Variational principles and generalized Hill's bounds in micromechanics of linear peridynamic random structure composites

We consider a static problem for statistically homogeneous matrix linear peridynamic composite materials (CMs). The basic feature of the peridynamic model considered is a continuum description of a material behavior as the integrated non-local force interactions between infinitesimal particles. In contrast to these classical local and non-local theories, the peridynamic equation of motion introduced by Silling (J Mech Phys Solids 2000; 48: 175-209) is free of any spatial derivatives of displacement. Estimation of effective moduli of peridynamic CMs is performed by generalization of some methods used in locally elastic micromechanics. Namely, the admissible displacement and force fields are defined. The theorem of work and energy, Betti's reciprocal theorem, and the theorem of virtual work are proved. Principles of minimum of both potential energy and complimentary energy are generalized. The strain energy bounds are estimated for both the displacement and force homogeneous volumetric boundary conditions. The classical representations of effective elastic moduli through the mechanical influence functions for elastic CM are generalized to the case of peridynamics, and the energetic definition of effective elastic moduli is proposed. Generalized Hill's bounds on the effective elastic moduli of peridynamic random structure composites are obtained. In contrast to the classical Hill's bounds, in the new bounds, comparable scales of the inclusion size and horizon are taken into account that lead to dependance of the bounds on both the size and shape of the inclusions. The numerical examples are considered for the 1D case.