Effective properties and local fields

Local fields

The projector $p$ or its supplement $q$ completely determines the properties of a laminate; $q$ depends on the normal $n$ and the type of the equilibrium (see Table 1)

To compute the local fields, we mention that the fields in the layers linearly depend on the mean field ${ v}_{0}$:

$${ v}_{1}=K_{1 }{ v}_{0}, \quad { v}_{2}=K_{2 }{ v}_{0}.$$

where $K_{1}, K_{2}$ are the matrix coefficients of magnification of the mean field in the layers.

Conditions 4 and 5 allow for determination of $K_{1}, K_{2}$:

$$K_{1} = I -m_{2}{ q} { M}, \quad K_{2}= I +m_{1}{ q} { M}.$$ (6)

where

$${ M} =[ { q}^{T} (m_{1} D_{2}+ m_{2} D_{1}) { q} ] ^{-1} { q}^{T}(D_{2}- D_{1})$$ (7)

Effective properties tensor

Knowing the fields in the layers, one easily calculates the effective properties tensor. Substituting 6 into 2, one obtains the formula for the effective properties tensor $D_{\mbox{\footnotesize {lam}}}$ of a laminate:

$$D_{\mbox{\footnotesize {lam}}}=m_{1} D_{1}+m_{2} D_{2} - m_ {1} m_{2} (D_{2}- D_{1}){ q} { M}$$ (8)

This formula for the effective properties of laminates is valid for any linear equilibrium.

Symbolically, we rewrite the representation 8 as a mapping $L$

$$D_{\mbox{\footnotesize {lam}}}={ L}(D_{1},D_{2},m_{1},{ n})$$ (9)

than links the tensor of effective properties $D_{\mbox{\footnotesize {lam}}}$ with the tensors the parent materials $D_{1},D_{2}$, and the parameters of the structure: volume fraction $m_{1}$ and normal ${ n}$ to layers. Go the examples in the Workshop