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

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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.