chen hahn efm1993 stress and deformn beh mode i 3d elpl crack tip fields

8/13/2019 Chen Hahn EFM1993 Stress and Deformn Beh Mode I 3D Elpl Crack Tip Fields

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Enginedng Fmcwe M echanicsVol. 44, No. 6, pp. 895-912, 1993 0013-7944/93 s6.00 0.00Printed in Great Britain. 0 1993 Pergamon Fkss Ltd.

STRESS STRUCTURE AND DEFORMATION BEHAVIOROF MODE I THREE-DIMENSIONAL CRACK IN

ELASTIC-PLASTIC STATEX. M. CHEN and H. G. HAHN

L.ehrstuhl fi% Teehnische Mechanik, Universitat Kaiserslautern, Postfach 3049, Geb. 44,6750 Kaiserslautern, GermanyAhatraet-The effects of load and geometry on the stress structure of a Mode I three-dimensional crackare investigated by means of finite element method. The functions of plastic deformation and stresstriaxiality constraint, during the failure process, are then analyzed. It is found that three regions, namelythe plane strain similar zone ZI, the plane stress similar zone ZIII and the transition layer between themZII, exist in front of the crack tip; three-dimensional deformation behavior is different from that intwo-dimensional states even in the ZI and ZIII zones. It is also revealed that the failure form and positionof a Mode I three-dimensional crack will be determined by both plastic strain and stress triaxiaiity.

INTRODUCTIONIT IS well known that the stress and strain fields of a Mode I crack are three-dimensional in nature,especially when plastic deformation is notable. However, they are different from those in generalthree-dimensional cases because of the particularities of specimen geometry and applied load. Someinvestigations on this subject show that Mode I cracked bodies can be treated as plane strain orplane stress problems using Hutchinson-Rice+Rosengren (HRR) solutions [l, 21 so long as the sizesof specimens are in agreement with certain requirements [3,4].Generally speaking, the stress and strain fields of a Mode I crack are constituted of three parts,the one similar to the plane strain solution, the one similar to the plane stress solution and theone with thr~-dimensional behavior. The work done by Narasi~a~ and Rosakis [S] indicates thatthe plane strain field prevails in the interior of a lo-mm-thick three-point bend specimen very nearthe crack tip, and plane stress conditions are approached for distances from the crack tip exceedingabout half of the specimen thickness. A further study by Narasimhan et al. 161 which considersa damage accumulation model accounting for void nucleation, growth and coalescence, givessupport to the conclusion mentioned above. In [7], it is proved that for a Mode I thin plate, theplane strain condition, a,/(~, + o,,~) = 0.5, is restricted to r/t G 0.01 ahead of the crack tip; anearly plane stress condition exists everywhere except in the region near the crack tip. similarresult is obtained by Horn and McMeeking [8] using large deformation three-dimensional finiteelement analysis for a blunting Mode I crack in a thin elastic-plastic sheet. A three-dimensionalfiuite element analysis done by Kikuchi and Yano [9] is aimed at both compact tension {CT) andcenter cracked tension (ET) specimens. It points out that for CT specimens, the stress and~spla~ent fields agree very well with HRR solutions while the thicknesses are larger than thoserecommended by the standard of fracture toughness testing; the HRR field exists only in the regionnear the mid-section of the specimens when the thicknesses are little smaller than the recommendedvalue. There is no HRR field in the CCT specimen in their investigation. However, a study by Malikand Fu [lo] of a CCT specimen by means of MOL (method of line) shows that an approximateplane strain condition occurs immediately ahead of the crack tip on the mid-plane.

It has been recognized that the variation of stress structure and the loss of HRR singularityof a Mode I crack are due to different stress constraints, and they may be described by a stresstriaxiality parameter. A plane strain state exists where stress triaxiality is large. On the contrary,a plane stress state corresponds to low triaxiality. Stress triaxialities for three-dimensional fieldsare between the two limits. ODowd and Shih [l 11suggest characterizing a family of crack-tip fieldsby a triaxiality parameter, called the Q-family. A Q-value is used to order the ~nst~int of crackgeometry. Then, a two-p~ameter fracture criterion is in~odu~d~ It is shown by Sun et al. f12] thatthe critical value of J-integral to predict crack initiation is concerned with stress triaxiality.EPM4/6 E 895


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896 X. M. CHEN and H. G. HAHN

There are two purposes in the present work. One is to describe load and geometry effects ona three-dimensional stress structure; the other is to analyze the functions of plastic deformationand stress triaxiality constraint during the failure process.

The three-dimensional stress and strain fields of CT specimens with different thicknesses areobtained using finite element method and the stress structure is presented graphically. It is verifiedthat three regions, with plane strain similar behavior, plane stress similar behavior and a transitionlayer between them, exist in front of the crack tip. Two essential factors, plastic strain and stresstriaxiality, are investigated based on the three-dimensional stress and strain fields. It is found thatthree-dimensional deformation has its own characteristics, differing from corresponding two-dimensional ones even in the plane strain and plane stress similar regions. Failure prediction heredepends on a void growth ratio criterion, which takes account of plastic deformation and stresstriaxiality simultaneously. It is revealed that the failure form and position will be determined byboth of these two important factors.

STRESS STRUCTURE IN FRONT OF THE CRACK TIPIt has been known by inferences from numerous studies that plane stress (for relatively thinspecimens) or plane strain (for relatively thick specimens) prevails simply in the area far away from

the crack tip. However, the near tip stress field is complicated.The solution of a Mode I three-dimensional crack (Fig. 1) in an elastic-plastic state is summedup as a boundary value problem of a differential equation of fourth order. There is obviously nodirect way to solve it. A previous investigation of the authors [13] was made combining analyticalmethod with finite element simulation and the stress structure of a Mode I three-dimensional crackwas obtained qualitatively, It is pointed out in this work that three regions can be divided in thethickness direction, namely, a plane strain similar zone ZI, a transition layer ZII and a plane stresssimilar zone ZIII.Such a stress structure is described graphically in Fig. 2, which implies that the fringe linesconnecting the three regions are curves instead of straights. The shapes and sizes of the regionsare concerned with applied load, specimen geometry, material characteristics and other serviceconditions as well.The three regions, ZI, ZII and ZIII, usually exist simultaneously. ZI might be, in some cases,very small compared with the other two. Further analysis will be done in the following.

NUMERICAL PROCEDURECT specimens with three different thicknesses, B = 4 mm, B = 8 mm and B = 24 mm aremodeled for numerical simulation. The three-dimensional stress and strain fields are calculated by

means of finite element method using the ADINA [14] program.The material chosen is 2024-T351 aluminum. It behaves with a power-law hardeningcharacteristic with n = 11, a = 0.8. The relationship between stress and strain is described by amulti-linear approximation during the computational procedure.

Fig. 2. Region division of Mode I three-dimensional crackFig. 1. Mode I threedimensional crack and the coordinates. (thin specimen).


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Stress structure of Mode I 3-D crack 897

Fig. 3. Finite element mesh of CT specimen.

Only one quarter of a specimen is taken into consideration due to symmetry. There are fourelement layers through the half-thickness for the specimen with B = 4 mm; six layers for that withB = 8 mm and eight layers for that with B = 24 mm. Element layers become thinner as the freesurface is approached.The meshes are shown in Figs 3 and 4.Twenty-noded isoparametric elements with 2 x 2 x 2 Gauss integration for the stiffness areadopted. Degeneration elements are employed for those directly in front of the crack tip. TheBroyden-Fletcher-GoldfarbShanno (BFGS) method is used to solve the equilibrium equationsduring the performance, which brings credible precision and saves computer running time. Stressand strain distributions under two-dimensional plane stress and plane strain conditions are alsoanalyxed in order to compare them with the corresponding three-dimensional results.

Fig. 4. Mesh in front of crack tip.


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898 X. M. CHEN and H. G. HAHNSTRESS STRUCTURE CHARACTERISTICS

It has been mentioned above that three regions can be defined in front of the crack tip, whichare named the plane strain similar zone ZI; the transition layer ZII; and the plane stress similarzone ZIII. Detailed results obtained from finite element simulation are shown in Figs 5-7, whichdescribe the effect of applied load and specimen geometry on stress structure.

The stresses and strains obtained by FEM are output by Gauss integration points. The pointswith plane strain similar behavior satisfy0.45 < a,/(o,* f cyv) < 0.5, c,,, 8, < 1;

and those with plane stress similar behavior satisfy:I%r/(cxx + q,,)l < 0.05, cxr, CYZ 1.

Further analyses show that the plane strain similar region ZI exists directly in front of the cracktip. It forms in the vicinity of 8 = x/2, mid-section z/@/2) = 0, and expands as applied load andspecimen thickness increase. In a plane where z/@/2) = const, the shape of ZI appears to be a fan.It always has its maximum area in the plane z/@/2) = 0.

For a given thickness B = const, region ZI grows with loading. The maximum radialsize of ZI, for the CT specimen with B = 8mm, is 0.1 when P/P, 0.6nd 0.25 when P/P, 1.0P,s the maximum P during loading). Similar conclusions are obtained by investigating otherspecimens.Under a given load P/Pr const, the thicker the specimen is, the larger its plane strain similar

region will be. As P/P, 1 O, the maximum radial size is 0.1 for the specimen with B = 4mm, 0.25for B = 8mm and 0.4 for B = 24mm.

. with pstress loobehavior

$.77*

Z/(9/2)= 0.9sP/Pr=O.6

-100 -50 0 50 looX/t Jo/q,)Fig. 5(a)


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Stress structure of Mode I 3-D crack 899

l- . with p.stress 100behavior Z/W2)4.02

f. P/Pr=O.8D with pstrain -)behavior Y*

. . * * * ., --* . . . := . . , . f.. . . *. , . .. :: . . *. ., . .

, . * ..*. * * 8 .. . s +. :-

::: . . . a. : =. . . . . . . .

-100 -50 0 50 lo(X/t Jo/so)

with p.stress 100.behavior

2.Z/(8/2)=0.8P/P,= 0.8. with p.strain -)

behavior Y*

X/t Jo/%)=with p.stress 100 I

behaviorf.

ZAB/2)= O.QP/Pr = 0.8_)Y2-

-100 -50 0 50 lO(X/l Jo/so)

. with p.stress 100behavior

?.Z/(8/2)4.02

with pstrain ,O P/P,= 1 o.behavior 2:>

. with p.stress lO(1 Z/(8/2)=0.6behavior P/P,= 1.0. ?with p.strain ybehavior T>1.,, ..:-. . .

-100 -50 0 50 100X/( JO/~)

-100 -50 x/(JoD/60) 50 100

-100 -50 0 50 lO(X/( Jo/%)Fig. 5(b)

Fig. S(a) and (b). Stress structure of B = 4 mm CT specimen.


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X. M. CHEN andH. G. HAHN

behavior

with pstress 100behavior?-with p.strain

behavior ,oYS.

z/W2)=0.6PiP,=O.6

with pstress 100behavior

?.,o7r

-100 -50 0 50 1CXX/CJo/%)Fig. 6 a)


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I with pstrrss loobehavior 2

4#V?J=O.EI. with p.strain

-100 -SO 0 50X f J o a~l

X/t J&o) -1UO -so 0 50X/t J,/%,l


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I behavior :

-100 -50 0 50 100X/t Jo/coo)

-100 -50 X/LLobI 50 100

-with p.stress loorbehavior 3

9Yz

f/03/2)= 0.9sP/4.=0.6

Fig. 7(a)


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- 100 -50 0xm dc@

. wEth p.strain b&WV&t- 3

100 -50 a sa lotX/f Jo&


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904 X. M. CHEN and H. G. HAHNThe plane stress similar region ZIII prevails in a much larger area than does region ZI. It

reduces as applied load and specimen thickness increase and vanishes, first of all, near 9 = nmid-section. However, the stress structure is plane stress similar while the free surface z/(B/2) = 1 Ois approached no matter how load and thickness change. It can be predicted that there is a certainthickness, B,, related to material character, within which only one plane stress similar state exists.It is suggested here that B, should be the thickness of the shear lip.

PLASTIC ZONE AND PLASTIC DEFORMATIONThe plastic zone and the plastic deformation of a three-dimensional CT specimen, even in its

ZI and ZIII regions, are not the same as those under corresponding two-dimensional conditions.Details are shown in Figs 8 and 9, which show the plastic zones of CT specimens with differentthicknesses.For a given thickness, the plastic zone expands with the increase of applied load. There areonly a few distinctions, under low load, among the zones in planes where z/(8/2) = const. Thisis because the stress structure, from mid-section to free surface, has not yet changed much. Thedistinctions become notable as the formation of regions ZI, ZII and ZIII at a higher load level.

Specimen thickness is an essential factor, which affects the shape and size of the plastic zone.Under a given load, the plastic zone of a thin specimen is larger than that of a thick one. On theother hand, the plastic zone of a thin specimen does not change so much, from mid-section to freesurface, as that of a thick one does because of the relative uniformity of its stress structure.Comparisons between three- and two-dimensional results are shown in Figs 10 and 11. It canbe seen here that the plastic zone of the plane strain condition is smaller than that of the

I- 100 0.025?. Z/(8/2)=0.6 -----, 0.99 -----TN P/Pr=0.6

Go0 -_ __-50 0 50 lotX/t Jo/o,,)

100 0.025 -?. Z/(8/2)=0.6 -----.,o 0.99 ---.-._Y> P&=0.6

0.025 -it. Z/(9/2)=0.6 ._---,o 0.99 -----Y* P/P,=l.O

-100 -50 0 50 1ocX/t Jo/ )Fig. 8. Plastic zone of CT specimen with B = 4 mm.


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Stress t&ructurcof Mode i 3-D crack 905

100 0.025 -?- Z/ 8/2)=0.6 0.99 -----..___,_3: P/Pr=O .6

0.025 -Z/ 8/21=0.6 -.__*

0.99 ___-_

PIP,= 0.8

-100 -50 0 50 lO(X/f Jo/ )

-100 -50 0 50 101X/t JoledFig. 9. Plastic zone of Cl specimen with B = 24 mm.

three-dimensional mid-section. This behavior is more notable for larger applied loads and thinnerspecimens. The plastic zone of the plane stress condition is usually larger than that of three-dimen-sional free surface, especially for specimens with greater thicknesses.

Figures 12 and 13 show the distributions of effective plastic strain along 9 = const, which helpto analyze the deformation characteristics of CT specimens.Plastic deformation, in the plane stress condition, starts and has its maximum at 9 = 0, whichleads to the formation and growth of a plastic zone nearby. In the plane strain condition, plasticdeformation is always significant in the vicinity of 8 = a/2, so that its plastic zone grows here, Theplastic mm ofa~~ension~ CT specimen forms near 9 = z f2, as does that of the plane strainstate. It grows more cluickly in the area of 0 6 9 < nf2 than in s/2 6 9 < 71,similar to the planestress state, especially in planes near the free surface.


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904 X. hf. CHEN and H. G. HAHN100 3-D mid-section -2 plane strain -.---2 P/Pr=O.6Yt

-100 -5 ) 0 50 101X/l JO/SO)

3-D free surface -______

3

-100 -50 0 50 100X/i J&s&100 3-D mid-section -2 plane strain ____--s P/R= 1.0>

. . .___ _y_

-100 -50 0 50 lO(XA Jo/oo)Fig. 10. Comparison with two-dimensional results (B =

loo 3-D mid-section -2 plane strain ---___3 P&+=0.6:

I

3-D frw surface -plane stress -_____P/Pr=O.6

-100 -50 0 50 100X/t Jo/%)

loo 3-D mid-section -$ plane strain ______7 P/Pr=l.OK>

loo 3-D frsa surface -I -$-lane stress ---___7 P/P,=145

-100 -50 50 100Fig. 11. Comparison with two-dimensional results (B =_.4 mm). 24 mm).


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Stress structureof Mode I 3-D crack

mid-section ~_ __ plane strain ------- 1.0

a

P 0.02 .

0.00 a/4 r/2 3~12 1

Fig. 12. Distribution of effective plastic strain for B = 4 mm.

It should be noticed, in the three-dimensional condition, that the magnitude of effective plasticstrain ep at the free surface is not always greater than that at mid-section. An example is given forthe CT specimen with B = 24 mm. The value of IZ,, t the free surface is greater than the L,, atmid-section only if 9 < 9, (8, < n/2). This means that in the vicinity where plastic deformationreaches its peak value, c+, s more notable at mid-section because of the higher level of stresses.However, the plastic zone grows more slowly at mid-section for a stronger constraint. Such aphenomenon can be seen more obviously for thicker specimens, which are generally underthree-dimensional conditions. It is not very typical for thinner specimens.

DISTRIBUTION OF STRESS TRIAXIALITY PARAMETERThe failure feature of a cracked body is substantially related to its stress constraint status,

which can be described by the stress triaxiality parameter R,.

where a,,, is the average stress and a, is the equivalent stress.It has been known that plane strain deformation is accompanied by higher stress triaxialitythan is plane stress deformation. Moreover, the R, f the three-dimensional stress field varies withinthe range provided by two-dimensional states.Figures 14 and 15 show the curves of R,, versus 9.

In a plane where z = const, the stress triaxiality constraint is larger as the ligament (8 = 0)is adjoined for both three- and two-dimensional conditions. In the thickness direction, R, ecreasesfrom mid-section to the free surface.


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908 X. M. CHEN and H. G. HAHN0.04

0.03mid-section -plans strain .-----. 1*0i-- P/Pr=O.8

f 0.02

0.01

0.00 n/4 n/2 3a/230.04

--fr.e.ywrface +---o,03 1.0

rp/q=o.a

0.6

1

3Fig. 13. Distribution of effective plastic strah for /3 = 24 mm.

mid-section -plane strain _-----. 1.0r- P&=0.8

4x

3.0a?

2.0

1.0

0.0

free surface -plane strsss -------

x/4 n/2 3x/2aFig. 14. Distribution of stress triaxidty parameter for B = 4mm.

(3

(b)


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0 0 n/4 lr/2 3~12 II3

T. free surface -plane stress -------

3 0

0 0 lt f4 r/2 3n/2a

Fig. 15. ~st~b~tion of stress triaxidity parameter for B = 24mm.

The stress triaxiality parameter is not very sensitive to applied load, especially when plasticdeformation is notable. This feature can be seen in Fig. 14b, from the three-dimensional curvesin Fig. 14a, as well as from the three-dimensional curves near 9 = n/2 in Fig. 15a.

The three-dimensional R,- curves at mid-section coincide well with those in the plane strainstate only if the specimen is thick enough (here with B = 24 mm), However, the agreement amongthe curves at the free surface and in the plane stress state is not restricted by a similar rule.

FAILURE PRED~~~NThe void growth ratio parameter Vg s employed in the present work to predict the failure of

Mode I t~imensional cracked bodies, The ~r~s~nding criterion is written as:vg= v,,.

VW s the critical void growth ratio, which has been tested as being a material constant independentof stress triaxiality [ 1Sj, and V, is expressed as:V, = tpexp(3&/2). (1)

Both plastic deformation and stress triaxiality effects are taken into consideration in this expression.It seams, from eq. (1), that R,, plays a more important role than c,, does because V, is theex~~cn~~ function of it. However, this is not absolutely true because of the in~siti~ty of stresstriaxiality to applied load. An inference can be drawn, from the features of % and c,,, that for agiven thickness B, the increment of void growth ratio is caused mainly by plastic defo~ation butthe position where maximum IJd takes place will be determined by both of these factors.


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plane strain ----_..0.39

0.2

free surface -plane stress ----..r

n/4 lc/2 3~12 n

Fig. 16. Distribution of void growth ratio for B = 4 mm.

mid-sectian -0.3 piane strain ._---_. 1.0

90.2

Fig. 17. Distribution of void growth ratio for B = 24 mm.


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Stress structure of Mode I 3-D crack 911In the thickness direction, the R, f the actual three-dimensional state is a monotone

decreasing function in the range 0 G z/@/2) < 1. Furthermore, V,, itself, also decreases monoto-neously from mid-section to free surface. In a plane where z = const, as well as in two-dimensionalconditions, the stress triaxiality parameter shown in Figs 14 and 15 increases as the ligament isapproached; and the R,,--?Jcurves are relatively stable in the range 0 9 < n/4.The effective plastic strain tP in the plane stress state has its maximum near 9 = 0 as does &.There is no doubt that the highest V, will take place here. The ~~-8 curves in th~dimens~on~and plane strain conditions have their peak values near 9 = nJ2. The highest Vg, in such eases, takeplace in the range of 7t/4 d 9 6 n/2. Details are shown in Figs 16 and 17.By comparing the three-dimensional V8-9 curves with the two-dimensional ones, the followingconclusions can be obtained. (1) For a given CT specimen with B = const, the highest value of voidgrowth ratio will appear at a mid-section of the three-dimensional state. (2) Under a given nominalapplied load, the V, of a thinner specimen is greater than the V, of a thicker one. Furthermore,failure will start in the range of n/4 < 9 f x/2, the mid-section of a three-dimensional CT specimen;the thinner the specimen is, the more serious the failure will be.

CONCLUSIONSThe stress and strain fields of a Mode I cracked body are actually thr~-dimensional but differfrom general thr~-dimensional ones because of the particular load and geometry conditions. Three

regions, namely the plane strain similar zone 21, the transition layer ZII, and the plane stress similarzone ZIII, can be defined according to the stress characteristrcs,A plane strain similar zone ZI exists directly in front of the crack tip. It starts in the vicinity

of 9 = n/2, mid-section, and expands with increments of applied load and specimen thickness, Theplane stress similar zone ZIII prevails in a much larger area than does ZI. It reduces with incrementsof load and thickness. However, there is a layer with thickness B, ear the free surface for CTspecimens. The stress field within it is always plane stress similar, no matter how the load andthickness change. This layer is suggested as being the shear lip of the specimen.The plastic zone of a CT specimen varies in the thickness direction and the variation issi~i~cant for thicker specimens. It is larger, at mid-s~tion~ than that in the plane strain state, butsmaller, at the free surface, than that in the plane stress state. In a plane where z = const, the shapeof the plastic zone looks like that under the plane strain condition because of the similarity of theirfp-8 curves.

The stress triaxiality constraint decreases, in the thickness direction, from mid-section to freesurface. It has higher magnitude near the specimen ligament and is relatively stable in the range0 6 9 G n/4. The stress triaxiality parameter R, s not very sensitive to applied load but is sensitiveto specimen thickness.

The void growth ratio criterion may be used to predict the failure of Mode I three-dimensionalcracked bodies. For a CT specimen with thickness B, the increment of void growth ratio Vg iscaused mainly by plastic deformation, but the position where highest V, takes place will bedetermined by both plastic defo~ation and stress triaxiality constraints. Actually, the three-dimen-sional situation is more dangerous than corresponding two-dimensional ones, especially for thinnerspecimens. It is pointed out that failure will start in the range of 7~/4 G 9 6 x/2, mod-s~tion, whereV, reaches its highest value.Acknowledgement-This work was supported by the Alexander von Humboldt Foundation.

REFERENCESJ. W. Rutchinson,J. &C/I. Phys. Solids 16, 13-31 (1968).3. R. Rim and G. F. Rosengren, J. Mech. P&L So& 16, 1-12 (1968).C. F. Shih and M. D. German. Int. J. Fracture Me 17. 2743 (1981).Sun Jlan, Deng Zengjie and TuMmgjing,Erzgttg FracttueMech. 7, 6%680 (B90).R. N~~rnh~ and A. J. Rosakis, J, appt. Mech. 57, 607-617 (1990)R. Nation, A. 1. Rosakis and B. Moran, SM 89-5, ~tifo~a Institute of Technology, CA (1989).T. Nakamura and D. M. Parks, J. ;cdech. Phys. So/& 38,787-8 13 (1990).C. L. Horn and R. M. McMeeking, in& J. Fracture 45, 103-122 (1990).

EFM 44/b-F


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912 X. M. CHEN and H. G. HAHN[9] M. Kikuchi and K. Yano, In Coll ected Work s of Japanese Society of M echanical Engineeri ng, Volume 55, No. 516(1989-g), Paper No. 88-1428 (1989).[lo] S. N. Malik and L. S. Fu, ht. J. Fract ur e 18, 45563 1982).[I I] N. P. GDowd and C. F. Shih, Family of crack-tip fields characterized by a triaxiality parameter: part I--structureof fields. J. M ech. Phy s. Soli ak 39, 989-1015 (1991).[12] Sun Jun, Deng Zengjie, Li Zhonghua and Tu Mingjing, Engng Fractur e M ech. 36, 321-326 (1990).[13] X. M. Chen, N. S. Yang and Z. X. Guan, Act a M ech. Soli da Si ni ca 3, 263-287 1990).

[14] ADINA Engineering, Inc., Automatic dynamic incremental nonlinear analysis, Report AE 83-5 (1983).[15] C. Q. Zheng, L. Zhou and J. M. Lui, The criterion of critical void growth rate and its application. Proceedings ofZCM S, Beijing, pp. 213-218 (1987).

Recei ved 8 July 1992)

chen hahn efm1993 stress and deformn beh mode i 3d elpl crack tip fields

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