200 12 Introduction to Damage Mechanics of Composite Materials 12.3 Local Approach In this approach,damage is introduced in the first step of the formulation using two independent damage tensors for the matrix and fibers.However, more damage tensors may be introduced to account for other types of damage such as debonding and delamination.The two steps involved in this approach are shown schematically in Fig.12.2.One first introduces the fourth-rank matrix and fiber damage effect tensors Mm and M,respectively,as follows: M M Fig.12.2.Schematic diagram illustrating the local approach for composite materials ak=Mk ok,k=m,f (12.13) The above equation can be interpreted in a similar way to (12.8)except that it applies at the constituent level.It also represents the damage trans- formation equation for each constituent stress tensor.In order to derive a similar transformation equation for the constituent strain tensor,one applies the hypothesis of elastic energy equivalence to each constituent separately as follows: =产：， k=m,f (12.14) In using(12.14),one assumes that there are no micromechanical or con- stituent elastic interactions between the matrix and fibers.This assumption is not valid in general.From micromechanical considerations,there should be interactions between the elastic energies in the matrix and fibers.How- ever,such interactions are beyond the scope of this book as the resulting equations will be complicated and the sought relations may consequently be unattainable.It should be clear to the reader that (12.14)is the single most important assumption that is needed to derive the relations of the local ap- proach.It will also be needed later when we show the equivalence of the over- all and local approaches.Therefore,the subsequent relations are very special cases when (12.14)is valid. Substituting(12.13)into (12.14)and simplifying,one obtains the required transformations for the constituent strain tensor as follows: 本=Mk-r:ek,k=m,f (12.15)200 12 Introduction to Damage Mechanics of Composite Materials 12.3 Local Approach In this approach, damage is introduced in the first step of the formulation using two independent damage tensors for the matrix and fibers. However, more damage tensors may be introduced to account for other types of damage such as debonding and delamination. The two steps involved in this approach are shown schematically in Fig. 12.2. One first introduces the fourth-rank matrix and fiber damage effect tensors Mm and Mf , respectively, as follows: Fig. 12.2. Schematic diagram illustrating the local approach for composite materials σ¯k = Mk : σk , k = m, f (12.13) The above equation can be interpreted in a similar way to (12.8) except that it applies at the constituent level. It also represents the damage trans￾formation equation for each constituent stress tensor. In order to derive a similar transformation equation for the constituent strain tensor, one applies the hypothesis of elastic energy equivalence to each constituent separately as follows: 1 2 εk : σk = 1 2 ε¯ k : ¯σk , k = m, f (12.14) In using (12.14), one assumes that there are no micromechanical or con￾stituent elastic interactions between the matrix and fibers. This assumption is not valid in general. From micromechanical considerations, there should be interactions between the elastic energies in the matrix and fibers. How￾ever, such interactions are beyond the scope of this book as the resulting equations will be complicated and the sought relations may consequently be unattainable. It should be clear to the reader that (12.14) is the single most important assumption that is needed to derive the relations of the local ap￾proach. It will also be needed later when we show the equivalence of the over￾all and local approaches. Therefore, the subsequent relations are very special cases when (12.14) is valid. Substituting (12.13) into (12.14) and simplifying, one obtains the required transformations for the constituent strain tensor as follows: ε¯ k = Mk−T : εk , k = m, f (12.15)