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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 ${ v}_{0}$:

\begin{displaymath}
{ v}_{1}=K_{1
}{ v}_{0}, \quad { v}_{2}=K_{2
}{ v}_{0}.
\end{displaymath}

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}
$:

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

where
\begin{displaymath}
{ M} =[ { q}^{T}
(m_{1} D_{2}+ m_{2} D_{1}) { q} ] ^{-1} { q}^{T}(D_{2}- D_{1})
\end{displaymath} (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:

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

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

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

\begin{displaymath}
D_{\mbox{\footnotesize {lam}}}={ L}(D_{1},D_{2},m_{1},{ n})
\end{displaymath} (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


next up previous
Next: Laminates of a rank Up: Simple laminates Previous: Continuity of field components
Andre Cherkaev
2001-07-31