The variational principle for probabilistic measure and Hashin–Shtrikman bounds

The paper is a review of Hashin–Shtrikman type bounds for effective moduli of conductivity and elasticity of polycrystals and composites written from the perspective of the variational principle for probabilistic measure. The results for such bounds are rederived in probabilistic terms. Remarkably, in probabilistic terms the Hashin–Shtrikman approach gets especially simple form. Besides, a clear distinction arises between the basic assumption, the choice of the trial field, and the simplifying assumptions, like geometrical isotropy, physical isotropy, texture isotropy, etc. We filled out several gaps. First, we derive an integral equation to be solved to get the bounds when the simplifying assumptions do not hold. Second, we extend the bounds for polycrystals with the cubic symmetry of crystallites to all thermodynamically possible crystallites; previously such bounds were found for crystallites with special elastic properties. One practical outcome considered is the derivation of approximate formulae for the temperature dependence of effective elastic moduli. Third, for crystallites with non-cubic symmetries, we formulated algebraic variational problems to be solved numerically to obtain the bounds, and solved these problems for several materials.