A penny-shaped cohesive crack model for material damage

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306 G. Wang, S.F. Li / Theoretical and Applied Fracture Mechanics 42 (2004) 303–316where I (2) = d ij e i e j is the second-order identitytensor. By the averaging theorem, it is obvious thatmean macro stress, defined as R m ¼ 1 R 3 kk, equalsapplied remote stress,R m ¼ r 1ð8ÞIn cylindrical coordinate, the traction conditionson the remote boundary oV 1 and symmetric displacementboundary condition are expressed asr zz j oV 1¼ r hh j oV 1¼ r rr j oV 1¼ R m ð9ÞFig. 2. A penny-shaped cohesive crack in representativevolume element (the shaded region: cohesive zone—yieldedring).Chen and Keer re-examined the problem, and theyobtained general solutions for a penny-shapedcohesive crack under mixed-mode loading [5,6].On the other hand, however, the problem hasnot been thoroughly examined from micromechanicsperspective. For example, the connectionamong the onset value of cohesive strength, microyield stress in an RVE, and remote macro stresshas not been made. By examining a cohesivepenny-shaped crack in an RVE, the study providesa link among cohesive strength, yield stress of virginmaterial, and remote stresses on the boundaryof an RVE, which provides foundation for ensuinghomogenizations.Consider a three-dimensional penny-shapedDugdale crack of radius a with a ring-shaped cohesivezone with width b a in an RVE, which maybe viewed as an infinite isotropic space by ‘‘amicro-observer ’’ inside the RVE.Let the outward normal to crack surface parallelto Z(X 3 ) axis (see Fig. 2) and uniform triaxialtension stress is applied at the remote boundaryof the RVE, i.e.r 1 ¼ r 1 I ð2Þð7Þu z ðr; h; 0Þ ¼0 for b 6 r; 0 6 h 6 2p ð10ÞThe stress distribution on the crack surface andcohesive zone isr zz ðr; h; 0Þ ¼r 0 Hðr aÞfor 0 6 r 6 b; 0 6 h 6 2pð11Þwhere R m is the remote stress, H(r a) is theHeaviside function, and r 0 is the materialÕs cohesivestrength, the onset value for crack opening,and it is different from the micro yielding stress.The problem can be solved via superposition oftwo sub-problems: a trivial problem—an intactRVE in uniform triaxial tension state, i.e. "x2V,r ð0Þzzr ð0Þrz¼ r ð0Þhh¼ r ð0Þrh¼ rð0Þ rr¼ R m ð12Þ¼ rð0Þ zh ¼ 0 ð13Þand a crack problem—an RVE with a center crackthat is subjected to the following boundary conditions(see Fig. 3):r ðcÞzz j oV 1¼ r ðcÞhh j oV 1¼ r ðcÞrr j oV 1¼ 0 ð14Þr ðcÞzz ðr; h; 0Þ ¼ R m þ r 0 Hðr aÞfor 0 6 r 6 b; 0 6 h 6 2pð15Þu ðcÞzðr; h; 0Þ ¼0 for b 6 r; 0 6 h 6 2p ð16ÞFor the crack problem, the crack opening displacementcould be solved as8>:2p2p1 v l 1 v l R mR mpffiffiffiffiffiffiffiffiffiffiffiffiffiffib 2 r 2pffiffiffiffiffiffiffiffiffiffiffiffiffiffib 2 r 2R bpffiffiffiffiffiffiffiffiffiffiffiffiffipr 0 ta2 a 2 = ffiffiffiffiffiffiffiffiffiffiffiffiffi t 2 r 2 dt ; 0 < r < aR bpffiffiffiffiffiffiffiffiffiffiffiffiffipr 0 tr2 a 2 = ffiffiffiffiffiffiffiffiffiffiffiffiffi t 2 r 2 dt ; a < r < bð17Þ
G. Wang, S.F. Li / Theoretical and Applied Fracture Mechanics 42 (2004) 303–316 307In the yield ring (z = 0 and a < r < b), stress distributionsare found asr ðcÞzz¼ r 0 R m ð18ÞFig. 3. Illustration of superposition of cohesive crack problem.rr¼ 1 þ 2v R m 2þ 1 2v2r ðcÞ1 þ a2r 2þ 2v r 0ð19Þr ðcÞhh ¼ 1 þ 2v 2þ 1 2v2r ðcÞrz¼ r ðcÞrhR m1a 2r 2þ 2v r 0ð20Þ¼ rðcÞ zh ¼ 0 ð21Þwhere PoissonÕs ratio v*, to be treated as eitheroverall or matrix material property, is unspecifiedat this moment, which may depend on the laterhomogenization procedures.To ensure the stresses at crack tip to be finite,the size of the cohesive zone a/b, macro stressR m , and the cohesive stress r 0 are related throughthe following expression:ab ¼sffiffiffiffiffiffiffiffiffiffiffiffiffiffiR 2 m1r 2 0orR mr 0¼sffiffiffiffiffiffiffiffiffiffiffiffiffi1a 2b 2ð22ÞDenote the projection of traction-free crack surfaceonto X 1 X 2 plane as X 1 , and the projectionof the cohesive zone (ring shape) as X 2 . The totalvolume of crack opening by a single cohesive crackis the integration of crack opening displacementFig. 4. Projection domain of crack surface and cohesive zone.over the entire projection area, X = X 1 [ X 2 (seeFig. 4). It is readily to show thatZ Z ZV c ¼ ½u z ŠdS ¼ ½u z ŠdS þ ½u z ŠdSXX 1 X 2¼ 8ð1 v Þhib 3 R3l m r 0 ð1 ða=bÞ 2 Þ 3=2¼ 8ð1 v Þa 3 R mqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3l 1 ðR m =r 0 Þ 2ð23Þwhere the crack opening displacement, COD, isdefined by½u z Š¼u þ zu z¼ 2u z ð24ÞIf one is mainly interested in inelastic deformationof quasi-brittle materials, it may be assumed that
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A penny-shaped cohesive crack model for material damage

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