B. F. Sorensen, R. Talreja the cross-section area of the matrix is En∈n(x)dx f2s27 V22 61-a2E(1 and the total volume of the cell is where the strain is defined to be zero at the stress free state at room temperature. The factor 2 in the (11) denominator before the integral is to account for the integration being only over half the volume of the unit cell shown in Fig. 4(from the fibre end to 3.4 Multiple matrix cracking the centre of the matrix block) During multiple matrix cracking one crack Likewise, the strain energy density in the fibres formed within each volume cell(Fig. 4). Thus during full slip is(by eqns 8 and 11) the energy absorped per unit volume of the com poSite is =4∫1Ee1()dx=1+ GaM=Um(l-f) V22 where Gm is the critical energy release rate of the fsr3lσus (18) matrix, and eqns(10)and(ll)have been used Er 2 Er 3.5 Fibre/matrix debonding where the composite strength is given by the fail- For each unit cell there is debonding over the ure strain(Appendix) surface A,, such that the energy absorbed due to fibre/matrix debonding per unit volume of the Ite Is σ=f1E+15x+/o严 (13) 3.7 Interfacial frictional sliding where Gab is the critical(mode II) energy release The energy dissipation due to frictional sliding can rate of the interface, and eqns(9)and(11)have be assessed by considering the difference in dis been used. Note that an advantage of this placement fields between the matrix and fibre at approach is that the estimates of Ume and Ua are the end state( the maximum applied stress at the independent of the load level at which matrix point of failure)and the initial state (prestressed cracking and debonding take place with the residual stresses), i. e. from the difference in the displacement fields of fibre and matrix of 3.6 Strain energy in matrix and fibre half the volume cell The strain energy densities due to residual stresse wsca I vm(, x)-ve(o, x) 2T a T, dx,(20) =4m1(a=1-(m,4 y 2 E 2 E where vm and v are the displacement fields of the and matrix and fibre at the applied stress a, defined to Ars 1(ors) 2(1-f)2(o be zero before loading (i.e, v and vm are zero (15) when the residual stresses act alone). The factor 2 y 2 Er E in front of the integral is to account for the inte- force balance of the residual stress com- gration being only over the half volume of the ponents in the axial direction unit cell. Using the displacement fields derived in the Appendix, the energy dissipation due to slid fo es +(1-o m=0, (16) ing is found by combining eqns(11),(19)and (20), and eqns 8 and 1l have been used. The strain nergy density in the matrix during full sliding Ws=I s(6,+0 ma ∫(4 along the fibre/matrix interface can be derived Em" ts EmE from the strain state in the matrix(Appendix), using eqns(10) and(ll) (21)1052 the cross-section area of the matrix is and the total volume of the cell is ra2s V=_ f . 3.4 Multiple matrix cracking During multiple matrix cracking one crack is formed within each volume cell (Fig. 4). Thus the energy absorped per unit volume of the com￾posite is B. F. Stirensen, R. Talreja (10) (11) urn, = G,A, = G, (l-j), v s (12) where G,,, is the critical energy release rate of the matrix, and eqns (10) and (11) have been used. 3.5 Fibre/matrix debonding For each unit cell there is debonding over the surface A,, such that the energy absorbed due to fibreimatrix debonding per unit volume of the composite is u,, = GdbAs = 2fGdb ) V a (13) where Gdb is the critical (mode II) energy release ‘rate of the interface, and eqns (9) and (11) have been used. Note that an advantage of this approach is that the estimates of U,,,, and U,, are independent of the load level at which matrix cracking and debonding take place. 3.6 Strain energy in matrix and fibre The strain energy densities due to residual stresses are @I) = 4lP 1 ht3= 1 -f (gF)= -__=- m V 2 E,,, 2 Em’ (14) and Q. _ 4s 1 G-C>=_ (1-f)’ hi?)= f , V 2 Ef 2f Ef (15) where the force balance of the residual stress com￾ponents in the axial direction, fu? + (1 -f)aF = 0, (16) and eqns 8 and 11 have been used. The strain energy density in the matrix during full sliding along the fibre/matrix interface can be derived from the strain state in the matrix (Appendix), using eqns (10) and (1 l), s/2 @III _ A m m s 1 E,E~,(x) dx = -i f2 _? 7f ,(17) VI2 O 2 6 1-f a2 E,,, where the strain is defined to be zero at the stress free state at room temperature. The factor 2 in the denominator before the integral is to account for the integration being only over half the volume of the unit cell shown in Fig. 4 (from the fibre end to the centre of the matrix block). Likewise, the strain energy density in the fibres during full slip is (by eqns 8 and 11) sl2 Af @‘;I = - I LE&)dx=L at+ VI2 o 2 2 fEf f s= 72 ----s_- ‘52~ 6 a2 Ef 2 Ef a ” (18) where the composite strength is given by the fail￾ure strain (Appendix) CT, =ft,E,++ +faf’““. (19) 3.7 Interfacial frictional sliding The energy dissipation due to frictional sliding can be assessed by considering the difference in dis￾placement fields between the matrix and fibre at the end state (the maximum applied stress at the point of failure) and the initial state (prestressed with the residual stresses), i.e. from the difference in the displacement fields of fibre and matrix of half the volume cell, s/2 W,k> = -$ \ I v,,,h 4 - vf hx> I 2~ a T dx, (20) 0 where v, and vf are the displacement fields of the matrix and fibre at the applied stress a, defined to be zero before loading (i.e, vf and v, are zero when the residual stresses act alone). The factor 2 in front of the integral is to account for the inte￾gration being only over the half volume of the unit cell. Using the displacement fields derived in the Appendix, the energy dissipation due to slid￾ing is found by combining eqns (1 l), (19) and (20)7