Damage tolerant ceramic matrix composites In many cases the stress-strain curve does not 3. 2 Basis of unit cell model show non-linearity immediately before failure, In this paper the strain densities and frictional slid indicating that the presence of Stage IV (distributed ing are calculated from a simple shear-lag model fibre failure)can be neglected. Thus, our analysis Similar concerns only energy uptake in Stages L, II and stresses are added( details are given in the appendix II, with Stage Ill extended until failure. In Stage I The model is a simple one-dimensional analysis, i.e the energy uptake per unit volume of the compos- only the changes in the axial stresses and strains ite, the area under the stress-strain curve up to E= are considered, and the interfacial friction shear Emen, is denoted U. According to the principle of stress Ts is assumed to be constant throughout the virtual work (or the first law of thermodynamics) experiment. This is a good assumption for the the energy uptake U is equal to the increase in accuracy of modelling presented here since it has strain energy density in the matrix and fibres, p been found that the effects of fibre poisson's con nd qr, respectively, where superscript 0 and i traction and roughness more or less cancel out. 26.27 denote the start and end of Stage I, respectively Initially, i.e. in the virgin state, the composite is free from external stresses, but has axial residual stresses, ores and o mes in the fibre and matrix, due Ul=ode=Im+i-oom-e (4) to a thermal expansion mismatch between fibre and matrix. Before fibre/matrix debonding has taken In Stage II the external work per unit volume is, place, there is no slip between fibre and matrix at again, according to the principle of virtual work, the interface. During multiple matrix cracking the equal to the sum of the changes in strain energy fibre and matrix debond during the loading, the density of fibre and matrix, the fracture energy of matrix retracts and, relatively, the fibres slide along multiple matrix cracks, fibre/matrix debonding the interface monotonically(Fig 4). It is assumed and the energy dissipation due to interfacial fric- that the fibres slide symmetrically inside the matrix tional sliding in Stage II. Denoting superscript II cylinder from both ends, such that there is sliding for the end of Stage II (e= fm ACk), the energy along the entire interface except at the middle of uptake in Stage II is the matrix cylinder(0<x<s/2, Fig. 4), where sticking friction is assumed due to symmetry(the condition TAI =o de=m +d- m-1+Um+UJb+wJl. (5) 3.3 Geometrical considerations where Wsstands for the frictional energy dissipa A unit cell model is depicted in Fig. 4. The cylin drical volume cell represents a single fibi tion per unit volume in Stage Il, Ume is the energy radius, a, surrounded by a matrix cylinder. The consumed due to the formation of matrix crack and Uab is the debond energy per unit volume. length of the cell is s, corresponding to the matrix Assuming that the deformation of fibre and matrix crack spacing, and the outer radius of the matrix cylinder is such that the fibre volume fraction is f continues in Stage Ill, the cncrgy uptake in Stage The cross-section area Ar of the fibre is d e=wu U=σd∈=φ+φ-φ-φ+W,(0 The interfacial surface area A is A where superscript Ill stands for the end stat when localization sets in(∈=∈u), and w is the frictional energy dissipation in Stage Ill. This gives the total energy uptake as U=U+Um+UⅢ=φⅢ+φ-Φ。-φ? Umc Uab+ Wsl where Ws is the total energy dissipation per unit Matrix volume due to frictional sliding in Stages II and t←a→0m III. Note that u can be calculated from the stress and strain states in the initial and end states. The frictional energy dissipation can also be calculated from the strain distributions in the end state assuming the fibre sliding to occur monotonically Fig. 4. The axisymmetric unit cell used in the analysisDamage tolerant ceramic matrix composites 1051 In many cases the stress-strain curve does not show non-linearity immediately before failure, indicating that the presence of Stage IV (distributed fibre failure) can be neglected. Thus, our analysis concerns only energy u.ptake in Stages I, II and III, with Stage III extended until failure. In Stage I the energy uptake per unit volume of the compos￾ite, the area under the stress-strain curve up to E = E i”i, is denoted U’. According to the principle of vrtual work (or the first law of thermodynamics) the energy uptake U’ is equal to the increase in strain energy density in the matrix and fibres, @,,,I and @f’, respectively, where superscript 0 and I denote the start and end of Stage I, respectively E 1,“: U’= ade=@;+@;-@0,-@Of. s (4) 0 In Stage II the external work per unit volume is, again, according to the principle of virtual work, equal to the sum of the changes in strain energy density of fibre and matrix, the fracture energy of multiple matrix cracks, fibreimatrix debonding and the energy dissipat:ion due to interfacial fric￾tional sliding in Stage II. Denoting superscript II for the end of Stage II (E = gmCACK), the energy uptake in Stage II is where II’,,” stands for the frictional energy dissipa￾tion per unit volume in Stage II, U,, is the energy consumed due to the formation of matrix cracks and U,, is the debond energy per unit volume. Assuming that the deformation of fibre and matrix continues in Stage III, the energy uptake in Stage III is where superscript III stands for the end state, when localization sets in (E = E,), and WsllI1 is the frictional energy dissipation in Stage III. This gives the total energy uptake as &I, + udb + Kl, (7) where I+‘,, is the total energy dissipation per unit volume due to frictional sliding in Stages II and III. Note that U can be calculated from the stress and strain states in the initial and end states. The frictional energy dissipation can also be calculated from the strain distributions in the end state, assuming the fibre sliding to occur monotonically. 3.2 Basis of unit cell model In this paper the strain densities and frictional slid￾ing are calculated from a simple shear-lag model, similar to Aveston et al.;’ the effect of residual stresses are added (details are given in the Appendix). The model is a simple one-dimensional analysis, i.e. only the changes in the axial stresses and strains are considered, and the interfacial friction shear stress 7s is assumed to be constant throughout the experiment. This is a good assumption for the accuracy of modelling presented here, since it has been found that the effects of fibre Poisson’s con￾traction and roughness more or less cancel out.26,27 Initially, i.e. in the virgin state, the composite is free from external stresses, but has axial residual stresses, afres and (T c in the fibre and matrix, due to a thermal expansion mismatch between fibre and matrix. Before fibre/matrix debonding has taken place, there is no slip between fibre and matrix at the interface. During multiple matrix cracking the fibre and matrix debond during the loading, the matrix retracts and, relatively, the fibres slide along the interface monotonically (Fig. 4). It is assumed that the fibres slide symmetrically inside the matrix cylinder from both ends, such that there is sliding along the entire interface except at the middle of the matrix cylinder (O<x<s/2, Fig. 4), where sticking friction is assumed due to symmetry (the condition for full slip is given in the Appendix). 3.3 Geometrical considerations A unit cell model is depicted in Fig. 4. The cylin￾drical volume cell represents a single fibre of radius, a, surrounded by a matrix cylinder. The length of the cell is s, corresponding to the matrix crack spacing, and the outer radius of the matrix cylinder is such that the fibre volume fraction isf. The cross-section area A, of the fibre is Af =ra2. (8) The interfacial surface area A, is A, = 2z-as, (9) S I\ II \\ II - -..-.-Ju- II Matrix , \ r------- L&L ,_-sA 1 ,1 I I I TV_ 7-7 Fibre / p % s/2 Fig. 4. The axisymmetric unit cell used in the analysis