B. F. Sorensen, R. Talreja 28. Marshall. D. B. Oliver, W.C. Measurements of interface as the matrix retracts. After application of facial mechanical properties in fiber-reinforced opposites. J. Am. Ceram Soc., 70(1987)542-8 the external force the strains in the slip region are 29. Bischoff. E. Ruhle. M. Sbaizero, o. Evans A. g with reference to the coordinate system in Fig 4, fiber-reinforced lithium aluminum silicate glass- J.Am. Ceran.Soc.,72(1989)741-4 n(,x)=2了x 0≤x≤l()(A3) 30. Bonney, L A.& Cooper, R. F, Reaction layer interfaces in Sic-fiber-reinforced glass-ceramics: a high-resolution scanning transmission electron microscopy analysis. J m. Ceram.Soc,73(1990)2916-26 31. Cho. C. Holmes. I. w. Barber. L.R. Estimation of ∈r(o,x) interfacial shear in ceramic composites from friction fE 0≤x≤l(o).(A4) ng measurements. J. Am. Ceram. Soc., 74(1991)280 32. Cao, H. C. Thouless M, D, Tensile tests ot matrix composites: Theory and experiments. J. A In these expressions, Is is the sliding length along Ceran.Soc,73(19902091-4 the interface. measured from the matrix crack 33. Cao, H. C. Bischoff. E. Sbaizero. O. Ruhle. M. Evans (ig. 4). The sliding length can be found as the A. G. Marshall, D. B. brennan J. J. Effect of inte position x along the interface where the fibre and faces on the properties of fiber-reinforced ceramics. J. Am. Ceram. Soc, 73(1991)1691-9 matrix strain changes are identical, E -Eres 34. Ricca N. Guette. A. Camus. G.& Jouin. J M. Sic Em(ls-Emes, giving (ex PCS)MAS composite with a BN interphase: micro structure, mechanical properties and oxidation resistance In High Temperature Ceramic Matrix Composites, HT- 4( o)1-a Em_or s Em ones E, CMCI, eds R. Naslain, J. Lamon D. Doumeingts fr、 E TE T E woodhead UK(1993)455-62. 35. Khodakovskaya, R. Y.& Pavlushkin, N as also found by Pryce and Smith. From eqns nd properties of metastable quartzitic phases n the SiOz-AL2Or-MgO-TiO, system. Inorg (A5)and (29)it follows that there is sliding along terface(s s/2), whe 36. Martin, B, Benoit, M.& Rouby, D, Interfacial stress exceeds a characteristic value, OS(Subscripts strength in fibre reinforced ceramic matrix co indicate Full Slip) nvolving positive radial thermal misfit. Scripta Mater,28(1993)142933. 7. Suo, Z, Bao, G,& Fan, B, Delamination R-curve phenom- s了sE ena due to damage, J. Mech. Phys. Solids, 40(1992)1-16 38. Rice, J.R., a path inde fEr Er Em l-f a Er em mate analysis of strain concentrations by notches and Above this stress level. the stress and strain states 39. Rousseau Davidson. D. L. Campbell, J. B, The in the matrix do not change. This is a consequence micromechanics of ambient temperature cyclic of the assumption in the model that Ts does not fatigue in a composite of CA ceramic rel change When the terms of the residual stresses with Nicalon fibers J. Com 40. Pryce, A. W.& Smith, P.A cracking in unido: are dropped in eqns(A5)and(a6) they become ctional ceramic matrix composites under quasi-static and cyclic loading. Acta Metall. Mater, 41(1993)1269-81. strains gives the strain densities in eqns 17 and 18 A2 Displacement fields during loading Appen The displacement ficlds at a stress o(aftcr multiplc matrix cracking and debonding) are, for the case of A 1 Strain fields full sliding, then found by integration of the strains Before matrix cracking and interfacial debonding the composite exhibits residual stresses, e.g. due to dvm=∈m(,x) thermal expansion mismatch. We define the Em dx(a) strains to be zero at stress free state at room tem erature. Then the strain fields in matrix and and fibres due to the residual stresses are (a,x) dx, (A8) Er where the last terms represent the displacements due to the release of the residual stress during debonding, eqns(Al)and(A2). In eqns(A)and (A2) (A8)Vm(x=s/2)and v(x=s/2)are the displacements at the centre of the matrix block. They are iden When matrix cracking and fibre/matrix debonding tical, since there is no sliding at the centre of tI take place frictional sliding takes place along the matrix block, and cancel out, when inserting into1058 B. F. Smensen, R. Talreja 28. 29. 30. 31. 32. 33. 34. 35. 36. 31. 38. 39. 40. Marshall, D. B. & Oliver, W. C., Measurements of inter￾facial mechanical properties in fiber-reinforced ceramic composites. J. Am. Ceram. Sot., 70 (1987) 542-8. Bischoff, E., Rtihle, M., Sbaizero, 0. & Evans A. G., Microstructural studies of the interface zone of a BC￾fiber-reinforced lithium aluminum silicate glass-ceramic. J. Am. Ceram. Sot., 72 (1989) 74145. Bonney, L. A. & Cooper, R. F., Reaction layer interfaces in Sic-fiber-reinforced glass-ceramics: a high-resolution scanning transmission electron microscopy analysis. J. Am. Ceram. Sot., 73 (1990) 291626. Cho, C., Holmes, J. W. & Barber, J. R., Estimation of interfacial shear in ceramic composites from frictional heat￾ing measurements. J. Am. Ceram. Sot., 74 (1991) 2808-8. Cao, H. C. & Thouless M. D., Tensile tests of ceramic￾matrix composites: Theory and experiments. J. Am. Ceram. Sot., 73 (1990) 2091-4. Cao, H. C., Bischoff, E., Sbaizero, O., Rtihle, M., Evans, A. G., Marshall, D. B. & Brennan, J. J., Effect of inter￾faces on the properties of fiber-reinforced ceramics. J. Am. Ceram. Sot., 73 (1991) 1691-9. Ricca, N., Guette, A., Camus, G. & Jouin, J. M., Sic (ex. PCS)/MAS composite with a BN interphase: micro￾structure, mechanical properties and oxidation resistance. In High Temperature Ceramic Matrix Composites, HT￾CMCI, eds R. Naslain, J. Lamon & D. Doumeingts. Woodhead UK (1993) 455-62. Khodakovskaya, R. Y. & Pavlushkin, N. M., Structure and properties of metastable qua&tic phases in sitalls in the SiO,-Al@-MgO-TiO, system. Inorg. Mater., 3 (1967) 1662-8. Martin, B., Benoit, M. & Rouby, D., Interfacial sliding strength in fibre reinforced ceramic matrix composites mvolving positive radial thermal misfit. Scripta Metall. Mater., 28 (1993) 1429-33. Suo, Z., Bao, G., & Fan, B., Delamination R-curve phenom￾ena due to damage. J. Mech. Phys. Solidr;, 40 (1992) 1-16. Rice, J. R., A path independent integral and the approxi￾mate analysis of strain concentrations by notches and cracks. J. Appl. Mech., 35 (1968) 379-86. Rousseau, C. Q., Davidson, D. L. & Campbell, J. B., The micromechanics of ambient temperature cyclic loading fatigue in a composite of CAS glass ceramic reinforced with Nicalon fibers. J. Comp. Tech. Res., 16 (1994) 115-26. Pryce, A. W. & Smith, P. A., Matrix cracking in unidi￾rectional ceramic matrix composites under quasi-static and cyclic loading. Acta Metall. Mater., 41 (1993) 1269-81. Appendix A.1 Strain fields Before matrix cracking and interfacial debonding the composite exhibits residual stresses, e.g. due to thermal expansion mismatch. We define the strains to be zero at stress free state at room tem￾perature. Then the strain fields in matrix and fibres due to the residual stresses are and (Al) WI When matrix cracking and fibre/matrix debonding take place frictional sliding takes place along the interface as the matrix retracts. After application of the external force the strains in the slip region are, with reference to the coordinate system in Fig. 4, f E,(ff, x) = 2 - -? 7s Olxll,(a) l-f a E,,, (A3) and c Ef(o; x) = - fEf -2 5 7s Olxll,(a). a 4 (A4) In these expressions, Z, is the sliding length along the interface, measured from the matrix crack (Fig. 4). The sliding length can be found as the position x along the interface where the fibre and matrix strain changes are identical, ~&Z&fres = E,(&-E?, giving lb) _ 1-f c Em afres Em + ai’= 4 (A5) - [ f7, 4 --- -- -~ 7s Ec 1 , a 2 7s 4 as also found by Pryce and Smith.40 From eqns (A5) and (29) it follows that there is sliding along the entire interface (I, = s/2), when the applied stress exceeds a characteristic value, a,, (subscripts indicate Full Slip) res gFS *f res -=_ -E!L+L~7,3_ fEf Ef Em 1-f aEf& * w-9 Above this stress level, the stress and strain states in the matrix do not change. This is a consequence of the assumption in the model that rS does not change. When the terms of the residual stresses are dropped in eqns (A5) and (A6), they become identical to the results in ref. 31. Inserting the strains gives the strain densities in eqns 17 and 18. A.2 Displacement fields during loading The displacement fields at a stress o (after multiple matrix cracking and debonding) are, for the case of full sliding, then found by integration of the strains, v,(x=si2) 62 a .r _ lxx 1 1 dv, = 1 1 E,,,(cT, x) - “+ 1 dx (A7) J bs-4 ,“L ‘%I] and 1 dx, 648) where the last terms represent the displacements due to the release of the residual stress during debonding, eqns (Al) and (A2). In eqns (A7) and (A8) v,(x=s/2) and v,(x=s/2) are the displacements at the centre of the matrix block. They are iden￾tical, since there is no sliding at the centre of the matrix block, and cancel out, when inserting into