Dynamic steady crack growth in elastic-plastic solids—propagation of strong discontinuities

Pergamon

J. Mech. Phys. Solids, Vol. 42, No. 11, pp. 1817-1848, 1994 Copyright © 1994 ElsevierScienceLtd Printed in Great Britain. All rights reserved 0022-5096/94 $7.00+ 0.00

0022-5096(94) 00041-7

D Y N A M I C STEADY CRACK GROWTH IN ELASTICPLASTIC SOLIDS--PROPAGATION OF STRONG DISCONTINUITIES A. G. VARIASt and C. F. SHIH$ tShell Research, Billiton Research B. V., Postbus 40, 6800 AA Arnhem, The Netherlands and SDivision of Engineering, Brown University, Providence, RI 02912, U.S.A. (Received 24 December 1993)

ABSTRACT The near-tip field of a mode I crack growing steadily under plane strain conditions is studied. A key issue is whether strong discontinuities can propagate under dynamic conditions. Theories which impose rather restrictive assumptions on the structure of an admissible deformation path through a dynamically propagating discontinuity have been proposed recently. Asymptotic solutions for dynamic crack growth, based on such theories, do not contain any discontinuities. In the present work a broader family of deformation paths is considered and we show that a discontinuity can propagate dynamically without violating any of the mechanical constitutive relations of the material. The proposed theory for the propagation of strong discontinuities is corroborated by very detailed finite element calculations. The latter shows a plane of strong discontinuity emanating from the crack tip (with its normal pointing in the direction of crack advance) and moving with the tip. Elastic unloading ahead of and/or behind the plane of discontinuity and behind the crack tip have also been observed. The numerical investigation is performed within the framework of a boundary layer formulation whereby the remote loading is fully specified by the first two terms in the asymptotic solution of the elasto-dynamic crack tip field, characterized by K~ and T. It is shown that the family of near-tip fields, associated with a given crack speed, can be arranged into a one-parameter field based on a characteristic length, L v which scales with the smallest dimension of the plastic zone. This extends a previous result for quasi-static crack growth.

1. INTRODUCTION In recent years several investigators have reported numerical studies and asymptotic solutions for dynamic crack growth in ideally plastic and linear or power law hardening materials. However, broad agreement between these studies could not be found. Indeed, several asymptotic solutions for plane strain mode I dynamic crack propagation in an incompressible, isotropic elastic-perfectly plastic solid can be found in the literature. It is not our intention to provide a detailed review of the existing solutions on the above subject in this work. Instead, we will point out certain features in existing solutions of dynamic crack growth, which are helpful to understanding our approach to the problem as discussed in this paper. A nice discussion of available asymptotic solutions for dynamic crack propagation under mode I plane strain conditions is given by Leighton et al. (1987). They have also provided a direct comparison of their solution for an incompressible elastic1817

1818

A.G. VARIASand C. F. SHIH

perfectly plastic material with those obtained by Slepyan (1976), Achenbach and Dunayevsky (1981) and Gao and Nemat-Nasser (1983). Every solution noted above shows bounded stresses at the tip. Strains are also bounded in the solutions presented by Slepyan, Achenbach and Dunayevsky, and Leigthon et al. ; these solutions are also continuous. In contrast, the solution of Gao and Nemat-Nasser is discontinuous and the strains are logarithmically singular. A feature common to all the above solutions is the absence of elastic unloading behind the crack tip. In this respect, none of the asymptotic solutions reduce to the corresponding quasi-statically growing crack result in the limit of vanishing crack speed (cf. Drugan, Rice and Sham, 1982). The absence of elastic unloading is puzzling since the numerical full-field solutions of Lam and Freund (1983, 1985) indicate the presence of elastic unloading behind the crack tip at high crack speeds. Lam and Freund have compared their finite element results with the analytical solution of Achenbach and Dunayevsky, which is valid for small crack speeds compared to the elastic shear wave speed. They found agreement for the in-plane angular stress distribution; however, no agreement was observed between the strain distributions. The latter provides a critical comparison since strain depends sensitively on crack speed while stress is only weakly dependent on crack speed. Therefore, no definite conclusions could be drawn from the above comparisons. Dynamic crack growth in a linear-hardening material under mode I plane strain conditions was investigated by Achenbach, Kanninen and Popelar (1981). The asymptotic solution, obtained by a variable separation method, established that the ratio of the (elastic-plastic) tangent modulus to the Young's modulus plays an important role. Solutions could be found when the above ratio is larger than a critical value and the latter increases with increasing crack speed. When the hardening is sufficiently small (or alternatively when the crack speed is high) the governing equations in the plastic zone become singular at a certain angle because of the change in character of the equations from elliptic to hyperbolic (see also Achenbach, Burgers and Dunayevsky, 1979). A physical interpretation for the loss of ellipticity is provided by considering a quasi-Rayleigh wave speed in which the Young's modulus of the material is replaced by its (elastic-plastic) tangent modulus; the latter serves as the material's characteristic wave speed at high strains. An improved solution to the same problem was provided by t~stlund and Gudmundson (1988) who considered the possibility of plastic reloading near the crack flanks, a feature that was neglected by Achenbach and co-workers. Ostlund and Gudmundson also encountered difficulties related to the change in character of the governing equations. In summary both studies suggest that, due to the change in the character of the governing equations, discontinuous solutions could be allowed when the hardening falls below a (crack speed dependent) critical value. In any case, elastic unloading was distinctly evident in both solutions. As already mentioned, Lam and Freund (1983, 1985) have reported finite element full-field solutions for dynamic growth of a tensile crack in an elastic-plastic solid obeying Mises yield condition and Prandtl-Reuss flow rule. Certain features of their solution for the perfectly plastic material are noted below. Elastic unloading was evident at crack tip speeds larger than about 0.3 times the elastic shear wave speed (V t> 0.3cs). For V = 0.3~).4cs unloading began at 90 ° away from the direction of crack advance (Fig. 1 in Lam and Freund, 1985, and Fig. 3 in Lam and Freund, 1983). At V = 0.5Cs unloading began at an angle smaller than 90 ° (Fig. 4 in Lam and

Propagation of discontinuitiesduringcrack growth

1819

Freund, 1983). Furthermore, a very narrow strip of plastic activity, separating the two elastic sectors, can be seen at 90 ° from the crack plane; relatively large changes in shear stress and (normal) radial stress are also observed at 90 ° . These features should be viewed as tentative since the mesh employed by Lam and Freund was not sufficiently refined to resolve strong variations in the field. Nonetheless, the above features are consistent with our numerical full field solutions which are discussed in Sections 3 .and 5. Deng a n d Rosakis (1991, 1992) have reported finite element fullfield solutions for steadily growing cracks in hardening and nonhardening elasticplastic solids under plane stress conditions. They have noticed that when crack tip speed is high and hardening is sufficiently low, "kinks or strong signs of slope discontinuity" in polar stress components art and Go appeared at 90 ° from the crack plane. Here again the question could be raised: are these features related to the presence of a propagating plane of strong discontinuity (across which a jump occurs in one or more components of stress and/or particle velocity)? In Section 3 we present small scale yielding solutions for a mode I crack growing steadily under dynamic plane strain conditions. The finite element full-field solutions, obtained with highly refined meshes, strongly suggest the existence of strong discontinuities ; the presence of elastic unloading zones in the near-tip region is unmistakable. Strong discontinuities are discussed in Section 4 and it is shown that a strong discontinuity can propagate at a speed different from an elastic wave speed if conditions for yield are satisfied along a part of the deformation path within the discontinuity. This result is strengthened by a credible example in Appendix A where it is shown that a strong discontinuity can propagate dynamically while still complying with the material's mechanical constitutive assumptions. These findings taken together provide a compelling case for rethinking some of the results pertaining to the propagation of strong discontinuities. In Section 5 we present numerical solutions obtained from a modified boundary layer (MBL) formulation whereby the remote loading is fully specified by the first two terms in the asymptotic solution of the elasto-dynamic crack tip field. This approach provides a convenient means to investigate the effects of crack geometry on the elastic-plastic near-tip fields of a growing crack. The solutions to MBL problems display features similar to those discussed in Section 3 thus providing further support for the existence of strong discontinuities.

2.

BOUNDARY VALUE PROBLEM

2.1. Modified boundary layer formulation Consider a mode I tensile crack growing steadily under plane strain conditions. The crack speed is V. Both cartesian co-ordinates xl, x2 with Xl aligned in the direction of propagation and polar coordinates r, 0 with 0 = 0 corresponding to the line ahead of the crack tip are introduced with their common origin at the tip of the crack and moving with the tip. Within the framework of an infinitesimal displacement gradient theory, we pose the modified boundary layer problem whereby remote tractions are

1820

A.G. VARIAS and C. F. SHIH

given by the first two terms of the elastic asymptotic stress field for a dynamically growing crack (e.g. Freund, 1990; Freund, 1976; Rice, 1968). Thus as r---, ov K,

'

O'ij - .//~r~i j (0, V/cs)-~- T~li~lj ,

(2.1)

where Cs = ~ is the elastic shear wave speed and p the material density. For I completeness, the functions ]E~j(0, V/cs) and the pertinent displacement gradients are given in Appendix C. Observe that K,, T as well as V are unchanging for steadystate conditions to prevail. Calculations have been performed for 0 < V/cs <~0.3 and - 1 ~ T/ao <~ 1.

Let ao be the material's yield stress in tension. Dimensional analysis shows that all field quantities should depend on r/(Ki/ao) 2, 0, T/ao and V/cs as well as dimensionless combinations of material parameters. For example, stress fields are of the form

(r

)

~,j = ~og~j Kl~012, O; r/~0, V/cs .

(2.2)

The dependence ofg~j on dimensionless combinations of material parameters is understood. 2.2. Material constitutive behavior The elastic-plastic response of the material is described by an infinitesimal displacement gradient J2-flow theory. The material hardens isotropically according to a power law relation {] - \a0]

a0

-

2(1 +v) a0

for

~>a0.

(2.3)

Here ~ = w / ~ (a~ja~j) ~/2 is the effective stress, aTj = aij-(1/3)akk6ij is the deviatoric stress and ~P is the accumulated effective plastic strain (d~ p = N / / ~ (d/~p dsp) l/2). The hardening exponent, N, has the range 0 ~< N ~< 1. The material is elastic when N = 1 and it is elastic-perfectly plastic when N -- 0. Results presented in this paper are for E = 300a0, v = 0.3 and N = 0, 0.1 and 0.2. Some results for a nearly incompressible material (v = 0.49) are also included. 2.3. Numerical method for steady-state analysis We employ the numerical procedure of Dean and Hutchinson (1980) and Parks et al. (1981), which was extended by Lam and Freund (1985) to dynamic steady-state problems. Improvements which increase the accuracy of the method were discussed by Varias and Shih (1993). The implementation of numerical procedure is discussed in Appendix B. Several checks were performed for the purpose of assessing the accuracy of the numerical solutions. As a first test, calculations were carried out for crack tip speed equal to 0.001 times the elastic shear wave speed--the present solution matched our previous solution for quasi-static crack growth (Varias and Shih, 1993) up to five

Propagation of discontinuities during crack growth

1821

significant digits. Additional checks on the quality of the numerical solutions were also made. For crack speeds under discussion we have verified that symmetry conditions at the crack front and traction-free conditions at the crack faces are satisfied to a high degree of accuracy. Satisfaction of the latter condition is a stringent test of the quality of the solution since stresses are calculated by integrating the constitutive relations along material trajectories parallel to the crack line and discretization errors at the crack tip are carried over to regions behind the crack tip. Our tests show that momentum-balance iterations provide an effective method for reducing numerical integration errors to an acceptable level. Based on our numerical testing we believe that a high degree of accuracy is achieved in the numerical solutions that are reported in this paper.

3.

A D I S C O N T I N U I T Y A T 90 ° F R O M C R A C K L I N E

Figure 1 shows the active plastic zone for a compressible (v = 0.3) nonhardening material, under T = 0 and crack speed V/cs = 0.2. The plastic zone size, measured by the normalized plastic wake height hpw/(K~/tro)2, is only slightly greater than that of a quasi-statically growing crack [see Fig. 2(a) in Varias and Shih, (1993)]. However, the shape of the plastic zone is quite different and the unloading sector behind the dynamically propagating crack tip is much larger. Two unloading sectors appear directly above the crack tip and are separated by a plane (xl ~ 0) of intense plastic activity. On closer scrutiny we found that the plastic dissipation is confined to a single line of elements (one element thick) along 0 = 90 ° in the Xl-X2 plane. We experimented with different element sizes in this region (in one case the element size was reduced

%

.o. o

.o.o5

o.oo

o.o6

o.,o

x,/(K,/oo)~ Fig. 1. Active plastic zone for mode I dynamic crack growth in a compressible elastic-perfectly plastic material under plane strain conditions (v = 0.3). Crack tip is located at the origin of the co-ordinate system. The crack faces are parallel to the x,-axis extending along xl < 0.

1822

A. G. VARIAS and C. F. SHIH

by a factor of five) but the plastic dissipation is still confined to a single line of elements oriented along 0 = 90 °. We can gain a better understanding of the field in the neighborhood of the plane of plastic dissipation by examining the variation of stress and deformation of a material point as it moves along its trajectory, i.e. along a line parallel to the crack faces. Figure 2(a) shows the variation of effective plastic strain, ~P, of a material point as the crack tip passes by; ~P suffers a finite increment when the material point is directly above the crack tip. Carefully checking over all the elements above the crack tip and its vicinity we found that the strain increment occurred between the element directly in front of the crack tip and the element immediately behind. That is, the dashed line in Fig. 2(a) connects ~ P ( x I = -l/2) and E P ( x I = +l/2), where I is the element size. We reanalyzed the same problem using a mesh with near-tip elements which are five times smaller. Again, over the two elements adjacent to x~ = 0, the effective plastic strain increased by the same amount. That is, the same increment in ~P took place over a distance which is five times smaller. Because of the refined mesh and attendant improvement in accuracy, the solutions displayed unloading sectors (near 0 = 90 °) extending to the crack tip. These results and additional results to be discussed next point to the presence of a plane of strong discontinuity (emanating from the crack tip with its normal pointing in the xl direction) propagating with the crack tip. Figure 2(b) shows the variation of total strains along a particle trajectory. Jumps in ell and e12 occur directly above the crack tip while e22 is continuous. This result is consistent with the discontinuity conditions for a compressible material which will be discussed in Section 4.1. Figure 2(c) show the variation of total strains when the material is nearly incompressible (v = 0.49) ; e12 suffers a discontinuity across x~ = 0 while Ell appears to be continuous. This result is also consistent w~th the general jump conditions discussed in Section 4.2 where the observation is made that Ell must be continuous when the material is fully incompressible (v -- 0.5). The variation of the shear stress try2 and the hydrostatic stress trh along a particle trajectory are shown in Fig. 3(a,b), respectively. Jumps across x ~ - - 0 can be seen. We have carried out calculations for several crack tip speeds (0 < V/cs <~0.3), hardening exponents (0 ~< N ~< 1) and T-stresses ( - 1 ~< T/tro <~ 1). In every solution involving discontinuous field quantities, the plane of discontinuity includes the crack tip and its normal points in the direction of crack propagation. Figure 4(a) shows the distribution of ~P along a material trajectory line for three values of N and VIes = 0.2. Note that the jump magnitude diminishes as N increases ; for N = 0.2, representive of a high hardening material, the solution is continuous. Figure 4(b) shows the variation of ~P for several values of V for N -- 0.1 which is representative of a low hardening material. Jumps develop at the larger crack speeds, V/c~ = 0.2 and 0.3, and the jump magnitude increases with crack speed. Broad features of our numerical solutions are easily understood from plots of active plastic zone. Figure 5(a) shows the active plastic zones for N = 0.1 and V/cs = 0.2; the plane of discontinuity, along x~ = 0, is easily seen. Figure 5(b,c) shows the active plastic zones for N = 0.2 and V/cs = 0.2 and 0.3. The field appears to be smooth for V/cs--0.2. However the field for V/cs= 0.3, shown in Fig. 5(c), displays a discontinuity plane along x~ = 0.

Propagation of discontinuities during crack growth

(a)

1823

0.03

N=0 v=03 0.02

0.01

x,/(K,/cr ,) 2 = 0 . 0 1 0.00

....

'. . . . .

-0,02

(b)

0.0~0 0.015

' ....

-0.01

' ....

0.00

0.01

x./~,/~°)~ =o.oi ,-~

0,02

N=O .,

v=0.3

0.010 0.005 0.000

~//v/c,=02

-0.005

cst

--0.010 -0,015

. . . .

'

-0.02

(c)

0,015 0,010

. . . .

'

-0.0!

. . . .

'

x./~,/~0)~ =o.oi /'\,

\

0.000

-0.015

----.-.w,,./ I

-o.o~

i

i

v--0.,,

I

"SS

-0.010 i

[

f

"

-0.005

).02

N=o

0.005

co

....

0,01

0.00

I

.i

-o.ot

|

•

•

I

,

,

v/% = 02 ,

o.oo

,

!

,

o.ot

,

,

.

o.o~

x4(r,/~,F Fig. 2. Strain distributions along a line parallel to the crack faces. The dotted lines connect the strain values just before and after the crack tip. (a) Effective plastic strain (v = 0.3). (b) Total strain (v = 0.3). (c) Total strain (v = 0.49).

A. G. V A R I A S a n d C. F. S H I H

1824

~

0.1

(a)

0.0 -0.1 -0.2 b"~ -0.3

x,/(K,/a,)2 0.01 =

N=0

-0.4 -0.5 t

-0.6

-0.02

,

e

I

,

-0.01

,

l

,

|

. . . .

0.00

t

. . . .

0.01

0.02

Xt/(Z,/ao) 2 (b)

2.5

=o.ot 2.0 o

%

-

1.5

-

1.0

N=0 /cs=02

0.5

....... -0.02 -0.01

' ........ 0.00 0.01 z

0.02

Fig. 3. Stress d i s t r i b u t i o n s a l o n g a line parallel to the c r a c k faces (v = 0.3). The d o t t e d lines connect the stress values j u s t before a n d after the c r a c k tip. (a) Shear stress. (b) H y d r o s t a t i c stress.

In previous studies, Drugan and Shen (1987), Leighton et al. (1987) and Shen and Drugan (1990) have argued against the existence of moving surfaces of strong discontinuity in the problem considered in this paper. Their argument hinges upon certain assumptions which may be too restrictive. In the following section the conditions that should be satisfied by a dynamically propagating discontinuity are reviewed. This is followed by a discussion of a general structure of admissible deformation paths within a discontinuity. A certain deformation path which is in excellent agreement with the numerical results presented in Figs 1-5 is also considered

4.

DYNAMIC PROPAGATION OF STRESS AND STRAIN DISCONTINUITIES

Consider a plane of discontinuity, Z, moving through a solid with normal velocity V > O. The cartesian co-ordinate system translates with the discontinuity surface. The

Propagation of discontinuities during crack growth (.)

1825

o.o3o x e / ( K , / a ° ) e = 0.01

~N... -/c.=02

...... 0.025

'~

0.020

" "'~'~i

0.015

0.010

.... -0.02

-0.01

0.00

0.01

x,/(r,/~o) ~ (b)

0.030

x,/(l(,/ao)~ =0.01 . . . . . .

0.025

--0.1

v/e,

0.020 el_

-

-

o

~

i\\i \\

0.015

......

o~

I

0.010

.........

0.005

-0.02

,..'R.'N

-0.01

0.00

x,/(K,/<,o)~

0.01

Fig. 4. Effective plastic strain distributions along a line parallel to the crack faces (v = 0.3). The dotted lines connect the strain values just before and after the crack tip. (a) Effect of hardening. (b) Effect of crack speed.

xraxis is normal to Z and points in the direction of propagation; x2 and x3 lie on the plane of discontinuity. The magnitude of the jump in a field quantity, g, across 37, is defined as the difference between the values of g directly ahead of and immediately behind Z. The jump is denoted as follows :

M = g+-g-,

(4.1)

where

g±

=-

lim g x! ~ 0 +

(xl, x2, x3).

(4.2)

Here and throughout the paper, tensor components with respect to the moving cartesian co-ordinate system are indicated by either Latin or Greek indices. Latin indices have range 1, 2, 3 and Greek indices 2 and 3 only, thus referring to tensor components in planes parallel to Z. Both Latin and Greek indices obey the summation convention.

1826

A . G . V A R I A S and C. F. S H I H

(a) ~

-0.10

-0.05

0.00

0.05

0.10

x , / ( z, / ~° ) " (b)

-0.10

-0.05

0.00

0.06

0,~

0.05

0.10

x,/(K,/a0) 2 (c)

-0.10

-0.05

Fig. 5. Active plastic zones (v = 0.3). (a)

0.05 x,/(K,/a,)"

V/cs= 0.2, N = 0.1. (b) V/cs= 0.2, N = 0.2. (c) V/cs= 0.3, N = 0.2.

Propagation of discontinuities during crack growth

1827

In the next section we restate the jump conditions derived from balance of linear momentum, material coherency and compatibility under the assumption of infinitesimal displacement gradient. A detailed derivation has been given by Drugan and Shen (1987) and Drugan and Rice (1984) for dynamically and quasi-statically moving discontinuities, respectively.

4.1. Discontinuity conditions The conservation of linear momentum applied across the moving surface of discontinuity E results in the well-known condition ~a li ~ = -- p V~l)i~ ,

(4.3)

where ~)iis the particle velocity. Material coherency throughout the deformation requires ~ui~ = 0,

(4.4)

where u~ is the displacement vector. Continuity of displacements, combined with the assumption that displacement gradients exist in the neighborhood of Y. and tend to finite limits as Z is approached, implies

[8u,/dx~ = 0.

(4.5)

An immediate consequence of the above relation is the continuity of strain components in the plane of E. Relation (4.4) also implies that the partial time derivative of displacement at a point on E is continuous (~8ui/dt~ = 0). Therefore the jump in particle velocity across E is given by

~vi~ = -- V~du,/dx, l.

(4.6)

The above relation is obtained by considering the material time derivatives of the displacement directly ahead of and behind E. The strain discontinuities are related to the discontinuities of particle velocities and stresses through the following relations :

~ttj~ = - (6~16jk + fijl 6~k)~Vk~/(2 V),

(4.7a)

~'ij~ : (6il (~jk "~- (~jl 6~k)~a~k~/(2p VZ),

(4.7b)

where 6~j is the Kronecker delta. Relation (4.7a) is derived from (4.6) and straindisplacement relations ; (4.7b) is obtained by using (4.3) in (4.7a). 4.2. Restrictions on the deformation path within the discontinuity The jump relations presented in Section 4.1 are obtained independently of constitutive assumptions. In the following discussion, we consider only those deformation paths within the discontinuity that obey the conditions in Section 4.1 as well as the mechanical constitutive assumptions specified below. It is assumed that the total strain increment within E admits an additive decomposition into elastic and plastic parts :

1828

A.G. VARIASand C. F. SHIH p deij = deije + dg~j.

(4.8)

The stresses are linearly related to the elastic strains. For simplicity isotropic elastic behavior is considered:

aij = Lij~t~, = { 26~j5~, + tt(6<6j~ + 6~6jk) }e~,,

(4.9)

where 2 and # are the Lam6 constants. Material yielding is governed by Mises yield criterion. For the most part, our analysis will be carried through for a nonhardening material with yielding governed by

/'3 \t/2 = (~a:ja:j) = o'0.

(4.10)

Comments on strain hardening are held off to a later section where the numerical results for hardening materials are presented. Plastic flow obeys the Prandtl-Reuss equations which are 3 d~p d~p = ~ o.~-a~j.

(4.11)

The above constitutive assumptions supply several additional relations between stress and strain jumps. Plastic incompressibility within E implies ~e~?k~= 0.

(4.12)

From (4.5), (4.9) and (4.12) we obtain the following relation between stress and strain jumps

~(Tij] =

~(~ij-'1-2#~e}~.

~11

(4.13a)

The jump in the hydrostatic stress must also satisfy the relation ~ k ~ = (32 +

2~/)[811 ~.

(4.13b)

The jumps in the elastic and plastic normal strain components in the plane of Y~ must have equal absolute values [see (4.5) and (4.8)] : ~e~2~ = --[~P2~,

(4.14a)

[[e~3~ = -- ~gP3~.

(4.14b)

a similar relation between I-el,~ and [ePI~ can be shown for fully incompressible deformation. Finally the jumps in the remaining three components of the elastic and plastic strains are given by the following relations : pv

~

#--PV 2 ,

(4.14c)

1

(4.14d)

~, ~ = 1

Propagation of discontinuitiesduring crack growth

1829

Observe that as V ~ O, ~e],~ becomes small compared to ~eP,~, while the ratio [ ~e], ~ I/I ~eP~El does not change significantly. 4.3. Velocity of propagation The jump relations derived thus far can be used to determine the propagation speed of the discontinuity, V, in terms of the jumps in the stresses and strains across Z. From (4.8) and (4.9) we obtain [a,j~ = L,jk,([ek,~-- [[e~,~),

(4.15)

Substituting (4.7b) in (4.15) and taking i = l, we get: (L,j,,, - p V 2(~jm)~fflm~ = PV2L ljkl~g~l~"

(4.16)

The relation in (4.16) provides three expressions for the velocity, V: [_ (~-{-2#)[0"II~ l '/2 V = LP@h ,~ +2/~[eP,~)J

[ v =

+l,l

[ /~la'3I Lp@,

(4.17a)

l l''

(4.17b)

l '/2

(4.17c)

Now suppose that ~e11~ ~ 0. Then all as well as hydrostatic stress [see also (4.13b)] suffer jumps across Z. Relation (4.17a) states that the jumps in ~r,, and hydrostatic stress can travel at a speed different from the longitudinal elastic wave speed only if ~e~P,~is nonvanishing. Similarly ~e,,~ ~ 0 implies ~tr,~ ~ 0. Then from (4.17b, c) we may conclude that a jump in shear stress can travel at a speed different from the elastic shear wave speed only if the respective plastic shear strain is discontinuous. When both normal and shearing plastic strains are discontinuous, relations (4.17a, b, c) provide the same value for V. Therefore, we have shown that a discontinuity satisfying the conditions in Sections 4.2 and 4.3 can propagate at any speed, the speed being dependent on the jump magnitude of the field quantities. Drugan and Shen (1987) and Leighton et al. (1987) imposed an additional restriction on possible deformation paths within the discontinuity for the stated purpose of enforcing the maximum plastic work principle. They assumed that if the jump in a quantity is zero, then that quantity remains constant along any admissible deformation path through the discontinuity. Under this additional assumption, they showed that it is not possible for a discontinuity to propagate at a speed other than one of the elastic wave speeds unless the yield condition is satisfied by all stress states traversed along the path. They also proved a stronger result : plastic discontinuities cannot propagate under dynamic plane strain deformation of fully incompressible, elastic-ideally plastic materials obeying Mises yield condition. Their conclusions are indeed correct within the family of deformation paths that they considered. However, we believe that the additional assumption made by Drugan and Shen and Leighton et al. rules out certain

1830

A. G. VARIAS and C. F. SHIH

deformation paths that comply fully with the principle of maximum plastic work. To make our point, we offer in Appendix A an example of a deformation path which does not belong to the family of paths considered by the above investigators, but which still satisfies every mechanical constitutive assumption discussed in Section 4.2. The example provides a cogent argument for extending the theory proposed by Drugan and Shen and Leighton et al. In this work, we do require alterations of stress and strain within the discontinuity to obey the same governing equations and constitutive assumptions as they must outside the discontinuity. Specifically, Mises yield condition and the Prandtl-Reuss flow rule are inviolable. However, we do not impose the additional assumption made by Drugan and Shen and Leighton et al. In effect, a larger family of admissible deformation paths through Z is included in our work. One path within this larger family of paths is discussed in the next section. This path is fully consistent with the numerical results for mode I dynamic crack propagation in nonhardening materials (discussed in Section 3). 4.4. A deJormation path which accommodates stress and plastic strain discontinuities We seek admissible deformation paths within a discontinuity which allow for significant alterations in both stresses and plastic strains. Attention is focused on plane strain conditions, where field quantities do not depend on the x3 co-ordinate. For the present we direct attention to elastic-perfectly plastic materials. Suppose the material is at yield. Then it must satisfy Mises yield condition which is very restrictive for a nonhardening material; accordingly, significant stress alterations cannot be accommodated. Therefore, we may argue that the stress jump can amount to an appreciable portion of a0 only if a part of the deformation path is purely elastic. Indeed, we can expect the stress change in the purely elastic part of the deformation path to dominate the total stress variation across Y~. On the other hand, particle velocity can undergo significant changes only in the elastic-plastic part of the deformation path due to the contribution of the plastic deformation. A path that may produce sizable jumps in both stress and strain is one where the ( + ) state is below yield and the ( - ) state is at yield. Paths with such ( + ) and ( - ) states are observed in all discontinuous numerical solutions for the perfectly plastic material. Figure 6 shows the stress path of a material point as the crack tip passes by. The stress path, indicated by the open circles, begins at the top left edge of the plot ; the solid line is the Mises yield locus. It can be seen that the stress state just ahead of the plane of discontinuity, circle labeled by (+), is below yield. The stress state immediately behind the plane of discontinuity, circle labeled by ( - ) , is at yield. A proposed path which satisfies all conditions presented in Sections 4.1 and 4.2 is indicated by the dash line. However, this path is not included in the family of paths considered by Drugan and Shen (1987) and Leighton et al. (1987) for reasons already noted. Observe that although ~e22~ = 0, e22 may not be constant along the proposed path within Z. Generally, de22 = (1/E){da22--v(dO'l! q-do'33)} 5~ 0, in the purely elastic part of the path and

Propagation of discontinuities during crack growth

1831

x,l(K'l"*)2=o.oz T

-o.v

N--0 u = 0.49

-0.8

/

j b

-0.9 +) -1.o

....

-1.0

ed

, . . ,".

-0.5

"-.".-:'7.

0.0

0.5

1.0

Fig. 6. Stress path of a material point in the near tip region as it moves parallel to the crack plane (given by the open circles). The solid line corresponds to the Mises yield locus. The circles labeled ( + ) and ( - ) correspond to the stress states just before and after the plane of discontinuity, k = e0/x/3.

de22 = (1/E){da22 --v(dall +da33)} + (d~P/ao){a22 - (1/2)(all + a33)} ~ 0, in the elastic-plastic part of the path. We have noted that alterations in stresses are small in the elastic-plastic part of the path. Therefore, the stress state within the elastic-plastic part of the path is nearly equal to the ( - ) state at the end of the path. This observation and (4.1 l) implies that 3

~02 ~P~.

(4.18)

Our numerical results satisfy (4.18)--the left- and right-hand sides of (4.18), calculated from the finite element solutions, differ by no more than 2%. Recall that jumps in the normal strain components in the plane of Z obey ~e~2 ~ = --~E2P2~ and [e~3~ = -[e~3~ [see (4.14a, b)]. By virtue of this result and the fact that the jump in the effective plastic strain is assumed to be large compared to the yield strain, ao/E, for the problem under consideration, we conclude that [e2P2~ << [~P~,

(4.19a)

~e3P3~<< ~P~.

4.19b)

Relations (4.19a, b) combined with the Prandtl-Reuss relations lead to ai-l ~ a~2 ~ a33,

(4.20a)

and (7 o

I tri-2l ~ : - .

#3

(4.20b)

1832

A.G. VARIAS and C. F. SHIH d

-0.10

-0.05

0.00

0.05

0.10

x,/(K,/~,)* Fig. 7. Active plastic zone (N = 0, v = 0.3, V/cs= 0.15).

That is, the stress state at the end of the path is nearly that of a hydrostatic stress superimposed on pure shear. Our numerical results for the ( - ) state (for example, see Fig. 6) are fully consistent with (4.20). The calculated values for a~, az2 and a73 differ by less than 2% ; also, the calculated value for a~2 is within 2% of the shear yield stress. To conclude this discussion, we note that strict equality a m o n g the three normal stresses, al~, a22 and a33, along the elastic-plastic part of the path implies (for both compressible and incompressible materials) that the discontinuity can propagate at a speed different from the elastic wave speed only when ~akk~ = 0. Though an admissible path, this is not the path followed by the solution discussed in Section 3 which has a j u m p in the hydrostatic stress. Next, we consider how discontinuity propagation speed V can affect the structure of the discontinuity. For this purpose we examine relations (4.14). We have already noted that as V decreases the j u m p in the elastic shearing strain becomes smaller relative to the j u m p in the plastic shearing strain. Likewise, the shearing traction j u m p also decreases, tending to zero in the limit of vanishing crack speed. Consequently, we can expect that as V decreases the elastic part of the path becomes smaller so that the ( + ) state approaches yielding. This provides an explanation for the reduction in size of the elastic unloading sector ahead of the discontinuity for the lower crack speed shown in Fig. 7 (compare Figs 7 and 1). We now consider strain-hardening effects on jumps across Z. For a nonhardening material, it was noted that the yield condition imposes strong restrictions on the amount of stress alterations. I f strain-hardening is allowed then the yield condition is not so restrictive. In this case it is possible to accommodate both stress and plastic strain discontinuities along a deformation path, while the material is always at yield. Indeed, this is observed in a low hardening material N = 0.1 and V/cs= 0.2. Numerical calculations performed with a very fine near-tip mesh revealed a region close to the tip where the two elastic sectors are separated by a discontinuity much like the

1833

Propagation of discontinuities during crack g r o w t h

N=0.1, T/o'.=O, v/o-.=02

~2

% x

-0.0~

-0.005

0.000

0.005

0.010

x,/(Ktfir,)~ Fig. 8. Active plastic zone near the crack tip (N = 0.1, v = 0.3,

V/cs =

0.2).

discontinuous field that prevails in a nonhardening material. The jump magnitude is diminished as the distance from the crack tip becomes greater. At a certain distance, our numerical solution shows that both stress and plastic strain jumps are accommodated while the material is at yield in both ( + ) and ( - ) states. This behavior can be seen in Fig. 8 (the transition to one elastic sector occurs at distance x2 ~ 0.01

(K, Icro)2). 5.

EFFECT OF T-STRESS--A CHARACTERISTIC LENGTH

For stationary cracks several investigators have shown that the T-stress, the second term of the asymptotic series for the stress field in a linear elastic material (Williams, 1957) exerts a strong effect on the size of the plastic zone and near-tip stress triaxiality (Larsson and Carlsson, 1973 ; Rice, 1974 ; Betegon and Hancock, 1991 ; O'Dowd and Shih, 1991, 1992). T-stress effects on steady-state quasi-static crack growth have been examined by Varias and Shih (1993). They concluded that, under contained yielding conditions, the near-tip fields conform to a one-parameter field based on a characteristic length, Lg. Both loading and crack geometry enter the near-tip field distribution only through L v which therefore scales the intensity of the near-tip field. The spatial extent of validity of the one-parameter near-tip field is of the order of the plastic zone size directly ahead of the tip. The present study reveals that a one-parameter field, as noted above, also applies to a dynamically propagating crack. Specifically, the dynamic KI and T scale the neartip field of a crack propagating steadily at a speed V only through a characteristic length, Lg. In the dynamic problem, the ratio LJ(Ki/ao) 2 depends on both T/tro and V/ Cs,

1834

A . G . VARIAS and C. F. SHIH

-0.08

,.0.04

0.00

0.04.

0.08

x,/(K,/~,)~ Fig. 9. Active plastic zone (N = 0.1, v = 0.3, V/cs = 0.2). Effect of positive T-stress.

to'0/" Based on the above characteristic length the near-tip field has the form :

aoaq= E~j

, O; Cs/ '

e,: = E0

,0 ;

Lg

,0 ;

(5.3)

/30

U,.

,

(5.4)

where e0 = ao/E and the nondimensional functions Eij, E 0 and U~ depend additionally on dimensionless combinations of material parameters. When distances are normalized by Lg, spatial variations of the fields associated with T/ao <~0 (for fixed V/cs) obey a c o m m o n pattern over - n ~< 0 ~< n and r/Lg ~< 1.4. Fields associated with T/go > 0 exhibit the same pattern in the forward sector [ 01 < n/2 and r/Lg ~< 1. For [ 0[ /> ~z/2 the field does not obey the c o m m o n pattern as it is affected by the large rotation of the plastic zone to back sector, illustrated in Fig. 9. A characteristic length can be defined using any of the field quantities in (5.2)(5.4). In this paper, a definition of Lg based on the effective plastic strain directly ahead of the crack (0 = 0 °) is adopted :

:

~p.

Lg

__

, T , V ~/

2' 0 . . . . .

\(K, tcro)

ao cs /

/

rpot2

~P[ - - 2 , 0

\(K, Icro)

o

V\

;0,--/,

cs /

(5.5)

Propagation of discontinuitiesduring crack growth

1835

Table 1. Normalized characteristic length, Lff(Kffao)2 for V/cs = 0.2 N

T/ao 1.00 0.50 0.00 -0.50 -0.75 1.00 -

0 0.007504 0.01479 0.02274 0.01824 0.01270 0.006629

O.1 0.008980 0.01528 0.01873 0.01506 0.01143 0.006845

0.2 0.01161 0.01662 0.01787 0.01483 0.01202 0.008077

where rp0 is the plastic zone size at 0 = 0 °. One may use any point within the domain of validity of (5.2)-(5.4) for the definition of Lg. The definition in (5.5) proves to be convenient because it provides a length comparable to the plastic zone size ahead of the tip, rp0. A similar definition for Lg was used for the quasi-static steady-state crack growth problem (Varias and Shih, 1993). Table 1 lists the values of Lg/(Ki/ao) 2 for V/cs = 0.2, - 1 ~< T/ao ~< 1 and N = 0, 0.1 and 0.2. These values differ slightly from those given by Varias and Shih (1993) for quasi-static crack growth. The weak dependence of Lg on V is a consequence of the weak dependence of ~P along 0 = 0 ° on V which is shown in Fig. 10(a). By contrast, if we had chosen a definition of Lg based on the radial variation of ~Palong 45 °, the values of/2g would depend sensitively on V; this can be deduced immediately from the results displayed in Fig. 10(b). It bears emphasizing that for a given crack speed, the near-tip field distribution does not depend on T-stress when distances are normalized by Lg; moreover, the validity of this statement does not depend on the way Lg is defined, as long as it is defined within the domain of validity of (5.2)-(5.4). The distribution of near-tip field quantities is examined next. Numerical solutions for N = 0, 0.1 and 0.2 have been obtained for several crack speeds and T-stresses. Every solution conforms to the general structure presented above. In the interest of space only results for N = 0.1 are reported. Some results for the N = 0 case are discussed.

5.1. Negative T-stress Figure 11 (a) shows the distribution of effective plastic strain along the crack line (0 = 0 °) for Vies = 0.2 covering a broad range of negative T-stresses. Here, distances are normalized by (/(i/o'0) 2. It can be seen that plastic strain displays strong dependence on T-stress--the strain level decreases as T-stress decreases. In Fig. 11 (b), distances are normalized by Lg--observe that the different distributions collapse to a single distribution in accordance with relations (5.2)-(5.4). The above behavior is also observed along other angles and for other hardening exponents and crack speeds investigated. The variation of the hoop stress along the crack line for the same crack tip speed and range of T-stresses is presented in Fig. 12(a) ; distances are normalized by (Ki/a0) 2.

1836

A . G . VARIAS and C. F. SHIH

4.0

(a)

\~

O=O*

vii

3.0

N=0.1

~

T/Go=O.

2.0

I~

1.0 o o

,

....

0.00

0.01

0.02

0.03

0.04

r/tX,k,o) ~ 0.010

0.0o8 0.006

,~

\\\\

o'l

............

~A;'~, "\\x"\.\N

.

~

.

.

-

.

o~.

.

0.3

-

0=4

0.002 N=O.I

" ~

T/oo=O. 0.000

0.00 0.01 0.02 0.03 0.04 ~/(K,l,,of . . . .

'

. . . .

'

. . . .

'

'

•

,

,

Fig. 10. Effective plastic strain distribution for several crack speeds. (a) Along the crack line. (b) Along 0 = 45 °.

It can be seen that the stress level decreases significantly with large negative Tstress. Again the different distributions collapse to a single curve, when distances are normalized by Lg as shown in Fig. 12(b). The above scaling also works for the other stress components and radial lines of different angles. It is noteworthy that the features displayed in Figs 11 and 12 resemble those for the quasi-static steady-state crack problem [see Figs 5 and 8 in Varias and Shih (1993)]. Of course, the one-parameter field for the dynamically propagating crack considered here depends additionally on the crack speed V. It is found that stress and effective plastic strain levels, at fixed normalized distance from the tip, decrease as V increases. The angular stress variation at r = 0.5 Lg is shown in Fig. 13(a,b) for N = 0.1 and N = 0; included for comparison purposes is the analytical stress distribution for quasi-static crack growth (Drugan, Rice and Sham, 1982). Scaling distances by L~ collapses all solutions (in the range 0 ~< T/fro <~ 1) into one distribution. It is easily

Propagation of discontinuities during crack growth

%

4.0

(a)

~i.~.

0--0"

i\-\

N=o.1

3.0

I~

1837

\-\

2.0

\, \,',...\ "~. *+°'*+

1.0

--1.

0.0 0.00

%

(b) I~

0.01

0.~

r/(r,/oof

0.03

0.114

4.0

~i

3.0

O=0* N=0.1

2.0

1.0

o.o

•

0.0

°

0.5

. . . . .

1.0

r/S,g

P.0

1.,5

Fig. 11. Effective plastic strain distribution along the crack line for several T-stresses. (a) Distances are normalized by (KUtr0)2. (b) Distances are normalized by Lg.

seen that the jump magnitudes are equal for all values of T. Similar plots were generated for several other distances from the crack tip ; the one-parameter scaling is found to apply for distances up to 1.4 Lg. To summarize---the jump magnitude across E decreases as the distance from the tip increases and the jump magnitude does not depend on T-stress when distance is normalized by Lg:

~aO~= Ao( r V) tr0

~gg,~

{r

o_ V\

{r

: E~jt~--~g,90 ,~)--Eijt~gg,90

o+ V\

,~).

(5.6)

The jump magnitude depends on N as well. For a given crack speed and normalized distance ahead of the tip, the jump magnitude increases as N becomes small---compare Fig. 13(a) with Fig. 13(b). In Fig. 13(b) the angular distribution of the asymptotic field for quasi-static crack growth is also shown. Note that an increase in crack speed brings about a reduction in the stress level ahead of the tip.

1838

A . G . V A R I A S and C. F. S H IH

(a)

4.0 0 = 0

3.5

°

N=0.1

~

v/c,=0.2

3.0 b 2.5

-0.5

2.O

,

,

0.00

,

,

I

j

,

,

,

0.01

I

,

, ~ ,

0.02

I

i

i

,

,

0.03

-0.75 0.04

r/(K,/#o)s (b)

4.0

35

v/es = 0.2 2.5

-IST/oo~0 2.0

. . . .

0.0

i

. . . .

0.5

I

. . . .

1.0

I

,

1.5

,

,

,

2.0

r/L Fig. 12. H o o p stress distribution along the crack linc for scveral T-strcsscs. (a) Distances arc normalized by (Ki/ao)2. (b) Distances arc normalized by Lg.

Crack opening profiles for several negative T-stresses and V/cs = 0.2 are shown in Fig. 14(a,b). The distinct profiles in Fig. 14(a), where distances are normalized by (Ki/a0) 2, collapse into a single profile in Fig. 14(b) when distances are normalized by Lg, as could be anticipated from (5.4). Figure 15 shows crack profiles for crack speeds in the range 0 ~< V/ca <<,0.3, but for T = 0 ; here distances are normalized by (K1/a0) 2. Note that a common profile is obtained. 5.2. Positive T-stress The one-parameter field discussed thus far in connection with negative T-stresses is also valid for positive T-stresses ; however, the field applies within a smaller domain. Figure 16(a,b) shows the radial distribution of the effective plastic strain ahead of the tip (0 = 0 °) for T/ao = 0, 0.5 and 1, N = 0.1 and V/cs= 0.2. The distributions of the hoop stress along the same direction and for the same remote load conditions, crack

Propagation of discontinuities during crack growth

(a)

1839

4.00

:.............

v/~-02

:

"\

-x'~Va.,~o

:

......... k" ........

Z.75

•N = 0 15~

-1.00

-"° : r/L s = 0 . . . . . . .~" -,.-°°° .... ' .... ' .... ' .... ' .... 0.0 20.0 6 0 . 0 9 0 . 0 1 2 0 . 0

~12 '

....

150.0

180.0

0

(b)

a.0 . . . . . . . - ~ , • : 2.0 ~

g

1.0

~

"

t

quuistatlc growth \ X ...... v/c.: O2, \%-~ -I~T/'~.~:0

T

l

~

t~ 0.0

f

f

~

r/L z = 0.8 --1.0

....

i ....

0.0 30.0

,,,i

60.0

i ....

i ....

I,,,~

90.0 120.0 150.0 180.0

0 Fig. 13. Angular stress distributions for negative T-stresses (r/L~ = 0.5). All distributions, including those which suffer discontinuities, collapse onto a single curve. (a) Hardening material, N = 0.1. (b) Nonhardening material, N = 0.

tip speed and hardening are presented in Fig. 16(c,d). It can be seen that both the effective plastic strain and the hoop stress exhibit strong dependence on T-stress when distances are normalized by (Ki/a0)2; both field quantities decrease as T-stress increases. When distances are normalized by Lg the distinct distributions merge into what appears to be a single curve for each field quantity. Under positive T-stress, our calculations show that a one-parameter scaling is applicable for r/Lg ~< 1. This is smaller than the spatial range of validity for negative T-stress. Positive T-stress produces strong perturbations to the field behind the crack tip. This is clearly seen from the plot of the active plastic zone in Fig. 9 for T/ao -- 0.5. A related effect is the increase in the plastic strain level in the region behind the tip. The higher straining results in crack tip profiles which differ slightly from that obtained for Tfiro = 0. The crack profiles for T/cro = 0, 0.5 and 1 are shown in Fig. 17. Note that distances are normalized by (Kx/tr0)2. We should point out that (5.2)-(5.4) are

A. G. VARIAS and C. F. SHIH

1840

ro ~,

io.o

~ =o.

(a)

N=O.I

"'--~L'-'o.5..'b,,v/~,=o2

8.0

~,

6.0

',<:....

4.0 2.0

,

0.0

,

-0.04

(b)

,

,

I

,

-0.03

,

,

,

I

,

,

-0.02

,

,

x,/(z,/o07

0.05 ~

I

.

.

.

.

-0.01

i.O0

N=0.1

0.04~

2

0.03

0.02

0,01

-I ~:T/'ao~:0 0.00

....

~ ....

-2.0

-1.5

~ ....

-1.0

~

....

0.0

-0.5

x~/Lg

Fig. 14. Crack profiles for V/cs = 0.2. (a) Distances are normalized by (Kd~ro) 2. (b) Distances are normalized by Lg.

T.O

10.0

""x N=0.1

0.0

•

6.0

4.0

~

r/~o=O.

2.0

0:; v/cs~ 0.3 0.0

. . . . . . . .

.

-0.04

-0.03

-O.Oe

x,/~,/,~o)'

-0.01

O.O0

Fig. 15. Crack profiles for a range of crack speeds.

1841

Propagation of discontinuities during crack growth

%

y,

4.0

~

(~)

4.0

(c)

8=0o

0=0 °

.

3.0

~

N=O.I v/c~=02

3.5

.=°.L

2.0

3.0 b

1.0

1. \ 0.0

2.5

.....

. . . . . . .

o,oo

o.o,

o.o~

o.o3

2.0

o.o,

,

,

,

,

0.00

I

.

.

.

.

I

i

0.0~

0,01

i

i

,

I

,

0.03

,

,

i~

0.04

~/(Z,loo)~

%

4.0

4.0

~~.

(b)

e=O*

3.0

"••__--••_,

(d) 3.5

N=0'102

3.0

2.0

N~O.I

~.,,

2.5

1.0

O~T/ao~ I 0.0 0.0

0.5

1.0

1.5

o,o

2.0

0,5

r/r.g

I.O r/Lg

1,5

2.0

Fig. 16. Effective plastic strain and hoop stress distributions along the crack line for positive T-stresses. (a), (c) Distances are normalized by (K[/tr0) 2. (b), (d) Distances are normalized by Lg.

-~ 4~

3.o~

10.0

~_

N=O.I

8.0

~

4.0

2.0

0.0

.

.

.

.

.

.

'

.

.

.

.

.

.

.

-0.04 -0.03 -0.02 -0.01

'

'

0.00

Xa/(K,/Qo)' Fig. 17. Crack profiles for positive T-stresses. Distances are normalized by (Kl/ao) 2.

1842

A.G. VARIAS and C. F. SHIH

valid over very small distances in the back sector. This feature has also been discussed in the case o f quasi-static crack growth.

ACKNOWLEDGEMENTS This investigation is supported by the Office of Naval Research through Grant N00014-90J-1380 and a Grant from the David Taylor Research and Development Center funded by the Nuclear Regulatory Commission. This work and a preliminary version of the manuscript was completed during A. G. Varias' post-doctoral year at Brown University (January-December 1992). A. G. Varias also acknowledges support from Shell Research to complete the writing of the paper.

REFERENCES Achenbach, J. D. and Dunayevsky, V. (1981) Fields near a rapidly propagating crack-tip in an elastic-perfectly plastic material. J. Mech. Phys. Solids 29, 283-303. Achenbach, J. D., Burgers, P. and Dunayevsky, V. (1979) Near-tip plastic deformation in dynamic fracture problems. Nonlinear and Dynamic Fracture (ed. N. Perrone and S. Alturi), AMD-35, pp. 105-124. ASME, New York. Achenbach, J. D., Kanninen, M. F. and Popelar, C. H. (1981) Crack tip fields for fast fracture of an elastic-plastic material. J. Mech. Phys. Solids 29, 211-225. Beteg6n, C. and Hancock, J. W. (1991) Two parameter characterization of elastic-plastic crack tip fields. J. Appl. Mech. 58, 104-110. Deng, X. and Rosakis, A. J. (1991) Dynamic crack propagation in elastic-perfectly plastic solids under plane stress conditions. J. Mech. Phys. Solids 39, 683-722. Deng, X. and Rosakis, A. J. (1992) A finite element investigation of quasi-static and dynamic asymptotic crack-tip fields in hardening elastic-plastic solids under plane stress. Int. J. Fract. 57, 291-308. Dean, R. H. and Hutchinson, J. W. (1980) Quasi-static steady crack growth in small-scale yielding. Fracture Mechanics: 12th Conf., A S T M S T P 700, pp. 383-405 American Society for Testing and Materials. Drugan, W. J. and Rice J. R. (1984) Restrictions on quasi-staticaUy moving surfaces of strong discontinuity in elastic-plastic solids. Mechanics o f Material Behavior: The D. C. Drucker Anniversary Volume (ed. G. J. Dvorak and R. T. Shield), pp. 59-73 Elsevier Science B.V., Amsterdam. Drugan, W. J. and Shen Y. (1987) Restrictions on dynamically propagating surfaces of strong discontinuity in elastic-plastic solids. J. Mech. Phys. Solids 35, 771 787. Drugan, W. J., Rice, J. R. and Sham, T.-L. (1982) Asymptotic analysis of growing plane strain tensile cracks in elastic-ideally plastic solids. J. Mech. Phys. Solids 30, 447-473. Freund, L. B. (1976) The analysis of elastodynamic crack tip fields. Mechanics Today (ed. S. Nemat-Nasser), Vol. 3, pp. 55-91. Pergamon, Oxford. Freund, L. B. (1990) Dynamic Fracture Mechanics. Cambridge University Press, New York. Gao, Y. C. and Nemat-Nasser, S. (1983) Dynamic fields near a crack tip growing in an elasticperfectly-plastic solid. Mech. Mater. 2, 47-60. Lam, P. S. and Freund, L. B. (1983) Elastic-Plastic Finite Element Analysis o f Steady State Dynamic Crack Growth in Plane Strain Tension. Materials Research Laboratory, Brown University, MRL E-149. Lam, P. S. and Freund, L. B. (1985) Analysis of dynamic growth of a tensile crack in an elastic-plastic material. J. Mech. Phys. Solids 33, 153-167. Larsson, S. G. and Carlsson, A. J. (1973) Influence of non-singular stress terms and specimen

Propagation of discontinuities during crack growth

1843

geometry on small-scale yielding at crack tips in elastic-plastic materials. J. Mech. Phys. Solids 21, 263-277. Leighton, J. T., Champion, C. R. and Freund, L. B. (1987) Asymptotic analysis of steady dynamic crack growth in an elastic-plastic material. J. Mech. Phys. Solids 35, 541563. O'Dowd, N. P. and Shih, C. F. (1991) Family of crack-tip fields characterized by a triaxiality parameter--I. Structure of fields. J. Mech. Phys. Solids 39, 989-1015. O'Dowd, N. P. and Shih, C. F. (1992) Family of crack-tip fields characterized by a triaxiality parameter--II. Fracture applications. J. Mech. Phys. Solids 40, 939-963. Ostlund, S. and Gudmundson, P. (1988) Asymptotic crack tip fields for dynamic fracture of linear strain-hardening solids. Int. J. Solids Struct. 24, 1141-1158. Parks, D. M., Lam, P. S. and McMeeking, R. M. (1981) Some effects of inelastic constitutive models on crack tip fields in steady quasistatic growth. Adv. Fract. Res., 5th Int. Conf. on Fracture (ed. D. Francois) Cannes, France, 5, pp. 2607-2614. Rice, J. R. (1968) Mathematical analysis in the mechanics of fracture. Fracture (ed. H. Liebowitz), Vol. 2, pp. 191-311. Academic Press, New York. Rice, J. R. (1974) Limitations to the small scale yielding approximation for crack tip plasticity. J. Mech. Phys. Solids 22, 17-26. Rice, J. R. and Tracey, D. M. (1973) Computational fracture mechanics. Numerical and Computer Methods in Structural Mechanics, (ed. S. J. Fenves et al.) pp. 585-623. Academic Press, New York. Shen, Y. and Drugan W. J. (1990) Constraints on moving strong discontinuity surfaces in dynamic plane-stress or plane-strain deformations of stable elastic-ideally plastic materials. J. Appl. Mech. 57, 569-576. Slepyan, L. I. (1976) Crack dynamics in an elastic-plastic body. Izv. Akad. SSSR, Mekh. Tverdo#o Tela 11, 144-153. Tracey, D. M. (1976) Finite element solutions for crack-tip behavior in small-scale yielding. J. Enyn# Mater. Technol. 98, 146-151. Varias, A. G. and Shih, C. F. (1993) Quasi-static crack advance under a range of constraints-steady-state fields based on a characteristic length. J. Mech. Phys. Solids 41, 835-861. Williams, M. L. (1957) On the stress distribution at the base of a stationary crack. J. Appl. Mech. 24, 109-114.

APPENDIX A : AN EXAMPLE OF A STRONG DISCONTINUITY WHICH PROPAGATES DYNAMICALLY WITHOUT VIOLATING MAXIMUM PLASTIC WORK INEQUALITY Drugan and Shen (1987) have proven that a strong discontinuity can propagate at a speed different from one of the elastic wave speeds only when yield condition is satisfied throughout the stress path across the discontinuity. They also considered the deformation of an incompressible elastic-ideally plastic material, satisfying Mises yield condition, under plane strain ; they proved that no strong discontinuity can propagate dynamically at a speed different from an elastic wave speed if the maximum plastic work inequality is to be satisfied. A proof was also given by Leighton et al. (1987). The proofs of the above statements were based on certain restrictions on the structure of the path through the discontinuity. These restrictions are given by relations (3.1) and (3.2) in the paper of Drugan and Shen. Leighton et al. have imposed the same restrictions by assuming that "if the jump in a quantity is zero then that quantity remains constant along the strain path through the discontinuity". In the following we consider plane strain deformation of a fully incompressible elasticideally plastic material obeying Mises yield condition. We examine a dynamically propagating discontinuity and will show that there is a deformation path through the discontinuity which satisfies the Prandtl-Reuss flow rule and therefore complies with the maximum plastic work inequality. The restrictions of Drugan and Shen will not be applied. It will also be shown that

1844

A.G. VARIAS and C. F. SHIH

the discontinuity path can have a purely elastic as well as an elastic-plastic part. This discontinuity can travel at any speed, the speed being dependent on the magnitude of the stress and strain jumps. Consider the following discontinuity : =

=

=

0,

[0-,2~ ¢ 0, (A1)

0-11 = 0"22 = 0-33,

where x2-x3 is the plane of discontinuity and x~ is the direction of propagation. From relations (4.5) and (4.7b) we conclude that

Also from stress-elastic strain relations we get

and therefore

For the nonzero shear jumps the following relations apply : ~,2~ = 2pV2~,2~ = 2/t~e]2].

(A5)

So far we have made no assumptions concerning the structure of the deformation path other than requiring the structure to obey strain decomposition and Hooke's law. In what is to follow a structure of the path is discussed. We consider cases which do not belong to the family of paths treated by Drugan and Shen. The discontinuity path is assumed to have a purely elastic part where the shear stress is changing and an elastic-plastic part where the stress remains constant while the effective plastic strain is changing. The beginning of the purely elastic part of the path is the ( + ) state while the end of the elastic-plastic part is the ( - ) state. According to the assumptions for that particular path, 0-0

0-~-2

V/~

(A6)

as derived from the Mises yield condition. Also from the Prandtl-Reuss relations we get : = 5 o--~-0-,2=

~P~.

(A7)

Note that (A4) can also be derived from the flow rule. Relations (A5) and (A7) have to be satisfied simultaneously. Combining these relations we obtain the speed of the discontinuity in terms of the jump quantities : V=

[.

"~-0-I2~

11/2

Lp([0-, 2]] + p, ,v/~ [~p]~)j

(a8)

"

Note that the speed depends on jump magnitudes of both the shear stress and the effective plastic strain. The above path does not satisfy the restrictions of Drugan and Shen. According to Drugan and Shen, the relation d0-12 = 2 p V 2 d812

Propagation of discontinuities during crack growth

1845

should apply throughout the path. One may observe that the above relation cannot be satisfied if the jump in the effective plastic strain is nonzero. When [~P~ ¢ 0 we have dCrl2 = 2/~d~12,

(A9)

dal2 = 0,

(A10)

in the purely elastic part and

in the elastic-plastic part of the path. Moreover, the Prandtl-Reuss flow rule and Mises yield condition are satisfied within the elastic-plastic part of the path. Therefore we have shown that the above deformation path through the discontinuity satisfies all material constitutive assumptions as it should. We end this discussion by noting that although the above is an admissible path, it is not the path followed by the solution to the boundary value problem under investigation. The solution to the latter problem exhibits a discontinuity in hydrostatic stress (see Fig. 3 and discussion in Section 3) while the example admissible path discussed in this Appendix does not. We should emphasize that a jump in hydrostatic stress does not exclude the case where the ( - ) state is strictly a hydrostatic stress superimposed on pure shear.

APPENDIX B : NUMERICAL

PROCEDURE

The numerical algorithm of Dean and Hutchinson (1980) and Parks et al. (1981), as extended by Lain and Freund (1985) to dynamic steady-state problems, was employed for the solution of the problem described in Section 2. The approach, based on infinitesimal displacement gradient theory, is considered adequate for the solution of steady-state crack growth problems since finite deformation effects are expected to be confined to a zone which is small compared to the length scales of interest. In the following all dependent variables are referred to the moving cartesian co-ordinate system with origin centered at the crack tip as defined in Section 2. Under steady-state conditions the equations of momentum balance take the following form :

O0"iJ

2 02ui

O--~j= pV ~x~ "

(B1)

Multiplying both sides of (B.1) by a virtual displacement and integrating over a volume V0 of the material, one may derive:

IVofO

~

. : ~u, ~ ( & ) ) . .

t6u,aonj--pl~6ui~xlnl~d + fvofe,D~jkte5 d Vo.

(B2)

D~kt is the elastic stiffness tensor. Equation (B.2) is solved by an iterative procedure. The initial estimate for e~ is generated by parameter tracking. The elastic solution (e~ = 0) is used to begin the iteration for the high hardening case. Once a convergent solution for this case is obtained, the distribution of ~ is then used to begin the iteration for a slightly lower hardening material. In this manner solutions for the full range of N are obtained. The left-hand side of (B.2) produces a global stiffness matrix, K, based on the elastic properties of the material and the crack tip speed. Therefore, the formation and triangular decomposition of K is performed once during the initial elastic solution. This decomposed K is used throughout the remaining computations for the crack tip speed under consideration. The applied tractions, determined from (2.1), and the displacement gradients, which correspond to

1846

A.G. VARIAS and C. F. SHIH

that field (see Appendix C), contribute to the first integral on the right-hand side of (B.2). This force vector is also formed during the initial elastic solution for the prescribed values of K~, T and V and is then unchanged. The second integral on the right-hand side of (B.2) is evaluated for an estimate of the plastic strains ; this vector changes at every iteration. One can now solve for the displacements and compute the total strains. The elastic stress-strain relation is used for the calculation of the stresses in the elastic region. Stresses in the elastic-plastic region can be calculated by integrating along trajectory lines of material particles, which are simply straight lines parallel to the crack faces. Under steady-state conditions an observer moving with the tip sees an unchanging field. Therefore, an increment of stress associated with a material point, due to an increment of the crack length da, is related to the xj-gradient : (B3)

d~rij = - d a c~aij/c~xl .

A similar relation also applies to total strain rates. Therefore, along a particle trajectory, the stress variation is related to the strain variation : ,~ij/~X1 ep

ep

(B4)

= D ijkl O~.kl/OX 1 .

'

Dii~l is the elastic-plastic stiffness : Dijkl = ~ v

6ik6j,+ l _ - ~ v f i i 6 k , -

1 + (2/3)H'(1 + v ) / E

nijnk~ .

(B5)

where nij is the unit vector normal to the yield surface, defined as in Rice and Tracey (1973) and taken as being constant between two successive integration points along x~. M is the elastic fraction of the strain increment, admitting values in the range 0 <~ M <~ 1. H' is the hardening rate, which is taken as the tangent of the stress-strain curve at the initial state of each integration step. Numerical experimentation has shown that this value of H' provides a more accurate description of the material response when unloading is imminent than a value based on the secant between the initial and an estimate of the final state (Varias and Shih, 1993). (When unloading is imminent H' is nearly unchanging so that the initial state does provide an accurate estimate.) More details on the above integration for nonhardening and hardening materials can be found in Rice and Tracey (1973) and Tracey (1976), respectively. Once the stresses have been calculated, the plastic strains can be determined by subtracting the elastic from the total strains. The numerical procedure is repeated until convergence is achieved. A coarse version of the finite element mesh is shown in Fig. B1 ; the quadrilateral elements are made of four cross-triangles. The strip of refined mesh parallel to the crack face is sized so that it always contains the active plastic zone and plastic wake. The height of the strip ranges from 0.5~).85% of the outer mesh radius, where the two-term elastodynamic crack-tip field is applied. A region of highly refined mesh surrounds the crack tip. It is sized to contain the plastic zone directly ahead of the tip and the unloading sectors behind the tip ( - 1 ~< xt/rpo <~ 1, 0 <~ x:/rpo <~ 1). This region has 4600 6000 quadrilateral elements; the actual number of elements used depends on the T-stress. All elements are square shaped and of equal size, say E. About 50 elements cover the linear distance rp0 ahead the crack, i.e. E ~ 0.02rp0. Depending on the value of T, f ranges from 0.8 x 10 - 7 to 0.4 × 1 0 - 4 of the distance to the outermost boundary. Another 5000 or so quadrilateral elements are used to connect the region of highly refined mesh to the remote boundary.

APPENDIX C : ASYMPTOTIC FIELD OF A MODE I CRACK DYNAMICALLY PROPAGATING IN AN ELASTIC MATERIAL UNDER PLANE STRAIN CONDITIONS The first two terms of the asymptotic expansion of the elastodynamic mode I crack tip field under plane strain conditions are given by the following relation (see Freund, 1990) :

1847

Propagation of discontinuities during crack growth

(a)

I I

i

ii

....................................., :::::::::::::::::::::::

I I |

I 1 ~

......... ! . . . . .

-i,'~

] "

/ -'-"

~ ~ ~ ~iiiiiiiiiiiiiiiiii~i~i~iiiiiii~:~i ;.J

/

crack tip

Cb)

Fig. BI. (a) Coarse version of near-tip mesh. The active plastic zone and plastic wake are contained within the strip of refined mesh parallel to the crack face. (b) Far-field mesh. Tractions corresponding to the first two terms of the elasto-dynamic asymptotic solution are applied along the outer boundary.

KI

l

o'q -- x/~r~.,ij(O, V/cs) + T6lfilj.

(C1)

The angular stress functions are 1 f

Y~I22

1

I

cosz0d cosz0s) (1+~)( l + 2 ~ d - ~ s ) - - 4 ~ d '

= -

~-(1 +~,~2)2 ~os~O,~ 1 ,~ ( 1 -

oos½O 0 ,

40q~a ~

(C2b)

x/Y~ J

s'n O t where D=4~d~--(l+~)

2,

(C3a)

1848

A.G. VARIAS and C. F. SHIH

7d = ~/1--(VsinO/Cd) 2, tan0d = ad tan 0,

(C3b)

~,~ = x/1--(VsinO/G) 2, tan0~ = a~tan0,

(C3c)

•d = ~/1--(V/%) 2, ~, = x / 1 - ( V / c , ) 2.

(C3d)

c, = ~ and cd = ~ are the elastic shear and longitudinal wave speeds, respectively. Under steady-state conditions, particle velocity coincides with the displacement gradient along the x~ direction multiplied by - V . The xrderivatives of the displacements, which correspond to the first two terms of the asymptotic expansion, are

au,

K,

c~x,

#D~/2nr t 0u2_

~xj

r..

cos½0

~(1 + Cts) ~

x/gd

%K~ I(l+

#Dx/2nr (

o

cos½O,

- Zas% - ~ - ~

x/7~ )

2 sin½0d sin½0,) cq ) ~ -- 2 - - }

~/7~

~

1-v

+ - ; - - T,

ZlJ

"

(C4a)

(C4b)