Damage tolerant ceramic matrix composites 1059 eqn(20). Inserting eqns(A3)and(A4)into(A7) and and(a8)gives M==02-(x Na=0,x)=vx=s/2) E,(4a (A9) E and v0,x)=H(x=s2)(y2 Inserting these displacements gives the frictional energy dissipation, Ws(eqn 28) A4 Overall composite strain The maximum composite strain during loading EJE八(2 can be estimated by the average elongation of the unit cell, using eqn A4(assuming full sliding) Inserting these displacements into eqn 20 gives the frictional energy dissipation(eqn 21) A3 Displacement fields during unloading yela u- JEr 2a Er er After fracture, as the specimen unloads the sliding direction changes, as the fibres contract and slide (A13) into the matrix. Then, the sign in front of T, changes in eqns(A3)and(A4). When the speci- The strain offset after unloading is (full sliding men is completely free of external load the dis- placement fields are (Al4) v(G=0,x)=v(x=s/2) I-f Em( 4a a on noting that the sign of in front of Ts is changed (All) and o vanishes. Subtracting these two strain values gives the strain change during unloading En(2 i.e. the left hand side of eqn 2Damage tolerant ceramic matrix composites 1059 eqn (20). Inserting eqns (A3) and (A4) into (A7) and (A8) gives v,(x) = v,(x=,y/2) - + I \ (A9) and I ^ ^\ l+(a, x) = vXx=s/2) -I- : $ -x’ + f i I a I (AlO) Inserting these displacements into eqn 20 gives the frictional energy dissipation (eqn 21). A.3 Displacement fields during unloading After fracture, as the specimen unloads the sliding direction changes, as the fibres contract and slide into the matrix. Then, the sign in front of 7S changes in eqns (A3) and (A4). When the speci￾men is completely free Iof external load, the dis￾placement fields are vZ(a = 0, x) = vZ(x=s/2) + - _ I \ (Al 1) X and $(a = 0, x) = vKx=s/2) - 5 5 -c + f l 1 a (Al21 *es Uf s -l-- 1 x . Ef 2 Inserting these displacements gives the frictional energy dissipation, WSl* (eqn 28). A.4 Overall composite strain The maximum composite strain during loading can be estimated by the average elongation of the unit cell, using eqn A4 (assuming full sliding), sl2 2 E” = - Ef(q,X)- s s 0 F&-v” ’ ‘s afreS . fEf 2a Ef Ef (A13) The strain offset after unloading is (full sliding) E* = s 7, ufres 2a Ef Ef ’ (A14) on noting that the sign of in front of 7S is changed and u vanishes. Subtracting these two strain values gives the strain change during unloading, i.e. the left hand side of eqn 25