International Journal of Plasticity, Vol. 6, pp. 389-414. 1990
0749-6419/90 $3.00 + .00
Copyright © 1990 Pergamon Press pie
P,knted in the U.S.A.
FINITE ELEMENT ANALYSIS OF STATIC AND DYNAMIC
FRACTURE OF BRITTLE MICROCRACKING
SOLIDS.
P a r t 3: S t a t i o n a r y a n d R a p i d l y P r o p a g a t i n g
Cracks Under Dynamic Loading
Y. Tox and S.N. ATLURI
Georgia Institute of Technology
A b s t r a c t - T h e continuum constitutive modeling for rate-dependent fracture of brittle
microcracking solids, which was described in Part 1 of this paper, is applied to the finite element analysis of stationary and rapidly propagating macrocracks under dynamic loading. The
microcrack toughening effect is discussed, along with the influence on it of the size of the
microcracked process zone and the various parameters in the microcrack density evolution equation, through the observation of the behavior of the general crack-tip energy-release parameter,
the T* integral.
1. INTRODUCTION
The continuum constitutive modeling for rate-dependent fracture of brittle microcracking solids was presented in Part 1 of the present study (Tox & ATLURI [1990a]), which
is an extension of the self-consistent modeling employed by CHARALAMSIt)ESand
McMEEICIr~G [1987] tO the rate-dependent problem including viscoplasticity. This modeling has been applied to the finite element analysis of stationary and quasi-statically
growing cracks under static loads in Part 2 (Tox • ATLtrRX [1990b]), in which the
microcrack toughening effect has been discussed by observing the behavior of the general crack-tip energy-release parameter, the T* integral. In the present report, the same
constitutive modeling is applied to the analysis of stationary and rapidly propagating
cracks under dynamic loads, where the strain rate-effect plays a more important role
than in the quasi-static analysis. In the present case also, the microcrack toughening effect is discussed, with the influence on it of the size of the microcracked process zone
and the various coefficients in the microcrack density evolution equation, through the
observation of the behavior of the crack-tip energy-release parameter, the T ° integral
for propagating cracks.
In Section 2 some remarks are made on the finite element calculations. In Section 3,
a problem previously analyzed by C'm~r~ [1975], which deals with a rectangular plate with
a centrally located crack subjected to a tensile step loading, is solved as an example of
dynamic stationary crack problems. In the following Section 4 a problem previously analyzed by BROBERO[1960], which deals with crack propagation with a constant speed
in an infinite body under constant tension, is analyzed as an example of dynamic crackpropagation problems. Note that C'i-mr~ [1975] and BROaE&O[1960] deal with linear elastic solids, whereas in the present work unmicrocracking as well as microcracking type
nonlinear materials are treated. Both problems are standard problems in dynamic fracture mechanics, and the finite difference solution and an analytical solution are available, respectively, to check the presently calculated solutions for unmicrocracking
389
390
Y. To~ and S. N. ArLURI
materials. Emphasis is placed on the difference of the solutions for unmicrocracking and
microcracking materials. Especially the T* integral values are compared to discuss the
microcrack toughening effect. Section 5 contains concluding remarks.
II. F I N I T E ELEMENT CALCULATIONS
The constitutive modeling proposed in Part 1 has been implemented in the two-dimensional explicit elasto-viscoplastic finite element code contained in the finite element textbook (OWEN ~ HINTON [1980]), in which four-noded bilinear quadrilateral isoparametric
elements are used with a 2 x 2 Gaussian quadrature, and the explicit central difference
time integration scheme is employed with lumped mass matrices.
In the present analysis, the crack-tip energy-release parameter for dynamically propagating cracks, the T*-integral (ATLURI [1986]), is calculated at each loading step by
using the following domain integral expression, in order to discuss the microcrack toughening effect:
T* = fr~ [ ( W + T)nl - ti(Oui/dxj)]dF
= --fA-A~ [(W + T) (OS/Oxt) - aij (Oui/Oxt) (3S/3xi)ldA
- :A
[ o ( w + T)/Oxl
-
aij(OEij/OXl)
--
(1)
piii(Oui/Oxl)}SdA,
where
xi = a system of Cartesian coordinates such that xl is along the crack axis, x2
normal to the crack axis
an arbitrary small loop surrounding the crack tip
Z ~ = an area surrounded by F~
G = an arbitrary loop outside/'~ surrounding the crack tip
A = an area surrounded by Fy
t~= the components in the xi direction of the traction on the contour/'~
the components in the xi direction of displacements
Ui
/'/1 ~ the component along the xt direction of the unit outward normal to the
contour /'5
W = the total stress-working density per unit volume defined by
W=
fo
oo d~o
(2)
T = the kinetic energy per unit volume defined by
T = (1/2)oftifti
(3)
S = an arbitrary continuous function (see Nmtsrmov ~ ArLtr~ [1987]) such that
S= 1 on~
and
S=0onFy.
(4)
T* integral has been chosen as a crack-tip parameter since it is applicable to dynamic
as well as static problems associated with any kind of material nonlinearities including
unloading. Details of the way to calculate this integral can be referred to in the preceding
report (ToI & ATLURI [1988b]).
Element analysis of solid fracture
391
lit. A STATIONARY CRACK UNDER DYNAMIC LOADING
III. 1 Problem description
A dynamic problem previously defined by CHEN [1975] has been chosen as a numerical example o f dynamic stationary crack analysis. A rectangular plate in plane strain
condition with a centrally located crack as shown in Fig. 1 is considered. The outer edge
of the plate is loaded dynamically by a uniform tension P ( t ) with a Heaviside-function
time dependence. The finite element analysis is conducted both for microcracking and
nonmicrocracking plates, and the obtained solutions are compared with each other in
order to discuss the microcrack toughening effect.
The material constants are assumed as follows:
Young's modulus: E = 0.2 x 10 ]2 ( N / m 2) [initial]
Poisson's ratio: ~ = 0.3 [initiall
Critical stress for microcrack initiation: ac -- 0.15 × 10 ~° ( N / m 2)
Microcracking rate with stress: A = 0.74 × 10 -9 (mE/N)
Saturated value of microcrack density: ~s = 0.37
Viscosity coefficient for microcrack density evolution: 71 = 0.1 x l0 -5 (l/sec).
p[t]
Y
X
4cm
0.48cm
~e
2cm
II
Pit]
P[t]
0 . 4 X 10 ° N//m2
L
0
TIME
Fig. 1. Dynamically loaded rectangular plate with a centrally located crack.
392
Y. Toz and S. N. ATLURI
The elastic properties are the same as the original values in CHEN [1975], and the material constants concerning microcracks have been assumed so as to cause a "large-scale"
microcracking near the macro-crack-tip.
III.2 Results of Jqnite element analysis
Figure 2 is the mesh subdivision for the right upper quarter of the plate shown in
Fig. 1, and is characterized by:
Total number of elements = 550
Total number o f nodes = 598
Total number of degrees o f freedom = 1153.
The time interval 0 - 14 t~sec is analyzed with 700 steps by using a constant time increment of 0.02/~sec which is based on the Courant's stability condition.
!
CRACK TIP
Fig. 2. Finite element mesh subdivision.
Element analysis of solid fracture
393
Figure 3 shows the evolution of zones o f microcracking near the stationary macrocrack-tip at various instants of time. [Note that a finite time elapses before the stresswaves reach the crack-tip from the loaded end.] Note that during the time t = 3.2/~sec
and t = 7 #sec, the microcrack zone progressively increases. The unloading near the
macro-crack-tip takes place beginning at t = 7 #sec and, consequently, the process zone
and the microcrack density distribution remain constant after that time. In this figure,
the microcrack density is plotted for each element using the maximum value for four
Gaussian points within each element.
Figure 4 shows the time-history of the equivalent stress at the Gaussian point closest
to the crack-tip, as computed for the material models with microcracking and without
microcracking, respectively. Beginning around t = 4 #sec, when microcracking with a
significant density (0.1 _< ~ _< 0.2) occurs, with an attendant moduli reduction, the equivalent stress of the chosen Gauss point for the microcracking case is significantly lower
than in the unmicrocracking case.
The distributions of the equivalent stress along the crack line (ahead of the crack-tip)
at t = 6 #sec, for the microcracking case as well as for the case when microcracking is
presumed to be absent, are shown in Fig. 5. While the stress in the unmicrocracked solid,
asymptotically close to the crack-tip, displays a (l/x/~) type behavior, such is not the
case for the microcracked solid. As also seen from the damage zones near the crack-tip
at t -- 6 #sec, presented in Fig. 3, none of the damage zone (at least within the accuracy
0000000000000000000000
0000000000000000000000
0000000000000000000000
0000,*0000000000**0000
0000,*0000000000,*0000
0000000000000000000000
0000000000000000000000
0000000000000000000000
0000000000000000000000
000,*000000000000,*000
00,***0000000000,***00
00**22000000000022**00
00**22000000000022**00
00,***0000000000,***00
000,*000000000000,*000
0000000000000000000000
t=3.2[~]
0000000000000000000000
0000000000000000000000
0000000000000000000000
0000110000000000110000
0000110000000000110000
0000000000000000000000
0000000000000000000000
0000000000000000000000
0000000000000000000000
0.***000000000000.***0
0,,II,0000000000,
I I**0
0**1330000000000331**0
0**1330000000000331**0
0.*11.0000000000.1
0,***000000000000,***0
0000000000000000000000
t=7.O[.~]
t=4.0[~sec]
0000000000000000000000
0000000000000000000000
000.00000000000000.000
000.11000000000011.000
000.11000000000011.000
000.00000000000000.000
0000000000000000000000
0000000000000000000000
l~lto
0000000000000000000000
0,***000000000000,***0
0,,II,0000000000,II**0
0**1330000000000331**0
0**1330000000000331**0
0.*11.0000000000.11~0
0.***000000000000.***0
0000000000000000000000
t=5.O[u~]
*
0.0 < ~ < 0.1 (initiation)
1
0.1 & ~ < 0.2
2
0.2S ~ <0.3
3
0.3 •
•
~ <0.37
~ = 0.37 (satur4tion)
0000000000000000000000
O~O00000000000~It~t*O
o,.11.oooooooooo.11..o
o..13aoooooooooo~1..o
1~.0
0..133OOOOOOOOO0
0"**11.0000000000.11t~'~0
0.***000000000000.***0
0000000000000000000000
t : 11.0 C/~'c]
Fig. 3. Development of the zone of microcracks near the crack lip.
394
Y. Tol and S. N. A'fLURI
50
.....
40
UNMICROCe..a~(]~
MICROCRACKED
~
30
,"\ /
I)
J
I
/
/
I
\.,~
\
~" 2 0
10
_
f
,/
I/"
0
2
I
I
4
I
I
6
riME
I
I
8
I
.I
I
10
ti
12
I
14
[,~]
Fig. 4. Time history o f oy at (X, Y) = (0.25 cm, 0.01 cm).
of the finite element mesh in Fig. 2) has reached microcrack-saturation level, and thus
the entire damage zone is behaving as a nonlinear inelastic solid as per Fig. l(a) of Part l
o f this paper. This further explains the reasons for the nonexistence of a (I/~T) type
stress behavior near the crack-tip in the present microcracking case. The significant reduction in stress near the crack-tip in the microcracking case can be labeled as the phenomenon o f crack-shielding in the dynamic case. The K / f a c t o r that can be extracted
from the computed stress solution for the unmicrocracked case, shown in Fig. 5, is
6.3 x 107 N / m 3/z. The Kt factor estimated from the computed T* integral for the unmicrocracked case (see Fig. 6) is 8.5 x 107 N / m 3/2. This discrepancy is due to the error o f finite element d i s c r e t i z a t i o n - a finer mesh than shown in Fig. 2 is needed to
capture the exact stress distribution for a macrocrack in an unmicrocracked plate. For
the case when microcracking is ignored, the variation o f K t ( t ) can be estimated more
accurately, using the computed T* integral as defined in eqn (1), through the relation
KZt = ( T * E u ) / ( 1 - ~2). The thus estimated K t ( t ) for the hypothetical unmicrocracked
plate is shown in Fig. 6, w h e r e / ( t = K z / P ( t ) ( ~ r a ) ~/2 (a: half crack-length). These results are found to agree excellently with the finite-difference results of Ct-tEN [1975], thus
lending credibility to the presently computed T* values for both the hypothetical unmicrocracking case as well as for the actual microcracking case.
Figure 7 shows the calculation path in the (oe - ~) plane [ae is the equivalent stress,
and/~ the microcrack density] at the Gaussian point nearest to the crack-tip. Due to the
rate-effect on microcracking, the calculated microcrack density is always smaller than
in the case o f neglecting the rate-effect, for a given stress-level, except during unloading. Two unloadings can be seen in this figure, one of which corresponds to the local
Element analysis of solid fracture
395
40
\
\
~----
UNMJCROCRACKED
- -
MICIIOCRACXED
\
\
30
\
\\
\\
2O
3
m
0
10
l
I
I
I
0.25
0.30
0.35
0.40
x
[,,~]
Fig. 5. Equivalent stress distribution on Y = 0.01 cm.
2.. I
2.4 [
2.0
1.6
1.2
0.8
0.4
2
4
6
8
TIMe [~ sec]
Fig. 6. Time history of/~t.
10
12
14
396
Y. Toz and S. N. ATLUR!
0.5
------
WITHOUT
• WITH
0.4
RATE EFFECT
RATE EFFECT
/
Iz
es
0.3
i i / ' ~ . . . ~
v
ll/l~--""J
Gg
u
0.2
v_
:E
o.1
0
////
r"
15
20
25
STRESS [10eN/m 2]
EOUIVALENT
Fig. 7. Relation between microcrack density and equivalent stress at (X, Y) = (0.25 cm, 0.01 cm).
peak at around 4.3 #sec, and the other corresponds to the maximum peak at around
6.2 #sec, as observed in Fig. 6.
The energy-flux to the crack-tip, under the crack-tip stress intensity of magnitude
shown in Fig. 6, depends on the elastic material properties E and v. For purposes of the
ensuing arguments, we consider two sets of elastic moduli: (i) in the first set, we take
(E and v) to be those of the unmicrocracked virgin material labeled here as (Eu and
vu); and (ii) for the second set we take E and v to be those of the microcrack-saturated
material near the crack-tip, labeled here (Es and ~s). That no zone of saturated
microcrack density appears in Fig. 3 is only the result of an insufficiently fine finite element mesh. However the material stress-strain curve shown in Fig. 1 of Part 1 of this
paper indicates that microcrack saturation should occur at finite stress levels. Thus, in
reality, a small zone of material asymptotically close to the crack-tip may be assumed
to behave linear elastically with moduli £~s and ~s. The dynamic energy-release rate to
the crack-tip (for incipient growth of a
for these two sets of moduli
are labeled respectively as 7"* and T*, where:
stationarycrack)
(1
=
-
v ,2
,)K1
2
E,
(5)
and
( I - ~sZ)K/z
(6)
Element analysis of solid fracture
397
2.0
T'm
T'u
1.5
calculated for mlcrocracklng
malericd [1 - F l ] [K tip ] l / i s
: [1 - vu l ] Kl2/l u
T" s : [1 - PS i ] K i 2 / l l
X
,1.1
1.0
oMd
/
I,-,
_z
\k
\
I
L
0.5
I
0
I
2
I
4
6
8
10
12
I
14
TIME b,,K]
Fig. 8. Time histories of T* integral for the stationary crack.
The energy fluxes T~ and T~* are plotted as functions of time in Fig. 8. Also plotted in
Fig. 8 is the actually computed T* [labeled here as T~] using the definition o f eqn (1).
As explained in Part 2 o f this paper dealing with static loading, the fact that T* <
Ts* at all times during the present dynamic loading, as seen from Fig. 8, implies the
microcrack toughening effect in dynamic fracture.
To delineate the effect o f rate o f microcracking on microcrack-toughening in Fig. 9
are plotted the ratio of the T*(t) values computed directly using eqn (l) for various
values o f ~/, to the pseudo-energy-release rate 7"*. As is to be expected, a smaller value
o f ~ causes a greater toughening o f its peak value and vice versa.
In order to gauge the effect of the size of the microcracked process zone, another two
analyses have been carried out, assuming a smaller or a larger critical stress for
microcrack initiation in order to induce a larger or a smaller damaged zone near the
macro-crack-tip. The value o f the ratio o f the directed computed Tin* for three values
o f oc = 0.5 x 109 N / m 2, 1.0 × 109 N / m S and 1.5 x 109 N / m z to the pseudo-energyrelease rate Ts* are plotted as functions of time in Fig. 10.
IV. ANALYSISOF FAST DYNAMIC CRACK PROPAGATION
IV. I Problem description
A model problem o f linear elastodynamic crack-propagation as defined by BROBERG
[1960] has been chosen as a numerical example for the present dynamic crack propagation analysis in a microcracking solid. This problem is illustrated in Fig. 11, and consists o f an infinite body in equilibrium with a uniform uniaxial tension, oy = a, prior
Element analysis of solid fracture
399
I
I
!I
•
0.36
I
0.34
i! :/
0.32
C
E
L
0.30
tJ
0.28
oc : O.lO x lo '° EN/.~]
0¢ : 0.15 X101° rN/m2]
0¢ : 0 . 2 0 X 1 0 lo [N//m2]
0.26
I
0
I
2
I
I
4
I
I
I
6
I
8
I
I
I
10
I
12
Fig. 10. Time histories of Tg/Ts* for different ae values.
In the present analysis the following values have been chosen:
cL/Cs = 2.
The material constants assumed are as follows:
Young's modulus: E = 0.123 x 1012 ( N / m 2)
Poisson's ratio: v = 0.3333
Critical stress for microcrack initiation: ~c = 0.2 × 109 ( N / m z)
Microcracking rate with stress: A = 0.74 × 10 -s (mZ/N)
Saturated value o f microcrack density: ~s = 0.37
Viscous coefficient for microcrack density: ~ = 0.1 × 10 -5 (l/sec)
Density o f mass: p = 3110 (kg/m3).
14
400
Y. Tol and S. N. A'rtcRl
IIIIt
Illll
tlr
Fig. 11. Broberg's problem.
The crack propagation speed and the wave velocities are as follows:
CL = 0.770(cm/~zsec)
Cs = 0.385(cm/#sec)
(C/Cs) = 0.2, 0.4, and 0.6.
IV.2 Finite Element Results
A sufficiently large rectangular-shaped area is assumed for numerical analysis, in order
to avoid the effects of reflected waves during the analyzed time interval. Figure 12 shows
the mesh subdivision o f the upper right quarter o f the whole area, whose data are as
follows:
Total number o f elements = 1296
Total number of nodes = 1369
Total number o f degrees of freedom = 2737 (initial value).
The crack propagation has been simulated by gradually releasing the nodal restraining force when the crack tip reaches that node. There are some elaborate techniques to
deal with the dynamic crack propagation (ATLtrRx [1986]), however, the simplest technique has been used here, since the primary purpose o f the present simulation is to gain
Element analysis of solid fracture
401
Illllll
i:i?, ~
h,i!
I
I
t 1
i, i i ill I
iiI!iiiiji 1t I
I!il i :! '
hi !!t!1
Iil!llilHillllllll
I
IfltlIHIIHIIfllIHUl I
::::::::::::::::::::::::::
. . . .
;;;;]11
t
i
I
Fig. 12. Finite element m e s h subdivision (one quarter).
a physical understanding of the phenomena including the microcrack toughening. The
entire propagation length of 3.08 cm (20 element length) has been analyzed using the
following numbers of time steps and time increments: 400 steps with At = 0.1 #sec for
C/Cs = 0.2; 200 steps with At = 0.1 /~sec for C/Cs = 0.4; and 200 steps with At = 0.667
/~sec for C/Cs = 0.6.
The analytical solution for the dynamic stress-intensity factor at the crack-tip, which
propagates with a constant velocity c starting from a zero initial crack length is given
by (BROBERG [1960]):
Kt = K(c)l(~,
(8)
where K(c) is the so-called velocity factor, a n d / ~ / i s the so-called static factor, which,
in this case is: / ( / = o q ~ - ( t ) where a ( t ) is the current half crack length, a(t) = ct
where c is the velocity of crack propagation. BROnERG'S [1960] solution assumes that
the solid is homogeneous, isotropic, and linearly elastic. In the present problem however, there is a zone of microcrack-saturated material [which behaves linearly elastically]
right near the propagating crack-tip; this is surrounded by a zone of inelastic material
in which microcracking is still taking place, and the third zone of the solid consists once
again of linear elastic isotropic material with constants Eu and pu corresponding to the
virgin, unmicrocracked material. To understand the effect of microcracking on dynamic
402
Y. Tol and S. N. ArLL'~I
crack propagation, we compute the dynamic energy-release rate for a propagating crack,
for the imposed Kt as in eqn (8), corresponding to two sets of elastic material constants: (i) Eu, 1,,, of the uncracked virgin material; and (ii) E~ and ~ of the microcrack
saturated material. The energy-release-rates for these cases are labeled here as T* and
7"*, respectively. Using the well-known expressions for energy-release for propagating
cracks, we write:
T,7 = [AI(c, Eu,~,~)K~I/(2G.)
(9)
T~ = [At(c,E~,~)K~J/(28~),
(1o)
where
At(c) =/3j(l - t3~)/D(c)
D(c) = 4fljfl2 - (1 +/3~) 2
~
= 1 -
(c/cL)2;
~
= 1 -
(c/cs)'.
(ll)
Note that At(c, Es, ~s) in eqn (10) has the negative value for the given material in the
case of C/Cs > 0.55 when the crack speed c is larger than the Rayleigh wave speed. In
order to avoid this situation, ~s and C/CL in eqn (10) are assumed to keep the initial values for the unmicrocracking solid in the present calculations. The imposed Kt is exactly
in equilibrium with the given loading under these assumptions. Whichever definition
of AI in eqn (10) is used for the case of C/Cs < 0.55, the following discussions are not
influenced. The "exact" values of T~ and T~* as computed from eqns (9) and (10) are
plotted as straight lines in Figs. 13(a), (b), and (c) for the values of (C/Cs)= 0.2, 0.4,
and 0.6, respectively. The numerically computed results, using the algorithm presented
in Section 2 of this paper, for T~ of the uncracked material are also plotted as T* in
Fig. 13(b) for the case (C/Cs) = 0.4. That Tu (exact) and T* (numerical) agree well indicates the numerical accuracy of the present finite element procedure of modeling dynamic crack propagation. The directly computed value Tin* for the microcracking
material, using the algorithm presented in Section 2, is also plotted in Fig. 13(b). That
T~ is much smaller than T~* at all times indicates the toughening effect, caused by the
material nonhomogeneity, especially the zone of inelastic microcracking material surrounding the microcrack-saturated material immediately near the crack-tip in dynamic
crack propagation in the presently considered class of materials. While the results presented in Fig. 13(b) correspond to the case of (C/Cs) = 0.4, similar results were also
noted for the cases (C/Cs) = 0.2 [Fig. 13(a)] and (C/Cs) = 0.6 [Fig. 13(c)], thus indicating the general phenomenon of toughening induced by microcracking during dynamic
crack propagation. Figures 13(a)-(c) demonstrate a rather complicated influence of the
velocity of crack propagation on the quantitative values of microcrack-toughening. This
is discussed further later.
Figure 14 indicates the equivalent stress distribution ahead of the propagating cracktip at t = 15 ~sec; for the two material models of (i) linear elastic isotropic solid without microcracking and (ii) with microcracking for the case of far-field a = 108 N/m 2
and C/Cs = 0.4. Considerable reduction in the magnitude of crack-tip stress intensity can
be observed in Fig. 14.
Element analysis of solid fracture
1.5
-
403
•
T°m calculated for microcracking
material
..
--
T°u = A0 [c]Ka2/2Gu [Broberg] . /
-.
T-,-- A, [c]K,~/2~,_././. /
x~
/
1.0
_z
mI
~"
0.5
' " ~ ' / ' Q oIO • • • 910 •
(a) c/cs = 0.2
0
5
I
10
15
20
~a [ x 0.154 cm]
2.0
14m calculated for microcracking
material
T'u : AI [c ]KI2/2G [Brob~rg]
E
1.5
Z
i
I"~s = Al[c ]K1212~-~s
o
T"
×
calculated for unmicrocracking
material
.J
J
1.0
O
LM
z
j-
t--
J"
0.5
•~
.~6.-~6
,,-..,--~"o
.~"
(b) c/cs = 0.4
0
o~a-Q-~,~
5
-eO
~ ~ • • • • "
10
15
20
Aa [X 0.154cm]
Fig. 13. Time histories o f T* integral for the propagating crack
(a = 0.10 × 109 (N/mZ),)1 = 0.1 x 10 -3 (l/see)) (Figure c o n t i n u e d o n o v e r l e a f )
404
Y. T o [ a n d S. N. ATLURI
2.0
T'm
T',, : A, [,] K,2/2Ou [BroOms]
1.5
IN
E
Z
qr
o
×
calculated for mkrocracking
material
r',
: A, [,] K,2/2~,
1.0
../
0LM
j "
I.-
Z
m
°
0.5
i I
(c)
C/Cs =
/
0
0.6
5
10
Aa [ × 0 . 1 5 4
15
20
cm]
Fig. 13 c o n t i n u e d .
0.8
&
0.6
UNMICROCRACKED
[
I
I
A
M4
In
0.4 _
MICROCRACKED
I
0.2
M4
0
I
1.0
I
1.25
I
1.5
= X/a
Fig. 14. E q u i v a l e n t stress d i s t r i b u t i o n s at t = 15 ~sec.
(c/cs
= 0.4, o -- 0.10 × 10 9 ( N / m 2 ) , 7/ = 0.1 × 10 -5 ( l / s e e ) )
Element analysis of solid fracture
405
Figure 15 shows the microcracking process zone near the propagating crack-tip along
with the microcrack wake zone for the case o = 10 s N / m 2 and (C/Cs) = 0.4 and ~/=
10-6/sec. It is seen that as steady-state conditions are reached at about t = 15/~sec, the
material near the crack-tip reaches microcrack saturation and starts behaving elastically,
with reduced moduli. At this point, the crack-tip fields may be assumed to have elastic
singularities; and the velocity factor K(c) in eqn (8) may be assumed to correspond to
that in the elastic case.
Figure 16 shows the comparison between contours of the displacement normal to the
crack line, for undamaged and damaged materials. In plotting these contours, the range
between the m a x i m u m displacement and the minimum displacement is subdivided into
20 equal subranges, and the ranges denoted by 0-9 are plotted for the two cases at each
node. A comparison of Figs. 16(a) and 16(b) shows that there is very little difference
0000000000000000000000000
0000000000000000000000000
0000~00000000000000000000
01k~l110000000000000000000
0~1110000000000000000000
0000~00000000000000000000
0000000000000000000000000
0000000000000000000000000
t=S.O[usec]
000000000000000000000
000000000000000000000
O000~t~O00000000
0~1112223~000000000
0~11122238~000000000
oooo~/~Ir~/~t~ooooooooo
ooooooooooooooooooooo
ooooooooooooooooooooo
0000
0000
0000
0000
0000
0000
0000
0000
t : 10.0 I'/~sec]
0000000000000000000000000
0000000000000000000000000
0 0 ~ 0 ~ ~ : k ~ l l l ~ 0 0 0 0 0 0 0
o~lll2228aaaoeel~ooooooo
o~ll1222zzzzoool~ooooooo
oo~o~t/~ot~/~tr~lll~oo0oooo
o000000000oo0ooooooooo000
ooooooooooo00o0ooooooo0oo
t = 15.0 [/~sec]
0.0 <
~ < 0,1
1
0.1 ~; ~ < 0 . 2
2
0.2 s: ~ < 0.3
3
0.3 •
•
(initiation)
~ < 0.37
~ = 0.37
(saturation)
Fig. 15. Development of the zone of microcracks near the crack tip.
(c/c, = 0.4, a = 0.10 x 109 (N/m2), T/= 0.1 x 10-s (l/see))
406
Y. Tot a n d S. N. A'rLUR~
999999999
88888
7
9999999
8888
77777
9999
8888
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77777
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66
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8
777 86
55 444
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77
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777 66 55 44
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77
66 55 4 33
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7
7
66 5 4 33
22222
77
6
44 3 222
7
7777
6 5 4 3 22
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7
777
86 5
3 2
1111111
7 777 6 55 3
11
777777
55
321
000000
777 77
6 554
O(~X~K)OO0
777 7 6 6
0
77
66655
0
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0
777 7 6 6
0
777 77
8 554
000000000
777777
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321
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7 777 6 55 3
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777
68 5
3 2
1111111
7
7777
6 5 4 3 22
1111
77
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44 3 222
7
7
66 5 4 33
22222
77
66 55 4 33
2222
777 6 6 55 44
33
77
6
5
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8
777 86
55 444
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8
777
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5
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(a) uruicrocrackea
99999999999
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7
99999999
88888
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999999
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8888
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555
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7
6
55
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66 55
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77
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77
6 55 4
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7 7 6
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7777
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7777
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6 5 432t 00000000
7777
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7
5
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77
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7777
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77 7777
6 5 4321 0 0 0 0 0 0 0 0
7777
8 5 43
1 000000
7777
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2 t
777 6 6 54 3 2 1 1 1 1 f l
777 66 5 4
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777 66 5 4 3
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7
(b) microcracked
Fig. 16. Contours of Crynear the crack tip at t -- 15 #sec.
(c/cs = 0.4, g = 0.10 x l09 (N/m2). r/-- 0.1 × l0 -s (1/sec))
in the far-field; however a considerable difference exists in the wake-zone. Note that the
applied far-field tension f o r all the results in Figs. 13, 14, 15, and 16 has been o =
lOSN/m 2.
In the following we discuss the q u a n t i t a t i v e effects o f crack-speed, applied far-field
stress level, a n d the rate (viscosity parameter) o f m i c r o c r a c k f o r m a t i o n , o n the microcrack t o u g h e n i n g .
F o r the applied stress o f o = 10 s N / m m 2, a n d viscosity coefficient 71 = 1 0 - 6 / s e e ,
the microcrack d a m a g e zone as "steady-state" c o n d i t i o n s are o b t a i n e d [at t = 30 ~sec,
Element analysis of solid fracture
407
Aa = 2.31 cm in the case (C/Cs) = 0.2; at t = 15/~sec, Aa = 2.31 cm in the case (C/Cs) =
0.4; and at t = lO ~ s e c , / t a = 2.31 cm in the case (C/Cs) = 0.6], are plotted in Fig. 17 for
various values of crack-speed (C/Cs). In Fig. 17, each element in the finite element mesh
near the crack tip, as in Fig. 15, is assigned a different symbol, depending on the level of
000000000000~0~00000000
0000000000000~00000000
0 0 0 0 0 0 0 0 0 0 ~ 0 ~ ~ 0 ~ 0 0 0
0 0 0 0 0 0 0 0 0 ~ ~ ~ ~ 0 0 0
0 0 0 0 0 0 0 0 ~ ~ ~ ~ 0 0 0
O 0 0 ~ ~ l
I 1122211~0000
0~1122~33310100211~0000
0~112233333@Q0@0211~0000
0 0 0 ~ ~ I
I 1122211~0000
0 0 0 0 0 0 0 0 ~ ~ ~ ~ 0 0 0
0 0 0 0 0 0 0 0 0 ~ ~ ~ 0 0 0
0 0 0 0 0 0 0 0 0 0 ~ 0 ~ ~ 0 ~ 0 0 0
0000000000000~00000000
000000000000~0~00000000
(a) c / c s = 0 . 2
0000000000000000000000000
0000000000000000000000000
0000000000000000000000000
0000000000000000000000000
0000000000000000000000000
0 0 ~ 0 ~ ~ ] ~ ] ~ 0 0 0 0 0 0 0
0~]]]2223333~e~0000000
0~I
112223333©@@I~0000000
O 0 ~ O ~ t ~ ~ ] ] ~ O 0 0 0 0 0 0
0000000000000000000000000
0000000000000000000000000
0000000000000000000000000
0000000000000000000000000
0000000000000000000000000
(b) clcs =0.4
0000000000000000000000000
0000000000000000000000000
0000000000000000000000000
0000000000000000000000000
0000000000000000000000000
0 0 0 0 0 0 ~ ~ ~ 0 0 0 0 0 0 0 0
~i'~l
1112222222~00000000
~~II112222222~00000000
O00000~~'~'~:~tO0000000
0000000000000000000000000
0000000000000000000000000
0000000000000000000000000
0000000000000000000000000
0000000000000000000000000
(c) c/cs = 0.6
2:
0.0 < ~ < O.l (initiation)
I:
0.1 ~ ~ < 0.2
0.2 • ~ < 0.3
3:
0.3 • ~ < 0.~
@:
= 0.37
(~ration
at oR G~ian
e:
= 0.~
(~ration
at ~ r e
mint)
t~n o~ ~ian
~in~)
Fig. 17. Microcrackcd zones M r d i f ~ r e m crack velocities.
(a = 0.10 × 109 (N/m2), ~ = 0.1 X 10-s (l/sec))
408
Y. Tol and S. N. ATLURI
microcracking. It is seen that the microcrack damage zone is largest for the lowest crackvelocity, and decreases progressively as the crack-speed increases. Also, the lower the
crack-speed, the sooner in time is the level of microcrack saturation reached near the
crack-tip. Also, the velocity factors K(c) in eqn (8), and At(c) in eqns (9, 10) are valid
for only linear elastic material behavior near the crack tip. For linear elasticity, K(c) decreases with crack-speed, reducing the zero as c = cR, the Rayleigh wave speed. In the
presence of a nonlinear inelastic material behavior near the crack-tip, the velocity factors
for energy-release rate are unknown. In the microcracking solid, the nature and size
o f the nonlinear zone near the crack-tip depends on the crack-speed itself as seen from
Fig. 17. However, if the "steady-state" values [at Aa = 3.08 cm in Fig. 17] of (T,7,/T~,)
(normalized values of T~) are compared, it is seen that this ratio decreases as (C/Cs) increases from 0.2 to 0.4, but the ratio (T,~/T~,) increases again as (C/Cs) increases from
0.4 to 0.6. For (C/Cs) = 0.2, an elastic zone develops near the crack-tip, at steady-state
conditions, as seen from Fig. 17. Thus, the near-tip field in the case of (C/Cs) = 0.2, at
steady state, can be seen to have an elastic singularity, and the velocity factor in eqn (8)
can also be estimated by the near-tip elastic properties. Thus, one may estimate the energy-release-rate T * from the equation:
T~ = [At(c,E~,~s)lKJp/(20~),
(12)
where
A, = 3111 - 3 ~ ] / [ 4 3 , 3 2 -
(l + 322)2 ]
321 =
1 -
1 -
(c/eL)2;
3~ =
(C/Cs) 2
(13)
c 2 = [Es(l - ~s)l/p(l + ~)(1 - 2~s)
c~ = ff~s/[2p(l + ~s)]Thus, Al(c, Es, Ps) used in eqn (12) is different from that used in eqn (9) where (Eu and
vu) correspond to the virgin undamaged material. It can easily be calculated for the
present case o f material properties,
A~(c,E,,~,)
Al,(c, Eu,Vu)
= 1.41
for c = 0.2Cs
(14)
ffts/G = 2.44.
By curve fitting the asymptotic stress field in the unmicrocracked case, and in the
microcracked case, at t = 37 #sec, for (C/Cs) = 0.2, it is seen that (Ktip/KD = 0.5.
Thus, evaluating T,~ f r o m eqn (12), and T* from eqn (9), we compute
T_._~ ~. 0.86.
r~
(15)
Element analysis of solid fracture
409
2.0
1.5
0
C/Cs: 0.6
•
C/Cs: 0.4
0.6
0.5
C/Cs: 0.2
i: I
I-
E 1.0
r-
O•OOuo °°Ooo
0
A
" ee
°
~
0 ,,o
ee
°°°e~
0.5
&g~
ooo
0,4
i~
0.3
~"
•
0.2
d$
O.1
&
/x t3
t3
I
i
I
5
10
15
20
AO I X 0.154¢m]
Fig. 18. Time histories of T* integral for different crack velocities.
(or = 0.10 × 1 0 9 (N/m2), ~ = 0.1 x 10-5 (l/sec))
This estimation o f Tm --- 0.86T~ agrees very well with the directly computed result at
steady-state, as shown in Fig. 18.
When (C/Cs) is held constant at 0.4 and ~ = 10-6/sec, the shapes o f the microcrack
damage zones for various values o f applied far-field stress, tr = 0.5 x 10 s N / m m 2, 108
N / r a m 2, and 1.5 x 10 s N / m m 2, respectively, are shown in Fig. 19, when steady-state
conditions are reached [Aa = 2.31 cm]. As may be expected, the size of the damage zone
increases as e increases. As a increases, so is the level o f saturation of microcracks near
the crack-tip; and more and more material near the crack-tip once again starts behaving linear elastically (with reduced moduli). Thus, it is likely that the velocity factor
Az(o) [which depends on the elastic behavior (and the attendant moduli or the lack
thereof, near the crack tip] itself is affected by the applied stress e, for a given c. Because o f these complicating influences, it seems from Fig. 20 that for a given (C/Cs) and
r/, as steady-state conditions are reached, the normalized value o f crack-tip energy release rate, T*, first decreases as a increases from 0.5 to 1.0 x l0 s N / m m 2 and again increases as o increases f r o m 1.0 to 1.5 x l0 s N / m m 2 when a = 1.5 x l0 s N / m 2, the
material near the crack-tip again behaves elastically. Thus, 7"* may be estimated, assuming elastic (saturated microcracking) behavior, as in eqn (12). For (C/Cs) = 0.4, the
estimation for (T*/T*) turns out to be about 1.14, as steady state conditions are attained. This estimate agrees quite well with the directly computed ratio o f (T*/T*)
shown in Fig. 20 at steady-state conditions.
For a rLxed-valued o f (C/Cs) = 0.4 and applied stress o f o = 10 s N/m z, the shapes of
the damage zones for values o f , / = 10 -5, 10-6, and 10-7/sec, respectively, are shown
in Fig. 20, when steady-state conditions are reached (t - 15 #sec). Note that as r/de-
410
Y. Tot and S. N. ATLUa[
000OOOOO00000OO0
000000000
0000000000000000
000000000
0000000000000000
O00000000
00000OO000000000
000OOO000
0000000000000~00
000000000
00OO00000
O 0 0 0 ~ ~ t ~ l l l l
O 0 0 0 ~ ~ l
l 11 0 0 0 0 0 0 0 0 0
0000000000000~00
000000000
0000000000000000
000000000
0000000000000000
000000000
0000000000000000
O00000000
0000000000000000
000000000
(a) a = 0.05 X 109 (N/m 2)
000000000000OO00000000000
0000000000000000000000000
00000000OO000000000000000
0000000000000000000000000
O 0 ~ O ~ ~ ~ l l l ~ 0 0 0 0 0 0 0
0~1112223333@@@I~0000000
0~II12223333@@@1~0000000
O 0 ~ O ~ ~ ~ l l ~ O 0 0 0 0 0 0
O000000000000000000000000
0000000000000000000000000
0000000000000000000000000
0000000000000000000000000
(b) (~ : 0.10 X 100 (N/m2)
0000000000000000~~0000
0 0 0 0 0 0 0 0 0 0 0 0 ~ ~ ~ 0 0 0 0
o o o o o o o o o ~ o ~ ~ ~ o o o
o o ~ o ~ ~ I
1111111~oo
~ ~ I I 1 2 2 2 2 2 3 3 2 2 1 ~ 0 0
t112@O@lldHli~3211~000
~ ~ I I 1 2 2 2 2 2 3 3 2 2 1 ~ 0 0
ooooooooo,,o,,~,~,~k~ooo
oooooooooooo****~**oooo
0000000000000000
~0000
(C) or = 0.15 X 109 (N/m2)
/i < 0.1
(initiation)
1:
0.1 :~ ~ < 0.2
3:
0.3 ~ ~ < 0.37
~:
0.0 <
9.:
0.2 ~i ~ < 0.3
@:
,~ = 0.37 (saturation at one Gaussian point)
0:
/~ = 0.37 (saturation at more than one Gaussian points)
Fig. 19. Microcracked zones for different a values.
(c/cs = 0.4, 7/= 0.1 x 10-~ (1/sec))
creases, the magnitude o f ~ increases for a given oe. F r o m Fig. l(a) o f Part 1, it is seen
that as 7/increases, the m a g n i t u d e o f strain (as well as stress) at which m a c r o c r a c k i n g
reaches saturation also increases; in fact as ~ increases, the strain-range over which
microcracking occurs (from initiation to saturation) also increases. F r o m Fig. l(b) o f
Part 1 it is also clear that as ~/increases, for a given stress level, the level o f microcracking
Element analysis of solid fracture
411
2.0
0
(r : 0 . 0 5 x 1 0 ' [ N / m 2]
o"
0.10 x 109 r N / m 2]
0
0.15 X 10 9 ['N/m 2]
. 0.6
1.5
A
0.5
O
O
m~
,E
0.4
O
1.0
•
iw
,E
0 0
A
0.3
n
~@@@@mO00 00
A
ooo
0.5
oe - 0.2
O.1
0
I
I
I
5
10
15
TIME
20
[/~$ec ]
Fig. 20. Time histories of T* integral for different a values.
(c/cs = 0.4, T/= 0.1 x 10-5 (I/sec))
decreases, and vice versa. These phenomena explain the effects seen in Fig. 21: as 71decreases from 10 -5 to 10-7/sec, for the given applied stress a, the level of microcracking increases, and more material near the crack-tip reaches microcrack saturation and
behaves elastically. Also for larger values of T/( - 10 -5/sec) the size o f the damage zone
is larger; however the intensity of microcracking is smaller. As noted earlier, the velocity
factor in the energy release rate expression is affected by the material behavior near the
crack-tip. This explains the result in Fig. 22 that as steady-state conditions are reached,
the value of (T,~/T~) decreases first as ~ decreases from 10-5/sec to 10-6/sec, and increases again as ~ decreases further from 10 -6 to 10-7/sec. For ~ = 10-7/sec, at steadystate, when elastic conditions prevail near the crack-tip, (T~,/T~) can be estimated as
explained in eqn (12). This estimate is ( T ~ / T ~ ) --- 1.14 which agrees quite well with the
directly computed result shown in Fig. 22 (at steady-state).
V. CONCLUDING REMARKS
The stationary and rapidly propagating cracks, under dynamic loading, brittle
microcracking solids have been analyzed by the finite element method. The obtained results can be summarized as follows:
1. The crack-tip integral T* should be used as the parameter describing the severity
o f the crack tip in the discussion o f the microcrack toughening effect, because it
412
Y. T o t a n d S. N. ArLURI
00000000000000~0000000000
0000000000~~000000000
O000000~O~t"k~O00000000
oooo~tutt~~~ooooooo
oo0.mt~..~t~~ooooooo
li~lrO~O0OOO
O#f~"Jg~k1~l
11111~0~00000
************************
oooo~t,~tr~~~lt~ooooooo
ooooooo~o~~ooooooooo
oooooooooo~t~tut~ooooooooo
oooooooooooooo~oooooooooo
(a) 7/ = 0.1 X 10-4 (I/sec)
0000000000000000000000000
ooooooooooooooooooooooooo
ooooooooooooooooooooooooo
ooooooooooooooooooooooooo
oo,o**********]l],OOOOOOO
o.tll|222aaaaooo|.ooooooo
0,,111222333300@1,00000o0
o o ~ o ~ n t ~ t ~ t ] ] ] ~ o o o o o o o
ooooooooooooooooooooooooo
ooooooooooooooooooooooooo
oooo0oooooooooooooooooooo
ooooooooooooooooooooooooo
(b) 7/ = 0.1 x I0-5 (1/see)
0000000000000000000000000
0000000000000000000000000
0000000000000000000000000
0000000000000000000000000
0000~0000~~l
11~0000000
1,oooooo
1~000000
1~0000000
0000000000000000000000000
0000000000000000000000000
0000000000000000000000000
0000000000000000000000000
(c) 7/ = 0.1 X 10-s (1/sec)
~:
0.0 < ~ < O.l
2:
0.2 ~
0:
~ = 0.37
(saturation at one G~ssian point)
0:
~ = 0.37
(saturation at =ore than one C~ussian points)
~ < 0.3
(initiation)
1:
O.l ~ ~ < 0.2
3:
0.3 ~
~ < 0.37
Fig. 21. Microcracked zones for different ~ values.
(c/cs
= 0.4, o = 0.10 X 10 9 ( N / m 2 ) )
is applicable to dynamic as well as quasi-static problems including any type of
materially nonlinear behaviors, and unloading.
2. The microcrack toughening effect exists in dynamic fracture of brittle microcracking solids. However, the magnitude of this toughening depends in a rather complicated way on the applied far-field stress, velocity of crack-propagation, and the
viscosity coefficients in the microcracking evolution equation.
Element analysis of solid fracture
413
2.0
1.5
13
7I : 0 . 1 X '10"" [1/1~:3
•
n = o.1 x lo-" [ 1 / , ~ : ]
0.6
0.5
,.
= o.1 x
[1/,,..]
0
0
o3
E
0.4
0
1.0
O0 @oO 8 n O ouBO°O
~6
-0.3
~ee
&
A
•
00
/~
• 000
00_
0.2
0.5
0.1
I
I
I
5
10
15
20
TIME b , , - , ]
Fig. 22. Time histories of T* integral for different ,1 values.
(c/cs = 0.4, o = 0.10 x 109 (N/m2))
3. The "velocity factor" in the energy-release-rate expression depends simply on the
material properties near the propagating crack-tip. When microcracking reaches
a saturation near the propagating crack-tip, under specific conditions of applied
stress, velocity of crack-propagation, and ~, the velocity factor is as if for a linear elastic material with the reduced moduli corresponding to the microcracksaturated material.
Acknowledgments-The support of this work by the U.S. Army Research Office, under the Balanced Technology Initiative, is gratefully acknowledged. It is a pleasure to thank Ms. Deanna Winkler for her assistance
in preparing this paper.
REFERENCES
1960
1975
1977
1980
1986
1987
1987
BROBERG,K.B., "The Propagation of a Brittle Crack," Arkiv fur Fysik, 18, 159-192.
CHErt,Y.M., "Numerical Computation of Dynamic Stress Intensity Factors by a Lagrangian Finite
Difference Method (The HEMP Code)," Engineering Fracture Mechanics, 7, 653-660.
ABERSON,J.A., A,',rDERSOn,J.M., and KING, W.W., "Dynamic Analysis of Cracked Structures Using
Singularity Finite Elements," in SHIrl, G.C. (ed.), Elastodynamic Crack Problems, Noordhoff International Publishing, 1977, pp. 249-294.
OWEN,D.R.J. and HtrcroN, E., Finite Elements in Plasticity; Theory and Practice, Pineridge Press
Ltd., 1980.
ArLt~i, S.N. (ed.), Computational Methods in the Mechanics of Fracture, Vol. 2 of Computational
Methods in Mechanics, North-Holland, Amsterdam.
Cm~a.~tamEs, P.G. and McMEEKIr~6, R.M., "Finite Element Method of Crack Propagation in
a Brittle Microcracking Solid," Mechanics of Materials, 6, 71-87.
Nmlsmcov, G.P. and Arttr~t, S.N., "An Equivalent Domain Integral Method for Computing CrackTip Integral Parameter in Non-Elastic, Thermo-Mechanical Fracture," Engineering Fracture Mechanics, 26, 851-867.
414
1990a
1990b
Y. Tol and S. N. A'rLURI
Tol, Y. and A'rttrRl, S.N.,
Microcracking Solids (Part
169-188.
Tot, Y. and A'rLURI, S.N.,
Microcracking Solids (Part
Plasticity, 6, 271-288.
"Finite Element Analysis of Static and Dynamic Fracture of Brittle
1: Formulation and Simple Numerical Examples)," Int. J. Plasticity, 6,
"Finite Element Analysis of Static and Dynamic Fracture of Brittle
2: Static and Growing Macro-Cracks Under Static Loading)," Int. J.
Center for Computational Mechanics
Georgia Institute of Technology
Atlanta, Georgia 30332-0356
(Received 25 October 1988; in final revisedforra 20 April 1989)