VARIATIONAL FORMULATION ON EFFECTIVE ELASTIC MODULI OF RANDOMLY CRACKED SOLIDS

摘要

Formulation of variational bounds for properties of inhomogeneous media constitutes one of the most fundamental parts of theoretical and applied mechanics. The merit of rigorously derived bounds lies in them not only providing verification for approximation methods, but more importantly, serving as the foundation for building up mechanics models. A direct application of classical micromechanics theories to random cracked media, however, faces a problem of singularity due to a zero volume fraction of cracks. In this study a morphological model of random cracks is first established. Based on the morphological model, a variational formulation of randomly cracked solids is developed by applying the stochastic Hashin-Shtrikman variational principle formulated by Xu (}. Eng. Mech., vol. 135, pp. 1180-1188, 2009) and the Green-function-based method by Xu et al. (Comput. Struct, vol. 87, pp. 1416-1426, 2009). The upper-bound expressions are explicitly given for penny-shaped and slit-like random cracks with parallel and random orientations. Unlike previous works, no special underlying morphology is assumed in the variational formulation, and the bounds obtained are applicable to many realistic non-self-similar morphologies.