1778 B. K Ahn et al Matrix Fiber Crack extending to z Debond Penetrati Matrix T lri=o Fiber tafO,r0 Crack Or}=t:,r)=0 Fig. 2. Debond and Penetration at fiber/matrix interface in energy for the deflected crack, Wa. The energy release rates are then calculated from the energy differences and the area of crack growth as G=("=H)1n4)Gn=(H4-w)1(r Fig 3. Schematic of ADM concentric cylinders. where ry is the fiber radius. The dependence of Ga/Gp cracks then simply define a particular boundary condi- versus a and B is then used as the deflection criterion tion along the surface of any region. The elasticity pro- th The crack extension is taken to be 0002 rf, blem within each region subject to appropriate 0-01 rr and 0-025 rf. To investigate the effect of different boundary conditions is solved using the Re eisner varia crack extensions in two types of cracks, we also fix ap at tional equation, as discussed briefly below 0-01 rand adopt various ad of 0-025 r, 0-01 rand 0-002 Reissner has shown that minimizing the functional rr. We use a 1% fiber volume fraction to approach the semi-infinite matrix crack limit. and a 40% volume fraction to model a realistic composite J=Fd-|rds:F=与+与)-W(5) 2.2 Numerical technique: axisymmetric damage model The axisymmetric damage model(ADM) was developed by Pagano to predict the stress and displacement dis- with respect to both stresses and displacements leads to tributions in composite constituents as well as the strain both the field equations and boundary conditions of lin- energy and energy release rates for cracked composites ear elasticity theory. In eqn(5), w is the complementary in which damage modes include fiber breaks, annular energy density, T; is the prescribed traction, and o; and 5 cracks in coatings or matrix, and debond cracks at are the stress and displacement components, respec- interfaces. Concentric cylindrical/annular elements are tively, in Cartesian coordinates. A comma after a sub used to model the fiber and matrix materials. Additional script represents a derivative with respect to the concentric annuli (radii r1, r2,.)can be introduced, as indicated coordinate, and Einsteins summation conven well as longitudinal sections(=1, 22,...), if necessary for tion is understood. V is an arbitrary volume enclosed by computational purpose, as indicated schematically in the entire surface S, while S is the portion of the Fig 3. This discretization forms regions bounded by ri boundary on which one or more traction components and ri+I and =; and E+I. The annuli, or layers, and are prescribed. The body forces have been neglected in sections are chosen so that cracks introduced into thethe formulation In the adm the stress field in each metry always lie along the boundaries of sections annular region is assumed to be one where oe and o: are (for transverse cracks)or layers(for longitudinal cracks) linear in the radial coordinate r, while the forms of and span one or more complete sections/ layers. The or and Tr are chosen to satisfy the axisymmetricenergy for the de¯ected crack, Wd. The energy release rates are then calculated from the energy di€erences and the area of crack growth as Gp ˆ…WpÿWr† =…r2 f ÿ…rfÿap† 2 † ; Gd ˆ…WdÿWr† =…2rfad† …4† where rf is the ®ber radius. The dependence of Gd/Gp versus  and  is then used as the de¯ection criterion with ad ˆ ap. The crack extension is taken to be 0.002 rf, 0.01 rf and 0.025 rf. To investigate the e€ect of di€erent crack extensions in two types of cracks, we also ®x ap at 0.01 rf and adopt various ad of 0.025 rf, 0.01 rf and 0.002 rf. We use a 1% ®ber volume fraction to approach the semi-in®nite matrix crack limit, and a 40% volume fraction to model a realistic composite. 2.2 Numerical technique: axisymmetric damage model The axisymmetric damage model (ADM) was developed by Pagano to predict the stress and displacement dis￾tributions in composite constituents as well as the strain energy and energy release rates for cracked composites in which damage modes include ®ber breaks, annular cracks in coatings or matrix, and debond cracks at interfaces. Concentric cylindrical/annular elements are used to model the ®ber and matrix materials. Additional concentric annuli (radii r1, r2, ...) can be introduced, as well as longitudinal sections (z1, z2, ...), if necessary for computational purpose, as indicated schematically in Fig. 3. This discretization forms regions bounded by ri and ri+1 and zj and zj+1. The annuli, or layers, and sections are chosen so that cracks introduced into the geometry always lie along the boundaries of sections (for transverse cracks) or layers (for longitudinal cracks) and span one or more complete sections/layers. The cracks then simply de®ne a particular boundary condi￾tion along the surface of any region. The elasticity pro￾blem within each region subject to appropriate boundary conditions is solved using the Reissner varia￾tional equation, as discussed brie¯y below. Reissner7 has shown that minimizing the functional J ˆ … V FdV ÿ … S0 TiidS; F ˆ 1 2 ij…i;j ‡ j;i† ÿ W …5† with respect to both stresses and displacements leads to both the ®eld equations and boundary conditions of lin￾ear elasticity theory. In eqn (5), W is the complementary energy density, Ti is the prescribed traction, and i and i are the stress and displacement components, respec￾tively, in Cartesian coordinates. A comma after a sub￾script represents a derivative with respect to the indicated coordinate, and Einstein's summation conven￾tion is understood. V is an arbitrary volume enclosed by the entire surface S, while S0 is the portion of the boundary on which one or more traction components are prescribed. The body forces have been neglected in the formulation. In the ADM, the stress ®eld in each annular region is assumed to be one where  and z are linear in the radial coordinate r, while the forms of r and rz are chosen to satisfy the axisymmetric Fig. 2. Debond and Penetration at ®ber/matrix interface in axisymmetric geometry. Fig. 3. Schematic of ADM concentric cylinders. 1778 B. K. Ahn et al