12.4 Final Remarks 201 The above equation implies a decoupling between the elastic energy in the matrix and fibers.Other methods may be used that include some form of coupling but they will lead to complicated transformation equations that are beyond the scope of this book. Substituting (12.13)and (12.15)into (12.1)and simplifying,one obtains: ok =Ek ck,k=m,f (12.16) where the constituent elasticity tensor E is given by: Ek =Mk-1 Ek Mk-T,k=m,f (12.17) Equation(12.16)represents the elasticity relation for the damaged con- stituents.The second step of the formulation involves transforming (12.17) into the whole composite system using the law of mixtures as follows: 0=cmam+cJσf (12.18) where cm and cf are the matrix and fiber volume fractions,respectively,in the damaged composite system.Before proceeding with (12.18),one needs to derive a strain constituent equation similar to(12.4).Substituting (12.10)and (12.15)into (12.4)and simplifying,one obtains: ek =Ak:E,k=m,f (12.19) where the constituent strain concentration tensor A in the damaged state is given by: Ak MkT Ak M-T,k=m,f (12.20) The above equation represents the damage transformation equation for the strain concentration tensor. Finally,one substitutes (12.11),(12.16),and (12.19)into (12.18)and sim- plifies to obtain: E=cm Em Am+clEf A (12.21) Equation(12.21)represents the elasticity tensor in the damaged composite system according to the local approach. 12.4 Final Remarks In this final section,it is shown that both the overall and local approaches are equivalent elastic composites which are considered here.This proof is performed by showing that both the elasticity tensors given in (12.12b)and (12.21)are exactly the same.In fact,it is shown that (12.21)reduces to (12.12b)after making the appropriate substitution.12.4 Final Remarks 201 The above equation implies a decoupling between the elastic energy in the matrix and fibers. Other methods may be used that include some form of coupling but they will lead to complicated transformation equations that are beyond the scope of this book. Substituting (12.13) and (12.15) into (12.1) and simplifying, one obtains: σk = Ek : εk , k = m, f (12.16) where the constituent elasticity tensor Ek is given by: Ek = Mk−1 : E¯k : Mk−T , k = m, f (12.17) Equation (12.16) represents the elasticity relation for the damaged con￾stituents. The second step of the formulation involves transforming (12.17) into the whole composite system using the law of mixtures as follows: σ = cmσm + cfσf (12.18) where cm and cf are the matrix and fiber volume fractions, respectively, in the damaged composite system. Before proceeding with (12.18), one needs to derive a strain constituent equation similar to (12.4). Substituting (12.10) and (12.15) into (12.4) and simplifying, one obtains: εk = Ak : ε, k = m, f (12.19) where the constituent strain concentration tensor Ak in the damaged state is given by: Ak = MkT : A¯k : M−T , k = m, f (12.20) The above equation represents the damage transformation equation for the strain concentration tensor. Finally, one substitutes (12.11), (12.16), and (12.19) into (12.18) and sim￾plifies to obtain: E = cmEm : Am + cfEf : Af (12.21) Equation (12.21) represents the elasticity tensor in the damaged composite system according to the local approach. 12.4 Final Remarks In this final section, it is shown that both the overall and local approaches are equivalent elastic composites which are considered here. This proof is performed by showing that both the elasticity tensors given in (12.12b) and (12.21) are exactly the same. In fact, it is shown that (12.21) reduces to (12.12b) after making the appropriate substitution