DISCUSSION
is m u c h more complicated and leads to slowly convergent or even
divergent expressions for the stress resultants.
2 T h e circular support's action on the plate is equivalent to
a line load, and so the discontinuity in the elastic constants certainly make the deflection function singular at r = a. This is reflected in the fact that two different expressions are needed to describe the deflection function in the region 0 < r < 0 3 , and that
the functions wi and w 2 are n o t analytic continuations of each
other across r = a, even for the uniform plate.
3 T h e second equation following (S) is only intended to indicate one convenient w a y for obtaining Vhv*.
If x,y;
X\,yi; and
x-i,y-2 are coordinate systems in the plane with different origins,
and V 2 , V r , and V22 denote Laplace's operators in these coordinate systems, it is clear that
V 2 ( / + g) = V 2 / + V2!? = V i 2 / +
the p r o b l e m in the intensity of the local stress field. Because of
the coordinate system e m p l o y e d in the paper, this conclusion is
n o t readily observable.
After lengthy manipulations, it can be shown that the local
stress distribution in a cracked plate on an elastic foundation m a y
be expressed in polar f o r m as follows:
2(3
>_ i u z 3 +r ov
~Z
+ v) A ( 2 r ) ' / ' L 7 + v
5 +
3v
ICz
*
v)
h(2r)
7 + i>
2(3 +
7 +
ao =
v
2(3 +
4 T h a t the longer expression gives the shorter can be checked
using the following intermediate results:
4( log r +
V ! r log r cos 6 = 2 cos
5 +
7 +
0
(2)
0
COS
v
(1)
—
2.
v
2(3 +
3v
v)
Kiz
i>) h{2r)1'Kiz
v) /i.(2r)'/= L
_
.3d
sm — + sin
6'
j
36
1
v
6
sm — —
sm —
2
7 + v
2
IC2Z
36
1 -
h(2r)l/l
v
6
cos —
2
5 + 3v
0 (/•'••)
(3)
U p o n a detailed examination of the authors' results, it should
be pointed o u t t h a t the effect of elastic foundation does n o t alter
the qualitative character of the ordinary bending stresses near
the crack point in an unsupported plate. Specifically, the c i r c u m ferential stress variation in equations (60) through ( 6 2 ) of the
paper should be the same as that obtained for the plate w i t h o u t
an elastic foundation. T h e foundation modulus only enters into
1 By D . D . Aug, E. S. Folias, and M. L. Williams, published in the
1 9 6 3 , i s s u e o f t h e J O U R N A L OF A P P L I E D M E C H A N I C S , v o l .
where z is the thickness coordinate measured f r o m the middle
plane of the plate. T h e formal appearance of these equations is
indeed in agreement with that derived in an earlier paper b y
Williams [ l ] 4 for a plate w i t h o u t elastic support. In making
comparison with equations ( 6 0 ) through (62) of the paper, it
should be noted that the angle 6, measured f r o m the line of crack
extension, is the c o m p l e m e n t of the angle ip. I n addition, the real
constants K i and K 2 are related to the complex constants P 0 a n d
Qo, respectively, b y the following expressions
27^(3 + f)(l +
r„
A-i =
wrr,
lA ' •ft-2 =
—
i)Ghr-
2tt'- /; (3 + y ) ( l ^rp;
lA ' -
n
p»
i)Gh\-
vo
(4)
(5)
Here, l i t and K2 m a y be regarded as the crack-tip stress-intensity
factors [2] for s y m m e t r i c and skew-symmetric bending-stress
distributions, respectively.
F r o m these definitions of K ^ i = 1, 2), the authors' results for
the two limiting cases of interest m a y also be rearranged into the
f o r m previously stated, equations ( 4 ) and (5).
T h e y are as
follows:
G. C. SIH2 and D. E. SETZER.3 This paper is a valuable addition
to the existing literature on the analysis of stress distribution near
a crack point. I t is of importance in the discussion of the remaining strength of bodies containing cracks.
T h e authors are
to be complimented n o t only for having solved a difficult crack
problem, b u t also for indicating a possible relation between the
solution of an elastically supported flat plate and a plate with
initial curvature.
This information suggests possibilities for
extending some of the current fracture-mechanics theories to shelllike structures.
30,
TRANS. A S M E , vol. So, Series E , pp. 2 4 5 - 2 5 1 .
Associate Professor of Mechanics, Lehigh University, Bethlehem,
Mem. A S M E .
Instructor of Mechanics, Lehigh University, Bethlehem, Pa.
Journal of Applied Mechanics
5 + 3v
+
0(/'=)
6/r,
The Bending Stress in a Cracked Plate
on an Elastic Foundation 1
3
=
5 +
5 L e t us consider the case of the load being inside the circular
support. F o r a vanishing rigidity of the exterior in comparison to
the rigidity of the interior, the interior part deflects as if the exterior part were absent, and w, becomes the deflection function for
a simply supported plate. T h e exterior part, however, is f o r c e d
to deflect so as to maintain continuity in the radial slope at
r = a. M o r e o v e r , the radial bending m o m e n t at r = a required
to deflect the exterior part causes only negligible deflections in the
interior part because of the vastly different rigidities. T h u s ( 2 8 )
constitutes the solution for the exterior part with the following
b o u n d a r y conditions at r = a: (a) Zero deflection, (fc) radial
slope equal to t h a t of the simply supported interior plate (Reissner's solution). A similar interpretation can be given to (38).
2
Tre
2(3 +
V 2 /' 2 = 4.
Pa.
3v
2(3 +
1),
V- log r = 0,
June,
36
cos —
2
r
>r)'/= L
h(2r)
+
1
V-rg
In addition, it makes 110 difference whether Ai 2 and A 2 2 are expressed in their Cartesian or polar forms, and the equation in
question follows directly f r o m (5).
V 2 ?' 2 log r =
3d'
3 + 5p
. 36
sin — —
5 + 3v
r
KM Z
v)
e
TT
2
cos
Case i:
X —»• 0 b u t a ^ 0
K1 =
K,
=
-
Case vi: A ^ 0 b u t a •
Ki
IC2 =
=
-
-
(1 -
-
(1 (1 +
(1
i)Gh
V
(4a)
) { a ^
i)Gh
v0
v){air)1/-
(on)
\a
0
Gh
(2ir\)'/-
[~3 +
v'
Ll
v.
-
m0
(46)
iGh
(27rX)'/!
1 -
m
v
I n connection with the Griffith-Irwin theory of fracture, it
m a y be stated that the onset of rapid crack extension will correspond to reaching some critical values of the combination of K1
and Kn for a given material. Presumably, equations ( 4 ) and ( 5 )
m a y be e m p l o y e d directly in the fracture analysis of plates on
elastic foundations.
4
Numbers in brackets designate References at end of discussion.
Copyright © 1964 by ASME
JUNE
1 9 6 4 / 365
Downloaded 09 Dec 2011 to 128.189.213.133. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
DISCUSSION
As a specific example, the authors' p r o b l e m of a cracked
rectangular strip, subjected to bending m o m e n t M* and uniform
normal loading qo, m a y be used. Substituting equations ( 7 9 )
and ( 8 0 ) of the paper into equations (46) and (56), it is f o u n d
that
K, = cry*f(\y*/V'2)
IC2
=
"(3 + y ) ( l -
v)'
2TTX
(6)
Nre
=
ki sin 8 cos — + k2 cos — (3 cos 0 — 1 )
2(2r)•
+
0(r'/!)
C o m p a r i n g equations ( 8 ) through ( 1 0 ) with the authors' e q u a tions ( 1 1 7 ) through (119) of the report [3], i t i s observed that the}'
are indeed identical if
0
A'i —
N o t e that qo does n o t appear in the soluwhere <ry* =
tion of Ki. This result is as expected, since a uniform compression of the plate on an elastic foundation does n o t affect the
singular stress components near the crack point.
2nr'/;(l +
;
i)\2
27^(1 +
i)X2
ft
1 o
(ID
QO
(12)
6M*/h2.
Aside f r o m its importance in fracture theories, equation ( 6 )
m a y be conveniently used to c o m p u t e the stresses along the line
of crack extension. T o r instance, inserting equation ( 6 ) into
( 2 ) , it gives
+
0(x^-)
This result differs f r o m that of equation ( 8 3 ) obtained in the
paper.
I t follows then t h a t b y these considerations equation
( S o ) should be of the f o r m
1
<ry(x,0)
As a final point of interest, it is worthwhile noting t h a t the
simultaneous extension-bending stress field in a shallow spherical
shell with a radial crack follows immediately f r o m the observations made in the earlier part of this discussion. A t the same
time, the discussers would like to take this o p p o r t u n i t y to clarify
the interpretation of s o m e of the results given b y the authors in a
previous report [3], which is intimately related to the work in this
paper. In the report, it was n o t clear that the shell solution for
increasingly large curvature tends toward the solution for initially
flat plates.
In the introduction to the paper, the effect of initial curvature,
R, is shown to be qualitatively equivalent t o having an elastic
foundation with modulus k for a flat plate. Hence, the bending
part of the shell solution is identical to the supported-plate solution u p o n recognizing the equivalency relation k = Eh/R2
=
DX 4 . This implies that equations ( 1 ) through ( 3 ) m a y also represent the bending stresses in a shell, where Kt(i = 1, 2 ) are n o w
dependent u p o n the initial curvature, R.
Similar results for the m e m b r a n e part of the p r o b l e m m a y be
observed also. Restricting attention to a small region around
the crack tip, the Reissner [4] shallow-shell equations show that
b o t h VJ and F satisfy basically the same t y p e of differential equation.
M o r e o v e r , the stress solution given b y equations ( 1 )
through ( 3 ) reveals that for small values of r, V4u> = 0, and thus
F must also behave as the biharmonic function in this vicinity.
B y specifying the free crack surface conditions of vanishing
normal and tangential membrane stresses
b2F
o,
Nr0
a ( 1
=- -
diA
— j = 0,
= ±7r
(7)
the local membrane stresses m a y be written
r =
i
r
2 ( 2 ^
L
e
e o s
i"
( 3
-
as 5
cos 8) + h sin -
(3 cos 8 +
~ 2(2r)'/> [
kl
COS — V(1
9
cos 8) — k2
0(r'/!)
1)
(S)
+
0(r'/!)
(9)
F o r details, the reader may refer to a paper b y Williams [5],
366
/
J U N E
19 6 4
k2
—
TTT:
'/3
X
Similar to the bending solutions, ki and k2 m a y be regarded as the
crack-tip stress-intensity factors f o r the s y m m e t r i c and skewsymmetric portions of the membrane stress field, respectively.
While the local angular distribution of the extension-bending
stress field in a spherical shell appears to be identical with those
obtained b y superimposing the separate extensional and bending
stress of a flat plate, it must be emphasized that the stress-intensity factors k{(i = 1, 2) and K{(i = 1, 2) for the shell p r o b l e m
are interconnected through the loading.
M o r e precisely, if a
plate is initially curved, a stretching load will generally p r o d u c e
b o t h types of stress-intensity factors; namety, kt and Kit and
similarly a bending load will also yield both kt and K(, i.e., the
curvature causes stretching to introduce bending and vice versa.
I t is n o w clear that as the initial curvature of the shell becomes
increasingly large, the circumferential stress distribution near the
crack point remains unchanged, while the stress-intensity factors
kj and K i approach the separate solutions for the initially flat
plate.
I n closing, the discussers wish to congratulate the authors for
the excellent piece of analytical work. T h e results of the paper
are m o s t useful to those interested in the field of fracture m e chanics.
References
1
of
M . L. Williams, " T h e Bending Stress Distribution at the Base
a Stationary
Crack,"
J O U R N A L OF A P P L I E D
MECHANICS, vol.
28,
TRANS. A S M E , vol. 83, Series E , 1961, p p . 7 8 - 8 2 .
2 G. C. Sill, P. C. Paris, and F. Erdogan, " C r a c k - T i p StressIntensity Factors for Plane Extension and Plate Bending Problems,"
J O U R N A L OF A P P L I E D
Series E, 1962, pp.
MECHANICS,
vol.
29, TRANS.
ASME,
vol.
84,
306-312.
3 D. D . Aug, E. S. Folias, and M . L. Williams, " T h e Effect of
Initial Spherical Curvature on the Stresses Near a Crack Point,"
G A L C I T S M 62-4, California Institute of Technology.
4 E. Reissner, "Stress and Displacements of Shallow Spherical
Sliell-1," Journal of Mathematics and Physics, vol. 25, 1946, pp. 80-85.
5 M . L. Williams, "Oil the Stress Distribution at the Base of a
Stationary
Crack,"
JOURNAL
OF
APPLIED
MECHANICS,
vol.
24,
TRANS. A S M E , vol. 79, 1957, p p . 1 0 9 - 1 1 4 .
Authors' Closure
T h e authors wish to thank the discussers for their interest in
the paper and for calling attention to corrections for equation (82),
and hence ( 8 3 ) and ( 8 5 ) which follow. Incidentally, it is b e lieved that the discusser's equations (46), (56), and ( 6 ) should be
corrected b y the multiplication factor 2 and three fourths on the
right-hand side of the respective equations.
I n connection with the report ( R e f . [3] of the p a p e r ) to which
the discussers refer, consisting of the p r o b l e m of a shallow spherical shell containing a semi-infinite crack, there are, as in the case
of a cracked plate on an elastic foundation, three special cases of
interest:
( i ) a->- 0,
3 sin 8 cos —
2
(10)
X ^ 0
(ii)
X
0
(iii) a -*• 0 ,
X
0
F r o m our solution we can only
recover the first t w o limits.
Transactions of the ASME
Downloaded 09 Dec 2011 to 128.189.213.133. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm
DISCUSSION
T h i s is characteristic of mixed-boundary-value problems, which
are solved b y the Wiener-IIopf technique. In order t o obtain
the third case we must relax our b o u n d a r y conditions at infinity.
T h u s the m e t h o d used m a y , for some loadings, prevent one f r o m
recovering the flat-sheet behavior f r o m that of the curved sheet.
F o r these reasons, and as discussed orally during the presentation,
the problem of a shallow spherical shell containing a finite crack
has now been solved using a different mathematical a p p r o a c h
( E . S. Folias, " T h e Stresses in a Spherical Shell Containing a
C r a c k , " G A L C I T S M 63-20, N o v e m b e r , 1963, doctorate dissertation).
I n closing, the authors wish to again emphasize the distinction
between the Kirchhoff and Reissner t y p e bending b o u n d a r y c o n ditions [Refs. [3] and [7] of the p a p e r ) and urge caution in the
use of stress-in tensity factors which, in contrast to the extensional
solution, have been deduced f r o m a local ( b e n d i n g ) stress distribution incorporating incomplete b o u n d a r y conditions.
subject to given compressor and turbine efficiencies, a given turbine-inlet temperature and given compressor-inlet conditions.
T h i s m a x i m u m efficiency is compared with the maximum efficiency obtainable b y utilizing an ideal gas with a constant
specific heat as a working substance. I t is shown that, for low
turbine-inlet temperatures, the m a x i m u m thermal efficiency can
be increased b y more than a factor of 2 b y choosing a dissociating
gas with an o p t i m u m value of the dissociation energy. However,
it is, of course, f o u n d that this higher efficiency is achieved only
at the expense of emploj'ing considerably higher pressure ratios.
Although the thermal efficiency is n o t the onlj' parameter of importance in heat-engine design, the authors have certainly justified their conclusion that the dissociating working fluid merits
further study with a view toward application.
Force Singularities of Shallow
Cylindrical Shells 1
The Influence of the Equilibrium
Dissociation of a Diatomic Gas on
Brayton-Cycle Performance 1
R. SABERSKY.2 This paper is an extension of the authors' previous work in which they analyzed the effect of changes in specific
heat of the working substance on the efficiency of the B r a y t o n
cycle. T h e results of the present paper show that the effect of
chemical equilibrium on this efficiency m a y , under certain conditions, be significant. T h e effect comes about, presumably, b y
the dissociation causing changes in the heat capacity in such a
w a y that the B r a y t o n cycle approaches the Carnot. cycle more
closely.
T h e range in which significant improvements are indicated is
limited, however; t o v e r y low overall temperature ratios (T 3 /T\),
and to a restricted range of the initial temperatures. In order t o
estimate whether or not the indicated i m p r o v e m e n t in efficiency
is of practical significance, it will be necessary to carry o u t the
indicated calculations for specific substances and to c o m p u t e the
actual temperatures which would occur in the C3'de.
Furthermore, one m a y also point out that if the temperature
ratios in question are small, changes in c o m p o n e n t efficiency also
lead to large percentage changes in efficiency. Facts of this t y p e
will have to be taken into account in any optimization s t u d y ,
particularly as one could imagine that the c o m p o n e n t efficiency
might be affected b y the initial temperature T,, which has to be
selected within certain limits in order to take advantage of the
shifting chemical equilibrium.
F. A. WILLIAMS.3 This is the second of t w o papers concerning
the m a x i m u m theoretical thermal efficiency of the B r a y t o n cycle
for a working substance with a variable specific heat. T h e first
paper dealt with an ideal gas with partially excited vibrational
m o d e s ; this paper concerns Lighthill's model of an ideal dissociating d i a t o m i c gas. I t m a y be worth emphasizing that chemical
and t h e r m o d y n a m i c equilibrium is postulated throughout the
cycle in both analyses. T h e thermal efficiency is maximized,
1
By T . A. Jacobs and J. R. Lloyd, published in the June, 1963,
issue
of
the
JOTJBNAL
OF
APPLIED
MECHANICS,
vol.
30,
TRANS.
A S M E , v o l . 8 5 , Series E , p p . 2 8 S - 2 9 0 .
California Institute of Technology, Pasadena, Calif.
Division of Engineering and Applied Physics, Harvard
versity, Cambridge, Mass.
2
3
Journal of Applied Mechanics
Uni-
W. FLUGGE2 and D. A. CONRAD. 3 T h e author is to be congratulated for his success in finding a solution for the concentrated force in terms of integrals of products of cylindrical and
circular functions. W e attempted several years ago to find such a
solution, but were unsuccessful. T h e author, referring to our
paper 4 states that we concluded " t h a t unlike the thermal singularities, the force singularities are not expressible in terms of
moderately simple solutions of field equations." In actuality, we
developed a particular set of singular solutions to the shallow-shell
equations, interpreted t w o of them as thermal singularities, and
showed that the force singularity was not included among them.
I t was not implied or stated t h a t other sets of solutions, perhaps
containing the force singularities, did n o t exist. Our o n l y conclusion was that we did not find such a solution and were therefore forced t o use other methods in dealing with the concentrated
force.
In a later note, 5 we called attention to a convenient method for
the calculation of shallow shells in general and presented a solution for the concentrated force on the cylinder. I t would have
been of interest t o see comparable numerical results using the new
solution. It would appear that the series approach is more convenient for direct calculation, b u t that the author's solution m a y
p r o v i d e s o m e advantages for use as a Green's function.
Author's Closure
I thank Professor Fliigge and D r . Conrad for their kind c o m ments and remarks. I would like to point out that partly what
I had hoped to be emphasized in the paper was the organic relation
between the singular solutions, that is, the fact that once a
singular solution of a partial differential equation is known in
general, other singularities can be identified b y differentiation or
integration.
1
By A. Jahanshahi, published in the September, 1903, issue of the
JOURNAL
OF A P P L I E D
Series E , p p .
MECHANICS,
vol.
30, TRANS.
ASME,
vol.
S5,
342-346.
2 Professor of Engineering Mechanics, Stanford University, Stanford. Calif. Mem. A S M E .
3 Senior Staff Engineer, Hughes Aircraft Company, Los Angeles,
Calif. Assoc. Mem. A S M E .
4 W. Fltlgge and D . A. Conrad, "Thermal Singularities for Cylindrical Shells," Proceedings of the Third U. >S. National Congress of
Applied Mechanics, 1958, p. 321.
6 W . Fltlgge and D. A. Conrad, " A Note on the Calculation of
S h a l l o w S h e l l s , " JOURNAL
OF A P P L I E D
MECHANICS, vol.
26,
TRANS.
A S M E , vol. 81, Series E, 1959, p. 6S3.
JUNE
19 6 4 / 367
Downloaded 09 Dec 2011 to 128.189.213.133. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm