EQUIVALENT QUANTITIES

Three basic conservation laws govern the behavior of an inhomogeneous anisotropic medium and its equivalent homogeneous counterpart. An arbitrary region V in three dimensions is shown in Figure . The region is bounded by surface S. The quantities we want to preserve in our equivalent medium are: total energy change, total volume change and total mass.

The volume, shown in Figure , changes its shape in response to the application of some external force on its exterior surface. The total energy during this process is the sum of the elastically stored energy in volume V, the total surface energy on boundary S, the kinetic energy and the energy provided by sources within the medium:

$$E_{\rm tot} = \oint_V { {{\epsilon}(r)}_{} : {{\sigma}(r)}_{... ...u}(r)}_{} \cdot {{F}(r)}_{} } dS + E_{\rm kin} + E_{\rm source}$$ (1)

where u the displacement vector and F is the vector of force applied to the exterior surface S. The contribution of energy in response to a force acting on the exterior of the medium is calculated by the dot product of displacement and force integrated over the whole exterior surface. The material has a stiffness that may depend on spatial variables. The strain ${{\epsilon}(r)}_{}$ in the medium is proportional to the stress ${{\sigma}(r)}_{}$ present. So part of the energy is elastically stored in the medium. It can be calculated by taking the double dot product of stress and strain and integrating it over the whole volume. In the static case we neglect kinetic energy $E_{\rm kin}$, which is equivalent to waiting until the medium reaches a state of equilibrium. Omitting the $E_{\rm source}$ term implies having no elastic sources within the medium. The total volume change $\delta V$ in response to an application of an external force is  

$$\delta V = \oint_V d\delta V$$ (2)

The total mass m of the medium after deformation is  

$$m = \oint_V \rho~dV$$ (3)

The equivalent homogeneous medium should have the following desired properties. There should be no change in the total mass. The total volume change should be identical to the homogeneous medium. When external forces are applied, the change in the sum of surface and deformation energy should be the same for the homogeneous and heterogeneous medium. On interior boundaries we want welded contacts; internal interfaces are not allowed to slip. This specification is important, since it governs the later choice of boundary decomposition.