202 12 Introduction to Damage Mechanics of Composite Materials First,one needs to find a damage transformation equation for the volume fractions.This is performed by substituting (12.8)and (12.13)into (12.5), simplifying and comparing the result with(12.18).One therefore obtains: ck1=dM-1:Mk,k =m,f (12.22) where I is the fourth-rank identity tensor.Substituting (12.17)and (12.20) into (12.21)and simplifying,one obtains: E=(cm Mm-:Em Am+oM:Ef A:M-T (12.23) Finally,one substitutes (12.22)into (12.23)and simplifies to obtain: E=M-1:(mm:Am+E时：Af):M-T (12.24) It is clear that the above equation is the same as (12.12b).Therefore, both the overall and local approaches yield the same elasticity tensor in the damaged composite system. Equation(12.24)can be generalized to an elastic composite system with n constituents as follows: E=M- *E: :M- (12.25) The two formulations of the overall and local approaches can be used to obtain the above equation for a composite system with n constituents.The derivation of(12.25)is similar to the derivation of(12.24)-therefore it is not presented here and is left to the problems. In the remaining part of this section,some additional relations are pre- sented to relate the overall damage effect tensor with the constituent damage effect tensors.Substituting(12.3)into (12.5)and simplifying,one obtains the constraint equation for the stress concentration tensors.The constraint equa- tion is generalized as follows: ∑B=14 (12.26) k= where 14 is the fourth-rank identity tensor.To find a relation between the stress concentration tensors in the effective and damaged states,one substi- tutes (12.8)and (12.13)into (12.3)and simplifies to obtain: ok=Bk:0,k=1,2,3,,n (12.27) where B*is the fourth-rank stress concentration tensor in the damaged con- figuration and is given by: B=Mk-:k:M,k=1,2,3,n (12.28)202 12 Introduction to Damage Mechanics of Composite Materials First, one needs to find a damage transformation equation for the volume fractions. This is performed by substituting (12.8) and (12.13) into (12.5), simplifying and comparing the result with (12.18). One therefore obtains: ckI4 = ¯ckM−1 : Mk , k = m, f (12.22) where I4 is the fourth-rank identity tensor. Substituting (12.17) and (12.20) into (12.21) and simplifying, one obtains: E =  cmMm−1 : E¯m : A¯m + cfMf−1 : E¯f : A¯f  : M−T (12.23) Finally, one substitutes (12.22) into (12.23) and simplifies to obtain: E = M−1 :  c¯ mE¯m : A¯m + ¯cfE¯f : A¯f  : M−T (12.24) It is clear that the above equation is the same as (12.12b). Therefore, both the overall and local approaches yield the same elasticity tensor in the damaged composite system. Equation (12.24) can be generalized to an elastic composite system with n constituents as follows: E = M−1 : n k=1 c¯ kE¯k : A¯k  : M−T (12.25) The two formulations of the overall and local approaches can be used to obtain the above equation for a composite system with n constituents. The derivation of (12.25) is similar to the derivation of (12.24) – therefore it is not presented here and is left to the problems. In the remaining part of this section, some additional relations are pre￾sented to relate the overall damage effect tensor with the constituent damage effect tensors. Substituting (12.3) into (12.5) and simplifying, one obtains the constraint equation for the stress concentration tensors. The constraint equa￾tion is generalized as follows: n k=1 c¯ kB¯k = I4 (12.26) where I4 is the fourth-rank identity tensor. To find a relation between the stress concentration tensors in the effective and damaged states, one substi￾tutes (12.8) and (12.13) into (12.3) and simplifies to obtain: σk = Bk : σ, k = 1, 2, 3,...,n (12.27) where Bk is the fourth-rank stress concentration tensor in the damaged con- figuration and is given by: Bk = Mk−1 : B¯k : M, k = 1, 2, 3,...,n (12.28)