Y-F. Liu et al./ Mechanics of Materials 29(1998)111-121 ence the fracture behavior of the composite model omposites containing an elliptic hole. Mech. Mater. 19, 239 The following conclusions were obtained (1)The distributed spring model was solved based Gao. Y-C. Mai. Y-W. Cotterell. B. 1988. Fracture of fiber n a discrete numerical method. Influence of such reinforced materials. J. Appl. Math. Phys. (ZAMP)39, 550- microstructural parameters as interface debonding Gupta, V,Yuan,J,Martinez, D,1993.Calculation,measure- toughness, sliding friction at the debond interface, ment, and control of interface strength in composites. J. Am the thermal stress and fiber volume fraction on the Ceran.Soc.76,305-315 bridging effect was studied quantitatively utchinson, J.W., Jensen, H M, 1990. Models of fiber debonding (2)Crack propagation was simulated and ex- and pullout in brittle composites with friction. Mech. Mater. 9, pressed in terms of R-curve. Influences of the above Lawn, B R, 1975. Fracture of Brittle Solids. Cambridge Univer- major microstructural parameters on the curve were sity Press, Cambridge studied. It was revealed that larger debonding tough Liu, Y.F., 1995. Doctoral dissertation, The University of Tokyo ness. frictional stress and fiber volume fraction and iu,Y F, Kagawa, Y, 1996. Analysis of debonding and frictional smaller thermal stresses provided larger initial slope sliding in fiber-reinforced brittle composites: basic problems. Mater Sci Eng. A 212, 75-87 of the R-curve, while maximum fracture resistance Luo, HA, Ballarini, R, 1994. The effects of anisotropy on the was obtained at an intermediate value of the these inear behavior of bridged cracks in long strips. J. Mech parameters. These results were discussed to under Solids42,14l-157 stand physical implications regarding fracture mech- D.B., 1992. Analysis of fiber debonding and sliding anisms and material design of fiber reinforced ce experiments in brittle matrix composites. Acta Metall. 40 ramIcs Marshall, D B, Cox, B N, 1985. The mechanics of matrix crack- ing in brittle-matrix fiber composites. Acta Metall. 33, 2013- References Marshall. D B. Cox. B N. 1987. Tensile fracture of brittle matrix osites: influence of fiber strength. Acta Metall. 35, Aveston, J, Cooper, G.A., Kelly, A, 1971. Single and McCartney, L N, 1987. Mechanics of matrix cracking in a racture. In: rties of Fiber Composites. Proc. of NPL brittle-matrix fiber-reinforced composites. Proc. R. Soc. Lon- IPC Science and Technology Press, Surrey, pp. 15-26 don,Ser.A409,329-350 Bao. G. Suo. Z. 1992. Remarks on crack-bridging Meda, G, Steif, P.S., 1994. A detailed analysis of cracks bridged Appl. Mech. Rev. 45, 355-366 y aligned, fibers: I. Limiting cases of short and long cracks. J. Mech Budiansky, B, Amazigo, J.C., 1989 hening Phys. Solids 42, 1293-1321 frictionally constrained fibers. J. Mech. Phys. Solie 93 Parthasarathy, T.A., Marshall, D B, Kerans, R., 1994. Analysis of the effect of interfacial roughness on fiber debonding and a iding in brittle matrix composites. Acta Metall. 42, 3773 fiber-reinforced ceramic composite containing a crack-like 784 flaw. J. Mech. Phys. Solids 42, 1-19 Rice, J.R., 1968. A path independent integral and the approximate Budiansky, B, Hutchinson, J.w., Evans, A, G, 1986. Matrix nalysis of strain concentration by notches and cracks. J. Appl fracture in fiber-reinforced ceramics. J. Mech. Phys. Solids 34 Mech.35,379-386 67-189 Sigl, L.S., Evans, A.G., 1989. Effect of residual stress Budiansky, B, Evans, AG, Hutchinson, J.W.,1995.Fiber-ma tional sliding on cracking and pullout in brittle matrix trix debonding effects of cracking in aligned fiber ceramic ites. Mech. mater.8. 1-12 composites. Int. J. Solids Struct 32, 315-328 Sih, G.C., 1985. Handbook of Stress Intensity Factors. Lehigh Ceramic Source. 1989. J. Am. Ceram Soc. 6. T8 University, Bethlehem Cox, B N, 1993. Scaling for bridged cracks. Mech. Mater. 15, Sneddon, I N, Lowengrug, M, 1968. Cracks Problems in the Classical Theory of Elasticity. Wiley, New York. Cox, B N, Marshall, D B. 1991. Stable and unstable Thouless, M D, Evans, A.G., 1988. Acta Metall. 26, 517. bridged cracks in various specimens. Acta Metall. 39, 579-589. Xia, Z.C., Hutchinson, J W, Evans, A.G., Budinasky, B, 1994 Cox, B N, Marshall, D B, 1994. Concepts for bridged cracks On large scale sliding in fiber-reinforced composites. J. Mech. fracture and fatigure. Acta Metall. 42, 341-363 hys. Solids 42. 1 139-1158 Cox, B.N., Rose, L.R.F, 1994. Time- or circle-dependent crack Yuuki, R, Liu, Y.F., 1994. Evaluation of debond length based on YLg 1995 On the strenth of fher reinforced energy release rate. Trans. Jpn. Soc. Mech Eng. A 60, 1951Y.-F. Liu et al.rMechanics of Materials 29 1998 111–121 ( ) 121 ence the fracture behavior of the composite model. The following conclusions were obtained. Ž . 1 The distributed spring model was solved based on a discrete numerical method. Influence of such microstructural parameters as interface debonding toughness, sliding friction at the debond interface, the thermal stress and fiber volume fraction on the bridging effect was studied quantitatively. Ž . 2 Crack propagation was simulated and ex￾pressed in terms of R-curve. Influences of the above major microstructural parameters on the curve were studied. It was revealed that larger debonding tough￾ness, frictional stress and fiber volume fraction, and smaller thermal stresses provided larger initial slope of the R-curve, while maximum fracture resistance was obtained at an intermediate value of the these parameters. These results were discussed to under￾stand physical implications regarding fracture mech￾anisms and material design of fiber reinforced ce￾ramics. References Aveston, J., Cooper, G.A., Kelly, A., 1971. Single and multiple fracture. In: Properties of Fiber Composites. Proc. of NPL. IPC Science and Technology Press, Surrey, pp. 15–26. Bao, G., Suo, Z., 1992. Remarks on crack-bridging concepts. Appl. Mech. Rev. 45, 355–366. Budiansky, B., Amazigo, J.C., 1989. Toughening by aligned, frictionally constrained fibers. J. Mech. Phys. Solids 37, 93– 109. Budiansky, B., Cui, Y.L., 1994. On the tensile strength of a fiber-reinforced ceramic composite containing a crack-like flaw. J. Mech. Phys. Solids 42, 1–19. Budiansky, B., Hutchinson, J.W., Evans, A.G., 1986. Matrix fracture in fiber-reinforced ceramics. J. Mech. Phys. Solids 34, 167–189. Budiansky, B., Evans, A.G., Hutchinson, J.W., 1995. Fiber–ma￾trix debonding effects of cracking in aligned fiber ceramic composites. Int. J. Solids Struct. 32, 315–328. Ceramic Source, 1989. J. Am. Ceram. Soc. 6, T8. Cox, B.N., 1993. Scaling for bridged cracks. Mech. Mater. 15, 87–98. Cox, B.N., Marshall, D.B., 1991. Stable and unstable solutions for bridged cracks in various specimens. Acta Metall. 39, 579–589. Cox, B.N., Marshall, D.B., 1994. Concepts for bridged cracks in fracture and fatigure. Acta Metall. 42, 341–363. Cox, B.N., Rose, L.R.F., 1994. Time- or circle-dependent crack bridging. Mech. Mater. 19, 39–57. Cui, Y.L., 1995. On the strength of fiber reinforced ceramic composites containing an elliptic hole. Mech. Mater. 19, 239– 249. Gao, Y-C., Mai, Y-W., Cotterell, B., 1988. Fracture of fiber reinforced materials. J. Appl. Math. Phys. ZAMP 39, 550– Ž . 572. Gupta, V., Yuan, J., Martinez, D., 1993. Calculation, measure￾ment, and control of interface strength in composites. J. Am. Ceram. Soc. 76, 305–315. Hutchinson, J.W., Jensen, H.M., 1990. Models of fiber debonding and pullout in brittle composites with friction. Mech. Mater. 9, 139–163. Lawn, B.R., 1975. Fracture of Brittle Solids. Cambridge Univer￾sity Press, Cambridge. Liu, Y.F., 1995. Doctoral dissertation, The University of Tokyo. Liu, Y.F., Kagawa, Y., 1996. Analysis of debonding and frictional sliding in fiber-reinforced brittle composites: basic problems. Mater. Sci. Eng. A 212, 75–87. Luo, H.A., Ballarini, R., 1994. The effects of anisotropy on the nonlinear behavior of bridged cracks in long strips. J. Mech. Phys. Solids 42, 141–157. Marshall, D.B., 1992. Analysis of fiber debonding and sliding experiments in brittle matrix composites. Acta Metall. 40, 427–441. Marshall, D.B., Cox, B.N., 1985. The mechanics of matrix crack￾ing in brittle-matrix fiber composites. Acta Metall. 33, 2013– 2021. Marshall, D.B., Cox, B.N., 1987. Tensile fracture of brittle matrix composites: influence of fiber strength. Acta Metall. 35, 2607–2619. McCartney, L.N., 1987. Mechanics of matrix cracking in a brittle-matrix fiber-reinforced composites. Proc. R. Soc. Lon￾don, Ser. A 409, 329–350. Meda, G., Steif, P.S., 1994. A detailed analysis of cracks bridged by fibers: I. Limiting cases of short and long cracks. J. Mech. Phys. Solids 42, 1293–1321. Parthasarathy, T.A., Marshall, D.B., Kerans, R.J., 1994. Analysis of the effect of interfacial roughness on fiber debonding and sliding in brittle matrix composites. Acta Metall. 42, 3773– 3784. Rice, J.R., 1968. A path independent integral and the approximate analysis of strain concentration by notches and cracks. J. Appl. Mech. 35, 379–386. Sigl, L.S., Evans, A.G., 1989. Effect of residual stress and fric￾tional sliding on cracking and pullout in brittle matrix compos￾ites. Mech. Mater. 8, 1–12. Sih, G.C., 1985. Handbook of Stress Intensity Factors. Lehigh University, Bethlehem. Sneddon, I.N., Lowengrug, M., 1968. Cracks Problems in the Classical Theory of Elasticity. Wiley, New York. Thouless, M.D., Evans, A.G., 1988. Acta Metall. 26, 517. Xia, Z.C., Hutchinson, J.W., Evans, A.G., Budinasky, B., 1994. On large scale sliding in fiber-reinforced composites. J. Mech. Phys. Solids 42, 1139–1158. Yuuki, R., Liu, Y.F., 1994. Evaluation of debond length based on energy release rate. Trans. Jpn. Soc. Mech. Eng. A 60, 1951– 1958