- Email: [email protected]
On the J-integral concept for elastic-plastic crack extension
Nuclear Engineering and Design 131 (1991) 157-173 North-Holland
157
On the J-integral concept for elastic-plastic crack extension Huang Yuan
a
and Wolfgang Brocks b
a Institute of Materials Research, GKSS-Research Centre, Geesthacht, Germany b Federal Institute for Materials Research and Testing (BAM), Berlin, Germany Received 6 May 1990; revised version 16 January 1991
In the present paper, the path-dependence and variations of the J integral in the near field during ductile crack extensions were investigated analytically and numerically. It is shown that the J integral is a monotonic function of its integration contour, when it becomes path-dependent. Especially, the J integral evaluated directly at the crack tip approaches zero. Our finite element studies confirm this theoretical prediction. At last, we discuss the J integral from viewpoint of the energy balance and the path-dependence is related to the energy dissipation in the vicinity of the crack tip.
I. Introduction
The J integral [1] was introduced for the evaluation of crack problems more than twenty years ago, which showed the applications of the J integral in cases beyond the limits of linear elastic fracture mechanics based on the K concept. An important theoretical background of the J integral as an elastic-plastic fracture criterion for crack initiation was supplied by the asymptotic solution of the crack tip field [2,3], the well-known HRR-theory. It was shown that under the deformation theory of plasticity there exists a field at the crack tip which is uniquely characterized by the J integral. The J integral represents remote loading on the crack tip field and is the stress intensity factor, as does K for a linear elastic crack field. Furthermore, it was proven that the J integral is equivalent to the energy release rate in elastic materials. Thus, two completely different concepts meet in the J integral, a concept of global energy balance and a concept of intensity of the local field singularity. Numerical studies [4,5] based on the incremental theory of plasticity and large deformations confirm that there is a zone around the crack tip where the J integral dominates, although, in a strict sense, the assumptions of the HRR-theory are violated. It means, before the crack initiates, the crack tip field is loaded nearly proportionally and the deformation theory is still applicable there. Although the stress and deformation fields around the crack tip are changed strongly after the crack initiation, Hutchinson and Paris [6] argue, if the crack growth is restricted to a small amount and the J resistance curve increases rapidly enough, the J integral can control the crack growth. Actually, maintaining J dominance of the stress and deformation field in the crack tip vicinity would mean that neither elastic unloading nor non-proportional loading may occur there, which is incompatible with any ductile crack growth at all. Thus, conditions for 'J control' may just admit of a sufficiently small amount of crack growth with practically adequate accuracy, but they are not suitable means for a well-founded continuum theory of ductile crack growth. Numerical computations [7,8] show that, as soon as the crack initiates, the environment of proportional loading around the crack tip no longer exists and thus, the J integral loses its path-independence. The J integral can become path-dependent immediately after the crack initiation and the path-dependent region spreads rapidly, which depends on the initiation value of J and possibly on specimen geometries. Asymptotic analyses of elastic-plastic materials [9-12] exhibit that, comparing to H R R solution, the singularity at a growing crack tip decreases and in this case the 0029-5493/91/$03.50 © 1991 - Elsevier Science Publishers B.V. All rights reserved
H. Yuan, |4( Bn)cks / J integral for elastic-plastic crack extension
158
energy release rate vanishes in elastic-plastic crack growth, cf. [13-15]. Strifors [16] shows that only in asymptotically elastic crack extensions the uniqueness condition for entropy balance equation in the vicinity of the crack tip is satisfied. There are numerous experimental and numerical results which support J as fracture parameter for ~a small amount' of crack growth and some authors try to extend this concept to large crack extensions theoretically through introduction of some so called path- or path-area-independent integrals, see e.g. [17-19]. In the present paper we discuss the crack tip field and the J integral for quasi-static crack growth theoretically, to examine changes of the J integral in ductile crack extension and its thermodynamic background and to investigate the limits of the energy form integrals in continuum fracture mechanics. Without loss of generality, our discussions here are restricted to a 2-dimensional cracked body and quasi-static crack propagations, only.
2. Crack tip field and the local integral in ductile crack growth In the vicinity of a singular crack tip, kinematics of continuum mechanics are applied to derive asymptotic expressions for the field of particle velocity and for material time derivative of stresses and strains. Let x , be Lagrangean coordinates which are bound to material particles of the reference configuration, and so, the Eulerian ones which are connected with the moving crack tip, generally =
t),
(2.1)
where the Greek indices represent different coordinates and take the values 1 and 2 in 2-dimensional crack problems, w~ is the vector of the velocity of crack propagation in Lagrangean coordinates and t is the time variable. Especially, if the Eulerian coordinates are linked to the propagating crack tip and move together with the tip (2.2)
~ =x, - w,t.
In the vicinity of the crack tip the stresses and strains can be determined in the form %¢ = o',0(~: ,, t),
e~0 = e,,0(~,, t).
(2.3)
On the right hand side of the equations all stress and strain functions are described in the Eulerian coordinates. In the coordinates of eq. (2.2) the stresses and strains become explicitly time-dependent, if and only if the crack propagates non-steadily. The material derivatives are then given by d"8
d% 0 d~- -
0% 8 w~%#,~(a~, t) + --~--(~:~, t)
i~t~
de,b dt
ae~0 w,e~0,,(~:K, t) + - ~ t (scK, t).
(2.4)
Here ( ),~ represents a derivative with respect to the spatial coordinate x v and each duplicated Greek index in a tensor product means summation of all components. The second terms of the partial derivatives in eq. (2.4) describe the changes with time of the fields at the crack tip and do not explicitly depend on the velocity of crack extension. If the crack propagates continuously, i.e. the velocity of crack extension is differentiable with respect to time, and the vicinity of the crack tip is homogeneous, the local change rates with time are of lower order of singularity than the spatial derivatives and, hence, cannot influence the governing singularity of stresses and strains at the crack tip. This implies also that these singularities are the same for both steady-state and non-steady-state crack extension processes. From the
H. Yuan, W. Brocks / J integralfor elastic-plasticcrack extension
159
viewpoint of asymptotic analysis of the crack field the solutions of the steady-state crack extension [9-12] are also an approximation of non-steady-state crack extension. Especially, if the second terms of eq. (2.4) vanish, that is, if the crack propagates steadily, then we have for mode I do-~ dt
d
ao-,t3 ~SCl
(~:~, t),
d%~
d
dt
0%~ 0~:1
(~:~ t), '
(2.5)
where t~ is the crack propagating velocity. The material derivatives are replaced by their gradients, as all asymptotic analyses before have assumed. From these solutions it is also to be supposed that any crack extension will change the singularities of stress and strain fields fundamentally, compared with the singularity of a stationary crack as given by the H R R theory. This change of singularities occurs regardless whether or not the crack grows in a steady-state manner. As is well known, the J integral of Rice [1] becomes path-dependent after the crack initiates. To investigate the path-dependency of the J integral in the near field theoretically, we define a path-areaindependent integral in the crack tip field, similar to the J integral by Rice [1], L( ro) =
-
(2.6)
ds,
and W = ftltr~t3E,~t3 ' dt,
(2.7)
where F, is shown in fig. 1 and e is its characteristic dimension. It is supposed that at time t = t 0, o-~ = %~ = 0. To distinguish J, from the J integral of Rice, we call it the local integral. It is easy to see that the J, integral as defined is identical to the J integral of Rice if and only if J is path-independent which requires the well-known postulates on the material behaviour. If the J integral becomes path-depenent, e.g. in ductile crack extensions, J,(F,) is made path-area independent with respect to any F by introducing an area-integral-term,
(2.8)
J.( F.) = fr(Wnl -n.(r.t3ut~,l ) d s - f V _ v ( W , 1 - OVa/3"ot/3,1) dW,
where V - V, is the integration domain between F and F,, see fig 1. From Eq. (2.6), the local integral is generally a functional of its integration contour which represents the path-dependence of the J integral. In the next section we are going to discuss the variation of J~ in the near field during crack extension.
) X2
XZ
Fig. 1. Integrationcontours.
F
fl. Yuan, W. Brocks / J integralfor elastic-plasticcrackextension
160
3. On path-dependence of the J integral in ductile crack growth Imagine two arbitrary integration contours, denoted F and F~ here, see fig. 1, in the near field where Eq. (2.4) holds. We suppose w 1 = ~ and w 2 0 in the following discussions. From Eq. (2.4) we have =
8W
0e,~ - -
8x i = o-~ Ox1
+H(r,
tO, t),
(3.1)
where
H(r, O, t) = -~ -~ oo'~,~ d t - % ~
at ]'
(3.2)
describes the changes with time of the strain energy density in the Eulerian coordinates. If the crack propagates steadily, i.e. if the state of the stress and strain fields is independent of the crack extensions, H(r, to, t) = 0. From the analysis of eq. (2.4) the function H must have a weaker singularity than second order, that is
H ( r , O, t) = o ( r - 2 ) .
(3.3)
Thus, the difference of the local integral between the two integration contours F and F~, cf. eq. (2.8) is
L(r)
-
L(r,) = fv-vH ( r ,
O, t ) d V ,
(3.4)
which vanishes if both integration contours approach to the crack tip. Regardless of the state of ductile crack growth, the local integral is asymptotically path-independent, cf. [20]. Especially, if the crack propagates steadily, the J, integral is exactly path-independent in the whole steady state field due to H=0. From the asymptotic path-independence, we have the uniqueness of the limit value of the local integral at the crack tip. We consider two different models of the crack tip here. (1) Suppose the crack tip is a mathematical point and the stress and strain distributions in the vicinity of the tip are controlled by the asymptotic solution [9-12]. All known solutions show a decrease of the singularities of stresses and strains in the tip field during elastic-plastic crack extensions. These singularities cannot maintain a finite value of the local integral, cf. eq. (2.6). The local integral is equal to zero if its integration contour approaches to the tip, as Rice [15] and Nguyen [13,14] pointed out.
Fig. 2. Evaluationof the J, integral at the tip.
H. Yuan, W. Brocks / J integralfor elastic-plastic crack extension
161
(2) Suppose a blunting crack tip with an arbitrarily small initial root radius, 6 ~ 0, see fig. 2, cf. [1]. We assume there are no loads on the crack surfaces F 8, i.e.
rf~U~,l ds = 0,
(3.5)
and the local integral can be calculated on the surface directly,
L(r~) = frW
dx 2.
(3.6)
On the other hand, following eqs. (3.5) and (3.1)
f / ~u~, 1 d s
=
fv,~ O~t/3,1
dV=
-Oxi --,dV.
(3.7)
Thus, the local integral on the contour F is
J,( F) = [jrw
dx2 -
fv( ~x,~W- H) dV.
(3.8)
The second term in the area integral is a correction to ensure path-independency. It follows from eqs. (3.4) and (3.8) that
L(r.)= frW dx2- fvWx dV.
(3.9)
Since the integration contour in eq. (3.9) is arbitrary in the field discussed here, we consider, e.g. a rectangular contour centered at the crack tip with four corners ( - ~ o , - % ) , (~o, -~o), (~0, % ) and ( - ~ 0 , T0), respectively, it follows
srW d x 2= So- n ° W ( - ~ 0 ,
xz) d x z = J
-+",,W(f0,
fv~xl dV=
x2)-W(-sC°'x2)]
- -
x2)
71o
dxz +
W(-fo, xz) dx2,
(3.10)
~ll
and -no W(~:°
dx2'
(3.11)
that is J~(Fs) = 0.
(3.12)
Thus the limit value of the local integral at the crack tip vanishes in both cases for E -o 0. It results in, cf. eq. (3.4), that the local integral on an arbitrary contour F can be replaced by an area integral
L( F) = fH(r, ,9, t) dV.
(3.13)
From eq. (3.13) it is to see clearly that the local integral describes the changes with time of the crack tip state in Eulerian coordinates. If the crack propagates steadily, i.e. if a steady state field at the crack tip exists, the local integral is equal to zero in the whole field. We are going to study the variation of the local integral outside the crack tip. Suppose the crack field, where the local integral is concerned, consists of two parts, a plastic loading zone and an elastic
H. Yuan, I~E Brooks / J integral for elastic-plastic crack extension
162
unloading zone. According to asymptotic analysis of the steady state crack extension [9], the plastic reloading is restricted to a quite small region near to the crack flanks and plays little role in the computation of the local integral, if the integration contours do not come to the crack flanks. The elastic region outside the plastic loading and the elastic unloading zone cannot influence our results, as the local integral is always path-independent there. Suppose the particle (x~) in the plastic loading zone is being unloaded at time t = ~-(x~),
(3.14)
and the equation of the boundary curve
x 1 =Xl(X2, t),
(3.15)
at any time t. The total strain rate is decomposed in an elastic and a plastic part •e -p ~,~(x~, t) = e,~(x~, t) +e~t~(x ~, t),
(3.16)
if we restrict our discussions to power-law hardening materials, then follow
~t3(x~, t) =
(l+v) E
q~(x~, t) =
. u cr.t3 - -~d~,6,t 3,
(3.17)
-f ~2-~e~s~
for t <_.~(x~)
0
for t > ~'(x~),
(3.18)
where s,,t3 is the stress deviator and ~r~ the effective stress defined by ~r~ = 3s~t3s~tJ2. E and u are the elastic modulus and Poisson's ratio, respectively. C is a material constant characterizing strain hardening and 8,t 3 is Kronecker's symbol. Similarly, we can decompose the strain energy function defined by eq. (2.7)
W(x~, t) = W~(x~, t) + WP(x~, t),
(3.19)
with 1
We(x~, t) ~- ~ { ( 1
WP(xK, t ) =
+ v)o'.o(r,,t3 - v(r~;2},
2C
~r~+l(x~, t),
3 E ( n + 1)
~7+'(x~, ~),
(3.20)
for t ~(x~).
(3.21)
From eq. (2.4) the plastic strain rate in the plastic loading zone can be represented by
q~(xK, t) =
_
w~,,~.~(~K, t) + ~oe~t~ --(~, o
t).
(3.22)
The first term on the right hand side in eq. (3.22) describes the changes due to crack extension and the second term represents the local changes of the plastic strain in the field at the crack tip. It can be assumed that the second term is continuous for crack extension in homogeneous materials, otherwise, it could result in a sudden change of the total plastic strain in Eulerian coordinates. It follows that due to
H. Yuan, W. Brocks / J integral for elastic-plastic crack extension
163
continuous plastic strain rate the gradient of the plastic strain must also be continuous in the whole crack field. By substituting eq. (3.18) into (3.22) it follows in the plastic loading zone
/
(
t)
so 7 o(x ,t)
=0. (3.23)
Similar to eqs (2.4) and (3.22), the first set of brackets in eq. (3.23) reveals the influences from the crack extension, the second set does not depend on the crack growth velocity explicitly and represents the change of the crack tip state. In steady state crack extension the second set vanishes since the crack tip field becomes invariant with respect to time. On the other hand, from the knowledge of crack tip field analysis [9,10,16] we can hope all terms in the second set near to the crack tip will be much smaller than the ones in the first set. Therefore, we can assume asymptotically C
eP~,v(x ,, t)
0
t) ----0, for t <_r ( x . ) .
- --~o'n-2safl-~---O'e(XK, /2;
(3.24)
OX7
From eqs. (3.24) and (3.21) follows a
a
--WP(x, ~Xr
'
P t) = tGt3(x , , t ) - -~ e~t3(x .,t), O~
for t < ~-(x,).
(3.25)
for t < r( x~),
(3.26)
We integrate eq. (3.17), then
e~t3( x ~, t)
(l+v)
v
E
o',t3 - -~o'rr~,t3,
and substitute eqs. (3.21), (3.26) and (3.25) with 3' = 1 into eq. (2.8), so the second integral on the right hand side vanishes. It means that the J integral is path-independent in the plastic loading zone. In the elastic unloading zone the increment of the plastic strain is zero, see eq. (3.18). It follows for t ___~'(x~) that
Ox--~W(xK, t) -~G~(xK, t)e~t3,v(x~, t) 0 3xv
W°(x~, r) - % , ( x ~ ,
t ) e P , , ( x ~ , r) = {tr~t3(x~, ~') - % , ( x ~ , t)},Pt3.~(x ~, ~').
(3.27)
As discussed for eq. (3.22) above, the gradient of plastic strain during crack extension is continuous in the whole crack field. Thus, from eq. (3.24) with t = r(x~) we have 0
--W( Oxv
x~, t) -tr~t3( x~, t)e,~t3,~,(x,,, t) 0
= f ( x ~ , t, r)~xvtre(x ~, r),
for t < r ( x x ) ,
where
f ( x~, t, r) = C o ' 2 - 3 s ~ ( x ~, ~'){s~(x~, "r) - s~,( x~, t)}.
(3.28)
H. Yuan, W Brocks / J integralfor elastic-plastic crack extension
164
i
elasticunloadingzone
\(Zo,hz)
/(Z,,hz)
I7 crack \ ! r v,~ / , ___ ~ _ l ( z " h ' / ~ a s t i c z ° n e x
x2
J
Fz
plasticloadingzone
xl
.,- .....
ao
~ "
~
,aa
. . . .
) I
t
Fig. 3. Crack tip field and integration contours.
With help of the unloading condition
s,~(x~, r)s,,~(x~, "r) < s,~(x~, t)s~a(x~, t)
we have
f(x~, t, r) > O.
(3.29)
Consider two arbitrary integration contours around the crack tip, /'1 and /'2, the difference of the local integral between /'1 and /'2 is equal, see eq. (2.8) 0 Je(/'2)
- J,(/'l)
t, T)-~--~
=-vf'l f(xK'2
r)
dV,
(3.30)
where the integration area Vt2 locates between the two contours F 1 and F z. Equation (3.29) exhibits the relation between the local integral and the gradient of current yield stress in the elastic unloading zone. Since the distribution of the current yield stress is unknown, we cannot generally predict the behaviour of the local integral from Eq. (3.30). We consider two rectangular integration contours for mode I case as shown in fig. 3 which were used in finite element studies, see section 4. Suppose that the points with coordinates (-)Co, h~) are located behind the initial crack tip, so that generally the yield stress on the particle near to the crack tip is larger than one far from the tip, i.e. ge(Xo, h,, r)<_ ge(xl(h~, r), h~, z) holds. Then, from eq. (3.30) the difference of the J, integral between the two contours, with help of mean-value theorem for integration, is
J,(r2) - J , ( r , )
:
,,
x,(x2,
x2,
x:,
dx:, (3.31)
where
¢1 E [)c0,
Xl(X2,
z)], ~'2 ~ [hi, h2].
It means that the local integral is an increasing function of the contour height. Since the local integral vanishes at the crack tip, thus, the local integral calculated on the rectangular contours is a monotonic function with a lower limit zero and an upper limit Jff, the so called far-field J integral.
4. Finite element investigation of ductile crack extensions
For further studies of the J, distribution around the crack tip, five specimens from an aluminium alloy (see table 1) were investigated by finite element method [21], i.e. two Compact Tension specimens (CT), two Center Cracked specimens (CCT) and one Single Edge Cracked Bend specimen (SECB). The material properties are the following: elastic modulus E = 71600 M N / m 2, yield stress % = 317 M N / m 2 and ultimate tensile stress o~u = 440 M N / m 2. Aaexp in table 1 denotes the maximum amount of crack growth reached in the experiment. The referenced experimental data for these specimens (loading and J-resistance curves) have been obtained by Schwalbe and Hellmann [22].
165
H. Yuan, W. Brocks / J integralfor elastic-plastic crack extension
Table 1 Geometries of numericallyinvestigatedspecimens No.
Specimen
1 ~2 ~3 ~4 5
CT CT CCT CCT SECB
W (mm) 50 100 50 100 50
B (mm) 5 20 5 20 5
a0 (mm) 25.1 71.0 25.3 74.0 25.5
Aacxp (mm) 14.44 17.15 8.58 8.18 11.90
The experiments were simulated by following the load line displacement records and all calculations base on plane stress models and Prandtl-Reuss flow theory. The crack growth was modeled by using the node release technique without node shifting. The node forces were slowly released at each node in a simultaneous, nonlinear way to model an increment of crack growth. No singularities in the crack tip element were included. The FE-program and the FE-mesh used here were validated through a number of numerical computations and comparisons with A D I N A calculation, see [21,23]. A typical finite element mesh in the near field is shown in fig. 4. The sizes of the elements directly at the tip are 0.25 mm × 0.8 mm. All computations take the same mesh in the near field. The meshes far from the crack tip for specimens ~ 2, ~ 3 and ] 5 are shown in Fig. 5a-c. VLL stands for load line displacement which were measured in experiments by Schwalbe and Hellmann [22] and VL is the displacement of the controlling curves in numerical simulations which was calculated from VLL curves by preceding FE calculations. More detailed FE-results about geometry effects on the crack tip opening displacement, J integral and the crack tip opening angle in ductile crack extensions were reported in [21]. The J, integral was evaluated according to the definition in Eq. (2.6). Numerical integration was performed through linear interpolation of integrands between Gaussian points. To investigate the variation of the Jr integral and effects of specimen geometries, six integration contours were defined for every specimen, see fig. 6 for CT specimen ~ 1. The contour Ftip was located on the tip directly and moved with the tip during crack extension, so that the correspondent contour of the integral values covers timely varying material particles. Its dimensions are 1.0 mm in height and 0.8 mm in length. The related J, value is denoted as Jtip. The contours F 1 to F 4 are defined in the near field, as denoted in ref. [24], and differ from one another mainly by their height h, cf. fig. 3. Their width is larger than the maximum crack extension allowed in the FEM calculation, Aamax--- Aaexp, SO that they contain the crack tip at all times. Our numerical experiments have shown that the length of the integration contours does not affect the evaluated Jr integral value, as predicted in section 3. The corresponding Jr values are denoted with
IIIIIIIIIIIIIII
IIIIIIIIIIIIIII
---~11 m,~ N
Fig. 4. Detail of mesh refinement in the vicinityof the growingcrack tip.
H. Yuan, W. Brocks / J integral for elastic-plastic crack extension
166
VL
I V LL 14
a
"1
(a) specimen #2 (CT)
VL
--~ VL
c.;
,V LL
\\\ \ \
!///~-/
///
••i!
I II1,,~
"~,~,~,~,,,,
I,
a
-I
I 1/11 I I///
VL
.i
'""
V LL a
m
(c) specimen #5 (SECB) (b) specimen #3 (CCT) Fig. 5. Element meshes in the far-field.
Jnf,i(i = 1. . . . . 4). The contour Fff is allocated in the so-called far field where the J integral is still path-independent for large crack extensions. The J integral evaluated is called Jff. The finite element computations agree with experimental records quite well, cf. [21]. As known, the J integral is path independent before the crack initiates, that is, the J, integral is identical with the J integral. After'crack initiation, the further increase of the Jff curve with crack extension, A a , depends on the specimen geometry, see fig. 7. Moreover, the J integral loses its path-independence rather quickly, see
H. Yuan, W. Brocks / J integral for elastic-plastic crack extension
167
F# IJ~ )
I, F IJ,r.e~) r3 .(J,) .3
I
! I i
\ r4 (J,~.4)
! Crack
VLL
-t~ mm
~!. I
r.
ao
L~.ai~~- -
1 ap ( Jtio
W
). ,!
Fig. 6. Upper half of CT-specimen ~ 1 with integration contours for the local integral.
Fig. 8a, b. The ./,-resistance curves evaluated in the near field rise much less than Jn, especially, they become constant after a transition phase, as [18,24] reported earlier. Hutchinson and Paris [6] hoped, that under the so called conditions for J controlled crack growth, HRR-theory still holds, so that the J integral can dominate in some zone around the crack tip. If we allow for a crack growth of 6% of the ligament width, i.e. a = A a / ( W - a 0) = 0.06, we will obtain deviations of the J~ integral from the far-field J, Jff - Jnf,1 S~
(4.1)
Aa=ct(W-ao)
Jff
which are summarized in table 2. As the choice of the contour F 1 is arbitrary, the deviation will be even larger if a smaller contour is defined, which explains the inequality sign in eq. (4.1). Obviously, the deviations of some specimens are much larger than 10% which is the admissible deviation supposed in [24]. If we allow for a 10% deviation of J, from Jff, the absolute and relative crack extensions under this condition are denoted as Aal0 and al0 in table 2, respectively. The permitted relative crack growths, al0, do not reveal clear and unambiguous differences between bend specimens, ~ 1, ~ 2, ~ 5, and tension ones,
300]
#I
t'*~
:::'.:'.:::.. ......
o
. . . . . . . . .
0
•
#5
,,/ff
i
5
. . . . . . . . .
i
10
. . . . . . . . .
15
,an (ram) Fig. 7. Effects of specimen geometries on the J n f and Jfr values.
168
H. Yuan, W. Brocks / J integral for elastic-plastic crack extension 300,
200.
I00,
"'~"~., o ,
.
~ ,.~ ... J~P.,,,.~ .-- ." .--',---.----.-- .---.~'"~-~. .~ 5
10
An (ram) 300
(b)
~ 200
~
100
~ ~,,."
,
f
3
' Jnf,1
.........
0
0,0
.
.
.
.
2,5 z
.
.
.
5,0
.
i
.
.
.
,~
.
(ram)
i
7,5
(b) specimen3 Fig. 8. Path-dependency of the J integral in ductile crack extensions.
3, ~ 4. A crack growth of 6% permitted for bend specimens in ref. [24] may yield a path-dependency of J as defined of more than 20% and, hence, seems not to be strict enough. The effects of specimen geometries on the J, values evaluated on the contour F 1 are not so distinct as on the Jff curves, see fig. 7. For large crack extensions all J,e,~ curves come nearly together, the geometric effects are much smaller than directly after crack initiations. In addition, they become independent of Aa after a transition phase, as mentioned before, which should be postulated for any
Table 2 P a t h - d e p e n d e n c y o f the local J integral Specimen
~1
~2
~3
~4
~5
s [,,=o.o6 > (%) zlalo _<(mm) Aalo
21 0.5
16 1
16 1
23 1.2
23 0.75
O/IO W- a0
0.020
0.034
0.040
0.016
0.029
H. Yuan, W.. Brocks / J integral for elastic-plastic crack extension
169
1,010 0,75 0,5]0 0,2I5 0,00
. . . . . . . . .
o,o
~
.
z,5
.
.
.
.
.
.
.
.
,
.
.
.
.
.
.
.
.
.
5,0
~
. . . . . . . . .
7,5
1o,o
height h (ram)
Fig. 9. J distributionin the near field. crack growth controlling parameter in steady state. From this evidence, the J integral or similar ones evaluated in the near field have been suggested as fracture parameter for ductile crack extensions in [18,19]. However, fig. 9 explains that this apparent loss of geometry dependence of J,f resistance curves is mainly due to the fact that in general J, decreases with decreasing radius of its contour F,, as predicted in section 3. Thus, any geometry dependence must necessarily diminish if the limiting value for all curves is zero. Jtip evaluated on a contour Ftip attached to the moving crack tip appeared to be zero, anyway, see fig. 8a, confirming the conclusion in ref. [15] that there is nothing like an energy release rate for elastic-plastic crack extensions.
5. Energy balance in ductile crack extension The thermodynamic principles must also be obeyed in the tip region during crack extension. Without loss of generality we consider all thermodynamic variables to be continuous during the whole physical process. Then the principle of balance of energy in its local form is = cr~,v~,o + h - q~,~.
(5.1)
We denote the specific internal energy with e, the specific heat absorption with h, the heat flux with q~ and the rate of deformation with v,,. Let us consider parts of a cracked body, V and V,, according to fig. 1 for the two-dimensional case. n , stands for the unit normal vector on the boundaries F and F~. The part V, contains the crack tip which moves through the body with a velocity we. The internal energy of the part V - V, is
(5.e)
E ( V - I,I,) = fv-v, e dV, and the kinetic energy K(V-
1 V,) = 2 f - v ,
dV,
(5.3)
H. Yuan. HLBrocks / J integralfor elastic-plasticcrack extension
170
where p is the mass density. From Reyno[d's transport theorem we have the rate of the internal energy
e(v-v,)=fv-v~Odv-ffwon°as,
(5.4)
and the rate of the kinetic energy
K( V - V,) = f , _
~fo~,.v.w~n~ds. 1
pu~u, d V -
(5.5)
It is assumed that all integrands have the necessary continuity properties. The mechanical power from volume loads f , and surface traction t, is defined by
M(v)=fv~L dV+ f~,v°t° d,,
(5.6)
where ~V denotes the surface of the volume V. The heat power is defined by
Q ( V ) = f~h dV+ ~,qo, no, as.
(5.7)
According to the law of conservation of energy, thermo-mechanical processes in an arbitrary part V of a body are governed by
E ( V - V,) + I ( ( V - V,) + ,~(,) = M ( V ) + Q ( V ) ,
(5.8)
where ~(E) is the energy dissipation rate in V, which contains the crack tip. From eqs. (5.1), (5.8) and the equilibrium equations of continuum mechanics we have 1
This is the time rate of change of surface energy in the crack tip vicinity introduced in ref. [16]. Regardless of material behaviour and crack growth velocity, eq. (5.9) describes the energy flux through F~ to the tip. Generally the integral value depends on its integration contour. Its difference between two integration contours is the energy dissipation rate in this region. We introduce a dissipation rate functional of energy
(e
+
where 6 and e are characteristic sizes of integration contours F~ and F~, and F~ is situated within F~ completely. Physically this functional describes the difference in energy flux between both integration contours, that is, the dissipation in the area between the two contours. From this physical significance /-/(S, e) >_0
(5.11)
holds and it must be a monotonically increasing functional of 6 and monotonically decreasing functional of e. With the Gaussian theorem eq. (5.10) becomes
n ( ~ , ,) = fv' - ~
[-eiwiq-o-ij~ij ] ,
dV,
(5.12)
where the integration domain V~ - V, is situated between F~ and F,, and the effects of heat conduction, kinematic energy and volume loads are neglected. Comparing to eq. (3.24), eq. (5.12) reveals the
H. Yuan, W. Brocks / J integral for elastic-plastic crack extension
171
relationship between path-dependency of the J integral and the energy dissipation in the vicinity of the crack tip. If and only if the divergence of the integrand in eq. (5.10) vanishes, i.e. the J integral becomes path-independent, no energy dissipation exists in Vn - V,. Especially, if the divergence in the crack tip vicinity is equal to zero, i.e. 11(6, 0) -- 0, then ~(•) is path-independent for arbitrary F8 and represents the energy flux to the crack tip. In mode I case the crack extends on the uncracked ligament, suppose at speed d. The energy release rate is G = limE---,o
(5.13)
d
Neglecting the effects of heat conduction, kinematic energy and volume loads in eq. (5.9) and appling Eq. (2.4) to the velocity vector, we then obtain G -- lim f
(en 1 -Ui,lorijnj)
ds,
(5.14)
which is the local integral evaluated at the tip, see eq. (2.6), where e = W holds. The local integral, J,, is equivalent to the energy release rate G. As known, the J integral of Rice is equal to the energy release rate, if no energy dissipation exists around the crack tip. To study the energy dissipation in the crack vicinity, we introduce a functional 1 1. ~ ( • ) = ~-11(•, 0) = ~-,~(•) - G.
(5.15)
The functional qt(e) represents the energy dissipation rate in the region around the tip and due to its physical significance it must be a monotonically increasing functional of its integration contour. On the other hand, it is to be seen, that from the definition of the local integral J, for • -~ 0, ~ ( • ) is equivalent to the area-integral in eq. (2.7). Thus, from energy balance we come again to the prediction on the local integral behaviour in the near field. Generally speaking, the medium around the tip absorbs energy during crack extension through inelastic deformation and material degradation, that is ~ ( e ) >_riG,
(5.16)
where E is an arbitrary real number. A necessary condition for a non-trivial but bounded energy release rate is that the integrands in Eq. (5.14) must exactly have a first order singularity. Unfortunately, all asymptotic solutions of steady crack extension inelastic-plastic materials show that the singularities of stresses and strains cannot meet this condition, i.e., the energy release rate is equal to zero, thus ~(E) = tiqt(e).
(5.17)
This means that all energy supplied by external loads is spent by plastic deformations and material degradation in the vicinity of the crack tip, in the so-called process zone, whereas the crack extension itself does not need any surplus energy, as Rice [15] pointed out. Thus, the energy balance equations do not provide any answer to the phenomena of ductile crack growth, though they undoubtedly must also hold in the vicinity of the crack tip. Help may come by introducing special constitutive equations [25,26] instead of Von Mises plasticity to account for the continuing process of damage ahead of the crack tip and thus not needing any 'energy release rate' or 'crack driving force'.
172
14, Yuan, W. Brocks /.1 integral for elastic-plastic crack extension
6. Conclusions In the present paper the J integral behaviour in the vicinity of a crack tip in ductile crack extension is discussed. It is shown by a mathematical analysis of crack tip fields that the J integral is monotonically increasing function of its integration contour, if it becomes path-dependent. The local integral is asymptotically path-independent during ductile crack extensions, while in the steady-state field it becomes exactly path-independent. No finite lower limit exists in the crack tip field, i.e. J, vanishes with vanishing contour radius e. Our FE-computations have confirmed this prediction and show that the J integral may lose its path-independence immediately after crack initiation. From the point of view of an energy balance the path-dependence is connected to the energy dissipations in the crack tip region. The local integral describes the energy dissipated in the vicinity of the tip. The analysis points out generally that the energy description of ductile crack extension may lead to paradoxical conclusions due to the restrictions of the fracture mechanics model of a singular crack tip in a continuum. Thus all other path- or path-area-independent integrals which have been supposed continuously by some authors for elastic-plastic crack growth cannot bring any new informations about the crack tip state.
Acknowledgements This work was supported by the German Research Foundation (DFG) under contract numbers Br 521/2-1 and Br 521/2-2. The authors thank Dr D. Hellmann of GKSS for the experimental data used for the numerical computations here.
References [1] J.R. Rice, A path-independent integral and the approximate analysis of strain concentration by notches and cracks, J. Appl. Mech. 35 (1968) 379-386. [2] J.W. Hutchinson, Plane stress and strain fields at a crack tip, J. Mech. and Phys. Solids 16 (1968) 13-31. [3] J.R. Rice and G.F. Rosengren, Plane strain deformation near a crack tip in a power-law hardening material, J. Mech. and Phys. Solids 16 (1968) 1-12. [4] W. Brocks and J. Olschewski, On J-dominance of crack-tip fields in largely yielded 3D structures, Int. J. Solids Struct. 22 (1986) 693-708. [5] R.M. McMeeking, Finite deformation analysis of crack tip opening in elastic-plastic materials and implications for fracture initiation, J. Mech. and Phys. Solids 25 (1977) 357-381. [6] J.W. Hutchinson and P.C. Paris, Stability analysis of J-controlled crack growth, in Elastic-Plastic Fracture: ASTM STP 668, eds. J.D. Landes, J.A. Begley and G.A. Clarke (1979) 37-63. [7] W. Brocks, H.H. Erbe and H. Yuan, Numerical study of stable crack growth, Failure Analysis Theory and Practice, Vol. A, ed. E. Czoboly (1988) pp. 205-210. [8] W. Brocks and H. Yuan, Numerical investigation on the significance of J for large stable crack growth, Eng. Fract. Mech. 32 (1989) 459-468. [9] J.C. Amazigo and J.W. Hutchinson, Crack tip field in steady crack-growth with linear strain-hardening, J. Mech. and Phys. Solids 25 (1977) 81-97. [10] P. Ponte Castafieda, Asymptotic fields in steady crack growth with linear strain hardening, J. Mech. and Phys. Solids 35 (1987) 227-268. [11] W.J. Drugan, J.R. Rice, and T.-L. Sham, Asymptotic analysis of growing plane strain tensile cracks in elastic-ideally plastic solids, J. Mech. and Phys. Solids 30 (1982) 447-473. [12] Y.C. Gao, X.-T. Zhang and K.-C. Hwang, The asymptotic near-tip solution for Mode III crack in steady growth in power law hardening media, Int. J. Fract. 21 (1983) 301-317.
H. Yuan, W. Brocks / J integral for elastic-plastic crack extension
173
[13] Q.S. Nguyen, A thermodynamic description of the running crack problem, in Three-Dimensional Constitutive Relations and Ductile Fracture, ed. S. Nemat-Nasser, (1981) pp. 315-330. [14] Q.S. Nguyen, An energetic analysis of elastic-plastic fracture, in Defect Assessment in Components-Fundamentals and Applications, eds. J.D. Blauel and K.-H. Schwalbe, (1991) pp. 75-85. [15] J.R. Rice, The mechanics of quasi-static crack growth, Proc. 8th US National Congress of Applied Mechanics, ed. R.E. Kelly, (1979) 191-216. [16] H.C. Strifors, Thermomechanical theory of fracture: kinematical and physical principles, Report 27, Department of strength of materials and solid mechanics, Royal Institute of Technology, Stockholm (1980). [17] S.N. Atluri, Path-independent integrals in finite elasticity, with body forces, inertia and arbitrary crack-face conditions, Eng. Fract. Mech. 16 (1982) 341-364. [18] F.W. Brust, R. Nishioka, S.N. Atluri and M. Nakagaki, Further studies on elastic-plastic stable fracture utilizing the T* integral, Eng. Fract. Mech. 22 (1985) 1079-1103. [19] T. Nishioka and H. Yagami, Invariance of the path independent T* integral in nonlinear dynamic fracture mechanics, with respect to the shape of a finite process zone, Eng. Fract. Mech. 31 (1988) 481-491. [20] L.B. Freund; Energy flux into the tip of an extending crack in an elastic solids J. Elasticity 2 (1972) 341-349. [21] W. Brocks and H. Yuan, Numerical studies on ductile crack growth, in Defect Assessment in Components-Fundamentals and Applications, eds. J.D. Blauel and K.-H. Schwalbe, (1991) pp. 19-33. [22] K.-H. Schwalbe and D. Hellmann, Correlation of stable crack growth with the J-integral and the crack tip opening displacement, effects of geometry, size, and material. GKSS Report 84/E/37, GKSS-Forschungszentrum Geesthacht. [23] S. Bartmann, W. Brocks and H. Yuan, Bruchmechanische Parameter zur Beschreibung yon stabilem RiBwachstum in duktilem Material, in Proc. of 21st 'Vortragsveranstaltung des DVM-Arbeitskreises Bruchvorg~inge', ed. Ch. Berger (1989) 527-537. [24] C.F. Shih, H.G. de Lorenzi and W.R. Andrews, Studies on crack initiation and stable crack growth, in Elastic-Plastic Fracture, ASTM STP 668, eds. J.D. Landes, J.A. Begley and G.A. Clarke (1979) 65-120. [25] D. Krajcinovic and D. Sumarac, Micromechanics of the damage process, in Continuum Damage Mechanics: Theory and Applications, eds. D. Krajcinovic and J. Lemaitre (Springer-Verlag, 1987) p. 135-194. [26] J. Lemaitre, A continuous damage mechanics model for ductile fracture, Trans. ASME. J. Eng. Mater. and Technol. 107 (1985) 83-89.
157
On the J-integral concept for elastic-plastic crack extension Huang Yuan
a
and Wolfgang Brocks b
a Institute of Materials Research, GKSS-Research Centre, Geesthacht, Germany b Federal Institute for Materials Research and Testing (BAM), Berlin, Germany Received 6 May 1990; revised version 16 January 1991
In the present paper, the path-dependence and variations of the J integral in the near field during ductile crack extensions were investigated analytically and numerically. It is shown that the J integral is a monotonic function of its integration contour, when it becomes path-dependent. Especially, the J integral evaluated directly at the crack tip approaches zero. Our finite element studies confirm this theoretical prediction. At last, we discuss the J integral from viewpoint of the energy balance and the path-dependence is related to the energy dissipation in the vicinity of the crack tip.
I. Introduction
The J integral [1] was introduced for the evaluation of crack problems more than twenty years ago, which showed the applications of the J integral in cases beyond the limits of linear elastic fracture mechanics based on the K concept. An important theoretical background of the J integral as an elastic-plastic fracture criterion for crack initiation was supplied by the asymptotic solution of the crack tip field [2,3], the well-known HRR-theory. It was shown that under the deformation theory of plasticity there exists a field at the crack tip which is uniquely characterized by the J integral. The J integral represents remote loading on the crack tip field and is the stress intensity factor, as does K for a linear elastic crack field. Furthermore, it was proven that the J integral is equivalent to the energy release rate in elastic materials. Thus, two completely different concepts meet in the J integral, a concept of global energy balance and a concept of intensity of the local field singularity. Numerical studies [4,5] based on the incremental theory of plasticity and large deformations confirm that there is a zone around the crack tip where the J integral dominates, although, in a strict sense, the assumptions of the HRR-theory are violated. It means, before the crack initiates, the crack tip field is loaded nearly proportionally and the deformation theory is still applicable there. Although the stress and deformation fields around the crack tip are changed strongly after the crack initiation, Hutchinson and Paris [6] argue, if the crack growth is restricted to a small amount and the J resistance curve increases rapidly enough, the J integral can control the crack growth. Actually, maintaining J dominance of the stress and deformation field in the crack tip vicinity would mean that neither elastic unloading nor non-proportional loading may occur there, which is incompatible with any ductile crack growth at all. Thus, conditions for 'J control' may just admit of a sufficiently small amount of crack growth with practically adequate accuracy, but they are not suitable means for a well-founded continuum theory of ductile crack growth. Numerical computations [7,8] show that, as soon as the crack initiates, the environment of proportional loading around the crack tip no longer exists and thus, the J integral loses its path-independence. The J integral can become path-dependent immediately after the crack initiation and the path-dependent region spreads rapidly, which depends on the initiation value of J and possibly on specimen geometries. Asymptotic analyses of elastic-plastic materials [9-12] exhibit that, comparing to H R R solution, the singularity at a growing crack tip decreases and in this case the 0029-5493/91/$03.50 © 1991 - Elsevier Science Publishers B.V. All rights reserved
H. Yuan, |4( Bn)cks / J integral for elastic-plastic crack extension
158
energy release rate vanishes in elastic-plastic crack growth, cf. [13-15]. Strifors [16] shows that only in asymptotically elastic crack extensions the uniqueness condition for entropy balance equation in the vicinity of the crack tip is satisfied. There are numerous experimental and numerical results which support J as fracture parameter for ~a small amount' of crack growth and some authors try to extend this concept to large crack extensions theoretically through introduction of some so called path- or path-area-independent integrals, see e.g. [17-19]. In the present paper we discuss the crack tip field and the J integral for quasi-static crack growth theoretically, to examine changes of the J integral in ductile crack extension and its thermodynamic background and to investigate the limits of the energy form integrals in continuum fracture mechanics. Without loss of generality, our discussions here are restricted to a 2-dimensional cracked body and quasi-static crack propagations, only.
2. Crack tip field and the local integral in ductile crack growth In the vicinity of a singular crack tip, kinematics of continuum mechanics are applied to derive asymptotic expressions for the field of particle velocity and for material time derivative of stresses and strains. Let x , be Lagrangean coordinates which are bound to material particles of the reference configuration, and so, the Eulerian ones which are connected with the moving crack tip, generally =
t),
(2.1)
where the Greek indices represent different coordinates and take the values 1 and 2 in 2-dimensional crack problems, w~ is the vector of the velocity of crack propagation in Lagrangean coordinates and t is the time variable. Especially, if the Eulerian coordinates are linked to the propagating crack tip and move together with the tip (2.2)
~ =x, - w,t.
In the vicinity of the crack tip the stresses and strains can be determined in the form %¢ = o',0(~: ,, t),
e~0 = e,,0(~,, t).
(2.3)
On the right hand side of the equations all stress and strain functions are described in the Eulerian coordinates. In the coordinates of eq. (2.2) the stresses and strains become explicitly time-dependent, if and only if the crack propagates non-steadily. The material derivatives are then given by d"8
d% 0 d~- -
0% 8 w~%#,~(a~, t) + --~--(~:~, t)
i~t~
de,b dt
ae~0 w,e~0,,(~:K, t) + - ~ t (scK, t).
(2.4)
Here ( ),~ represents a derivative with respect to the spatial coordinate x v and each duplicated Greek index in a tensor product means summation of all components. The second terms of the partial derivatives in eq. (2.4) describe the changes with time of the fields at the crack tip and do not explicitly depend on the velocity of crack extension. If the crack propagates continuously, i.e. the velocity of crack extension is differentiable with respect to time, and the vicinity of the crack tip is homogeneous, the local change rates with time are of lower order of singularity than the spatial derivatives and, hence, cannot influence the governing singularity of stresses and strains at the crack tip. This implies also that these singularities are the same for both steady-state and non-steady-state crack extension processes. From the
H. Yuan, W. Brocks / J integralfor elastic-plasticcrack extension
159
viewpoint of asymptotic analysis of the crack field the solutions of the steady-state crack extension [9-12] are also an approximation of non-steady-state crack extension. Especially, if the second terms of eq. (2.4) vanish, that is, if the crack propagates steadily, then we have for mode I do-~ dt
d
ao-,t3 ~SCl
(~:~, t),
d%~
d
dt
0%~ 0~:1
(~:~ t), '
(2.5)
where t~ is the crack propagating velocity. The material derivatives are replaced by their gradients, as all asymptotic analyses before have assumed. From these solutions it is also to be supposed that any crack extension will change the singularities of stress and strain fields fundamentally, compared with the singularity of a stationary crack as given by the H R R theory. This change of singularities occurs regardless whether or not the crack grows in a steady-state manner. As is well known, the J integral of Rice [1] becomes path-dependent after the crack initiates. To investigate the path-dependency of the J integral in the near field theoretically, we define a path-areaindependent integral in the crack tip field, similar to the J integral by Rice [1], L( ro) =
-
(2.6)
ds,
and W = ftltr~t3E,~t3 ' dt,
(2.7)
where F, is shown in fig. 1 and e is its characteristic dimension. It is supposed that at time t = t 0, o-~ = %~ = 0. To distinguish J, from the J integral of Rice, we call it the local integral. It is easy to see that the J, integral as defined is identical to the J integral of Rice if and only if J is path-independent which requires the well-known postulates on the material behaviour. If the J integral becomes path-depenent, e.g. in ductile crack extensions, J,(F,) is made path-area independent with respect to any F by introducing an area-integral-term,
(2.8)
J.( F.) = fr(Wnl -n.(r.t3ut~,l ) d s - f V _ v ( W , 1 - OVa/3"ot/3,1) dW,
where V - V, is the integration domain between F and F,, see fig 1. From Eq. (2.6), the local integral is generally a functional of its integration contour which represents the path-dependence of the J integral. In the next section we are going to discuss the variation of J~ in the near field during crack extension.
) X2
XZ
Fig. 1. Integrationcontours.
F
fl. Yuan, W. Brocks / J integralfor elastic-plasticcrackextension
160
3. On path-dependence of the J integral in ductile crack growth Imagine two arbitrary integration contours, denoted F and F~ here, see fig. 1, in the near field where Eq. (2.4) holds. We suppose w 1 = ~ and w 2 0 in the following discussions. From Eq. (2.4) we have =
8W
0e,~ - -
8x i = o-~ Ox1
+H(r,
tO, t),
(3.1)
where
H(r, O, t) = -~ -~ oo'~,~ d t - % ~
at ]'
(3.2)
describes the changes with time of the strain energy density in the Eulerian coordinates. If the crack propagates steadily, i.e. if the state of the stress and strain fields is independent of the crack extensions, H(r, to, t) = 0. From the analysis of eq. (2.4) the function H must have a weaker singularity than second order, that is
H ( r , O, t) = o ( r - 2 ) .
(3.3)
Thus, the difference of the local integral between the two integration contours F and F~, cf. eq. (2.8) is
L(r)
-
L(r,) = fv-vH ( r ,
O, t ) d V ,
(3.4)
which vanishes if both integration contours approach to the crack tip. Regardless of the state of ductile crack growth, the local integral is asymptotically path-independent, cf. [20]. Especially, if the crack propagates steadily, the J, integral is exactly path-independent in the whole steady state field due to H=0. From the asymptotic path-independence, we have the uniqueness of the limit value of the local integral at the crack tip. We consider two different models of the crack tip here. (1) Suppose the crack tip is a mathematical point and the stress and strain distributions in the vicinity of the tip are controlled by the asymptotic solution [9-12]. All known solutions show a decrease of the singularities of stresses and strains in the tip field during elastic-plastic crack extensions. These singularities cannot maintain a finite value of the local integral, cf. eq. (2.6). The local integral is equal to zero if its integration contour approaches to the tip, as Rice [15] and Nguyen [13,14] pointed out.
Fig. 2. Evaluationof the J, integral at the tip.
H. Yuan, W. Brocks / J integralfor elastic-plastic crack extension
161
(2) Suppose a blunting crack tip with an arbitrarily small initial root radius, 6 ~ 0, see fig. 2, cf. [1]. We assume there are no loads on the crack surfaces F 8, i.e.
rf~U~,l ds = 0,
(3.5)
and the local integral can be calculated on the surface directly,
L(r~) = frW
dx 2.
(3.6)
On the other hand, following eqs. (3.5) and (3.1)
f / ~u~, 1 d s
=
fv,~ O~t/3,1
dV=
-Oxi --,dV.
(3.7)
Thus, the local integral on the contour F is
J,( F) = [jrw
dx2 -
fv( ~x,~W- H) dV.
(3.8)
The second term in the area integral is a correction to ensure path-independency. It follows from eqs. (3.4) and (3.8) that
L(r.)= frW dx2- fvWx dV.
(3.9)
Since the integration contour in eq. (3.9) is arbitrary in the field discussed here, we consider, e.g. a rectangular contour centered at the crack tip with four corners ( - ~ o , - % ) , (~o, -~o), (~0, % ) and ( - ~ 0 , T0), respectively, it follows
srW d x 2= So- n ° W ( - ~ 0 ,
xz) d x z = J
-+",,W(f0,
fv~xl dV=
x2)-W(-sC°'x2)]
- -
x2)
71o
dxz +
W(-fo, xz) dx2,
(3.10)
~ll
and -no W(~:°
dx2'
(3.11)
that is J~(Fs) = 0.
(3.12)
Thus the limit value of the local integral at the crack tip vanishes in both cases for E -o 0. It results in, cf. eq. (3.4), that the local integral on an arbitrary contour F can be replaced by an area integral
L( F) = fH(r, ,9, t) dV.
(3.13)
From eq. (3.13) it is to see clearly that the local integral describes the changes with time of the crack tip state in Eulerian coordinates. If the crack propagates steadily, i.e. if a steady state field at the crack tip exists, the local integral is equal to zero in the whole field. We are going to study the variation of the local integral outside the crack tip. Suppose the crack field, where the local integral is concerned, consists of two parts, a plastic loading zone and an elastic
H. Yuan, I~E Brooks / J integral for elastic-plastic crack extension
162
unloading zone. According to asymptotic analysis of the steady state crack extension [9], the plastic reloading is restricted to a quite small region near to the crack flanks and plays little role in the computation of the local integral, if the integration contours do not come to the crack flanks. The elastic region outside the plastic loading and the elastic unloading zone cannot influence our results, as the local integral is always path-independent there. Suppose the particle (x~) in the plastic loading zone is being unloaded at time t = ~-(x~),
(3.14)
and the equation of the boundary curve
x 1 =Xl(X2, t),
(3.15)
at any time t. The total strain rate is decomposed in an elastic and a plastic part •e -p ~,~(x~, t) = e,~(x~, t) +e~t~(x ~, t),
(3.16)
if we restrict our discussions to power-law hardening materials, then follow
~t3(x~, t) =
(l+v) E
q~(x~, t) =
. u cr.t3 - -~d~,6,t 3,
(3.17)
-f ~2-~e~s~
for t <_.~(x~)
0
for t > ~'(x~),
(3.18)
where s,,t3 is the stress deviator and ~r~ the effective stress defined by ~r~ = 3s~t3s~tJ2. E and u are the elastic modulus and Poisson's ratio, respectively. C is a material constant characterizing strain hardening and 8,t 3 is Kronecker's symbol. Similarly, we can decompose the strain energy function defined by eq. (2.7)
W(x~, t) = W~(x~, t) + WP(x~, t),
(3.19)
with 1
We(x~, t) ~- ~ { ( 1
WP(xK, t ) =
+ v)o'.o(r,,t3 - v(r~;2},
2C
~r~+l(x~, t),
3 E ( n + 1)
~7+'(x~, ~),
(3.20)
for t
(3.21)
From eq. (2.4) the plastic strain rate in the plastic loading zone can be represented by
q~(xK, t) =
_
w~,,~.~(~K, t) + ~oe~t~ --(~, o
t).
(3.22)
The first term on the right hand side in eq. (3.22) describes the changes due to crack extension and the second term represents the local changes of the plastic strain in the field at the crack tip. It can be assumed that the second term is continuous for crack extension in homogeneous materials, otherwise, it could result in a sudden change of the total plastic strain in Eulerian coordinates. It follows that due to
H. Yuan, W. Brocks / J integral for elastic-plastic crack extension
163
continuous plastic strain rate the gradient of the plastic strain must also be continuous in the whole crack field. By substituting eq. (3.18) into (3.22) it follows in the plastic loading zone
/
(
t)
so 7 o(x ,t)
=0. (3.23)
Similar to eqs (2.4) and (3.22), the first set of brackets in eq. (3.23) reveals the influences from the crack extension, the second set does not depend on the crack growth velocity explicitly and represents the change of the crack tip state. In steady state crack extension the second set vanishes since the crack tip field becomes invariant with respect to time. On the other hand, from the knowledge of crack tip field analysis [9,10,16] we can hope all terms in the second set near to the crack tip will be much smaller than the ones in the first set. Therefore, we can assume asymptotically C
eP~,v(x ,, t)
0
t) ----0, for t <_r ( x . ) .
- --~o'n-2safl-~---O'e(XK, /2;
(3.24)
OX7
From eqs. (3.24) and (3.21) follows a
a
--WP(x, ~Xr
'
P t) = tGt3(x , , t ) - -~ e~t3(x .,t), O~
for t < ~-(x,).
(3.25)
for t < r( x~),
(3.26)
We integrate eq. (3.17), then
e~t3( x ~, t)
(l+v)
v
E
o',t3 - -~o'rr~,t3,
and substitute eqs. (3.21), (3.26) and (3.25) with 3' = 1 into eq. (2.8), so the second integral on the right hand side vanishes. It means that the J integral is path-independent in the plastic loading zone. In the elastic unloading zone the increment of the plastic strain is zero, see eq. (3.18). It follows for t ___~'(x~) that
Ox--~W(xK, t) -~G~(xK, t)e~t3,v(x~, t) 0 3xv
W°(x~, r) - % , ( x ~ ,
t ) e P , , ( x ~ , r) = {tr~t3(x~, ~') - % , ( x ~ , t)},Pt3.~(x ~, ~').
(3.27)
As discussed for eq. (3.22) above, the gradient of plastic strain during crack extension is continuous in the whole crack field. Thus, from eq. (3.24) with t = r(x~) we have 0
--W( Oxv
x~, t) -tr~t3( x~, t)e,~t3,~,(x,,, t) 0
= f ( x ~ , t, r)~xvtre(x ~, r),
for t < r ( x x ) ,
where
f ( x~, t, r) = C o ' 2 - 3 s ~ ( x ~, ~'){s~(x~, "r) - s~,( x~, t)}.
(3.28)
H. Yuan, W Brocks / J integralfor elastic-plastic crack extension
164
i
elasticunloadingzone
\(Zo,hz)
/(Z,,hz)
I7 crack \ ! r v,~ / , ___ ~ _ l ( z " h ' / ~ a s t i c z ° n e x
x2
J
Fz
plasticloadingzone
xl
.,- .....
ao
~ "
~
,aa
. . . .
) I
t
Fig. 3. Crack tip field and integration contours.
With help of the unloading condition
s,~(x~, r)s,,~(x~, "r) < s,~(x~, t)s~a(x~, t)
we have
f(x~, t, r) > O.
(3.29)
Consider two arbitrary integration contours around the crack tip, /'1 and /'2, the difference of the local integral between /'1 and /'2 is equal, see eq. (2.8) 0 Je(/'2)
- J,(/'l)
t, T)-~--~
=-vf'l f(xK'2
r)
dV,
(3.30)
where the integration area Vt2 locates between the two contours F 1 and F z. Equation (3.29) exhibits the relation between the local integral and the gradient of current yield stress in the elastic unloading zone. Since the distribution of the current yield stress is unknown, we cannot generally predict the behaviour of the local integral from Eq. (3.30). We consider two rectangular integration contours for mode I case as shown in fig. 3 which were used in finite element studies, see section 4. Suppose that the points with coordinates (-)Co, h~) are located behind the initial crack tip, so that generally the yield stress on the particle near to the crack tip is larger than one far from the tip, i.e. ge(Xo, h,, r)<_ ge(xl(h~, r), h~, z) holds. Then, from eq. (3.30) the difference of the J, integral between the two contours, with help of mean-value theorem for integration, is
J,(r2) - J , ( r , )
:
,,
x,(x2,
x2,
x:,
dx:, (3.31)
where
¢1 E [)c0,
Xl(X2,
z)], ~'2 ~ [hi, h2].
It means that the local integral is an increasing function of the contour height. Since the local integral vanishes at the crack tip, thus, the local integral calculated on the rectangular contours is a monotonic function with a lower limit zero and an upper limit Jff, the so called far-field J integral.
4. Finite element investigation of ductile crack extensions
For further studies of the J, distribution around the crack tip, five specimens from an aluminium alloy (see table 1) were investigated by finite element method [21], i.e. two Compact Tension specimens (CT), two Center Cracked specimens (CCT) and one Single Edge Cracked Bend specimen (SECB). The material properties are the following: elastic modulus E = 71600 M N / m 2, yield stress % = 317 M N / m 2 and ultimate tensile stress o~u = 440 M N / m 2. Aaexp in table 1 denotes the maximum amount of crack growth reached in the experiment. The referenced experimental data for these specimens (loading and J-resistance curves) have been obtained by Schwalbe and Hellmann [22].
165
H. Yuan, W. Brocks / J integralfor elastic-plastic crack extension
Table 1 Geometries of numericallyinvestigatedspecimens No.
Specimen
1 ~2 ~3 ~4 5
CT CT CCT CCT SECB
W (mm) 50 100 50 100 50
B (mm) 5 20 5 20 5
a0 (mm) 25.1 71.0 25.3 74.0 25.5
Aacxp (mm) 14.44 17.15 8.58 8.18 11.90
The experiments were simulated by following the load line displacement records and all calculations base on plane stress models and Prandtl-Reuss flow theory. The crack growth was modeled by using the node release technique without node shifting. The node forces were slowly released at each node in a simultaneous, nonlinear way to model an increment of crack growth. No singularities in the crack tip element were included. The FE-program and the FE-mesh used here were validated through a number of numerical computations and comparisons with A D I N A calculation, see [21,23]. A typical finite element mesh in the near field is shown in fig. 4. The sizes of the elements directly at the tip are 0.25 mm × 0.8 mm. All computations take the same mesh in the near field. The meshes far from the crack tip for specimens ~ 2, ~ 3 and ] 5 are shown in Fig. 5a-c. VLL stands for load line displacement which were measured in experiments by Schwalbe and Hellmann [22] and VL is the displacement of the controlling curves in numerical simulations which was calculated from VLL curves by preceding FE calculations. More detailed FE-results about geometry effects on the crack tip opening displacement, J integral and the crack tip opening angle in ductile crack extensions were reported in [21]. The J, integral was evaluated according to the definition in Eq. (2.6). Numerical integration was performed through linear interpolation of integrands between Gaussian points. To investigate the variation of the Jr integral and effects of specimen geometries, six integration contours were defined for every specimen, see fig. 6 for CT specimen ~ 1. The contour Ftip was located on the tip directly and moved with the tip during crack extension, so that the correspondent contour of the integral values covers timely varying material particles. Its dimensions are 1.0 mm in height and 0.8 mm in length. The related J, value is denoted as Jtip. The contours F 1 to F 4 are defined in the near field, as denoted in ref. [24], and differ from one another mainly by their height h, cf. fig. 3. Their width is larger than the maximum crack extension allowed in the FEM calculation, Aamax--- Aaexp, SO that they contain the crack tip at all times. Our numerical experiments have shown that the length of the integration contours does not affect the evaluated Jr integral value, as predicted in section 3. The corresponding Jr values are denoted with
IIIIIIIIIIIIIII
IIIIIIIIIIIIIII
---~11 m,~ N
Fig. 4. Detail of mesh refinement in the vicinityof the growingcrack tip.
H. Yuan, W. Brocks / J integral for elastic-plastic crack extension
166
VL
I V LL 14
a
"1
(a) specimen #2 (CT)
VL
--~ VL
c.;
,V LL
\\\ \ \
!///~-/
///
••i!
I II1,,~
"~,~,~,~,,,,
I,
a
-I
I 1/11 I I///
VL
.i
'""
V LL a
m
(c) specimen #5 (SECB) (b) specimen #3 (CCT) Fig. 5. Element meshes in the far-field.
Jnf,i(i = 1. . . . . 4). The contour Fff is allocated in the so-called far field where the J integral is still path-independent for large crack extensions. The J integral evaluated is called Jff. The finite element computations agree with experimental records quite well, cf. [21]. As known, the J integral is path independent before the crack initiates, that is, the J, integral is identical with the J integral. After'crack initiation, the further increase of the Jff curve with crack extension, A a , depends on the specimen geometry, see fig. 7. Moreover, the J integral loses its path-independence rather quickly, see
H. Yuan, W. Brocks / J integral for elastic-plastic crack extension
167
F# IJ~ )
I, F IJ,r.e~) r3 .(J,) .3
I
! I i
\ r4 (J,~.4)
! Crack
VLL
-t~ mm
~!. I
r.
ao
L~.ai~~- -
1 ap ( Jtio
W
). ,!
Fig. 6. Upper half of CT-specimen ~ 1 with integration contours for the local integral.
Fig. 8a, b. The ./,-resistance curves evaluated in the near field rise much less than Jn, especially, they become constant after a transition phase, as [18,24] reported earlier. Hutchinson and Paris [6] hoped, that under the so called conditions for J controlled crack growth, HRR-theory still holds, so that the J integral can dominate in some zone around the crack tip. If we allow for a crack growth of 6% of the ligament width, i.e. a = A a / ( W - a 0) = 0.06, we will obtain deviations of the J~ integral from the far-field J, Jff - Jnf,1 S~
(4.1)
Aa=ct(W-ao)
Jff
which are summarized in table 2. As the choice of the contour F 1 is arbitrary, the deviation will be even larger if a smaller contour is defined, which explains the inequality sign in eq. (4.1). Obviously, the deviations of some specimens are much larger than 10% which is the admissible deviation supposed in [24]. If we allow for a 10% deviation of J, from Jff, the absolute and relative crack extensions under this condition are denoted as Aal0 and al0 in table 2, respectively. The permitted relative crack growths, al0, do not reveal clear and unambiguous differences between bend specimens, ~ 1, ~ 2, ~ 5, and tension ones,
300]
#I
t'*~
:::'.:'.:::.. ......
o
. . . . . . . . .
0
•
#5
,,/ff
i
5
. . . . . . . . .
i
10
. . . . . . . . .
15
,an (ram) Fig. 7. Effects of specimen geometries on the J n f and Jfr values.
168
H. Yuan, W. Brocks / J integral for elastic-plastic crack extension 300,
200.
I00,
"'~"~., o ,
.
~ ,.~ ... J~P.,,,.~ .-- ." .--',---.----.-- .---.~'"~-~. .~ 5
10
An (ram) 300
(b)
~ 200
~
100
~ ~,,."
,
f
3
' Jnf,1
.........
0
0,0
.
.
.
.
2,5 z
.
.
.
5,0
.
i
.
.
.
,~
.
(ram)
i
7,5
(b) specimen3 Fig. 8. Path-dependency of the J integral in ductile crack extensions.
3, ~ 4. A crack growth of 6% permitted for bend specimens in ref. [24] may yield a path-dependency of J as defined of more than 20% and, hence, seems not to be strict enough. The effects of specimen geometries on the J, values evaluated on the contour F 1 are not so distinct as on the Jff curves, see fig. 7. For large crack extensions all J,e,~ curves come nearly together, the geometric effects are much smaller than directly after crack initiations. In addition, they become independent of Aa after a transition phase, as mentioned before, which should be postulated for any
Table 2 P a t h - d e p e n d e n c y o f the local J integral Specimen
~1
~2
~3
~4
~5
s [,,=o.o6 > (%) zlalo _<(mm) Aalo
21 0.5
16 1
16 1
23 1.2
23 0.75
O/IO W- a0
0.020
0.034
0.040
0.016
0.029
H. Yuan, W.. Brocks / J integral for elastic-plastic crack extension
169
1,010 0,75 0,5]0 0,2I5 0,00
. . . . . . . . .
o,o
~
.
z,5
.
.
.
.
.
.
.
.
,
.
.
.
.
.
.
.
.
.
5,0
~
. . . . . . . . .
7,5
1o,o
height h (ram)
Fig. 9. J distributionin the near field. crack growth controlling parameter in steady state. From this evidence, the J integral or similar ones evaluated in the near field have been suggested as fracture parameter for ductile crack extensions in [18,19]. However, fig. 9 explains that this apparent loss of geometry dependence of J,f resistance curves is mainly due to the fact that in general J, decreases with decreasing radius of its contour F,, as predicted in section 3. Thus, any geometry dependence must necessarily diminish if the limiting value for all curves is zero. Jtip evaluated on a contour Ftip attached to the moving crack tip appeared to be zero, anyway, see fig. 8a, confirming the conclusion in ref. [15] that there is nothing like an energy release rate for elastic-plastic crack extensions.
5. Energy balance in ductile crack extension The thermodynamic principles must also be obeyed in the tip region during crack extension. Without loss of generality we consider all thermodynamic variables to be continuous during the whole physical process. Then the principle of balance of energy in its local form is = cr~,v~,o + h - q~,~.
(5.1)
We denote the specific internal energy with e, the specific heat absorption with h, the heat flux with q~ and the rate of deformation with v,,. Let us consider parts of a cracked body, V and V,, according to fig. 1 for the two-dimensional case. n , stands for the unit normal vector on the boundaries F and F~. The part V, contains the crack tip which moves through the body with a velocity we. The internal energy of the part V - V, is
(5.e)
E ( V - I,I,) = fv-v, e dV, and the kinetic energy K(V-
1 V,) = 2 f - v ,
dV,
(5.3)
H. Yuan. HLBrocks / J integralfor elastic-plasticcrack extension
170
where p is the mass density. From Reyno[d's transport theorem we have the rate of the internal energy
e(v-v,)=fv-v~Odv-ffwon°as,
(5.4)
and the rate of the kinetic energy
K( V - V,) = f , _
~fo~,.v.w~n~ds. 1
pu~u, d V -
(5.5)
It is assumed that all integrands have the necessary continuity properties. The mechanical power from volume loads f , and surface traction t, is defined by
M(v)=fv~L dV+ f~,v°t° d,,
(5.6)
where ~V denotes the surface of the volume V. The heat power is defined by
Q ( V ) = f~h dV+ ~,qo, no, as.
(5.7)
According to the law of conservation of energy, thermo-mechanical processes in an arbitrary part V of a body are governed by
E ( V - V,) + I ( ( V - V,) + ,~(,) = M ( V ) + Q ( V ) ,
(5.8)
where ~(E) is the energy dissipation rate in V, which contains the crack tip. From eqs. (5.1), (5.8) and the equilibrium equations of continuum mechanics we have 1
This is the time rate of change of surface energy in the crack tip vicinity introduced in ref. [16]. Regardless of material behaviour and crack growth velocity, eq. (5.9) describes the energy flux through F~ to the tip. Generally the integral value depends on its integration contour. Its difference between two integration contours is the energy dissipation rate in this region. We introduce a dissipation rate functional of energy
(e
+
where 6 and e are characteristic sizes of integration contours F~ and F~, and F~ is situated within F~ completely. Physically this functional describes the difference in energy flux between both integration contours, that is, the dissipation in the area between the two contours. From this physical significance /-/(S, e) >_0
(5.11)
holds and it must be a monotonically increasing functional of 6 and monotonically decreasing functional of e. With the Gaussian theorem eq. (5.10) becomes
n ( ~ , ,) = fv' - ~
[-eiwiq-o-ij~ij ] ,
dV,
(5.12)
where the integration domain V~ - V, is situated between F~ and F,, and the effects of heat conduction, kinematic energy and volume loads are neglected. Comparing to eq. (3.24), eq. (5.12) reveals the
H. Yuan, W. Brocks / J integral for elastic-plastic crack extension
171
relationship between path-dependency of the J integral and the energy dissipation in the vicinity of the crack tip. If and only if the divergence of the integrand in eq. (5.10) vanishes, i.e. the J integral becomes path-independent, no energy dissipation exists in Vn - V,. Especially, if the divergence in the crack tip vicinity is equal to zero, i.e. 11(6, 0) -- 0, then ~(•) is path-independent for arbitrary F8 and represents the energy flux to the crack tip. In mode I case the crack extends on the uncracked ligament, suppose at speed d. The energy release rate is G = limE---,o
(5.13)
d
Neglecting the effects of heat conduction, kinematic energy and volume loads in eq. (5.9) and appling Eq. (2.4) to the velocity vector, we then obtain G -- lim f
(en 1 -Ui,lorijnj)
ds,
(5.14)
which is the local integral evaluated at the tip, see eq. (2.6), where e = W holds. The local integral, J,, is equivalent to the energy release rate G. As known, the J integral of Rice is equal to the energy release rate, if no energy dissipation exists around the crack tip. To study the energy dissipation in the crack vicinity, we introduce a functional 1 1. ~ ( • ) = ~-11(•, 0) = ~-,~(•) - G.
(5.15)
The functional qt(e) represents the energy dissipation rate in the region around the tip and due to its physical significance it must be a monotonically increasing functional of its integration contour. On the other hand, it is to be seen, that from the definition of the local integral J, for • -~ 0, ~ ( • ) is equivalent to the area-integral in eq. (2.7). Thus, from energy balance we come again to the prediction on the local integral behaviour in the near field. Generally speaking, the medium around the tip absorbs energy during crack extension through inelastic deformation and material degradation, that is ~ ( e ) >_riG,
(5.16)
where E is an arbitrary real number. A necessary condition for a non-trivial but bounded energy release rate is that the integrands in Eq. (5.14) must exactly have a first order singularity. Unfortunately, all asymptotic solutions of steady crack extension inelastic-plastic materials show that the singularities of stresses and strains cannot meet this condition, i.e., the energy release rate is equal to zero, thus ~(E) = tiqt(e).
(5.17)
This means that all energy supplied by external loads is spent by plastic deformations and material degradation in the vicinity of the crack tip, in the so-called process zone, whereas the crack extension itself does not need any surplus energy, as Rice [15] pointed out. Thus, the energy balance equations do not provide any answer to the phenomena of ductile crack growth, though they undoubtedly must also hold in the vicinity of the crack tip. Help may come by introducing special constitutive equations [25,26] instead of Von Mises plasticity to account for the continuing process of damage ahead of the crack tip and thus not needing any 'energy release rate' or 'crack driving force'.
172
14, Yuan, W. Brocks /.1 integral for elastic-plastic crack extension
6. Conclusions In the present paper the J integral behaviour in the vicinity of a crack tip in ductile crack extension is discussed. It is shown by a mathematical analysis of crack tip fields that the J integral is monotonically increasing function of its integration contour, if it becomes path-dependent. The local integral is asymptotically path-independent during ductile crack extensions, while in the steady-state field it becomes exactly path-independent. No finite lower limit exists in the crack tip field, i.e. J, vanishes with vanishing contour radius e. Our FE-computations have confirmed this prediction and show that the J integral may lose its path-independence immediately after crack initiation. From the point of view of an energy balance the path-dependence is connected to the energy dissipations in the crack tip region. The local integral describes the energy dissipated in the vicinity of the tip. The analysis points out generally that the energy description of ductile crack extension may lead to paradoxical conclusions due to the restrictions of the fracture mechanics model of a singular crack tip in a continuum. Thus all other path- or path-area-independent integrals which have been supposed continuously by some authors for elastic-plastic crack growth cannot bring any new informations about the crack tip state.
Acknowledgements This work was supported by the German Research Foundation (DFG) under contract numbers Br 521/2-1 and Br 521/2-2. The authors thank Dr D. Hellmann of GKSS for the experimental data used for the numerical computations here.
References [1] J.R. Rice, A path-independent integral and the approximate analysis of strain concentration by notches and cracks, J. Appl. Mech. 35 (1968) 379-386. [2] J.W. Hutchinson, Plane stress and strain fields at a crack tip, J. Mech. and Phys. Solids 16 (1968) 13-31. [3] J.R. Rice and G.F. Rosengren, Plane strain deformation near a crack tip in a power-law hardening material, J. Mech. and Phys. Solids 16 (1968) 1-12. [4] W. Brocks and J. Olschewski, On J-dominance of crack-tip fields in largely yielded 3D structures, Int. J. Solids Struct. 22 (1986) 693-708. [5] R.M. McMeeking, Finite deformation analysis of crack tip opening in elastic-plastic materials and implications for fracture initiation, J. Mech. and Phys. Solids 25 (1977) 357-381. [6] J.W. Hutchinson and P.C. Paris, Stability analysis of J-controlled crack growth, in Elastic-Plastic Fracture: ASTM STP 668, eds. J.D. Landes, J.A. Begley and G.A. Clarke (1979) 37-63. [7] W. Brocks, H.H. Erbe and H. Yuan, Numerical study of stable crack growth, Failure Analysis Theory and Practice, Vol. A, ed. E. Czoboly (1988) pp. 205-210. [8] W. Brocks and H. Yuan, Numerical investigation on the significance of J for large stable crack growth, Eng. Fract. Mech. 32 (1989) 459-468. [9] J.C. Amazigo and J.W. Hutchinson, Crack tip field in steady crack-growth with linear strain-hardening, J. Mech. and Phys. Solids 25 (1977) 81-97. [10] P. Ponte Castafieda, Asymptotic fields in steady crack growth with linear strain hardening, J. Mech. and Phys. Solids 35 (1987) 227-268. [11] W.J. Drugan, J.R. Rice, and T.-L. Sham, Asymptotic analysis of growing plane strain tensile cracks in elastic-ideally plastic solids, J. Mech. and Phys. Solids 30 (1982) 447-473. [12] Y.C. Gao, X.-T. Zhang and K.-C. Hwang, The asymptotic near-tip solution for Mode III crack in steady growth in power law hardening media, Int. J. Fract. 21 (1983) 301-317.
H. Yuan, W. Brocks / J integral for elastic-plastic crack extension
173
[13] Q.S. Nguyen, A thermodynamic description of the running crack problem, in Three-Dimensional Constitutive Relations and Ductile Fracture, ed. S. Nemat-Nasser, (1981) pp. 315-330. [14] Q.S. Nguyen, An energetic analysis of elastic-plastic fracture, in Defect Assessment in Components-Fundamentals and Applications, eds. J.D. Blauel and K.-H. Schwalbe, (1991) pp. 75-85. [15] J.R. Rice, The mechanics of quasi-static crack growth, Proc. 8th US National Congress of Applied Mechanics, ed. R.E. Kelly, (1979) 191-216. [16] H.C. Strifors, Thermomechanical theory of fracture: kinematical and physical principles, Report 27, Department of strength of materials and solid mechanics, Royal Institute of Technology, Stockholm (1980). [17] S.N. Atluri, Path-independent integrals in finite elasticity, with body forces, inertia and arbitrary crack-face conditions, Eng. Fract. Mech. 16 (1982) 341-364. [18] F.W. Brust, R. Nishioka, S.N. Atluri and M. Nakagaki, Further studies on elastic-plastic stable fracture utilizing the T* integral, Eng. Fract. Mech. 22 (1985) 1079-1103. [19] T. Nishioka and H. Yagami, Invariance of the path independent T* integral in nonlinear dynamic fracture mechanics, with respect to the shape of a finite process zone, Eng. Fract. Mech. 31 (1988) 481-491. [20] L.B. Freund; Energy flux into the tip of an extending crack in an elastic solids J. Elasticity 2 (1972) 341-349. [21] W. Brocks and H. Yuan, Numerical studies on ductile crack growth, in Defect Assessment in Components-Fundamentals and Applications, eds. J.D. Blauel and K.-H. Schwalbe, (1991) pp. 19-33. [22] K.-H. Schwalbe and D. Hellmann, Correlation of stable crack growth with the J-integral and the crack tip opening displacement, effects of geometry, size, and material. GKSS Report 84/E/37, GKSS-Forschungszentrum Geesthacht. [23] S. Bartmann, W. Brocks and H. Yuan, Bruchmechanische Parameter zur Beschreibung yon stabilem RiBwachstum in duktilem Material, in Proc. of 21st 'Vortragsveranstaltung des DVM-Arbeitskreises Bruchvorg~inge', ed. Ch. Berger (1989) 527-537. [24] C.F. Shih, H.G. de Lorenzi and W.R. Andrews, Studies on crack initiation and stable crack growth, in Elastic-Plastic Fracture, ASTM STP 668, eds. J.D. Landes, J.A. Begley and G.A. Clarke (1979) 65-120. [25] D. Krajcinovic and D. Sumarac, Micromechanics of the damage process, in Continuum Damage Mechanics: Theory and Applications, eds. D. Krajcinovic and J. Lemaitre (Springer-Verlag, 1987) p. 135-194. [26] J. Lemaitre, A continuous damage mechanics model for ductile fracture, Trans. ASME. J. Eng. Mater. and Technol. 107 (1985) 83-89.