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.

本文从概率测度变分原理的角度回顾了多晶和复合材料的有效电导率和弹性模量的 Hashin-Shtrikman 型界限。此类界限的结果以概率术语重新推导。值得注意的是，从概率的角度来看，Hashin-Shtrikman 方法的形式特别简单。此外，基本假设、试验场的选择和简化假设，如几何各向同性、物理各向同性、纹理各向同性等之间存在明显的区别。我们填补了一些空白。首先，我们推导出一个积分方程，当简化假设不成立时，需要求解该积分方程以获得界限。其次，我们将具有微晶立方对称性的多晶的范围扩展到所有热力学上可能的微晶；以前发现具有特殊弹性特性的微晶的这种界限。考虑的一项实际结果是推导有效弹性模量与温度的关系的近似公式。第三，对于具有非立方对称性的微晶，我们制定了需要数值求解的代数变分问题以获得界限，并针对几种材料解决了这些问题。