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

Abstract

Bounds are here derived for the effective bulk modulus in heterogeneous media, denoted by k ∗ 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 ∗ k* are derived in terms of the correlation functions ⟨ μ ′ ( 0 ) k ′ ( r ) k ′ ( s ) ⟩ \left \langle {\mu ’(0)k’(r)k’(s)} \right \rangle and ⟨ [ k ′ ( r ) k ′ ( s ) / μ ( 0 ) ] ⟩ \left \langle {\left [ {k’(r)k’(s)/\mu (0)} \right ]} \right \rangle where μ ′ \mu ’ and k ′ k’ are the fluctuating parts of the shear modulus μ \mu and the bulk modulus, k k , respectively. Explicit calculations are given for two phase media when μ ′ ( x ) = 0 \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.