An effective-medium theory for the electrical conductivity of Ohmic dispersions taking explicit account of particle shape and spatial distribution independently is available from the work of Ponte Castaneda and Willis [J Mech Phys Solids 43:1919–1951, 1996]. When both shape and distribution take particular “ellipsoidal” forms, the theory provides analytically explicit estimates. The purpose of the present work is to evaluate the predictive capabilities of these estimates when dispersions exhibit dissimilar particle shape and distribution. To this end, comparisons are made with numerical bounds for coated ellipsoid assemblages computed via the finite element method. It is found that estimates and bounds exhibit good agreement for the entire range of volume fractions, aspect ratios, and conductivity contrasts considered, including those limiting values corresponding to an isotropic distribution of circular cracks. The fact that the explicit estimates lie systematically within the numerical bounds hints at their possible realizability beyond the class of isotropic dispersions.

中文翻译：

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具有不同颗粒形状和分布的分散体的电导率的显式估计与数值界限

从 Ponte Castaneda 和 Willis [J Mech Phys Solids 43:1919–1951, 1996] 的工作中可以获得独立地明确考虑颗粒形状和空间分布的欧姆分散体电导率的有效介质理论。当形状和分布都采用特定的“椭球”形式时，该理论提供了分析明确的估计。当前工作的目的是评估当分散体表现出不同的颗粒形状和分布时这些估计的预测能力。为此，通过有限元方法计算的涂层椭球组合的数值界限进行了比较。发现估计值和界限在考虑的整个体积分数、纵横比和电导率对比范围内表现出良好的一致性，包括那些与圆形裂纹各向同性分布相对应的极限值。显式估计系统地位于数值范围内的事实暗示了它们可能超出各向同性色散类别的可实现性。