SUN and SINGH: MULTIPLE MATRIX CRACKING =3盒m+E居+2)2( The expressions for calculating量 in composites were proposed for both bonded and Taking the pertinent material parameters and prop- frictionally-coupled interfaces erties from Table 2. the saturation stress o. calcu- lated from equation(18)is about 190 MPa if one Acknowledgements-Author takes a equal to 2 because it provides a closer value Gadkareen of Corning Glass Works for supply of the of matrix crack spacing for these composites show. glass powder, and Rollie Dutton for information about ing debonding. In case of the frictionally-coupled National Science Foundation through Grant #DMR interface, ad is reduced to slide, which is normally 930787 negligible. Equation(19) then can be simplified as REFERENCES amul+ (20) 1. Sun, Y. J. and Singh. R. N, Advances in Ceramic- Matrix composites Ill, Ceramic Transactions, 1996. 74 f we take a=1.34. a statistically-derived par- ameter for matrix crack spacing in composites with 2. Prewo, K. M, Journal of Materials Science, 1988, 23 a pure frictional interface [7, 9], then equation(20) 2745. with these assumptions gives the same prediction as 3. Briggs, A. and Davidge. R. w. Materials Science and Zok [7 that the saturation stress is about 30% 4 Llorca, J I and Singh, RN,J.Am.CeramSoc higher than FMC stress cFMc, which can be approximated as=omu 5. Steif. P. S. and Schwietert. H. R. Ceram. Eng. Sci. Proc,1990,11,1567. 6. Aveston, J. and Kelly, A. Journal of materials 5. SUMMARY Science,1973,8,352. 7. Zok, F.w. and Spearing, S. M, Acta metall. 1992, n this study, the role of the interfacial debonding he multiple matrix cracking was investigated. It 8. Weitsman, Y. and Zhu, H,J. Mech. Phys. Solids, was shown that the debond initiated when the first 9. Curtin. W.A. Acta metalL. 1993. 41. 1369 matrix cracking occurred in a composite with a 10. Hutchinson, J. W. and Jensen, H. M, Mechanics of weakly-bonded interface which is desirable for Materials. 1990.9. 139 toughening. The debond length increased linearly I1. Marshall, D. B, Acta Metall., 1992, 40, 427 ith the applied stress and then saturated as the 12 Budiansky, B, Evans, A.Gand Hutchinson,J.w debonded zones from either sides of the uncracked 13. Li s. Shah. S P. Li. Z. and Mura. T. Imt,,. solids matrix block approached each other. The matrix Structures,1993,30,.1429 crack density also increased with the applied stress, 14. Aveston, J,Cooper, G. A. and Kelly, A, National and saturated at some critical stress level. the ysical Laboratory Conference, U.K., Nov. 4th, 1971 matrix crack spacing was not uniformly distributed Cox, B. N. and Evens. A. G, cta indicating their dependence on the distribution of 16. Budiansky, B, Hutchinson, J.W. and Evans,A.G flaws and faw size J. Mech. Phvs. Solids. 1986. 34. 167 The relationship between debond length and 17. Sutcu, M. and Hillig, W.B., Acta metall, 1990, 38 2653. applied stress was examined by both the force bal- 18. Singh, R. N. and Sutcu, M, Journal of Materials ance and energy balance approaches in the case of multiple matrix cracking. For a very short debond 19. Marshall. D. B. and Oliver. W. C.J. Am. Ceram length, both approaches gave an estimate close to Soc., 1987, 70, the experimental data. But, for a long debond 20 Kerans, R Parthasarathy, T. A.,J. Am length, the energy balance provided a better fit. It 21. Dutton.R.e was also found that the saturation stress for matrix 22. Sun, Y. Ceran.Soc.,1996,79,865 R. N. Materials Science and cracking was determined by the interaction between Engineering, in reviewss ˆa 2 Ec Em smu‡ 1 2Em a2 s2 muEcEfVf ‡4E2 ms2 d ÿ
1=2 …19† Taking the pertinent material parameters and prop￾erties from Table 2, the saturation stress ss calcu￾lated from equation (18) is about 190 MPa if one takes a equal to 2 because it provides a closer value of matrix crack spacing for these composites show￾ing debonding. In case of the frictionally-coupled interface, sd is reduced to sslide, which is normally negligible. Equation (19) then can be simpli®ed as ss ˆa 2 Ec Em smu 1 ‡ EfVf Ec  1=2 ! …20† If we take a = 1.34, a statistically-derived par￾ameter for matrix crack spacing in composites with a pure frictional interface [7, 9], then equation (20) with these assumptions gives the same prediction as Zok [7] that the saturation stress is about 30% higher than FMC stress sFMC, which can be approximated asEc Em smu: 5. SUMMARY In this study, the role of the interfacial debonding on the multiple matrix cracking was investigated. It was shown that the debond initiated when the ®rst matrix cracking occurred in a composite with a weakly-bonded interface which is desirable for toughening. The debond length increased linearly with the applied stress and then saturated as the debonded zones from either sides of the uncracked matrix block approached each other. The matrix crack density also increased with the applied stress, and saturated at some critical stress level. The matrix crack spacing was not uniformly distributed, indicating their dependence on the distribution of ¯aws and ¯aw size. The relationship between debond length and applied stress was examined by both the force bal￾ance and energy balance approaches in the case of multiple matrix cracking. For a very short debond length, both approaches gave an estimate close to the experimental data. But, for a long debond length, the energy balance provided a better ®t. It was also found that the saturation stress for matrix cracking was determined by the interaction between multiple matrix cracking and interfacial debonding. The expressions for calculating the saturation stress in composites were proposed for both bonded and frictionally-coupled interfaces. AcknowledgementsÐAuthors are grateful to Kishore Gadkareen of Corning Glass Works for supply of the glass powder, and Rollie Dutton for information about composite processing. This research was supported by National Science Foundation through Grant #DMR- 9307877. REFERENCES 1. Sun, Y. J. and Singh, R. N., Advances in Ceramic￾Matrix composites III, Ceramic Transactions, 1996, 74, 141. 2. Prewo, K. M., Journal of Materials Science, 1988, 23, 2745. 3. Briggs, A. and Davidge, R. W., Materials Science and Engineering, 1989, A109, 363. 4. Llorca, J. I. and Singh, R. N., J. Am. Ceram. Soc., 1991, 74, 2882. 5. Steif, P. S. and Schwietert, H. R., Ceram. Eng. Sci. Proc., 1990, 11, 1567. 6. Aveston, J. and Kelly, A., Journal of Materials Science, 1973, 8, 352. 7. Zok, F. W. and Spearing, S. M., Acta metall., 1992, 40, 2033. 8. Weitsman, Y. and Zhu, H., J. Mech. Phys. Solids, 1993, 41, 351. 9. Curtin, W. A., Acta metall., 1993, 41, 1369. 10. Hutchinson, J. W. and Jensen, H. M., Mechanics of Materials, 1990, 9, 139. 11. Marshall, D. B., Acta Metall., 1992, 40, 427. 12. Budiansky, B., Evans, A. G. and Hutchinson, J. W., Int. J. Solids Structures, 1995, 32, 315. 13. Li, S., Shah, S. P., Li, Z. and Mura, T., Int. J. Solids Structures, 1993, 30, 1429. 14. Aveston, J., Cooper, G. A. and Kelly, A., National Physical Laboratory Conference, U.K., Nov. 4th, 1971. 15. Marshall, D. B., Cox, B. N. and Evens, A. G., Acta metall., 1985, 33, 2013. 16. Budiansky, B., Hutchinson, J. W. and Evans, A. G., J. Mech. Phys. Solids, 1986, 34, 167. 17. Sutcu, M. and Hillig, W. B., Acta metall., 1990, 38, 2653. 18. Singh, R. N. and Sutcu, M., Journal of Materials Science, 1991, 26, 2547. 19. Marshall, D. B. and Oliver, W. C., J. Am. Ceram. Soc., 1987, 70, 542. 20. Kerans, R. J. and Parthasarathy, T. A., J. Am. Ceram. Soc., 1991, 74, 1585. 21. Dutton, R. E., J. Am. Ceram. Soc., 1996, 79, 865. 22. Sun, Y. J. and Singh, R. N., Materials Science and Engineering, in review. SUN and SINGH: MULTIPLE MATRIX CRACKING 1667