Statistical homogenization of particulate composites within strain gradient elasticity

Abstract This work aims to generalize the method of conditional moments (MCM), a statistical approach developed for classical elasticity, to the case of strain gradient elasticity and to evaluate the effective properties of random particulate composites within the simplified strain gradient elasticity theory. The problem of finding the effective constants is reduced to a system of stochastic differential equations based on the fundamental equations of linear elasticity, which are solved by the method of conditional moment functions. Closed-form expressions for the effective moduli of a composite with an isotropic matrix and randomly distributed spherical inhomogeneities are derived. The effective properties obtained within strain gradient elasticity depend not only on the properties of constituents, their volume fraction, shape, and distribution of inhomogeneities, but also on their size, which is impossible in the frame of usual classical elasticity. As a numerical example, it is considered a special case of a composite with an isotropic matrix and randomly distributed spherical inhomogeneities, where the materials of the matrix and the inhomogeneities are second gradient media. Variation of the normalized bulk and shear moduli of the particulate composite material as a function of volume fraction of inhomogeneity and as a function of inhomogeneity size is evaluated, analyzed, and compared in the context of other theoretical predictions.