STATIC EQUIVALENT MEDIA

When we wish to calculate a static equivalent homogeneous medium we must consider exactly what we mean by an equivalent medium. In the paper by Schoenberg and Muir the following requirements were given. The equivalent medium should match the following properties of the stack of layers,

1.

Thickness

2.

Mass

3.

Displacement of top of stack in response to a given surface traction.

4.

Total force acting across planes perpendicular to the layers.

In static equilibrium we have the additional property that the continuous stress and strain components ${\sigma}_{N}$ and ${\epsilon}_{T}$ are constant throughout the stack.

In the following equations the properties of the equivalent medium will be denoted by the superscript equiv. The layered medium can be a general layered medium whose properties vary continuously as a function of z.

Condition (1) is satisfied by the equation,

$$z^{equiv} = \int dz$$ (7)

Condition (2) is satisfied if,

$$\rho^{equiv} \int dz = \int \rho(z) dz$$ (8)

The total displacement of the top of the stack is given by the integral of the normal strains over the thickness of the stack. The total force acting perpendicular to the layering is given by the integral of the tangential stresses over the thickness of the stack. I.e. to satisfy conditions (3) and (4) we must preserve the integral of the discontinuous components over the thickness of the stack.

$$\int \pmatrix{{ {\sigma}_{T} }^{equiv} \cr { {\epsilon}_{N}... ... = \int \pmatrix{ {\sigma(z)}_{T} \cr {\epsilon(z)}_{N} \cr} dz$$ (9)

Since the normal components of the stresses and the tangential components of the strains are constant within the medium, we are able to obtain the integrated quantities, as an integral over the material properties with the stresses and stains pulled out of the integral.

$$\int \pmatrix{ {\sigma(z)}_{T} \cr {\epsilon(z)}_{N} \cr} dz... ...z)}_{NN} \cr} dz \pmatrix{ {\epsilon}_{T} \cr {\sigma}_{N} \cr}$$ (10)

The homogeneous equivalent medium has constant medium properties so that,

$$\int \pmatrix{{ {\sigma}_{T} }^{equiv} \cr { {\epsilon}_{N} ... ... }^{equiv} \cr } \pmatrix{ {\epsilon}_{T} \cr {\sigma}_{N} \cr}$$ (11)

These conditions on the behavior of the homogeneous equivalent medium give a unique solution for that medium.

$$\begin{eqnarray} z^{equiv} & = & \int dz \nonumber \\ \rho^{equiv} & = & \frac{... ...{} }^{equiv} & = & \frac{1}{z^{equiv}} \, \int {{\bf X}(z)}_{} dz \end{eqnarray}$$ (12)