of stress as given. Let us assume that the line of the cut
corresponds to the positive semi-axis x ; the normal stresses, g ( x ) , applied
at the surface of the cut in the second state of stress, represent the difference
between the stresses applied a t the surface of the cut in the actual field,
G ( x ) , and the stresses at the place of the cut, P ( x ) , corresponding to the first
state of stress.
Applying Muskhelishvili’s method [18] to the analysis of the second state
of stress, we obtain the relations determining stresses and displacements
(34
(3.2)
+
ax(2) a,(2)= 4 Re @(z),
u p - ia:;
= @(z)
+ Q(2) + - 2 ) @ 3 ,
(2
This convenient method of reducing the load to a load distribution over the
discontinuity surface was developed in the most general form by H. F. Bueckner [33].
MATHEMATICAL THEORY OF EQUILIBRIUM CRACKS
71
x = 3 - 4v,
+
where z = x
i y ; U,(~),U~(~),CT~,!
are the components of the stress tensor
of the second state of stress; d2),d2)are the displacement components
along the x and y axes corresponding to the second state of stress;
,u = E / 2 ( 1
Y) is the shear modulus, E is Young's modulus, and v is
Poisson's ratio. The analytical functions p,u,@,Qare expressed by formulas
+
(3.4)
[VFy
@ ( z ) = . n ( z ) = p'(z) = o ' ( z ) = w
1
~,
2ni v z
0
(3.5)
A t the cut ( x 2 0, y = 0 ) and its prolongation ( x
following relations hold :
< 0, y = 0) the
Using known formulas for limiting values of a Cauchy-type integral a t
the ends of the contour [19], we obtain an expression for the tensile stresses
near the end of the cut along its prolongation,
(3.7)
where s1 is the small distance of the point considered from the end of the cut
(Fig. 6 ) . Similarly, we have for the distribution of normal displacements of
points a t the surface of the cut near its end
where s2 is the distance of a surface point of the cut from its end, and negative
and positive signs correspond to the upper and lower faces of the cut,
respectively (Fig. 6 ) .
72
G . I. BARENBLATT
This result also fully elucidates the distribution of normal tensile stresses
and normal displacements near the contour of an arbitrary surface of normal
discontinuity. Indeed, the following formulas are readily obtained from
relations (3.7) and (3.8):
where uy is the tensile stress at a point of the body a small distance s1 away
from the contour of the discontinuity surface, lying in the osculating plane
to the contour of the discontinuity surface through the point 0; N is the
4"
a
f"
b
f"
0
C
X
FIG. 7.
stress intensity factor, a quantity dependent on the acting loads, on the the
configuration of the body and of the discontinuity surfaces in it, and on the
coordinates of the point 0 considered; G(0) is the magnitude of the normal
stress applied to the discontinuity surface at 0 (Fig. 6 ) ; s2 is the small
distance of a point of the discontinuity surface from its contour. Depending
on the sign of N , there are in general three possibilities.
If N > 0 , an infinite tensile stress acts a t the point 0. The shape of the
deformed discontinuity surface and the distribution of normal stresses uy
near the point 0 are represented in Fig. 7a.
If N < 0 , then an infinite compressive stress acts at the point 0; the
shape of the deformed discontinuity surface and the distribution of stresses
near 0 are represented in Fig. 7b. The opposite faces of the crack overlap
in this case, and it is quite evident that this case is physically unrealistic.
Finally, if N = 0 , the stress acting near the contour is finite and tends
to the normal stress applied at point 0 of the contour if 0 is approached.
Thus the stress ay is continuous at the contour, and the opposite faces of the
discontinuity surface close smoothly (Fig. 7c).
MATHEMATICAL T H E O R Y O F E Q U I L I B R I U M CRACKS
73
The investigation of the stress and strain distribution near the edge of
the surface of normal discontinuity was begun by Westergaard [44, 131
and Sneddon [14, 151 and continued later by the author [40], by Williams
[17], and by Irwin [45-471. In view of the character of the stress states
considered in [14, 15, 45-47] results were obtained only for the
case N > 0.
2. Stresses and Strains Near the Edge of an Equilibrium Crack
The results obtained in the preceding section pertain to an arbitrary
surface of discontinuity of normal displacement. We now show that, for
an equilibrium crack, N = 0 a t all points of its contour.
FIG. a.
Consider a possible state of the elastic system, which differs from the
actual state of equilibrium only by a slight variation in the form of the
crack contour in a small vicinity of the arbitrary point 0 (Fig. 8). The new
contour is a curve that encloses the point 0 lying in the plane of the crack.
This curve is tangential to the former contour of the crack at points A and B
close to 0 ; everywhere else the contours of all the cracks remain unchanged.
In view of the closeness of the points of tangency A and B to the point 0,
the initial contour of the crack at the portion A B can be considered as
straight. The distribution of normal displacements of the points of the new
crack surface and the distribution of tensile stresses a t these points prior
to the formation of the new crack surface are, according to the above, given,
to within small quantities, by
(3.10)
v = T
4(1 - y 2 ) N
E
~
Vh-y,
a,=
N
VY
- 1
where N is the stress intensity factor a t the point 0.
The energy released in the formation of the new crack surface, which is
equal to the work required to close this new surface, is given by
74
G . I . BARENBLATT
b
(3.11)
h
- 2(1 - va),N2
-
hdx =
E
2(1 - v ~ ) T c N ~ ~ S
E
a
where 6s is the area of the projection of the new crack surface on its plane.
The condition of equilibrium of the crack requires that 6 A vanishes;
this together with (3.11) implies that N = 0. Thus we arrive a t a very
important result characterizing the structure of cracks near their contours :
1. T h e tensile stress at the contour of a crack i s finite.
2. The o#posite faces of a crack close smoothly at its contour.
I t appears, therefore, that contrary to Griffith’s conception the form of
a crack near its edge is as represented in Fig. 4. Since the only acting forces
at the surface of a crack near its contour are forces of cohesion, it follows
from (3.9) that the tensile stress at the crack contour is equal to the intensity
of forces of cohesion at the contour. In particular, if there are no forces of
cohesion, the tensile stress at the crack contour is equal to zero.
The condition of finiteness of stresses and smooth closing of the opposite
faces a t the edges of a crack was first suggested as a hypothesis by
S. A. Khristianovitch [38], to serve as a basic condition that determines the
position of the crack edge. The proof of this condition given above follows
[60] mainly. Formula (3.11) for the case of plane stress was first proved by
Irwin [45, 461 irrespective of finiteness of stresses and smoothness of closing
(see also the review by Irwin [47] and the paper by Bueckner [33]). The
early paper by Westergaard [44] contains a statement concerning the absence
of stress concentration at the end of a crack in brittle materials like concrete,
but the condition of finiteness of stress that appears in this work was not
connected with the determination of the size of the crack.
We have confined ourselves here to the examination of cracks of normal
discontinuity only for simplicity of treatment. Analogous reasoning, in
particular the proof of finiteness of stress a t the crack edge, can be extended
without any substantial changes to cover the general case in which also the
tangential displacement components have a discontinuity a t the crack
surface.
3. Determination of the Boundaries of Equilibrium Cracks
The conditions of finiteness of stresses and smooth closing of a crack a t
its contour permit us to formulate the problem of equilibrium cracks for a
given system of loads acting upon the body: for a given position of initial
MATHEMATICAL THEORY OF EQUILIBRIUM CRACKS
75
cracks and a given system of forces acting upon the body, it is required to
find stresses, deformations, and crack contours in the elastic body so as to
satisfy the differential equations of equilibrium and the boundary conditions,
and to insure finiteness of stresses and smooth closing of the opposite faces
at the crack contours.
We shall illustrate the solution of this problem by an elementary example
of an isolated straight crack in an infinite body under all-round compressive
stress q a t infinity and with concentrated forces P applied at opposite points
of the crack surface (Fig. 9).
The solution of the equilibrium equations satisfying the boundary conditions can be obtained by Muskhelishvili's method [18] for an arbitrary crack
length 21. Stresses and displacements
are expressed by formulas (3.1)-(3.3)
with
@(z) =
4c2
(3.12)
z
='(c4 +
+).
FIG.9.
Evidently, equilibrium equations and boundary conditions do not determine
the length of the crack. The distributions of stresses uy at the prolongation
of the crack and normal displacement v of points of the crack surface near
its edge are given by
(3.13)
Finiteness of stress and smooth closing of the crack at its ends are assured
simultaneously by the condition
(3.14)
1=-,
P
n4
which determines the crack size under given loads P and q.
Let us now attempt to determine the size 21 of an isolated straight crack
in an infinite body stretched by uniform stress Po at infinity in the direction
perpendicular to the crack. If the crack surface is assumed to be free of
76
G . I . BARENBLATT
stress, then one can easily show that the tensile stress a t the prolongation
of the crack near its edge depends on the distance s1 as follows:
Po v 1 .
(3.15)
UY =--
V2S, ’
hence it appears that for no 1 the stress uy will be finite a t the crack end and
there does not exist an equilibrium crack! This paradoxical result is due to
the fact that we did not take into account the molecular forces of cohesion
acting near the crack edges and thus did not completely account for the
loads acting upon the body. The consideration of these forces and the
definitive formulation of problems in the theory of equilibrium cracks of
brittle fracture are discussed in the following section.
IV. BASICHYPOTHESESAND GENERALSTATEMENT
OF THE PROBLEM
OF
EQUILIBRIUM
CRACKS
1. Forces of Cohesion; Inner and Edge Regions; Basic Hypotheses
In order to construct an adequate theory of cracks of brittle fracture,
it is necessary t o supplement the model of a brittle body by considering the
molecular forces of cohesion acting near the edge of a crack at its surface.
It is known that the intensity of forces of cohesion depends strongly on the
distance. Thus, for a perfect crystal the intensity f of forces of cohesion
acting between two atomic planes a t the distance y from each other is zero
if y is equal to the normal intermolecular distance b. With y increasing up
to about one and a half of b, the intensity f grows and reaches a very high
maximum f,
VETo/b E/10; after that it diminishes rapidly with
further increase of y (Fig. 10). Here E is Young’s modulus, and To is the
surface tension related to f ( y ) by the formula
-
N
m
r
b
The maximum intensity fm defines the theoretical strength, i.e. the strength
of a solid if it were a perfect crystal. The actual strength of solids is usually
several orders of magnitude lower because of defects of crystal structure.
For amorphous bodies the relation between the intensity of forces of cohesion
and the distance has qualitatively the same character.
Data at present available, which confirm the above character of the
relation between the intensity of forces of cohesion and the distance, lead
MATHEMATICAL THEORY OF EQUILIBRIUM CRACKS
77
to the following conclusion. I t has long been known that the strength of
thin fibers exceeds considerably that of large specimens of the same material
[62, 631. Experiments carried out recently with filamentary crystals of some
metals revealed an exceptionally high strength approaching the theoretical
value [63]. I t is supposed that this phenomenon is due to the relatively
small amount of structural defects in thin fibers and filamentary crystals.
Furthermore, numerous direct measurements of the intensity of molecular
forces of cohesion for glass and silica [64-661 were made recently. The
FIG.10.
FIG.11. I-inner region, ZZ-edge region.
very elegant method used in this kind of measurements is based on the
application of a regenerative microbalance and was suggested and employed
by B. V. Deryagin and I. I. Abrikosova [64, 651. However, these direct
measurements deal with very great distances y compared to the normal
intermolecular distance and thus determine only the end of the falling
branch of the curve f ( y ) . A macroscopic theory for forces of cohesion at such
distances was developed by E. M. Lifshitz [64] and was found in good
agreement with the results of these aforementioned measurements. The
relation f ( y ) , if y equals several normal intermolecular distances, seems to
be beyond any strict quantitative theory and difficult for the direct experimental determination at present. A description of available attempts to
estimate the relation f ( y ) at such distances and, consequently, the theoretical
strength can be found in [67, 63, 681.
The distance between the opposite faces of a crack varies from magnitudes
of the order of the intermolecular distance near the crack edge to sometimes
rather great magnitudes far from the edge. I t is therefore convenient to
divide the crack surface into two parts (Fig. 11). The opposite faces in the
first part - the inner region of the crack - are a great distance apart, hence
their interaction is vanishingly small, and the crack surface can be considered
free of stresses caused by the interaction of the opposite faces. The opposite
faces of a crack in the second part are adjacent to the crack contour - the
edge region of che crack - and come close to each other so that the intensity
78
G. I. BARENBLATT
of the molecular forces of cohesion acting on this part of the surface is great.
Of course, the boundary between the edge and inner region of the crack
surface is conventional to a certain extent. For very small cracks there may
be no inner region of the crack a t all.
Since the distribution of the forces of cohesion over the surface of the
edge region is not known beforehand, a substantial part of the loads applied
to the body is not known. I t is thus impossible to handle the problem of
cracks directly in the way it was stated in Chapter 111. But the following
method of solving problems of cracks is possible in principle: the distance
between the opposite faces of a crack is found at each surface point as a
function of the unknown distribution of forces of cohesion over the surface.
Assuming the relation f ( y ) between forces of cohesion and distance as given,
a relationship can be obtained which determines the distribution of forces
of cohesion over the crack surface.
Such an approach is not practicable. First, the relation f ( y ) is not known
to a sufficient extent for a single real material. Even if it were known, the
problem would constitute a very complex non-linear integral equation, the
effective solution of which presents great difficulties even in the simplest
cases.*
Attempts were made to prescribe the distribution of forces of cohesion
over the crack surface in a definite manner, but these attempts cannot be
considered sufficiently well founded.
For sufficiently large cracks, consideration of which is of principal
interest, the difficulty connected with our lack of knowledge of the distribution of forces of cohesion can be avoided without making any definite
assumptions concerning this distribution. In this case the general properties
of the relation between forces of cohesion and distance allow the formulation
of two basic hypotheses which not only simplify essentially the further
analysis, but permit the determination of contours of cracks, although the
forces of cohesion are finally altogether excluded from consideration as
loads acting upon the body.
First hypothesis: The width d of the edge region of a crack i s small compared
to the size of the whole crack.
This hypothesis is acceptable because of the rapid diminution of forces
of cohesion with the increase in the distance between the opposite faces of
* In papers of M. Ya. Leonov and V. V. Panasyuk [69, 701 the relation f ( y ) is approximated by a broken line, and on the basis of this approximation a linear integral equation for the normal displacements of the crack surface points is derived. It is solved
approximately, the representation of the solution being not quite successfully selected
SO that the form of the crack at its end appears wedge-shaped with a finite edge angle.
In fact, as was shown above, the edge angle must be zero. The shortcoming of these
papers lies also in t h e application of the results obtained by the methods of mechanics
of continua to cracks whose longitudinal dimensions are only of the order of several
intermolecular distances.
MATHEMATICAL THEORY OF EQUILIBRIUM CRACKS
79
a crack. Of course, there exist micro-cracks to which this hypothesis cannot
be applied. However, as the width d of the edge region is quite small, the
hypothesis is already valid for very small cracks and certainly for all macrocracks. Nevertheless, the width d is considered to be sufficiently great
compared to micro-dimensions (for instance, compared to the lattice constant
in a crystalline body), so that it is permissible to employ the methods of
continuum mechanics over distances of the order of d.
Second hypothesis: The form of the normal section* of the crack surface
i n the edge region (and consequently the local distribution of the forces of cohesion
over the crack surface) does not depend on the acting loads and is always the
same for a given material under given conditions (temperature, composition
and pressure of the surrounding atmosphere and so on).
When the crack expands, the edge region near a given point, according
to the second hypothesis, moves as if it had a motion of translation, and the
form of its normal section remains unchanged. This hypothesis is applicable
only to those points of the crack contour where the maximum possible
intensity of forces of cohesion is reached; an expansion of the crack occurs
then a t this point with an arbitrarily small increase in the loads applied to
the body.
Equilibrium cracks, on whose contour is a t least one such point, will be
called mobile-equilibrium cracks to distinguish them from immobile-equilibrium cracks which do not possess this property, i.e. do not expand with an
infinitesimal increase in the load. Thus the second hypothesis and all
conclusions based on it are applicable to reversible cracks as well as to
irreversible equilibrium cracks, which formed a t the initial rupture of a
brittle body in the process of increasing the load. I t is not applicable to
cracks which result from equilibrium cracks existing at some greater load
by diminishing that load; nor can it be applied, to artificial cuts made without
subsequent expansion.
The second hypothesis is suggested by the fact that the maximum
intensity of the forces of cohesion is so very great and exceeds by several
orders of magnitude the stresses which would arise under the same loads
in a continuous body without a crack. Therefore it is possible to ignore the
change of stress in the edge region when loads vary and, consequently, the
corresponding variation of the normal sections.
These two hypotheses reformulate the results of the qualitative analysis
of the brittle-fracture phenomenon carried out by a number of investigators
beginning with Griffith. They are the only assumptions concerning the
forces of cohesion which underlie the theory presented below and appear
in this explicit form in [56,571.
= intersection
'
with a plane normal to the crack contour.
80
G. I . BARENBLATT
2. Modulus of Cohesion
The body considered is assumed to be linearly elastic up to fracture.
The elastic field in the presence of cracks can then be represented as the sum
of two fields: a field evaluated without taking into account forces of cohesion
and a field corresponding to the action of forces of cohesion alone. Therefore
the quantity N entering in formulas (3.15) and, as was proved, equal to zero
can be written as N = N o N,, where the stress intensity factor N o corresponds to the loads acting upon the body and to the same configuration
of cracks without considering forces of cohesion, and the stress intensity
factor N , corresponds to the same configuration of cracks and forces of
cohesion only.
According to the first hypothesis the width d of the edge region acted
upon by forces of cohesion is small compared to the crack dimensions on the
whole and, in particular, to the radius of curvature of the crack contour a t
the point considered. In determining the value of N , we may thus assume
that the field belongs to the configuration discussed in Section I I I , l , i.e. to
an infinite body with a semi-infinite cut, with symmetrical normal stresses
being applied to the surface of the cut. Hence it follows from (3.7) that
+
m
d
where G(t) is the distribution of forces of cohesion different from zero only
in the edge region 0 t d .
According to the second hypothesis, the distribution of forces of cohesion
and the width d of the edge region at those points of the contour, where the
intensity of forces of cohesion is a maximum, do not depend on the applied
load; the integral in (4.2) represents then a constant characterizing the given
material under given conditions. This constant will be denoted by K :
< <
.=IT.
d
G(t)dt
(4.3)
0
I t was termed the modulzls of cohesion since this quantity characterizes the
resistance of the material to an extension of its cracks, caused by the action
of forces of cohesion. As will be shown below, the quantity K is the only
characteristic of the forces of cohesion, that enters in the formulation of the
problem of cracks.
The dimension of the modulus of cohesion is:
(4.4)
[ K ] = [ F ][ L ] - 3 / 2= [MI [ L ] - ' / 2 [ T ] - 2 ,
MATHEMATICAL THEORY OF EQUILIBRIUM CRACKS
81
where [I;],[I.],[MI, and [TI denote the dimensions of force, length, mass,
and time, respectively. Constants of a similar dimension are encountered
in the contact problem of the theory of elasticity [71, 72, 731. It is no
coincidence, that there exists a profound connection between the contact
problem and problems in the theory of cracks of brittle fracture; it seems
that this was first pointed out in the papers of Westergaard [44, 131.
3. T h e Boundary Condition at the Contour of a n Equilibrium Crack
For points of the contour of an equilibrium crack, a t which the maximum
intensity of cohesion is reached, the second hypothesis is applicable, and
(4.2) may be written as
1
(4.5)
N , = -- K ;
7c
considering that N = 0, we obtain
The boundary condition a t contour points of an equilibrium crack, a t
which the intensity of forces of cohesion is maximal, can also be formulated
as follows: on approaching these points, the normal tensile stress a,, a t the
points of the body lying in the crack plane, if calculated without taking into
account forces of cohesion, tends to infinity according to the law
(4.7)
cry =
-
K
+ 0(1),
Vs
where s is the (small) distance from the contour point considered. Satisfying
(4.6) a t least a t one point of the contour is the condition that the crack is in
the state of mobile equilibrium.
One should not connect, in general, the reaching of the state of mobile
equilibrium by the crack with the onset of its unstable rapid growth and
still less with complete fracture of the body. A mobile-equilibrium crack
may be either stable or unstable. Only in case of instability is the condition
for the onset of rapid crack propagation given by (4.6). However, not even
in this case is complete fracture of the body unavoidable, since the transition
from the unstable state of equilibrium to the other, stable one, is possible.
Numerous examples illustrating various possibilities will be discussed in the
following chapter.
82
G. I. BARENBLATT
If a crack is irreversible and there are points on its contour where the
intensity of forces of cohesion is less than maximal,* then the second
hypothesis is not applicable a t such points. Since cohesive forces that act
in the edge region of the crack surface are smaller near such points than those
acting near points of the type considered above, it follows from (4.2) that
- N , < K l h ; and since N o = - N,, we have for these points
(4.6 a)
K
No<--.
n
As the load increases, forces of cohesion in the edge region grow; they
compensate the increase in the load and insure finiteness of stress and smooth
closing at the crack contour. However, the crack does not expand at a given
contour point until the forces of cohesion become maximal. The second
hypothesis now becomes applicable, and condition (4.6) is satisfied.
In determining the form of contours of equilibrium cracks, conditions
(4.6) and (4.6a) permit us to exclude the forces of cohesion altogether from
the consideration of the loads acting upon the body. Instead, we work with
their overall integral characteristic, the modulus of cohesion. Special
estimates show [57, 581 that the influence of molecular forces of cohesion
on the stress and displacement field is essential only in the neighbourhood
of the edge in a region of the order of magnitude d . Forces of cohesion thus
determine the structure of cracks near their ends, and the forms of crack
contours depend on them only through the integral characteristic K .
4. Basic Problems in the Theory of Equilibrium Cracks
The basic problem in the theory of equilibrium cracks can be stated in
its most general form as follows. A certain system of initial cracks and a
process of loading the body, i.e. a system of loads acting upon the body,
dependent on one monotonously increasing parameter A, are given. The
value of I for the initial state may be assumed as zero. I t is required to
determine the form of the crack surfaces and to find the distribution of
stresses and strains in the body corresponding to any il > 0. The process
of varying the load is supposed to be sufficiently slow so that dynamic
effects need not be considered.
When the symmetry of body, loads, and initial cracks insures the possibility of developing a system of plane cracks and the extensional loads
grow monotonously with increasing A, the configuration of cracks in the body
is determined by the current load only and not by the whole history of the
* For instance, contour points of non-expanded cuts or of cracks formed from
cracks which existed under a greater load when the load is diminished.
MATHEMATICAL THEORY OF EQUILIBRIUM CRACKS
83
process of loading, as it is in the general case. In this case the problem is
formulated as follows (it will be called problem A ) . In a body bounded by
a surface Z contours of an initial system of plane cracks I', are given (Fig. 12;
the plane of the drawing is the plane of the cracks). It is required to find the
enclosing the
elastic field and the contours of a system of plane cracks
contours To (and perhaps coinciding with them partially) corresponding to
a given load, i.e. to a given value of 2.
r
This problem reduces mathematically to the following one. I t is required
to construct the solution of the differential equations of equilibrium of
elasticity theory in the regon bounded by plane cuts with contours 'I
and by the body boundary Z under
boundary conditions corresponding
to the given load. The contours I'
must be determined so that condition (4.6) is satisfied at points of
these contours not lying on
and
condition (4.6a) at points of
lying
FIG. 12.
on I',.
@
r,,
r
If the cracks are reversible or if the applied loads are sufficiently great
so that the contours T do not coincide with I', at any point, then the form
of the initial contours is of no importance. It is then possible, without
prescribing the initial cracks, to formulate directly the problem of determining the contours I' of equilibrium cracks of a given configuration so that
condition (4.6) is satisfied at each point of
Here we assume that the initial
cracks are such that they are compatible with the realization of the given
configuration of cracks when the load increases. This problem will be called
problem B.
r.
I t may happen that a solution of either of the above stated problems
does not exist. If this happens, it has quite a different significance for the
problems A and B. If no solution of problem A exists this means that the
applied load exceeds the breaking load, hence its application causes fracture
of the body. The limiting value of the parameter 2 up to which the
solution of problem A exists, corresponds to the breaking load. The
determination of the breaking load for a given configuration of the initial
cracks and a given process of loading presents an important problem in the
theory of cracks. Non-existence of the solution of problem B signifies that,
whatever initial cracks may be within a given configuration, they will not
expand under a given load, hence the applied load is too small. In such
cases the conventional description of the state would be that mobileequilibrium cracks do not form under the given load.
84
C. I. BARENBLATT
5. Derivation of the Boulzdary Condition at the Contour of a n Equilibrium Crack
b y Energy Considerations
Molecular forces of cohesion so far have been considered as external
forces applied to the surface of the body. This was necessary for analysing
the structure of cracks near their ends.
‘If only boundary conditions are to be obtained, another approach can
be employed which considers the forces of cohesion as internal forces of the
system. On the basis of this approach, the idea of which goes back to Griffith
[3,41, a relation between the modulus of cohesion and other characteristics
of the material will be obtained.
As before, let there be a certain configuration of equilibrium cracks in
a brittle body and consider as in Section II1,2 a possible state of the elastic
system, which differs from the real one only by a variation in the crack
contour near a certain point 0 (Fig. 8). However, unlike Section 111,2,
the characteristic size of the new area of the crack surface is assumed to be
large compared to the dimension d of the edge region, though small compared
to the size of the crack as a whole; according to the first hypothesis
(Section IV,1) such an assumption is permissible. Under this assumption
forces of cohesion can be considered merely as forces of surface tension, and
a certain amount of work must be done to overcome these forces in increasing
the crack surface. The influence of forces of cohesion on the stress and strain
fields can be neglected since it is essential only in the neighbourhood of the
crack edge, whose dimension is of the order of the width of the edge region.
The work SA required for the transition from the actual state to a virtual
one is equal to the difference between the corresponding increment in surface
energy SU and released elastic energy SW:
(4.8)
SA = SU - SW.
For the actual state of an elastic system to be an equilibrium state,
6 A must vanish, hence
(4.9)
6U = 6W.
Quite similarly to Section III,2 an expression for SW is obtained:
(4.10)
sw = 2(1 - v2)nNo2SS
E
where N o is the value of the stress intensity factor at the point 0 calculated
without taking into consideration the forces of cohesion. Formula (4.10) in
a somewhat different form was established by Irwin [4547].
If the form of the edge region near a given point of the contour corresponds to the maximum intensity of forces of cohesion, then, according
85
MATHEMATICAL THEORY OF EQUILIBRIUM CRACKS
to the above, in forming a new crack surface the edge region is displaced
without deformation; the work against the forces of cohesion per unit of
newly formed surface is then constant and equal to the surface tension To.
Therefore, 6U = 2T0SS, because two surfaces form in rupture. Together
with (4.9)and (4.10),we have
No =
(4.11)
ETo
n(1 - v2) *
Comparing (4.11)and (4.6),we obtain a relationship correlating the
modulus of cohesion K , defined independently by (4.3),with the surface
tension To and the elastic constants of the material E and Y :
(4.12)
6 . Experimental Confirmation of the Theory of Brittle Fracture:
Quasi- Brittle Fracture
After Griffith's work [3, 41 many investigators attempted to carry out
experimental verifications of the theory of brittle fracture. We cannot
Spherical bulbs
Cylindrical tubes
21
inches
D
inches
Po
psi
POv'
0.15
0.27
0.54
0.89
1.49
1.53
1.60
2.00
864
623
482
366
237
228
251
244
21
inches
D
inches
psi
0.25
0.32
0.38
0.28
0.26
0.30
0.59
0.71
0.74
0.61
0.62
0.61
678
590
526
655
674
616
240
232
229
245
243
238
analyse all this work here in detail and shall dwell only on several of the most
characteristic papers, referring for details and discussion of other numerous
investigations to the special publications [62, 55, 74-78].
Griffith's paper [3]gives descriptions and results of the following experiments. Cracks of various length 21 were placed on spherical glass bulbs and
cylindrical tubes, whose diameter D was sufficiently great so that a special
verification showed no influence of the diameter on the experimental results.
After the tubes and bulbs had been annealed to relieve residual stresses
86
G. I. BARENBLATT
produced by making the cracks, they were loaded from the inside by hydraulic
pressure up to fracture. The breaking stress Po corresponding to each crack
length 21 was measured.
According to the foregoing theory it appears that the breaking stress Po
at which a given crack becomes unstable
(onset of mobile equilibrium) can depend
only on the crack length 21 and the modulus
of cohesion K . From dimensional analysis
[79] it follows that Po = a K / v c where a is
a dimensionless constant. Consequently,
+,Vdmust be constant for a given material
(in full accord with (2.1)).
"
Griffith's experiments, which are tabulated here, confirm the constancy of this
quantity and thus the foregoing theoretical
scheme.
FIG. 13.
The remarkably elegant experiments of
1. Steel indentor.
2. SDecimen.
Roesler [53] and Benbow [54], in which
3. Steel support.
stable conical cracks were produced, are of
special interest for the confirmation of the
theory of brittle fracture. The scheme of these experiments is presented
in Fig. 13; the photograph of conical cracks in fused silica, borrowed
'
FIG. 14.
from Benbow's paper [54], is given in Fig. 14. The cracks were formed
by the penetration into a specimen of glass [53] and fused silica [54]
of a cylindrical steel indentor with a flat end. In accordance with the
MATHEMATICAL THEORY OF EQUILIBRIUM CRACKS
87
above, the diameter s of the base of a conical crack can depend only on
the diameter do of the indentor base, the force P pressing the indentor, the
modulus of cohesion K , and Poisson’s ratio v. Since the correct formulation
of the corresponding problem of elasticity theory does not include Young’s
modulus, it should not be included in the number of determining parameters
of the crack problem. Dimensional analysis yields
(4.13)
where p is a dimensionless function of its arguments.
I
1x10~
I
10x10~
I
100~
107p
FIG. 15.
Experiments carried out with indentors of three diameters on eleven
glass specimens [53] confirm well the existence of the universal relation (4.13).
A t large values of P, when the first argument of the function q becomes
vanishingly small, self-similarity takes place, and the following relationship
holds :
(4.14)
Fig. 15 represents a graph, taken from Benbow’s paper [54], of the s ( P )
relation according to data from experiments with fused silica carried out
under conditions corresponding to the self-similar regime. As can be observed,
these experiments give a conclusive proof of the validity of relation (4.14)
and confirm thereby the above scheme.
The experiments described were carried out with materials which can
be considered as perfectly brittle. This refers especially to fused silica.
Benbow [54] presents certain facts indicating that the mechanism of crack
88
G . I . BARENBLATT
formation in fused silica is closer to being perfectly brittle than it is in glass:
cracks in glass grow for a long time under constant load, whereas in fused
silica their size is established quickly and then remains unchanged; after
removal of the load, cracks in glass remain distinctly visible, but in silica
they are imperceptible, etc. However, the significance of the theory of brittle
fracture greatly exceeds what should be the limits of its applicability to
those comparatively rare materials that are perfectly brittle. Experimental
investigations show that when cracks appear some materials, which behave
as highly plastic bodies in common tensile
tests, fracture in such a way that plastic
deformations, though present, are concentrated in a thin layer near the crack
surface.
D. K. Felbeck and E. 0. Orowan [28]
carried out experiments on fracture of
low-carbon steel plates with a saw-cut
crack under conditions corresponding to
Griffith’s scheme of uniform extension.
Experimental results are in good agreeFIG.18.
ment with Griffith’s formula, but the
surface-energy
-- density exceeds by about
three orders of magnitude the surface tension of the material investigated. It was found in good agreement with the specific work of
plastic deformations in the layer near the crack surface, which was determined by independent measurements.
On the basis of this and similar experimental results Irwin [23] and
Orowan [24] advanced the concept of quasi-brittle fracture, which permitted
an important extension of the limits of applicability of the theory of brittle
fracture. Here the theory of brittle fracture covers the case when the plastic
deformations are concentrated in a thin layer near the crack surface. The
energy T required to form the unit surface of a crack is expressed as the sum
of the specific work against the forces of molecular cohesion To (= surface
tension) and the specific work of plastic deformation T,:
(4.15)
T = To
+ T,.
A formal extension to quasi-brittle fracture is made as follows (Fig. 16,
the plastic deformation zone near the surface is shaded). Imagirfc the whole
plastic region cut out and shift the crack end to the end of the plastic region.
This can be done, if the forces exerted by the plastic zone upon the elastic
zone are considered as external forces applied to the crack surface. After
that the previous reasoning remains unchanged, if the plastic zone is assumed
as thin and use is again made of the hypothesis concerning the invariability
MATHEMATICAL THEORY OF EQUILIBRIUM CRACKS
89
of the edge region (which here includes the boundary of the elastic and plastic
zones). The modulus of cohesion is now expressed as
(4.16)
where G,(t) is the distribution of normal stresses acting on the boundary
of the elastic and plastic zones.
When the contribution of molecular forces of cohesion to integral (4.16)
can be ignored in comparison to the contribution of stresses that act in the
region ahead of the actual crack end and have the order of magnitude of the
yield point stress a,, we obtain an estimate for the modulus of cohesion:
(4.17)
Note that the value of a, at the yield point near the crack end may
differ from that a t the yield point obtained in tensile tests with large
specimens.
The concept of quasi-brittle fracture is somewhat related to the concept
of the “plastic particle” a t the ends of notches with a zero radius of curvature,
advanced in a classical monograph by H. Neuber [80].
In the following we shall speak of cracks of brittle fracture, bearing in
mind the possibility of extending the results to the case of quasi-brittle
fracture. Of course, in this latter case it is necessary to take into consideration the irreversibility of cracks of quasi-brittle fracture.
7. Cracks in Thin Plates
If the state of stress can be assumed to be plane, then all relations derived
previously for the case of plane strain hold also for thin plates, if only E is
replaced by E(l - v2) and the modulus of cohesion is assumed to have some
other value K,. Repeating the derivation of formula (4.12)for the plane
stress state we obtain
(4.18)
K I 2= n E T .
The experiments show that the surface energy density T in the case of
quasi-brittle fracture increases with a reduction in the plate width [48],
which is due to a broadened plastic-strain zone near the crack surface. An
approximate theoretical analysis of this phenomenon was attempted by
I. M. Frankland [81].
Bearing in mind the complete analogy of the analysis of plane stress and
plane strain we shall in the following consider only plane strain.
90
G. I. BARENBLATT
V. SPECIALPROBLEMS
I N THE THEORY
OF EQUILIBRIUM
CRACKS
This Chapter deals with solutions of special problems in the theory of
cracks available at present. A few of the examples have illustrative character, but most problems presented are interesting in themselves.
1. Isolated Straight Cracks
In this and the following section isolated mobile-equilibrium cracks are
examined, and all along the contour the maximum intensity of forces of
cohesion is assumed to prevail. The problem reduces here to the determination of crack contours corresponding to a given load so that condition (4.6)
is satisfied at these contours, and it represents a particular case of problem B
formulated above. I t is supposed that the initial cracks guarantee the possibility of producing such cracks ; the necessary requirements for the initial
cracks in the cases of reversible and irreversible cracks follow readily from
the solutions obtained.
Let us consider an isolated straight mobile-equilibrium crack extending
along the x-axis from x = a to x = b in an infinite body subject to plane
strain. Let p ( x ) be the distribution of normal stresses, which arise a t the
place of the crack in a continuous body under the same loads. This distribution is computed by the usual methods of elasticity, and we may consider it
as given. I t may be shown by using Muskhelishvili’s solution [2, 181 that
tensile stresses near the crack ends calculated without taking into account
forces of cohesion become infinite according to the law ay = N / V s + . . .,
where
h
I.
are the values of the stress intensity factors for points a and b, respectively.
Satisfying condition (4.6) at these points, we obtain relations that determine
the coordinates of the crack ends a and b :
b
(5.2) 1p(d
a
ve
-~
x - a dx = K v b - a,
[p(x)
v
E
d x =K
V G .
In particular, if the applied load is symmetrical with respect to the crack
middle, where we place the origin of coordinates, then - a = b = 1, and
Eqs. (5.2) become one relation determining the half-length of the crack 1 :
MATHEMATICAL T H E O R Y OF EQUILIBRIUM CRACKS
91
(5.3)
Note that (5.2) and (5.3)represent finite equations, since f i ( x ) is a given
function. These equations determine the position of the ends of an isolated
straight-line mobile-equilibrium crack under a given load, if this load
guarantees that such a crack can exist.
FIG.17.
A method to calculate the strain-energy release rate aWlal for a symmetrical isolated crack was indicated by K. Masubuchi [82]. He proposed a
trigonometrical representation of stresses $(x ) and displacements v of points
of the crack surface,
m
(5.4) f i ( x ) = E
41
2
m
sin d
l
n A n x
I
v =4 2AnsinnB,
n=l
x =lcosO.
n= 1
As was shown by Masubuchi,
(5.5)
aw
- -ai
m
nE
2 (nAn)2.
qi -t q z n= 1
Equating this expression to 4 T , where T is the surface energy density,
a relation between the applied stresses and the crack size can be obtained,
though in a form far more complicated than (5.3).
Let us now look a t a few examples. A crack may be kept open by a
uniform tensile stress Po applied a t infinity. As already pointed out, this
92
G . I . BARENBLATT
problem was first treated by Griffith [3, 41.
equation (5.3) yields
In this case p ( x )
Po and
Relation (5.6) appears in Fig. 17 as the dotted line. One sees that the
size of a mobile-equilibrium crack diminishes with increasing tensile stress,
which is indicative of the instability of mobile equilibrium in this case.
Despite this instability the size I defined by (5.6) has a physical meaning:
If there is a crack of length 21, in a body, to which constant tensile stress Po
is applied at infinity, then a t I,< I this crack does not expand (and closes
if it is a reversible crack) while at I, > I it grows indefinitely. Thus, the
equilibrium size is in a certain sense critical (this will be discussed in more
detail in Section V,3). It is obvious that instability of mobile equilibrium in
this case fully corresponds to the substance of the matter and, contrary to
the opinion expressed by Frenkel [5], is not connected with Griffith’s incorrect
ideas about the geometry of the crack ends.
If stresses vanish at infinity, and if a crack is maintained by a uniformly
distributed pressure applied over a part of its surface (0 x
I,) while the
remaining part of the crack surface (I,
x
I ) is free of stress, then the
half-length of the mobile-equilibrium crack I is given by the relation [58]
< <
< <
(5.7)
This relation is shown in Fig. 17 by the solid lines which may be
obtained from each other by a similarity transformation. It is evident that
the opening of a crack, i.e. the appearance of a free segment, is possible
provided 1, is not less than the corresponding size of a mobile-equilibrium
crack kept open by a uniform tensile stress a t infinity, Po, which is determined
b y (5.6). Therefore all the solid lines (Fig. 17) must start from the dotted line.
A limiting case of (5.7) is of interest. It occurs when p , tends to infinity
Const = P. This corresponds to a crack
and I, tends to zero, while 2p,l,
kept open by concentrated forces applied at opposite points of its surface.
The half-length of the crack is then given by
Note that (5.6) and (5.8) may be obtained, disregarding the value of the
numerical factor, by dimensional analysis. For example, the size of a crack
maintained by concentrated forces is determined only by the magnitude P
of these forces and the overall characteristic of the forces of cohesion, K .
It is obvious that the modulus of elasticity and Poisson’s ratio do not enter
MATHEMATICAL THEORY OF EQUILIBRIUM CRACKS
93
in the number of determining parameters, since the corresponding problem
of the theory of elasticity is naturally formulated only in terms of stresses.
Considering the dimensions of P and K , we see that it is possible to set up
only one combination with the dimension of length from these quantities,
namely the ratio P 2 / K 2 ,and no dimensionless combination exists. Thus
the length of a mobile-equilibrium crack must be proportional to P 2 / K 2 ,
and the coefficient of proportionality a universal constant [cf. 791.
Let now a crack be maintained by two equal and opposite concentrated
forces P , whose points of application are separated by L along the common
line of action of the forces; the crack is supposed to be perpendicular to the
line of action of the forces and located symmetrically [ 5 8 ] .
FIG.18.
The distribution of tensile stresses at the place of the crack in a continuous body is in this case given by
(5.9)
(the origin of coordinates is taken in the middle of the crack). Using (5.3),
we obtain the relation determining the crack size in the form
(5.10)
A plot of PIKVZ versus the relative length of the crack IIL for v = 0.25
is shown in Fig. 18. As can be seen, at P > Po two lengths of a mobileequilibrium crack correspond to each value of P , the smaller decreasing and
the greater increasing with increasing P. States of mobile equilibrium corresponding to the smaller equilibrium length are unstable ; the corresponding
branch of the load-length diagram in Fig. 18 is shown by the dotted line.
States corresponding to the greater length are stable (solid line in Fig. 18).
94
G. I . BARENBLATT
The smaller size I, is the critical size at a given load P ; initial cracks present
in the body and smaller than 21, do not expand under the action of applied
loads of magnitude P (in case of reversible cracks they close), and those
which are greater expand until the crack reaches the second (stable) equilibrium size.* At P < Po equation (5.10) has no solution. This means that,
whatever length of the initial crack we take, it will not develop into a mobileequilibrium crack at the given load. The size of a mobile-equilibrium crack lo
different from zero corresponds to the critical value of forces Po.
FIG. 19.
An interesting problem concerning the influence of riveted stiffeners on
crack propagation was treated by J. P. Romualdi and P. H. Sanders [ 5 2 ] .
This problem is schematized by the authors as follows (Fig. 19). An infinite
plate is stretched by a uniform stress Po in the direction perpendicular to a
crack. The action of the rivets and the stiffeners is represented by two
symmetrically located pairs of opposite concentrated forces equal in magnitude to P ; they are considered as given.
Substituting the corresponding stress distribution in (5.3)and working
out? the elementary though somewhat cumbersome integrals, we obtain the
relation between the applied load and the half-length of an equilibrium
crack 1:
+
+ A z ( A +12(1
B-2)vA-B+2
1-v
AvA--2
(5.11)
2(1
+ v ) ( 2 B- A -___4)
Aq/A-B+2
9 --,Y
0 - L
1
l=z,
;
4y02
+
yo
+
( 1 + v ) ( B A ) ( 2 B- A - 4)
A 3 ( A B - 2) V A - B T ]
+
B = y o z + l z + 1,
+-nv% .’
A =VBZ-412.
* Note that, because of dynamic effects accompanying the expanding of the initial
cut, the crack actually may “overshoot” the stable equilibrium state to %me extent.
This will be discussed later in more detail.
t Computation of the integrals and numerical calculations for the graphs in Fig. 20
were made by V. Z. Parton and E. A. Morozova.
MATHEMATICAL THEORY OF EQUILIBRIUM CRACKS
95
The results of the calculations are plotted in Fig. 20 for Y = 0.25,
P / K V z = 0.2 and for several values of the parameter yo/L. As is seen,
mobile-equilibrium cracks are unstable in the absence of stiffeners. The
influence of the stiffeners shows itself first of all in an increase of the size
of a mobile-equilibrium crack at a given load and, as an especially important
feature, in the appearance of stable states of mobile equilibrium at sufficiently
small yo/L, i.e. when rivets are spaced closely enough. The appearance of
stable states of mobile equilibrium changes considerably the character of the
crack expansion (see details below).
1.0
L
05 -
t
FIG.20.
The authors observed experimentally the transition of cracks from
unstable mobile-equilibrium states to stable ones; their experiments, carried
out with aluminium alloy plates in the presence and absence of stiffeners,
reveal a considerable increase in size of mobile-equilibrium cracks in the
presence of stiffeners at the same value of Po. In [52] the stress intensity
factor a t the crack ends was also determined experimentally for several stable
and unstable mobile-equilibrium states. In the absence of stiffeners, measurements of the stress intensity factor were made by the direct method,
i.e. by diminution of tensile stresses near the crack ends (at distances
obviously large compared to the size of the crack-edge region). In the
presence of stiffeners the stress intensity factors were measured indirectly.
The values of these factors were found to coincide except in two cases when
they were smaller by approximately 15 per cent. However, these two tests
carried out with one and the same specimen, with a stable crack in one case
96
G . I . BARENBLATT
and unstable crack in the other, gave values of the stress intensity factor
close to each other. (A somewhat lower value of this factor at the end of
the stable crack can be explained by the considerable dynamic effects which,
according to the authors, occur in the transition from the unstable state to
the stable one.) Thus it may be supposed that the deviation observed is due
to some peculiarity of the specimen. Altogether, these experiments confirm
directly the proposed general scheme.
This discussion can be readily extended to straight cracks in an anisotropic
medium, placed in the planes of elastic symmetry of the material. The
problem of a straight crack in an orthotropic infinite body subjected to a
uniform stress field was treated by T. J. Willmore [21] and A. N. Stroh [83].
In [83], the results of [16] were also extended to cover the case of a straight
crack in an anisotropic body under an arbitrary stress field, and the stress
intensity factors at crack ends were found for this problem. Paper [84]
brings the solution of the general problem concerning a straight mobileequilibrium crack in an orthotropic body subjected to an arbitrary stress
field symmetrical with respect to the line of the crack.
2. Plane Axisymmetrical Cracks
If a disk-shaped mobile-equilibrium crack of radius R is maintained in
an infinite body by an axisymmetrical load, tensile stresses near the crack
contour calculated without taking into account forces of cohesion tend to
infinity according to the law
R
(5.12)
where p ( 7 ) is the tensile-stress distribution at the place of the crack in a
continuous body subjected t o the same loads. According to the general
condition (4.7),the equation determining the radius of a mobile-equilibrium
crack R is
R
(5.13)
This equation was established in [56, 571. Its derivation is based on the
application of the method of Fourier-Hankel transforms, developed by
I. N. Sneddon [14, 151 for solving axisymmetrical problems of elasticity.
In particular, if a mobile-equilibrium crack is kept open by a uniform tensile
97
MATHEMATICAL THEORY OF EQUILIBRIUM CRACKS
stress at infinity Po, then p ( r )
is given by
Po and the radius of an equilibrium crack
(5.14)
This problem was first solved by R. A. Sack [20] by the energy method;
his method is quite similar in principle to Griffith's [3, 41 treatment of the
corresponding plane problem.
If there is no tensile load at infinity, and if the crack is kept open by a
uniformly distributed pressure Po over a part of its surface (0 Y
ro)
while the remaining part of the crack surface (ro Y
R ) is free, then the
radius of the mobile-equilibrium crack is found from the relation
< <
< <
(5.15)
Here, just as in the plane case, the radius of the loaded part of the crack
surface ro must not be less than the critical radius for a given pressure Po,
which is defined by (5.14). In particular, if a disk-shaped crack is maintained
open by equal and opposite concentrated forces P applied a t its surface,
then the radius of a mobile-equilibrium crack is determined by the formula
(5.16)
Relations (5.14) and (5.16) can be obtained, except for the numerical
factor, from dimensional analysis (cf. (5.6) and (5.8)).
If a disk-shaped crack is kept open by equal and opposite forces P whose
points of application are 2L apart along the common line of action, then the
radius of a mobile-equilibrium crack R is determined from the equation
(5.17)
P
KL3I2
-
L2
The above solutions were obtained in [56, 571 ; the interpretation of the
relations obtained is quite similar to the corresponding cases for a straight
crack.
3. T h e Extension of Isolated Cracks Under Proportional Loading;
Stability of Isolated Cracks
The problem of this section is a special case of problem A. A complete
investigation is carried out for symmetrical loading and initial cracks,
straight and disk-shaped cracks being considered simultaneously. An
example of a problem concerning the growth of an unsymmetrical initial
98
G . I. BARENBLATT
crack is given, which illustrates the general procedure of solving this problem.
Under proportional loading the tensile stresses at the place of the crack,
but in a continuous body subjected to the same load, are proportional to the
loading parameter I ; hence p ( x ) = A / ( x ) and p ( r ) = A/(,f(r)in the cases of
straight and disk-shaped cracks, respectively. Introducing the dimensionless
variable 5 equal to x/1 and r / R in these cases, respectively, one obtains
relations (5.3) and (5.12) in the form
p
=q(c),
(5.18)
where q ( c ) is defined respectively by
and c denotes, respectively the half-length 112 or the radius R.
Thus the relation of the crack length to the parameter I of proportional
loading is completely determined by the length of the initial crack and by
the function q ( c ) , corresponding to a given load distribution. Certain properties of the function q ( c ) can be obtained under the most general assumptions. Omitting the case of a crack maintained by concentrated forces
applied at its surface, let us suppose that the crack is kept open by any loads,
in particular, by concentrated loads applied inside the body and perhaps
by distributed loads applied a t the crack surface. In this case the functions
p ( x ) , p ( r ) , and, consequently, / ( c E ) are obviously bounded. For small c we
obtain from (5.19), respectively:
Suppose that the tensile loads applied to the body on each side of the
crack are bounded and, for definiteness, equal to AP. Then the following
relations are valid :
m
(5.21)
5
m
@(x)dx= IP,
5
-m
0
m
m
P
f(c6)dE = 2c '
MATHEMATICAL THEORY OF EQUILIBRIUM CRACKS
99
Eqs. (5.21) and (5.19) yield asymptotic representations for the functions ~ ( c )
when c
00 :
-+
(5.22)
Thus, under the assumptions made, p(c) tends to infinity when c + O
and c- 00. Owing to the boundedness of f(c[), the integrals in expressions
(5.19) do not become infinite a t any c, therefore p(c) vanishes nowhere and,
consequently, has at least one positive minimum, one falling branch, and
one rising branch. If the forces applied to the body on either side of the
crack are not bounded, then the function p(c) may not have rising branches
and, consequently, minima. This happens in particular in case of a uniform
tensile stress field when p = lfio and
(5.23)
for a straight and axisymmetrical crack, respectively.
By definition, an equilibrium crack is stable if no (sufficiently small)
change in its contour produces forces which tend to move the crack further
away from the disturbed state of equilibrium. It is evident that immobileequilibrium cracks are always stable. For stability of a mobile-equilibrium
crack it is necessary that its size should grow with an increase of the loading
parameter I . Suppose indeed that the corresponding size of a mobileequilibrium crack c grows with increasing load. If the crack size is diminished
without changing the load ( I = const), the crack extension force will be
greater than it was in equilibrium. Therefore the equilibrium is disturbed,
and the crack tends to widen under the action of the excess force. Conversely,
if the crack size is slightly increased compared with its equilibrium size,
then the equilibrium is disturbed in the opposite direction, and the crack
tends to close, if it is reversible.* If near a given equilibrium state the
equilibrium crack size c diminishes with an increase of I , then it is obvious
that its small change under a constant load will produce forces favouring
further departure from the equilibrium state. The corresponding equilibrium
state will be unstable. Hence the equilibrium state of a crack is stable, if
for given c and 1 the following condition is satisfied:
(5.24)
adc> o .
* If the crack is irreversible, then with an increase in its size no reverse closing
takes place, but no further expansion of the crack takes place either. Equilibrium is
attained in this case because of diminution of forces of cohesion acting in the edge
region of the crack.
100
G . I . BARENBLATT
Differentiating (5.18) with respect to I , we find
(5.25)
Thus the condition for stability of the state of mobile equilibrium is
y'(4 > 0,
(5.26)
and only those states of mobile equilibrium are stable which correspond to
rising portions of the curve y(c).
4
cz
c
0 Go
GI
c2
FIG.21.
Now we have everything that is necessary for the complete investigation
of the extension of an isolated symmetrical crack under proportional loading.
Let a function y(c), such as shown in the graph of Fig. 21, correspond to a
given system of loads applied to the body and consider first the case when
p(c)+ 00 as c+ OQ (Fig. 21a). Such a case occurs in particular when the
loads applied on both sides of the crack are bounded. Let the dimension of
the initial crack 2c, correspond to an unstable brach of y(c). Then the
crack length remains constant with increase of I , until I reaches the magnitude, for which the initial crack of size 2c, becomes one of mobile equilibrium. Since the mobile equilibrium is unstable, the crack begins to expand
under constant load, until it reaches the nearest stable mobile-equilibrium
state. With further increase of I the crack size grows continuously, until the
load corresponding to a maximum of y(c) is reached, then changes again
in a stepwise manner when the transition to another stable branch takes
place, after which it grows continuously with increasing A. The path of the
point representing the change of the crack is indicated by the number 1
in Fig. 21a.* Let now the size of an initial crack 2c, correspond to a stable
* Owing to dynamic effects that occur in this transition, the crack may overexpand
a little beyond the size of the stable mobile-equilibrium crack corresponding to the
given load (apparently that happened in the experiments described in paper [ S Z ] ) .
In this case, a further increase in the load leaves the length unchanged up to reaching
mobile equilibrium, after which the crack starts t o lengthen further. Naturally, the
purely static theory considered here cannot describe these dynamic effects; the corresponding parts of the graph in Fig. 21a are dotted and designated by the number 1'.
MATHEMATICAL THEORY OF EQUILIBRIUM CRACKS
101
branch of ~ ( c ) . The crack size now remains unchanged up to the load a t
which it reaches mobile equilibrium, after which it increases continuously.
The path of the representative point is indicated by the number 2 in Fig. 21a.
In the case considered, no fracture of the body occurs for any values of the
parameter 1. If 1 is less than its critical value (corresponding to the lowest
of the minima of ~ ( c ) ) then
,
great as the size of the initial crack may be,
it does not expand under a given load. The size of the mobile-equilibrium
crack corresponding to this critical value of A is finite.
This means in particular: if a crack is kept open by forces applied inside
the body and perhaps by loads distributed over the crack surface, and if the
forces applied on each side of the crack are bounded, then there exists a
critical value of the parameter 1; for all values of 1 greater than the critical
one there exists a t least one stable and one unstable state of mobile equilibrium.
Let us now turn to the case when ~ ( c-+) 0 as c + 00 (Fig. 21b). If the
size of an initial crack 2c, corresponds to a stable branch of ~ ( c )then
,
the
crack does not expand until a load is reached at which its state becomes a
mobile equilibrium. After that, the crack grows continuously with increasing
A, until a value of 1 is reached that corresponds to a maximum. If this
I-value is exceeded, the solution of the problem does not exist any longer,
and fracture of the body occurs. The path of the representative point is
indicated by the number 1 in Fig. 21b. If the size of an initial crack 2c,
corresponds to the right-hand unstable branch of p(c), then no expansion
of the initial crack occurs with increasing 1,until a value of 1 is reached for
which the state of the initial crack becomes a mobile equilibrium. The
slightest exceeding of this value of 1 causes complete fracture of the body.
If the size of an initial crack 2c3 corresponds to the left-hand unstable branch
of the curve ~ ( c ) then
,
for c3 < co the crack develops in the same manner as
in case 2 ; for c3 > co the development of the crack is similar to case 1 in
Fig. 21a before reaching a maximum, after which the body fractures.
The investigation of other forms of the curve ~ ( c can
)
easily be carried
out by combining the cases considered. We see that the knowledge of the
)
it possible to describe completely the behavior of a
function ~ ( c makes
symmetrical isolated crack in an infinite body under proportional loading.
In the case of reversible cracks, a change in the crack size can be traced by
means of the graph of p(c) also for a non-monotonous variation in the load.
I t is of interest to note that in this case a decrease in the load produces a
stepwise diminution of the crack size, but this happens, in general, when
critical equilibrium states are passed that are different from those corresponding to an increase in the load.
Recently, L. M. Kachanov [84a] carried out an investigation generalizing
the previous treatments so as to cover the case of the time-dependent modulus
of cohesion. This investigation is of basic importance in connection with
the problems of so-called “stress rupture”.
102
G. I . BARENBLATT
The analysis carried out in the present section is based on [59].
Consider now the solution of a problem concerning the extension of an
unsymmetrical initial crack in one simple case. Let a straight initial crack
with the end coordinates x = - a, and x = bo be given in an infinite
unloaded body (for definiteness assume b,< a,) and let equal and opposite
concentrated forces P be applied at opposite points of the crack surfaces,
say, at x = 0. The magnitude of the force P plays the role of the loading
parameter. According to (5.l ) , the values of the tensile-stress intensity
factors No at x = - a and x = b are, respectively,
When P < PI,where
5.28)
both factors N , and Nb are less than K l n so that the crack expands neither
to the right nor to the left. A t P = P, the factor Nb becomes equal to K / n ,
I
FIG.22.
mobile equilibrium is reached and the end b begins to move to the right.
The advance depends on the magnitude of the applied force according to
the relation
(5.29)
As long as P < P,, where
2
(6.30)
5
= 2a,,
K2
we have N a < Kln, and the left end does not move. At P = P2,we have
b = a,, a symmetrical crack in mobile equilibrium, and a t P > P, the
-
MATHEMATICAL THEORY OF EQUILIBRIUM CRACKS
103
development of the crack continues according to (5.8). The development
of the initial crack with changing P is plotted in Fig. 22.
4. Cracks Extending to the Surface of the Body
If a crack extends to the surface of the body, it becomes difficult to
obtain effective analytical solutions. Mapping of the corresponding region
on a half-plane cannot be carried out by means of rational functions, and
Muskhelishvili’s method does not make it possible to obtain solution in
finite form. Therefore it is necessary to resort to numerical methods in
analysing such problems.
A number of numerical solutions have been derived up to now; the
mobile-equilibrium states are unstable in all analysed cases.
FIG.23.
0. L. Bowie [22] treated the problem of a system of k symmetrically
located cracks of equal length extending to the free surface of a circular cut
in an infinite body (Fig. 23). The body is stretched at infinity by the allround stress p,. Bowie employed Muskhelishvili’s method for calculating
stresses and strains. To obtain the solution in effective form, the author
used a polynomial approximation to the analytical function mapping the
exterior of the circle with adjacent cuts on the exterior of the unit circle.
For the determination of the dimensions of mobile-equilibrium cracks Bowie
used directly Griffith’s energy method and computed the strain-energy
release rate. Numerical calculations were made for cases of one crack and
two diametrically opposite cracks. To obtain sufficient accuracy of calculations it proved necessary to retain about thirty terms in the polynomial
representation of the mapping function. The numerical results for the cases
k = 1 and R = 2 obtained by Bowie are shown in Fig. 24. It follows from
these computations that at LIR > 1 the tensile stress for two cracks with
a circular cavity is very close to the tensile stress for one crack of length
104
G . I . BARENBLATT
2(L + R),so that the influence of the cavity proper is almost unnoticeable.
Furthermore, in the case of small crack lengths the conditions of mobile
equilibrium are obviously determined by the tensile stresses directly a t the
3-
I
I
I
I
2 -\
I
0
FIG.24.
1
I
1
2
3 a
1
uniaxial tension, - - - all-round tension.
surface of the circular cavity. As is known, in case of uniaxial extension the
highest tensile stress a t the boundary of the cavity is equal to 3p0 and in
case of all-round extension 29,. Thus the
ratio
of equilibrium loads in these cases
a
b
Y
should approach 213, and this is found
in agreement with Bowie's calculations.
The problem of a straight crack ending on a straight free boundary of the
half-space (Fig. 25) was treated independently by L. A. Wigglesworth [86] and
G. R.Irwin [51] using different methods.
Wigglesworth [85] investigated the
case of an arbitrary distribution of norFIG.25.
mal and shearing stresses over the faces
of the crack. For a symmetrical distribution of stresses he reduced the problem to an integral equation for the
complex displacement w ( x ) = zc(x) + i v ( x ) of points of the crack surface:
'w
I
(5.31)
X
MATHEMATICAL T H E O R Y OF EQUILIBRIUM CRACKS
105
+
Here L ( x , t ) is a singular integral operator and p ( x ) = a ( x ) it(%);a(x) is
the distribution of normal stresses; t ( x ) is the distribution of shearing stresses.
Equation (5.31)is solved in the paper by an integral-transform method.
Detailed calculations are made for the case when the surfaces of the crack
and boundary are free of stresses, the tensile stress Po being applied a t
infinity parallel to the boundary of the half-space.
For stresses near the crack end the author obtains in this special case the
following relations :
hence we find a t the prolongation of the crack (@ = z)
ax = a,, = 0.793 Po
(5.33)
PI
ax,,= 0,
which together with (4.6)gives the expression for the length of the mobileequilibrium crack in the form
1=
(5.34)
K2
1.61,.
K2
n2(0.793)Po
Po
Irwin [51] investigated only the last special case. He represented the
unknown solution as the sum of three fields. The first field corresponds to
I, y = 0) in an infinite body subjected to constant
a crack (- 1 x
tensile stress Po a t infinity, the second field corresponds to the same crack
under normal stresses Q ( x ) symmetrical with respect to the x and y axes
and applied at the crack surface, the third field corresponds to a half-space
x 0 without crack, a t the boundary of which ( x = 0) the distribution of
normal stresses P ( y ) , symmetrical with respect to the x axis, is given.
Satisfying the boundary conditions at the free boundary and the crack
surface, Irwin obtained for P ( y ) and Q ( x ) the system of integral equations
< <
I
(5.35)
m
xy2
0
d y =Q(x),
106
G. I. BARENBLATT
which he solved by the method of successive approximations. The first
approximation yields a relation for the length of the mobile-equilibrium
crack 1 :
(5.36)
I=
K2
2K2
= 1.69 7 ,
7c2 1.0952P02
Po
which differs, as is seen, insignificantly from the more exact relation (5.34).
H. F. Bueckner [50] treated a problem of one straight crack reaching
the boundary of a circular cavity in an infinite body. No stress is applied at
infinity and at the boundary of the cavity, the surface of the crack is free
of shearing stresses, normal stresses are applied symmetrically and vary
according to a given law - p ( x ) . Such a form of the problem arises in the
analysis of rupture of rotating disks. Like Wigglesworth [85], Bueckner
proceeds independently from a singular integral equation for the lateral
displacements of points of the crack surface. He considers a one-parameter
family of particular solutions of this equation, corresponding to certain
special distributions p , ( x ) . In the general case it is recommended to represent
P ( x ) as a linear combination of P , ( x ) :
n=m
n=o
the coefficients a, are determined by the least-square method or by collocation. The factor of stress intensity at the crack end No is expressed in terms
of the coefficients a,.
If the length of the crack is far less than the radius of the circular cavity,
then we have in the limit the previous particular case of a straight boundary.
As it follows from Bueckner’s calculations in this particular case when
P = Po = Const, the expression for the length of a mobile-equilibrium crack is
(5.38)
1=
2K2
Pa
= 0.159 0 ,
7c2 1.132$02
K2
which is in good agreement with (5.34) and (5.36).
In [50] Bueckner also treated a problem of a crack reaching the surface of
an infinitely long strip of finite width under an arbitrary load, symmetrical
with respect to the line of the crack (Fig. 25b). He showed that it is possible
to replace with a high degree of accuracy the integral equation occurring
in this case by one with a degenerated kernel. The numerical solution
obtained by Bueckner in the special case when the load is produced by
couples M , applied on both sides of the crack at infinity, gives the relation
between the length of a mobile-equilibrium crack and the load; it is represented by the curve in Fig. 26.
MATHEMATICAL THEORY OF EQUILIBRIUM CRACKS
107
As has already been pointed out, in all cases discussed in the present
section mobile-equilibrium cracks are unstable. Thus, when loads increase,
extension of an initial crack does not take place until it reaches mobile
equilibrium, after which the body fractures. In these problems the load a t
which an initial crack reaches mobile
equilibrium coincides with the breaking
load, which is in general not true.
In the paper by D. H. Winne and
B. M. Wundt [32] some of the solutions
presented in this section were employed
for the analysis of fracture of rotating
notched disks, and of notched beams in
I
I
I
I
bending. The experiments conducted
0
a5 I
by Winne and Wundt, analysed on
L
the basis of these calculations, revealed
FIG.26.
close coincidence of the values of surfaceenergy density T (or, which amounts
to the same, of the moduli of cohesion K ) determined from the angular
speed, a t which fracture of rotating notched disks occurs, and from the
loads a t which fracture of notched beams in bending occurs. This confirms
that the quantities T and K are characteristics of the material and do not
depend on the nature of the state of stress.
I
I
5 . Cracks near Boundaries of a Body; Systems of Cracks
Crack development in bounded bodies possesses some characteristic
peculiarities. Difficulties of a mathematical character do not allow us to
a
b
i
I
f l t
0
1
0
t
I
h
FIG. 27.
carry out here as complete an investigation as in the case of isolated cracks.
However, the qualitative features and some of the quantitative characteristics of this phenomenon can readily be elucidated in connection with the
108
G . I. BARENBLATT
simplest problems that yield to analytical solution. Let us examine first of
all the problem of a straight crack in a strip of finite width (Fig. 27a). The
crack is assumed to be symmetrical with respect to the middle line of the
strip, and the direction of its propagation is normal to the free boundary.
The load keeping the crack open is considered symmetrical with respect to
the line of the crack and the middle line of the strip.
In solving the problem we use the method of successive approximations
developed by D. I. Sherman [86] and S. G. Mikhlin [87]. As the first
approximation we take the solution of a problem in the theory of elasticity
for the exterior of a periodical system of cuts (Fig. 27b). Denoting again
by $(t) the distribution of tensile stresses, which would be at the place of
the cracks in a continuous body under the same loads, we obtain the equation
determining the half-length of a mobile-equilibrium crack 1 in the form
(5.39)
m-t
-m
where t = sin (nt0/2L),m = sin (n1/2L). In the particular case represented
in Fig. 27, when the crack is maintained by equal and opposite concentrated
forces P with points of application 2s apart along their common line of
action, (5.39) becomes
+ 1) sin (nl/L)
V8(a2
(5.40)
u(2a2
+ 1) cosha
+
%(a2 l)m,
I’
where a = sinh a/m, u = ns/2L. When s = 0 (concentrated forces applied
a t the crack surface), (5.40) reduces to
(5.41)
Let us also quote the relation between the size of a mobile-equilibrium
crack and the load for the case of a uniform tensile stress at infinity, P/2L,
(5.42)
__ K?Z -
v
nl
2
-cot n
2L
Relation (5.40) for various u is presented in Fig. 28. The solid and dotted
lines denote, as usual, stable and unstable branches. As is seen, for u 2 ucw 0.5
there are no stable branches, hence for distances between points of application of forces exceeding 2Lln m 0.64 L mobile-equilibrium cracks are always
unstable. Quite similarly to the analysis in Section V.3 (extension of an
MATHEMATICAL THEORY OF EQUILIBRIUM CRACKS
109
isolated crack under proportional loading) the graph in Fig. 28 makes it
possible to describe completely the extension of any symmetrical initial
crack when the load increases.
The present analysis is based on papers [58, 881. The solution of the
corresponding problem in the theory of elasticity for the case s = 0 was
obtained by Irwin [45]. The problem of a periodical system of cracks under
uniform loading a t infinity was solved by Westergaard [13] and independently by W. T. Koiter [89].
t
0
\ N
\
\
0.5
1
10 T
L
FIG.28.
In the first approximation only the shearing stresses vanish at the lines
of symmetry (shown by the dotted lines in Fig. 27b, which correspond to
the boundaries of the strip); the normal stresses are different from zero.
To obtain the second approximation, the first approximation is addcd to
the solution for an uncracked strip, at the boundaries of which the normal
stresses are given; their distribution is chosen in such a manner as to compensate the normal stresses at the boundary obtained in the first approximation. Now the boundary condition is no longer satisfied at the crack surface.
To obtain the third approximation, the second approximation is added to
the solution for the exterior of a periodical system of cuts, a t the surface of
which the distribution of normal stresses is equal to the difference between
the given stresses and those obtained in the second approximation, and
so on.
110
G. I. BARENBLATT
Special estimates obtained in [88] show that for stable mobile-equilibrium
states the considerations of the second and subsequent approximations leads
to corrections of the order of 2.5-3 per cent in the above relations. This
permits us to confine ourselves to the first approximation.
In addition to these problems (the periodical system of cracks and the
system of radial cracks ending in a circular cavity), several other problems
of systems of cracks have been treated; they deal with straight cracks
located along one straight line. Mathematical methods developed by
Muskhelishvili [go, 181, D. I. Sherman [91], and Westergaard [13] permit
FIG.29.
the reduction of any such problem to quadratures. Let us here consider the
simplest example: it is the problem of the extension of two collinear straight
cracks of the same length in an infinite body, stretched by a uniform stress
p at infinity (Fig. 29). This problem was treated by Willmore [21]; it also
occurs in a paper by Winne and Wundt [32] (the authors refer to a private
communication by Irwin). According to the solution presented in [21],
the sizes of the cracks remain unchanged at p < pl, where
(5.43)
,
a
a=-<l.
b
Here K’, E’ are standard notations of elliptic integrals.
At p = 9, the cracks attain an unstable state of mobile equilibrium,
after which the inside edges of the cracks join and form a crack of length 2b.
The further extension of the crack depends on whether the bracketed
expression in (5.43) is greater or less than unity. If it is less than unity,
which happens for a < 0.027, the size of the crack resulting from the joining
of the inside edges is less than the size of the mobile-equilibrium crack
corresponding to the load p,. In this case the crack remains unchanged up
to the load p , = VTK/n]/b,after which the body fractures. If it is greater
than unity, complete fracture of the body occurs immediately upon reaching
the load p,. Assuming b - a = 21 and making b - + 00 in (5.43), we obtain
in the limit (5.6), as expected. The solution given in [32] leads to the
same qualitative results. However, it cannot be accepted as correct because
MATHEMATICAL T H E O R Y OF EQUILIBRIUM CRACKS
111
it is based on the erroneous expressions of the stress intensity factors
given in [47].
The case of two identical cracks maintained open by concentrated forces
applied at their surface was treated in [88]. A complete investigation of the
general case of symmetrical loading for a system of two cracks can be carried
out quite similarly, with expressions for the stress intensity factors a t the
crack ends x = a and x = b
h
h’, =
v2
x Va(b2- a*)
b
[ f i ( t ) tt2 v-Ea2d t - C]
b
(5.44)
As is seen from these examples, collinear cracks “weaken” each other
and reduce their stability. Ya. B. Zeldovitch noticed that in the case
FIG.30.
of a “chess-board’’ pattern of cracks (Fig. 30) the inverse phenomenon
occurs. As the calculations show, even for uniform normal loads a t the
crack surfaces, mobile-equilibrium cracks may become stable for a certain
mutual position.
We consider briefly the so-called “size effect” in the brittle fracture of
bounded bodies. Take similarly shaped bodies, which differ only in the
characteristic size d and in the characteristic scale of the applied extensional
load S (it is supposed that macroscopic cracks present in the bodies are also
geometrically similar). In brittle fracture, the value S = So that corresponds
to fracture depends only on the characteristic size of the body d and the
112
G. I. BARENBLATT
modulus of cohesion K . There is only one way to form a characteristic
having the dimension S from the quantities K and d , and it is impossible
to make any dimensionless combinations. Therefore the following simple
rdations govern the magnitude of the breaking load:
where So has the dimension of a force, of a force distributed along a line
(as, for instance, a concentrated force in plane strain), and of a stress, respectively. The quantities ei are constants for a given geometrical configuration
of the body. About the fracture of geometrically similar bodies a great
deal of experimental data is a t present available, which permits clarification
of the limits of applicability of the theory of brittle fracture. Detailed
information on this topic can be found in a paper by B. M. Wundt [92], and
some new results have been presented by S. Yusuff [93].
6. Cracks in Rocks
The investigation of crack extension in rock massifs is of great interest
in theoretical geology. Cracks can form in rocks because of various causes
of tectonic character, but also because of some artificial actions (mining
excavations, hydraulic fracture of oil-bearing strata, etc.),
In connection with the theory of
hydraulic fracture of an oil-bearing
stratum a number of problems of the
theory of cracks have been treated,
among them the problem of the vertical
crack: A crack in an infinite space
subjected to all-round compressivepressure q at infinity is maintained open
by a flowing viscous fluid injected into
it (Fig. 31). Themain peculiarity of the
FIG. 31.
problem is that the fluid does not fill
the crack completely: there is always
a free part of the crack on both sides of the wetted area. The pressure Po
in the flowing fluid troughout the wetted area of the crack can be considered
constant in first approximation. Indeed, a t the end of the wetted area an
abrupt narrowing of the crack takes place, and almost all of the pressure
drop will occur there. The problem is called so, because the actual fissure,
idealised by this problem, is located in a vertical plane, and q represents
the lateral pressure of the rocks. In comparison with the action of lateral
rock and fluid pressures the action of the forces of cohesion may be
“ 2
113
MATHEMATICAL T H E O R Y OF E Q U I L I B R I U M CRACKS
neglected, as estimates show.* Condition (5.3) determining crack sizes
becomes
1
hence
1 =I,
(5.47)
21-l
[ sin
. - .
The expression for the maximum half-opening of the crack vo is
(5.48)
As calculations show, for values 1,/1 close to unity which are usually
encountered in practice, the opening of the crack is almost constant all
along the wetted area of the crack; the crack closes rapidly along the free
part. - This problem of the vertical crack was first stated and solved in
a paper by Zheltov and Khristianovitch [38].
The problem of the horizontal crack [40] is stated as follows. In a heavy
half-space at a certain depth N a horizontal disk-shaped crack is formed by
injecting viscous fluid as before; the surface of the crack is again divided
R,) and a free part (Ro< Y
R ) , and the
into a wetted part (0 r
fluid pressure p in the wetted part may again be considered as constant.
Forces of cohesion, as in the preceding case, are neglected. Under the
assumption that the depth of the crack position H is sufficiently great, the
boundary condition at the boundary of the half-space need not be taken into
account. The condition of finiteness of stresses a t the crack contour yields
in this case
<
< <
(5.49)
where y is the specific weight of the rock. For the volume of the injected
fluid one obtains
(5.50) T/ =
4(1 - y 2 ) # R 3
(2)
[
2
z
fp - , v(z)= z 3 - - - -
3
3
3(1
+
z
v-)]'
* The condition that forces of cohesion be negligibly small is K/gvl<< 1. It is in
general not satisfied in laboratory scaling.
114
G . I. BARENBLATT
In practice, z = Ro/R is close to unity so that it is possible to use the
asymptotic form of (5.50)
4(1 - v2)$R9
v2(1 - z) [l
3E
~
(5.51)
L' =--
+ v2(1 - z) - 3(1 -41.
The maximum half-opening of the crack is determined by the formula
(5.52)
Thus, if the depth of the crack position, the fluid pressure, and the
specific weight of the rock are known, Ro/R can be found according to (5.49).
Then the crack radius is obtained from (5.51) and a knowledge of the total
volume of the injected fluid V , after which the determination of the remaining
parameters does not encounter any difficulties.
In [40, 411 problems were also treated concerning horizontal cracks in
a radially varying pressure field caused by the higher lying rocks. Under
certain conditions a complete wetting of the crack surface (i.e. the absence
of a free part) may in this case occur.
Yu. P. Zheltov [43] proposed an approximate method for solving the
problem of the horizontal crack in a radially varying vertical pressure field.
A comparison between the results obtained by this method and the exact
solutions for certain cases showed quite satisfactory agreement.
By using the method of successive approximations Yu. A. Ustinov [94]
estimated the influence of the free boundary in the problem of the horizontal
crack. If the depth is larger than twice the crack radius, the influence of
the free boundary is negligibly small.
The problem of a crack formed by driving a horizontal wedge of constant
thickness into a heavy space was treated in [39].
The solution of the problem of the vertical crack was extended by Zheltov
[42] to cover the case when the rock is permeable and the injected fluid
flows through the rock.
PROBLEMS
I N THE THEORYOF CRACKS
VI. WEDGING;DYNAMIC
1. Wedging of an Infinite B o d y
Wedging is formation of a crack in a solid by driving a rigid wedge into it.
The most characteristic property of the wedging of a brittle body is that
the wedge surface never comes in complete contact with the body: there is
always a free portion in the front part of the wedge; ahead of the wedge
a free crack forms, which closes at some distance from the edge of the wedge
(Fig. 32).
MATHEMATICAL THEORY OF EQUILIBRIUM CRACKS
115
It appears that the problem of wedging of an infinite body by a fixed
wedge [39, 58, 951 is the simplest to formulate among problems of this kind;
it yields to an effective exact solution by the methods of elasticity theory
and gives a qualitative idea of wedging under more complex conditions.
Let a uniform, isotropic brittle body be wedged by a thin, symmetrical,
perfectly rigid semi-infinite wedge with thickness 212 a t infinity (Fig. 32).
In front of the wedge a free crack forms, which closes smoothly a t a certain
- /
c/ B'
Y
"/////TI
FIG.32.
point 0; the position of the point 0 with respect to the front point of the
wedge C is not known beforehand and must be determined in the course of
solving the problem. If the wedge has a rounded front part (Fig. 32a). the
position of the points of departure of the crack surface from the wedge,
B and B', is not prescribed and must also be determined in the course of
solving the problem. If the wedge has a truncated front part (Fig. 32b)
as e.g. in the case of a wedge of constant thickness, the position of the points
of contact is quite definite; they coincide with the corners of the wedge
front. I t is evident that the stress at the points of departure is in this case
infinite. We shall a t first assume that there is no friction a t the surface
of contact between wedge and body.
The field of elastic stresses and strains satisfies the usual equations of
static elasticity in the exterior of the crack. In view of the assumed
slenderness of the wedge, the boundary conditions may be transferred from
the crack surface proper to the x-axis. Without considering forces of cohesion,
the boundary conditions are
uxy=o,
(6.1)
v = f f ( x - ZJ,
u,=O
uxy =
(O,<X<l,,
0
y=0),
(12 Q x < 00, y = 0);
116
G. I . BARENBLATT
here uy,uZyare the stress-tensor components; I, and 1, are the distances of
the point 0 from the edge of the wedge and from the points of departure
B, B ' ; f ( t ) defines the wedge surface in a system of coordinates with origin
a t the front point of the wedge; the positive and negative signs correspond to the upper and lower faces of the cut, respectively.
As is seen, the problem of wedging is a peculiar combination of the
contact problem in the theory of elasticity [18,72,73] and the problem of the
theory of cracks.
The position of the points of departure of the crack surface from the
wedge in the case of a wedge with rounded edge, and the position of the
point of closing with respect to the edge are determined from the following
conditions:
1. Stresses at the points of departure must be finite. For the contact problem
a similar condition was first suggested as a hypothesis by Muskhelishvili
[96, 181 and independently by A. V. Bitsadze [97]; it was proved in [6l].
2. Stresses at the crack edge are finite or, which i s the same, the opposite
faces of a crack close smoothly at its end. Since the intensity of forces of
cohesion at the crack edge is maximal, stresses near the crack edge calculated
without taking into account forces of cohesion must tend to infinity according
to (4.7).
The problem of wedging is a mixed problem of the theory of elasticity.
For its solution it is convenient to consider the singular integral equation for
the compressive force acting on the face of the wedge, uy = - + ( x ) . If + ( x )
is known, the determination of the elastic field obviously reduces to the
solution of the first boundary-value problem in the theory of elasticity for
the exterior of a semi-infinite straight-line cut, which can be found by
Muskhelishvili's method ([18], 95). This solution gives an expression for
the lateral displacements at points of contact between wedge and crack
surface :
where 5 =.vG, and the root may assume positive and negative values for
displacements of the upper and lower face. The second condition (6.1) yields
the fundamental integral equation of the problem :
MATHEMATICAL THEORY OF EQUILIBRIUM CRACKS
117
which can be shown to be equivalent to the singular integral equation
obtained from (6.3) by differentiation with respect to c :
and the condition
where h = f ( b o ) . By using the methods for singular integral equations
developed in the monogra.ph by Muskhelishvili ([19], Chapter 5) the solution
of equation (6.4) can be found in the form:
where A is an indefinite constant.
The integral in (6.6) does exist in view of the finiteness of f ( m ) = h, and
it tends to zero as x-+ bo ; this together with (6.5) determines the value of
the constant A :
(6.7)
A =
Eh
2(1 - v2) *
For finiteness of stress at the points of departure x = I 2 in case of a
wedge with rounded edge, it is necessary and sufficient that the bracketed
expression in (6.6) vanishes at x = 1,. This gives one equation for the
determination of 1, and I,:
Now the following expression for the tensile stresses a t the prolongation
of the cut results from the solution:
t - x
118
G. I .
BARENBLATT
Together with (4.7) it leads to
Relations (6.8) and (6.10) are finite equations which determine the
unknown constants I, and 1,.
In the particular case of constant wedge thickness f ( t ) h, condition
( 6 4 , which is no longer valid, is replaced by the relation 1, = I,, and (6.10)
gives the following expression for the length of a free crack in front of a
“square” wedge :
(6.11)
In 1951 other special forms of the wedge are also treated such as a wedge
rounded-off with a small radius of curvature and a wedge rounded-off
according to a power law. Investigation of the first example shows that
I
I
I
FIG. 33.
roundness affects slightly the length of the free crack in front of the wedge.
I n [95] also a case when Coulomb friction acts on the faces of the wedge is
treated.
In [84] wedging of an anisotropic body by a semi-infinite rigid wedge is
studied.
I. A. Markuzon [98] treated a problem of wedging an infinite body by
a wedge of finite length 2b (Fig. 33). In case of constant thickness of the
wedge 2h, the relation between crack length 21 and wedge length 2b, other
things being equal, is as represented in Fig. 34 (1, is the length of a free
crack for an infinite wedge defined by (6.11))-
MATHEMATICAL THEORY OF EQUILIBRIUM CRACKS
119
In [98] the influence of a uniform compressive or tensile stress at infinity
on the length of a free crack, when the wedge is of finite length, was also
investigated.
Relation (6.11) can be used for the experimental determination of the
modulus of cohesion K . For that purpose a wedge is driven into a plate of
the testing material, the wedge being substantially more rigid than the plate.
The length L of the resulting free crack is measured. The modulus of cohesion
can then be found according to the formula
(6.12)
The wedge must be sufficiently long in order to eliminate the influence
of the plate boundary, and it should be driven in, until the distance between
the wedge end and the crack end stops varying with further displacement
of the wedge. The plate must be wide and sufficiently thick so that the
state of stress essentially corresponds to plane strain. To insure a straight-line
form of the crack, it is necessary to compress the specimen in the direction
of crack propagation. This is recommended by Benbow and Roesler [9].
(It can be shown that (6.11) and (6.12) remain unchanged in this case.)
2. Wedging of a Strip
In strict formulation, problems concerning the wedging of bounded
bodies are very difficult to solve. Up to now there are but a few approximate
solutions, based on the application of the approximations of simple beam
theory.
The first of these solutions was obtained by I. V. Obreimov [8]; as a
matter of fact, this work was the first investigation in which wedging was
considered. In connection with his experiments on the splitting of mica,
Obreimov examined the case when a strip being torn off has small thickness
and only one-point contact with the wedging body (Fig. 35). In order to
establish a relation between the surface tension of mica and the parameters
of the crack shape, Obreimov applied to this problem the methods of strength
of materials, considering a shaving as a thin beam. The theoretical part of
the work of Obreimov is not free from shortcomings. Later, corrections were
introduced into these calculations in the book by V. D. Kuznetsov [99] as
well as by M. S. Metsik [lo] and N . N . Davidenkov [12]. In addition Metsik
improved the experimental procedure of [8]. Application of the approximations of thin-beam theory for the determination of the crack length is
justifiable in some cases. However, these approximations cannot be applied
to describe the form of the crack surface in the immediate vicinity of its edge,
even if the distribution of forces of cohesion in the edge region is explicitly
120
G. I. BARENBLATT
considered, as it was done by Ya. I. Frenkel [5]. The fact is that the longitudinal dimension of the edge region cannot be assumed to be large
compared to the shaving thickness; hence a shaving cannot be considered
as a thin beam in the region where the forces of cohesion are acting.
To illustrate the approximate approach based on the methods of simple
beam theory, we discuss the paper by Benbow and Roesler [ 9 ] in more
detail. Note that in this work possibilities and limits of applicability of the
above approach are most clearly pointed out.
The fdllowing statement of theproblem is considered (Fig. 36). A strip
of finite width b is wedged symmetrically so that the crack passes along the
i
FIG. 35.
1. a Body being wedged.
1. a Wedge.
FIG. 36.
1. a Body being wedged.
2. Grips.
middle line of the strip. At the end of the strip, compressive forces Q / 2
are applied to insure straight crack propagation; the wedging force P
produces a crack length 1 and initial width h.
Having obtained an expression for the strain energy from dimensional
considerations, the authors write the equilibrium condition for the crack in
the form
(6.13)
T
E
ha
l
so that for a given material the quantity h2/1 is uniquely determined by the
quantity bll. The experiments made with specimens of two different plastics
[ 9 ] give a conclusive proof of the existence of such a one-one relation.
For small b/Z, i.e. for long cracks, it is possible to obtain an asymptotic
form of relation (6.13) by considering both halves of the strip as thin beams
fixed a t the section corresponding to the crack end. The expression for the
strain energy of the strip is in this case
(6.14)
U = 3h2B/13,
where B = E J is the stiffness of the beam, J = nb3/96, and n is the transverse
thickness of the beam. The surface energy of the crack is, evidently, 2Tnl.
MATHEMATICAL THEORY OF EQUILIBRIUM CRACKS
121
In the mobile-equilibrium state the variation of surface energy corresponding
to a small variation of the crack length 61 is equal to the corresponding
variation of strain energy of the strip. Hence it follows that
(6.15)
- -au
=2Tn
T - 3h2b3
---.
E
6414
or
ai
By comparing the second formula (6.15) with (6.13), an asymptotic
expression for @(b/l) can be found as b/l ---t 0 :
b 3
3
(6.16)
Q=&)
*
From (6.15) an expression for the length of an equilibrium crack is
obtained :
4
4
(6.17)
vc
Thus in this case the length of the crack is proportional only to
whereas in the wedging of an infinite body by a semi-infinite wedge the length
of the crack is proportional to h2 (cf. (6.11)).
Relation (6.15) was used by Benbow and Roesler for the determination
of the surface-energy density of the plastics investigated. The careful
experimentation and the scrupulous evaluation of the sources of possible
errors and of their magnitude are remarkable.
In the recent review of J. J. Gilman [ll] a detailed summary and a
bibliography of experimental investigations on wedging can be found.
3. Dynamic Problems in the Theory of Cracks
Considerable attention is nowadays given to questions of dynamics of
cracks. A detailed consideration of these questions is beyond the scope of
the present review; we shall confine ourselves here to a brief information
about basic results achieved in theoretical investigations of dynamics of
cracks.
In the paper by N. F. Mott [36] the crack-expansion process is treated
in the case of an isolated straight crack in an infinite body subjected to a
uniform field of tensile stresses &* On the basis of dimensional analysis
Mott obtained an expression for the kinetic energy of a body,
(6.18)
d = kpl2V2pO2/E2,
* Unlike [36J, plane strain is here considered rather than the state of plane stress.
122
G . I. BARENBLATT
where p is the density of the body, 1 the half-length of the crack, V the rate
of crack expansion, and k a dimensionless factor which Mott considered
constant and left indefinite. Adding to the static-energy equation (2.1)
the derivative with respect to I of the kinetic energy (6.18) and assuming the
remaining terms in (2.1) to be the same as in Griffith’s static problem,
Mott found the rate of crack expansion
(6.19)
where I , is the half-length of the mobile-equilibrium crack defined by (5.4).
Thus, as the crack propagates, its extension rate increases, approaching the
limit
(6.20)
The ultimate rate constitutes, according t o Mott, a certain part of the
longitudinal wave propagation velocity. In this reasoning the use of the
static expression for the decrease of the strain energy W remains unfounded.
Moreover, the quantity k in (6.18) and (6.19) need not be constant in general;
it may depend on l/l*, V/cl and other dimensionless combinations.
E. Yoffe [loo], using the exact formulation of dynamic elasticity theory,
investigated the problem of a straight crack of constant length, moving with
constant velocity in an infinite body stretched by uniform stress a t infinity.
Notwithstanding the somewhat artificial character of the problem, an
important result was obtained in this paper, which has quite a general
meaning: If the crack propagation rate becomes greater than a certain
critical rate, the direction of crack propagation is no longer the direction
of maximum tensile stress, and the crack begins to curve. The magnitude
of the critical rate Vl is about 0.4 cl, where c1 is the longitudinal wave
propagation velocity in the given material (the ratio Vl/cl depends slightly
on Poisson’s ratio v of the material).
D. K. Roberts and A. A. Wells [ l o l l made an attempt to evaluate the
constant k , which remained indefinite in [36]. Using the value of k obtained,
they found the ultimate crack expansion rate close to that found by Yoffe.
However, their estimate, based as it is on the solution of a static problem,
is too rough; and since the straight-line direction of crack propagation in [ l o l l
was assumed as certain, the close agreement between the critical rate found
by Yoffe [loo] and the ultimate rate obtained in [ l o l l must be considered
as incidental.
If the straight-line direction of crack propagation is somehow insured
(for instance, by a large compression of the body in the direction of crack
propagation or by the anisotropy of the material), then the maximum rate
MATHEMATICAL THEORY OF EQUILIBRIUM CRACKS
123
of crack propagation coincides with the velocity of propagation of Rayleigh
surface waves in the given material, which is about 0.6 cl.
The fact that the ultimate rate of crack propagation coincides with the
Rayleigh velocity was first stated by A. N. Stroh [l02]. The heuristic proof
given in that paper amounts to the following. Stroh correctly notes that the
ultimate rate of crack propagation does not depend on the surface energy
of the body, and he assumes the surface energy to be zero. Proceeding from
this, Stroh is led by energy considerations to the conclusion that the tensile
stress near the crack end (on its prolongation) is equal to zero. Thus the crack
may be thought of as a disturbance moving on a stress-free surface, which
can propagate only with the Rayleigh velocity. In fact, from Stroh’s
reasoning it may only be concluded that the tensile stress at the very contour
of the crack is equal to zero. From this fact, however, it does not follow
that the rate of crack propagation is equal to the Rayleigh velocity, as can
be seen from the following simple example. Take a body subjected to allround compressive stress at infinity and wedged by a semi-infinite wedge
as in Fig. 32, moving with infinitely small velocity. Forces of cohesion and,
consequently, surface energy are assumed to be zero. In view of the infinitesimal velocity of the wedge, dynamic effects are insignificant, hence,
according to Section 111,2, the tensile stress at the crack end must vanish.
,4t the same time, the rate of crack propagation is equal to the velocity of
the wedge, i.e. it is also infinitesimal.
By arguments based on the analysis of exact solutions of the dynamic
equations of elasticity, the conclusion that the ultimate rate of crack propagation is equal to Rayleigh velocity was drawn independently and simultaneously by several authors. I. W. Craggs [lo31 considered steady propagation of a semi-infinite straight crack with symmetrically distributed normal
and shearing stresses applied on a part of the crack surface adjacent to the
edge. In a paper by Dang Dinh An [lo41 a non-steady field of stresses and
strains was investigated, acting in an infinite body with a semi-infinite crack,
along the surface of which symmetrical concentrated forces normal to the
crack surface begin to move suddenly away from the edge with constant
velocity. Paper [95] examines the wedging of an infinite isotropic brittle
body by a semi-infinite rigid wedge of arbitrary form, moving with constant
velocity. In [84] a similar problem is treated for a case of an anisotropic
body. B. R. Baker [lo51 considers a non-steady distribution of stresses
and strains in a solid with a semi-infinite crack, at the surface of which
constant normal stress is applied at the initial moment, after which the
crack begins to expand with constant velocity.
From the various problems treated in these papers the following general
result was obtained which led to our earlier conclusion: when the
characteristic rate involved in the problem approaches the Rayleigh
velocity, peculiar resonance phenomena arise. Note that the appearance
124
G . I . BARENBLATT
of resonance when the Rayleigh velocity is approached is not specific for the
problems of cracks: the investigation of the problem of a punch moving
along the boundary of a half-space, carried out by L. A. Galin [72] and
J. R. M. Radok [log], reveals [95] that the same resonance phenomena
occur, when the velocity of the punch approaches the Rayleigh velocity.
I t appears that the limiting character of the Rayleigh velocity is most
directly illustrated by a problem of wedging. Obviously the maximum
possible rate of crack propagation can be reached in wedging a body by a
moving wedge. The analysis of this problem shows [95] that with increasing
velocity of the wedge the length of the free crack in front of the wedge
decreases and tends to zero when the Rayleigh velocity is approached.
For larger wedge velocity a free crack does not form in front of the wedge.
Hence the maximum rate with which a crack can expand is equal to Rayleigh
velocity.
K. B. Broberg [107, 1081 treated the problem of a uniformly expanding
crack of finite length in an infinite body subjected to a uniform tensile
stress field. The solution obtained by Broberg is an asymptotic representation for great values of time of the solution of the problem treated by Mott [36]
and Roberts and Wells [ l o l l . However, unlike [loll, Broberg’s solution
was obtained on the basis of the exact methods of the dynamic theory of
elasticity. Independently of [102-104, 57, 95, 1051 and in full accord with
the results of these investigations, Broberg obtained that the rate of crack
expansion in his problem, equal to the ultimate rate of crack expansion in
the problem considered in [36, 1011, coincides with the Rayleigh velocity.
Note the papers by B. A. Bilby and R. Bullough [lo91 and F. A.
McClintock and S. P. Sukhatme [110] which treat uniformly moving cracks
of finite and infinite length, respectively, a t the surface of which symmetrical
shearing stresses parallel to the crack edge were applied. Instead of plane
strain we have in this problem what is often called anti-plane strain: one
displacement component, parallel to the crack edge, is different from zero.
The investigation of such cracks reduces to the solution of a single
wave equation (reducing to Laplace’s equation for equilibrium cracks).
Cracks under anti-plane strain conditions are of considerable interest, being
the simplest model for which an effective solution is possible for many problems, which are intractable for cracks under plane-strain conditions because
of the great mathematical difficulties.
An analysis of the dynamics of crack propagation on the basis of the
approximations of the simple beam theory was carried out by J. J. Gilman
[ l l ] and J. C. Suits [ l l l ] .
MATHEMATICAL THEORY OF EQUILIBRIUM CRACKS
125
ACKNOWLEDGEMENT
The author is very grateful t o Prof. Ya. B. Zeldovitch and Prof. Yu. N. Rabotnov
(USSR Academy of Sciences) and Dr. S. S. Grigorian for the invariable interest and
attention given t o his work on cracks and for a number of valuable advices. He recalls
with appreciation the valuable discussions with Prof. S. A. Khristianovitch (USSR
Academy of Sciences). The author considers it his pleasant duty to express his sincere
thanks t o Prof. G. Kuerti (USA) and Prof. G. G. Chernyi for the amiable assistance
in writing this review. Credit is also given t o I. A. Markuzon who assisted the author
in compiling the bibliography.
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Phys. Solid5 8, 66-75 (1960).
104. DANGDINHAN, Elastic waves by a force moving along a crack, J . Math. and
Phys. 38, 246-256 (1960).
105. BAKER,B. R., Dynamic stresses created by a moving crack, Paper presented at
the X t h Intern. Congr. A p p l . Mech., Stresa (1960).
106. RADOK,J. R. M., On the solutions of problems of dynamic plane elasticity,
Quart. A p p l . Math. 14, 289-298 (1956).
107. BROBERG,
K. B., The propagation of a brittle crack, Paper presented at the X t h
Intern. Congr. A p p l . Mech., Stresa (1960).
108. BROBERG,
K. B., The propagation of a brittle crack, Arkiu for Fysik 18. 159-129
(1960).
109. BILBY,B. A,, and BULLOUGH,
R., The formation of twins by a moving crack,
Phil. Mag., V I I ser. 45, 631-646 (1954).
F. A., and SUKHATME,
S. P., Travelling cracks in elastic materials
110. MCCLINTOCK,
under longitudinal shear, J . Mech. Phys. Solids 8, 187-193 (1960).
111. SUITS,J. C., Cleavage. ductility and tenacity in crystals. Discussion, i n “Fracture”
(B. L. Anderson, and 0th.. eds.), pp. 223-224. Wiley, New York, 1959.
This Page Intentionally Left Blank
Plasticity Under Non-Homogeneous Conditions
BY W . OLSZAK. J . RYCHLEWSKI AND W . URBANOWSKI
Institute ofFundamenta1 Technical Problems. Polish Academy of Sciences. Warsaw. Poland
Page
.
I Physical Foundations . . . . . . . . . . . . . . . . . . . . . . . .
132
132
1. Introductory Remarks . . . . . . . . . . . . . . . . . . . . . .
2 . Plastic Non-homogeneity of Real Bodies . . . . . . . . . . . . . . 133
3. Definition and Classification of Non-homogeneous Elastic-plastic Bodies . 148
I1. Plane Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . .
151
1. Introductory Remarks . . . . . . . . . . . . . . . . . . . . . .
151
2 . Basic Assumptions and Equations . . . . . . . . . . . . . . . . . 151
155
3. The Possibilities of Solving . . . . . . . . . . . . . . . . . . . .
4. Equations of Equilibrium of a Non-homogeneous Body in Curvilinear
156
Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . .
5 . The Geometry of Slip Lines and Trajectories of Principal Stresses . . . 160
6 Biharmonic States of Equilibrium . . . . . . . . . . . . . . . . . 165
7. Analytical Solutions in Particular Cases . . . . . . . . . . . . . . 167
8. Approximate Solutions . . . . . . . . . . . . . . . . . . . . . .
174
9 Inverse and Semi-inverse Methods . . . . . . . . . . . . . . . . . 177
.
.
I11. Particular Solutions . . . . . . . . . . . . . . . . . . . . . . . . .
183
183
1. General Method . . . . . . . . . . . . . . . . . . . . . . . . .
2 . Axially Symmetric Problems . . . . . . . . . . . . . . . . . . . .
185
3. Spherically Symmetric Problems . . . . . . . . . . . . . . . . . . 189
189
4. Torsion of Prismatic Bars . . . . . . . . . . . . . . . . . . . . .
5 Rotating Circular Disc . . . . . . . . . . . . . . . . . . . . . .
190
.
. . . . . . . . . . . . . . . . 190
V . Limit Analysis and Limit Design . . . . . . . . . . . . . . . . . . . 191
1. One-dimensional Structural Elements . . . . . . . . . . . . . . . . 191
195
2. Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
197
3. Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4. Minimum-weight Design . . . . . . . . . . . . . . . . . . . . . .
199
VI . Propagation of Elastic-plastic Waves in a Non-homogeneous Medium . . 201
203
VII . Other Problems . . . . . . . . . . . . . . . . . . . . . . . . . . .
1. Physically Non-Linear Bodies . . . . . . . . . . . . . . . . . . . .
203
2. Loose and Cohesive Granular Media . . . . . . . . . . . . . . . . 204
3. Assumption of Non-homogeneity as a Method of Solving Homogeneous
Plastic Problems . . . . . . . . . . . . . . . . . . . . . . . . .
205
IV . Elastic-plastic Non-homogeneous Plates
References
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
131
206
132
W.
OLSZAK, J . RYCHLEWSKI AND W. URBANOWSKI
I. PHYSICAL
FOUNDATIONS
1. Introductory Remarks
'
The theory of plasticity represents one of the branches of Continuum
Mechanics which is in a particularly rapid development. The progress
consists in both a careful analysis of the basic equations and physical relations, and an extension of the range of problems and effective solutions
covered by this theory. In recent years one of the points of interest has
been formed by the problems connected with bodies, whose plastic properties
are functions of the position of the point under consideration P , i.e. with
bodies exhibiting plastic non-homogeneity ; the investigations take usually
also the plastic anisotropy into account, i.e. the dependence of the plastic
properties on the direction considered.* This progress is mainly due to the
growing demands of engineering applications, which involve ever increasing
parameters such as higher pressures, temperatures, velocities, and greater
spans, the requirement of minimum weight of the structure playing also
an important part. Properties of materials which have so far been disregarded
must now be taken into consideration.
The present survey deals with problems of theoretical and experimental
research related t o plastically non-homogeneous media.
In real media the non-homogeneity of mechanical properties may be
caused by numerous phenomena and its nature may be very diverse. Consequently, it is expedient to state precisely what kind of non-homogeneity
will be considered in this review.
First of all it is evident that a universal property of bodies occuring in
practice is their microscopic non-homogeneity. It is well known that there
have been many successful1 attempts to include microscopic non-homogeneity
in the structure of Continuum Theory. Our considerations, however, do not
concern this type of non-homogeneity. Without going into more precise
definitions and specifications, let us state that the object of our interest is
the macrosco~icnon-homogeneity taking place in regions whose linear dimensions are comparable with the characteristic dimensions of the body under
consideration.
Moreover, it is known that in most materials used in practice a random
distribution of mechanical properties is encountered; sometimes the variations are considerable and due to various phenomena, (as, for instance, the
technological processes involved, etc.). This feature of materials is the
subject of a developing new science - the statistical theory of strength of
* A t the International IUTAM Symposium in Warsaw, 1958, 18 papers were devoted
to these problems, [74]. Some of them will be quoted on several occasions and their
results taken into account in the present survey article.
PLASTICITY UNDER NON-HOMOGENEOUS CONDITIONS
133
materials - which deals with stochastically non-homogeneous media, their
mechanical properties being random variables. Here, however, we shall
consider well-defined distributions of non-homogeneity, the mechanical
properties being definite functions of position.
I t may be noted that the first attempts in this direction were made
about twenty-five years ago (see W. Olszak, [73 e-j]) ; they represented
a comparatively general approach, based on the assumption of both nonhomogeneity and anisotropic structure of the body. The scientific and
engineering importance of these properties were emphasized ; moreover,
these papers, in which, in particular, the static problems of thick-walled
pressure pipes, gun barrels, tunnel linings, galleries, and mine shafts were
examined [73 e-j], resulted in some effective solutions for comparatively
simple systems in the case of non-homogeneous curvilinear anisotropy of
the polar and cylindrical types.
In these publications the postulate of simultaneous occurrence of the
critical plastic state a t all points of the body (for instance, over the entire
thickness of a pressure pipe) was also formulated; it was shown that this
effect can be achieved by a suitable choice of the field of non-homogeneous
properties in an isotropic or anisotropic body.*
2. Plastic Non-Homogeneity of Real Bodies
Prior to rigorous definitions and equations, it is necessary to grasp the
actual physical conditions connected with mechanically non-homogeneous
bodies. As already mentioned, we are mainly concerned with the plastic
non-homogeneity,
The non-homogeneity of mechanical properties may be caused, for
instance, by the following phenomena: the influence of flow of elementary
particles; the action of temperature gradients; a non-homogeneous hardening
of the material; different types of surface working; a non-homogeneity
of the composition, etc. In certain cases, it may be created by ourselves
(plastic working, surface and radiation working, etc.), in other case5 it
occurs independently (for instance, in soils).
We now proceed to consider some typical causes, devoting more attention
to phenomena which are new and less familiar.
* A method was also indicated for a practical realization of this theoretical assumption [73 f], [73 h]. [73 i], [73 j].
Analogous problems under more general assumptions have been treated recently
(see Chap. I11 of this survey). The following general remark m a y be of interest here.
The plastic non-homogeneity can result in effects which, if compared t o those involved
in plastic homogeneity, may exhibit essential quantitative and even qualitative differences. Many such results were obtained in the course of the investigations presented
in t h e papers which are quoted in this survey.
134
W . OLSZAK; J . RYCHLEWSKI A N D W . URBANOWSKI
A . Plastic non-homogeneity due to the action of neutron flow. The problems
of plastic non-homogeneity become particularly important in connection
with investigations on the influence of flow of elementary particles on the
mechanical properties of materials. Numerous experimental investigations
have proved that the action of neutron flow is especially strong. The sources
of neutron flows of various intensities and energies are provided above all
by nuclear reactors.
The action of radiation of considerable energy on solid bodies is a very
complicated phenomenon and depends on the one hand on the composition
and energy of the radiation and on the other hand on the composition of
the matter subjected to the radiation. A physical analysis of the mechanism
of radiation can, for instance, be found in [17], [30],[6]. We shall present
here a general description of phenomena that lead to changes of mechanical
properties of materials undergoing neutron radiation.
Neutrons, as particles without charge, bring about structural changes
only by direct interaction with nuclei. Passing through the crystal structure,
the neutron is subject to elastic impacts with atoms (ions) of the material,
and a transfer of energy to the latter occurs. The energy needed for displacing
an atom from its place in the cristal lattice depends on the kind of material
and its structure, the average being 20-25 eV. The neutrons in flows we
are interested in possess various energies, the average being of the order
of 1-2 MeV. The probability of displacing an atom from the atom site
of the lattice is therefore enormous. Moreover, the displaced atom has
such a large energy that it displaces other atoms and a chain reaction occurs.
The displaced atoms as a rule become interstitial. Thus, along the way of
a neutron there arises a large quantity of pairs of defects of crystal structure
of the type “vacancy-inclusion’’ (so-called Frenkel’s defects). I t was found
that a radiation of sufficient intensity can result in displacing as many as
l0-20% of atoms. Frenkel’s defects play an important role in the change
of mechanical properties of materials. From the viewpoint of the theory
of dislocations it can be stated that the above defects create additional
obstacles on the motion of dislocations. This effect has a considerable
influence on the plastic and strength properties of the material.
In addition to the above effect the creation of displacement zones is
of considerable importance, [127], [6]. In the final stage of the motion
of a neutron, when it has lost a notable part of its energy, the displaced
atoms have a smaller energy. Then the number of impacts among the atoms
increases and is so large that practically all neighbouring atoms are set in
motion. There arises a local thermal spike (covering several tens of A)
estimated a t about a few thousand degrees. I t is accompanied by a violent
evaporation and an instantaneous hardening of the structure (in about
10-10 sec), The resulting non-homogeneities of the crystal structure are
called displacement zones. They obviously influence the mechanical
properties, for instance by a dislocation mechanism.
PLASTICITY UNDER NON-HOMOGENEOUS
CONDITIONS
135
According to [el, in the case of light metals the creation of Frenkel’s
defects is decisive, while in the case of heavy metals the displacement zones
are of utmost importance.
Other phenomena have also an influence on the mechanical properties.
In some cases (for instance, for materials of large effective cross sections
for some processes, if the neutron flux density is large) we have to take into
account nuclear reactions caused by the neutrons, first of all nuclear changes.
Quantitative changes of mechanical properties of materials depend on the
quantity of radiation measured by the number of neutrons passing through
1 cm2 of the surface of the body. Denoting by n the number of neutrons
per unit volume of the flow, by v its average velocity, the intensity of the
neutron flow is nv (in neutron/cm2sec) and the quantity of radiation is
nvt (in neutron/cm2).
There already exists a considerable amount of experimental data
concerning the influence of neutron flow on the mechanical properties of
various materials. These continuously growing data can be found in various
journals devoted to physics and metallurgy, and in review articles [52], [ l l l ] .
We especially note the excellent review of V. S. Lensky [57 b], dealing
wholly with changes of mechanical properties of materials (274 references) .*
It should be observed that the experimental data are by no means complete
and that there are no extensive generalizations. The investigations have
been carried out under various conditions, and frequently cannot be
compared. Moreover, for obvious reasons most of the investigations performed by physicists are devoted to pure elements and often to crystals. All these
facts make it difficult to formulate general rules; however, certain most
interesting conclusions seem to be already safe.
a) Numerous investigations exhibited changes of elastic properties of
materialsundertheinfluence ofneutron bombardment, [17], [52], [50], [59]. In
some cases significant changes of the elastic modulus E were noted, for instance
a three-fold increase in the case of graphite, when nvt = 1020 neutron/cm2,
[126]. In the case, however, of structural materials, the changes of E , even
for large quantities of neutrons, are of the order of 5% and can be neglected
in static calculations. On the other hand, the damping factor undergoes
a significant change [57 b].
b) Of much greater importance from our standpoint is the increase of
the yield limit and the strength of metals, observed by many investigators.
The reasons for this phenomenon have been mentioned before. The results
of J. C. Wilson and R. G. Berggren are of particular value [125].
* Many valuable informations are to be found in [I1 I]. This book, however, appeared
only when this survey article was already in print and, therefore, could not be reviewed
here.
136
W. OLSZAK, J , RYCHLEWSKI AND W. URBANOWSKI
I I I I I
2
3
4
10 qaneutron/cm
5
6
7
FIG.1. Influence of neutron flux on the shear yield limit for monocrystals of copper, [46].
100000
J
C
c
0
0
I
60000
I
0
t
60000
'I
Y
4
F
&
2 40000
*
20000
I
2
Jnkgra Ted Neutron Flux ().IMeV)
-
FIG. 2. Change of the yield limit due to neutron bombardement for four stainless
steels, [125].
PLASTICITY U N D E R NON-HOMOGENEOUS C O N D I T I O N S
137
The increase of the yield limit of a copper crystal, [46], is shown in
Fig. 1. Already when nvt = lox8 neutron/cm2, a tenfold increase of the
yield limit was observed. Let us note the typical shape of the curve: first
a large increase, and subsequently a kind of saturation.
425000
100000
0
r
2
I
I
4
6
Megrated Neutron Flux (nvt; > I MeV)FIG.3. Influence of neutron flux on the plastic properties of silicon-carbon steel, [125].
Evidently, the results for structural metals are more interesting. The
results of investigations for a few types of stainless steels, [l25],are presented
in Fig. 2. We note a 2 to 2.5-fold increase when nvt = 8.1Ox9neutron/cm2, and
the same trend of the phenomenon.
The increase of the yield limit and strength for a silicon-carbon steel
A-212B is shown in Fig. 3.
Plastics used in reactor engineering exhibit a far greater variety of
phenomena under neutron bombardment. For instance, the yield limit
for some plastics increases quite strongly, while for others it decreases or
remains unaltered.
138
W.
OLSZAK, J . RYCHLEWSKI AND W . URBANOWSKI
c) In some cases a change of the nature of the whole stress-strain diagram
under the action of neutron flow was observed. A typical example is provided
by the curve for nickel, [21], [7], shown in Fig. 4. The material with a
hardening characteristic approaches, as a result of neutron radiation, a
material with a rigid-plastic characteristic (curve B ) .
0
20
30
40
True stmin ,% ---+
FIG.4. Change of the stress-strain graph for nickel due to radiation in a reactor (A ordinary sample, B - radiated sample, N loaon/cm2, > 1 MeV, T m SO'C), [7], [21].
The results of investigations [125] for an annealed stainless austenite
steel of type 347 (Fig. 5 ) are most interesting. Curve A corresponds to the
material before the action of neutrons, the velocities of loading being 0.01
and 0.05 per min. Curves B (velocity of loading 0.01 per min) and C (velocity
of loading 0.05 per min) correspond to the material which acquired the
quantity of neutrons nvt = 7.8
neutron/cm2. We observe that the
material, which originally exhibited strong hardening effects even without
yield platform, approached a perfectly plastic model after the action of
a neutron flow.
d) Radiation effects on materials are, as a rule, of permanent nature.
Some effects of this kind can only be eliminated by a thermal working.
We have stated before that the changes caused by neutron flows are
quantitatively dependent on the intensity and time of radiation. Since
the distribution of neutrons in the radiated body is non-homogeneous,
the neutron bombardment results in a body the properties of which are
functions of position; the body, therefore, is non-homogeneous, [57 b],
[44 a]. Thus there arises a necessity of determining this non-homogeneity.
Suppose that we know from experiment the function which describes
the variation of a property under consideration
(1.1)
K = K(uzvt)
under conditions of a homogeneous radiation (e.g. Figs. 3, 4). We assume
that the nature of this function is the same at all points in the case of non-
139
PLASTICITY UNDER NON-HOMOGENEOUS CONDITIONS
homogeneous distribution of neutrons*. I t is then evident that the mechanical
non-homogeneity is known if the spatial distribution of neutrons in the
body is known. We now devote some attention to this problem.
120000
c
80000
t
O
T
1ooooo
14
H
I-
6oom
@"
G
- 40000
- looooo~-+
20000
0
4
8
Total Elongation, per cent
,I2
E
FIG.5 . Change of the tension graph for stainless austenite steel due to neutron bombardement. [125].
I t is known that the basis of the neutron transport theory is the Boltzmann
equation. The relevant theory includes effective integration methods for
this equation and various approximate procedures. In our case the essential
point is to assume a sufficiently exact and sufficiently effective approximaThis assumption is not very accurate; we shall, however, make i t for the time being.
140
W . OLSZAK, J . RYCHLEWSKI AND W . URBANOWSKI
tion. This, obviously, depends on many factors, namely the distance of
the sources, the nature of the medium, of the flow, et.c. We have to apply
different approaches, for instance, to structural elements of a reactor located
near the sources and to problems of radiation working. It seems, however,
that many problems of interest can be treated by the constant cross-section
approximation. Moreover, let us observe that in our approach the angular
distribution is of no importance, the only relevant distribution of neutrons
being their spatial density. Consequently, it is convenient to deal with
the corresponding integral equation, instead of the integro-differential
equation. If the scattering is isotropic and the body is homogeneous and
non-re-entrant, the constant cross-section approximation in the case of
stationary processes yields the relation, [15],
where the following notations have been adopted: c, mean number of
secondaries per collision; I , total mean free path; r, position vector, p(r),
neutron flow density, s(r), source strength. Solving this integral equation,
we obtain the distribution of the mechanical non-homogeneity
(1.3)
.
K = K [p(r) t ] ,
For many, perhaps even for most of the cases, the diffusion approximation
is entirely satisfactory. Making the same assumptions as before, and,
moreover, assuming that the sources do not significantly depend on position,
we have instead of the integral equation the partial differential equation
(1.4)
where L is the diffusion length, depending on the neutron energy.
Solving the appropriate boundary value problem for this equation, we
obtain the density of the neutron flow and hence the distribution of the
mechanical non-homogeneity from Eq. (1.3).
In the case of non-stationary processes the solution of the appropriate
equations leads to the following result:
I
(1.5)
The fact that the energies of neutron flows we are interested in are of the
order of 1-2 MeV calls for the expressions (1.2) + (1.5) to be handled
PLASTICITY UNDER NON-HOMOGENEOUS CONDITIONS
141
with some care. The numerical data should be taken from investigations
concerning fast-neutron reactors.
The foregoing considerations will now be illustrated by an elementary
example, which, however, is of considerable importance. Suppose that
the semi-space x 3 0 is subject to a parallel neutron flow the intensity of
which is independent of time. We assume that no generation of neutrons
occurs in the body (absence of nuclear reactions). Making use of the theory
of diffusion we have
whence
Let us finally note that in view of the phenomenon of saturation, mentioned in connection with Figs. 1 and 2 , for large times of radiation the
non-homogeneity of the medium should decrease.
In some cases the non-homogeneity due to direct action of a neutron
flow is superimposed on the non-homogeneity due to the associated increase
of temperature.
Let us also recall that the states of stress and strain are subject to other
processes accompanying the neutron bombardment, first of all volume
changes, [loo].
The great sensitivity of plastic and strength properties to the influence
of neutron flows constitutes undoubtedly a really striking phenomenon for
specialists in the field of plasticity. The practical aspects are also of
considerable importance. Many modern structures are exposed to neutron
radiation; their strength analysis has to take into account the resulting
non-homogeneity. Moreover, there arise many interesting phenomena in
connection with the possibility of applying radiation-working to improve
the properties of the material. All these aspects will strongly influence the
development of the theory of plastically non-homogeneous materials.
B. T h e action of tempevatuve gvadients. The influence of temperature
gradients on the mechanical properties of materials is a well-known
phenomenon. Most of the available data concern the behaviour of metals
in variable temperature fields.
Without going into details of the physical analysis, we shall present
some facts concerning the influence of temperature on the yield limit upl
and the strength uUlt.
Fig. 6 represents the change of strength of some of the most frequently
used Soviet structural steels, investigated by M. I. Zuyev and others, [128].
The dependence of the yield limit and strength of the steel O H N l M 0-65
142
W. OLSZAK, J. RYCHLEWSKI AND W. URBANOWSKI
FIG.6. Influence of temperature on the strength for some of the most frequently used
structural steels, [128].
t°C
-
10
FIG.7. Influence of temperature on the yield limit and strength for steel O H N l M , [44 b].
PLASTICITY UNDER NON-HOMOGENEOUS CONDITIONS
143
on the temperature is shown in Fig. 7, 144 b]. Similar graphs for an aluminumalloy-clad sheet 2024- T3 is presented in Fig. 8, [as]. In the case of uranium,
a decrease of the yield limit was observed, from upr= 25.10-s psi a t room
temperature, to upz= 18.10-3 psi for 600" F, and a decrease of strength
from uWrt
= 90.10-3 psi at room temperature to uult= 32.10 psi for 600" F,
[25 b]. The same results for Thorium are the following: upzfrom 27 to 12,
uultfrom 38 to 22. The tensile strength in terms of temperature for materials
extensively applied in reactor structures is given in Fig. 9, [lo].
TotFIG.8. Influence of temperature on the strength, yield limit and extension for aluminumalloy-clad sheet, [28].
It follows from the graphs 6 and 7 that in the range of temperatures,
in which structural transformations do not yet occur, we observe significant
variations of strength properties. Starting from 300-400" C we note a
very fast decrease of the yield limit and of the stength. As a rule this is
accompanied by a significant trend of these characteristics (cf. Fig. 7 and
also Fig. 8) ; the properties of the material approach those of a perfectly
plastic body.
Incidentally, let us observe that alumothermic chromium makes an interesting exception from the above rules; its strength increases from 470 kg/cm2
for 20" C to 1000 kg/cm2 for 1100" C, [43].
The modulus of elasticity E of metals decreases with the increase of
temperature, whereas the Poisson ratio v increases.
144
W. OLSZAK, J . RYCHLEWSKI AND W . URBANOWSKI
Many empirical formulae have been proposed to describe the dependence
of mechanical properties on temperature. For some metals the formula
(1.8)
K , = K,e- a(Ta - TL)
may be used.
A decrease of temperature leads for numerous steels to an increase of
the yield limit andtensilestrength, [124]. Certain other phenomena also occur,
for instance the brittle fracture investigated by T. Pelczydski, [go]. According
1
Temperature OCFIG.9. Influence of temperature on the tension strength for reactor structural
materials, [lo].
to the data presented by G. V. Uzhik for the steel 1010 the yield limit
increased from 2300 kg/cm2 at the temperature 17" C to 7200 kg/cm2 at
- 1'37" C ; this was accompanied by an increase of the tensile strength
from 3700 kg/cm2 to 7300 kg/cm2.
The influence of temperature on the mechanical properties of other
structural materials may be different for various groups of materials. For
PLASTICITY UNDER NON-HOMOGENEOUS CONDITIONS
145
instance, ceramic materials, concretes etc., in the range of temperatures
resulting in no structural changes (chemical processes, sintering, etc.) do
not generally exhibit significant changes of the mechanical properties. In
contrast to this, plastics are very sensitive to temperature variations.
When the distribution of temperature in the body is inhomogeneous,
the body with mechanical properties depending on temperature becomes
elastically and plastically non-homogeneous. These non-homogeneities are
determined similarly to the case of neutron flows.
Solving the appropriate boundary value problem for the heat conduction
equation, [70],
A P T - CP-
aT
at
+ W = 0,
where A is the heat conduction coefficient, c the specific heat, p the density,
and W = W ( P ) the intensity of heat sources per unit time and volume,
the mechanical non-homogeneity is obtained in the form
(1.10)
K =K[T(P,I)].
Observe that in some cases it is also necessary to take into account the
thermal non-homogeneity, [116] (for instance the dependence of A on P as
the result of the action of variable T ) .
I t should be noted that the influence of temperature can result in
mechanical non-homogeneity of two kinds, namely: a) that existing while
the temperature is variable; b) that arising after the temperature field has
become homogeneous, due to temperature peaks which led to structural
changes.
Still a different topic is formed by problems in which high time temperature gradients occur (thermal shocks, etc.).
C.. Non-homogeneous strain-hardening of metals. Consider a body made
of a material exhibiting strain-hardening. If the body be subjected to forces
resulting in a plastic deformation, then in the case of a non-homogeneous
stress field this process leads to a body of variable yield limit; this is a
typical plastically non-homogeneous body. We do not dwell here on these
problems but regard them as fairly well-known in literature.
Non-homogeneities of this kind are very frequently encountered in
practice. They occur as the result of technological processes of plastic
working: cold rolling, drawing, forging, etc. The distribution of such nonhomogeneities is determined by considering the history of the loading pr6cess
a t all points; this constitutes a problem of plasticity with strain-hardening.
For torsion, this problem was examined by R. Hill, [34 b]. I. Berman
examined the plastic non-homogeneity of a pipe manufactured by means
of cold working from a sheet, [4]. W. Truszkowski found experimentally
146
W. OLSZAK, J. RYCHLEWSKI AND W. URBANOWSKI
a plastic non-homogeneity in the “neck” of a rod subjected to large plastic
deformations in tension, [123]. Test pieces subjected to a dynamic load may
also exhibit a non-uniform distribution of plastic properties, which was confirmed by D. B. Taylor, [121], and J . D. Campbell, [8].
D. Surface working. In the field of machine elements and structures, a
procedure is frequently applied which we call surface working; it consists
in applying a hardening effect to the surface of a manufactured element.
This working is carried out in many ways; for instance, the processes
considered in A, B, and C can be applied. The diffusional processes (nitriding,
cementation, cyaniding, metallization) lead to a change of the properties
of the body up to a depth of 2-3 mm and even more. Sometimes the working
by means of high frequency currents is being used. Another effective method
is shot peening.
These procedures of surface working produce, besides other effects, a
change of mechanical properties, and hence effects of non-homogeneity
caused by the elementary structure that varies with the depth. On the
surface, a high strength layer is created, which gradually passes into a
homogeneous weaker core (cf. M. T. Huber, [40a]).
The non-homogeneity due to surface working concerns as a rule the
plastic properties, the elastic properties remaining essentially unaltered.
E. Complex structwres. Frequently the mechanical non-homogeneity
is due to the non-homogeneous composition of the body. A typical example
are soils which usually are media of variable properties (see Chap. VII).
A required type of non-homogeneity may in practice also be obtained
in another way, namely by designing “composite” structures, well-known in
engineering practice and based on the co-action of two or more “components”
possessing different mechanical properties. Typical examples are reinforced
concrete structures with variable amounts of steel reinforcements, sandwich
and layered structures, etc.
An approximate analytic model of such a structure may be a body,
whose mechanical properties vary continuously in such a way that the
variable average properties of the actual material are imitated. The first
suggestions of a theoretical nature and the related practical solutions
employing the possibilities contained in this idea were given in the papers
mentioned above, [73 f]- [73 h], [73 j]; the objects of applications were
thick-walled high-pressure pipes of reinforced concrete, thick-walled linings
for mine shafts, tunnels and galleries of variable density of reinforcement.
In [73 i], the same author applied an analogous idea to gun barrels.
Obviously, there exist other causes of non-homogeneity, not dealt with
here. Thus the mechanical properties of many bodies depend, e.g., on the
degree of humidity, on gradual chemical changes, etc.
PLASTICITY UNDER NON-HOMOGENEOUS CONDITIONS
147
One of the possible non-homogeneity parameters of averaged mechanical
properties may also be a variable thickness of a plate or shell. In fact, the
limit analysis of load-carrying structures of variable thickness is a particular
case of the general theory of non-homogeneous bodies.
In addition, it may be observed that actual engineering structures are,
as a rule, characterized not only by the above considered non-homogeneity
(artificial or not); in fact, another typical property is their anisotropy
(material or structural). This concerns first of all the reinforced-concrete
structures mentioned before, in which the steel reinforcement is always
arranged in certain particular directions. However, the anisotropy may
also be due to processes of plastic deformations or plastic working, as well
as to structural arrangements (e.g. ribs), etc.* Thus in order to approach
the real conditions as closely as possible, the majority of papers (by Polish
authors in particular) take into account not only the non-homogeneity of
the material but also its anisotropy. This is also true for the first of the
above mentioned papers on these problems [73 f ] - [73 i].
The theory of plastically non-homogeneous media is faced with two
groups of tasks. First, it is necessary to develop sufficiently effective methods
of determination of the states of stress and strain, when the non-homogeneity
of the medium is given. Secondly, we must be able to determine certain
postulated distributions of non-homogeneity (e.g., optimum distribution),
if the loading is given; in other words, it is necessary to create sound
theoretical foundations of the methods in use (surface working, plastic or
thermal working, complex structures) and future methods (e.g. radiation
working) of improvement of the properties of structures.
As regards machine parts, N. 1. Popov and Ya. B. Fridman, [93], have
indicated a possibility of creating the required non-homogeneity (mainly
in metallic elements) by means of a suitable mechanical or thermal treatment.
Elastic-plastic bending of beams with surface effects due to such a treatment
were investigated by A. T. Poletzky, [92].
N. D. Sobolev and Ya. B. Fridman, [114], considered the interaction
of two scalar fields: the field of reduced stress and that of the local yield
limit (more strictly, the local strength). They introduced coefficients
* Experimental investigations (for instance those of A . Krupkowski [53], P. M.
Naghdi and J. C . Rowley [SS],L. Hu and J. Marin 1391) have proved t h a t even initially
isotropic materials acquire anisotropic properties in the course of plastic deformation.
Hence, if we extend the theory to include the class of anisotropic bodies, we should
specify the type of anisotropy considered - whether only initial anisotropy or also
anisotropy caused by plastic deformation is taken into account. The principal difficulty
in taking into account the change of anisotropy during plastic deformation consists
in the choice of the function Y determining the plastic potential. No ways of general
solution of this problem have so far been proposed, our object therefore will not be
quite as general, and the initial anisotropy will be the only one taken into account.
148
W . OLSZAK, J . RYCHLEWSKI A N D W. URBANOWSKI
characterizing the degree of utilization of the strength, both the ultimate one
and that related to the reduced stress distribution.
All these facts which prove that the non-homogeneity of mechanical
properties of a material can have a strictly determined (and regular) nature,
call for an investigation of its influence on the states of stress and strain
of the systems under consideration.
We note that as a consequence of the above considered processes there
may, in the bodies considered, arise stresses which essentially influence their
subsequent response to loadings. The problems related to this phenomenon
are outside the scope of this survey and will not be discussed here.
3. Definition and Classification of Non-Homogeneous Elastic-Plastic Bodies
The relations between the strain and stress tensors and their time
derivatives are expressed by the so-called constitutive equations, the
mathematical formulations of which contain also certain quantities (moduli)
that characterize the mechanical properties of the material independently
of the states of strain and stress.
The material characteristics describing the behaviour of the body in the
elastic range will be called its elastic moduli, the set of which will symbolically
be denoted by Me'.
It is known that the elastic moduli MeJcan be regarded as components
of a certain tensor (tensor of elastic moduli). Such an approach has many
merits, such as the possibility of expressing the constitutive equations in
a form invariant with respect to coordinate transformations.
Thus, in an elastically homogeneous body the invariants of the tensor
of the elastic moduli are independent of the position P ( x j ) . Such bodies
are described by the set of moduli Md'. Now, non-homogeneous elastic
bodies are characterized by a tensor field of moduli, the invariants of which
depend on position; the corresponding set of moduli will be denoted by
Mpel (see [73 k], [73 11, [71], [82 h] and [82 i].
If elastic-plastic bodies are investigated the problem requires introduction
of suitable additional concepts.
The phenomena occurring in the material when the yield limit is reached
will be described by means of certain quantities which, by analogy, will
be called the plastic moduli of the material and will symbolically be denoted
by Me'.
Let the generalized concept of plastically non-homogeneous bodies be
related to bodies the yield condition of which has the form
(1.11)
F(Ji;MpP') = 0,
where 1,denote the invariants (i = 1, 2, 3) of the stress (or strain) tensor
and MpPrthe set of the plastic moduli; in view of the assumed non-homogene-
PLASTICITY UNDER NON-HOMOGENEOUS CONDITIONS
149
ity the moduli M,P' depend on position P ( x i ) . In the simplest case of a
perfectly plastic body the set of plastic moduli is reduced to a single function
K pP'. *
Thus, the approach is similar to that for the case of elastic bodies. The
plastic moduli Mp' may be regarded as the components of a tensor (the
tensor of plastic moduli).+ If its invariants are independent of position
P ( x , ) , we shall refer to plastically homogeneous bodies (the set of moduli
being then denoted by MOP'),whereas plastically non-homogeneous bodies
require introduction of the concept of a tensor of plastic moduli whose
invariants are functions of the coordinates of P(xi) (the corresponding set
of moduli being then denoted by MpP')
The different types of bodies, which can be represented by theoretical
models on the basis of the above concepts, may be classified according to
various principles.
In view of engineering applications it seems that the classification
presented in the papers [73 k], [73 I], (the C-classification) may prove
particularly useful; accordingly, the more important types of bodies may
be classified as follows:
(I) elastically and plastically homogeneous bodies (M;', and Mop!
respectively) ; symbol C,, :
(11) elastically homogeneous but plastically non-homogeneous bodies
(M;' and MpP', respectively); symbol C o , p ;
(111) elastically non-homogeneous but plastically homogeneous bodies
(MPC'and MOPL,respectively); symbol C P , , ;
(IV) elastically and plastically non-homogeneous bodies (Mpe'and Mpp',
respectively) ; symbol Cp,p.
The order of the groups is, generally speaking, such that each body is
of a more general type than the preceding one.
In the most general group (IV) two cases are to be distinguished:
(IV,) the moduli Mpcland MpP' depend on position P and are independent
of each other; the corresponding symbol is C p ~ ;
(IV,) the moduli Mpc' and MpP' depend on position P but remain (as
implied by experimental data and certain theoretical consideration, see,
* It is interesting to note t h a t there exists a possibility of representing the workhardening type of a n elastic-plastic body (for any form of the e,o-graph) by means
of a model of a perfectly plastic non-homogeneous body; the same holds for the
explanation of some phenomena known from experiment, for instance "the Bauschinger
effect"; this problem has been treated by G. Colonnetti. [12], and H. F. Bohnenblust
and P. Duwez, [ 5 ] . See also the footnote 15b in [73 k].
t However, this tensor must satisfy certain additional conditions which are discussed
in the papers of W. Olszak and W. Urbanowski, [82 hJ, [82 j].
160
W. OLSZAK, J . RYCHLEWSKI AND W. URBANOWSKI
e.g. [72], [73 k], [73 I], [82 h], [82 i]) cozqbled in a certain manner; this can
symbolically be expressed by the relation
(1.12)
MpPt = f(MPet) ;
the corresponding symbol is C p 7 ( = C,).*
The theory of classes (1)-(111)can be deduced from the theory of class
(IV) by introducing successively stronger limitations on the generality of
the basic assumptions.
Another classification (the S-classification) that holds essentially for
work-hardening bodies is given in a paper of W. Olszak [73 k], [73 11, where
it is proposed on the basis of the scalar function characterizing the workhardening properties (Prager’s function, [94a], [94b]). In addition to
this, the paper [73 k] presents relationships between the classifications
of the non-homogeneity in accordance with the (C) and (S) schemes.
An essential point is the special role played by class I1 (symbol C,,p).
The importance of this class is readily observed, if we bear in mind the
physical foundations exposed in Sec. 2. Metals subjected to the action of
neutron flow are typical examples of bodies possessing marked nonhomogeneity of plastic properties, non-homogeneity of the elastic properties
being insignificant. The same holds for the non-homogeneity of metals as
a consequence of work-hardening, surface working, and temperature variations in certain ranges.
From the standpoint of mathematics, it is important to select properly
the functions describing the non-homogeneity ; this selection is also strongly
influenced by the physical considerations of Sec. 2. As a rule we shall assume
that these functions are bounded and sectionally continuoust.
* The actual nature of the coupling evidently depends on the nature of the causes
that lead to non-homogeneity of the properties. If a cause of measure A leads to a
change of the elastic and plastic properties
MpI = ,)@?I
Me’ = $ ( A )
and, say, the functions q and $-’ are single-valued, we have a one-to-one coupling
MPI = q[$-’(M6’)],
f = fp * $-1.
Since, on the one hand, there is a great number of various bodies and of causes of
non-homogeneity, and, on the other hand, these phenomena are very little investigated,
it is difficult to state more definite views on this topic.
f The condition of continuity (and also of differentiability of the functions M p etc.)
will be used in subsequent considerations; they are specified in greater detail in the
papers referred to.
PLASTICITY UNDER NON-HOMOGENEOUS CONDITIONS
151
11. PLANESTRAIN
1. Introductory Remarks
Problems of plane strain form a branch of the theory of plasticity with
comparatively old traditions and numerous results. The theory of plane
strain is particularly advanced for homogeneous perfectly plastic bodies ;
the character of the basic equations has been analyzed, important general
theorems have been proved, integration methods have been developed,
and a number of exact and approximate solutions are available. This state
is reflected in the treatment of the theory of plane strain in the monographs
by A. A. Ilyushin, R. Hill, V. V. Sokolovsky, W. Prager and P. G. Hodge,
and L. M. Katchanov.
For a body with variable plastic properties the situation is quite different.
The plastic non-homogeneity influences essentially both the mathematical
side of the problem and its physical sense. The character of the fundamental
equations changes and becomes complex, and some of the familiar theorems
and notions cease to hold. These difficulties and also the circumstance that
the interest in these problems is comparatively recent, result in the fact
that even the theory of an incompressible perfectly plastic body is only in
the first phase of development. Integration methods are still undeveloped,
and there are not many solutions available.
It would, therefore, be premature and even impossible to give such a
systematic and deep analysis of these problems as was given by H. Geiringer
[29] for the classical case.* Therefore, our survey will, of necessity, have
a somewhat fragmentary character. I t will discuss prospective possibilities
of development rather than the developments themselves. A number of
problems, among them essential ones, will be omitted or only touched upon.
Thus, for instance, we shall confine ourselves to the analysis of the statical
side of the problem, with questions related to deformations being only
mentioned.
Our attention will mainly be focused on the differences between plastically
non-homogeneous and homogeneous bodies.
2 . Basic Assumptions and Equations
In principle, a rigid-plastic, incompressible, anisotropic, plastically
non-homogeneous medium without strain-hardening will be considered.
Several remarks will also be devoted to plastically non-homogeneous bodies
having other properties.
By classical case we mean that of homogeneity.
152
W. OLSZAK, J . RYCHLEWSKI AND W. URBANOWSKI
Plane strain, which is the object of this Chapter, will be defined
starting from the two-dimensional case which is the more general one. In
the case of non-homogeneous (isotropic or anisotropic) bodies, the twodimensional problem ceases to be trivial, as was shown in Ref. [79].
A two-dimensional state may be defined in four ways. Starting from
the most general, they are as follows:
1) dEii depend on the coordinates x1,x2 only, the stress tensor Ta being
still a function of all the three variables, T,, = T,(x,,x2,x3);
2 ) in addition to what was said under 1, we have
3) in addition to what was said under 1 and 2, we have
4) in addition to the above conditions, the non-vanishing components
of the stress tensor are exclusively the normal and shear components parallel
to the basic plane, and u33.
The plane strain problem is thus seen to be a special case of the more
general two-dimensional strain problem. Therefore a clear distinction will
be made between these two definitions in the non-homogeneous theory.
The fundamental equations may be specialized for various types of
anisotropy (monoclinic, orthotropic, transversally orthotropic, cubically
anisotropic, and isotropic).
In the case of two-dimensional stress, analogous definitions can be given
for all the possible particular cases (and various anisotropy types).*
In Cartesian coordinates the familiar fundamental equations for the plane
strain are as follows.
The tensor field of stresses uii = aii(x,y) is determined by the relations
(2.1)
= a,
0;x
=0
o6a = B k J X X
+
.yy)
;
the equations of equilibrium (body forces being omitted) are
(2.2)
0xx.x
+
uxy.y
= 0,
uxy,x
+
oyy,y
= 0;
the yield condition is
(2.3)
( u x x - ayJ2
+42,
= 4K2(x,y).
The tensor field of strain rates dij = dii(x,y) satisfies the relations
(2.4)
d,, = dlY = dg*
=0
* An interesting case of quasi-plane stress in an elastic semi-space subject to the
action of temperature on the boundary has been given by W. Nowacki, [70].
PLASTICITY UNDER NON-HOMOGENEOUS CONDITIONS
153
and the incompressibility condition
+
ixx iyy
= 0.
(2.5)
Its relation to the vector field of displacement rate is
(2.6)
8xx = vx,x,
8yy = vy,y,
gxy
= S(vx.y
+
vy,x).
The stress tensors and strain-rate tensors are related by the condition
2i.q
(2.7)
-
8yy - 8xx
2 u ~ =~ctg 28,
u y y - uxx
where 8 is the angle between the tangent to the trajectory of maximum
stresses (the slip line) and the x-axis.
Assuming the five functions uxxruyy,uxy,v,,vy
to be unknown, we have
five equations for their determination. They are the Eqs. (2.2), (2.3) and
(2.8)
(2.9)
(vx,y
+
vy,x)(uyy
-uxx)
+
2(Jxy(vx,x
- %y)
= 0,
+
= 0.
vx,x
yy,y
This system of equations was obtained first by B. de Saint-Venant,
[lo4 a ] ; the only difference in our case is that we have a function of x,y
instead of a constant on the right-hand side of (2.3). This circumstance,
however, will be seen to have serious consequences, thus making the entire
problem quite different from the classical one.
The boundary conditions are formulated, in general, in terms of stresses
and velocities. If the boundary conditions in the stresses are such that
we can determine the field uii from Eqs. (2.2) and (2.3), the vector field v
will be found from (2.8), (2.9), and from the boundary conditions in the
displacement velocities. The form of the latter enables us to formulate the
following simple theorem :
Let us consider two bodies, geometrically identical but mechanically
different, i.e. characterized by plastic non-homogeneities of different types,
K,(x,y) and K,(x,y). Let these two bodies become entirely plastic under
the action of two different systems of forces. As a result the stress field a,,(')
will be realized in the body (1) and us?)in the body (2). If now (a) the boundary
conditions in the displacement velocities stresses are the same for both
bodies, and (b) the relation
(2.10)
holds (i.e. the slip lines coincide), then the same velocity field will be obtained
in both bodies, i.e. v(') = v ( ~ ) .
154
W . OLSZAK, J . RYCHLEWSKI A N D W . URBANOWSKI
In the case of homogeneous bodies, this theorem reduces to the familiar
possibility of superposition of a hydrostatic pressure.*
Let us add that the system ( 2 4 , (2.9) is linear and the same as in the
classical case. The differences between the non-homogeneous and the classical
case lie in the first group of our equations.
Thus, the statical side of the problem i.e. the system of equations (2.2)
and (2.3) will be the main object of our interest. In other words, the object
of our discussion will be statically admissible stress fields in plastically
non-homogeneous bodies.
There are various types of approach to a system of two partial differential
equations of the first order (2.2) and an algebraic condition of the second
order (2.3). As is known, the two classical methods for homogeneous bodies
are :
(a) (B. de Saint-Venant, [lo4 b]) : The equilibrium equations (2.2) have
to be satisfied identically by the Airy function, thus leading from (2.3) to
the following equation for its determination
(2.11)
(a,yy
-Q x J 2
+ 4Qpy = 4K2.
This is a difficult non-linear equation. An ingenious solution was
obtained in this way by L. A. Galin [27].
(b) (M. LCvy, 1581): The yield condition is satisfied identically by the
trigonometric substitution
(2.12)
uXx= u - K sin 20,
uYy= u
+ K sin 20,
uxy= K cos 20,
thus leading from the equilibrium equations to two partial differential
equations for the two functions u,0 which are now sought. This way has
led to remarkable achievements in the theory of plane strain of a perfectly
plastic, incompressible, homogeneous body.
There are also a number of other methods (see for instance H. Geiringer
~91).
For the time being, we shall follow the best explored way (b). Other
possibilities will be mentioned later. Substituting (2.12) into (2.2),we have
+ O,y sin 20) = K,xsin 20 - K,ycos 20,
u ,+
~ 2K( - O,x sin 20 + O,y cos 20) = - K,xcos 20 - K , ysin 20.
u,. - 2K(0,, cos 20
(2.13)
These equations have been given by A. Kuznetzov [55a]. Treated as
equations for a,@ with a given function K(x,y), these equations are quasi-
* In the theory of plane strain of elastically non-homogeneous bodies, the same
stress fields may be, in general, related to an infinite number of different strain fields
(cf. [80 c]).
PLASTICITY UNDER NON-HOMOGENEOUS CONDITIONS
155
linear, non-homogeneous. Clearly, the terms containing the derivatives of
K do not influence the form of the characteristics, which are still identical
with the slip lines.
The system of equations (2.13) differs in an essential way from its
particular case when K = const. I t constitutes an irreducible system [55 a].
In other words, no change of variables of the type
(2.14)
is capable of reducing this system to a linear one. The proof has been given
by A. I. Kuznetzov (see also Sec. 4). Let us write the characteristic system
corresponding to (2.13):
+
du - 2KdO - K,%dy K,,dx = 0,
(2.15)
on the slip line
of the first family
dy = tgOdx;
da
(2.16)
+ 2KdO + K,,dy - K,,dx = 0,
dy = - CtgOdx.
on the slip line
of the second family
I t can be shown that these equations have no integrable combinations
(cf. Sec. 3). If, however, it is assumed that the change of variables (2.14)
reduces the system (2.12) to a linear system and that x,y, say, are its
characteristic variables, then it follows that
(2.17)
fl(x,y,u,O)= const
fz(x,y,u,O)= const
constitute integrable combinations of (2.15), (2.16). This contradiction
proves the theorem. Therefore, the plane-strain problem of a plastically
non-homogeneous body is essentially non-linear (irreducible to a linear
one). The solution of such a problem encounters serious mathematical
difficulties.
In conclusion of these introductory remarks let us observe that Eqs.
(2.13) are also valid, of course, for a strain-hardening body, in particular
for a primarily non-homogeneous strain-hardening body. In this case K is
not a function given beforehand, but depends on the deformation process.
3. The Possibilities of Solving
As already noted, the system of equations describing the plane strain
equilibrium of a plastically non-homogeneous body offers serious difficulties
in the forms (2.2), (2.3) as well as (2.13) or (2.11). In particular, the technique
156
W. OLSZAK, J . RYCHLEWSKI AND W. URBANOWSKI
developed for the homogeneous case cannot be applied to the case of the
Eqs. (2.13). Thus, for instance, the Kotter-Hencky relations are no longer
valid, no solution in the form of simple waves can be obtained etc. The
question arises, what are the prospects and actual possibilities of solving
problems of non-homogeneous bodies.
Analytical methods, Analytical integration methods of irreducible nonlinear partial differential equations have not yet been developed. The
difficulties encountered in the considerably simpler homogeneous case do
not seem to predict that exact integration methods for actual boundaryvalue problems will be found soon, simple particular cases being of course
excluded. It may be expected that even if appropriate methods are developed
in the future the solutions will have a rather complex form (cf. the integration
by the Riemann method in the homogeneous case),
Afifiroximate meth.ods. The most important of these methods is of course
the numerical (or graphical) method of characteristics, which is very versatile
and has an “algorithmic” character. However, it gives no possibility of
analysing the influence of each particular parameter on the solution. Therefore
other approximate methods have been devised, especially for the case of
weak non-homogeneity (B. A. Druyanov, A. I. Kuznetzov). Variational
methods have not yet been applied, but their application is not expected
to be different from the homogeneous case.
Inverse and semi-inverse methods. I t seems that one of the typical features
of non-homogeneous problems is at present the preponderance of inverse
and semi-inverse methods, which offer remarkable possibilities of obtaining
closed-form solutions in a relatively simple manner. These methods are
important, first, because they enable us to examine the qualitative influence
of plastic non-homogeneity ; second, because they may constitute a measure
of efficacy of the approximate methods; and, third, because in view of the
hyperbolic character of the system of equations they can represent parts
of the solution in more complex cases.
In Secs. 6 and 7 the results hitherto obtained by means of approximate
and inverse methods will be discussed in greater detail.
4. Equations of Equilibrium of a Non-homogeneous Body in Curvilinear
Coordinates
In what follows it will be more convenient to use the equilibrium equations
of a plastically non-homogeneous body referred to an arbitrary system of
orthogonal curvilinear coordinates, in particular to the system of slip lines
and trajectories of principal stresses.
PLASTICITY UNDER NON-HOMOGENEOUS CONDITIONS
157
Preparatory to this section and the next, let us briefly recall the most
important facts of the geometry of nets. Consider a curvilinear orthogonal
net, related to a fixed Cartesian reference frame by
(2.18)
y = r(a,P).
x = x(a,P)
(The expression d i n e will be applied to a line P = const and vice-versa;
the labels correspond to the lines considered.)
The Lam6 parameters
(2.19)
dsa
H --
' - da
2
+ Y , ~ ) " ~ , Hp =-las,
= (x,p + y;p)l/'
dP
=( x , ~
2
2
are functions of a,P and satisfy the relations
+
(2.20)
= 0.
&),p
They determine the net within the indeterminacy of displacements and reflections. These quantities have no direct geometrical sense, because they depend
on the way in which the parameters of the net are chosen. For
(2.21)
a = a(a*),
P = P(P*)
we have
(2.22)
Ha* = a,a** Ha,
H p = P,p * Hp.
An isothermal net is a net with a certain parametrization (called isothermal
parametrization), for which we have
(2.23)
Ha = Hp = H .
The necessary and sufficient condition for the net described by Ha,Hp
to be isothermal is
(2.24)
Ha = A (a) * B(P).
-
HP
The geometric quantities are determined by Ha,Hp in the following
way: for the angle of inclination rp of the vector tangent of an a-line with
respect to the x-axis we have
(2.25)
158
W. OLSZAX, J . RYCHLEWSKI AND W. URBANOWSKI
the curvatures of the net are
These quantities are positive for the configuration illustrated by Fig. 10.
It can easily be seen that
X
FIG. 10. Positive directions of the lines a$.
Four coordinate systems will be introduced
C ; x,y
Cartesian coordinates ;
L ;ar,P
arbitrary (that is, not related to any physical phenomena) fixed
orthogonal curvilinear coordinates;
P ; y,S
orthogonal curvilinear system of trajectories of principal stresses ;
S; 5 , ~
orthogonal curvilinear system of slip lines.
The equations describing the equilibrium of a plastically non-homogeneous
body in plane strain, referred to the L-system, have the form [69]
PLASTICITY UNDER NON-HOMOGENEOUS CONDITIONS
159
Applying the substitutions of M. Lkvy
a
, = a - K sin 24,
app = a
+ K sin 24,
Gap = K cos 2$,
where 4 is the angle between the tangents to the a- and (-lines, we have
Ha K,p cos 2 4 = 0,
- K,a sin 24 + HP
(2.31)
HP K , cos 2 4 + K,p sin 2 4 = 0.
+Ha
If L coincides with C, we have, of course, the Eqs. (2.13), 4 8.
If t,h ~ n / 4the
, system L coincides with P, and Eqs. (2.31), referred to
the trajectories of principal stresses, have the form
If t,b E 0, the system L coincides with S, and the equations (2.31),referred
to the slip lines, are
(2.33) a,t
Ht
+ 2 KHt,,
A+
K,rl = 0,
Hrl
Hrl
-
a,,,
H
= 0.
+ 2K;Hrlt
Ht + -K t
Since, obviously, the net of slip lines constitutes a natural system of
coordinates for plane strain, some remarks will be devoted to Eq. (2.33).
In the homogeneous case when K = const, we obtain immediately,
making use of (2.25) with pl = 8, the Kotter-Hencky integrals
and the relation
(2.35)
8,crl = 0
together with (2.27) determines geometric properties of the slip lines known
as the first and the second Hencky theorem.
160
W. OLSZAK, J. RYCHLEWSKI A N D W. URBANOWSKI
I t is seen from (2.33) that none of the above facts take place in the case
of a plastically non-homogeneous body. Moreover, in the general case of
K # const, no relations exist that constitute a generalization of (2.34).
Certain integral relations along the slip lines could only be obtained by
postulating additional relations. If there exists a function F such that
then
(2.37)
0
+F f(q),
=
0
- F = g(5).
This can happen only in very special cases of the form of the slip lines
and of K(c,q). Treating the relations (2.36) as partial differential equations
for K , we can write the conditions, which determine admissible classes of
slip lines. I t is found, for instance, that by requiring K to be determined
up to a constant, we obtain among other relations [In (Hc/Hq)],cq
= 0,
which, in view of (2.24),is equivalent with the isothermal property of the net.
5 . The Geometry of Slip Lines and Trajectories of Principal Stresses*
I t is known that not every orthogonal curvilinear net can play the role
of slip lines in a homogeneous body. Only Hencky-Prandtl nets satisfying
the familiar Hencky theorems expressed by (2.35) and (2.27) are admissible.
In a non-homogeneous body this problem becomes more complex.
Let us write the compatibility condition of the system (2.31) with respect
to u. On differentiating and subtracting we obtain
(2.38)
+
HP
Ha
cos 214 *K,uu sin 294 S K , ~-P
cos 214 * K J ~
2HU
2HP
~
~
+
+ [ A , cos 214 + A , sin 2 # ] K , , + [B, cos 214 + B, sin 21,hlK,~+
+ [(C, + C,) cos 2# + (C, + C,) sin 2I41K = 0
where
* In this Section, mainly the results of Ref. [80 a] are discussed.
161
PLASTICITY UNDER NON-HOMOGENEOUS CONDITIONS
This equation relates, in any fixed system L , the plastic non-homogeneity
function K(cr,P) to a geometric quantity, #(cr,P), connected with the net
of slip lines or, in view of # =
n / 4 , with a geometric quantity connected
with the net of trajectories of principal stresses. Thus it characterizes the
geometric properties of the slip lines and trajectories of principal stresses.
This equation must hold if the stress field is to satisfy, in addition to the
equilibrium equations (2.28), the yield condition (2.29). I t will be called
the geometric condition of plastic state.
If the system L coincides with C, then $ f 8, and the geometric condition
of plastic state has the form
x+
w,,, - K , ~ ,cos) 28 + 2 ~ , , sin
, 281 +
(2.39)
-
-
4 K , , [COS
28 8,y - sin 28 O,,]
2 K { 2 cos 28
+ 4 K , , [COS28 - 8,, + sin 28 - 8,y]+
[e,,, + (e,,)' - (8Jz] + sin 28 - [- 48,, 8,y + O,,, - O,,,]) = 0.
The form of (2.38) is most interesting when expressed directly in the net
of trajectories of principal stresses or the net of slip lines. If $ ~ n n / 4the
,
system L coincides with P, and we have
If $ = 0, the system L coincides with S, and we have
(2.41)
+ 2 K [(G)
HC .c
(%),?I
=O.
Eqs. (2.40), (2.41) relate the plastic non-homogeneity function K with the
geometry of the net of trajectories of principal stresses and of slip lines
expressed by means of Lam6 coefficients and subject to the restrictions
(2.20).
162
W. OLSZAK, J. RYCHLEWSKI A N D W. URBANOWSKI
If the form of the non-homogeneity function K = K(a,P) is given, the
equation (2.38) determines the class of nets admissible as slip lines and
the class of nets admissible as trajectories of principal stresses. If we have
K = K ( y , d ) , the admissible nets of trajectories of principal stresses are
determined by (2.40); if we have K = K(C,q),the admissible nets of slip
lines are determined by (2.41).
Let us consider for instance K = const. I t can easily be seen that only
the last terms remain in (2.38), (2.39), (2.40), (2.41). Therefore the nets
admissible as trajectories of principal stresses in a homogeneous body are
subject to the restriction
(2.42)
[In(H,Ha)I,,d = 0,
whereas those admissible as slip lines must satisfy
(2.43)
Upon confronting the latter result with (2.25), it is seen that this is
equivalent to the Hencky theorems, as could be expected. Therefore, the
latter constitute a particular case of the geometric condition of plasticity
for K = const.
If K # const, the direct geometric sense of the relevant restrictions is
difficult to grasp (i.e. to express in terms of net geometry) even in the simplest
cases.
Another approach to the geometric condition of plasticity is also possible.
Taking for instance (2.40),we can, for a given class of nets or a given net
y,6, seek the class of non-homogeneity for which they are admissible. Thus,
for instance, for nets or trajectories of principal stresses subject to (2.42)
we have
(2.44)
and, in addition to K = const, we have an enormous variety of other nonhomogeneity types.
Every orthogonal net can be considered to constitute a net of slip lines
(or trajectories of principal stresses) in a body of an appropriate nonhomogeneity type* or, even more, for a large class of such bodies. In other
words, the set of nets admissible for slip lines (or trajectories of principal
stresses) has no common geometric features to distinguish them from other
orthogonal nets.
* In this section, not all the conditions for which the relations in question take
place are considered (boundary-value problems are not discussed).
163
PLASTICITY UNDER NON-HOMOGENEOUS CONDITIONS
We proceed now to discuss certain particular cases of Eqs. (2.40), (2.41).
A. Trajectories of principal stresses. Eq. (2.40) can be written in the
convenient form
(2.45)
+ (In Hd),yK,df [In (HyHd)],&
K,yd -k (In Hy),dK,y
0.
If K = K(y,S) is given, we have for Hy,Hd the non-linear system of
equations, (2.45), (2.20). The inverse problem is somewhat simpler, and
some elementary examples will be given.
A Cartesian system (H,= Hd = 1) may constitute a net of trajectories
of principal stresses only when
(2.46)
K = K,(4
+ KAY).
This is a hint for the solution of boundary-value problems with rectilinear
edges.
The polar net (Hr = 1,H, = Y ) may be admitted as a net of trajectories
of principal stresses only when
(2.47)
Let us consider a class of nets with the property that one of the families,
the a-lines for instance, is composed of straight lines. In view of x, = 0
from (2.26), we have Ha = A ( a ) . The parameters can be chosen so that
Ha = 1. Then, HB = BI(P)a B,(P) from (2.20). For the Lam6 parameters
we obtain
+
(2.48)
Hy = 1,
Hd = y
+ D(a),
and by virtue of (2.26) we have
(2.49)
Ra = Ha = f D(S),
D(S) = R d l y = 0.
The geometric condition of plastic state takes the form
(2.50)
Hence
(2.51)
We observe that in the class of nets for K = const only the polar net
may constitute a set of trajectories of principal stresses.
164
W. OLSZAK, J. RYCHLEWSKI A N D W. URBANOWSKI
The geometric condition of plastic state takes a particularly simple form
for isothermal nets in isothermal parametrization. Substituting
K =T*h
(2.52)
h = H-I,
where
we have
h T,yd - h,y5 T = 0,
(2.53)
where, from (2.20);
P h = 0.
(2.54)
We see at once that the following expression* is a particular integral:
T=C-h
(2.55)
or
K=H-2.
Another immediate observation is that
involves
(2.57)
and vice-versa.
The geometrical sense of (2.56) is evident.
From (2.26) we have
x y = h2’(d),
constant on the y-lines;
= hl’(y),
constant on the &lines;
xd
therefore the net is composed of two families of circles. Conversely, if the
net is of that type, (2.56) follows from (2.26). Thus, if the trajectories of
principal stresses are two families of circles, then
(2.58)
K =H - l W
+ g(41
and vice-versa. Thus, for instance, for a bipolar net we have
(2.59)
K = (cos y
+ cash 4 [f(r)+ g(41.
For logarithmic spirals
(2.60)
, p - l n - =r y y ,
a
p+ln-=
rd
a
* This integral has been obtained by S. Zahorski directly from (2.40).
PLASTICITY UNDER NON-HOMOGENEOUS CONDITIONS
165
we have
(2.61)
and the equation determining the admissible non-homogeneity types reduces
to the telegraphist's equation
(2.62)
TP!3
+tT
=0
(a particular integral is, among others, the case of K = const, T = h-1).
B. Slip lines. No detailed analysis can be given here (cf. [80 a]), since this
is a considerably more difficult problem than the foregoing case. The
investigation of the existing possibilities is more essential, because the slip
lines are the characteristics for the equations (2.13).
Let us observe incidentally that the possibility of a generalization of
the Kotter-Hencky equations considered in Sec. 3 exists only if
zK,tt
+ K , c Hc
[ p .t) + 2%]
Hc + ZK(%)
Ht , t =0,
(2.63)
Ht
- K,qq
Hll
+ K,q[(") + 2
21+ @)
2K
Hll
= 0,
,rl
which is a serious restriction compared to the general case (2.41).
An instrument for the analysis of the geometry of slip lines is available
in the form of the geometric condition of plastic state (2.38).
More detailed information on the questions* discussed in the present
section can be found in [SO a].
6. Biharmonic States of EquilibriGmt
A tensor field aii satisfying, in addition to the equilibrium conditions
and the yield condition, the equation
(2.64)
P2(Uxx
+
Uyy) =
pv2n= 0,
will be called a bihannonic state of plastic equilibrium.
* Such considerations arise in connection with statical problems of originally
homogeneous bodies exhibiting strain-hardening. Natural reference frames for the nonhomogeneity appearing in the course of the process of deformation of such bodies are
the nets 5.17 and 7.6,
f This section contains chiefly the results of Ref. [80 b].
166
W. OLSZAK, J . RYCHLEWSKI AND W. URBANOWSKI
Such states are, of course, not typical for problems of plasticity and
must be of exceptional character (particularly so in the homogeneous case).
However, the analysis of such states is useful from many points of view.
Let us recall, for instance, that one of the most elegant solutions of the
classical theory - that obtained by L. A. Galin - is connected with such
a state.
Biharmonic states of plastic equilibrium have interesting geometric
features of the trajectories of principal stresses and of slip lines. To see this,
let us write the biharmonic condition in the system L :
(2.66)
Substituting the derivatives of the equilibrium equations (2.31), we obtain
the biharmonic condition in the form of a relation resembling (2.38), except
that sin 2# and cos2# as well as cos 2# and - sin 2# are interchanged.
By comparing these two relations it is seen that the geometric condition
of biharmonicity of plastic equilibrium is expressed by the two relations
If the system L coincides with C, # = 0 , we find
+
+
+ qYy- e , % ~ 0,
K,,~ 2 ~ ,e,%
,
2 ~ ,qY
,
K ( - 4 0 , qY
~
(2.67)
+
K , , ~- K , ~ , 4~,,e,,
=
+ 4 ~ , , e , , + 4~[e,,, + (e,,)~- (e,,)zi
= 0.
For the trajectories of principal stresses and for the slip lines (and also for
any other net maintaining a constant angle with the principal direction)
we obtain
(2.68)
Thus, for biharmonic states of equilibrium the form of the equations
relating K(a,P) with H,(a,P),Hp(a,P)is the same in all nets forming a constant
angle with the net of trajectories of principal stresses.
PLASTICITY UNDER NON-HOMOGENEOUS CONDITIONS
167
In particular we obtain for K = const the following necessary geometric
condition of biharmonicity: every net making a constant angle with the
net of trajectories of principal stresses (including the net of trajectories
of principal stresses and that of slip lines) must satisfy the following system
of equations resulting from (2.66) and (2.20):
(2.69) a)
[In (HaH/)],a)= 0,
b)
(%),a
= 0,
(%),u
c)
= 0.
These are therefore Hencky-Prandtl nets of a particular type, the additional
restriction being a). I t can be shown, [80 b], that these nets constitute
logarithmic spirals (with the polar and the Cartesian net as particular cases).
For isothermal nets in isothermal parametrization the biharmonicity
conditions have the form
hT,aB - h,aBT = 0,
(2.70)
h.( T,aa - T,pp) - (h.,aa - h,pp)T = 0.
It is easy to find a K satisfying these relations for each particular net.
Thus, for instance, if a bipolar system is to be a net of trajectories of principal
stresses, 12 = cosh u
cos v , and if the state is to be biharmonic, this is
possible only for the following non-homogeneity types :
+
K = (cosh u + cos v)(C, sin u + C2cos u + C, sinh v + C, cosh v ) ,
(2.71)
K = (cosh u + cos v)(C, sinh u + C, cosh u + C, sin v + C, cos v ) .
7 . Analytical Solutions in Particular Cases
The integration of the equilibrium equations may in the case of special
distributions of plastic non-homogeneity be performed in a closed form.
Several relatively simple cases will be given here.
A, Two simple solutions in Cartesian coordinates have been obtained by
A. I. Kuznetzov
a) Let the semi-plane, bounded by the line y = 0, be acted upon by the
forces, [55 a, b],
uyy= - p = const,
(2.72)
uxy= 1 = const.
Assuming that K is a function of y only, we obtain an elementary selution
of the Cauchy problem for the system (2.2) from (2.3) in the form
(2.73)
uyy =
-p,
uxy = t ,
uxx =-
p f V P ( y )- t 2 .
It should be assumed in addition that It1 < K ( y ) in the entire semi-plane
including the edge.
168
W. OLSZAK, J. RYCHLEWSKI AND W. URBANOWSKI
The slip lines can easily be found:
For t = 0, these are straight lines y = & x -t C; this possibility is foreseen
in Eq. (2.41). If K increases with y , the lines (2.74) approach their rectilinear
asymptotes.
Observe that the occurrence of the plastic state in the entire region is an
assumption (independent of the relation between the quantities P,t,K), which
requires the realization of the stress pattern following from (2.73) in sections
x = const (cf. for instance the case of t = p = 0).
FIG. 11. Compression of a non-homogeneous plastic layer between two rough rigid
plates.
b) Let us consider, [55 c], the known problem of compression of a
perfectly plastic layer, Fig. 11 (I >> h) between two rough rigid plates. It
is assumed again that K = K ( y ) . The boundary conditions are
y = - It,
uxy= - n K ( - h),
0
< Iz < 1.
Assuming that uxyis independent of x , we obtain for the system (2.2),
(2.3) the solution
uxy= ay 4-b,
(2.76)
oXx= - ax - c
uyy= - ax
- c,
+ 2 V K 2 ( y )- (ay + b ) 2 .
PLASTICITY UNDER NON-HOMOGENEOUS CONDITIONS
169
From (2.75) we have
and from the equilibrium condition of an element of width x ,
li'
c = - V K 2 ( y )- ( a y + b ) 2 d y .
h
(2.78)
-h
The stress distribution is a generalization of the known solution found by
Prandtl.
Solving the system (2.3), (2.9) for the velocities with the boundary
condition
v y = 'f U
(2.79)
for
y = ih,
we obtain
vy =
Y
u--,
h
(2.80)
V
The constant C will be obtained from the incompressibility condition
(231)
- r v x d y =2U(l- x).
-h
The velocity distribution (2.80) generalizes the known solution by Nkdai.
Ref. [55c] contains a discussion of the possibility of preservation of
rigid zones adjacent to the plates.
Making use of the result obtained, the author, following A. A. Ilyushin,
[42 b], studied an approximate approach to flow problems of a plastically
non-homogeneous material between curved surfaces.
B. In polar coordinates some results have been obtained in cases when
K is a function of the radius only or of the angle o$y.
a) For
(2.82)
K =K ( r )
170
W. OLSZAK, J. RYCHLEWSKI AND W. URBANOWSKI
the stress distribution around a hole has been discussed in detail in [82 a, b]
(cf. Chap. 111) and generalized by A. I. Kuznetzov to the case when ur, # 0.
b) Assuming that
K = K(p),
the plastic equilibrium of a wedge (Fig.12) has been studied by A. Sawczuk
and A. Stepieri, [lo5 d], [llO]. This is a one-dimensional problem. In the
components of the stress deviator
(2.83)
(2.84) Str = ~r - S,
spq
- S,
UP,
srq = ~ r q ,
+
s = &(urt a),
FIG.12. Wedge loaded by side pressure.
the equilibrium equations (2.28) in polar coordinates can be written in the
form (since no quantity depends on Y )
(2.85)
%,,
+ 2s,,
=0,
s,p,p
+ 2st, +
S?, = 0.
Bearing in mind that s,,: :,s,
s,, = 1: - 1: 0, one obtains the yield condition
(2.86)
s,
2
+
2
Srp =
qp).
The boundary conditions are seen from Fig. 12.
Substituting sp, of (2.85) into (2.86), we obtain the non-linear ordinary
partial differential equation
(2.87)
Srp.,, =
f [K2(p)- s&]'/2.
The types of solution and the possibility of the appearance of discontinuities
were discussed.
The determination of strain rates was reduced to the integration UP the
equation
(2.88)
PLASTICITY UNDER NON-HOMOGENEOUS CONDITIONS
171
For linear non-homogeneity
m = 2 ( K , - K0)a-l
(2.89)
a
2
K=m(a-p)+Ko,
-<p<a
the solution of (2.87) was found in the form of a power series:
srp=-Ko(2p+
(2.90)
[
-m
p2--
KO
sip = - KO 2(a - p)
4
3
3 p +
m
+(a KO
...)
4
p)2 - - ( a - p ) 3 +
* * -
(successive terms can be found from a recurrence formula). The remaining
components are also expressed in the form of series.
From the relevant equations it follows that the line p = a/2 is a
discontinuity line.
From the condition
+ + 4; = 2 ~ p q ,
(2.91)
urr
the ultimate load is found:
(2.92)
p = 2 K 0 [ 2 ( i ) ' + "3. 2 KO
( E 72
-z(Ey
3 2
+...I.
Certain numerical results are shown in Fig. 13. The value of K = J b 2
will be discussed in Sec. 9.*
C. A. I. Kuznetzov has generalized L. A. Galin's solution, [55 a, b],
of the case of the infinite plate with a circular hole for the boundary conditions
* In the paper [20d], B. A. Druyanov solved the case when
K = K , eZnlpll
where K , and a stand for constants. I t may be noted that in both cases the nonhomogeneity is symmetrical with respect t o the axis of the wedge, which facilitates the
solution.
172
W. OLSZAK, J . RYCHLEWSKI AND W . URBANOWSKI
Requiring that the state of equilibrium of the plastic part be biharmonic
(the semi-inverse approach), he assumes that
(2.94)
(:T
K=K(cQ)+dK -
The admissibility of this non-homogeneity type may be verified in the
light of the general considerations of Sec. 5 (cf. [80a]).
-n
FIG.13. Limit loading for wedge whose yield limit is a linear symmetric function of
angle, [lo5 d].
The results obtained by means of L. A. Galin’s method are as follows.
The plastic zone is bounded by an ellipse the semi-axes of which are
(2.95)
c1,2 = 4 1 i P )
173
PLASTICITY UNDER NON-HOMOGENEOUS CONDITIONS
where
(2.96) c = a e x p
__p+Q
(4K(m)
-1 +-- P
2
2K(m)
----,
p=--.Q - - P
2 KA( m
K ))
2K(m)
The stresses in the elastic region are determined by the relations
where
(2.98)
z=x+iy
,
FIG.14. Boundary of the plastic zone and the graph tmax
I ,= 0 and tmax I = 0 for nonhomogeneous disc with circular hole, [55 a].
The stresses in the plastic region are
r
u,,=-f~+2K(m)ln-+fK
a
(2.99)
upp= - p
a,, = 0.
+ 2K(m) l n j l +In:)
+
+
g),
174
W . OLSZAK, J . RYCHLEWSKI A N D W . URBANOWSKI
Fig. 14 shows the boundaries of the regions and diagrams of tmaX/K(m)
on the x-axis and the y-axis in the following cases
(2.100)
AK
Roo)=-0.5,
AK
jqoo) = 1,
AK = 0,
for the conditions
(2.101)
p =0,
Q = 3K(m),
P = 2.6K(m).
D. Problems of a non-homogeneous elastic-plastic strain-hardening body
require a different approach and are not discussed here. Let us only mention
the paper by W. Olszak, J. Murzewski and J. Golecki, [77 a], devoted to
the problem of a semi-plane loaded by a concentrated force. The Epdiagram
is assumed in the form of two intersecting straight lines (linear strainhardening). The three parameters determining this relationship are assumed
to depend on the depth coordinate. Closed-form and approximate solutidns
have been obtained for incompressible and compressible material, respectively. However, a characteristic discontinuity at the boundary between
the elastic and plastic domains appears; this problem is discussed separately.
8 . Approximate Solzltions
For problems with a "weak" non-homogeneity approximate closed-form
solutions may be sought on the basis of the known solution of the classical
problem. This approach is founded on the intuitive expectation that a
weak non-homogeneity should not considerably disturb the stress and
strain-rate field. It should, however, be noted that this expectation has
no mathematical foundation and recalled that for problems of perfect
plasticity instability of solutions may occur, [42 a].
A. The problem of a punch pressed into a body of weak non-homogeneity
has been studied by B. A. Druyanov by means of the perturbation method.
He solved the problem of a smooth punch, [20 a], and a rough punch, [20 b],
and a broad and narrow semi-strip [20 c]. Assuming that
(2.102)
K = Ko(l - &)Y/*,
H
h =-,
a
where 2H is the depth of the strip, 2a the breadth of the punch, y the depth
coordinate ( y < 0), Eqs. (2.13) are obtained in the form*
w,=+ cos 29, * v,.
(2.103)
+ sin 29, = ln(12h.- cos 29,,
E)
+ sin 29, pl,. - cos 29, = In(12h- E ) (sin 29, - 2w).
*
* The author takes o = u / 2 K , q = f3 + n/2.
175
PLASTICITY UNDER NON-HOMOGENEOUS CONDITIONS
Assuming that E = (KO- K h ) / K Uis sufficiently small, we seek the
solution in the form
m
m
(2.104)
w =
2
p=
W$,
0
2 pr&i,
#
0
which leads to the system of equations
(2.105)
+
+
..
+ sin 2p0 p,,, - cos 29, pt,y= F Z , ( ~ , , W. .,po,o~o),
,.
w , , ~ cos 2p0 p,,% sin 231, * pz,y= Fl,(pl,,w,. ,po~mo),
9
*
*
where Flr,F2$are linear functions of p),,w,.
f
X
FIG.15. Approximate characteristic net for E > 0 and the velocity distribution on the
free boundary.
_ _ - _ classical problem, ___ non-homogeneous problem
h = 10)
( E = 0.5.
The systems (2.105) are hyperbolic and have common characteristics,
known from the solution of the classical problem (i = 0). All systems are
linear for i > 0. Starting out from the continuity condition for to and p
on the characteristics, the author formulates the boundary-value problems
for these equations. The equation of the characteristics is also sought in the
form
m
(2.106)
y =
2
Y,(X)&i.
0
The problem of the velocities is stated in a similar way, and an analogous
linearization is obtained.
In Refs. [20 a] and [20 b] expressions are obtained for all the required
quantities in the second approximation (order E ~ ) . Figs. 15, 16 give the
176
W. OLSZAK, J. RYCHLEWSKI AND W. URBANOWSKI
results of Ref. [20]. For the normal pressure under the punch (with E = 0.5,
h = lo), the maximum correction of the first approximation in relation to
the zero order approximation is S.9%, that of the second 4.9%.
In a paper devoted to the problem of a punch pressed into a narrow
strip on a rigid foundation, it is shown that the equations of all approximations
may be reduced to the telegraphist’s equation and therefore integrated in
principle by quadratures. In view of the necessity of heavy computations,
the author stopped at the first approximation, expanding the Bessel functions
HA
-%
I
2
FIG. 16. Stress distribution under the punch
1 - ayo/K0,
2 - ~,t/Ko,
3 - 10t,y/Ko
(E = 0.5,
h = 10)
under the integrals in power series. The computations for given values of
the parameters were performed by means of the digital computer “Strela”
in the numerical computation centre at Moscow University.
The perturbation method could probably be applied to solve some
problems. However, the work required in comparison to the simpler
numerical method of characteristics may prove so large that it will no
longer be justifiable by the desire to obtain a solution in analytic form.*
B. A certain linearization of the problem of weak non-homogeneity has
been proposed by A. I. Kuznetzov, [55 a, b]. His idea consists in superposing
on the tensor field oiii(o), constituting the solution of the classical problem,
a small correction in form of a tensor field aii(l), having the character of a
* In the paper [ZOe], the same author gave the solution for the punch indentation
problem with a n exponential type of non-homogeneity (numerical solution by the
method of characteristics).
PLASTICITY ‘UNDER NON-HOMOGENEOUS CONDITIONS
177
self-equilibrating system of stresses and satisfying a certain linearized
“yield condition”. Within this framework the theory of equilibrium of a,?)
is developed. However, the proposed method of linearization is open to
objections (cf. [lo3 a]).
C. For non-homogeneous elastic-plastic bodies satisfying the HenckyIlyushin equations, a generalization of the algorithm of the Ilyushin method
of elastic solutions has been proposed (V. S. Lensky [57 b], cf. Chap. 111).
This method should be particularly useful for weak non-homogeneity types.
9. Inverse and Semi-Inverse Methods
The survey by P. F. NemCnyi [67] was devoted to the results and prospects
of the development of inverse and semi-inverse methods in continuum
mechanics. For plastically non-homogeneous bodies these problems will
be treated in detail in [lo3 b]. We shall confine ourselves here to a few
remarks and examples.
First let us formulate a remark which seems to be almost trivial. Consider
a body with boundary conditions expressed in stresses. Any statically
admissible field aii(0)(i.e. a field satisfying the equations of equilibrium and
the boundary conditions) may serve as a statical solution for a certain type
of plastic non-homogeneity. I t suffices to find the corresponding K from
(2.3). Such a field aii(0)may, for instance, be taken over unchanged from
any branch of continuum mechanics.
Observe that a similar approach to problems of plane stress and plane
strain of elastically non-homogeneous bodies is much less trivial, [SO c] ;
Rather than to a definite non-homogeneity type, every field aii(o) corresponds
to a vast class of these types.
For a plastically non-homogeneous body, the solution of classical theory
of elasticity may, for instance, be taken for ajj(0).From the viewpoint of the
theory of plasticity, in this way only a very narrow class of solutions may
be obtained, namely biharmonic solutions.
There are still various approaches possible ; for instance Airy’s stress
function Q may be used. Take SZ as a function satisfying at the edge ([69]
for instance) the equation
f
J
(2.107) Q = - x pz(s)ds
0
S
S
S
J
+ Y py(s)ds + ( p y ( s ) ~ -( ~P) z ( s ) ~ ( s ) ) d s .
0
0
Then K is obtained from Eq. (2.11). In some cases, it is more convenient
to use directly the stresses (cf. the example A below).
The above remarks are obvious and their importance is limited. However,
in view of a great number of possible solutions, a number of valuable results
178
W.
OLSZAK, J . RYCHLEWSKI AND W. URBANOWSKI
can be found in addition to the very special and less interesting ones (for
instance, for K = 0, the incompressible liquid, in regions of hydrostatic
states).
The complete statical problem is formulated by giving a) external loads,
b) geometric and mechanical properties of the body, c) the stress field. All
these quantities may be prescribed with various degrees of definiteness.
Thus, for instance, the geometric form may be left indeterminate to the
extent of a continuous transformation, the load may be taken from a certain
class, the stress field may be determined by prescribing some of its components
or, for instance, the trajectories of principal stresses etc. This variety of
possibilities is typical for the general structure of the inverse and semiinverse methods.
We shall not endeavour to give a general definition of the inverse or semiinverse methods (cf. the remarks on this subject in [67]). For the problems,
in which we are interested, the method must include at least the following
two features: a) K is not given or is given only with a certain indeterminacy;
b) certain properties of the field are assumed a priori.
If the form of the trajectories of the principal stresses is assumed beforehand, then 0 = 0 ( x , y ) is known in Eqs. (2.13). The system of equations for
u and K thus obtained is hyperbolic, and its characteristics coincide with
the trajectories of principal stresses as can easily be seen.* I t is convenient
to take the equations directly in this form, that is (2.32). The boundary
conditions cannot be arbitrary here.
A. An illustration of the efficacy of the simple approach discussed above
is furnished by the results for a wedge, [105d], Fig. 12.
Taking the elastic solution, we obtain the uninteresting result
(2.108) K ( q ) = I d a= K 0 [ 2ctgaa(l - cos 2y)
+ 1 - 2 ctgor sin 21711.
Taking now, as a statically admissible system of stress fields in the
inverse method, the expressions
(2.109)
a
s=-~(;z”+2z)d~+s(0),
0
with the function z satisfying the boundary conditions of Fig. 12, we have
(2.110)
K2(q) = i ( t ” ) 2 $ 2 2 .
* This fact has been pointed out by W. Szczepihski.
PLASTICITY UNDER NON-HOMOGENEOUS CONDITIONS
179
For instance, for
I
- 2K,v,
(2.111)
we have
(2.112)
FIG. 17. Eccentric cylinder subject to interior pressure and a force in the slit.
where
B. A more difficult problem is treated in a paper by W. Olszak and
S. Zahorski, [83 c]. The object of the paper is the plane quasi-static problem
of static flow of an eccentric ring (Fig. 17) as determined by two parameters,
the ratio of the radii p ( t ) and the ratio ~ ( tof) the eccentricity to the outer
radius :
(2.114)
p=-
C
a
(O<p<l),
e
q=T
(O<q<l-p).
It is assumed that the ring is made of a perfectly plastic material, showing
non-homogeneity of certain definite types, and that it is under conditions
180
W . OLSZAK, J. RYCHLEWSKI A N D W. URBANOWSKI
of plane strain. The “quadratic” yield condition is assumed (Huber-Mises)
adapted to non-homogeneous bodies.
The ring is always kept in equilibrium under the internal pressure # ( t ) ,
the external pressure q(t),and the force S(t) being the resultant of the normal
forces in the slit. The one-parameter problem ( p = const or q = const)
is a particular case of the two-parameter case under consideration.
The assumptions are as follows (semi-inverse method) : a) the trajectories
form a bipolar net; the two cylindrical surfaces (= circles) are coordinate
lines; b) the state of plastic equilibrium is biharmonic.
The problem is considered in a system of moving bipolar coordinates
by applying the inversion transformation. The analytic function
(2.115)
maps the eccentric ring in the original system of polar coordinates O(R,@)
onto the concentric ring in the inverted system I(r,p),where the minus sign
corresponds to the “internal inversion” and the plus sign to the “external
inversion”.
The corresponding relations between the stress field in the 0-system
and the I-system [if the existence of the biharmonic stress function w ( Y , Q ) )
is assumed] take the form
(2.116)
ZRT =
1
C2
+
- - tr;(r2 h2
2hr COST),
constituting a generalization of the relations obtained by J. H. Michell
in 1902, [60].
The yield condition is assumed in the form
(2.117)
(OR
+
- a d 2 4t&
= 4[K(P,t)I2.
where
(2.118)
K(P,t) = K(t)R(P).
Five possible types of plastic non-homogeneity are considered, I, 11,
111, IV, and V, of which the last is a linear combination of the former four
K(P,t)= K(1) [-f x2(4R,(P) A W ) R , ( P )fp2(1)Rs(P)f Y2(1)R,(P)l
(2.119)
PLASTICITY UNDER NON-HOMOGENEOUS CONDITIONS
181
and takes in the inverted system I a form depending on the radial coordinates
rli and r, (for the “internal” inversion) and rIe and re (for the “external”
inversion) :
By a judicious choice of the functional coefficients ~ ~ ( t ) , A ~ ( ( t ) and
,p~(t),
v2(t)we can approach the real properties of the plastic material in a satisfactory manner.
Starting from the stress function
which is bihannonic in ri and re, we are able to satisfy the yield condition
(2.117) for a general plastic non-homogeneity type described by (2.119).
The stress components will be obtained, according to (2.116), in the
form
+ D,(2hi2 In - + hi2)+ E0(2S2In re - re2+ he2),
ri2
Y,
+ D,(2hi2 In + ri2 + 3hi2 - 4r;hi cos yi) +
+ EO(2he2In re + re2+ 3S2 + 4rchecos vC),
Y;
ZRT = 0
(2 * 122)
-
-
where the functions A,,~,,8,,. . .,Go, and also the pressure difference
n(t)
= p ( t ) - q(t), under which the ring is in equilibrium, are determined
from the boundary conditions. Also the force S(t) in the slit is determined
together with its moment about the inversion pole.
182
W . OLSZAK, J . RYCHLEWSKI AND W. URBANOWSKI
In the case of the one-parameter problem ( p = 0.5), diagrams of all
the mechanical quantities appearing in the paper have been made for a
set of values of the parameter ~ ( t and
) of the ratio of the radii R / d . Thus,
for instance, the diagram of K ( P ) = k(t)R,(P) is represented in Fig. 18.
This case corresponds to a “practically” homogeneous material with timevariable yield point.
The yield condition, expressed in terms of principal stresses (net of
curvilinear coordinates), becomes linear. Using this, the authors were able
FIG.18. Klk plotted against the ratio Rld for different values of the parameter 77.
to superpose the several types of non-homogeneity and the corresponding
solutions obtained, so as to adapt the final result to the actual conditions
(e.g. homogeneity, or non-homogeneity of a specified character).
In addition to the problem of continued plastic flow, it proved possible
to solve the problem of a non-homogeneous eccentric cylinder, part of which
remains elastic, the rest being already plastified (cf. W. Olszak, [73 u], and
W. Olszak, 2. Mrbz [76]). Some singular properties of the corresponding
closed-form solution were demonstrated.
PLASTICITY UNDER NON-HOMOGENEOUS CONDITIONS
183
111. PARTICULAR
SOLUTIONS
1. General Method
In the Hencky-Ilyushin theory of small elastic-plastic strain the nonhomogeneity of elastic and plastic properties is manifested by the fact
that the form of the a;,&,-curvedepends on the point considered (see Fig. 19).
Various types of such non-homogeneities have been discussed on the basis
of the character of the a;,q-curve by W. Olszak [73 p]. In the case when
FIG. 19. Curve ci, ei for two different points.
the non-homogeneity is caused by neutron flux, an effective computation
method has been proposed by V. S. Lensky, [57 b] as a generalization of
the method of elastic solutions derived by Ilyushin, [42 a]. The applicability
of this method to cases of non-homogeneity due to other causes is obvious.
In the case considered, the basic laws of the theory of small elasticplastic strain have the form
(34
amn
- as,,
2a;
= - (Em, - Ed,*),
3 ~ i
where a;,&;are the stress and strain intensities, respectively, p is the neutron
flux, Go a certain value of the modulus G in the body e.g. the maximum
184
W. OLSZAK, J. RYCHLEWSKI AND W. URBANOWSKI
value, and K the bulk modulus. The meaning of the function R is similar
to that in the case of homogeneity. If the curve OA'B' in Fig. 19 corresponds
to the point where G is a maximum, then
AA'
a=-.
(3.4)
CA'
ConsideringR (which makes the problem non-linear) as a small parameter,
we shall generalize the method of elastic solutions in the following way.
Substituting in the equilibrium equations omnof (3.1) and bearing in mind
that
(3.5)
Emn = +(um,n
+
Gn.m),
we obtain equations analogous to the Lam6 equations of the theory of
elasticity, except that additional terms containing the functions R,v and
their derivatives are involved. Transferring the latter to the right-hand
side, we have
(34
(A, + p0) grad div u + p 0 P u = @ + IL,
where &,po denote the maximum values of the Lam6 moduli in the body,
@ the vector of the body force and R, a vector depending on !2 and
and
their derivatives. The boundary conditions are written in the form
+ T,
(3.7)
S, = T,
(3* 8)
u=v
on
S,,
on
S,,
where T, denotes the vector of the surface load prescribed on the surface-part
S,,V the vector of displacement given on the surface-part S,, and T, a
vector depending on 5 2 , ~and their derivatives. S, denotes the vector of
stress computed from u on the basis of Hooke's law (fictitious elastic
stresses) acting on a surface element of which the normal is v .
For the zeroth iteration we take the solution of Eqs. (3.6) for R
0,rp 3 0.
Solving this problem of classical theory of elasticity, we obtain u0,oin,E;,,
and calculate ~,O(x,y,z,t) In the regions ci0 < E ~ ,where E, = ~,(x,y,z,t)
is the limit of elastic strain, the function R is given (the distribution of
the modulus G being known) and independent of E $ . In the regions E,O > E ~ ,
the function Ro is computed from (3.2).
Having uo,Ro, we compute R,,T, and, substituting in (3.6), we face
the next problem of the theory of elasticity. Solving this we obtain u',&,,~~,,
and so on.
The above algorithm is very general, but it is seen to lead to very cumbersome computations. Its convergence in the classical case, for plates and
shells, has been proved by V. M. Panferov, [ 8 Q ] .
186
PLASTICITY UNDER NON-HOMOGENEOUS CONDITIONS
In conclusion of this section let us mention the paper by D. D. Ivlev,
[45 b], where a plastically non-homogeneous body is considered under the
conditions of so-called total plasticity, corresponding to the state of stress
on the edges of the Tresca prism.
2. Axially Symmetric Problems
In what follows we give a survey of the results obtained for axially
symmetric problems.
A. Isotropic, thick-walled cylinder. For the non-homogeneous case this
problem has been solved by W. Olszak and W. Urbanowski, [82 a] and
[82 b]. These authors assume that the modulus of elasticity in shear, G,
3
Q
b
r
Q
b
r
--L
Q
r
FIG.20. Various cases of propagation of the plastic region in a non-homogeneous thickwalled cylinder.
and the yield limit for shear, K , are functions of the radius only, G = G(r),
K = K ( r ) . Further assumptions are: incompressibility of the material in
the elastic and plastic ranges, plane strain with respect to the axis of
the cylinder, and an ideally plastic material.
I t was found that the necessary condition for the plastic region to start
from the interior surface of the cylinder and to expand towards its exterior
surface [case (l), Fig. 201 is that the function
186
W. OLSZAK, J. RYCHLEWSKI AND W. URBANOWSKI
(3.9)
be monotonically decreasing in the interval [a, b ] , where a denotes the
interior radius, and b the exterior radius. If f(r) = const, the whole of the
cross section becomes plastic simultaneously [case (a)]. For f ( r )monotonically
increasing, the plastic region would start from the external surface of the
cylinder [case (2)]. This region would expand with increasing load p ( t )
towards the internal surface of the cylinder. In Fig. 20, J,(r) denotes the
second stress invariant.
If f(r) is not monotonic in the interval [u,b], the radial distance, a t
which the first plastic deformation will occur, appears somewhere inside
the wall [case (3)].
Two critical values of the internal pressure p are found, the first, p,,
being characteristic of the appearance of the first plastic deformations in
the cylinder, the second, p,, being related to the phenomenon of full
plastification of the material in the entire system considered.
The authors considered in greater detail the first case (plastic zone starts
from the interior surface). They proved that the first critical pressure p ,
(corresponding to the appearance of the first plastic deformations) is equal to
(3.10)
K(4
p -2 G(a) a,[g(u) - g ( b ) ] ,
g(r) = -
{Far.
whereas the second critical pressure p , is given by
(3.11)
p , = h(b) - h(a),
In the particular case when f ( r ) =
h(Y) =2
5
dr.
r = const, we have
If p , < p < p,, the radius n(a < n < b) of the interface separating the
two regions (elastic and plastic) may be calculated as the root of the transcendental equation
(3.13)
The state of stress in the elastic (outer) region is determined by the
expressions
PLASTICITY UNDER NON-HOMOGENEOUS CONDITIONS
187
(3.14)
where
(3.15)
The state of stress in the plastic (inner) region is determined by the expressions
a, = h(r) - k(a) - p ,
(3.16)
+ 2K(r) - 9,
a, = h(r) - h(a) + K ( r ) - p .
at = h(r) - h(a)
A similar problem for non-homogeneous strain-hardening characteristics
has been studied by Lee Ming-hua and Pei Ming-li, [ 5 6 ] . The problem of
a quasi-static motion of a thick-walled circular cylinder made of a rigidplastic material and subjected to internal pressure p has been treated by
E. T. Onat, [ 8 5 ] . He obtained the relation
(3.17)
where 9’ stands for the rate of internal pressure, U for the boundary velocity,
Yo and Yo‘ for the yield stress and the slope of the stress-strain curve a t
the yield point, respectively. It follows that the quasi-static motion can
only be maintained by decreasing the pressure, if the non-dimensional rate
of hardening Yo’/Yois smaller than
Experimental investigations of thick-walled cylinders under internal
pressure have been carried out by M. C. Steel and J. Young [118] and M. C.
Steel and L. C. Eichberger [119]. The results show the appearance of an
irregular strain distribution over the cross-section, although its structure
was initially carefully prepared to be uniform. This phenomenon may be
explained as the effect of small non-homogeneities of another type than
the axially symmetrical distribution.
v%
B. Orthotropic thick-walled cylinder. This problem has been solved by
W. Olszak and W. Urbanowski, [82e] and [82f]. The authors assume
188
W. OLSZAK, J . RYCHLEWSKI AND W. URBANOWSKI
that the yield limits in the radial and the axial directions are the same
and constant, that is Qr = Q, = Qo = const. Similarly it is assumed that
the yield limit in shear Qrt in the plane normal to the axis of the cylinder
is constant, whereas the variable plastic properties are assumed to depend
on the variable yield limit in the circumferential direction Qt = Qt(r). Such
assumptions provide a fair approximation to the properties of thick-walled
reinforced concrete pipes with circumferential reinforcement.
The state of stress in the plastic region under the condition of plane
strain (E, = 0) is determined by the expressions
(3.18)
where
The constant C and the signs in the above expressions should be
determined from the conditions a t the boundary of the region.
The practical application of the result for thick-walled non-homogeneous
tubes is the object of the papers by W. Olszak [73 el, [73 f], [73 j], and
[73 q] where reinforced concrete tubes with variable amounts of reinforcement, such as are used in coal mines for pressures of 20 to 50 atm, are
considered.
C . Transversally isotropic non-homogeneous cylinder. B. R. Seth, [112],
takes the yield condition in the form
(3.19)
Tll - T33
+ K l k )( TI1 + T,2 + T33) K,(r)
=
where T,, >, T,, 2 T33 are the principal stresses referred to the strained
frame of reference. Assuming a linear stress-strain law for large deformations
and rotations he found the yield stress for tension and compression.
D. The Boussinesq problem for the semi-infinite space. The problem of
a non-homogeneous linear foundation subsoil was treated by I. BabuSka,
PI.
The problem of the elastic semi-space loaded by a concentrated force
has been treated by K. Hruban, [38 a] and [38 b]. The physical relatibn
between the stress intensity oi and the strain intensity E~ is assumed in the
parabolic form E~ = (ui/K)”, K and n denoting constants. Also Poisson’s
ratio is assumed to be constant. The results for various n are collected in
a table [38 a]. Under the provision of an “active” straining process these
solutions hold also for a non-homogeneous elastic-plastic body (on the basis
of Hencky’s “deformation” theory).
PLASTICITY U N D E R NON-HOMOGENEOUS CONDITIONS
189
3. Spherically Symmetric Problems
Problems of spherical symmetry are treated in the literature in a manner
analogous to axially symmetric problems ; therefore results will be indicated
only briefly.
W. Olszak and W. Urbanowski discuss in Refs. [82 c] and [82 d] the
problem of a thick-walled spherical shell with interior radius a and exterior
radius b made of an incompressible material ( v = 1/2), radially non,
to the action of internal and
homogeneous [G = G(r),Q = e ( ~ ) ]subjected
external pressures p and q, the difference I7 = p - q being a monotonically
increasing function of time t. Here Q = Q ( r ) denotes the yield limit in
tension. The results obtained do not differ qualitatively from those obtained
by the same authors for the thick-walled cylinder. The functions describing
the states of stress and strain are of course different.
The case of a thick-walled transversally isotropic sphere under internal
pressure was studied by B. R. Seth, [112], the assumptions being those
of Section 2.
Some problems of thermoplastic strain of the spherical shell have been
discussed by M. Rogoziliski, [1021.
4. Torsion of Prismatic Bars
For this problem A. I. Kuznetzov considered in the first part of his
paper [55 b] the state of full plasticity of the cross section, assuming that
the yield limit is an arbitrary point function and formulating the characteristics for the problem. It was found that in this case the characteristics
are also slip lines, but, in general, are not straight lines. The function obtained
by the author, a generalization of the “sand hill surface” to the case of
plastically non-homogeneous bars, may be interpreted as a characteristic
surface of the wave equation with variable velocity, inversely proportional
to the yield limit of the material. The slip lines are the extremum lines* on
this surface, its contour lines are the trajectories of the shear stresses, and
its ridges are lines (or points) of discontinuity of the stress field.
I t was also found that in the particular case when the characteristics
are straight lines, the yield limit varies along the normal to the contour.
* The slip line passing through the points M , and M , is a n extremum line of the
integral
190
W. OLSZAK, J . RYCHLEWSKI AND W. URBANOWSKI
The author succeeded also in effectively solving the problem in the
particular case of partial plasticity of a circular cross-section, if the yield
limit is a function of the radius only.
6. Rotating Circular Disc
The problem of determining the angular velocity o,for which the disc
becomes entirely plastic, was studied by M. Zyczkowski, [129 a], [129 b].
The assumptions are as follows: the material of the disc is perfectly
plastic; the yield limit Q is a function of the radius only, Q = Q(r),similarly
the specific gravity y = y ( r ) . In addition, it is assumed that the disc is
fully connected (without hole) Although the author has indicated how
the problem may be solved in the case of variable thickness, the results
are only given, in principle, for a disc of constant thickness.
Assuming that the values of the functions Q(r) and y ( r ) do not deviate
considerably from their mean values and introducing the yield condition
of maximum shear stress, the author obtained
1
(3.20)
0
where R denotes the exterior radius, p = r / R and g is the acceleration
due to gravity.
For Q(r) and y ( r ) varying in an arbitrary manner, the author indicated
a method for approximate solution of the problem based on the HuberMises yield condition.
IV. ELASTIC-PLASTIC
NON-HOMOGENEOUS
PLATES
The basic equation for elastic-plastic bending of non-homogeneous
plates of arbitrary form has been established in the papers by W. Olszak
and J. Murzewski [76 a]-[76 c] for arbitrary boundary conditions and for
various types of the yield condition. The analysis is concerned with those
particular cases of bending, for which the principal directions of stress
and strain coincide with those of orthotropy.
The following equations express the relations between the bending
moments m1,m2 and the curvatures a1,a2 of an elastic-plastic orthotropic
plate :
PLASTICITY UNDER NON-HOMOGENEOUS CONDITIONS
191
The elastic rigidities 8,,8,
and Poisson’s ratios v1,v2, appearing in these
equations, are functions of the coordinates of the point P of the middle
surface of the plate. Also the plastic moduli, appearing in the coefficients
x,w1,w2 may vary with the position of the point P . The latter are functional
coefficients defined by
x = hcl/h,
h =h(P)
for a plate of variable thickness
are “left-hand” limit values of principal stresses a t the distance
hcz/hfrom the middle plane of the plate.
The quantities x,wl,w2 may be expressed as functions of the principal
curvatures a1,a2. The corresponding values of the coefficients w1,w2 are
given, and the existence of the plastic potential in the “quadratic” form
is assumed (here the orthotropy of the material is taken into account).
Making use of the results of [76 a]-[76 c] the same authors applied in
[76 a], [76 d] the general equations of elastic-plastic bending of plates to
the problem of axially symmetric bending of such plates.
These references contain some numerical examples : (a) elastic-plastic
bending of an orthotropic ring with variable thickness, clamped along the
exterior periphery and acted on uniformly by a linear load on the interior
edge; this problem may be reduced to the corresponding problem of a
non-homogeneous ring. (b) elastic-plastic bending of a simply-supported
densely meshed circular grid-work ; this consists of radial and circumferential
beams and is uniformly loaded on the central platform; such a system is
adequately represented by a continuous model of a non-homogeneous
elastic-plastic plate with well-determined mechanical properties. (c) a
circular reinforced concrete plate with variable radial and circumferential
percentage of steel reinforcement.
a;l,aaCl
V. LIMITANALYSIS
AND LIMITDESIGN
1. One-Dimensional Structural Elements
A. Non-homogeneity function. For the purpose of limit analysis it is
convenient to distinguish between the “transverse” non-homogeneity and
the axial (longitudinal) non-homogeneity of a structural element. The
transverse plastic non-homogeneity corresponds to the variable yield-point
distribution across the thickness of a bar. Thus this type of non-homogeneity
influences both the dimensions and shape of the yield locus I; in the stress-
192
W. OLSZAK, J. RYCHLEWSKI AND W. URBANOWSKI
resultant space. However, the most important thing in this case is the
influence on the shape of the yield locus.
If the yield point uo* is a continuous function of the bar thickness
[ao* = ao(l f ) , say, with f ( z ) # 01, then the limit state of the rectangular
cross section is defined in pure bending by the following relations
+
F = M = Mo*,
(54
Mo* = uob
j+
[l
f(z)]zd~,
-H
where h stands for the width of the bar, and 2H for its depth. This type
of transverse non-homogeneity was applied to simple cases of beams by
a)
I
n
FIG.21. Bending of a rectangular cross section with a piece-wise linear non-homogeneity
distribution.
A. S. Grigoryev, [31], assuming that f ( z ) = f ( - z ) . Similarly in the case
of simultaneous bending and tension or compression, the parametric equation
of the yield locus is simply
H
I n
The elimination of the parameter zo from these equations yields the
corresponding interaction curve, Thus the plastic interaction curve or,
in general, the plastic interaction surface F is a function of the transverse
non-homogeneity .
If the non-homogeneity function is a step function, the corresponding
yield locus can be obtained in a similar way. P. G. Hodge, [35 b], applied
this approach to derive approximate interaction curves for a homogeneous
material. For a rectangular cross section and a piece-wise linear non-homogeneity distribution as shown in Fig. 21 a the interaction curve given in Fig. 21 b
PLASTICITY U N D E R NON-HOMOGENEOUS C O N D I T I O N S
193
has been obtained in Ref. [35b]. The material is supposed to obey the
Coulomb-Tresca yield condition. In Fig. 21 n and m stand for the properly
defined dimensionless axial force n and bending moment m. The concept
of step-wise transverse non-homogeneity is therefore useful in the linearization
of limit analysis problems. The influence of non-homogeneity on the shape
of the yield locus is evident.
The behaviour of structures made of imperfectly plastic material was
studied by J. Heyman, [33 d].
‘t
FIG. 22. Interaction surface for simultaneous bending, compression and shear of a
step-wise transversally non-homogeneous element.
A composite structural element made of materials characterized by
distinct yield properties (e.g. reinforced concrete) can also be considered
as step-wise transversally non-homogeneous. For such a case the yield
locus in simultaneous bending, compression, and shear has been derived by
A. Sawczuk and M. Janas, (1061. The octant of the interaction surface
is shown in Fig. 22. The details can be found in the reference.
The difference between the interaction curve in the m,n-plane for a
homogeneous and a non-homogeneous material becomes apparent simply
by comparison of Figs. 21 and 22. Thus the non-homogeneity influences
194
W. OLSZAK, J . RYCHLEWSKI AND W. URBANOWSKI
the solution of limit-analysis problems because it influences the shape of
the interaction surface F .
Thc case, when only the size but not the shape of the yield locus varies
with the longitudinal coordinate of a bar, corresponds to the axial (or
longitudinal) non-homogeneity. To this group belong, for example, the
cases of plastic beams of variable cross sections. Solutions of practical
FIG.23. Load carrying capacity of circular non-homogeneous arch with concentrated
load applied a t the arch center.
problems concerning the “variable rigidity” can be found in standard
textbooks on the limit analysis of beams and frames (cf. [3] and 135 c]),
since it appears that the methods of analysis are the same as in the case
of constant size of the yield locus.
B. Arches. The load carrying capacity of circular arches of transverse
non-homogeneous cross sections was studied by A. Sawczuk and M. Janas,
[106]. For a simple concentrated load applied at the arch center results
have been compared with the experimentally obtained values. The comparison is shown in Fig. 23.
PLASTICITY UNDER NON-HOMOGENEOUS CONDITIONS
195
2. Plates
A number of papers are concerned with the influence of non-homogeneity
on the phenomena accompanying the exhaustion of load carrying capacity
of different types of plates and shells. These are based on the general theory
of plastically non-homogeneous bodies (cf. [73 k]-[73 13).
In most cases the extremum theorems are used; in the kinematical
approach, an upper bound of the load intensity, which cannot be lower
than the actual intensity of the ultimate load, is determined or a lower
bound of this intensity is determined by considering the statically admissible
fields of internal forces.
Statical methods, often giving valuable insight, but generally involving
more complicated procedures, were applied to the analysis of systems exhibiting plastic non-homogeneity coupled with orthotropy in the papers by
J. Murzewski, [65], and A. Sawczuk, [lo5 a] and [lo5 b].
The problems connected with the determination of the field of internal
forces for non-homogeneous plates were treated on the basis of the HuberMises and Coulomb-Tresca yield conditions.
As an example let us mention Ref. [lo5 a] by A. Sawczuk, who studies
the problem of choosing the type of non-homogeneity so that the circular
plate, subjected to axially symmetric load, passes into the fully plastic
state a t once over the entire region. If the value of the limit moment is
denoted by N ( r ) and the shear force by Tv(r),then the necessary condition
may be expressed by means of the two inequalities
(5.3)
N>rT,,
dN
dr
-< T,.
The distribution of the limit moments for a circular plate of radius R
without a hole, simply supported a t the outer edge and subjected to the
load p = const whose the action is restricted to the annulus a r
R, is
< <
N 0 -- PR2
-(1 - a2 + 2a21n a),
4 ,
(5.4)
O<p<a
where p = r / R and a = a/R.
The distribution N ( r ) being known, we can choose in a suitable way the
thickness of the plate
196
W. OLSZAK, J . RYCHLEWSKI AND W. URBANOWSKI
(5.5)
or the corresponding yield limit, or also the distribution of the steel reinforcement in the plate, because then M , = M , = N ( r ) . For some types of load
distribution on circular isotropic plates whose non-homogeneity is determined
from the postulate of equal circumferential and radial moments,
-I+
P
I
(5.6)
W )=
0
O<p<r
If(f)Pdf.
O<r<R
0
examples of non-homogeneity functions are shown in Fig. 24.
FIG. 24. Non-homogeneity functions for plates for different load distributions.
Ref. [13] by H. Craemer may, to some extent, also be regarded as related
to the problem of plastically non-homogeneous plates. For a rectangular
plate, the author assumes the absence of twisting moments, which is possible
for a suitable choice of the non-homogeneity function. In this way he obtains
the following equation relating the moments to the load +:
(5.7)
where m = m, = L- my.
(g*
$)m
=
P,
PLASTICITY UNDER NON-HOMOGENEOUS CONDITIONS
197
A number of other particular cases have been considered as, for instance,
that of plates with linearly variable mechanical properties (A. Sawczuk,
[lo5 a ] ) ; here, the influence of this type of non-homogeneity on the
kinematics (system of facture lines) was examined. D. Niepostyn, [68],
studied the problem of the load-carrying capacity of non-homogeneous
plates of various types, based on the theory of fracture lines (kinematical
approach) as proposed by K. W. Johansen, [47].
Problems of optimum design of orthotropic plates lead to prescribed
non-homogeneity functions. This question was analyzed, among others,
by 2. Mrbz, [64 a ] ; the optimum distribution of reinforcement has been
found for circular reinforced-concrete plates.
The limit analysis of ribbed-plate structures was taken up. These may
be treated as structures with jump-like variability of non-homogeneity
(cf. A. Sawczuk and M. Kwiecifiski, [l08]); solutions were obtained by
means of both the kinematical and the statical approach.
Some other problems of the same kind have been treated for various
support conditions (cf A. Sawczuk, [lo5 a]) as, for instance, for simply
supported rectangular plates with one additional support within the plate ;
two-span plates; transversely ribbed plate strips; etc.
In Ref. [lo5 b], A. Sawczuk is concerned with the problem of limit
analysis and limit design of flat slabs, taking into consideration the conditions
of both the idealized “point” support and the actual support (the reaction
being distributed over a finite area). The existence of “voutes” adjacent
to the supporting columns is also taken into account. For various geometrical
parameters characterizing prismatic and conical voutes, formulae together
with corresponding diagrams and tables are given.
3. Shells
Some problems of limit states of shells have been solved as a result of
further investigations. Thus, the paper by W. Olszak and A. Sawczuk,
[81 a], is devoted to plastically non-homogeneous shells. The authors
considered an axially symmetric membrane shell passing into the plastic
state over its entire volume. The Coulomb-Tresca and Huber-Mises yield
conditions were considered. Assuming constant ratio of interior forces
directed along the parallel and the meridian ( N a and Np), that is
N a / N , = k = const, they determined the variability of the yield limit
necessary for those assumptions :
198
W. OLSZAK, J. RYCHLEWSKI AND W. URBANOWSKI
where y1,yZ are the radii of the principal curvatures, and 121 is the load
component normal to the middle surface of the shell. The component Z ( v )
is related to the component Y ( v )tangent to the meridian, by means of the
equation
(5.9)
2’
+ Z(l- K) cot + (1 + k)Y = 0.
Solutions are obtained for some typical loading conditions, Fig. 25.
In addition, the field of displacement increments is determined under the
assumption of uniform elongation increments. This assumption is discussed
in detail in paper [41 a].
FIG.25. Variable thickness of a membrane shell passing to the plastic state over its
entire volume.
Load carrying capacities of cylindrical tanks with axial non-homogeneity
were studied by A. Sawczuk and W. Olszak, [log]. It was found that for
linearly varying load intensity the total collapse of a cylindrical wall of a tank
occurs only for a very limited range of dimensions. In general partial collapse
occurs. However, by making a tank wall non-homogeneous in the direction
of the generators total collapse can be assured. Thus an efficient way of
design is found. If I Z ~stands for the properly defined circumferential yield
force a t the tank bottom and n, for the corresponding value a t the tank
top and if n varies linearly in-between, then total collapse is assuredif wl = 0.
The corresponding stress and displacement rate field are shown in Fig. 26.
PLASTICITY UNDER NON-HOMOGENEOUS CONDITIONS
199
Thus the influence of the longitudinal non-homogeneity on the mode of
collapse of a structure is found to be very important.
The collapse-load intensities for a non-homogeneous tank as shown in
Fig. 26 is simply
(5.10)
where mo stand for the dimensionless moment at the tank bottom and
c2 = 2L2/HR refers to the tank dimensions. Because of the influence of
non-homogeneity on the mode of collapse, the kinematical method of solution
of limit analysis problems has to be applied very carefully.
FIG.26. Stresses and displacement rates in a cylinder tank with axial non-homogeneity.
Some other questions concerning non-homogeneous tanks are discussed
in Refs. [lo91 and [lo?].
For stepwise transversally non-homogeneous shells, the yield locus,
similar to that of limited interaction (cf. P. G. Hodge, [35 b]), has been
derived by %. Mr6z, [64 b]. Here a discussion of types of failure depending
upon the actual plastic regime is presented.
4. Minimum Weight Design
Problems of minimum-weight design are to some extent related to those
of plastic non-homogeneity. Design with the requirement of lowest material
200
W . OLSZAK, J. RYCHLEWSKI A N D W . URBANOWSKI
consumption may in some cases lead to plastically non-homogeneous
structures. This is, for instance, the case if we design structures of variable
depth with the purpose of keeping the material consumption as low as
possible. Even with plastically homogeneous material this may lead to
point-variable integral quantities (taken over the cross section) which are
characteristic of the plastic behaviour of the different cross sections.
Some papers are devoted t o problems of economic design of one-dimensional types of structures such as beams, frameworks, and frames. Here
belong the papers by M. R. Horne, [37], J. Heyman, [33], J. Foulkes,
[22], and W. Prager, [94d]. As a rule they use the relation between the
unit weight of a steel beam and its section modulus.
The object of the papers by W. Prager, [94 d], and H. G. Hopkins and W.
Prager, [36], is the minimum weight design of discs and plates the material
of which is subjected to the Coulomb-Tresca yield condition and the associated
flow-rule. Use is made of the following postulate: The structure designed
for minimum weight is capable of failing in a number of different ways,
and for any mode of failure the load does work that must be equal to the
rate at which mechanical energy is dissipated in plastic flow.
The problem of minimum weight of shells was treated by E. T. Onat
and W. Prager, [86]. The problem of a cylindrical shell was analyzed by
W. Freiberger, [23] on the basis of the condition that the shell (which satisfies
the minimum-weight postulate) should be designed so that the number of
modes of collapse is infinitely great. As P. G. Hodge has pointed out, [35 a],
this postulate does not necessarily lead to a minimum-weight shell.
Anisotropic minimum-weight shells were investigated by M. Sh. Mikeladze,
The problem of minimum weight of a circular plate satisfying the HuberMises yield condition was studied by W. Freiberger and B. Tekinalp, [24],
who formulated the relevant variational problem.
The papers on minimum-weight design based on the general theorems
of limit analysis are in another group. D. C. Drucker and R. T. Shield,
[19], and D. C. Drucker, [18], discussed the criteria for determining the
upper and the lower bounds of the weight of a structure carrying a definite
load and subjected to definite boundary conditions. The condition of constant
rate of dissipation per unit volume of structure of minimum weight (or per
unit area of the middle surface of a plate) is of basic importance in the
theory of limit design. These principles were used by E. T. Onat,
W. Schumann, and R. T. Shield, [87], for the design of circular plates.
PLASTICITY UNDER NON-HOMOGENEOUS CONDITIONS
20 1
VI. PROPAGATION
OF ELASTIC-PLASTIC
WAVESI N A NON-HOMOGENEOUS
MEDIUM
There are two principal reasons for the development of theoretical
research on stress waves in non-homogeneous media.* The first is the necessity
of finding a theoretical explanation of certain experimental results obtained
for test-pieces subjected to several tests. The occurrence of permanent
deformation was observed here, modifying the yield limit in different ways
a t different stations and thus giving rise to non-homogeneity. The other
stimulus for the study of the influence of variable mechanical properties
on the propagation of stress waves came from dynamic tests of soils. A
soil is as a rule non-homogeneous, and the difference between its mechanical
properties a t different depths may be considerable.
The non-homogeneity type of a test-piece is essentially different from
that of a soil; it is the result of plastic strain and is called “strain-type
non-homogeneity”, while the non-homogeneity of a soil is of primary nature.
The first survey of the investigations concerned with the non-homogeneity
of the first type has been made by Kh. A. Rakhmatulin and G. S. Shapiro,
[99]. A general and broad analysis of the two trends has been given by
N. Cristescu in his survey, Ref. [14 b].
Investigations of the first type were started by Kh. A. Rakhmatulin,
[98 a], [98 b]. Two cases should be distinguished here: the yield limit can be
an increasing or a decreasing function along the test piece. This depends
on whether the test piece (a cylindrical bar, in general) has been subjected
to repeated impacts a t opposite ends or a t the same end. A particular nonhomogeneity type was studied by Kh. A. Rakhmatulin, who assumed that
the yield point a, = as(x)is the only point function, every other mechanical
property being uniform. In the general case of a curvilinear CJ,&-characteristic,
the solution in the loading region was determined by the method of characteristics. In the unloading region the possibility of using an inverse method
has been shown: the form of the curve representing the unloading wave
is assumed, and the deformation u, and the velocity ut are assumed to be
known on this curve. Then, the solution in the unloading zone can be
determined by solving a Cauchy problem. Assuming linear strain-hardening
Kh. A. Rakhmatulin, [98 b] obtained a closed-form solution.
The case of repeated loads acting on the end of a cylindrical test-piece
was studied by V. S. Lensky, [57 a] assuming linear strain-hardening.
* The importance of the problem is well illustrated by the fact t h a t recently
(September 1961) a special International Symposium on “La Propagation des
kbranlements dans les Milieux Hetkrogknes” in Marseille was held, organized by the
Centre National de la Recherche Scientifique (C.N.R.S.) and directed by Th. Vogel.
About 20 papers were read (theoretical and experimental). These, of course, could
not be reviewed here.
202
W. OLSZAK, J . RYCHLEWSKI A N D W . URBANOWSKI
The work of Kh. A. Rakhmatulin was continued for other physical relations
in the papers of S. D. Mokhalov, [63 a], [63 b].
The paper by S. Kaliski and J. Osiecki, [48], is devoted to the investigation of the propagation of stress waves in soils. By examining the behaviour
of the soil many authors observed that the unloading characteristic depends
in a definite manner on the maximum stress that is followed by the unloading
process. Assuming that the unloading is elastic, one should also assume
that Young's modulus of the unloading characteristic is a function of the
load ao(x) in the unloading wave. The stress-strain relation in the unloading
zone has then the form
(6.1)
a(x,t) = a&)
+ E(a0)[W)
-
&0(4I,
where eO(x) is the strain in the unloading wave. For such a relation, the
solution for the unloading region can be obtained by iteration in an inverse
way by assuming beforehand the form of the unloading wave. The problem
is reduced to the search for the Riemann function and the solution of a
Cauchy problem. I t has been pointed out in Ref. [48]that in many practically
important cases, for which the Riemann function can be obtained in an
accurate manner, the solution of the problem under consideration can be
obtained in closed form. P. Perzyna in Refs. [91 a] and [91 b] investigated
the solutions in loading regions in the general case of non-homogeneity.
In Ref. [91 a] it is shown that in the case of curvilinear physical a,&-relations
the solution may be obtained by the method of characteristics. The case
of linear strain-hardening has been considered in greater detail for the physical
relations
where a(%)and P ( x ) denote the variable elastic and plastic modulus, respectively, and E,(x) is the variable strain at the elastic limit. In the case of
sudden loading with a load increasing in time, two strong discontinuity
waves propagate in the body, representing the front of the elastic wave and
the front of the plastic wave, bounding two regions: the elastic region and
the plastic region. By satisfying the kinematic and dynamic conditions
and by making use of the relations along the characteristics, the solution
for the discontinuity waves has been obtained.
Ref. [91 b] is devoted to the determination of the solutions by iteration
in the loading regions, elastic and plastic. In these regions the problem is
reduced to the solution of two generalized Picard problems.
In the general case of non-homogeneity, paper [32] by R. Gutowski,
S. Kaliski and J. Osiecki is concerned with the analysis of the solutions
in the unloading regions. The physical relations are assumed in the form
PLASTICITY UNDER NON-HOMOGENEOUS CONDITIONS
203
The solutions are obtained by iteration in the inverse way, the form of
the unloading curve being assumed beforehand.
Certain general solutions concerning the problem of propagation of
elastic-plastic waves in non-homogeneous slender rods are given in Ref.
[14 a], and some generalizations to the case of finite strain are given in
Ref. [91 c].
The paper by J. Osiecki, [88 a], is devoted to a broad tentative application
of the results of theoretical investigations to the determination of the state
of stress in a soil in the case of dynamic load on the surface of the soil. The
second part, [88 b], brings a solution of the problem of reflection from a
rigid and a deformable wall under various physical relations and general
non-homogeneity of the soil. These results make it possible to apply the
theory to the computation of the deflection of a plane floor of an underground
structure.
The propagation of longitudinal stress waves in visco-elastic bars was
studied by V. N. Kukudyanov and L. V. Nikitin, [54]. The assumed stressstrain relation takes the influence of the strain rate into consideration, the
yield point being assumed to be variable along the bar. With arbitrary
time-variable stress or velocity being prescribed a t the end of a semi-infinite
bar, or with the end of the bar being subjected to the impact of a body of
finite mass, the solution of the boundary-value problem is reduced to a
mixed problem for the telegraphist's equation. Only a particular type of
non-homogeneity is considered, and the influence of the strain rate is taken
into consideration by using the simplest model of a body with relaxation.
VII. OTHER PROBLEMS
1. Physically Non-Linear Bodies
Paper [84] by W. Olszak and M. Zyczkowski is devoted to physically
non-linear, non-homogeneous bodies. The basic equations of this theory
were derived*, and a suitable classification of such bodies into five different
groups was proposed. Some simple cases of application of the derived
relations were also discussed.
The problem of torsion of a circular bar is solved in an effective manner.
The physical non-linearity is assumed to be of the parabolic type, and its
* For "active" straining processes, these equations may also serve as basic equations for non-homogeneous elastic-plastic bodies according to Hencky's "finite" ("deformation") theory of plasticity.
204
W. OLSZAK, J. RYCHLEWSKI AND W. URBANOWSKI
non-homogeneity is supposed to be a function of the coordinate Y (in
cylindrical coordinates r,p,z). The non-homogeneity law is assumed in the
form
(7.1)
f,(r) = 1
+ n -a,
Y
or in the form
where n is a numerical coefficient and a the exterior radius.
To a certain extent these assumptions are justified e.g. in the case of
cast bars. This holds in particular for the hyperbolic type of non-homogeneity
(7.2),because the variation of the properties of the material is more significant
in the neighbourhood of the surface than inside the bar. Assumptions
(7.1) and (7.2) make solutions in closed form possible.
In addition, the equations of longitudinal vibrations of a rod of a
physically non-linear and “longitudinally” non-homogeneous material were
derived.
2. Loose and Cohesive Granular Media
In view of many practical cases in which the physical properties of loose
and cohesive materials exhibit a marked variability depending on the position
of the point considered, the necessity of analyzing non-homogeneous loose
and cohesive granular media is evident.
Take for instance the angle of internal friction @, which explicitly depends
on the granularity. In loose materials, this angle decreases with decreasing
soil grain, and the same applies to the water content (cf. Fig. 27). The
value of the cohesiveness K can also vary in a similar manner. On the other
hand, a definite connection between the granularity and the depth z below
the surface has been observed in many particular cases (cf. e.g. [73 a]).
Thus a theory which is to be applied to problems of real soils can also require
essential corrections of the assumption of homogenity.
The monograph by V. V. Sokolovsky, [117], contains a complete theory
for homogeneous loose and cohesive materials. Problems of non-homogeneity
were approached in the papers by W. Olszak [73 r], [73 y]. This nonhomogeneity may be “natural” or “artificial”. In addition, there are
phenomena and processes which leave the mechanical properties of the
PLASTICITY UNDER NON-HOMOGENEOUS CONDITIONS
205
material unchanged, but result in effects that are equivalent to a change
of a homogeneous interior structure into a non-homogeneous one.
The above papers, [73 r] and [73 y], are based on the formal analogies,
which exist between the theory of plasticity and the theory of loose and
cohesive media in limiting states of their equilibrium. According to the
fundamental notions of the theory of non-homogeneous bodies, [73 k]
1
wFIG.27. Angle of internal friction in granular media as a function of moisture.
and [731], the basic equations of non-homogeneous loose and cohesive
materials were established, their classification into appropriate groups
proposed, and some plane problems discussed.
Some plane problems of the theory of limiting equilibrium of loose and
cohesive non-homogeneous isotropic media were discussed by C. Szymadski,
[120],and 2. Sobotka, [115].
3. Assumption of Non-Homogeneity as a Method of Solving Homogeneous
Plastic Problems
The notion of mechanical non-homogeneity may sometimes be used as
an auxiliary method for the effective solution of problems of elastic-plastic
equilibrium and plastic flow. One of the main difficulties in problems of
the theory of plasticity originates in the non-linear character of the basic
equations, which can be directly integrated in rare cases only. On the other
hand, special types of non-homogeneity functions can in certain cases be
considered with the aim of obtaining exact solutions of the basic equations.
If, in addition, free parameters are introduced into these functions, their
2 06
W.
OLSZAK, J . RYCHLEWSKI AND W. URBANOWSKI
values may afterwards be fixed in such a manner as to approach the given
real conditions as closely as possible (e.g., homogeneity or non-homogeneity
of a specified type).
It has already been observed by some authors (cf. e.g. A. M. Freudenthal,
H. Geiringer, [26]) that it is often more desirable to know the closed-form
solution for a sufficiently approximate physical model than to solve for
an exact physical model for which the corresponding results may be obtained
only for particular conditions only by elaborate numerical methods. Thus,
instead of rigorous physical assumptions and approximate solutions, we
introduce approximate physical assumptions for which we can obtain exact
solutions. Here, e.g., also the inverse and semi-inverse methods should
be mentioned (for plane strain problems, these were discussed in Chap. 11).
Such an approach is possible in problems of elastic-plastic equilibrium,
as shown, e.g. in papers [73 u], [73 XI, and in problems of quasi-static
plastic flow (cf. papers [83 c], [83 d]). In an analogous way, some previously
known approximate solutions have been verified, for instance, in the papers
[129a], [129b] by M. Zyczkowski.
Sometimes solutions can be sought in still another way, as exemplified
in some of the papers discussed in this survey: particular assumptions
for the form of the non-homogeneity function are introduced, chosen for their
simplicity and without reference to experimental evidence that such a type
of non-homogeneous model reflects actual mechanical properties of the
material in question. Nevertheless, the results thus obtained may provide
useful hints for different actual problems related to other mechanical
properties*.
Ref erelzces
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A , On the state of stress in a non-homogeneous foundation soil (in
Czech with Russian and German summaries), A p l i k . Matem. 5, 2 , 361-369 (1957).
2. BARTA,J., On the minimum weight of certain structures, Acta Techn. Acad. Scz.
Hungar. 18, 67-76 (1957).
3. BEEDLE,L. S., “Plastic Design of Steel Frames”, Wiley, New York, 1958.
4. BERMAN,
I., Expansion of thick-walled cylinders fabricated from cold bent plates,
J . A p p l . Mech. 3, 87 (1960).
5. BOHNENBLUST,
H . F., DUWEZ,P., Some properties of mechanical model of plasticity, J . A p p l . Mech. 16, 222-225 (1948).
6. BRINKMAN,
J. A., On the nature of radiation damage in metals, J . AppZ. P h y s .
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* A similar approach may be found in other fields of mechanics of deformable bodies
as, for instance, in the analysis of finite deformations (cf. [ l o l l ) , in some recent studies
of mechanical behaviour of granular media (cf. [la]), or in basic theoretical research
on plasticity (cf. [34 f]).
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7. BRUCH,C. A,, MCHUGH,
W. E., HOCKENBURG,
R. W., Variations in radiation
damage to metals, J . Metals 8, 1362 (1956).
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F., MASSONET,
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11. CHECHULIN,
B. B., Study on the micro-nonuniformity of the plastic deformation
of steel, NACA TN, No. 1411, Aug. 1957; A p p l . Mech. Rev. 8, 11, 431-432,
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?I la resistance des matkriaux. A n n . Inst. Techn. Bdt. Trav. Publ., No. 99 (1949) ;
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H., Ideal-plastische isotrope und orthotrope Platten bei Vollaus13. CRAEMER,
nutzung aller Elemente, Ing.-Arch. 3, 28, 151-158 (1955).
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B., “Neutron Transport Theory”, Univ. Press, Oxford, 1957.
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H., A review of some recent studies of the mechanical behavior
of granular media, A p p l . Mech. Rev. 6, 11, 259-261 (1958).
17. DIENES.G. J., VINEYARD,
G. H., “Radiation Effects in Solids”, Intersciences
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D. C., On minimum weight design and strength of non-homogeneous
18. DRUCKER.
plastic bodies, i n [74], 139-146.
19. DRUCKER.
D. C., SHIELD,R. T., a) Design for minimum weight, i n 9th Congrks
Intern. MCcan. Appl., Univ. Bruxelles 5, 212-222 (1957); b) Bounds on
minimum weight design, Quart. A p p l . Math. 3, 15, 269-281 (1957).
20. DRUYANOV,
B. A,, a) Indentation of the thick plastically non-homogeneous
plate by a rigid punch (in Russian), Izv. Akad. Nauk S S S R , Mekh. Machinoved.
8, 161-166 (1959) ; b) Indentation of thick plastically non-homogeneous plate
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107. SAWCZUK,
A., KONIG, J. A., Load-carrying capacity of orthotropic cylindrical
tanks loaded by loose materials (in preparation).
108. SAWCZUK,
A., K W I E C I ~ ~ SM.,
K I ,Load-carrying capacities of slab-beam structures
(in Polish; summary in English and Russian), Arch. I n i y n . Lqd. 5, 335-370
(1957).
109. SAWCZUK,
A,, OLSZAK,
W.. Methods of limit analysis of reinforced concrete tanks,
Symposium on Simplified Methods of Shell Analysis, Brussels, 1960 (in print).
110. SAWCZUK,
A., S T F P I E ~A,,
~ , Problem of non-homogeneous wedge under plane
plastic flow conditions (in Polish), Zesz. N a u k . Politechn. Warszaw., Budownictwo
(in print).
111. SERGEYEV,
G. YA., TITOVA,V. V., BORISOV.K. A,, “Metallurgy of Uranium
and Some Other Reactor Materials” (in Russian), Atomizdat, Moscow, 1960.
112. SETH, B. R., Non-homogeneous yield condition, in [74], 133-138.
K. N., a) Elastic-plastic state due to concentrated force acting
113. SHEVTCHENKO,
on a semi-plane (elastic-plastic problem) (in Russian), Dokl. Akad. N a u k . 1,
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214
W . OLSZAK, J . RYCHLEWSKI A N D W . URBANOWSKI
YA. B., On the strength of bodies with variable
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N. D., FRIDMAN,
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M., Thermal stresses in a spherical and cylindrical shell in the case
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V. V., “Statics of Granular Media” (in Russian), 2nd ed.. Moscow,
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L. C . , Non-homogeneous yielding of steel cylinders,
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D. B., The dynamic straining of metals having definite yield points,
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Nauk S S S R , Otd. Tekhn. Nauk 1 1 , 58-61 (1955).
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A S T M 55 (1955).
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A . I . , Effects of radiation on the physical properties and structure
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A . V., LuM. A . , DZUGUTOV,
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Some Elementary Problems in Magneto-hydrodynamics
BY RAYMOND HIDE* AND PAUL H . ROBERTS
.
Physics Department. King's College (University of Durham)
Newcastle-upon- Tyne. 1. England
Page
1.Introduction.
.
...........................
216
I1 Basic Equations of Magneto-hydrodynamics . . . . . . . . . . . . . 219
1 Continuity of Matter and Momentum . . . . . . . . . . . . . . . 219
2 . Thermodynamic Equations . . . . . . . . . . . . . . . . . . . .
220
3 . Electrodynamic Equations . . . . . . . . . . . . . . . . . . . .
221
4 . Range of Validity . . . . . . . . . . . . . . . . . . . . . . . .
222
.
I11. Electromagnetic and Mechanical Effects; Dimensionless Parameters . . . 224
224
1 Electromagnetic Effects . . . . . . . . . . . . . . . . . . . . .
2 . Mechanical Effects . . . . . . . . . . . . . . . . . . . . . . . .
228
3 Two-dimensional Theorem : Analogy with the Proudman-Taylor Result
230
for Rotating Fluids . . . . . . . . . . . . . . . . . . . . . . .
.
.
IV . Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . .
233
1 Methods of Deriving Boundary Conditions . . . . . . . . . . . . . 233
2. The Electromagnetic Boundary Conditions . . . . . . . . . . . . . 234
3 . The Mechanical Boundary Conditions . . . . . . . . . . . . . . . 238
4 . Small Departures from a Steady State . . . . . . . . . . . . . . 241
6. Boundary Conditions a t a Solid Insulating Surface
. . . . . . . . 243
.
V . Plane Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . .
244
1 . The General Effect of a Magnetic Field . . . . . . . . . . . . . . 244
246
2 . Alfvdn Waves . . . . . . . . . . . . . . . . . . . . . . . . . .
3 Sound Waves . . . . . . . . . . . . . . . . . . . . . . . . . .
249
253
4 The Effect of Dissipation . . . . . . . . . . . . . . . . . . . . .
6 Shock Waves . . . . . . . . . . . . . . . . . . . . . . . . . .
256
.
.
.
VI . Alfvdn Waves in Systems of Finite Extent . . . . . . . . . . . . . . 261
261
1 Orders of Magnitude . . . . . . . . . . . . . . . . . . . . . . .
2 Standing Waves in a Fluid Bounded by Conducting Planes . . . . . 282
.
.
VII . Gravity Waves: Rayleigh-Taylor Instability . . . . . . . . . . . . . 267
1 Introduction: Choice of Model . . . . . . . . . . . . . . . . . . 267
2 Solution of Model Problem . . . . . . . . . . . . . . . . . . . .
288
.
.
* Now at Department of Geology and Geophysics. Massachusetts Institute of
Technology. Cambridge 39. Massachusetts .
t Now a t Yerkes Observatory (University of Chicago). Williams Bay. Wis'consin
.
216
216
RAYMOND H I D E AND PAUL H. ROBERTS
. . . . . . . . . . . . . . 270
I X . Steady Flow between Parallel Planes . . . . . . . . . . . . . . . . 274
X. Flow due t o a n Oscillating Plane: Rayleigh’s Problem . . . . . . . . . 286
1. Definition of Problem : Dimensionless Parameters . . . . . . . . . . 286
2. The Formal Solution. . . . . . . . . . . . . . . . . . . . . . .
288
3. Discussion of Some Limiting Cases . . . . . . . . . . . . . . . . 292
VIII. Gravitational Instability : Jeans’ Criterion
4. Rayleigh’s Problem
. . . . . . . . . . . . . . . . . . . . . . .
296
XI. Steady Two-dimensional Inertial Flow in the Presence of a Magnetic Field 300
Appendix A : The Hydromagnetic Energy Equation . . . . . . . . . . . . . 305
Appendix B : Relativistic Magneto-Hydrodynamics . . . . . . . . . . . . . 31 1
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . .
313
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
313
I. INTRODUCTION
Magneto-hydrodynamics - or hydromagnetics - is the study of the
flow of an electrically conducting fluid in the presence of a magnetic field.
Nowadays it is often regarded as a part of “plasma physics”. This article,
which has developed largely out of a final honours course given to physicists,
deals with a number of mathematically elementary problems in the subject
which are nevertheless far from trivial physically, and which contain pitfalls
for the unwary. Thus, our principal objective is to provide an introduction
to hydromagnetics in which the physical wood is not obscured by mathematical trees.
Although hydromagnetics is based on the equations of classical hydrodynamics and electromagnetism, no serious investigations were carried out
in this field until comparatively recently. This is because the parameter
which measures the strength of the coupling between the magnetic field and
the fluid flow, the so-called magnetic Reynolds number (see Ch. 111),
(where a is the electrical conductivity and p the magnetic permeability of
the fluid and L and U are, respectively, a length and a flow velocity characteristic of the system*) is so very small for most practical values of L , U ,
and ap that hydromagnetic phenomena cannot readily be produced in the
laboratory. On the other hand, owing to the vast length scale associated
with cosmical phenomena, even poorly conducting fluids moving quite slowly
are strongly coupled to any magnetic field that is present. It is hardly
surprising, therefore, that the origins of hydromagnetics are to be found in
* Rationalized M.K.S. units are used throughout.
ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS
217
the study of the magnetism of celestial bodies and are not associated with
any discovery, accidental or otherwise, made on the scale of a terrestrial
laboratory.?
The possession by the Sun of a general magnetic field was first inferred
in 1899 from the appearance of the corona during an eclipse. In 1908 Hale
discovered by direct measurement of the Zeeman splitting of lines in the
solar spectrum that sunspots are the seat of strong magnetic fields, of order
10-1 volt sec m-2 ( = 10-l weber m-2 = lo3 gauss), and subsequent observations, by several workers, of the general solar field show that its strength
at the surface is. about
volt sec m-2, and somewhat variable during a
typical solar cycle. It is now known that a number of stars possess quite
strong surface magnetic fields (0.1 to 1 volt sec m-2), some of which vary
quite rapidly, altering considerably, and in some cases reversing in direction,
in periods as short as a few days.
Nowadays it is known that cosmical magnetic fields are not restricted
to celestial objects such as the Sun, Earth, and stars. There seems to be a
general Galactic magnetic field of about
volt sec m-2. This f ield possibly
prevents the spiral arms from collapsing gravitationally and may also play a
key r6le in the processes leading to the formation of stars. Moreover, hydromagnetic interaction between this field and rapidly moving gas clouds in
interstellar space is the likely cause of the acceleration of cosmic rays to
stupendous energies. The importance of the r61e of hydromagnetics in
cosmical physics can be judged from the theoretical contributions stemming
directly from work on the origin of the magnetic fields of celestial objects,
heating of the solar corona, dynamics of solar rotation, theory of magnetic
storms, to name but a few astrophysical problems. Details of this work and
exact references can be found in several recent review articles [l, 2 , 3 , 4 , 5 , 5 6 ] .
Although the early development of hydromagnetics took place as a
branch of cosmical physics, in the past fifteen years or so there have been
several investigations of laboratory systems. Thus, the fact that a pressure
head results when a current is passed through a conducting fluid immersed
in a magnetic field has been exploited, in the electro-magnetic pump, in order
to circulate liquid metals in systems, such as nuclear reactors, where the
presence of moving machinery would pose severe technical problems.
Theoretical work on hydromagnetic flow along pipes (see Ch. IX) followed
the invention of the electro-magnetic pump and so did experimental work
using liquid metals [el. For practical reasons these experiments have been
confined to low values of R, when the coupling between the motion and
field is weak. I n such circumstances the principal effect of the field is to
enhance energy dissipation, through the agency of ohmic heating associated
t One wonders how ordinary fluid dynamics would have developed had no fluid
less viscous than thick treacle, say, been available to the experimenter.
218
RAYMOND HIDE AND PAUL H. ROBERTS
with the induced electric currents. In practice, ohmic dissipation may exceed
viscous dissipation by one or two orders of magnitude.
I t is the high density of liquid metals that limits the value of U which
safely can be attained in controlled experiments. However, by using media
of lower density, such as an ionized gas moving at much higher speeds, it
has been possible to work at higher values of R and thus secure tighter
coupling between the magnetic field and fluid motion. Under the controlled
coriditions of an ordinary shock tube [7] values of R of order unity have
been attained, and even higher values are associated with gas discharges of
very high current intensity, on which, unfortunately, it is notoriously
difficult to carry out reliable measurements.
Laboratory systems differ in an essential way from cosmical ones: fixed,
rigid conductors capable of carrying currents without moving, and/or
insulators capable of withstanding strong electrostatic potential differences
are always present in the former, but always absent in the latter. As a
consequence, there are in principle laboratory phenomena which have no
counterpart in cosmical physics. By the same token, even if an indefinite
range of R were available to the experimenter, many cosmical phenomena
could not be reproduced in laboratory models.
In the past ten years the study of high-current gas discharges and other
low-density plasma systems has been strenuously promoted by workers
aiming to produce the high temperatures required to induce thermonuclear
reactions in light elements such as deuterium [8]. The concomitant increase
in the number of physicists and electrical engineers, as well as mathematicians,
engaged in hydromagnetic research has been considerable. Now, as a result
of recent renewed interest in the problem of devising an economic method
of direct conversion of thermal to electrical energy, mechanical engineers
are beginning to show an interest in the subject [9], and there are aeronautical
engineers who see applications of hydromagnetics to the problems of propulsion and boundary layer control [lo]. On the more academic level, according
to current issues of their journals many applied mathematicians are now
examining the effects of a magnetic field on the “standard solutions” of
problems in ordinary fluid dynamics.
The make up of this article is evident from the table of contents. The
selection of topics arises largely from the authors’ personal interests and
limitations of space, and is arbitrary to that extent. We have endeavoured,
however, to ensure that the introductory sections, dealing with the basic
equations, dimensionless parameters and boundary conditions, and an
appendix dealing with the hydromagnetic energy equation, are reasonably
complete. It is hoped that the present survey will provide an introduction
to hydromagnetics for workers possessing some knowledge of continuum
mechanics but who are otherwise unfamiliar with the subject.
ELEMENTARY- PROBLEMS IN MAGNETO-HYDRODYNAMICS
219
11. BASICEQUATIONS
OF MAGNETO-HYDRODYNAMICS
1. Continuity of Matter and Momentum
The continuity and momentum equations governing the flow of a
Newtonian fluid of density p, coefficient of shear viscosity py, and coefficient
of bulk viscosity pc, relate the values of pressure ( p ) , fluid velocity (u) and
body force (F) a t a general point in space, coordinates (x1,x2, x3) a t time t.
They are, respectively,
(2.1)
3
= 9 + u - g r a d p = - pdivu,
Dt - at
and
1
(2.2)
In practice, fluid dynamicists distinguish between “compressible” and
“incompressible” fluids, corresponding, respectively, to whether or not
density changes associated with pressure variations in the fluid result in
significant dynamical effects, [ l l ] . Such effects are insignificant only when
the speed of fluid flow is much less than that of sound in the medium, and
when accelerations are slow compared with those associated with sound waves.
Under these conditions the continuity equation (2.1) simplifies to
(2.3)
div u = 0,
and the corresponding term of (2.2) vanishes.
In the case of a conducting fluid carrying an electric current, whose
density j is measured in amp m-2, in the presence of a magnetic field, whose
intensity B is measured in volt sec m--2, to the usual body force we must
add the Lorentz force j x B newton m-3. Thus
(2.4)
F=pgrad@+jxB,
where @ is the potential of external forces, such as gravity or any other force
causing gross acceleration.
The seven scalar equations to which (2.1), (2.2) and (2.4) are equivalent
contain fourteen unknowns, and we must therefore supplement them with
further mathematical relations. These relations stem from thermodynamic
and electrodynamic considerations.
220
RAYMOND HIDE AND PAUL H. ROBERTS
The thermodynamic relations (see Appendix A) comprise an equation of
state together with statements concerning irreversible molecular transport
processes, such as diffusion and thermal conduction, leading to entropy
changes. The electrodynamic relations are Maxwell’s equations together
with a statement concerning the dependence of the current on the electric
field present (e.g. Ohm’s law).
We shall assume in most of this article that the gravitational potential @
is a known function of the space coordinates. In one problem where variations of p affect ds (cf. Ch. VIII) to the equations of hydrodynamics, thermodynamics and electrodynamics must be added Poisson’s equation
V W = - 4?Lcp,
(2.5)
where G is the universal gravitational constant.
2. Thermodynamic Equations
To illustrate the thermodynamic relations required for a typical compressible fluid, consider a perfect gas for which
where W is the universal gas constant divided by the gramme-molecular
weight. Two extreme cases will be cited to illustrate the range of possibilities as regards entropy variations, without having to write down the
field equations governing molecular transport processes. These are the
isentropic case, in which changes of state are so rapid that transport processes
can be ignored, and the entropy per unit mass
s = cu log, (Pp-Y),
(2.7)
(where c, is the specific heat a t constant volume, q, is the specific heat at
constant pressure, and y
cp/c,) of a fluid particle remains constant, and the
isothermal case in which heat conduction is so effective that the temperature
of a fluid particle remains constant.
I t is convenient to consider two types of incompressible fluid, namely,
those which are barotropic and those which are baroclinic.
Barotropic incompressible fluids have uniform density and in consequence
there is no contribution of the gravitational part of F to the dynamic pressure
field. In the absence of hydromagnetic effects, hydrodynamical flow of a
barotropic fluid has to be generated by applying forces a t the bounding
surfaces of the fluid. When p is kept constant in equations (2.1) and (2.2)
there is no need to supplement them with thermodynamic relations. For
this reason the study of the flow of incompressible barotropic fluids is the
most highly developed branch of fluid dynamics.
=
ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS
22 1
Baroclinicity is associated with density variations, the action of gravity
(or of any other general acceleration) on which gives rise to buoyancy forces
(see the first term on the right-hand side of (2.4)).It is not generally possible
to maintain hydrostatic equilibrium in the presence of these forces and
hydrodynamic flow must ensue. Baroclinicity arises in a variety of ways;
it may be due to variations in temperature, chemical composition, or both.
Differential heating produces temperature variations from place to
place in a fluid and, as a result of thermal expansion, these are associated
with density variations. Thermal convection is one example of this type
of system, but there are also situations in which baroclinic effects inhibit
fluid motion.
If the heated incompressible fluid has a volume coefficient of thermal
expansion a, the equation of state is
where po is the density at T = 0. Entropy changes are taken into account
by including the equation of heat conduction which, in the absence of heat
sources within the fluid, has the form
(2.9)
pcp- Dt =
- pcp
+ u grad
*
= div [ K ~ grad
c ~ TI,
where the thermal diffusivity is K , and cp is the specific heat a t constant
pressure, ~ p being
c ~ identical with the coefficient of thermal conductivity.
When baroclinicity is due to variations in chemical composition, (2.8) has
to be replaced by a relationship between density and composition, and (2.9)
by the equation of diffusion of matter. A particularly simple example is
that of two immiscible liquids of different density, where baroclinic effects
arise only at their surface of contact and there is no diffusion of matter.
Baroclinic effects of this type can arise in gravity waves at the free surface
of a liquid (see, e.g. Ch. VII).
3. Electrodynamic Equations
Now we must write down equations relating current density j, magnetic
field B, electric field E (in volt m-l) and charge density 6 (in coulomb
m-9. If
(2.10a, b)
H G B/p amp m-l,
D E EE coulomb m--2,
222
RAYMOND HIDE A N D PAUL H. ROBERTS
where ,u and E denote, respectively, magnetic permeability and dielectric
constant,* these are the following.
(2.11)
curl H = j,
(2.12)
curl E = - aB/at,
(2.13)
div B = 0,
and
(2.14)
divD = 6.
(2.11) is Ampere's circuital law relating the magnetic field to its basic
source, the electric current, displacement current being neglected (see
Sec. 11. 4 below and also equation (B. 1.)); (2.13)expresses the fact that the
field is solenoidal (=source free), the isolated magnetic pole being amathematical fiction. (2.12)is Faraday's law of induction, which, in its differential form
conceals many subtle difficulties of interpretation brought out clearly in
relatively few standard texts [12]. (2.14) and (2.10b) relate the electric
field to the volume density of electric charge 6; (2.14)includes a statement
of the inverse square law of electrostatics.
A unit electric charge moving at velocity u relative to a magnetic field B
experiences in addition to a force E, a force u x B. Thus, if the conducting
fluid satisfies Ohm's law with conductivity u (in ohm-l m-I),
(2.15)
j = a(E
+ u x B),
(cf. equation A . 21 and Sec. 11. 4 below)
This completes our set of equations governing seven unknown vector
quantities, u, F, E, D, H, B, j, and four unknown scalar quantities, p , p, T
and 6, or twenty-five scalars in all. An integral energy relation based
on the differential equations of this section is derived in the Appendix A.
4. Range of Validity
Now we must consider the range of validity of our equations. For this
purpose it is convenient to introduce a typical flow speed, U , a speed
V G B / ( ~ u p ) based
' / ~ on the magnetic field strength (see equation 5.1) and
the ordinary sound speed, a,. If c is the speed of electromagnetic waves then
U2/c2,ao2/c2and V2/c2are measures of ordered kinetic energy, thermal energy
and magnetic energy respectively in terms of the rest energy of the fluid.
* For free space, ,u = p o = 4n * lo-' newton amp+ (henry m-l)
E = E,, = 8.86 x
lo-'* coulomba newton-' m-a (farad m-1) ;
(poeJ-1/2 = 2.995 x lo8 m s-l, the speed of light.
ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS
223
When U/c << 1 the D/Dt terms in (2.1) and (2.2) are non-relativistic
(see Appendix B) and, to the same approximation, although the effective
electric field.depends on the local frame of reference in which it is measured,
being (E u x B) in a frame relative to which the fluid moves with velocity
11, the magnetic field is frame independent. When ao/c << 1, that is to say,
when the root mean square speed of thermal motion is much less then the
speed of light, the relativistic correction P / c 2 to the density p (cf, equations
B.20 and B.21) is negligible.
If t is a characteristic period of time associated with a hydromagnetic
system, L being a typical length (see Ch. I) then according to (2.10, 2.11,
2.12, and 2.14),
+
so that when the time taken for energy to be transmitted across the system is
much greater then that taken for electro-magnetic waves to cross it, (more
precisely, as we shall see below in Sec. V.2, when I/ << c ) the neglect of
displacement current, aD/at, in equation (2.1l ) , of the electrostatic body
force 8 E in the expression for the body force (%.4),and of the advective
contribution 23u to j in equation (2.15) is justified. However, even when
our equations are non-relativistic in the sense that U / c << 1, ao/c << 1, when
I/ is not much less then c equations (2.4), (2.11), and (2.15) become
(2.4‘)
(2.11’)
(2.15‘)
F=pV@+j x B+6E,
curlH = j
+ aD/at,
j=6u+a[E+uxB].
The neglect of terms of order ( V / C )in~ equations (2.4’), (2.11’) and
(2.15’) is a convenient way of filtering out of a mathematical theory
terms representing electromagnetic waves. This is a useful procedure when
an insight into other effects is being sought, and has been used in most of
the problems dealt with below. In others, where the limit of very high
magnetic field strength is of physical interest (see Sec. V.3) terms of order
( V / C )and
~ higher are retained.
Plasma oscillations are automatically excluded by taking Ohm’s law
(equation 2.15’) to relate the current to the electric field [13].
Because E does not appear in the expression (2.4) for the body force,
some authors have erroneously ignored 6 in (2.14). This procedure misses
an essential point, that because of the magnitude of E (see (2.10))only very
low charge densities are needed to set up quite strong electric fields, and
although there are systems having geometries such that 6 is zero everywhere
there are others in which electrostatic charges play an essential rale.
224
RAYMOND HIDE AND PAUL H. ROBERTS
In addition to supposing that relativistic effects can be ignored, we have
assumed implicitly that the fluid can be regarded as a continuum with
isotropic transport coefficients Y, K and a. This is a valid procedure in
sufficiently dense media, but in the case of a tenuous fluid - and much
current work in the subject is concerned with low density plasmas continuum theory breaks down. It would take us too far away from our
main topic even to summarize how criteria for the validity of continuum
theory can be deduced by considering the forces on individual atomic particles
and thus formulating equations which tend to our continuum equations in
appropriate limits. A number of writers have considered this problem, and
we refer here to the original papers [13, 14, 15, 161.
111. ELECTROMAGNETIC
A N D MECHANICALEFFECTS
; DIMENSIONLESS
PARAMETERS
1. Electromagnetic Effects
Operate on (2.15) with curl, make use of (2.11, 2.12, 2.13) and the
vector identity curl (curl a) = grad (div a) - V2a, and thus find that
(3.1)
aB
- = curl (u x B)
at
+ APB,
where
(34
A 3 (puo)-l m2 sec-l.
According to (3.1) the rate of change of B depends on two agencies, namely,
motional induction, represented by curl (u x B), and ohmic dissipation due
to electrical resistance, represented by the term AV2B (cf. (2.15)). The
magnetic Reynolds number (see Ch. I, (1.1)) is
(3.3)
R = ayLU = LU/A- (curl (u x B)(/(APB(.
This number is so called because it is analogous to the ordinary Reynolds
number
(3.4)
R' EZ U L / v
which measures the ratio of inertial force p(u * V)u to viscous force p v P u
in the equation of motion (2.2). Indeed, if F is irrotational, the vorticity
vector
(3.5)
w
= curl u
ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS
225
satisfies the equation
(3.6)
ao
+
- = curl (u x o) v P o ,
at
(if the fluid is incompressible) which is mathematically identical with (3.1)
and has an analogous physical interpretation.
To illustrate the nature of the solution of (3.1) consider two extreme
cases, R = 0 and R = 0 0 . In the first of these, for which (3.1) becomes
i3B
at
-- - APB,
(3.7)
motional induction effects are absent. (3.7) is the well known diffusion
equation, acceptable solutions of which represent a decaying magnetic field,
with time constant t = L2/A. The circuit equivalent of such a system is
an inductance ( I ) and resistance ( Y ) in series, the time constant of which is
(I/Y)seconds. Observe that R is equal to t / ( L / U ) . Small R is the case when
the motion is so slow that free decay of currents due to ohmic losses prevents
any significant change in the magnetic field due to the fluid flow. In the
other case R = 0 0 , there is no ohmic dissipation and thus
aB_
_
curl (u x B) = 0.
at
This equation should be compared with that satisfied by the vorticity
vector, w in a homogeneous inviscid fluid (in the absence of a magnetic
field) namely
(3.9)
a o_
_
curl (u x o)= 0
at
(see equation (3.6)). Eq. (3.9) is the basis of Helmholtz’s celebrated
theorem in classical hydrodynamics [17], which states that in a homogeneous
inviscid fluid the circulation around any circuit comprising material particles
cannot change, and in consequence, the vortex lines move with the
fluid. Equation (3.8) shows that the magnetic flux through a region
bounded by any material circuit cannot change, and in consequence the lines
of magnetic force move with the fluid [l]. The concept of magnetic lines of
force being “frozen” into the fluid when R is infinite and, as we shall see
when we consider mechanical effects, acting as “elastic strings”, leads to
a valuable insight into hydromagnetic phenomena at high magnetic Reynolds
number. Even when the conductivity is not perfect, this concept is still
qualitatively valuable provided R >> 1. There will now be relative motion
between fluid particles and magnetic lines of force but this will be slight,
226
RAYMOND HIDE AND PAUL H. ROBERTS
of order ulR [ 5 ] . By analogy, in high Reynolds number flow (R’ i> 1)
satisfying (3.6), the order of magnitude of the speed with which fluid particles
can slip across vortex lines is ulR‘.*
Now we can be a little more precise about what is meant by an “inviscid
fluid” and a “perfect conductor”, idealisations which we often make when
effects of damping are of secondary interest. Respectively, these cases require
that both R‘ and H shouldapproach infinity. However, in some such problems,
it is important to consider the possibility that the final solution will depend
on a further parameter, the ratio
(3.10)
RIR’ = VIA.
For example, as is shown in Sec. IV.5, the boundary conditions to be
imposed may depend on this quantity.
The Prandtl number VIK[18], where K is the thermal diffusivity, resembles
the parameter VIA in form and these dimensionless quantities have some
analogous properties. v / K is the ratio between the Piclet number ULIK,
[18], and the Reynolds number, ULIr. ULIK is a measure of the magnitude
of the ratio of the convection term to the conduction term in the heat flow
equation (2.9). In problems of forced convection, the ratio of the natural
length scales of the velocity and temperature fields is determined by the
value of VIK.
In Table 1 are listed some typical values of o,A and t = L2/A together
with that value U , of U for which R = 1. At higher values of U than this,
motional induction effects are strong. While in cosrnical systems typical
flow speeds are much in excess of U,, in only one of the laboratory examples
cited (the last one) would it be practical for U to exceed U,.
In many idealized problems the magnetic field B can be regarded as
consisting of two parts, B, and b (say). B, is the inducing field which may be
due to currents flowing in external circuits or to currents within the system;
b is the induced field resulting from inductive interaction between u and R.
When B, is due to currents within the system, if these are freely decaying
with time constant to = L2/1,
in dealing with the theory of b the concept
of perfect conductivity is a valid approximation only when the time scale zl
associated with b is much less than to. If tl is the time taken for AlfvCn
waves (see Sec. 111.2 and Ch. 17, Eq. (5.1))to traverse a distance L , then the
requirement that z, >> tl is equivalent to the requirement that the Lundquist
number (3.16) should be very large (see below). In such problems, in which
the effects of the free decay of B, are not of primary interest, this assumption
of perfect conductivity is very useful because it automatically filters out
these effects and thereby reduces mathematical complexity. It is clear,
* For further discussion of the motion of lines of force, see [13].
TABLE1
Decay
wv
Time
L z / A (set)
104
107
10-2
10
10-1
103
105
10-1
1
10
10-7
1
10-3
02
1
104
106 ?
10'2
0-6
102
10-4
10
10'0
0-4
10
10-1
103
1013
104
1
103 ?
108
10'8
10-9
l?
10-'6?
10'8
10-9 ?
10-15 ?
lo21
10-10 ?
10-14 ?
1029
10-13 ?
d = (pr7-1
3
2 2
Ionized Hydrogen
10-1
105
10
Earth's Liquid Core
log
104
10-6 ?
Solar Granulation
log
6 x 103
Sunspots
107
4 x 103
Magnetic Variable Stars
lo9
1 06
Solar Corona
10s
log
Inter-planetary Gas
10"
105
Interstellar Gas
10'6
104
10-7
10-
I
10-2
N
%
X
3
3
f
3
n.
10-15
2
10 ?
2
n.
10-20
3
103 ?
10-21
10'6
1 0'6
1017
For Ionized Hydrogen:
n.
104
2
v
w
2 2 2 2 2 2
%
X
3
10
1
3
E: 2
v
373
2
I
m
tm
I-
10-1
0,
Liquid Sodium
m
10-1
G.1
293
Mercury
v o( p-1T5/2,
1 cc T - Y 2
u~copper
= 5 x 107 51-1 m-1
227
po = 432 x 10W7 newton ampp2
ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS
Density
p (kg m-?
P 2
Kinematic
Viscosity
v (m2s-l)
3
Temperature
T (OK)
Fluid velocity
corresponding to
magnetic
Reynolds number
equal to unity
U , (m sec-l)
I
Length
Scale
L (m)
E1ectr oMagnetic
Viscosity
1 (m2s-l)
228
RAYMOND HIDE A N D PAUL H. ROBERTS
however, that a complete picture cannot be obtained in this way. For
example, the theory of the setting up of B, in the first instance must take
into account departures from infinite conductivity, since a perfect conductor
completely excludes magnetic fields from its interior.
When B, is due to currents flowing in external circuits it is presumed
that these circuits are energized by some source (battery, dynamo, etc.).
Within the region of fluid under consideration, there is no current associated
with B, so that free decay of B, due to ohmic dissipation within the fluid
does not arise. However, there is the possibility of interaction between the
induced currents in the fluid with those in the external circuits, and this
gives rise to energy exchange between the two systems. This should be
borne in mind when dealing with energetics (see Appendix A). I n using
(A.10) to obtain an energy balance equation for the whole system, the
region of integration must include the external circuits.
Cowling’s Theorem. A simple but useful result follows immediately from
equations (2.11) to (2.15). According to the last of these, a t any neutral point
of B (i.e. where B = 0 ) ,j = aE.If the system possesses a closed line on which
B = 0, then, if the component of j along this line always has the same sense,
the line integral of E around this line is not zero. Hence the system cannot
be steady; in fact lines of force collapse on the neutral line a t a rate which
is independent of u. This result was first given by Cowling [19].
One system to which Cowling applied his theorem is the mechanism
proposed by Larmor [20] to account for a centred axial dipole field external
to the sun in terms of axisymmetric meridional motions in the solar interior.
Because the magnetic field lines lie in meridian planes, the system possesses
a t least one neutral line within the fluid, and this line will, of course, be a
circle about the axis of symmetry. Moreover by (2.11) j is azimuthal everywhere and has a constant non-zero component round the neutral line.
Thus, by Cowling’s theorem, Larmor’s mechanism cannot account for a
sleady magnetic field. Therefore, in the search for self-exciting dynamos,
it has been necessary to invoke more complicated forms of u and B (see
for example, 1211).
2. Mechanical Effects
Having examined the extent to which the motion u modifies the magnetic
field B, we now consider the extent to which B modifies u. Even when there
is strong coupling between u and B (i.e. when R >> 1) mechanical effects
will not be noticeable if the field is too weak. A useful measure of the relative
magnetic field strength is S, where
(3.11)
s2 = (+B2/p)l(4pU2)= B2IppU2,
ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS
229
which is the ratio of the magnetic energy to the kinetic energy of ordered
mass motion of the system, if B is a typical field strength. Another parameter,
(3.12)
where 2, is the average pressure in the fluid, measures the magnetic energy
in terms of the thermal energy (i.e. kinetic energy of random molecular
motion). Note that for a perfect gas, (2sz/y.S2)1/2
is equal to U divided by
the speed of sound, which is, of course, the Mach number.
is usually
called “beta” by plasma physicists and S-2 is sometimes called “dynamic
beta”. In the case of an incompressible fluid (for which the Mach number is
zero) vanishes and is therefore a redundant parameter.
In order to classify hydromagnetic phenomena in incompressible fluids
we distinguish two general cases, R << 1 and R >> 1, corresponding, respectively, to weak and strong coupling between u and B. Because in the weak
coupling case the externally impressed field B, is hardly affected by u, we
can take B = B,. Except in the cases of certain especially simple flow fields
and geometrical configurations of the boundaries, when E may practically
annul (u x B) (see (2.15)),
s-2
s
j N a(u x B,).
(3.13)
Hence, the magnetic term in the body force (see (2.4))
(3.14)
This force retards the motion and degrades the kinetic energy through the
agency of ohmic heating. If S is such that the so-called Hartmann number
(3.15)
M
B , L ( U / ~ Y=) S(RR’)l12
~/~
is much greater than unity, the dissipative force represented by (3.14) is
more powerful than viscous friction ; otherwise viscosity is largely responsible
for energy dissipation. In the field of a powerful electromagnet, waves on
the surface of mercury are attenuated a t a rate much in excess of that due
to viscosity acting alone and this effect is readily demonstrated. Some
insight into the subtle interplay between viscous and magnetic forces is to
be gained from analysing hydromagnetic flow between parallel planes as we
shall see when this problem is discussed in Ch. IX.
In contrast to the case of weak coupling ( R << 1) when the mechanical
effect of B on u is essentially dissipative, when the coupling is strong ( R >> 1)
the fluid effectively takes on “elastic” properties, the importance of which
depends on S. When S is small these “elastic” forces are insignificant.
(However, in the case of turbulent flow S may not remain small indefinitely
because in general the lines of magnetic force will increase in length as they
move with the fluid [22]. As a result B will increase a t the expense of U
230
RAYMOND HIDE AND PAUL H. ROBERTS
until a state of equipartition has been attained, and oscillations about this
equilibrium state may occur.) When S is large, however, the magnetic field
dominates. Because of the elastic properties of the magnetic lines of force
oscillations can occur in which the inertia is provided by the fluid and the
restoring force by the field. These oscillations give rise t o hydromagnetic
waves which travel with velocity V = B/(,up)'12 (see Sec. V.2), which were
first discovered theoretically by AlfvCn [23].
When R >> 1, the Lundquist number,
K
(3.16)
aLB(p/p)'12 = S R ,
which is a magnetic Reynolds number based on the AlfvCn speed, is the
appropriate parameter determining the degree of mechanical coupling
between the field and the motion.
3. Two-dimensional Theorem: Analogy with the Proudman-Taylor Result for
Rotating Fluids
In view of the number of papers in the recent literature which deal with
nearly uniform flow in a nearly uniform magnetic field [24, 25, 261, it is
instructive to consider whether any general statements can be made about
b where B, is supposed to be a uniform magnetic
such flows. Let B = B,
field parallel to the z axis, and b << B,; and let u = U, ul, where U, is
a uniform flow in the direction of unit vector 8 (say), and u1 << U,. For a
steady system, t o first order of smallness (2.11) and (2.15) lead to
+
(3.17)
+
U,x b - A c u r l b - B , x
ul=-EE,,
+
the electric field E having been taken as E, E, where E, = - ITo x €3.,
On taking the curl of (3.17) and using the fact that for a steady system
curl E, = 0 by (2.12), we find that
(3.18)
The equation governing the steady flow of an incompressible fluid may
be written (see (2.2))
(3.19)
(U *
P)u= - V I + Y P ~+U (pp)-l(B * V)B,
where
(3.20)
=PIP
+ B212tLp.
Applying (3.19) t o nearly uniform flow in the presence of a nearly uniform
field, to first order of smallness we have
ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS
23 1
(3.21)
where
(3.22)
171 = P l l f
+ Bob*lPup.
Equation (3.21) may be rewritten in the form
(3.23)
curl [U, x u1 - Y curl u1 - (pp)-lB, x b] = ITl.
Now eliminate ITl from (3.21) by operating on it with curl and find
(3.24)
(pp)-lB0a(cur1b)/az = p-lB0
= U, awlas - Y P ~ ,
where j = p-l curl b (see (2.11)) and w = curl u1 is the vorticity (see
(3.9.) and (4.34)).
Finally, by eliminating one or the other of the two variables w and j
between (3.18)and (3.24),it follows that both variables satisfy the differential
equation
Written in this form, it is easy to see that when S2>> 1 and S2RR' >> 1
(see (3.3), (3.4), (3.11)),the right-hand side of (3.25) can be ignored, and to a
first approximation
(3.26)
a2
- (w,j) = 0.
a22
S2>> 1 implies that the AlfvQ speed is very much greater than U,.
S2RR'>> 1 is equivalent to requiring that the Hartmann number (3.15) should
be much greater than unity; this is a weaker requirement than that of
high conductivity ( R >> 1) and low viscosity (R' >> 1).
Equations (3.25) and (3.26) are independent of the form of El and n1,.
When I7n1= 0, that is, when the total pressure, fil
B,b,/p, is constant,
but El # 0, it can be shown that
+
(3.27)
a more restrictive condition than (3.26), which it replaces. When, in addition
to VITl = 0, we require that El = 0,
(3.28)
232
RAYMOND HIDE A N D PAUL H. ROBERTS
where I and I I designate, respectively, components perpendicular and
# 0 and
parallel to the direction of the magnetic field. Finally, when
El = 0, it may be shown that
onl
(3.29)
The foregoing results are reminiscent of the so-called Proudman-Taylor
theorem governing slow, steady hydrodynamical flow of an inviscid, homogeneous, uniformly rotating (non-conducting) fluid [ 2 7 ] . This flow is twodimensional, having no variation in the direction of SZ, the basic rotation
vector, Coriolis forces being the dynamical constraints operating in this case.
If u is the flow velocity relative to a uniformly rotating frame of reference,
(2.2) still holds provided 2pSZ x II is added t o the left-hand side, and centrifugal effects are included in F (see (2.5)). On taking the curl of the resulting
equation, if SZ = (0, 0, Q), then, remembering that p is assumed unifom
and, since u = 0, j = 0,
(3.30)
2 Q a ~ I a z= vP(cur1 u )
+ curl (u x curl u ) ,
and in the limit of small viscosity (more precisely when the Ekman number
(v/2RLz)”2 is very small) and slow relative flow (small Rossby number,
UILR), where U is a typical flow speed and L a characteristic length, we
have the result
(3.31)
aulaz = 0.
Some writers have erroneously concluded that there is an exact parallel
between the hydromagnetic case and the rotating fluids case. According to
(3.26) and (3.31) this is not so.
Although the Proudman-Taylor theorem has been amply verified by
experiment, and work on the dynamics of rotating fluids now forms a large
and fascinating chapter of hydrodynamics, the experimental verification of
the hydromagnetic two-dimensional theorem has not been given.
It is instructive to consider flows which satisfy the condition aulaz = 0.
In the case of the flow caused by the uniform motion in the z direction of a
solid object immersed in a fluid of indefinite extent, a whole column of
fluid extending from the object to infinity in both upstream and downstream
directions partakes of the motion of the object and in consequence, the total
energy of the flow is infinite. In the absence of the constraint that au1a.z = 0
the total energy is finite (e.g. potential flow). Hence, while the latter flow
can be set up from rest in a finite time by the application of finite forces,
the former cannot. This can be important when one considers the mathematical uniqueness of solutions of steady state problems, and care has to
be exercised in the interpretation of such solutions. As Stewartson [28]
ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS
233
has emphasized the solution, if unique, should be the limit as t+ bo and
v + 0 (and/or A + 0 ) of some time dependent solution for a real fluid (v # 0,
A # 0 ) irrespective of the order in which one proceeds to these limits. The
danger of such pitfalls can be reduced by working not with t , v and A, but
with the appropriate dimensionless parameters which measure their values
(see above).
IV. BOUNDARY
CONDITIONS*
1. Methods of Deriving Boundary Conditions
In many situations, two media ( M , and M,) of almost uniform composition are separated by a relatively thin layer Y (of thickness 1, say) in which
there is a rapid and continuous transition between the two states. Also,
some of the physical variables (such as, for example, the normal component
D, of electric displacement D) change rapidly in 9.I t is clear that, rather
than treat 9as being of finite thickness, it would be theoretically simpler
to treat it as an abrupt interface S of zero thickness. However the basic
differential equations break down on such an interface since some quantities
(e.g. 6) are unbounded and likewise some derivatives ( e g the normal
derivative of D,) do not exist. Nevertheless it is necessary to establish relations between the fields on either side of S before the problem can be
solved uniquely. These are called “boundary conditions”. The form they
take depends on the relative magnitude of 1 compared with other length scales
of the system. In a viscous fluid, 1 must be small compared with the boundarylayer thickness, but, if viscosity is ignored, the boundary layer must be thought
of as being contained in Y no matter how small 1 is. Similarly in a fluid of
finite conductivity, 1 must be small compared with the thickness of the electromagnetic boundary layer (in which the eddy currents flow) but, if resistivity is ignored, the boundary layer must be thought of as being contained
in Y no matter how small 1 is. The boundary conditionsmay be derived in one
of two equivalent ways. Either the basic differential equations are integrated
across Y before the limit 1+ 0 is taken, or the integral equations (from which
the differential forms were originally derived) are used. We will adopt the
latter approach. Displacement currents are retained in the first instance,
but are neglected subsequently in taking the hydromagnetic approximations.
We use the following notation. We will define (locally Cartesian) coordinates x+, on S and locate a point Q near S by its shortest distance x, to S
Owing to the length of this section, in which it was felt desirable to discuss
boundary conditions in more detail than is customary, the principal results are given
in boxed equations.
234
RAYMOND HIDE AND PAUL H . ROBERTS
and the coordinates xq,x, of the foot of the normal from Q to S. We take
the normal vector n = n(x,,xs) to be directed from M , into MI and the
coordinates x,,xq,x, to be right-handed in that (alphabetical) order. It is
convenient to introduce an abbreviation DIV A defined by
DIV A = div A - [(n * grad) A] * n,
(4.1)
aAq/axq aA,/ax,.
Consider a point P of S: let it be xq = 0, x , = 0 for convenience. The
integrations we will perform are of two types:
(i) Integrations over the interior dV and surface dS of a “penny-shaped”
disk. The radius Y of the penny is small compared to the radii of curvature
of S at P and small compared to all the physical length scales except,
+
T”
/ MI
/
Mz
FIG1.
FIG.2.
+
possibly, the boundary layer thicknesses. Its thickness I,
1, is small
compared to Y . Its top surface lies in x, = 1, and its bottom surface in
x, = - I, initially. Since the disk is considered to be fixed in space and the
boundary may be moving (with velocity U , in the direction of its normal,
say), a t a later time dt, the top surface lies in x, = 1, - U,&, and the bottom
surface in x, = - I , - U,dt (see Fig. 1).
(ii) Integrations over the surface d S and round the perimeter dr of a
rectangle. The plane of the rectangle will be taken to be either x, = 0 or
xq = 0 and, in the former case, the rectangle is defined by its intersections
with x, = I,, xq = 0, x, = - I, and xq = M(>> 1, 12) (see Fig. 2). Again
M is small compared with the radii of curvature and all physical length
scales except, possibly, the boundary layer thickness.
+
2. The Electromagnetic Boundary Conditions
(a) Normal component of B.
Apply the equation (cf. (2.13))
(44
I
Beds =0
ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS
236
to the penny-shaped disk. We find
(4.3)
&'[B,(') - B,(')]
5
+ B dS +
*
O(73) = 0.
b
Here
(4.4)
B,(l) = limit B,,(Zl,O,O),
B,(') = limit B,,(- Z,,O,O)
Zl+O
l,+O
and b denotes the curved surface of the disk. Now B is everywhere bounded;
thus the integral over b can be made arbitrarily small compared with the
remaining terms of (4.3)by choosing I , and I, sufficiently small. Hence (4.3)
gives
(The left-hand side is an abbreviation for B,,(') - BL').)
(b) Normal component of D.
Apply the equation (cf. (2.14))
I
5
D a d s = 6dV
to the penny-shaped disk. The argument is essentially as in case (a), and
gives
(4.7)
where
7 = limit
dV+O
~
'I
6dV
7C7'
is the surface charge density on S: dimensions coulomb/m2.
(c) Normal component of j.
From considerations of conservation of electric charge, we find
(4.9)
5
at
In applying this to the penny-shaped disk, we must recognize that (in the
relativistic case) currents arising from convection of charge and (in the zeroresistivity case) eddy currents will flow in an infinitely thin layer a t the
236
RAYMOND HIDE AND PAUL H. ROBERTS
interface. Denoting this surface current by J = J(x,,x,) we can easily
show that
j * dS = - m2DIV J
(4.10)
+
O(y8).
b
Thus, by (4.9), we have
(4.11)
The first term on the right-hand side is negligible if displacement currents
are ignored; the second if eddy currents are ignored.
(d) Tangential components of E and H.
Apply the equation (cf. (2.12))
(4.12)
to an elementary rectangle of the type described in the first section. If the
rectangle lies in the plane x, = 0, we find that the value of the left-hand
side of (4.12) is
Hence, since the right-hand side of (4.12) only differs from M[E,I12 by a
second-order quantity, we have
I [E,]12 = U"[BSI,2. I
(4.14)
Similarly
I [ESl12
= - U,[B,]12.1
(4.15)
According to (4.14) and (4.15) the tangential components of E, measured
in a frame of reference which is locally moving with S, are continuous.
Similarly, from the equation (cf. (2.11')),
(4.16)
$
1
H . dr = (j + aD/at) dS
ELEMENTARY PROBLEMS I N MAGNETO-HYDRODYNAMICS
237
we find
(4.17)
I [Hq]12=
Js
- Ufi[Os]i2,
[Hs]12 =
- Jq
+
u n [Oq]i2.]
If p and E are continuous across S, (4.14), (4.15) and (4.17) require that
(4.18)
(1 - Ufi2/c2)[Eq]i2 = - PUnJq,
(4.19)
(1 - Un2/c2) [Hq]I2= J s ,
(1 - un2/c2)[Es]12= - PunJs,
(1 - un2/c2)[Hs1i2= - J q .
If displacement currents are neglected, (4.18) and (4.19) reduce to
(4.20)
(4.21)
[Hq1I2 = J s t
[Eq1I2= - P u n J q ,
[Hs1i2 = - Jq,
[Es1i2 = - PUUnJs.
Equation (4.20) may be combined with (4.5) to give
(4.22)
[HIl2 = J x n.
This completes the set of electromagnetic boundary conditions. They
are still valid if differentiated with respect to xq or x s or differentiated with
respect to t , following the motion of the boundary. They are therefore not
independent. For example, if we differentiate (4.7) with respect to t following
the motion of the boundary, or if we differentiate (4.6) with respect to t before
applying it to the disk, we find
(4.23)
- U,, [DIV DIl2 = a7
- a
at
Differentiating the first of (4.17) with respect to xs and the second with
respect to xq and subtracting, we find
(4.24)
[(curlH),,Il2 = - DIV J
+ U,, [DIV DIl2.
By subtracting (4.23) and (4.24), and using (2.11’), we recover (4.11).
Similarly, from (4.5) we find
(4.25)
and from (4.14) and (4.15) we find, by differentiating with respect to x,
and xq respectively,
(4.26)
[(curl E)n]i2 = - U,, [DIV BIl2.
Thus, using (2.12), we see that (4.25) and (4.26) are equivalent.
238
RAYMOND HIDE AND PAUL H. ROBERTS
In the zero resistivity case, (4.7) and (4.17) do not restrict the solutions
in M , and M , : they merely serve to determine q and J. Also, since
E = - u x B in this case, the two conditions which do restrict the solutions, namely (4.14) and (4.15), may be written:
(4.27)
[(un - Un)BqIi2= Bn[uq]i2,
[(un
- Un)Bsl,’ = Bn[21s112*
In the finite resistivity case, no eddy currents flow in 9 and the only
surface current is that due to displacement of surface charge q, i.e.
(4.28)
+u(~)]).
J = +q[u(’) + u ( ~ ) ] &qn(n* [u(’)
If displacement currents are ignored, J is negligibly small. Otherwise (4.28)
determines J and, by (4.14), (4.15) and (4.17), four restrictive boundary
conditions.
3. The Mechanical Boundary Conditions
(a) Normal component of u.
Apply the equation (cf. 2. 1)
(4.29)
at
to the penny-shaped disk. Assuming that p remains bounded everywhere
within the disk (i.e. excluding fictitious mass surface densities), the righthand side of (4.29) may be written
and (4.29) gives
i.e.
(4.32)
There are two main possibilities: either S is the contact surface between
two “immiscible” media, i.e. media unrelated physically or chemically, or M ,
and M , are composed of the “same” fluid in two different thermodynamic or
ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS
239
chemical states, the fluid particles crossing S from one to the other ( e g S is
a shock front or detonation front). In the former case, (4.32)simplifies to give
(4.33)
(b) Tangential components of u.
If we define the vorticity by
(4.34)
w
= curl u,
we have, by Stokes' theorem,
(4.35)
If we apply (4.35) to the rectangular circuits of Sec. IV.1 we find
(4.36)
[21J12 =
w,, [U5I12 = - w,, 0 = w,
Here W is the surface vorticity, dimensions m.sec-l. Its rBle may be clarified
if we compare the viscous and electromagnetic boundary layers as v - 0
and A + 0. . If v # O ( A # 0 ) the viscous (electromagnetic) boundary layer
is of finite thickness, but may be the seat of tangential shears (currents)
which are so large that the vorticity (current density) integrated across
the boundary layer does not vanish but tends to a limit W(J) as v + O ( A --+ 0).
Then, in the case v = O ( A = 0 ) ,as we have seen, the viscous (electromagnetic)
boundary layer is entirely contained within a, our rectangular path of
integration shown in Figure 2. x contains a finite integrated vorticity W
(current density J) no matter how small 1 may be. If however v # O ( A # O),a
contains none of the viscous (electromagnetic) boundary layer as the limit
I -,0 is taken, and the rectangular path of integration contains zero integrated
vorticity (current) and we must, therefore set W = O(J = 0 ) , i.e. in this
case we have
(4.37)
(c) Normal components of the stress tensor.
Let p:;"' be the total stress tensor in the fluid, that is the sum of the
mechanical stress tensor
and the electromagnetic stress tensor p,",:"'.
The equation of conservation of momentum (cf. (2.2), (2.4), and Appendix A)
may be written
pyh
(4.38)
Here EE x B is the momentum of the electromagnetic field and is negligible
when displacement currents are negligible, as is the electrostatic part of
Pfj"'. Apply (4.38) in the integrated form
240
RAYMOND HIDE AND PAUL H. ROBERTS
5
(4.39)
fit?t"'&
*7
"5 +
- - - [PU
at
7 -
to the penny-shaped disk.
E ( E x B ) ] d V-
I
pUiUjdSj
It is easily shown that
(4.40)
a5
z5
at
at
[pu
+ EE x B],dV = - nrzU, [ p ~ , , -] ~nr2U,
~
[sE,B, - EE,B,],~
+0 ( r 3 ) ,
[pu
+ EE x B],dV = - nrzU, [ p ~- ~nr2U,B,
] ~ [~ E E , ] ,+
~
+
nrzU, [ E E , B ~ I , ~W 3 ) ,
1
2
[PU + EE x B],dV = - nrzU, [ p ~+~~ G] Y ~~ U
~,B,[E
-E ~ ] ~ ~
at
+
.
I
nr2U,[~E,Bq]12 O(r3).
It therefore follows that
+ U,[EE,Bs - EEsB,11',
- Un)11' + UnBn [~Esll'[~EnBsll',
- [pus(u, - Un)]i' - Un&[~Eq11' + Un[~E,Bq11'.
WF'I1' = - [p%&
(4.41)
total
Ppnq 11 =
total 2
[fins 11 =
- U,)11'
[puq(Hn
These can easily be expressed in an alternative form which involves $yh,
J and 7. We will suppose, for analytical simplicity, that E and ,u are continuous across S. The last term on the right-hand side of the first of (4.41)
may be written
+
+
+
+ +Uw(Bs(l)+ Bs"))[~Eql,',
iUn&(EP(l) Eq('))[BSI1' - +U,E(E,(') ES('))[B,],'
(4.42)
- iUn(Bq(l) Bq"))[~Esl,'
which, on using (4.14), (4.15) and (4.17), is equal to
+
MEq(') Eq(2))
[EqI1'
+ M E s ( ' )+ El')) [EsI1' +
241
ELEMENTARY PROBLEMS I N MAGNETO-HYDRODYNAMICS
Similar results hold for the remaining equations of (4.41) and we find
~
~=
+
+
- [pun(un
~
1 ~ n ) 1 ~1 2 qi(En(1)
2
- [ ~ ~ 9 ( u-nUn)l,2
[f~,",~=
li~
+ q&(E9'')+
+ (arx
~ ~ ( 2 ) )
Eq''))
+ ~(z))),,,
+(~(1)
+ (J' x &(B(')+ B'"))9,
[pFhi1z
= - [ p u , ( ~ ,- u n ) 1 1 2 + v+(E,(~)
+ EP)) + (J' x i ( ~ ( 1 +) B ( S ) ) ) ~ ,
(4.44)
+
where J' = J
qU,n = (qU,,J,,J,).
In the absence of displacement
currents, all these results assume simpler forms: the terms with E and B
do not appear in (4.40) and the terms with q do not appear in (4.44).
The method we have chosen to derive (4.44) may seem unnecessarily
elaborate, but it introduces the mean fields (E(') E('))/2 and (B(') B(2))/2
appearing in the final result in a natural and unforced way.
In the case in which M , and M , are inviscid, the mechanical stress
tensor is diagonal and, if the fluids are "immiscible", the last two equations
of (4.44) give
+
(4.45) B,J = -
+
+ U,n x B(')) + (E(2)+ U,,n x B(2))}
x {(E(l)
If the fluids are perfectly conducting, (4.45) gives
(4.46)
+
IBn[J - i q ( ~ ( ' ) u@))
+ qU,,nIl2 = 0.1
If B, is zero, (4.27) and (4.46) are identically true, and the first of (4.44)
requires the total pressure P to be continuous across S. If B, is not zero,
u is continuous by (4.27) and J = q(u - U,n) by (4.46). I t follows by (4.14),
(4.15), (4.17) that B and E are continuous across S, and that therefore q
and J are zero. Also, the first of (4.44) requires that the pressure is continuous across S. Summarizing we have:
(a) If B, # 0, E, B, u and p are continuous, q and J are zero. These conditions are not independent and are all satisfied by making u and p
continuous.
(b) If B, = 0, u, and P must be continuous.
4. Small Departures from a Steady State
We are often faced by the question of whether a certain steady state is
stable or not. A necessary condition for stability is that the system be
stable against infinitesimal perturbations. Thus, denoting by B,, E,,
the magnetic field, electric field, . . . in the steady system, we examine
b, E = E,
e, . . . . If the amplitudes of
solutions of the form B = B,
these perturbations grow in time, we conclude that the system is unstable.
...
+
+
242
RAYMOND HIDE AND PAUL H. ROBERTS
The perturbations will, in general, involve a motion of the interface S
separating MI and M,, and, of course, B, E,.. . must satisfy on S the
boundary conditions we have derived in the preceding sections. However,
it is generally more convenient to apply conditions at So, the position of the
interface before the system is perturbed. For this reason, we will briefly
discuss the problem of translating the boundary conditions at S to equivalent conditions at So. Erect coordinates xq,xs on So, as before. Let the
equation of S be
(4.47)
xn = E ( x q , x s J ) *
Let P be the point (t,xq,xs) of S and Po the point (O,xqrxS),of So. The
magnetic field at P is, by supposition,
+
B ( P ) = Bo(trXq,Xs)
b(EtXq,%),
(4.48)
to first order. The direction of the unit normal to S at P is given by
(4.49)
N = (1, - allax,, - aEjax,).
Hence the normal component B N ( P )of B at P is
The unperturbed steady state satisfied the boundary conditions (4.5).
Hence Bm(P0) is continuous. Thus, since B N ( P )must be continuous also,
we require
(4.51)
where we have used double brackets to emphasize that the condition is
to be satisfied on So and not on S.
To first order, N, Q = (a(/ax,, 1, 0) and S = (@/ax,, 0, 1) form a righthanded triad of vectors at P on s. By (4.14) we see
i.e. to first order
(4.63)
ELEMENTARY. PROBLEMS IN MAGNETO-HYDRODYNAMICS
243
On expanding these quantities in terms of their values on So, we find
or
Since E, refers to a steady state, U,, = 0 and so, by (4.14), [ [Eoq]]12= 0.
Further, by (2.12), curl E, = 0. Thus (4.55) may be written
(4.56)
again expressing that the tangential electric field, in a frame moving locally
with S, is continuous. A similar result holds for eos. The other boundary
conditions of Sections IV.2 and IV.3 may be transformed into conditions
on So in exactly the same way.
6. Boundary Conditions at a Solid Insulating Surface
In determining the flow past insulating (or poorly conducting) surfaces
a t high Reynolds and magnetic Reynolds numbers, it is often convenient
to assume that, except in a thin boundary layer near the insulating surfaces,
the viscosity and resistivity are negligible. The problem is then divided into
two parts. First, the structure of the boundary layer is determined and the
“jump conditions” across it are discovered. Second a solution of the equations governing the flow in the main body of fluid is chosen to satisfy these
jump conditions. In the present section we will illustrate the first of these
processes (see [28]).
Consider the steady flow in a sufficiently small region of the boundary
layer to be laminar and in the xq direction, say. Since the boundary layer
is thin, we may consider that B and 11 vary much more rapidly with x, than
with xq or x,; i.e. ajax, >> a/&,, alax,. We may also assume that B, is
approximately constant and u, zero in the layer. I t follows that the basic
equations (3.1) and (2.2) reduce to
(4.57)
(Here x, is measured out of the fluid.) The solution to these is
244
RAYMOND HIDE AND PAUL H. ROBERTS
(4.58)
Bq = A
+ C exp
P'xn/(h)1/2,
(4.59)
where
(4.60)
(4.59) satisfies the condition that ztq vanishes at the surface of the insulator
Also (4.58) and (4.59) show that the thickness of the boundary
layer is of the order of ( A I J ) ~ / and
~ / V that
,
the changes in B, and ug across
it are
x, = 0.
(4.61)
i.e.
[ztg]12
(4.62)
);(
= B,
1 P'
1/2
[B,112
or more generally,
(4.63)
When ( 2 1 ~is) large, as it is as a rule (except invery tenuous media, see Table 1)
we may often (cf. Ch. XI) replace (4.63) by
(4.64)
[n x BIl2 = 0.
However, in the general case it is important to realize that even though the
limit R - 00, R'+ ce has been taken in the main body of the fluid, it is still
necessary to specify R ' / R = A/v (cf. (3.10)).
V. PLANE
WAVES
1 . The General Effect of a Magnetic Field
In the absence of a magnetic field, an ideal fluid cannot transmit shear
waves. I t can, however, transmit compressional waves, and these travel
with the same velocity in all directions. In the presence of a field, the
situation is radically different. We have seen in Ch. I11 that the lines of
ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS
245
force are “frozen” in a perfectly conducting fluid; that is, particles of the
fluid lying on P particular line of force at one time lie on that line of force
for all times. Following Faraday and Maxwell we may picture the lines of
force as mutually repelling elastic strings, whose tension per unit area of
cross-section is B2/2p. By the arguments of Ch. 111, their mass per unit
length and per unit area of cross-section is p. Thus, if they are stretched and
then released, a transverse wave will travel down them with a velocity of
vc(B2/2p)/p] = B / ( 2 p ~ p ) ~Thus,
/ ~ . in the presence of a field, a conducting
fluid can transmit a shear wave in the direction of the field. These waves
are often called “AlfvCn waves” after their discoverer AlfvCn [23]. A more
precise argument, which makes allowance for the repulsion of neighbouring
lines of force, shows that (neglecting displacement currents) their velocity V
is actually greater than that derived in the approximate argument above.
It is
v = B(pup)-”?
The repulsion between neighbouring lines of force has an important
effect on the transmission of compressional waves. Since this magnetic force
effectively increases the pressure driving sound waves across the field, the
velocity of compressional waves travelling across the field is enhanced.
Moreover the velocity of compressional waves is no longer the same in all
directions. In fact it will be shown that in a direction making an angle 8
with B their velocity is (neglecting displacement currents)
if their amplitude is infinitesimal. (a,,= velocity of sound for zero B ; see
Sec. 11.4.) The velocity in the direction of the field is unaffected by the
field, since the field does not resist the motion of particles of fluid along
the lines of force.
There are many accounts of the properties of AlfvCn waves in the nonrelativistic case: we mention a few: [l, 3, 4, 5, 23, 29, 30, 311. The relativistic
case is treated in [32] an& the rotating case in [21, 33, 341. The influence of
a magnetic field on sound waves is discussed in, among other places,
[31, 351. The effects of dissipation upon the propagation are described in
some of these references and also in [l, 4, 361. Magneto-hydrodynamic
shock waves are studied in [37,38,39]. In Sec. V.2 below we discuss transverse
AlfvCn waves and touch briefly on the effect of Coriolis forces and displacement current upon their propagation. In Sec. V.3 the effect of a magnetic
field on sound waves in a perfectly conducting inviscid fluid is considered.
In Sec. V.4 the effects of finite conductivity and viscosity on the results
of Sec. V.3 are discussed, and shock waves travelling perpendicular to the
magnetic field are considered.
246
RAYMOND HIDE AND PAUL H . ROBERTS
2. Alfvkn Waves
Consider a perfectly conducting inviscid fluid of infinite spatial extent
pervaded by a uniform magnetic field B,. Let a disturbance be generated
in this fluid, and let the magnetic field then be
B = B,
(5.3)
+ b.
First suppose that the fluid is incompressible; more precisely, if L and z
are a length and a time characteristic of the disturbance, we consider that
L / t << a,. It follows that
div u = 0.
(5.4)
(We do not suppose here that IbI is small compared with B,.) Suppose
that wave motion exists for which the total pressure is constant:
1
p +B2 = constant.
(5.5)
2P
Then, by (2.2) and (3.1), it may easily be shown that
ikl
1
at
PP
- + (us grad)u - - (B * grad)B = 0 ,
(5.7)
-
-
ab
+ (u grad)B - (B grad)u = 0.
at
-
From the second of these equations we see that ab,/at = 0 , where z is a
coordinate measured in the direction of B,. By supposition, B = B, and
u = 0 before passage of the wave. Thus, any parts of b and 11 which do not
depend on t must be excluded. It follows that
(5.8)
b, = U, = 0.
The equations (5.6), (5.7) governing these transverse waves admit two
types of solution
(5.9)
b(x,y,z,t)= b(x,y,z - Vt,O),
(5.10)
b(x,y,z,t)= b(x,y,z
+ Vt,O),
u = Vb/Bo,
u = - Vb/B,,
although they do not admit a linear combination of these solutions except
in the case of waves of infinitesimal amplitude. Solutions (5.0) and (5.10)
correspond to waves travelling parallel and antiparallel to the field B,,
respectively.
Even in the compressible case (when L / t << a, does not hold), the same
results can be established for small perturbations, i.e. when u<< V and
ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS
247
b << B,. For then, if we suppose that solutions satisfying (5.4) exist, p is
constant to first order, and, taking the divergence of the equations of motion,
we find that the Laplacian of the total pressure vanishes everywhere, i.e.
(5.5) must hold and by (5.8),R2 and RO2are equal to first order, and therefore,
by (5.5),the kinetic pressure is constant to first order as, therefore, is p. This
justifies the original supposition that solutions satisfying (5.4) exist.
Similar solutions can be derived even if displacement currents are included
in (2.11) and the space charge contribution 6E to the body force F is included
in (2.4). Then it may be shown that if, in place of (5.5) we assume
1
p +B2- -&E2= constant,
2P
2
(5.11)
1
the equation of motion may be written
= - grad(&E2)+6E-
(5.12)
where E = - u x B is the electric field and
6=divD=-&div(u
(5.13)
x B)
is the charge density. Again ab,/at is zero and in consequence (5.8) follows.
Assuming that solutions of the form b = xu exist, we find that the righthand side of (5.12) may be written as
:[ +
- &BO2-
(5.14)
(u * gradju]
.
Thus (5.12) can be written
(5.15)
(1
+ g)[$ + (u - grad)u - - grad B) = 0,
I(,",
and this has the same form as (5.6) if we choose
(5.16)
x = f (pp)'i2(1 + V2/c2)1/2.
Thus we have two sets of exact solutions
(5.17)
b(x,y,z,t)= b(x,y,z - Ct,O),
(5.18)
b(x,y,z,t)= b(x,y,z
where
(5.19)
+ Cl,O),
u = Cb/B,,
u = - Cb/B,,
248
RAYMOND HIDE A N D PAUL H. ROBERTS
The division of energy between the electric field energy B = &E2/2,the
magnetic field energy 2? = B2/2p and the kinetic energy density 9 = pu2/2
is determined by
(5.20)
2?=.9++,
(5.21)
B = (C/C)29?.
In the relativistic case (V-• w ) , this gives the familiar equipartition of
energy between the magnetic and electric fields in electromagnetic waves
which, in fact, (5.17) and (5.18) become in this case. In the non-relativistic
case (V/c<( l ) , (5.20) shows that the energy is equally divided between the
magnetic and velocity fields.
The waves are unattenuated and non-dispersive. In the presence of
rotation, this is not the case. In the non-relativistic case, the equation of
motion for the rotating system is
1
+
(us grad)u -- (B. grad)B + 2 8 x u = P
at
PP
au
-
(5.22)
where 8 is the angular velocity of the system. Suppose GI is parallel to B,.
If we assume that all quantities depend only upon t and z (or if we assume
that the system is only slightly perturbed from the equilibrium state), we
find that transverse waves are governed by
ab
au
-=B-,
at
O az
(5.23)
(5.24)
au _
_ _B,- -ab
at pp az
(5.25)
1
p +B2 =constant.
2 8 xu,
2P
These equations admit solutions in the form of undamped waves
(5.26)
u = (uux,uy,O)
exp i(wt - kz),
provided
(5.27)
w =
*R&
+ VW)1/2.
(a2
In highly rotating systems (Q >> V k ) , this dispersion relation requires
(5.28)
w = f 2R,
0 =
*
1,'%2/2R.
ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS
249
The first of these corresponds to the usual behaviour in the absence of a
field (the semi-diurnal tide in the case of the Earth). The other corresponds
to an AlfvCn wave whose phase velocity V2k/2!2 has been greatly reduced
(by a factor of VR/2!2) by the rotation. Both waves (5.27) have the same
group velocity
+ V2k2)-1/2.
V , =kV2(Q2
(5.29)
The partition of energy between magnetic and kinetic energy for the
AlfvCn wave solution is
93 = ( V k / ~ ) ~ 9 .
(5.30)
Thus, in a highly rotating system, the magnetic energy exceeds the kinetic
energy by a factor of (2!2/Vk)2.
In the case of the Earth’s core, for example, a realistic estimate of V is
40 cm sec-l (taking B = ,045 weber/m2).For this value V , = 5
cm sec-l
for k = 2n/3,000 km-l. Also, for the AlfvCn wave solution 93/95 3 lo4.
There are strong indications that 99/9>> 1 in the Earth’s core, although
%?/9
+ 100 might be a more realistic estimate.
-
The modifications necessary in order to include the effects of displacement
currents are easily carried out, and lead to identical results except that V
is replaced by V ( l
V2/c2)-1/2.
+
3. Sound Waves
We will assume that the compressional waves are of infinitesimal amplitude and that before the arrival of the wave
u =o,
B = B, = constant,
E =0,
j =0,
6 =0,
p = Po,
p = p,,
a = a,,
where a (= I / ( y P / p ) )is the velocity of sound* at density p and a t pressure p
in the absence of B,. During the passage of the wave, we will suppose that
+ Here we are regarding a as a variable replacing T.
250
RAYMOND HIDE AND PAUL H. ROBERTS
Henceforth the primes will be omitted, and we will work to first order in
b, f ~ p,
, a, u, E,j, 6. We may divide u into its irrotational and solenoidal
parts
(5.33)
u = - grad #
+ curl A,,,
and express B in the form
(6.34)
B = B, f curl At,.
I t is convenient to divide the solenoidal parts into their toroidal (F)and
poloidal (9’)parts by writing*
(5.35)
(6.36)
where 1, is a unit vector in the direction of B, and Bo = IBol.
Using (5.33) - (5.36) together with (5.7) and working to first order,
we find
(5.37)
Similarly, by (5.6), working to first order, we find
(6.39)
* The notion of toroidal and poloidal vector fields was first introduced by H. LAMB
(Proc. Lond. Math. SOC.18, 61-66, (1881)). In essence, Lamb showed that rather
than describe a vector field by its three scalar component fields V,, V y and V z , it is
sometimes more convenient to represent it by scalar fields #I, T and P related to V,, J’y
and V , by
P#I= div v;
(P- a y a 9 ) T = - (curl v),;
P(P- a2/a$)P = - (curl2 v),
It is then possible to express V in the form
V = grad4
+ curl TI, + curl2PI,
Here curl TI, is described as a toroidal field and curl2 PI, as a poloidal field. The curl
of a toroidal field curl TI, is the poloidal field curl* TI,, and the curl of a poloidal field
curl* PI, is the toroidal field curl (- V*P)l,. Similar expansions are possible, as Lamb
showed, in which I, is replaced by r, the radius vector drawn from the origin of
coordinates. In this case, the lines of toroidal fields lie entirely on surfaces of constant
Irl and, in fact, this led to the terminology “toroidal” by which they are now described.
ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS
251
(5.40)
Also by the equation of continuity (2.1) we have
aP =poV2c#J.
(5.41)
at
I t is evident from (5.37) to (5.41) that the parts of u and B involving the
toroidal fields 9 separate from the remainder involving B and 4. We
will not discuss the toroidal parts further as they have been fully treated
in the preceding section.
From (5.38) to (5.41), it is easily shown that B b , B,,, V2c#J,and p satisfy
+
+ VZ)V2 + ao2v2
at4
(5.42)
If the waves travel in the z-direction, (5.42) separates into
(5.43)
In fact, the poloidal and compressional parts of the solution separate:
the compressional part corresponds to a sound wave travelling along the
z-axis with velocity a,; the poloidal part to a shear wave of the type discussed
in Sec. V.2.
If the waves are propagated perpendicularly to B, (in the x-direction, say),
B,, vanishes and P b , V2c#J,and p satisfy
(5.44)
Thus, in this case the compressional waves travel with a velocity
(5.45)
+
[-_lu2
+
ao2
c
c2
V2
V2
(1 - Uo~/""~
=
[4'+ v2 (1+ V2/c2)
.
'
this is greater than their velocity a, in the absence of a field.
In the strong-field limit V >> c , V >> a,, (5.42) factorizes immediately
to give
(5.46)
($
- c2v2)($ - a,2-$)B
=o.
252
RAYMOND HIDE AND PAUL H. ROBERTS
I n fact, in this case, the waves are electromagnetic. In the non-relativistic
limit c )> V , c >> a,, (5.42) reduces to
(5.47)
This corresponds to two waves, whose phase velocities in a direction at
any angle 19 to B, are (see Fig. 3)
(5.48)
4 [ao2 + v2 + 2 a , ~cos e y 2
g [ao2+ v2 - 2a,v cos e y z .
For weak fields (V << a,) these velocities reduce to a, and V cos 8.
FIG.3.
Denoting by % the increment in internal energy density during the
passage of the wave, it can be shown that the division of energy between
internal, electric, magnetic and kinetic forms is such that
(5.49)
9+&=3?+%,
(5.50)
(5.51)
8 = (C/C)2@,
ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS
253
where C is the phase velocity of the wave. In the strong-field limit ( V + 00,
C- c) 9 and
are negligible compared to 8 and B, which become almost
equal. In fact, as we have already remarked, the waves are electromagnetic
in this case. In the non-relativistic case, the electrical energy is negligible
and by (5.49)
(5.52)
.9=37+@.
4. The Effect of Dissipation
We consider only the non-relativistic case, and adopt the formalism of
the last section, together with Ohm’s law in a form suitable for moving
conductors (2.15)
(5.53)
j = a(E
+ u x B),
and the relationship (see Appendix (A. 3) and (A. 4))
(5.54)
[p - p ([ - - v 3” ) divu] 6 i j - p ~(auis.i+a.i
prh=
auj)
relating the viscous stresses to the rates of strain.
The equations of electromagnetism (2.11) to (2.14) together with (5.53)
give, working to first order,
(5.55)
(5.56)
Similarly the equation of motion (2.2) gives, to first order,
(5.57)
(5.58)
(5.59)
+
where v’ = [ 4v/3.
These with the equation of continuity (5.41) complete the equations.
Thus, it follows that F,and
, Y bobey:
254
RAYMOND H I D E AND PAUL H . ROBERTS
(5.60)
while P,,, Pb, Vz$ and p satisfy (cf. (5.47))
(5.61)
Equation (5.60)is exact for solutions depending only upon 2 and t. Let
us consider solutions proportional to exp i(kz - ot). Taking k to be real,
we find o is determined from it by
(5.62)
+
w = - 8ik2{(A
Y) f
[(A - Y)' - 4V2k-2]1/2}.
Thus, for example [40],in an insulating box of side 1 (in the direction of B,)
the standing waves which decay away most slowly (k = x / l ) have a decay
time of
+
+
2P/x'(A Y),
if
= { 2 l z n Z { ( A Y) f [(A - Y ) -~ 4V212n-2]1/2}-1, if
'
I >nil - 4 / 2 V ,
1 < nil - v1/2V.
(6.63)
In the first case the waves die away as an oscillation of continually decreasing
amplitude; in the second they die away aperiodically.
If we take w to be real, we find k is determined from it by
k = &i(Av)-1/2{& [ V z - i o ( A 1 / 2 + y U 2 ) 2 ] 1 ( 2 f [v' - io(A1P - ~ 1 / 2 ) 2 ] 1 / 2 } .
(5.64)
For small I and v(i, e. <( V 2 / o )this gives
k =i Vl
( h ) - ' / 2 [ 1- i W ( 2
+ Y)/2V'],
or
k =
-+ wV-'[l + iw(A + v ) / 2 V 2 ] .
(5.65)
The first of these possibilities correspond to shear waves damped heavily
by viscosity and resistivity. Their decay length, in which the amplitude
diminishes by a factor e-l, is (cf. Sec. IV.5)
(5.66)
do = (Av)1/2/V.
ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS
255
The second corresponds to AlfvCn waves. Their decay length is
(5.67)
do = 2V3/02(1
+ v).
The result (5.67) may be understood by considering the decay of an AlfvCn
wave in a coordinate system moving with it. In this frame, the current
system is approximately stationary and its time of decay must be of the
order of 12/A, where 1 is the spatial scale of the disturbance. If v exceeds A,
the decay time will be governed by viscosity and 12/v would be a better
estimate. I t is therefore plausible that, in the general case, the decay time
is of the order of 212/(A v). In the present case I = V / w ,so this decay time
is 2V2/w2(A v ) . Now, in this time, the wave has travelled a distance of
2V3/w2(A v) which must, therefore, be the decay length do of the wave
in a frame fixed in the fluid. Further discussion of the attenuation of shear
waves is given in Chs. VI. and X.
For the compressional waves governed by (5.61) we will consider damping
only in the special case of waves travelling in the x-direction. As in Sec. V.3,
the B wave separates from the compressional wave in this case, and the
sound wave is governed by
+
(5.68)
+
+
[$-( a o 2 + v f $ ) $ ] [ $ - A 5 ] .
=V2&9’.
Again consider solutions proportional to exp i ( k x - ot). Taking k to be
real, we find that, for lightly damped sound waves (Ak,v’k << a, or V ) , w is
determined from it by
1
Av’k2
+
w = (ao2 Y2)l/’k
+ . . . ] - i k 2 [v’ + (ao2AV2
+ V 2 )+... -
(5.69)
Thus, for example, the standing waves in a box of side 1 which decay most
slowly (k = n/l) have a decay time of
+
(5.70)
z = 12/n2[vf
AY2(ao2
+
V2)-1].
If we take o to be real, we find that, in the slightly damped case
(Aw,v’w << ao2 or V 2 ) ,k is determined from it by
+ V2)-1/2
f k = o(ao2
(5.71)
+
AV’d
iw2(a02
...I +
+ V2)- 312 [v’ + (ao2AV2
+ V2)
The decay length do of these waves is
(5.72)
do = (ao2+ V 2 ) 3 / 2 / W Z [ V ’
+ AV’2(ao2+
V2)-1].
+...I.
256
RAYMOND H I D E AND PAUL H. ROBERTS
5 . Shock Waves
Now we examine compressional waves of finite amplitude in a particularly
simple case in which the effect of non-linearity can easily be included. We
no longer suppose that the medium is perfect but ignore dissociation and
ionization processes, i.e. we consider the gas to be completely ionized before
the passage of the shock. We also ignore all relaxation effects associated
with the vibrational and rotational energy of the molecules of the gas. We
consider plane shocks which are propagated in a direction perpendicular to
the applied magnetic field.
Gas ahead of shock
Temperature TI
density
p,
pressure p,
Gas behind shock
Shock front
(inframein
which it is
a t rest)
Temperature Ta
density pr
pressure p2
FIG.4.
Suppose the shock wave moves with a velocity u, in the negative
x-direction and that the field into which it moves is in the z-direction and of
strength B,. Transform to a frame moving with the shock-wave (see Fig. 4).
In this frame
u = (u,O,O)
(5.73)
B = (O,O,B),
where
(5.74)
u+ul,
B - B,,
p-p,,
p - + p l , T - T I , as
p-+p,,
p+p,,
x+-
00.
We will suppose that
(5.75)
u-u,,
B - B,,
T-+ T,,
as x +
+
00
For strong shocks the transition between states (5.74) and (5.75) takes place
in a distance of the order of a few mean free paths, and (5.74) and (5.75)
hold closely over large ranges of x .
ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS
257
Since, in the moving frame, the system is stationary, curl E must vanish.
This implies, using (2.15), that E , is constant and therefore
aB-1 _
- U B - u1B1,
(5.76)
pc a x
since E y - u,B, as x + - 00.
As X +
W,
aB/ax- 0 . Hence
+
(5.77)
u1Bl = U ~ B , .
By the equation of continuity ( 2 . 1 ) , we have
(6.78)
a(pu)/ax= 0,
i.e.
(5.79)
Pl"1
=P2%
In a perfectly conducting fluid (see Sec. 111. 1) according to (5.76) and
(5.78)
(5.80)
Blp = Bl/pl =L constant.
Thus in these circumstances the total pressure may be written
(5.81)
It follows that the velocity of waves of infinitesimal amplitude is
(5.82)
V(aPlap)s= V(.,2
+ V2),
in agreement with the calculations of Sec. V.3. (The suffix S in (5.82) means
constant entropy.)
Returning to the general case of non-zero transport coefficients, by (5.77)
and (5.79), ~ ' ~ , u ~ , p ~ , p ~are
, Brelated
, , B , to the strength of the shock q by
(5.83)
By the momentum equation (2.2),we have
(5.84)
Integration, using (5.83), gives
(5.85)
B2.
258
RAYMOND HIDE AND PAUL H. ROBERTS
Thus,
(5.86)
Next consider the energy equation (A.18). We have
i.e.
Eliminate next dp/dx from the second term on the left-hand side of (5.88)
by using (5.84), whence
(5.89)
Integrating, using (5.83), we find
plul (u,
(5.90)
+ +;
q)- ufi2
P
*
Hence
(5.91)
Equations (5.83), (6.86) and (5:91) are the necessary generalizations of the
well known Rankine-Hugoniot relations. If we introduce the total pressure
(5.81) and the total internal energy per unit mass
1
U* = U f -B2,
(5.92)
2PP
(5.86) and (5.91) become
(5.93)
Pl
+PP?
= p2
+
P2"a2,
ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS
259
P
p2
1
u,*+ 2
+
2I u12 = u2*+ +
- u22.
P1
P2
2
(5.94)
which are formally identical to the corresponding Rankine-Hugoniot results.
Solving equations (5.83), (5.86) and (5.91), using the fact that for an
ideal gas
U = W T / ( y- l),
(5.95)
p / p =W T ,
and
a2 = Y P / P
(5.96)
we find
(5.97)
where V12 = B12/ppl, V22= B22/pp2. We notice that (5.97) may be
rewritten
Thus
(5.101)
u12
2 a12 + v12
with equality only in the case of infinitesimally weak shocks (11 = 1) i.e.
sound waves (cf. (5.45)). Thus the shock velocity is supersonic with respect
to the undisturbed gas. Similarly it is easily shown that M~ is subsonic with
respect to the gas behind the shock front. Ry (2.7), (5.83), (5.95) and (5.99),
it is easily shown that the change in entropy across the shock is
{
W
log [{(Y+ 1 ) -~ (Y- 1)) + V M Y - 1 ~ -7 1 ) 3 / 2 4
V " Y + 1) - T ( Y - 1)1
(Y- 1)
A S =(5.102)
260
For
RAYMOND HIDE AND PAUL H. ROBERTS
= 1, A S
is zero. For 7 > 1
(5.103)
>
>
0,
i.e. a(AS)/aq 0, with equality only if 7 = 1. It follows that A S
with equality only if 7 = 1.
The foregoing results relate only the conditions prevailing at
00 and
- 00, and they are independent of the viscosity V I P , thermal conductivity
K P C ~ ,and electrical conductivity u of the gas. There are three characteristic
lengths based on the properties of the undisturbed gas, namely, I = vl’/al,
the mean free path between collisions, I‘ = K1cp,/Wyla,, a thermal penetration
depth, and I“ = ( ~ ~ , u u ~an
) - ~electromagnetic
,
penetration depth, which
together with the shock strength, 17, determine the character of the transition
between the states on either side of the shock. As can be seen directly from
(5.76), (5.85) and (5.90), the structure of the shock depends on the relative
magnitude of these three lengths. In practice, I’ >> I for an ionized gas.
For a highly conducting gas (one for which al,uv, >> l ) , a strong shock
has much the same general character whether a field is present or not: the
velocity, density, pressure and field, change rapidly in a region whose width
is of the order of 1. However, I‘ must govern the spatial variations of the
temperature, and preceding this region is a layer (thickness -1’) in which
the variables together with the temperature change more gradually. The
field and density are closely proportional throughout.
For a weakly conducting gas (I” >> I ) , strong shocks have a different
character. The velocity, pressure, density and temperature change rapidly
in a region whose thickness is of the order of 1’. However, I” must govern
the spatial variations of field and preceding this region is a layer (thickness
l”), in which these variables, together with the field, change more gradually [38].
+
-
I N SYSTEMSOF FINITE
EXTENT
VI. A L F V ~ WAVES
N
1. Orders of Magnitude
In the case of an isolated body of fluid, dissipation of energy is associated
with the attenuation of AlfvCn waves which happen to be present and, as
we have seen in Sec, 17.4, this depends on the electrical conductivity and
viscosity of the fluid. However, when the body of fluid is in contact with
ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS
261
solid conductors, currents associated with the AlfvCn waves crossing the
fluid will generally penetrate the solid conductor and thus give rise to
increased energy dissipation. In examining this effect, we will take v for the
fluid to be zero, and we will suppose that the electromagnetic time constant
tl = L2/A, associated with the ohmic dissipation within the fluid, is large
compared to the time constants t 2= L2/A’, associated with ohmic dissipation within the conducting shell bounding the fluid, and t3= L / V , the time
taken by AlfvCn waves to cross the body. Here A’ denotes the magnetic
diffusivity of the conducting shell, and L a typical dimension of the fluid
body. We may easily derive rough estimates of the order of magnitude of
the effect of dissipation in the conducting shell in two extreme cases, depending on whether t 2>> t3,or t 2<< t3.
First let t 2>> t3,and consider standing waves of wavelength comparable
with L. We may regard these standing waves as the resultant of progressive
waves and their reflections in the boundary. Since z2 >> t3,the reflected
and incident waves are almost in phase. Thus, the amplitude of the standing
wave will be of the same order of magnitude throughout the fluid, including
regions near its boundary. Now let B be a typical field strength. At the
boundary 3?, the incident waves produce an almost sinusoidal magnetic
field of frequency w = 2izV/L and by the customary electrodynamic skindepth arguments, fields and currents will be attenuated exponentially with
depth in the conductor, the skin depth being ( A ’ / 2 7 ~ w )=
~ /(~A ’ L / ~ ~ C ~ V ) ~ / ~ .
The rate of power dissipation per unit area of B will therefore be
~ ~ , U - ~ B ~ ( A ’ joule/m2
V / L ) ~ ’sec.
~ But the total energy of the incident wave
per unit area of B is of order p-lB2L joule/m2. Thus, the time constant
of the decay process is approximately ( I U - ~ B ~ L ) / ~ ~ ~ - ~f
B~(A’V/
(L3/A’V)1/2
= (tzt3)1/2.
Comparing this with zl,we see that dissipation in
the solid conducting shell cannot be neglected if z2 << z12/t3.
If t2<< t3,the amplitude of the standing wave is very small near the
boundary of the fluid since the incident and reflected waves are nearly in
antiphase. In fact, if B denotes a typical field strength far from the boundary,
the field strength a t the boundary is approximately (LV/A’)1/2B.(see
Sect. VI. 2 below) Thus, while the total energy of the incident wave
lying above unit area of the conducting shell is still of order p-lB2L joule/m2,
the rate of power dissipation per unit areaof shell is (LV/A’).~ ? C ~ - ~ B ~ ( A ’ V / L ) ~ /
joule/m2 sec. Thus, the time constant of the process is approximately
(A’L/V3)1/2
= (t33/t2)1/2.
Comparing this with zl,we see that dissipation in
the solid conducting shell cannot be neglected if z2 >> ~ ~ ~ / t ~ ~ .
Observe that, in the present context, the decision as to whether the
conducting shell is a “good” or “bad” conductor depends on whether t2>> ts
or t 2<< z3,and not upon whether t2>> z1 or z2 << tl.
262
RAYMOND HIDE AND PAUL H. ROBERTS
2. Standing Waves in a Fluid Bounded b y Conducting Planes
We will consider a very simple model of the process discussed above.
We will assume that the fluid transmitting the AlfvCn waves is inviscid,
incompressible, and lies between two walls z = + a and z = - a. In the
regions z > a and z < - a, we suppose that there lies a homogeneous
conductor, and that the whole system is immersed in an initially uniform
field B, in the z-direction (see Fig. 5 ) . We will examine the properties of
standing AlfvCn waves in the fluid, i.e. we suppose that
B = (b,O,B,),
(6.1)
z--a
4
u = (u,O,O).
z- +a
7
I/
/1
fluid:
solid:
density-p
maqnetic diffurivity -A’
maqnetic diffurivity-h
solid:
maqnetic diffurivity -h’
FIG. 5 .
The associated electric field is
E = (O,E,O),
(6.2)
where
(6.3)
E={
(B,u
+ Aab/az) in the fluid,
A’ablaz
in the solid.
I t is readily verified from Sec. 17.4, that in the fluid
and that in the solid
ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS
263
Let us suppose that b and zl are proportional to en'. Then, by (6.3) to (6.5),
we have
for IzI < a, while by (6.3) and (6.6),
(6.8) b = C exp [(z
+a ) v m ) ] ,
+a)Vm)],
E = C v m exp [(z
for z < - a
(6.9) b = C exp [ ( a - z ) V m ] ,
for z >
E =-C
Vm exp [(a
z) V m ] ,
+ a. The continuity of b and E requires
(6.10)
Thus, the eigenvalues of n are determined by
(6.11)
If the walls have zero conductivity, (6.11) gives
(6.12)
2anl V
V(1
+ n W 2 ) (2s + l ) n i ,
=
s = 0,1,2..
..
If s > ( 2 ~ V / n i l ) - ~the
' ~ ,mode is damped aperiodically in a decay time of
(6.13)
-&-
="[
2 v2 1
+
V
W
+
}
]
742s
1)
#
while, if s < (2aV/nil)-1/2, the mode is damped periodically with a time
constant of decay of
z = 8a2/ln2(2s+ 1)2.
(6.14)
These results (for s = 0, 2a = I, v = 0 ) agree with (5.63).
If the walls have infinite conductivity, (6.11) requires
(6.15)
2anlV
V(1
+ n W 2 ) asni,
-
s =1,2..
..
264
RAYMOND HIDE AND PAUL H. ROBERTS
If s > 2aV/n1, the mode is damped aperiodically with a decay time of
(6.16)
otherwise (if s < 2aV/n1), the mode is damped periodically with a decay
time of
The limit in which the conductivity of the fluid is infinite and all dissipation takes place in the walls is particularly interesting. Then for both A' = 0
and A' = 0 0 , the waves are not damped at all and there must be some
intermediate value for which dissipation is a maximum and for which the
maximum decay time of the wave is least. For 1 = 0, (6.11)may be written as
coth (vzu/V)= - (Vz/~1')1/2
(6.18)
or as
tanh (na/V)= - (~1'/Vz)l/~.
(6.19)
For brevity we shall write
(6.20)
an = V(- x fiy) = a ( -
x fi y ) / t 3 ,
and
(6.21)
= (av/a')l/2= (22/t3)1'2,
(Note: tl= a2/A, t 2= $/A', t3= a/V.)
We will consider only that mode for which x is least.
For small x, (6.18) gives by successive approximation
(6.22)
= 0.56419
= 1.57080
x - 0.40528 x2 - 0.11063 x3 - 0.09886 x6 + . . .,
+ 0.56419 x + 0.11063 x3 - 0.06744 x4 - 0.09886 2' + . . . .
(6.23)
For large
x, (6.19) gives by successive approximation
265
ELEMENTARY PROBLEMS I N MAGNETO-HYDRODYNAMICS
x-' + 0.50000 xF2 - 1.26260x-3 - 1.68961x-' + . . . ,
= 1.25331
(6.24)
= 3.14159 - 1.25331
x-' - 1 . 2 6 2 6 0 ~ +
- ~2 . 0 9 4 4 0 ~ - ~ +
1.68961~-~+
...
(6.25)
TABLE2
x
x
Y
x
x
Y
0
0.003
0.01
0.03
0.1
0.3
0.6
1.0
1.3
1.4
1.5
1.55
1.6
0
0.00169
0.00562
0.01674
0.05436
0.15029
0.26082
0.34863
0.37801
0.38255
0.38500
0.38554
0.38566
7~12= 1.57080
1.57249
1.57644
1.58772
1.62723
1.74034
1.90853
2.11898
2.25771
2.29954
2.33908
2.35800
2.37635
1.65
1.7
1.75
1.8
1.9
2
3
4
7
10
30
100
300
0.38541
0.38481
0.38390
0.38271
0.37957
0.37559
0.3 1752
0.26242
0.16512
0.11906
0.04118
0.01248
0.004 17
0
2.39415
2.41140
2.4281 1
2.44428
2.47509
2.50394
2.70773
2.81812
2.95984
3.01522
3.09977
3.12906
3.13742
3.14159 = n
m
Returning to the original non-dimensionless units, we see that the time
constant of the standing waves is
(6.26)
266
RAYMOND HIDE AND PAUL H. ROBERTS
< <
< <
in agreement with the qualitative discussions of Sect. I. For 0 x
3,
a Newton’s method based on (6.18) converges rapidly. For 0.3 x
bo,
a Newton’s method based on (6.19)converges rapidly. The results, computed
on the “Pegasus” computer of the computing laboratory of the University
of Durham, are presented in the table above, and lox and y are plotted
in Fig. 6. I t will be seen that x has a sharp maximum of approximately
0.38567 near x = 1.5916. Thus the longest free decay time t associated
with the system is never less than 2 . 5 9 3 ~ ~ .
4r
3-
1-
I-
0.01
0.I
10
.
1
0
I00
1000
x
FIG.6.
In the case of the Earth’s core, we may take.
I‘ I 4.5 x
m/s (based on an average poloidal field of 5 x lo-‘ weber/m2),
1 5 2.6 ma/s (based on u = 3 x lo5mholm,),
1‘ = 7.9 x lo3m2/s (based on u = lo2 mholm.),
a = 3.5 x 108m.
Thus x = 4.5 and so x = 0.25, and t I 9.8 years. Had we ignored dissipation
in the mantle, we would have found by (6.14) that t = 1.2 x lo6 years.
Note also that since x > 1, we must, in the present context, regard the
mantle of the Earth as being a good electrical conductor.
ELEMENTARY PROBLEMS I N MAGNETO-HYDRODYNAMICS
267
VII. GRAVITYWAVES: RAYLEIGH-TAYLOR
INSTABILITY
1. Introduction: Choice of Model
In general, the surface waves travelling along the interface between two
conducting fluids will bend the lines of force of any magnetic field present 151.
The reaction of the lines of force to this bending will affect the propagation
of the surface waves. In this section, we consider the simplest case of two
semi-infinite immiscible, incompressible fluids of densities p1 and p2 separated,
in the undisturbed equilibrium state, by an infinite plane horizontal interface.
We suppose the fluid of density p2 lies above the fluid of density pl. For
simplicity, we ignore viscosity and treat the fluids as perfect electrical
conductors.
In the absence of a field, there is a discontinuity of tangential fluid
velocity a t the interface between the fluids. For example, consider a wave
travelling in the (horizontal) x-direction. By continuity of fluid mass, the
fluid particles in the troughs are moving in the positive x-direction and
those on the crests in the negative x-direction. But the crests in the waves
of one fluid are the troughs in the waves of the other. Hence, there is a
discontinuity of velocity across the interface between them. Clearly, in the
presence of a field which has a non-zero component normal to the interface,
this discontinuity of velocity implies the existence of a discontinuity in the
tangential components of E,and this in turn implies the existence of a surface
current, across which the tangential components of R are discontinuous. But
this is impossible since it would imply a discontinuity of tangential stress,
giving rise to infinite acceleration. I t follows, therefore, that the perfect
fluid approximation is incompatible with gravity wave solutions of the
type that arise in the absence of a magnetic field. More precisely,
if o is the frequency of the wave and k its horizontal wave number, there
are no such solutions for which 1 and are negligible, compared to both V 2 / w
and u / k 2 . In a real fluid, there can be no discontinuities of 11 and B, but
these quantities change rapidly within a boundary layer separating the two
fluids. The relationship between the net change in B across this boundary
to the net change in u can only be found by studying the structure of the
boundary layer itself. This is found to depend, in an essential way, upon v/1.
We will not enter into a full discussion of this problem here because it is
treated in full elsewhere [41; see particulary 9 I11 A, B of this paper].
Instead we shall consider the simpler case [42] in which the prevailing
(uniform) field B, is everywhere tangential to the interface and in the
z-direction (say) and thus avoid the foregoing complications. We shall take
the upward vertical to be in the y direction, and initially let the density be a
general function of y .
I)
268
RAYMOND HIDE AND PAUL H. ROBERTS
2. Solution of Model Problem
The basic equations of the problem are (2.2), (3.8), (2.3), 2.13)
au
(7.1) -
at
1
+ (u - grad)u - (B grad)B = *
PP
aB
(7.2)
at + (u * grad)B - (B * grad)u = 0,
-
div B = 0,
(7.3)
DplDt = 0,
where 1, is a unit vector upwards and g is the acceleration due to gravity.
The steady state which satisfies these equations is
(7.4)
u = 0, B = Bo = constant,
5
p = Po = constant - g pay.
In the slightly perturbed state, write
(7.5)
B = Bo
+ b,
f~= $0
+ p’,
p = PO
+
~
‘
8
and neglect the squares and products of b,u,p’, and p‘ wherever they occur.
We will henceforth omit primes. We then find from (7.1) to (7.3)
(7.7)
(74
div b = 0,
where
and 5 is the displacement of the fluid particle from its equilibrium position: i.e.
(7.10)
u = ayat,
to first order. I t satisfies
(7.11)
d i v 5 = 0.
ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS
269
By the equation of continuity,
P = - EYaPolaY.
(7.12)
Thus, by (7.6), (7.7) and (7.11), we have
(7.13)
~
a% = - -grad
1
Bo2 a2c + -g aP0
6 +___
Eyly.
PPO az2
Po
at2
Po ay
Suppose now that pa is a step-function
(7.14)
Others of the physical variables will also have a step-function behaviour
across y = 0, but, by integrating (7.11) across an infinitesimal path crossing
y = 0, we see that
(7.15)
[EY112 =
0.
As in Ch. IV, this notation means
limit t y- limit Ey.
(7.16)
[tYl12 =
y++o
y+-0
Similarly, from (7.13)
(7.17)
[all2= f Y t Y [P112 = g M P 2 - P l ) .
Supposing that all quantities are proportional to exp i(Zx
we find, by (7.13) and (7.15),
+
I ~ Z- wt),
(7.18)
where
m22 =
+
f12),
Wm,< 0,
- (22 + n2),
Wm2> 0.
m12 = - (Z2
(7.19)
From the form of (7.18) and (7.19) it is evident that the amplitude of the
waves dies exponentially with distance from the interface, the scale length
of the attenuation being unaffected by the presence of a field. Condition
(7.17) requires
(7.20)
270
RAYMOND HIDE AND PAUL H. ROBERTS
where
+
(7.21)
VSZ= 2BOZ/P(fl Pz).
In the absence of gravity, (7.20) corresponds to A l f v h waves travelling
along the interface with a velocity V s appropriate to the mean density
(pl p z ) / 2 and with an amplitude which dies exponentially with distance
into each media. Gravity waves travelling in the x-direction are unaffected
by the field and it follows that the criterion for instability ( p z > PI) is
unaltered by the presence of a field. The phase velocity of gravity waves
travelling in the z-direction is increased by the field to
+
(7.22)
In astrophysical circumstances, the magnetic field may be associated
with a strong rotation Q. It is well known that, if 0 is parallel to g, it tends
to inhibit instability 1431. If S2 is perpendicular to g it promotes the instability of surface waves travelling in the direction of n and stabilizes
waves travelling perpendicular to Jz and g. In the case in which B, is
perpendicular to g and 51,it can be easily shown that, if 2QV > g, all waves
travelling in the direction of B, are stable no matter what their wavelength
and no matter what the difference in densities of the media may be: however,
waves in the S2 direction are always unstable. In the case in which B, is
parallel to S2 and the heavier fluid lies on top, waves in the direction perpendicular to SZ and g are stable provided their wave number is less that
Q 2 ( p z - pl)/g(pz pl), and it is likely that a sufficiently high viscosity or
resistivity would stabilize them completely. Waves travelling in a direction parallel to n are stable provided their wave number n exceeds
g(Pz - P l ) / ~ S 2 ( P Z Pl).
+
+
VIII. GRAVITATIONAL
INSTABILITY
: JEANS' CRITERION
A problem of astronomical interest is that of establishing the physical
conditions under which gravitational condensations of matter will arise in a
large mass of gas [44]. Jeans [45] considered this problem first and put
forward the criterion that the size L, of the condensation must exceed a
certain value L j usually called the Jeans' wavelength:
where a, is the local speed of sound and p,, is the local density of the gas.
Because of the importance of Coriolis forces and hydromagnetic forces in
ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS
27 1
cosmical physics, subsequent writers have extended Jeans’ analysis to
include their effect, and have shown, in fact, that Jeans’ criterion is independent of them [46]. Because the exact circumstances under which
Jeans’ method of analysis is acceptable are not immediately obvious in some
of the treatments which have been given, we shall enumerate these circumstances before deriving their results.
Imagine a large mass of gas of typical dimension Lo, in which the velocity
is uo, satisfying
Dug = - grad Po
ZT
+ po grad @,
where Po is the pressure and @j0 is the gravitational potential satisfying
V2@o = - 4nGpo.
(8.3)
In order to discuss the gravitational stability of this motion, we consider
small perturbations about the state characterized by po,uo,@o,~o.Let
@, Po pl,
the new values of these quantities be po pl, uo ul, Q0
respectively. By (8.2) and the equation of motion for (ug ul), we have,
to first order,
+
+
+
+
+
gradp, - pograd G1
(8.4)
Jeans effectively considered a system in free motion under gravity, i.e.
grad Po = 0, so that the right-hand side of (8.4) could be set equal to zero.
If to is the shortest time scale associated with the basic motion and tlthat
associated with the perturbation, when
(8.5)
we find
Thus (8.4) reduces to
(8.7)*
po au, = - grad 9,
at
+ po gradQ1.
Similarly, using (8.5), the equation of continuity (2.1) gives
aPl - - po div u,.
at
‘Writers who have followed Jeans’ approach have used (8.7) as a starting point.
272
RAYMOND HIDE AND PAUL H. ROBERTS
These, together with
and the appropriate thermodynamic relations, are the equations governing
the perturbation. Jeans simplified the problem still further by considering
one-dimensional isentropic disturbances in the z-direction, say, having the
harmonic form exp i(Kz - ol). He found that their frequency is
(8.12)
[ (EL)2]1/2,
-w_- &a, 1 2n
Ll
where L , = (2n/k)is their wavelength. For L,< L j , w is real; in this case,
the disturbances are propagated in the z directions with phase velocity
o L J Z n , which reduces to a, when L , <( L,. For L , > L,, o is imaginary
and the disturbance is aperiodic and increases exponentially with time;
after a time of t,, where
(8.13)
tl
=“?[($Y
a0
- 1/2
- 11
,
its amplitude has increased by a factor of e. Clearly tl is infinite when
L, = L,.
These results can be understood by the following rough argument.
Suppose a slight condensation occurs in the gas in a region 9i? of typical
dimension L,. Because of this condensation, any two halves of 92 attract
each other with a gravitational force of the order of
(8.15)
Thus, an increase in pressure of approximately
(8.16)
is required to act across the interface between them, in order to prevent
further condensation. If this exceeds the increase aO2pl in gas pressure
caused by the condensation, the region 9 will condense further, i.e. the
medium is gravitationally unstable if
ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS
273
(8.17)*
GL12PoPlln> UO2Pl.
This rough analysis makes it clear why Coriolis forces and a uniform
magnetic fields do not affect Jeans’ criterion. The magnetic field does not
influence and is not influenced by the motion of particles along the lines
of force. Thus, provided the length scale of the magnetic field is large
compared to L,, condensations of W along the lines of force are unaffected
by the field. Similarly, in a rotating system, condensations along the
lines of S2, the angular velocity, are not affected.
In rederiving Jeans’ criterion for a conducting gas in the presence of
a uniform magnetic field B,, we follow the notation of Sec. V.3, adding the
effects of self-gravitation. We omit suffixes 1 hereafter. We find that
(5.37), (5.39), (5.40) and (5.41) still hold. However,
replaces (5.38) and (neglecting displacement currents),
(8.19)
replaces (5.47). Thus the expressions (5.48) for the phase velocitias are
replaced by
(8.20)
where 0 is the angle between B, and the direction of propagation of the wave.
When these wave velocities are complex, the amplitude of the wave increases
without limit, i.e. the system is unstable for such waves. By (8.20), it is
seen that this happens when (8.17) is obeyed. This, therefore, remains the
criterion for instability, even in the presence of a field. In fact, for waves
travelling in the direction of the field, (8.19) separates into (cf. (5.43))
* Since the physical argument leading to (8.17) is independent of the detailed
model chosen, and is almost certainly correct, the conclusion, based on (8.13). is
acceptable. However, Jeans’ model is probably not the best approach, since there
are uncertainties in the value of t o that should be employed. The most serious
difficulty is that the value of t oimplied by (8.2) with grad Po = 0, never satisfies
(8.5) when tl is given by (8.13). These difficulties have not been discussed by
writers who start with (8.7), but have led other writers (cf. e. g. McCrea, [44; to
consider alternative models to avoid the present inconsistencies.
274
RAYMOND HIDE AND PAUL H. ROBERTS
As in the discussion of Sec. V . 3 , the vanishing of the second bracket is
uninteresting, and we find
(8.22)
(g-
a2
ao2-
a22
1
- 4nGpo B = 0,
showing that waves travelling in the direction of B, are unaffected by the
field. On the other hand, for waves travelling perpendicularly to the field,
the criterion for instability is
(8.23)
k(ao2
+ V2)ll2< (4nGpo)'12,
proving that the field stabilizes these waves.
IX. STEADYFLOW
BETWEEN PARALLEL
PLANES
In this section we consider steady laminar flow, in the x-direction (see
Fig. 7), of an incompressible conducting fluid along a rigid pipe of length c
and of rectangular cross-section 2a x 2b, in the special case when
C
-
I
FIG.7.
a << b and a << c and when the externally impressed magnetic field B,
is uniform S and directed parallel to the z-axis. Under these conditions,
except near the side walls a t y = b and at the entrance and the exit of
the pipe, the flow velocity, u, the induced electric field, E, current density j,
and magnetic field, b, will be independent of position coordinates x and y .
This simplifies the problem enormously.
As will be shown below, the parameter in terms of which the impressed
magnetic field has to be measured is the appropriate Hartmann number
(see (3.15))
(94
M = B,a(a/vp)1/2.
When M = 0 there are no hydromagnetic effects, so that E = j = b = 0.
The velocity is then given by
ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS
u = (U,,O,O) =
(%( $) .o,o)
1-
275
,
where - Po is the (constant) impressed pressure gradient along the pipe.
In the absence of any external field of force, such as gravity, having a
component in the z-direction, the pressure 9 is independent of z. The parabolic
velocity profile given by (9.2) satisfies the only two boundary conditions
arising in the absence of hydromagnetic effects, namely, that because there
can be no slipping of the fluid relative to the pipe walls,
(9.3)
u,(a) = ux(- a ) = 0 .
The mean velocity is
(9.4)
UoG-
2a
j
P,a
3pv
zC,dz=-.
-a
When M # 0, an induced current j = ( O , j , O ) flows and, because j ,
is a function of z only, the induced magnetic field b associated with j is in
b)
the x-direction (see (9.12)). Hence, the resulting body force j x (B,
(see (2.4)) has two components, - jybx in the z-direction and jyBo in the
x-direction. The first of these is offset by a pressure gradient in the z-direction which sets up elastic forces in the fixed rigid side walls of the pipe.
The second force, j,,Bo, cannot be offset by any hydrostatic pressure field
and in consequence it reacts on the flow field, causing a general retardation
of the motion and modifying the velocity profile.
+
In the non-hydromagnetic case, M = 0, having supposed that b >> a,
it is not necessary to be specific about the side walls because no boundary
conditions on the differential equation governing the flow field remote from
y = & 6 result from considerations of conditions near y = f b. It is
immaterial, for instance, whether the situation is regarded as the limiting
case of flow along a pipe of rectangular cross-section when alb tends to zero,
a remaining finite, or of that of flow along a pipe of annular cross-section
as the mean radius of curvature, (rl rz)/2, tends to infinity, the width
(r, - rl) of the annular remaining finite and equal to 2a. However, when
M # 0, although direct frictional effects of the side walls in y = f b can
still be ignored, because j is in the y direction, the manner in which the
current circuit is completed via the side walls and through conductors (if
any) external to the fluid has to be specified. Otherwise, there are insufficient
boundary conditions ta determine the mathematical problem uniquely.
+
If the side walls are not in electrical contact with one another outside
the fluid, the total current,
2 76
RAYMOND H I D E AND PAUL H. ROBERTS
(9.5)
I =(j,dz
-a
(ampere per unit length in the x-direction) must vanish. Hence, regions of
positive j , will have to join with regions of negative j , via regions in
the fluid near y = f b in which current flows parallel to the z-axis. Electric
charges present on the side walls at y = f b are associated with an electric
field E having a y-component only within the fluid.
Conductance /unit
N
a / b)
lenqfh
FIG.8.
A t the opposite extreme we have the case corresponding to perfect
electrical contact outside the fluid between the side walls in y = f b. Then
E , must vanish because otherwise I would be infinite. E , vanishes in the
annulus problem; otherwise the line integral of E around (say) a circle
parallel to both walls would not vanish, and according to Faraday’s law of
induction this is inconsistent with the supposition that the system is steady.
In general, if the side walls are connected externally via a conductor
having conductance N-l[aa/b] per unit length in the x-direction (see Fig. 8)
where N is a dimensionless parameter, by Ohm’s law applied to the external
circuit,
5
b
(9.6)
N-l(aa/b) E , d y
+ I = 0,
-b
(see (9.5)).
In deriving (9.6) use has been made of the fact that E , is independent
not only of x and y, but of z also (see equation (9.9b) below). Now make
further use of the fact that E , is independent of y and thus simplify (9.6) to
(9.7)
2aaE,
+ N I = 0.
ELEMENTARY PROBLEMS I N MAGNETO-HYDRODYNAMICS
277
The two extreme cases considered above correspond, respectively, to N + 00,
so that I = 0, E , # 0, and N = 0 so that E , = 0, I # 0.
Having discussed the boundary conditions, apply the equations of
Ch. 11. By (2.11)
by (2.12)
(9.9a,b)
and by (2.13)
(9.10)
dB* = o ;
whence
~
dz
B, = B,,
b, = 0.
Now combine (9.8) with (2.15) and make use of the fact that u, = (u,,O,O)
to find that
(9.11a, b, c)
- - - paE,;
dz
db,
- = p a [ E y - u, B,];
dz
0 =pu[E,
+ u,b,].
At this point, we introduce the assumption that i, = 0, whence, by (9.8),
b y is constant and this constant must vanish because by cannot be discontinuous at the side walls a t y = 3 b. Hence, by (9.8),
(9.12)
by (9.9), (9.11a) and (9.11~)
(9.13)
E = (O,E,,O),
where, by (9.9b) E , is independent of z, and by (9.10)
(9.14)
B = B,
+ b = (b,,O,B,).
Now combine (9.14) and (9.12) with the equation of motion, (2.2). The
z-component leads to
(9.15)
if gravitational effects are ignored (i.e. a@/az = 0, see (2.5)); this is the
hydrostatic pressure gradient which has to be offset by stresses in the side
278
RAYMOND HIDE AND PAUL H. ROBERTS
walls a t y = & b. The only other component of (2.2) of interest, the xcomponent, leads to
d2u,
1
db,
O = P o + p ~ ~ + - B
,u O d z
(9.16)
- t
where, as noted above, - Po is the imposed pressure gradient along the
pipe, being equal to the pressure drop between the two ends of the pipe
divided by the length c. Eliminate db,/dz between (9.11b) and (9.16), whence
(9.17)
0 = [Po
+ OBOE,] + pv d2uz
-- aBo2ux.
a22
'"n
0.6
0.4-
0.2 , M ,
10
0
20
,
,
30
FIG.9.
Because the term in square brackets is independent of z , this equation can
be integrated to give
(9.18)
where [ E z/a, and the no-slip boundary condition (9.3) has been introduced
to evaluate the constant of integration.
Now we must eliminate E , by making use of (9.7). First observe that by
(9.12) and (9.llb),
(9.19)
.
IY =
cosh MC
Po aE,Bo coshMC
6{(7 )
-
('- cosh M )}'
279
E L E M E N T A R Y P R O B L E M S I N MAGNETO-HYDRODYNAMICS
so that
Hence, (9.7) leads to
where U , is the value of U when M = 0 (see (9.4)). E , vanishes when N = 0 ;
otherwise, E , is always positive and opposes 11 x B,, as expected. The
variation of E , with M when N = 00 is illustrated by Fig. 9.
Now substitute for E , in (9.18) and find
+
cosh MC
2(N 1)
M ( M N tanh M )
(9.22)
+
In order to study the limiting forms of (9.22) first note that
cosh x = 1
+ -21 x 2
;
sinh x + x
X
(9.23)
1
cosh x = - e x
2
;
tanh x 1 1 - 2e-2x
;
1
sinh x +-ex
2
coth x A 1 $- 2e-2x
x>> 1.
.
When M is close to zero, (9.2) is a close approximation to zt, irrespective
of N . For the variation of u, with t at other values of M , see Fig. 10, which
illustrates two cases, corresponding to N = 0 and N = 0 0 . (As in Figs. 12
and 14, the profile on only one side of the plane of symnietry is given.)
In both cases, increasing M results in reduced u, everywhere, but the reduction is more pronounced near [ = 0 than elsewhere. This has the effect
of flattening the velocity profile. Evidently, these effects are much more
pronounced when N = 0 than when N = m, a result which is due to the
lesser restriction on the current flow in the former case than in the latter.
The average flow velocity is
(9.24)
u = uO M3 2 ( M +( NN t a ’)n h M ) [M - tanh M I ,
-
+
280
RAYMOND HIDE AND PAUL H. ROBERTS
(see (0.4)); the variation of CT with M for N = 0 and N = 00 is plotted
in Fig. 11.
Observe that, according to (9.21) and (9.24), when N = m
(9.25)
E , = B,U.
Case (b):N-O
FIG.10.
1.0
E L E M E N T A R Y PROBLEMS I N MAGNETO-HYDRODYNAMICS
FIG.11
FIG.12.
281
282
RAYMOND HIDE AND PAUL H . ROBERTS
The current density
(9.26)
-
I-
+
+
(1 1/N)
coshM5
(1/N tanh M / M ) cosh M
is plotted for a number of typical cases in Fig. 12. j , is always negative on
5 = 0, and is negative everywhere when N = 0. When N # 0, there are
regions near 5 = & 1 in which j , is positive; the higher the value of M the
thinner these regions become, and when M = oa, this return current in the
positive direction flows in a sheet of zero thickness at the wall.
FIG.13.
In Fig. 13 the variation of i,(O) with M is plotted for N = 0 and N = 00.
It is noteworthy that - j,(O) approaches the value Po/Boas M - 00 and
is 90% of this asymptotic value when M = 2.5 in the case N = 0, and when
M = 4.2 in the case N = 00. - Po/Bois just that value of j , required for
a static balance between the force j x B, and the impressed pressure gradient
along the channel.
The induced magnetic field b, can be found by integrating (9.26) with
respect to 5 (see (9.12)), giving
where A is a constant of integration.
further information.
To evaluate this constant requires
ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS
283
By a simple application of Ampere's circuital law,
(9.28)
bX(1) - b*(- 1 ) = PI,
(see (9.5) and (9.27)),and as I , given by
(9.29)
I =-
tanh M
BaaE,
~
tanh M
N
(see (9.7) and (9.21)),only vanishes when N = 00, b , ( l ) is not in general
equal to b x ( - 1 ) . Because bx must be continuous everywhere, and outside
Case (a): N-w
.20I
0
1.0
z/a
0.6
0.4
0
0.2
0
0.2
0.4
% Z I I I I l O0
z/a
1.0
3.6
0.6
FIG. 14.
the fluid, b , must be uniform, if bu and bl stand respectively for the uniform
values of b, outside the fluid in > 1 and [ < - 1 ,
(9.30)
bu = b x ( +
I),
61 = b x ( -
1))
whence, by (9.28),
(9.31)
b, - bi = pI
which shows that generally the system is not symmetrical in all respects
about the plane 2' = 0.
284
RAYMOND HIDE AND PAUL H. ROBERTS
The constant A depends on the properties of the external circuit, conductance per unit length N-l(ua/b). Let this conductor be made up of two
components in parallel, one in the upper space mainly a t >> 1 and the
0
5
1
0
15
20
30
asymptotic
1.0-
0
25
,
5
1
0
15
FIG.16.
20
25
M
value
,
30
ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS
286
other in the lower space mainly at 5 << - 1, having conductances aN-l(ua/b)
and (1 - a)N-’(ua/b) respectively. Now
(9.32)
b, = a p I ;
br = - (1 - a)pul;
and this together with (9.30) suffices to determine A in (9.27). The “annulus
problem” corresponds to a = 0, so that b, = 0, bz = - p1. Fig. 14
illustrates b, as a function of 5 for two cases, (a) N = 00 and (b) N = 0,
when a = 0.5. In both cases the profile is antisymmetrical about 5 = 0.
Although varying a does’nothing to the induced currents and adds only
a constant A to b,, this does not mean that a is unimportant. The magnetic
energy density depends on the square of the magnetic field strength, so that
the total energy in the system depends in general on the external circuits.
For this reason, one expects a to be of the greatest significance in nonsteady problems, one example of which would be the generation from a
state of rest (say) of the steady flow described here.
In conclusion, it is of some interest to evaluate the magnetic Reynolds
number (see (1.1)) defined as follows:
R = 2upaU.
(9.33)
On substituting for U from (9.24) we find that
(9.34)
R = 2upaUo
(N -k ’)
(M - tanhM)
\M2
( M + N tanh M )
(M
+ N tanh M )
The variation of R with M is plotted in Fig. 15. When N = 0 0 , R increases
indefinitely with M , in contrast to the behaviour when N = 0, when R
tends asymptotically to unity as M goes to infinity. This is related to the
difference in the distribution of current in the two cases (see Fig. 12). When
N = 0 the bulk of the current flow is in the main body of the fluid and j v
drops to zero at = & 1. On the other hand, when N = 0 0 , the return
current j , near 5 = & 1 flows in a layer of ever decreasing thickness as M
increases, and a magnetic Reynolds number based on this thickness tends
to infinity more slowly - logarithmically in fact - with M , and is
probably more significant physically than the one based on the distance 2a.
For references to the early work on flow between parallel planes, see [3].
Globe [47] has treated the problem of flow along an annular pipe in a radial
magnetic field and Shercliff [ 6 ] has considered the more difficult problem
of flow along a circular pipe in a uniform transverse magnetic field.
286
RAYMOND HIDE AND PAUL H. ROBERTS
X. FLOW
D U E TO AN OSCILLATING
PLANE:RAYLEIGH'S
PROBLEM
1. Definition of Problem: Dimensionless Parameters
In order to understand the effect of viscosity in modifying the motion
of a fluid in contact with vibrating solids, Stokes examined a particularly
simple case (see [48],p. 317). He supposed that an infinite plane located
at z = 0 executes harmonic vibration in a direction ( x , say) parallel to itself,
and that the (incompressible) fluid in contact with this plane at z = 0
occupies the whole of the region z > 0 and is a t rest at very large values of z.
Assuming that v is constant, and that no slip occurs between the fluid and
the vibrating surface, he showed that if the velocity of the vibrating plane is
(U, cos o t , 0, 0), where U,, and w , which are assumed to be constant, are,
respectively the velocity amplitude and angular frequency of the vibration,
the velocity at any point in the fluid, u = (u,,uy,u,) is given by
(10.1)
u, = U,exp
[ ($-3
[
($7
cos wt - z -
-z
,
uy = 0 ,
u, = 0.
According to this expression the velocity amplitude falls off exponentially
with distance from the plate, having dropped to e-lU, = 0.3679 U , at
a distance z = A where
A
(10.2)
(2v/~)l/~.
In addition to this variation of amplitude with z , there is also a variation
in phase, due to the inertia of the fluid. The wavelength associated with
this variation in phase is %A, at which distance from the plate the velocity
amplitude is eW2*U,= 0.0018 U,. The flow generated by the vibrating
plane can be regarded as a heavily damped plane shear wave, the coupling
between different layers being due to viscous friction. The tangential force
per unit area at any level can be obtained from the viscous stress tensor
(10.3)
This gives
p , =d
PVUO exp( (10.4)
5)[-
cos(ot -
$)+ sin (cot -
;)I
so that the tangential force per unit area acting on the plate is
(10.6)
pxz(O,t) = (pvU,/d)(sin o t - cos wt).
,
ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS
287
The negative of the second term in this bracket is in phase with the motion
of the plane and corresponds to a dissipative force tending to stop the motion.
The other term is x12 out of phase and represents an effective increase in
the inertia of the vibrating body due to the presence of the fluid.
The rate a t which internal stresses do work at any surface parallel to the
vibrating plane, in general p,,u, pz,,uy P,,z~, (see Appendix A), reduces
to p,,u, in this problem. Denoting this quantity by Q , we find by equations
(10.1) and (10.4) that
+
(10.6)
Q = Q(z,t) = -
+
~
which oscillates at double the frequency of the vibration about a non-zero
average value
(10.7)
~ ( z=
) - __
PVUO2exp
24
(- g).
The power input per unit area required to maintain the motion, which
will be denoted by P , must equal - Q(0,t) so that
(10.8)
The direction of energy transfer between the vibrating plane and the fluid
depends on the sign of P , which alternates, changing twice each half cycle.
However, the average value of P , given by
(10.9)
P = pvUO2/2A
is essentially positive, corresponding to a net energy transfer from the
vibrating plane to the fluid, where it is dissipated by viscous friction.
Rayleigh [48]made use of Stokes' result for the drag on the plane (see
Eq. (10.5)) in his investigation of the effect of the boundary layer on the
propagation of sound in tubes, and it is of interest to ask how, in a conducting
fluid, Stokes' result is modified by the presence of a magnetic field. The
obvious extension of Stokes' problem is to the case of a conducting fluid
in the presence of an impressed uniform magnetic field of strength B, in
the z direction, the formal solution of which is presented below in Sec. X.2
and discussed in Sec. X.3 (see also [34,491). In Sec. X . 4 the corresponding
aperiodic solution is derived (see also [36, 50, 511).
The overall behaviour of the system now depends on three parameters
(10.10)
288
RAYMOND HIDE AND PAUL H. ROBERTS
a,@, and y measure, in suitable units, the magnetic field energy, the electrical
resistivity, and the velocity amplitude of the oscillating plane. From
their definitions we can relate a,@ and y to the more familiar dimensionless
parameters of Ch. I11 provided we base these parameters on the characteristic length L = ( v / o ) ’ / ~which
,
is of the order of the boundary layer
thickness in Stokes’ problem. Thus we find the Reynolds number (3.4)
(10.11)
R’ f UoL/v= y ,
the magnetic Reynolds number (3.3)
(10.12)
R E U0L/I = y/p,
the Hartmann number (3.15)
(10.13)
M =- B,L(a/pv)1/2=
and the Lundquist number (see (3.16))
(10.14)
K E B,La(p/p)’/2 = 0 1 1 / 2 / p .
The fluid motion is no longer of the form given by equation (10.1). I n
the presence of the magnetic field, u, consists of two parts characterized by
different attenuation and phase factors. The form of these parts suggests
that they should be termed “velocity” mode and “magnetic” mode. The
relative amplitudes of these modes, their associated attenuation and phase
factors, and the induced magnetic and electric field depend on a,p and y.
These quantities also depend on the electromagnetic boundary conditions,
which in turn are determined by the electrical properties of the region z
0
not occupied by the fluid. We restrict attention to the case when this region
is filled by an insulator. There are no additional difficulties associated with
0 is a conductor.
the other case when the region z
Explicit solutions can be found in a sufficient number of cases, corresponding to different limiting values of a and p, to cover most situations
of physical interest. The results demonstrate quantitatively a complicated
interplay between hydromagnetic and viscous effects (see Sec. X.3).
<
<
2. The Fovmal Solution
We divide B into a uniform field B, = (O,O,Bo) and an induced field
b = (b(z,t),O,O):
(10.15)
B=B,+h.
The velocity field has only an x-component u = (u(z,t),O,O).According to
the analysis of Sec. V.4, zc and b are governed, without approximation, by
the linear equations
289
ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS
(10.16)
(10.17)
The boundary conditions necessary to determine the unique solution
from these equations are the following. The fluid must be at rest at large
distances from the plane and must not slip relative to the plane a t the plane
itself; i.e.
(10.18)
u(z+ 0)
= 0,
u ( z = 0) = u, cos ot.
The boundary conditions on the magnetic field require that 1, vanish a t
infinity and, (since z < 0 is occupied by an insulator), at the plane itself; i.e.
b(z+ )..
(10.19)
= 0,
a q z = oyat = 0.
In the steady flow problem of Ch. IX, it was necessary to specify the nature
of the external electrical connection between the fluid at y = 00 and
y = - 0. However, in the case of unsteady motion, the total current I
flowing in the y-direction must vanish for it vanishes initially (by supposition) and, were dIldt non-zero, the induced e.m.f. LdIjdt (where L is the
inductance of the external circuit) divided by J?,Eydy
would be infinite.
Thus (10.19) gives
+
b(z+ 0 )
= b ( z = 0) = 0.
(10.20)
Another way of looking a t this boundary condition is in terms of the magnetic
energy outside the fluid due to the current flowing within the fluid. If I
is non-zero, the magnetic energy outside the fluid is infinite and, indeed,
in setting up the steady state, this energy would have to be supplied by the
source. In an oscillating problem, if the current were non-zero, the power
required to drive the plane would have to be infinite.
To clarify the question further, we will consider the following simple
example. An insulating cylinder of radius a immersed in a conducting fluid
oscillates in a direction parallel to its axis and in the presence of a magnetic
field which is radial in the region Y > a. I t is clear that the oscillating plane
problem is a limiting form of this oscillating cylinder problem as a+ 0.
To preserve this similarity, we take cylindrical polar coordinates (z,B,x),
z being radial and x along the axis of the cylinder. The 8-increasing direction
corresponds to the y-decreasing direction of the plane problem. The oscillation generates an x-component of u and b, and a y-component of j and E.
The continuity of E at the surface of the cylinder requires that
(10.21)
- nu2
ab(z = a)
at
ab(z = a)
290
RAYMOND HIDE AND PAUL H. ROBERTS
If we assume that all quantities vary as eiWt in time (see (10.22)), according
to (10,21), when w # 0, the value of b on the cylinder vanishes as a+ m .
However, if o = 0, we can only conclude that ahlaz vanishes on the cylinder.
More precisely, we may only assume the truth of (10.20) when wa2/11>> 1.
If the criterion is not satisfied, the total current in the y direction need not
be zero. Thus, care must be exercised in interpreting the results of idealized
problems of the kind discussed in this section.
When the moving surface is that of a solid conductor rather than that
of an insulator, currents will be induced in this conductor and the behaviour
of the system will be modified significantly. It is then necessary to specify
carefully the relative motion between the source of the main magnetic field
and the vibrating solid. In the literature there has been a certain lack of
clarity on this point and some errors have been made. The difficulty seems
related to the correct application of the law of induction which is often
incompletely dealt with in standard texts of electromagnetism. The subtleties of this point have been considered by a few writers (see Sec. 11.3).
Since (10.16) and (10.17) are linear, we seek solutions of the form
(10.22)
(zc,b) oc exp (iwt - qz/L),
(L= (V/O)’/~),
where W ( q )> 0, by (10.18) and (10.20). On substitution, we find
(10.23)
(i - q 2 ) ( i - pq2) = uq2.
When u = 0, the roots of this equation are
(10.24)
q1 corresponds to Stokes solution in which the shear wave is attenuated in
. corresponds to electromagnetic skin currents which
a distance ( 2 ~ / w ) ’ / ~q2
are attenuated in a distance ( 2 2 / ~ ) ’ ’(although
~,
since u = 0, they are not
excited in this case). (10.23) has two roots q1 and q2 which tend uniformly
to (10.21) as u-+ 0. We can therefore, without ambiguity, term them the
“velocity mode” and “magnetic mode”, respectively. If u # 0 both these
modes are excited. Let
(10.25)
By equation (10.16), g, is related to f, by
(10.26)
(i - P4l2k1 = - Y41flt
ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS
and a similar equation relates g, and f 2 .
and (10.20), we have
291
Also, by the conditions (10.18)
+ f z = 1,
(10.27)
fl
(10.28)
g1+ g2 = 0.
From (10.23) it follows that
(10.29)
= ip- 112.
QlQ2
Solving equations (10.26, 27, 28) for f i , f 2 , g, and g, and using (10.29) as
a means of simplifying the final results, we find
(10.30)
VF41 - 4 2
fl =
[1
(10.31)
g1 =
+ VPI (41 -
’
fz =
42)
VP42 - 41
[1
+ VPI
(42
- 41) ’
1
- g2 = -
V F 11 + VPI(41 - 4 2 )
*
From these results,
and
Note that, on the plane itself,
(10.34)
and
e”t
(10.35)
E,(z = 0) = (UoBo)
1
~
+ va’
Thus, in the limit 8 - 0, there is a surface current on the plane, while the
electric field there is given by
(10.36)
E = - Uo x Bo.
In the limit p- 0 0 , both current and electric field tend to zero.
of the force driving the plane is the average,
The mean rate of work
over a cycle, of
(10.37)
P=-vp[U$]
E = O
9
292
RAYMOND HIDE AND PAUL H. ROBERTS
and, by (10.25),
+ +Uoe-ioo”,
U ( Z = 0)= +UOeiog
(10.38)
so that
(10.40)
1:
P = pyu,2 -H(f1ql + f2q2) + periodic terms.
(
L
Hence, using equations (10.29) and (10.30) and averaging,
(10.41)
3. Discussion of Some Limiting Cuses
Having given the formal solution, we now present the results in a number
of limiting cases. We are interested in low, moderate and high conductivity
( p >> 1, p = 1, p << l), and these cases will be designated A, B, and C.
In each of these cases we must consider first the effect of a weak magnetic
field, and then the effect of a strong field. We put 5 = z / L .
The results for case A are summarized in Table 3. Observe that a/P
turns out to be the appropriate measure of the magnetic field. In the weak
field case, although the magnetic mode of u, is associated with a slow fall-off
of that of the
with z , its amplitude is only a small fraction,
velocity mode. In the presence of a strong field, the phase and amplitude
factors of the velocity mode now depend strongly on a and Bo, the amplitude
of this mode of u, being only slightly less than in the absence of a magnetiac
field. The magnetic mode of u x is weak, the amplitude at z = 0 being
,8-1/2. According to the form of q8, this mode corresponds to an AlfvCn
wave damped by electrical resistance (see Sec. V.4)
Now consider case B, that of moderate conductivity, corresponding to
p = 1. The results are summarized in Table 4. Now it is a that measures
the strength of the impressed magnetic field. Observe that in the presence
of a weak field, in contrast with cases A and C, q1 and q2 contain terms of
order a1/2.The amplitude factors of each mode of u, are the same, namely,
0.5. In the strong field case, the phase factors of each mode correspond
to an AlfvCn wave. The velocity mode is much more rapidly attenuated than
the magnetic mode, and at moderate distances from the plane, the magnetic
mode dominates. The form of the attenuation factor of this mode shows
-
ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS
TABLE3
Case A.
P
1 : (u
<< (pv)-I)
Low Conductivity
EY
UOBOeiot
293
294
RAYMOND HIDE AND PAUL H. ROBERTS
TABLE 4
Case B.
p =1
(I? = (pv)-l)
Moderate Conductivity
ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS
TABLE5
Case C.
P<< 1
(n>> ( p 4 - l )
High Conductivity
1
i
2a3/2+.1/2
EY
UoBoeiWf
+T
a ..
q=
1 (l
.)
295
296
RAYMOND HIDE AND PAUL H. ROBERTS
that viscosity and electrical resistivity play equal parts in dissipating the
energy of the wave.
Finally, we consider the third case, C, that of high conductivity (fi << 1).
The results are summarized in Table 5. When the magnetic field is weak,
the velocity field is only slightly modified by it. The magnetic mode of 21,
is weak, having a small amplitude a t z = 0 and a high attenuation factor,
of order /?-1’2. There is no term in ul/’in the expressions for q1 and q2.
In the presence of a strong field, the velocity mode of 21% is characterized
by a small amplitude at z = 0 and rapid attenuation, the motion consisting
almost entirely of an Alfven wave, which, from the expression for y2 can be
seen to be damped by viscosity.
In all cases, the mean power P required to maintain the vibration has
to be increased in the presence of a magnetic field; in the case of a strong
magnetic field P is proportional to B,.
The possibility of detecting the effect of a magnetic field on the propagation of sound in a tube of mercury has been considered [34]. As p >> 1
for mercury, u/P is the appropriate measure of Bo. As
50/w,frequencies
of vibration as low as a few cycles per second would be needed to produce
any marked effect. In the kilocycle region the sound speed would be reduced
by something of the order of one per cent. The situation should be rather
more favourable if liquid sodium were used, because then it would be possible
to work at much higher frequencies.
-
4. Ra yleigh’s Problem
Instead of forcing the insulating plane z = 0 to oscillate, we will now
simply suppose that a t time 1 = 0 it is jerked into uniform motion with
velocity U,, in the direction of the x-axis. In the absence of a magnetic field,
this problem was first considered by Rayleigh [48] and is sometimes named
after him. The simplest method of recovering his result is by the method
of Laplace transforms (equivalent to Heaviside’s operational method), We
replace a/at by p , (10.17) then gives
where the superimposed bar distinguishes transformed from untransformed
quantities. This must be solved in conjunction with the boundary conditions
(cf. (10.18))
(10.43)
n(z =)..
= 0,
n(z = 0)= uo/fi.
ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS
297
Thus
(10.44)
Inverting the Laplace transform (see, for example, [52], p. 354, No. 29)
(10.45)
where erfc x is the complement of the error function:
m
(10.46)
In the case in which the magnetic field is non-zero, the operational
solution may be easily derived by replacing io by p in the analysis of
Sec. X.2. For example, by (10.25) and (10.30), we have
where
The method we use to invert (10.47) is a simple extension of a method employed by Roberts [36] to solve a modified form of the present
problem.
Express u(t) in the form
(10.49)
where
Then (see, for example, [52], p. 354, No. 27)
298
RAYMOND HIDE A N D PAUL H . ROBERTS
(10.51)*
{ ( A W l - v1/2sZ)esle+ ( v l k l - j11~2sz)es~z},
= O,(P)
+ %(P)’ say,
where v1 and v2 are the parts of v involving exp slz and exp szz, respectively, and
where ,8 = ~ V ( A V ) ’ / ~/ ( v).
A I t follows that
Now, by elementary methods (see, for example, [52], p. 353, Nos. 7 , 8)
(10.54)
and also (see, for example, [52], p. 356, No. 53)
* The analytical advantages of this transformation are somewhat offset by the
apparent dimensional inconsistencies it introduces. The reader should therefore take
heed that, since O(p’/i) is related to a ( p ) by (10.51), the direct Laplace inversion of
ti($) by a Bromwich integral involving ePt must lead to a function v ( t ) in which 1 has
the dimension (time)’/%. (See 10.57 and 10.60 below.)
ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS
(10.56)
['P V (li(P" 2 P2)"1'
-
-
299
esla N
where I, and I , are modified Bessel functions of the first kind of orders
zero and unity, respectively. Thus
where
1
(10.58)
6 = V(A1/2 - y1/2) (t' - zv- '/2),
(10.59)
5 = p [(t- t ' ) 2 + (A- 112 + Y- l l Z ) Z ( t - t ' ) ] l / Z .
In an exactly similar fashion
if
t <z A - ' ~ ~ ,
where C is defined as before and 5' is given by
(10.61)
Having calculated vl(t) and v&) in this way, we can now calculate ~ ( twith
)
the help of (10.49):
300
RAYMOND HIDE A N D PAUL H. ROBERTS
(10.62)
#”-
l/Z
,A- 1/2
Solutions for short times (i.e. for t << zv-lI2 for ul,and t << z I . - ~ / ~for u2)
may be quickly derived from (10.57), (10.60) and (10.62) or, indeed, directly
from the operational form (10.47) itself. Solutions for large times mav be
computed from (l0.57), (10.60) and (10.62) by a steepest descent approximation. (For details of this procedure applied to a similar problem, see
Roberts [36]. Chang and Yen [all also discuss approximations in some
detail .)
XI. STEADYTWO-DIMENSIONAL
INERTIAL
FLOWIN THE PRESENCE
OF A
MAGNETJCFIELD
We have seen in an earlier section (see (3.26)) that in the presence
of a very strong uniform magnetic field B, = (0,0,B,) (say) slow (i.e. S = m )
steady motion of a perfectly conducting (i.e. R = m ) fluid will be twodimensional, in planes perpendicular to B,. To gain an insight into this
theorem, this chapter deals with the problem of nearly uniform flow in the
x direction of an incompressible inviscid fluid across a strong magnetic
field in the z direction.
For steady flow, by (2.2)
(11.1)
p ( u . V)u = - V p
+ j x B.
Assume now that
\i.here U , and B, are constants and 21, v , w,b,, by, b,, e x , e y , e, are small
quantities. If the motion is two-dimensional, having no dependence on y
(i.e. ajay = 0 ) , we can introduce scalar functions 4 and +!I of x and z , where
which automatically satisfy (2.3) and (2.13). By (2.12) as aB/at = 0
(11.4)
ELEMENTARY PROBLEMS I N MAGNETO-HYDRODYNAMICS
301
By (2.15), when q--+0 0 , to first order of smallness
E , =0;
E , = U,B,
+ uB, + Uob, = E , + uB,+ U,b,;
E, = 0.
(11.5)
By (11.3) and (2.11)
j, = V2$
(11.6)
so that to first order, (11.1) becomes
(11.7a, b)
pU,- a2+ = - az ax
ax
+ B,V2$;
- p U a24
o w=-aP
a2
+
where V 2 denotes the operator (a2/ax2 a2/az2)).
By (11.4), E , is independent of x and z and is, therefore, constant.
According to (11.5),this constant must be U,B,, the undisturbed value of E,,
and equal to E,, so that
ey = uB,
+ U,b, = 0,
whence
(11.8)
The undisturbed situation in which the electric field E has one component,
in the y direction, equal to U,B,, and no electric current flows, is only possible
when there are insulating surfaces at y = f 00 on which charges can be
set up.
Now eliminate p between (11.7a) and (11.7b) and thus find
(11.9)
and with the aid of this equation and (11.8),
(11.10)
If
(11.11)
S 2 = B 20 lPPUO2I
(see (3.11)), (11.10) may be re-written as
(11.12)
302
RAYMOND HIDE AND PAUL H. ROBERTS
Consider solutions which are harmonic in the x-direction, namely
= eikx F ( z ) .
(11.13)
F ( z ) satisfies
(11.14)
which has solutions
(11.15)
F ( z ) = A eka
+ B e - + Ceq, + De-
gr,
where
q =ik/S
(1 1.16)
and A , B , C and D are constants to be determined by the boundary conditions of the problem. According to (11.3) and (2.11)
4 = ( A & + Be-kz + Ceqz + De-qx)eskx,
$ = - ' t S ( p / p k 2 ) 1 / 2 [ k A e k-z k B c k P+ qCdl - qDe-qx]eEkx,
+ qCeqa - qDe-q*)&,
w = - ik(Aekz + Be-kz + Cd* + De-q*)etkx,
b, = - iS(,uup/k2)1/2[k2Aek'+ k2Be-kE + qzCeq* + q2De-qz]eEkx,
u = ( A h k *- Bke-k*
(11.17)
b, = - S(,u~p)l/~
[KAe" - k B e - kz
j,, = v2$ = - k2 (1
+ qCeqz - qDe-q*]e'kx,
+ S-2)(p/,u)1/2[Ceg*- De-gl]eZkX.
Now consider the specific problem of flow along a channel with insulating
walls. Three of these walls, in y = b and z = d are plane and the
fourth is slightly wavy, occupying
+
(11.18)
z = f cos k x ,
the mean position being z = 0. If d << b, the flow can be regarded as twodimensional, and if / << d deviations from uniform motion in the x direction
will be slight.
As a result of the electric currents in the fluid, the magnetic field in the
insulating regions outside the fluid will be distorted. Introduce magnetic
"stream" functions,
and 2$ in the upper and lower regions, z > d and
z < f cos k x , respectively. These functions satisfy
ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS
303
(see (11.6)) since j = 0 outside the fluid. Solutions of (11.19), which remain
finite a t z = w and vary harmonically in the x direction with wave
number k , have the form
lt,b = ae- kz elk, ,
(11.20)
2t,b = pekr erkx,
where a and p are constants. The induced magnetic field components in
regions 1 and 2 are, respectively,
The constants A , B , C, D , a and p, are determined by the boundary
conditions, which we must now consider.
Since the fluid is assumed inviscid, the only requirement of u on z = d
and z = f cos k x is that the normal component should vanish, Hence w = 0
on z = d, so that
+ Be- + Ceqd + De- = 0
and d(feikx)/dx= w / ( U 0+ u ) on z = f cos k x , which, to first order, leads
(11.22)
Aekd
kd
qd
to ik/eikx= u / U o on z = 0 ; hence
A
(11.23)
+ B + C + D = -fU,.
Assuming that A/v >> 1 (see (4.64)) j, cannot be infinite, even a t the boundaries
of the fluid, so that b has to be continuous. The continuity of the normal
component of b on z = d gives b,(d) = ,b,(d), and of the tangential component, b,(d) = ,b,(d) ; whence,
(11.24)
S(p/p)1/2{Akekd
- Bke-kd
+ Cqeqd - Dqe-Qd}= ikae-kd,
and
+
(11.25) iS(p/,u)1/2{Ak2ekd Bk2e-kd
+ Cq2eqd+ Dq2e-qd} = k2ae-kd.
On the other surface, z = / cos k x , to first order of smallness, the continuity
of b leads to b,(O) = ,b,(O); b,(O) = ,b,(O); whence
(11.26)
S(p/p)1'2{kA- kB
+ qC - qD} = ikp,
and
(11.27)
+
iS(p/p)'/2{K2A K'B + q2C
+ q2D} = - k 2 p .
Equations (11.22) and (11.27) suffice to determine A , B , C, D , wci'nd p.
Eliminate a between (11.24) and (11.25) and p between (11.26) and (1.1.27)
and thus find
(11.28)
+ q(q + k)Ceqd+ q(q - k)De-qd = 0
2k2Aekd
304
RAYMOND HIDE AND PAUL H. ROBERTS
and
2k2B
(11.29)
+ q(q - k)C + q(q 4-k)D = 0.
Now eliminate B between (11.23) and (11.29), and, making use of (11.16) find
(11.30)
2A + C(2
+ i / S + 1/S2)+ D(2 - Z/S + 1/S2)= - ZfU,.
Eq. (11.28) may be re-written as
(11.31)
+ (i/S - 1/S2)Cerkd/S- (i/ S -k l/Sz)De-ikd/S= 0.
2Aekd
Eliminate B between (11.22) and (11.23); whence
(11.32)
A(ekd- e-kd)
+ C(e'kd/S- e-kd) + D(e-rkd/S- e-kd) = fe-kd U,.
A t this stage, it is convenient to simplify the problem by considering a
deep system, for which d is very much greater than 2 4 k . Eq. (11.32) then
reduces to
Aekd + Ce'kd/s + De- W S = 0.
(11.33)
In order to illustrate the behaviour of the solution, we shall consider two
extreme cases, S << 1 and S >> 1. In the first case, it may be shown that to
first order in S 2 the velocity potential is
(11.34)
4 = fU,
-
(1 - 2S2)e-k"+ 2S2
-1
sin k(z - d)/S
cos kx
sin kd/S
(reverting to real quantities). When S = 0 corresponding to B, = 0, we
recover the well-known solution
(11.35)
4 = - f U,e- kE cos kx,
which falls off quite rapidly with z. In the presence of a weak magnetic
field (S << 1) there is an additional term which varies harmonically with z .
Although the amplitude of this term at z = 0 is very small, being S2 of the
other term, it predominates a t great distances in the case considered. This
result is of limited physical interest because, in a real fluid, such rapid
spatial variations (the wavelength in the z direction is 2nS/k) would be
rapidly attenuated by friction.
In the case S << 1
so that
( 11.36)
j y = V2$ = - 2{k2Uo(f)
I/:! cos k(z - d ) / S
sin kd/S
sin kx.
ELEMENTARY PROBLEMS I N MAGNETO-HYDRODYNAMICS
306
h'ow consider the case of a very strong magnetic field (S >> 1). Equations
(11.30),(11.31),(11.33)and (11.23)then lead to
'="'
sin k(z - d ) / S
cos k x
sin kdIS
(11.37)
to zeroth order in S-l.
If S >> kd, then
(11.38)
4 = - fUO(1- z / d ) cos k x
and as kd >, 1, 4 hardly varies at all between z = 0 and z = 2 n / k , in contrast
to the rapid exponential decay in the other case, S <( 1. This result is in
accord with Sec. 111.3. The flow field is completely altered in character as
a result of the presence of the magnetic field, the energy of the disturbance
arising at the wavy wall being transmitted to considerable distances from
the wall.
Corresponding to d given by (11.37)we have
* "):(
(11.39)
'''cos k ( z - d ) / S sin k x ,
sin kd/S
=
and
(11.40)
j
-
Y-
-k2fU
&
' ( p r
cos k ( z - d ) / S .
sin k x .
sinkd/S
ENERGY
EQUATION
APPENDIXA : THE HYDROMAGNETIC
We have seen (Ch. 11) that, except for the "degenerate" case of incompressible flow, it is necessary to supplement the hydrodynamic and
electromagnetic equations by thermodynamic relations. However, strictly
speaking a fluid in mass motion cannot be in thermodynamic equilibrium,
and the application of relations which are valid only in thermodynamic
equilibrium is, at first sight, questionable. Provided the mean intermolecular
distance d is small compared with any macroscopic length scale L characteristic of the hydrodynamical flow, it is easy in principle to define density
p, velocity of mass motion u, and internal energy per unit mass U as point
functions of position. For example, to determine the value of p for a particular point P , we simply draw a sphere 9
' centred at P whose radius Y is
large compared to d but small compared to L ; we divide the mass A contained in 9
'by the volume 4nr3/3= V of 9.The resulting value, p, will
306
RAYMOND H I D E A N D PAUL H . ROBERTS
be a meaningful definition of the density at P since it must be insensitive to
the value of Y chosen (provided d << r << L ) and will not be subject to significant statistical fluctuations. Similarly, to define u, we divide the total
momentum of the particles in Y by A, and to define U we divide the energy
of the particles in 9,
measured in a frame moving with velocity u, by A.
Having defined p, u and U , we may define a “temperature” T , and
“entropy per unit mass” S, and “enthalpy per unit mass” W , and a “pressure”
fi simply by applying the usual thermodynamic relations as though they
were exactly valid for 9.For example, we can define T by
(A4
T =(y - l)U/g.
The values T , S, W , and fi obtained in this way are those which would be
appropriate to Y if, moving in a frame with velocity u, the content of Y
is isolated and allowed to come to thermodynamic equilibrium (keeping
the volume constant). Provided L >> 1, the mean free path between
collisions, we can choose Y so that 1 << Y << L. Then the thermodynamic
state of Y before isolation is close to the equilibrium state arising after
isolation and may be regarded as a small statistical fluctuation from it. We
may, therefore, apply the results of classical fluctuation theory to Y and
assign an entropy per unit mass S‘ to Y by means of the familiar (see,
for example [53], p. 11)
A S ‘ = k log R,
(A4
where SZ is the number of complexions appropriate to the state of Y before
isolation. S’ may be regarded as the “true” entropy at P ; it will in general
differ from the entropy S defined from Y after isolation. However, for
given p and U , the entropy of Y is a maximum when it is in thermodynamic
equilibrium: i.e. S - S’ is positive and of order (1/L)2S:Similarly, after
isolation, the mechanical stress tensor fiyh
for Y assumes the form appropriate to thermodynamic equilibrium ; viz. fib,,. Before isolation the
mechanical stress tensor will in general differ from fibii, and the difference,
(A.3)
fimech
$7
- fib,, = - fiyp = viscous stress tensor
will be of order (l/L)fi.For Newtonian fluids, fi? will depend only on the
instantaneous values of the space derivatives of u a t P and, since it is
symmetrical, the only possible first order combinations are
Thus fi?
must be a linear combination of these:
E L E M E N T A R Y P R O B L E M S I N MAGNETO-HYDRODYNAMICS
307
Here, the constants pv and pC are termed “the viscosity” and “the bulk
viscosity”. (The latter is zero for a monatomic gas; cf. [54], 5 2.5.)
Throughout this paper, we use the quantities T , S, W , p etc. defined
after the isolation of 9.As a consequence, relations such as
TdS = dU
+ pd(:).
1
TdS = dW - - d p ,
(-4.6)
P
hold true despite me fact that irreversible changes are taking place. This
is a consequence of the fact that if, in following the hydrodynamical flow,
a fluid element undergoes an expansion dV, the work it does on the surrounding fluid is, in a dissipative medium, less than PdV. I t follows that TdS
must exceed the total energy supply. In applying relationships such as
(A.5) and (A.6) it is, of course, not necessary to consider that d p , dW,dS, etc.
refer to changes following the motion. For example, we may consider dW(dS)
to be the difference between W ( S )a t two adjacent points of the fluid a t one
particular time. (A.6) then determines the corresponding changes d p in 9,
giving
grad p = p grad W - pT grad S.
(A.7)
Similarly, applying (A.5) to the same fluid particle at two adjacent times,
We will use this relation in the discussion of the hydromagnetic energy
equation below.
The total energy of all forms contained in a volume V (fixed in space) is
I[pU
V
1
1
+ %pu2
- p@ +
B2+
2P
-
where grad Q, is the gravitational force prevailing. To avoid the difficulties
that arise when one tries to treat the general case of a self-gravitating medium,
we restrict attention to a gravitational field of external origin to the fluid
and which is constant in time but not necessarily in space. Now let us
consider the energy budget. Bodily transport is responsible for an outward
flux of internal, gravitational and kinetic energy over the surface S of V ,
amounting to
308
RAYMOND HIDE AND PAUL H. ROBERTS
(dS outwards from V ) . Thermal conduction removes heat energy at a rate
-
1
grad T * dS.
S
The (Poynting’s) electromagnetic energy efflux is
5
(E x H) * d S .
S
This term must, of course, be evaluated in such a way that flux of electromagnetic energy between the fluid and external circuits is not overlooked.
Finally the rate at which mechanical stresses do work on the surrounding
medium is
5
ui
p y h dSj.
S
Thus
at
V
pcpK(grad T ) * dS
(A.lO)
S
s
S
s
If we apply the divergence theorem to the integrals on the right-had side
of (A.lO) and shrink the volume V to a point, we find
(A.ll)
+ div (pcptc grad T)- ax,a (ui$Fh).
~
ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS
309
By (A.3), this may also be written
-aE_ - - div Q ,
(A.12)
at
where
(A.13)
E = pU
1
1
B2 + -&E2
+ 1 pu2 - p@ + 2
2E.l
is the total energy density, and
(A.14)
Qi=
u2
+ W - @ - wjfir-
ar + -1 (E x B)i
PC~K-
8%
#u
is the total energy flux vector.
From Maxwell's equations, it is readily verified that
1
= - div-(E
(A.15)
x B) - j . E .
#u
Also, since we are assuming a@/at = 0,
a
- (p@) = @-
at
aP
at = - @ div p~ = - div (p@u)+ pu grad @.
Thus ( A . l l ) may be written
a
P D t ( U + +uz - @) = j - E + pu .grad @ + div (pcpcgrad T ) - -(u$Yh).
axi
(A.16)
The left-hand side represents the rate of increase of kinetic and internal
energy in V , making allowances for that advected over the boundary. The
first term on the right-hand side represents the difference between the rate
at which the electromagnetic field does work on the material particles in V
and the rate at which the mass motions generate electromagnetic energy
in V , i.e. it is the rate at which electromagnetic energy is converted
irreversibly into heat. The second term represents the rate a t which gravitational forces do work. The remaining terms on the right-hand side can be
interpreted as before.
Equation (A.16) may be cast into an alternative form by using the
momentum equation
310
(A.17)
.
RAYMOND HIDE AND PAUL H . ROBERTS
DU~
p-=-Dt
apyh+ (6E+ j x B)i + p grad @.
axj
Multiplying this equation scalarly by u and subtracting the result from
(A.16) we find
(A.18)
p-
DU
= - e i , p y h - 6u E
Dt
+ j (E+ u x B) + div (pcpK grad T ) .
*
Thus, by (A.3), (A.8) and the equation of continuity (2.1),
pT-
DS
Dt
=e,jpp+ (j - 6u).(E+ u x B)+
+ T div (‘7 grad T).
PCPK (grad T ) 2
(A.19)
~
T
~
The first term on the right-hand side of (A.19) may be written
(A.20)
and is essentially positive (or zero, in inviscid fluid). On assuming Ohm’s law
(A.21)
j =6u
+ a(E + u x B),
the second term on the right-hand side of (A.19) may be written
(A.22)
1
- (j - 6
(I
~ )or ~ a(E + u x B)2,
and is essentially positive (or zero, in a fluid of infinite electrical conductivity).
Of the last two terms, the first is essentially positive (or zero, in a fluid of
vanishing thermal conductivity) while the second is zero if divided by T
and integrated over a region over whose boundary no heat flows. Thus,
integrating over such a region
(A.23)
1g
P
dV t 0,
in agreement with the second law of thermodynamics.
If we ignore displacement currents, the term in E2 in equations (A.9,
10, 11, 13, 15) are negligible, as are the terms in 6 in equations (A.17, 18,
19, 21, 22). A full discussion of this non-relativistic case is given by Hide [55]
for an incompressible fluid.
ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS
311
APPENDIXB : RELATIVISTIC
MAGNETO-HYDRODYNAMICS
In Section 111.3 and throughout most of this article, we have adopted
the “quasi-equilibrium’’ approsimation to the electrodynamic equations
in which displacement currents are neglected, and the space charge contribution 6E to the body force F is ignored in the equations of motion. In Sections
IV and V.2, the full electrodynamic equations have been considered whenever
the resulting complications did not give rise to obscurities. We have referred
to such analyses as being “relativistic”, although this is not strictly correct;
a full relativistic treatment must also allow for the possibility that the
velocities, both random and ordered, of the material particles are comparable
with c, the velocity of light. In this appendix we briefly consider this
possibility for an ideal, perfectly conducting fluid. Nevertheless, it should
be noticed that, as far as perturbation analyses are concerned, there is no
contradiction in supposing that the frequencies are so large that displacement
currents are not negligible, and yet that the macroscopic fluid velocity
is small compared with c. Under these circumstances, the analyses presented
in sections IV and V.2 are valid provided that the r.m.s. thermal velocities
of the fluid particle are small compared with c. Even for a hot fluid, similar
results hold if the density p is replaced by p
p / c 2 (see below).
The full electrodynamic equations are
+
Let 11s introduce the customary devise of representing space-time as
a Euclidean &space in which the coordinators are (x,y,z,ict). Define the
electromagnetic stress tensor Fli by the array
(B.5)
0
H,
- HY
- icD,
-H,
0
H,
- icD,
HY
-HH,
0
- icD,
icD,
icDy
icD,
0
and the current 4-vector J by
312
RAYMOND HIDE AND PAUL H . ROBERTS
Then equations (B.l) and (B.4) may be rewritten
j i = aFii/axi,
(B.7)
and equations (B.2) and (B.3) may be rewritten
(B.8)
Also, pFiiJi i s a four-vector closely related to the force density acting on
a fluid element, since
(J3.9)
PFajJj = (BE
(B.lO)
+ j X B)a,
(a = 1,2,3),
PFajJj = ij * E/c.
(In these equations, and in what follows, the summation convention has
been adopted. Also, Latin letters have been used to denote suffixes which
range over 1 to 4, and Greek letters for those which range over 1 to 3.)
For a perfectly conducting fluid
E=-uxB,
(B.ll)
and (R.10) may therefore be rewritten as
(B.12)
~FajJ=
j iU * (jx B ) / c .
The electromagnetic energy 4-tensor is (cf. e.g. [55])
(B.13)
T!?m.) = / i FOiFaj - 7 dijFdFab *
$7
[
l
l
The electromagnetic stress tensor introduced in Section I V is c2T$jm.';the
Poynting's vector E x H is cTti".'; the energy density in physical space
is - T f i m . ) .
The energy tensor
[67], Ch. 8, eq. 178)
(B.14)
of the material particles is of the form (cf.
Ti?'' = pUiUj - Sij/c2,
where Sii is the mechanical stress tensor, U, is the 4-velocity dx,/ds of the
fluid and p is the rest density. In an ideal fluid Sij takes the form (cf. [55],
Ch. 8, eq. 206)
(B.16)
Sij = - fi(d;j
+ UiUj),
ELEMENTARY PROBLEMS IN MAGNETO-HYDRODYNAMICS
313
where fi is the pressure in the frame locally at rest with respect to the fluid.
The total energy tensor for the fluid when an electromagnetic field is present
is
(B.16)
T$7. . T!e,m“.)
+ T!Yt),
- :7
and the equations governing its motion is
(B.17)
aTiilaxi = 0.
For the process of reducing this equation to more familiar forms, the reader
is referred to [57]. The final results of reducing (B.16)and (B.17)to physical
space are the equations
aP
1 aP
(B.19)
x B),,
at
where 3/ =: (1 - u2/c2)-”9. These are the required relativistic generalizations
of the equations of motion and continuity. Note that, if u is small compared
with c, they reduce to
(B.20)
j x B,
(B.21)
These are identical to Euler’s equation and the equation of continuity
plc2.
provided p is replaced in these equations by p
+
ACKNOWLEDGEMENTS
We wish to record our thanks to Dr. I. D. C. Gurney for discussions relating to
Appendix A and to Professor P. A. Sweet for discussion relating to Ch. V I I and for drawing
our attention to a paper by McCrea [44].
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Clarendon Press, 1962.
57. SYNGE,J. L., “Relativity: the Special Theory”, North-Holland Publishing Co.,
Amsterdam, 1956.
ADDENDUM
TO:
Hypersonic Flow over Slender Bodies
Associated with Power-Law Shocks
BY HAROLD MIRELS
Note added in proof
Reference [49] is an important recent contribution to the theory of
hypersonic flow associated with slender power law shocks. The method of
“inner and outer expansions” is used therein to obtain uniformly valid
solutions far downstream from the blunt nose of slender bodies. The inner
expansion describes the flow in the entropy layer, the outer expansion
describes the flow external to the entropy layer and the two expansions are
combined to give a uniformly valid description of the entire far downstream
flow, Explicit expressions were obtained for the asymptotic shape of the body,
as x -+ m, corresponding to a given one- or two-dimensional blast wave.
The asymptotic body shape, corresponding to a power law shock, can also
be found by the methods of the present paper. The procedure is indicated
herein, not only for blast waves, but for all power law shocks where the
entropy layer effect must be considered (ie., [ y / ( y l)]< jl 1). Dimensional variables are used. The limiting case M + 00 and (R‘)2<( 1 is considered. The method used is to find the difference between the body shape as
indicated by zero order similarity theory (3.26a) and by the continuity integral
(5.9). For
<< 1 and a power law shock, 0 = (R’/Ri’)z/p,
the latter becomes
+
<
Let Vb.0 represent the body location according to zero order similarity theory.
The difference between (3.26a) and (A) is then
where the higher order term in (A) has been omitted. For (R’)2small, the
main contribution to (B) comes from the integration in the vicinity of 0 M 0
317
318
HAROLD MIRELS
(i.e., the entropy layer) and f can be replaced by the zero order wall pressure
in each integral. Substituting z = r9-a(R')zinto the second integral,
integrating by parts and taking the limit ( R ' ) 2 + 0 then gives
/b,o
where T( ) is the gamma function. The error in (C) is of the order of the
second term or the right hand side of (A). Equation (C) gives the asymptotic
body shape, as x --* (M,associated with a power law shock in the range where
the entropy layer is important (i.e., y / ( y 1)< fi f 1 as discussed in
Section VI) and is the desired result. The difference between the actual
body location, rb/R, and the zero order similarity body location, r ] b , o , goes to
zero as (R')2---r 0.
For fi = 1, equation (C) becomes
+
This is in exact agreement with the results of [49] which were obtained by
the method of inner and outer expansions. Thus Y b x2I3Y and N x'lzv for
the one dimensional and two dimensional blast waves, respectively. The
constants of proportionality are found from (D). The asymptotic body shape
is not similar to the shock shape, except for y = 1, due to the factor y in the
denominator of the exponents of x .
Equation (C) can be used to find the asymptotic body shape associated
with any given power law shock. The pressure distribution on the body is
found from the zero order similarity solution. This equation can also be
used to find the asymptotic shock shape and surface pressure distribution
associated with a given power law body (ie., direct problem) in the manner
described in Section VI.1.
Some recent work [50-521 might also be briefly mentioned. Van Hise [50]
has systematically studied the shock shapes and surface pressures associated
with a series of long slender bodies of revolution having varying nose
bluntness. The flow field was found, using the method of characteristics, for
air (perfect gas assumption) and helium at Mach numbers from 5 to 40.
Nose fineness ratios (base diameter to length) were varied from 0.4 to 4. Flow
parameters obtained from zero order and first order blast wave theory were
used as a guide to obtain good correlations of the surface pressure and shock
N
HYPERSONIC FLOW OVER SLENDER BODIES
319
shape. The correlations of Van Hise [50], and similar correlations of
Lukasiewicz [€ill, appear to be the most extensive presently available.
Inger [52] has examined the similitude requirements for nonequilibrium
dissociated diatomic gas flows over blunt nosed slender bodies a t hypersonic
speeds. (Recall that Cheng [24] treated equilibrium real gas flows.) The free
stream was taken to be in an arbitrary state of dissociative nonequilibrium,
as may exist in high temperature hypersonic tunnels. The flow downstream
of the shock was also taken to be out of equilibrium and the conditions under
which different flows are similar was examined. Such studies are expected
to become increasingly more important, both for correlation of ground tests
with flight and for more accurate estimates of the disturbed flow field
generated by high speed-high altitude vehicles.
Additional References
49. YAKURA,
J. K., A theory of entropy layers and nose bluntness in hypersonic flow,
Amer. Rocket SOC., Preprint 1983-61, 1961.
50. VANHISE, V., Analytic study of induced pressure on long bodies of revolution with
varying nose bluntness a t hypersonic speeds, N A S A TR R-78, 1961.
51. LUKASIEWICZ,
J., Blast-hypersonic analogy-theory and application, Amer. Rocket
SOC., Preprint 2169-61, 1961.
52. INGER,G. R., Nonequilibrium hypersonic similitude in a dissociated diatomic gas.
Douglas Report SM-38972, 1961.
This Page Intentionally Left Blank
Author Index
Numbers in parenthese are reference numbers and are included to assist in locating references when the
author’s names are not mentioned in the text. Numbers in italic refer to the paKe on which the complete reference
is listed.
A
Abrikosova, I. I., 77(64, 65), 727
Adamski, V. B.. 3(21), 16(21), 19(21),
20(21). 21(21), 23, 24(21), 26(21), 53
Alekseyenko, Yu. N., 135(50), 209
Alexandrov, A. P., 77(62), 85(62), 127
AIfvBn, H.. 217(I), 225(1), 230, 245(1, 23).
313, 314
Averbach, B. L., 85(78), 728
B
BabuSka, I., 188, 206
Baker, B. R., 123, 124(105), 729
Banos, Jnr.,A., 245(32), 375
Baradell. D. L., 42(43), 49(43), 54
Barenblatt, G. I., 67(39, 40, 41), 69(40,
56, 57, 58, 59, 60, 61). 73(40), 74(60),
79(56, 57), 82157, 58), 92(58), 93(58),
96(56, 57, 84), 97(56, 57), l02(59),
109(58, 88), 110(88), 111(88), 113(40),
114(39, 40, 41). 115(39, 58, 95),
116(61), 118(84, 95), 123(84, 95).
1241.57, 95), 726, 128, 129
Barta, J.. 206
Batchelor, G. K.. 229(22). 374
Beedle, L. S., 194(3), 206
Benbow, J. J.. 65(9), 68(9), 86(54), 87,
119, 120, 125, 127
Berggren, R. G., 135(125), 136(125),
137(125), 138(125), 139(125). 214
Berman, I., 145(4), 206
Bertram, M. H., 29(37), 42(43), 49(43),
51(47), 54
Bilby, B. A,, 124, 129
Bitsadze, A. V., 116, 129
Blewitt. T. H., 136(46). 137(46), 209
Boardman, A. D., 267(41), 375
Bogdonoff, S. M., 61(48), 54
Bohnenblust, H. F., 149, 206
Borisov. K. A., 135(111), 213
Born, M. Kun Huang, 77(67), 128
Bowie, 0. L.. 65(22), 103, 726
32 1
Brinkman, J. A , , 134(6). 135(6), 206
Broberg, K. B., 124, 129
Brocher. E. F., 18, 54
Bruch, C. A., 138(7), 207
Bueckner, H. F., 66, 68(33, 50), 70, 74,
106, 726, 127
Bullough, R., 124, 729
Bupp. L. P., 135(125), 214
c
Campbell, J. D., 146, 207
Campus, F., 207
Cap, F., 143(10), 144(10), 207
Carrier, G. F., 230(24), 314, 319(24)
Carslaw, H. S., 297(52), 298(52), 376
Casaccio, A., 49(44), 54
Chandrasekhar, S., 271(46), 316
Chang, C. C., 287(51), 300, 316
Chapman, S., 224(14), 314
Chechulin, B. B., 207
Cheng. H. K., 2(8). 3, 8, 29(23), 43, 44(24),
48, 49(23), 51(23), 5 3 , 319(24)
Cherepanov, G. P., 96(84), 109(88),
110(88), l l l ( 8 8 ) , 115(95), 118(84, 95).
123(84. 95), 124(95), 128, 129
Chernyi, G. G., 3, 34, 41, 42(15, 16, 22).
53
Cohen, C. B., 30(38), 54
Cole, J. D., 33, 38, 54, 245(35), 375
Colonnetti, G., 149, 207
Cowling, T. G., 217(2, 3). 224(14), 228,
245(3), 285(3). 313, 374
Craemer, H., 196, 207
Craggs, I. W., 123, 124(103), 129
Creager, M. 0.. 51(46), 54
Cristesen, N., 201, 203, 207
Cullwick, E. G., 222(12), 374
D
Dang Dinh An, 123, 124(104), 729
Davidenkov, N. N., 65(12). 85(75), 119,
125. 128
Davison, B., 140(15), 207
322
AUTHOR INDEX
de Hoffman, F., 245(37), 375
Deresiewicz, H., 206(16), 207
Deryagin, B. V., 77(64, 65). 727
de Saint-Venant, B., 153, 154, 273
Dimes, G. J . , 134(17), 135(17), 207
Dolder, K., 218(7), 374
Drozdovskii, B. A., 69, 85(55), 727
Drucker, D. C., 200, 207
Druyanov, B. A., 171, 174, 175(20a, 20b).
176(20), 207
Dungey, J. W., 217(4), 245(4), 374
Duwez, P., 149, 206
Dzugutov, M. Ya., 141(128), 142(128), 274
E
Eichberger, L. C., 187, 274
Elliot, H . A., 65, 66(35), 96(16), 725, 726
Elsasser, W. M., 245(29), 375
F
Faris, F. E., 138(21), 207
Fehlbeck, D. K., 65(28), 88, 726
Feldman, S., 3, 4(26), 49(26), 53, 54
Feller, W. V., 29(37), 54
Fletcher, J. F., 135(125), 274
Foulkes, J., 200, 207
Frankland, I. M., 89, 728
Freeman, N. C., 18, 19(31), 39, 54
Freiberger, W., 200, 207
Frenkel, Ya. I., 84(5), 66, 67(5), 92, 120,
125
Freudenthal, A. PI., 143(25b), 206, 207,
208
Fricke, W., 271(46), 376
Fridman, Ya. B., 69, 85(55), 727, 147,
212, 274
Frieman, E. A,, 245(30), 375
Golian, T. C., 3(23), 29(23), 48(23), 49(23).
50(23), 5 3
Gonor, A. L., 33, 54
Greenspan, H. P., 230(24), 374, 319(24)
Griffith, A. A , . 63, 64(3), 85, 92, 125
Grigorian, S. S., 3(19), 25(19). 26(19), 5 3
Grigoryev, A. S., 192, 208
Grodzovskii, G. L., 3, 5 3
Guderley, G., 26(36), 54
Guindin, I. A., 77(63), 727
Gutowski, R., 202, 208
H
Hafele, W., 19(34), 21(34), 28(34), 54
Hall, J . G., 3(23), 29(23), 48(23), 49(23),
51(23), 53
Hayes, W. D., 2, 3(11), 8(11), 18(11),
21(11), 31(11), 49(11), 52, 53
Helfer, H. L., 245(39), 375
Henderson, A. Jr., 51(45, 47), 54
Hertzberg, A., 3(23), 29(23), 48(23),
49(23), 50(23), 5 3
Heyman, J., 193, 200, 208
Hide, R., 218(7), 228(21), 245(21, 34),
254(40), 270(43), 287(34), 296(34),
310, 312(55), 374, 375, 376
Hill, R., 146(34b), 206(34f), 208
Hockenburg, R. W., 138(7), 207
Hodge, P. G., Jr., 192, 193(36b),194(35c),
199, 200, 208, 273
Hopkins, H . G., 200, 208
Horne, M. R., 200, 208
Howarth, L., 219(11), 374
Hruban, K., 188, 208
Hu, L., 147, 208
Huber, M. T., 146(40a), 208
Hundy, B. B., 198(4la), 208
I
G
Galin, L. A,, 81(72), 116(72), 124, 728.
154, 208
Garber, R. I., 77(63), 727
Gatewood, B. E., 143(28), 208
Geiringer, H . , 151, 154, 208, 208
Germain, P.. 245(39), 375
Gigon, J., 135(59), 209
Gilman, J . J.. 65(11), 121, 124, 725
Glasstone, S., 218(8), 374
Glen, J . W., 134(30), 208
Globe, S., 285, 376
Goldstein, S., 226(18), 374
Golecki, J., 174, 271
Ilyushin, A. A., 138(44a), 142(44b),
143(43, 44b). 169(42b), 174(42a), 183,
208
Inger, G. R., 318(52), 319, 379
Inglis, C. E., 62(l), 64, 66(1), 725
Ivelev, D. D., 185, 209
Irwin, G. R., 65(23, 25, 26), 68(47, 48, 51).
73(45, 46, 47), 74, 84, 88, 89(48), 104,
105, 109, 111(47), 726, 727
J
Jaeger, J. C., 297(52), 298(52), 376
Jamiston, R. E., 136(46), 209
Janas, M., 193, 194, 273
323
AUTHOR INDEX
Jeans, J. H., 270, 316
Johansen, K. W., 197, 209
Johnston, Patrick J., 51(45), 54
Jongh, J. G . V., 77(66), 728
K
Kachanov, L. M., 101, 128
Kakutani, T., 287, 316
Kaliski, S., 202(48), 208, 209
Kashdan, Ia. M., 19(35), 54
Katchanov, L. M., 209
Khristianovitch, S. A., 67(38, 39), 74, 113,
115(39), 126
Kies, J. A., 65(25, 26). 68(48, 49), 89(48).
126. 127
Klimenkov, V. I., 135(50), 209
Kochina, N. N., 3(20), 26(20), 5 3
Kogler, F., 209
Koiter, W. T.. 109, 129
Konig, J . A., 199(107), 213
Konobeyevsky, S. T., 135(52), 209
Krupkowski, A., 147, 209
Kubota, T., 2(9), 3, 11(10), 12(10), 17,
26(10), 29(10), 35(10), 51(10), 5 3
Kukudzhanov, V. N., 203, 209
Kulsrud, R. M., 245(30), 315
Kultigin, V. S., l41(128), 142(128), 214
Kutaycev, V. I., 135(52), 209
Kuznetsov, V. D., 119, 129
Kuznetzov, A. I., 154, 155(55a), 167(55a,
55b), 168(55c), 169(55c), 171, 173
(55a). 176, 189(55b), 209
Kwiecinski, M., 197, 213
L
Lamb, H., 225(17), 250(250), 314
Landau, L. D., 81(71), 128
Larmor, J., 228, 314
Latter, R., 13, 17, 54
Lee, Ming-Hua, 187, 209
Lees, L., 2(9), 53
Lehnert, B., 224(16). 245(33), 374, 315
Leibfried, G., 77(68), 128
Lensky, V. S., 135(57b),138(57b),143(43),
177, 183, 201. 208, 209
Leonov, M. Ya., 78, 128
LBvy, M., 154, 209
Liftshitz, E. M., 77(64), 81(71), 127, 128
Lighthill, M. J., 307(54), 316
Lin, C. C., 230(26), 314
Lin, S. C., 2(7), 53
Lindley, B. C . , 218(9), 314
Lovberg, R. H., 218(8), 314
Lubinskaya. M. A., 141(128), 142(128),
214
Ludford, G. S. S., 245(35, 39), 287(50),315
Lukasiewicz, J., 318(51), 319, 319
Lundquist, S.. 217(5), 226(5), 245(5),
267(5), 374
Lust, R., 245(39), 315
M
McClintock, F. A., 124, 729
McCrea, W. H., 270(44), 313, 316
Mc Hugh, W. E., 138(7), 207
Marin, J.. 147, 208
Markuzon, I. A., 118, 119(98), 729
Marshall, W., 245(38), 261(38), 315
Massonet, C., 207
Masubuchi, K., 91. 128
Mayer, G., 135(59), 209
Mel’nikova, N. S., 3(20). 26(20), 5 3
Metsik, M. S., 65(10), 119, 125
Michell, J. H., 180, 209
Mikeladze, M. Sh., 200, 209
Mikhlin, S. G., 108, 128
Mirels, H., 3(18), 4(29), 11(17), 12(17),
14, 21, 26(17, l a ) , 27(18), 28(18),
% ( l a ) , 30(18), 32(18), 33(18), 34(17),
35(17), 36(17), 39, 53, 54
Mokhalov, S. D., 202, 209
Mott, N. F., 66, 121(36), 122(36), 124(36),
126
Mr6z, Z., 182, 197, 199, 270, 211
Murzewski, J., 174, 190, 191(76a, 76d),
19.5, 210, 211
Muskhelishvili, N. I., 62(2), 65, 70,
71(19), 75, 90, 110, 116(18), 117, 125,
7 29
N
Naghdi, P. M., 147, 210
NemBnyi, P. F., 177, 178(67), 210
Neuber, H., 89, 128
Niepostyn, D., 197, 210
Nikitin, L. V., 203, 209
Novozhilov, V. V., 158(69), 177(69), 210
Nowacki, W., 145(70), 152, 210
Nowinski, J., 148, 150(72), 210
0
Obreimov, I. V., 64, 65(8), 68, 119(8), 125
Ogibalov, P. M., 138(44a), 142(44b),
143(44b), 208
324
AUTHOR INDEX
Olszak, W., 132(74), 133(73e, 73f, 73h,
73i. 73j). 146(73f, 73g, 73h, 73i, 73j),
147(73f, 73g, 73h, 73i). 148(73k, 731,
82h, 82i), 149(73k, 731, 82h, SZj),
150(73k. 731, 82h. 82i), 152(79),
154(80c), 160(80a), 165(80a, 80b),
167(Sob), 170(82a, 82b), 172 (80a),
174, 177(80c),179(83c), 182, 183, 185,
187, 189, 190, 195173k, 731). 197, 198,
199(109), 203, 204(73a), 205(73k,
731, 73r, 73y), 206(73u, 73x. 83c,
83d), 210, 211, 212, 213
Onat, E. T., 187, 200, 212
Orowan, E. O., 65(24, 27, 28), 88, 126
Osiecki, J., 202(48), 203(88b), 208, 209,
212
Ostanenko, A . V., 141(128), 142(128), 214
P
Pallone, A. J,, 2(8), 53
Panasyuk, V. V., 78, 128
Panferov, V. M., 184, 212
Parker, E. R., 85(74), 128
Pashkov, P. O., 85(76). 128
Pelczynski, T., 144, 212
Perio, P., 135(59), 209
Perzyna, P., 152(79). 201(9la, g l b ) ,
203(91c), 211, 212
Poletzky, A. T., 147, 212
Popov, N. A., 3(21), 16(21), 19(21), 20(21),
21(21), 23, 24(21), 26(2l), 53
Popov, N . I., 147, 212
Potak, Ya. M., 85(77), 128
Prager. W., 150, 200, 208, 212, 213
Pravdyuk, N. F., 135(52). 209
Primak, W., 213
Probstein, R . F., 3(11), 8(11), 18(11),
21(11), 31(11). 49(11), 53
Proudman, J., 232(27). 314
Pui, Ming-Li, 187, 209
R
Radok, J . R . M., 124, 129
Rakhmatulin, Kh. A., 201, 213
Rayleigh, Lord, 286(48), 296, 316
Rebinder, P. A., 66(34), 126
Remnev, Yu. I., 141(100), 213
Reshotko, E., 30(38), 54
Resler, Jr., E. L., 230(26), 314
Resler, E. L., 218(10).. 314
Rivlin, R . S., 208(101), 213
Roberts, D. K., 112(101), 124(101), 729
Roberts, P. H., 228(21), 245(21, 34, 36).
267(41), 287(34, 36), 296(34), 297,
300, 314. 315
Roesler, F. C., 65(9), 68(9), 86(53), 87(53),
119, 120. 125, 127
Rogozinski, M., 189, 213
Romualdi, J. P., 68(52), 69, 94, 95(52),
lOO(52). 127
Rowley, J . C., 147, 210
Rushbrooke, G. S., 306(53), 316
Rychlewski, J., 154(80c), 160(80a), 165
(Boa, Sob), 167(80b), 172(80a), 177
271, 213
( ~ O C103a),
,
Rzhanitsyn, A. R., 67(37), 126
S
Sack, R. A., 65(20), 66(20), 97, 126
Sakurai, Akira, 2, 3, 12(5), 21, 22(5), 5 3
Sanders, P. H., 68(52), 69, 94, 95(52),
100(52), 127
Sawczuk, A., 170, 172(105d), 178(105d),
193, 194, 195, 197, 198, 199(107, 109).
211, 213
Scheidig. A., 209
Schliiter, A., 224(15), 314
Schumann, W., 200, 272
Sears, W. R., 218(10), 314
Sedov, L. I., 3(12), 13, 16, 17(12). 26(12),
45, 53, 86(79), 93(79), 128
Sergeyev, G. Ya., 135(111), 213
Seth, B. R., 188, 189, 213
Shapiro, G. S., 201, 213
Shercliff, J. A., 217(6), 245(39), 285(6),
314, 315
Sherman, D. I., 108, 110, 128, 129
Shield, R. T., 200, 207, 212
Shtaerman, I. Ya., 81(73), 116(73), 728
Singer, A. R. E., 198(41a), 208
Smekal, A , , 64, 125
Smith, A. MO., 30(39), 54
Smith, H. L., 68(48, 49), 89(48), 127
Sneddon, I. N., 65, 73(14, 15), 96(16), 125
Sobelov, N. D., 147, 214
Sobotka, Z., 205, 214
Sokolovski, V. V., 204, 214
Sokolowski, M., 145(116), 214
Spitzer, Jr., L., 223(13), 224(13), 226(13),
314
Stanyukovich, K. P., 3, 53
Steel, M. C., 187, 214
Stepien, A., 170, 213
Stewartson, K., 230(25), 232, 243(28),
374, 315
325
AUTHOR INDEX
Stroh. A. N., 96(83), 123, 124(102), 728,
129
Suits, J. C., 124, 729
Sukhatme, S. P., 124, 729
Sychev, V. V., 3, 39, 40, 43, 44(25),
45(25), 53
Synge, J. L., 312(57), 313(57), 376
Szymaliski, C., 152(79), 205, 277, 274
T
Talwar, S. P., 267(42), 375
Taylor, D. B., 146, 274
Taylor, G. I., 2, 21, 52, 238(27), 374
Tekinalp, B., 200, 207
Teller, E., 245(37), 375
Thornton, P. R . , 3(18), 26(18), 27(18),
28(18), 29(18). 30(18), 32(18), 33(18),
53
Titova, V. V., 135(111), 273
Tournarie, M., 135(59), 209
Trella, M., 4(28), 49(28), 50(28), 5 4
Truszkowski, W., 146(123), 274
Turski, S., 150(72), 270
U
Urbanowski, W., 148(82h, 82i), 149(82h,
82j). 150(82h, 82i). 170(82a, 82b),
185, 187, 189, 277, 272
Ustinov, Yu. A., 114, 729
Uzhik. G. V., 144(124), 274
V
Vaglio-Laurin, R., 4(28), 49(28), 50(28),54
van de Hulst, H . C., 245(31), 315
Van Dyke, M. D., 2, 4(2), 6, 5 2
Van Hise. V., 318(50), 319, 379
Vas, I. E., 51(48), 54
Vineyard, G. H.. 134(17). 135(17), 207
Vinograd, M. I., 141(128), 142(128), 274
von Hoerner, S., 19(33), 54
van Mises, R., 209
W
Wells, A. A ,, 65(29, 30, 31). 68, 122(101),
124(lOl), 726, 129
Westergaard, H. M., 65, 68(13), 73, 74,
81, 109, 110, 725, 727
Wigglesworth, L. A., 104, 106, 728
Williams, M. L., 65, 73, 125
Williams, W. E., 245(32), 375
Willmore, T. J., 65(21), 96, 110(21), 726
Wilson, J. C., 135(125), 136(125), 137
(125), 138(125), 139(125), 214
Winne, D. H., 65(32). 68(32), 107, 110(32),
7 26
Wolf, K., 64(7). 725
Woods, W. K.,135(125), 274
Wundt, B. M., 65(32), 68(32), 107,
llO(32). 112, 726, 729
Y
Y a Kura, T. K. 317(49), 318(49), 379
Yen, J. T., 287(51), 300, 376
Yoffe, E., 122, 729
Young, J., 187, 274
Yusuff, S., 112, 129
Z
Zahorski. S., 179(83c), 272
Zakharov, A. I., 134(127), 274
Zheltov. Yu. P.. 67(38, 42, 43), 74(38),
113, 114, 726, 727
Zhurkov, S. P., 77(62), 85(62), 127
Zhukov, A. I . , 19(35), 54
Zuyev, M. I., 141, 142(128). 274
Zyczkowski, M., 190, 203, 206, 212. 274
Subject Index
A
E
Alfvbn waves, 245 ff.
in systems of finite extent, 261 ff.
Ampere’s law, 222
Angle-of-attack effects (hypers. fl.), 31 ff.
Effective electric field, 223
Energy considerations in crack-boundary
conditions, 84 f.
Energy equation in MHD, 305 ff.
Entropy layer (hypers. fl.), 43 ff., 48
Equilibrium cracks, basic hypotheses,
76 ff.
boundaries of, 74
boundary conditions, 81
stress and strain at, 73 f.
structure of, 69 ff.
Equivalent steady and unsteady flows, 7
B
Baroclinic, 220
Barotropic, 220
“Beta” in MHD, 229
Biharmonic states of equilibrium, 185 ff.
Blast wave, 12, 16 ff.
Blunt-nose effect, 33 f.
Boundary conditions in MHD, 233 ff.
at solid insulating surface, 243 f.
electromagnetic, 234 ff.
mechanical, 238
Boundary-layer effects (hypers. fl.), 29 ff.
Brittle fracture, 82
experimental confirmation of, 86 ff.
F
Faraday’s law, 222
Finite Mach number, 46
Flow between parallel planes (MHD),
274 ff.
Flow due to oscillating plane in MHD,
286 ff.
limiting cases, 292 ff.
Flow fields associated with power-law
shocks, 23 ff.
“Frozen” field lines, 225
C
“C-classification”, 149
Cohesion, forces of, 76 ff.
modulus of, 80 f .
Continuity integral (hypers. fl,), 37 ff.
“Counter pressure”, 34
Cowling’s Theorem, 228
Cracks extending to body surface, 103 f f .
in rocks, 112 ff.
in thin plates, 89 f.
isolated, 90 ff.
near body boundaries, 107 ff.
plane axisymmetrical, 96 f.
stability of, 97 ff.
Crack systems, 108 ff.
G
Granular media (plast.), 204 f.
Gravitational instability in MHD, 270 ff.
H
Hartmann number, 229
Hypersonic slender-body theory, 4 ff.
Hypersonic slender-body approximation,
38 f., 41 ff.
I
Immobile-equilibrium cracks, defined 79
Infinite Mach number (hypers. fl.), 39 f.,
43 ff.
Integral methods (hypers. fl.), 36 ff., 47 ff.
Inverse methods (plast.), 177 ff.
Isotropic cylinder (plast.), 185 ff.
D
Disc with circular hole, 173
Dissipation effects in MHD, 253 ff.
“Dynamic beta” in MHD, 229
Dynamic problems (th. of cracks), 121 ff.
326
327
SUBJECT INDEX
K
Ktitter-Hencky integrals, 159
L
Lagrangian formulation (hypers. fl.), 14 ff.
Limit analysis (plast.), 191 ff.
Lundquist number, 230
M
Magnetohydrodynamics, basic equations
of, 219 ff.
Magnetic Reynolds number, 216, 224
Minimum weight design, 199 f.
Mobile-equilibrium cracks, defined 79
Momentum integral (hypers. fl.), 40 ff.
N
Newtonian flow, 18 f.
Non-homogeneity function, 162
Non-homogeneity, macroscopic, 132
plastic, 133 ff.
Non-homogeneous elastic-plastic body,
148 ff.
equilibrium in curvilinear coordinates,
156 ff.
plates, 190 ff.
strain-hardening, 145 f.
0
Ohm’s law in MHD, 222
One-dimensional structural elements
(plast.), 191 ff.
Orthotropic cylinder (plast.), 187 f.
P
Particular solutions (plast.), 167 ff.
Physically non-linear bodies (plast), 203 f.
Plane strain, 151 ff.
possibilities of solving, 155 f.
Plane waves in MHD, 244 ff.
Plastic equilibrium of wedge, 170 ff.
Plastic layer, 168
Plastic state, geometric conditions of, 161
Plates (plast.), 195 ff.
Power law shocks, 8 ff.
perturbed, 26 ff.
Proudman-Taylor analogue in MHD,
230 ff.
Punch pressed into body, 174 ff.
Q
Quasi-brittle fracture, theory of, 65
R
Rayleigh velocity, 123
Rayleigh’s problem in MHD, 296 ff.
Related bodies, 3
Relativistic magneto-hydrodynamics,
311 ff.
Rotating disc (plast.), 190
S
“Second order” blast-wave theory, 34
Sedov formulation (hypers. fl.), 16
Self-similar solutions, validity of, 43 ff.
Self-similarity (cracks), 87
“Sharp-blow’’ solution, 19 ff.
Shells (plast.), 197 ff.
Shock waves in MHD, 256 ff.
Similitude in hypersonic flow, 8
Slip-line geometry, 160 ff.
Small departures from steady state
(MHD), 241 ff.
Stable cracks, 68
Standing waves in MHD, 262 ff.
Stream-function formulation (hypers. fl.),
13 f.
Stress intensity factor, defined 57, 72
Stress-strain diagram changed by
neutrons, 138 f.
Stress trajectories, 163 ff.
T
Temperature gradients, influence on
mechanical properties, 141 ff.
Three-dimensional problems (plast.),
183 ff.
Torsion (plast.), 189 f.
Two-dimensional state (plast.),definedl52
Two-dimensional inertial flow in MHD,
300 ff.
w
Waves in a non-homogeneous medium
(plast.), 201 ff.
Wedging, 114 ff.
of a strip, 119 ff.
Y
Yield limit changed by neutrons, 136
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