A penny-shaped cohesive crack model for material damage

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