Crack deflection/penetration criteria for fiber-reinforced ceramic matrix composites 1779 equilibrium equations. Then, all of the stress compo- the shear stresses in adjoining constituents are con- nents depend on r through known shape functions tinuous. We assume the tractions on the debond crack Using the assumed stress field in the reissner functional surface vanish as shown in Fig. 2, while in the elastic and appropriate boundary/continuity conditions across zone(perfectly bonded zone) both of the radial and the region boundaries leads to a set of algebraic equations axial displacements are continuous. We note that if and linear ordinary differential equations in z only for debond crack takes place at an interface, one may the unknown coefficients of the stress fields. For a body choose between maintaining continuity of the 8N+16 equations to solve.g and N annuli, there are stress or continuity ofweighted displacement'at the composed of a core cylinder crack tip(see Ref. 8 for details). Selecting shear stress At the coarsest level, regions can be chosen to fully continuity has been shown to lead to energy con- span existing cracks and interfaces. However, in order sistency, and this condition is used throughout this to improve the accuracy of the solution for the stress study. The boundary conditions used here on the sur- fields and energy release rates, it is possible to introduce faces of the axisymmetric cell are shown in Fig. 2. At the additional annular layers in the neighborhood of inter- outer boundary of the composite cylinder(r= Im), we faces, crack tips, and the other forms of stress con- apply the traction-free boundary conditions for simpli- centrations. Although similar to mesh refinement in city, but we also consider the radial displacement finite element analysis, the refinement here is primarily boundary conditions on a limited basis, for illustrative only in the radial coordinate, refinement in the axial purposes. To obtain the proper displacement boundary coordinate is only necessary for computational tract- conditions, we proceed as follows ability in the solution and not for accuracy. The accu- racy of the ADM calculation depends on the number 1. assume a crack-free uniaxially tensile loaded com- and the location of the boundaries of these regions posite with perfectly bonded interface and a stress- Since we will deal with small finite crack extensions. we free outer surface boundary; w in Fig. 4 the general mesh structure used for 2. obtain the constant radial displacement perpend three calculations of matrix crack only, penetrated cular to the cylinder outer surface in the composite crack, and deflected crack, which we performed to using the adm etermine 3. use the radial displacement as a boundary condi represents a model composite with a 40% fiber volume tion in a matrix-cracked composite with an imper fraction and a length of 10 rf. A schematic of the fect interface. As expected, the two different additional layers and sections used for the boundary conditions do not make any difference ad=ap=0-002ra case is illustrated. In addition to the in GaGn over all range of a in the case of 1% fiber physical material boundary at r/ry=l we employ 10 volume fraction additional annular layers at r/r=0.80.9, 0.93, 0.96,0.99 0.998.1-03. 1-07. 1. 2. An additional axial section at For 40% fiber volume fraction. we notice differences is used and a section at z= ad is also required for as large as 7% in the values of energy release rates from the deflection problem he two different boundary conditions, but the bound Regarding boundary conditions, the Reissner varia- ary condition effect diminishes considerably in the final tional principle can handle mixed boundary problems so calculations for Ga Gp. In Fig. 6, the results using the that both displacements and stresses can be specified. displacement boundary condition are shown for com- General boundary conditions which should be satisfied parison, and the difference by the two boundary condi at all interfaces are as follows. The radial stresses and tions appears negligible. Regarding the loading rm=1.58/ Matrix : Addition Fiber r=0 10-080.6-0.4.2000204060.81.0 al sections Fig. 5. GalGp versus a with various crack extensions for Fig 4. General mesh structure used in the adm modelequilibrium equations. Then, all of the stress compo￾nents depend on r through known shape functions. Using the assumed stress ®eld in the Reissner functional and appropriate boundary/continuity conditions across region boundaries leads to a set of algebraic equations and linear ordinary di€erential equations in z only for the unknown coecients of the stress ®elds. For a body composed of a core cylinder and N annuli, there are 18N+16 equations to solve.8,9 At the coarsest level, regions can be chosen to fully span existing cracks and interfaces. However, in order to improve the accuracy of the solution for the stress ®elds and energy release rates, it is possible to introduce additional annular layers in the neighborhood of inter￾faces, crack tips, and the other forms of stress con￾centrations. Although similar to mesh re®nement in ®nite element analysis, the re®nement here is primarily only in the radial coordinate; re®nement in the axial coordinate is only necessary for computational tract￾ability in the solution and not for accuracy. The accu￾racy of the ADM calculation depends on the number and the location of the boundaries of these regions. Since we will deal with small ®nite crack extensions, we show in Fig. 4 the general mesh structure used for the three calculations of matrix crack only, penetrated crack, and de¯ected crack, which we performed to determine the various energy release rates. Figure 4 represents a model composite with a 40% ®ber volume fraction and a length of 10 rf. A schematic of the additional layers and sections used for the ad ˆ ap ˆ 0
002rf† case is illustrated. In addition to the physical material boundary at r=rf=1 we employ 10 additional annular layers at r=rf=0.8,0.9,0.93,0.96,0.99, 0.998,1.03,1.07,1.1and1.2. An additional axial section at z ˆ rf is used and a section at z ˆ ad is also required for the de¯ection problem. Regarding boundary conditions, the Reissner varia￾tional principle can handle mixed boundary problems so that both displacements and stresses can be speci®ed. General boundary conditions which should be satis®ed at all interfaces are as follows. The radial stresses and the shear stresses in adjoining constituents are con￾tinuous. We assume the tractions on the debond crack surface vanish as shown in Fig. 2, while in the elastic zone (perfectly bonded zone) both of the radial and the axial displacements are continuous. We note that if a debond crack takes place at an interface, one may choose between maintaining continuity of the shear stress or continuity of `weighted displacement' at the crack tip (see Ref. 8 for details). Selecting shear stress continuity has been shown to lead to energy con￾sistency, and this condition is used throughout this study. The boundary conditions used here on the sur￾faces of the axisymmetric cell are shown in Fig. 2. At the outer boundary of the composite cylinder …r ˆ rm†, we apply the traction-free boundary conditions for simpli￾city, but we also consider the radial displacement boundary conditions on a limited basis, for illustrative purposes. To obtain the proper displacement boundary conditions, we proceed as follows: 1. assume a crack-free uniaxially tensile loaded com￾posite with perfectly bonded interface and a stress￾free outer surface boundary; 2. obtain the constant radial displacement perpendi￾cular to the cylinder outer surface in the composite using the ADM; 3. use the radial displacement as a boundary condi￾tion in a matrix-cracked composite with an imper￾fect interface. As expected, the two di€erent boundary conditions do not make any di€erence in Gd/Gp over all range of  in the case of 1% ®ber volume fraction. For 40% ®ber volume fraction, we notice di€erences as large as 7% in the values of energy release rates from the two di€erent boundary conditions, but the bound￾ary condition e€ect diminishes considerably in the ®nal calculations for Gd/Gp. In Fig. 6, the results using the displacement boundary condition are shown for com￾parison, and the di€erence by the two boundary condi￾tions appears negligible. Regarding the loading Fig. 4. General mesh structure used in the ADM model. Fig. 5. Gd/Gp versus  with various crack extensions for Vf ˆ 1%. Crack de¯ection/penetration criteria for ®ber-reinforced ceramic matrix composites 1779