Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



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

Authors: M. Beran and J. Molyneux
Journal: Quart. Appl. Math. 24 (1966), 107-118
DOI: https://doi.org/10.1090/qam/99925
MathSciNet review: QAM99925
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Abstract | References | Additional Information

Abstract: 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.

References [Enhancements On Off] (What's this?)

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Additional Information

DOI: https://doi.org/10.1090/qam/99925
Article copyright: © Copyright 1966 American Mathematical Society

American Mathematical Society