For random media of arbitrary microstructure, exact analytical solutions of the effective properties are unattainable, and so any rigorous statement about the effective properties must be in the form of rigorous bounds. To get variational bounds on effective properties, one must first express the effective parameter in terms of some functional and then formulate an appropriate variational (extremum) principle for the functional. We shall primarily deal with “energy” functionals. Once the variational principle is established, then specific bounds on the property of interest are obtained by constructing trial, or admissible, fields that conform with the variational principle. Specific bounds derived from trial fields are the subject of Chapter 21. In this chapter we will derive variational principles for the effective conductivity, effective elastic moduli, trapping constant, and fluid permeability.
KeywordsVariational Principle Effective Conductivity Energy Representation Stiffness Tensor Fluid Permeability
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