Estimations of energy-based criteria in nonlinear phenomena in peridynamic micromechanics of random structure composites

The basic feature of the peridynamic model considered is a continuum description of a material behavior as the integrated nonlocal force interactions between infinitesimal particles. In contrast to these classical local and nonlocal 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. For random structure composite materials (CMs) being considered, a linearized theory of bond-based peridynamics was proposed before for multiphase constituents of arbitrary geometry. It is demonstrated that many nonlinear phenomena (such as, e.g. plasticity, damage, and fatigue) in both local micromechanics and peridynamic one are described by energy-based criteria which are, in its turn, depend on the second moments of the local fields. Analyses of these nonlinear phenomena in peridynamics imply the availability of the direct numerical simulation that is only possible for CMs of deterministic structure (e.g. either a finite sample or periodic structure CMs). Consideration of random structure CMs with nonlinear phenomena requires an acceptance of some additional assumptions. The corresponding approaches in local micromechanics are well developed. A goal of the current study is bridging the gap between the linearized peridynamic micromechanics gathering momentum and a wide class of nonlinear phenomena for random structure CMs. It is performed by a straightforward generalization of the concept of the field second moments (used in energy-based criteria) developed before in nonlinear local micromechanics to their peridynamic counterparts. The method proposed is most advantageously applicable due to identity (with an accuracy to notations) of the operator forms of the new general integral equations (GIEs) for both the local micromechanics of random structure CMs and peridynamic one.