Use of classical variational principles to determine bounds for the effective bulk modulus in heterogeneous media

Bounds are here derived for the effective bulk modulus in heterogeneous media, denoted by $k*$, using the two standard variational principles of elasticity. As trial functions for the stress and strain fields we use perturbation expansions that have been modified by the inclusion of a set of multiplicative constants. The first order perturbation effect is explicitly calculated and bounds for $k*$ are derived in terms of the correlation functions $\left \langle {\mu ’(0)k’(r)k’(s)} \right \rangle$ and $\left \langle {\left [ {k’(r)k’(s)/\mu (0)} \right ]} \right \rangle$ where $\mu ’$ and $k’$ are the fluctuating parts of the shear modulus $\mu$ and the bulk modulus, $k$, respectively. Explicit calculations are given for two phase media when $\mu ’(x) = 0$ and when the media are symmetric in the two phases. Results are also included for the dielectric problem when the media are composed of two symmetric phases.