eon-7944/84 s3.00 + .@I
Pergamon Press Ltd.
&gina~
Fracnw MechanicsVol. 20, No. 2, pp. 19MO8, 1984
Printed in the U.S.A.
ON THE COMPUTATION OF MIXED-MODE K-FACTORS
FOR A DYNAMICALLY PROPAGATING CRACK,
USING PATH-INDEPENDENT
INTEGRALS .J;
T. NISHIOKAt and S. N. ATLURIt
Center for the Advancement of Computational Mechanics, School of Civil Engineering,
Georgia Institute of Technology, Atlanta, GA 30332, U.S.A.
Ah&act-The
path-independent integral J;, which has the meaning of energy release rate in
elastodynamic crack-propagation, is used to numerically obtain the mixed-mode dynamic stressintensity factors for a crack propagating in,a prescribed direction with a prescribed velocity.
Moving isoparametric (non-singular) elements are used to model crack propagation. Even though
J’ is a vector integral and hence is coordinate invariant, the desirability of using speci& coordinate
systems to improve the accuracies of the numerical solutions for K-factors is pointed out. Two
procedures for extracting the mixed-mode K-factors from the S integral for a propagating crack
are given. It is found that the component of J’ along the crack-axis, i.e. J;O, b always equal to or
greater than the product of a crack-velocity-function and the component normal to the crack-axis,
2;“. Several examples of a slanted crack are presented to demonstrate the practical utility of the
J’ integral. A discussion is also presented concerning the velocity factors for dynamic A-factors,
and energy release rate, in a finite body.
INTRODUCTION
THE SUWECT of the so-called path-independent integrals has received considerable attention due
to its several attractive features in application to fracture analyses. Although innumerable number
of papers have appeared in literature concerning the application of path-independent integrals in
fracture mechanics, most of these have been limited to pure Mode I cases. The application of
path-i~de~ndent
integrals in elastodynamic crack propagation is of fairly recent origin.
In Ref. [l] Atluri, through an elaborate discussion of conservation laws, has derived a
path-independent integral J for an elastodynamically propagating crack, which has the meaning
of rate of change of Lagrangean per unit crack extension. Later, through a simple modifi~tion
of integral in Ref. [I], the present authors[2] have derived a path-independent integral J’ which
has the precise meaning of the rate of energy release in elastodynamic crack propagation. The
integrals in Refs. [I, 21 were defined along spatially fixed paths. On the other hand, using paths
which move rigidly with the propagating crack-tip, Bui [3] has derived a path-ind~ndent
integral
which is also equivalent to the energy release rate. Also Kishimoto, Aoki and Sakata[4,5] have
derived a path-independent integral .?, for spatially fixed paths, which has the meaning of energy
release rate only for a stationary crack in a solid in dynamic equilibrium. The formulae relating
the path-inde~ndent
integrals J’, .?and J to dynamic stress intensity factors of a&the three fracture
modes were derived in Ref. [2].
For problems of elastodynamic crack propagation, analytical solutions can be obtained for only
a few special cases (such as infinite bodies), due to the inherent mathematical difficulties. In previous
papers[GlO], the present authors have developed a “moving” singular element procedure for the
analysis of unsteady crack-propagation
in finite bodies. The asymptotic general solutions were
embedded in the moving singular element, and hence the stress intensity factors were evaluated
directly and accurately. However, even while using a less sophisticated finite element model with
isoparametric elements near the crack-tip, the path-independent integrals may be conveniently used
to evaluate the crack-tip parameters such as the stress intensity factors as well as energy release
rate. Numerical studies using the moving singular element procedure [ 1l] and moving isoparametric
element procedure[l2] indicate that the use of the J’ integral is essential to a correct evaluation
of stress intensity factors. Special attention should be paid to the use of other path-independent
integrals which do not have the meaning of energy release rate. It has been shown[l2] that if one
uses a finite element method in which the singularity in the kinetic energy density for a propagating
TVisiting Assistant Professor.
JRegents’ Professor of Mechanics.
193
194
T. NISHIOKAand S. N. ATLURI
crack is not incorporated, the use of J^ and J integral may lead to substantially erroneous values
for the stress intensity factor.
The above cited numerical studies of the authors pertain to Mode I dynamic loading. In the
present paper, we consider mixed mode dynamic loading. However, the primary aim of the paper
is to illustrate the use of the path-independent integrals, and simple isoparamet~c crack-tip finite
elements, in the computation of mixed-mode dynamic stress-intensity factors. As such, attention
is not paid in the present paper to the problem of direction of crack-propagation under mixed-mode
loading. Rather, the admittedly hypothetical case of self-similar crack-propagation under mixed
mode loading is assumed. While the path-independent
J’ integral is vectorial and hence is
coordinate-invariant
in nature, the influence of the use of specific coordinate systems on the
accuracy of the computed components of vector J’ is illustrated. Several procedures for extracting
the mixed-mode dynamic stress-intensity factors using the vector J’ integral are also presented.
PATH-INDEPENDENT
INTEGRAL Jk UNDER MIXED MODE CONDITIONS
Consider a crack propagating with an instantaneous crack velocity C(t) in a linear elastic solid,
as shown in Fig. 1. The general solutions (eigen solutions) for stress and displacement, asymptotically close to the crack-tip, under unsteady as well as steady conditions were obtained in Ref.
[2] in a unified fashion for all the three fracture modes. The singular stress field and the
corresponding displacement field are given as [2]:
ao
WMC)
_
I’- J5
+
i
(1 +2~,~-B;i)li-~‘~cos~-~~~-~~2cos~j
2
K,,&(C)
-
*
i
0 4&(C)
822-‘fi
4
(1 + 2j!?,2- &qr* - V2sin zei + (1 + flz2)r2- l/2sin -21
- (1 + @?)r, -“2cos 5,
2
(la)
+$)r,“%os;
1
l
+ K,&‘(C)
J2n
(1 + /12)rl - li2 sin : - (1 + /?;)r* - ijz sin ;)
1
0 &4(C)
fr,2=2&r, - ‘I2sin $ - 2&r2 - It2sin ;
(
1
fi
+ KIBIIK?
2/?*r,-li2 cos y4 - (1 + 82’)’ _ ij2cos 4
Iz 1
i
28,
r2
fi
(lb)
(14
(ld
0: plane stress
i
WI(C)
43=
I
2v(fi,2 - B22)r,- ij2cos $
J%
1
i
+ &4I(C)
u1
rJ_--
P
-2v(B,2-8’).r,-‘!2sin~
i
fi
WI(C)
2
Jr
+ &&(C)
p
of)
7t
: plane strain
I
~S,Bz
pcos!i_-~
2
(1+/3,2)2
i/2cos 1e
21
4 _ 2(1 + 82’) r2*j2sin $
-2 ril/z sin T
JT R
1
(24
On the computation
of mixed-mode K-factors for a dynamically propagating crack
195
a(v-v,) = I- + l-, - r,
re = r,’ + r;
x,
xl
Fig. 1. Definitions of integral paths and coordinate systems.
(2b)
/II2 = 1 - C2/Cd2; /12’= 1 - C2/Cs2; B,(C) =
where rjeioj = x,’ + isix,’ (j = 1,2); i = fl;
(1 + /?:)/D(C); B,,(C) = 2/?,/D(C); &r(C) = 1//12; D(C) = 4&T12- (1 + /?,‘)‘; and p is the shear
modulus, C, and C, are the dilatational and shear wave velocities, respectively. The superscript ( )O
denotes the tensor components with respect to the local coordinate xko attached at the crack-tip
(see Fig. 1).
For a crack propagating, in a self-similar fashion, at an angle 0, measured from the global X,
axis as shown in Fig. 1, the energy release rate G can be expressed as[l]:
G = (C,/C)G, = G, cos 0, + G2 sin 0,
(3)
and
Gk = ljz
[( W + T)n, - tit&j dS
(4)
s r,
where C, and Gk are the components of crack velocity and energy release rate with respect to the
global coordinates X,; W and T are the strain and kinetic energy densities, respectively; n, the
outward normal direction cosines; ti the traction; and ( )k denotes a( )/8X,.
The components of the path independent integral which are equivalent to the components of
energy release rate were derived in[2]:
[(W + T)nk - tiui,J dS
J; = ljz
f r,
= lim
a-0
[(W + T)Q - tiui,J dS +
y_ y &&Ui,ks
PG,kId’ .
(5)
6
Specifically for a crack subject to prescribed traction 6 on the crack surfaces Tc + and r, - as shown
T. NISHIOKA
196
and S. N. ATLURI
in Fig. 1, the far-field expression for the J’ integral can be modified as:
-tiui,d
dS
r-0
{Ir +T)n,
J; = lim
[( W
[( w
+
+ T)Q
-
<ui,k]dS +
biii~i,k - ptiiri,k] d V .
J v- v,
J re
The body forces JI other than inertia forces can be included easily replacing pz&by (ptii -J).
We now consider the change of components of J’ under a coordinate transformation. Suppose
that the angle between the X, and xl0 axes is eo. The local coordinates x0 can be expressed by
Xf =
a,(e,)X,
(7)
where
cos
&j(4)
=
[
The same transformation
we have
-
e,
sin 0,
sine,
case,
1’
(8)
rule holds for the J’ integral given by eqns (5,6). Thus, from eqn (6),
s[(w+
!i, J[
iJ[
_&‘$1dV
+Jvv,
[p$!.$
1
Qr;-Q$,]dS
J;’ = lim
6-4 r,
0
=
lim
c-0
r
(W + T)n,” - to ax;o dS+
(W+T)+$‘$
rr
k
k
1
dS
k
Pb)
= a,&.
Thus,
J; = a,,&‘.
(94
Obviously, we have the same relations for energy release rate.
G,’ = a&,
Or
Gk =
a,GP.
(loa, b>
Substituting the singular stress and corresponding displacement fields given by eqns (1) and (2)
into eqn (9a), the relations between the J’ integral and the instantaneous stress intensity factors
can be expressed as[2]:
J;’ = Go= & {K%(C) + G,&(C) + G&(C))
J;O=G;=
-K,K”&(C)_
P
(114
(lib)
The crack-speed functions are given by
4(C) = A(1 - 822)/~(C)
(124
MC)
WV
= Ml - B22)lNC)
On the computation of mixed-mode K-factors for a dynamically propagating crack
4,,(C)
= VP2
A,(C)
=
197
WC)
(81- 82x1- 822)p/u2 + (1+ 82w
(w>t
2Ju
+ Ml
+ A + A)_
24
+
&2)
+ 82)
(124
It is of interest to note from Eqns (11) and (12) that
(13)
As shown in Ref. 121,the crack speed functions Ai, Ai1 and A, are all positive, therefore, we have
(14)
For a stationary crack C = 0, A,(O) = A,,(O) = AN(O) = (K.+ 1)/4, and (K = (3 - v)/(l + v) for plane
stress and (3 - 4v) for plane strain, v: Poison ratio). The value of ,/&&/A,
decreases from unity
to zero as the crack velocity increases from zero to Rayleigh wave velocity CR. Thus, for a
stationary crack, the relation (14) can be reduced to
PROCEDURES
FOR DETERMiNIN
G MIXED-MODE DYNAMIC STRESS INTENSITY
FACTORS
A global coordinate system is generally used in most computations such as the finite element
method, since it is the most convenient one to describe the geometry and boundary conditions.
Thus, we assume that the numerical solutions are obtained using a global coordinate system.
Since the asymptotic stress and displacement fields are given in terms of the stress intensity
factors, one can extract the stress intensity factors from the numerical solutions near the crack-tip.
In this procedure, the obtained solutions must first be transformed to the local crack tip coordinate
system, for example, xko as shown in Fig. 1. Then the crack opening ~spla~ments related to each
of the fracture modes can be expressed by
61= (uz”)+ - (uz”)- ,
s,,= (ulO)* - (24;)- f
&I = (2.4~)
+ - (24,“)
-
(16)
where ( )+ and ( )- denote the quantities at the upper and lower surfaces of the crack.
Using eqn (2), the stress intensity factors are evaluated by
(A4 = I, II, III).
(17)
Usually the K values calculated from eqn (17) depend on the distance from the crack-tip r, at which
the crack-opening displacement is considered. One can estimate unique value of stress intensity
factors through the extrapolation to r = 0 of the K values obtained from eqn (17).
However, there are several disadvantages in this procedure: (i) it requires a very fine finite
element mesh near the crack-tip; (ii) the computed K values depend on the extrapolation technique;
and (iii) in transient (dynamic) responses, since the crack opening displacements are localfy
disturbed, the extrapolation of K values may not be easy. For these reasons, the results obtained
through this procedure are not presented in this paper.
We now present two alternate procedures for dete~~ng
the mixed-mode dynamic stress
intensity factors for a propagating crack using the path-independent integrals.
(i) Procedure Z (directly from Ji”>
In two-dimensional cases, one can decompose the solution field into the defo~ations
of inplane
and antiplane type, which in turn are, respectively, related to Modes I and II, and the Mode III
behavior. Thus, the Mode III can be treated separately from the other modes. First, the J; integral
198
T. NISHIOKA
and S. N. ATLURI
for the Mode III, say (J&, is calculated using only o;,, oj2, u3 in eqn (6). The (1;) value is converted
to (J;O),r, by (J;*)i,i = (J&ii cos 8, + (J&i, sin B, (see eqn (SC)). The K,,, value is evaluated by
Km
=i
J-__?eA,,,(C)
WOhI.
The sign of Kni can be determined monitoring the sign of 6ni.
For the inplane mixed-mode problems, the J; integrals are transformed to J;O values. Then from
eqns (I 1 a, b), one may evaluate Ki and K,, as follows:
(Gm * [(4”)”- (AIAIIIA2N)(J;0)211’2)
>“2
(194
1’2
>
WW
(&
K,l= +(--,L!.lu, = It
A,,(C)
{(.y)
T
[(J;*)2- (A&*/A&)(J;O)y)
The signs of K, and Kn can be determined mo~to~ng the signs and ma~tudes
of S, and Sir. The
signs of K,, K,, correspond to the signs of 6i, &, respectively (Ki > 0, if 6, > 0, and so on). The
term in braces ( } in eqn (19a) is determined as:
[1
(204
if lWh/ < 14dS2j,take- [ 1
P-W
if (&Yilr
I&ilS2/, take +
with similar choices for Ku.
(ii) Procedure ZZ (decomposition method)
For static crack problems, Ishikawa et al. 1131 have developed a procedure for dete~ining
mixed mode stress intensity factors, in which the solutions for displacement, stress and strain are
decomposed into components which are symmetric and antisymmetric, respectively, with respect
to the x,* axis. This procedure was also employed in Ref. [4] for dynamic stationary crack problems
and later in Ref. [ 143 for dynamic crack propagation using the J^vs K relation which was derived
by the present authors[2]. We expand this procedure for determining stress intensity factors in
mixed mode dynamic crack propagation using the f’ integral.
As explained under Procedure I, the Mode III stress intensity factor can be evaluated separately
from the other modes, Thus, in the following, we describe the procedure for inplane mixed-mode
problems. If we use a displacement finite element model, only the displacement is required to be
decomposed as [ 131:
(21)
where
z.q(x,O,
x2”>= uj@(xlo,
- x2”).
The decomposed strains and stresses can be obtained by using the linear strain-displacement
stress-strain relations.
Substituting eqns (21), (23), (24), etc. into eqn (5) or (6), we have
(22)
and
On the computation
of mixed-mode K-factors for a dynamically propagating crack
199
where the near-field expression for JiMN (M,N = I, II) is given by
( WMN+ T”“)n,” - t?
J&, = lim
c-0
dS
(26)
S[r,
and
1
2
wMN = - &!!
‘I
(27)
IJ
TMN = ; p$$jiN.
Substitution
of the asymptotic
(28)
solutions given by eqns (1) and (2) into eqn (28) leads to
Jim= J;III=~
(2%
J;II = J;IIII= 0
(30)
and
(314
J;I*II=JJ;**=-
J;II,+JSIII=
41(C) p
2/l
I1
-
(31b)
(3lc)
Thus,
J;‘= J;I+ JiII
(32)
J;” = JiIII + J;,,,.
(33)
The stress intensity factors are directly converted from the J;,,, integral.
KM= f
JiM =
(W”+
(A4= I, II).
(34)
TM)n,‘-
+
(35)
Note that nlo = 0 on r,. If we consider a stress-free crack (6 = 0 on r,), the second term in eqn
(35) vanishes.
FINITE ELEMENT
ANALYSES
In all the present analyses, we use a material with the properties of p = 29.5 GPa, v = 0.286,
p = 2.45 x lo3 K g/ m3 under a plane strain condition. All cracks are assumed to be stress-free and
to propagate self-similarly.
T. NISHIOKA and S. N. ATLURI
200
Fig. 2. Finite element pattern for a plate with a slanted edge crack.
Static analysis of slanted edge crack
The geometry and loading conditions of the problem are shown in Fig. 2 together with the finite
element breakdown and three contour paths. Two types of elements, (i) quarter-point singular
elements and (ii) non-singular isoparametric elements, are alternatively used to model the crack-tip.
It is noted that the former elements do represent a (l/r) singula~ty in W near the crack-tip, while
the latter do not, unless a mesh much finer than that in Fig. 2 is employed.
For a static problem, the far-field path-independent integral given by eqn (6) is reduced to
(i)
J’O
1 =:
Wn,o-
Wd
@$)dS
I
dS +
>
In the above, the components
(W’ - W-)dS.
f
(36b)
r,
of J’ the local crack-tip coordinate
system are given. The
On the computation
of mixed-mode K-factors for a dynamically propagating crack
Static
Quoter
ah
201
Analysis
Point Elements
= 0.5
‘,
: ’
Id”
II
0.5
h-w
Lower
(e=tw)
(8=-r)
0.3
0.2
4r/o
04
0
01
0.1
02
04
0.3
r/o
05
-c-
IO’ t
&
5
.E
e
iz
16’“~
Lower
Upper
(e=-lr)
(e=+*)
lo-“1 .
0.5
,
0.4
a3
-
0.2
0.1
0
0.1
0.2
0.4
0.3
r/o
r/a
0.5
+
Fig. 3. Distribution of strain energy density along crack surfaces.
expressions in eqns (36 a, b) should be contrasted
(J,“) =
to those given by Budiansky and Rice[lS]
Wn: - t? %
dS.
I ax,"
(37)
Figure 3(a) shows the distributions of the computed W on the upper and lower surfaces of the
crack. It is clearly seen that the strain energy density is discontinuous at the crack surfaces, i.e.
(W’ - W-) # 0. This explains the reason as to why Budiansky and Rice’s definition of (J,“) is
not, in general, path independent. This has also been noted by Herrman and Herrman[lB]. The
decomposed strain energy densities which are used in Procedure II described in the foregoing are
also shown in Fig. 3(b). It is seen from Fig. 3(b) that both W’ and W” are continuous at the crack
surfaces. (Note that W # W’ + WI’, since strain energy is superposed quadratically.)
The computed KM (M = I, II) values are summarized in Tables 1 and 2. Since the JL integrals
Table 1. Normalized stress intensity factors for static slanted edge crack problem
(Qnrter-Point Singular Elements)
KII"X,G
KIIIOfi
Procedure I
1.275
0.569
Procedure II
1.285
0.547
Bowie & Freese [17]
1.27
0.58
T. NISHIOKA
202
Table
2. Normalized
stress intensity
(Non-Singular
and S. N. ATLURI
factors
for static
slanted
edge crack
problem
IsoparametricElements)
as computed in eqn (6) involve the effect of (W+ - W-) and (T+ - T-), numerically this is
reflected in the J;O values from which Ki and Ki, are evaluated in Procedure I. This is the reason
why the Procedure I is sensitive to the different modelings around the crack-tip, as seen in Tables
1 and 2. On the other hand, Procedure II for computing K, and K,, circumvents the problem with
( W + - W -) and (T + - T -) being non-zero, since no integration on r, is necessary for the case
of a stress-free crack. Therefore, the Procedure II is not sensitive to the different crack-tip
modelings, as also seen from the Tables 1 and 2.
The combined use of Procedure I and quarter-point singular elements around the crack-tips
gives excellent results as seen in Table 1. However, it was found in Ref. [12] that, for high-speed
crack propagation, C z=-0.3C,, a procedure based on moving non-singular isoparametric elements
gives better results than moving quarter-point singular elements, as compared to the analytical
solution obtained by Broberg[23]. Therefore, in the following dynamic analyses, we use the
non-singular isoparametric elements together with Procedure II.
(ii) Dynamic analysis of stationary slanted crack
The geometry and boundary conditions of the problem are shown in Fig. 4. This problem is
almost identical to the one solved by Kishimoto et al.[4], except that the length of the plate
(L = 44.8 mm) is slightly larger than in [4] (L = 44 mm). An impact tensile stress (Heaviside step
function) acts at the edge of the plate as shown in Fig. 4, which also shows the finite element
breakdown and the contour paths for the calculation of J’ integral.
The variations of the dynamic stress intensity factors Ki and Ki, are shown in Fig. 5. The
computed Ki and K,, values for different paths at t = 20 psec differ within f 0.6% from the average
f-t
t
t
+
t
t
W= 32mm
t
L = 44.8mm
e,
= 45”
0 cm e= 0.5 w
Fig. 4. Finite element
pattern
for a plate with a slanted
crack
subject
to impact
tension.
On the computation
of mixed-mode K-factors for a dynamically propagating crack
203
1.5
I
o K,
Present
A K,
(At =a25rsec)
0
-
Thou
ond
0
Lu
A
0
2
Static
Y” “iI-
0
K,
AA
AAAA
A
AAA
Static
t
(~sec)
Fig. 5. Variations of mixed-mode dynamic stress intensity factors of stationary crack.
values for the three paths plotted in Fig. 5. The times indicated by D, and D2 are, respectively,
the first arrival times of reflected dilational waves at the crack-tip, from the left and right
boundaries and from the lower boundary. It is seen that the present results are in good agreement
with the analytical solutions obtained by Thau and Lu[18] until the time when the analytical
solutions become invalid due to the arrival of the reflected waves at the crack-tip.
(iii) Analysis of dynamic crack propagation under mixed-mode loading
To simulate crack propagation, the present authors[l2,22]
have developed a moving isoparametric procedure for pure Mode I problems, in which the crack-tip is surrounded by the
moving elements in a 180” arc. This procedure is extended for mixed-mode crack propagation, in
which the crack-tip should be surrounded by moving elements in a 360” arc as shown in Fig. 2.
Twenty-four elements around the crack-tip, as shown in Fig. 2, move with the crack speed. The
elements neighboring the moving elements deform accordingly. The mesh pattern is readjusted
periodically to simulate a large amount of crack propagation.
A plate with a slanted edge crack shown in Fig. 2 was also used as an example for the analysis
of mixed mode dynamic crack propagation. In this case, the initial crack (a, cos 0, = 0.4 W) starts
propagating self-similarly in the plate subject to a time independent tensile stress 0 at the edges.
Three constant crack velocities of C = 0.2, 0.4 and 0.6 C, were considered in the present analysis.
The aim of the analysis is to find the mixed-mode K-factors for the prescribed motion (direction
as well as speed) of the crack.
The variations of normalized stress intensity factors during the crack propagations are shown
in Figs. 6-8. As seen from the figures, the normalized K,, values remain almost constant throughout
the crack propagation, while the normalized Ki values gradually decrease. The arrows in Figs. 6-8
indicate the times when the mesh pattern is readjusted to shift the moving elements continuously.
As seen from the figures, the far-field integral calculations were not affected by the mesh
readjustment procedure [7, 121.
The high-frequency oscillations in the normalized Ki solution, due to the high speed crack
propagation can be observed in Figs. 6-8. The magnitude of oscillation increases with the
increasing crack velocity. However, the normalized K,, values remain almost constant. This suggests
that the triangular part @ indicated in Fig. 9 vibrates vertically rather than horizontally with
T. NISHIOKA
204
and S. N. ATLURI
4
t
t
t
,.’
,’
/
)45O
/-o-
Kr
-A-
K,
c10.2 cs
a.5
as5
0.6
a case,/,
Fig. 6. Normalized stress intensity factors for propagating crack (C = 0,2Cs).
respect to the crack axis. This is explained by the fact that the vertical movement (solid line arrows
in Fig. 9) of the part @ can occur freely while the parallel movement is constrained by the ligament
part @.
Now, we investigate tht transient solution for the stress intensity factors due to the initiation
of crack propa~tion, As seen from Figs. 6-8, the dynamic stress intensity factors drop imm~iately
after the crack begins to propagate. The amount of decrease in the stress intensity factors depends
on the crack velocity.
As noted by several researchersjl9-211, the dynamic stress intensity factors KM(M = I, II, III)
Fig. 7. Normalized stress intensity factors for propagating crack (C = 0.4CJ.
On the computation
205
of mixed-mode K-factors for a dynamically propagating crack
t
t
0
-I
0.45
0.4
025
0.55
0.6
0 03s %/w
Fig. 8. Normalized stress intensity factors for propagating crack (C = 0.6CJ.
may be expressed as the product of velocity factors kM(C) and static factors KS, thus,
K,(t,
C) = k&K‘&)
(A4 = I, II, III).
(38)
The velocity factors R&C) (A4 = I, II, III) are given in Ref. [20] as
k,(C)
=
s
MC) = s
-’
u- cKJ
( >
(
c
1
- c
(1 - C/C,)“’
>
(l-C/CR)
(1 _ C/C,)‘/’
WW
WI
(394
Fig. 9. Vibration of crack surface (Part A) due to crack propagation.
T. NISHIOKA
206
and S. N. ATLURI
The static factors Kjl; depend on the current length of the crack, the applied load, the history of
crack extension, but not on the instantaneous crack velocity. As discussed in Refs. [20,21], KE are,
in general, not equal to the static stress intensity factors, KMslp,for a stationary crack of the same
length as the moving crack. The analytical expression for K$ is given in Ref. [20] as
dxlo
(M = I, II, III)
(40)
where da is the amount of crack extension at time t. PM(x,@) are the tractions distributed ahead
of the crack-tip (xi’> 0) prior to the crack extension (t = 0); and P,(x,O) = o%(x,O),
M-%“) = Q!I(XiO),&1i(x,o) = &x,“).
If we restrict our attention to the very early stage of the crack propagation (da/a 4 I), the
traction distribution PM(xio) in the vicinity of the crack-tip is adequately represented by
(M = I, II, III).
(41)
Substituting eqn (41) into eqn (40), one can obtain
Gxt> = Kwo (M = I, II, III).
(42)
Equation (42) is valid for both infinite and finite bodies.
Similarly, the dynamic energy release rates GYM,G: can be expressed as:
GfM = g,,,(C)GfM(t)
(M = I, II, III)
Gi’ = gdW:W
(43)
(44)
The velocity factors and static factors can be derived by substituting eqn (38) into eqns (11 a, b)
as
gM(c) =
k,l(C) * A,(C)/A (0) (M = I, II, III)
(M = I, II, III)
G:,(t) =
(46)
&v(C)= ~1(C)~**(C)~,(C)I~,(O)
G:(t) = - y
(45)
(47)
{Kjyt)Kfi(t)).
(48)
For the problem considered in this section, the static factors can be expressed in the form:
K; = Knr,,, = &&)cr~
(M = I, II)
(49)
where FMs,(M = I, II) are the normalized static stress intensity factors (or static correction factors
for a finite body). Thus, the velocity factors can be extracted from the numerical results as
kdc)=(~g’).(&)
&f(C) =
Jhf(t, Cl
6271u
Aw(0)F2
2P
W=I,
w = I,
II)
11).
(50)
(51)
Mst
It is noted that GYM= JiM. The numericahy
obtained
velocity factors are shown in Fig. 10,
On the imputation
0
of mixed-mode K-factors for a d~a~cally
Mode
I
A Mode II
-
Analytical
FEM
propagating crack
Solutions
-
SOhtiOnS
(Freund
, Kostrov,
Rose;
I
6.4
c/c,
(a)
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.6
I.0
C/CS
(b)
Fig. 10. Comparisons of velocity factors for dynamic stress intensity factor and energy release rate.
comparing the analytical solutions given in eqns (39) and (45). The numerical solutions for the
velocity factors were obtained using the KMand J&, values at the second time step (t = 2dt), since
the normalized K,, values become stable after the second time step. The FMstvalues in eqns (50)
and (51) were evaluated using the stress intensity factors at t = 0. As seen from Figs. 10 (a, b), the
velocity factors for Mode II are in excellent agreement with the analytical solutions, while the
numerical solutions for Mode I are slightly lower than the analytical solutions. This may be
explained by the vibration caused by the rapid crack propagation as mentioned earlier in Fig. 9.
SUMMARY AND CONCLUSIONS
The practical utility of the pan-inde~ndent
integral J’, which has the meaning of energy
release rate, for determining the mixed-mode dynamic K-factors for a propagating crack has been
demonstrated. The finite element method for the analysis of dynamic crack propagation used
presently is based on a procedure of moving isoparametric elements near the crack-tip. Even though
J’ is a vector integral, and hence is coordinate-invariant, the desirability of using specific coordinate
f the computed components
of J’ is demonstrated,
It was found
systems to improve the
W - IJ;*l always holds among the magnitudes
of the paththat the relation [&“I
independent
integral. For a stationary
crack (C = 0), this relation can be reduced to lJ;“I > IJ;‘I.
Two procedures for extracting the mixed-mode stress intensity factors from the J’ integral for
a propagating crack were also presented. Using the above mentioned procedures, dynamic stress
intensity factors for several cases of a slanted crack were obtained. The validity of the velocity
factors k&C) and gM(C), which are derived analytically for a crack in an infinite or ~mi-in~nite
body, was also numerically verified for a finite crack in a finite body.
Acknowledgements-The results were obtained in the come of research supported by the U.S. Office of Naval Research
under Contract NOOO14-78-C-0636.
This support, as well as the continual encouragement of Dr. Y. Rajapakse, is gratefully
acknowledged. It is a pleasure to thank Ms. J. Webb for her care in the preparation of this manuscript.
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(Received 8 June 1983; received for publication 9 November 1983)