Effective nonlocal behavior of peridynamic random structure composites subjected to body forces with compact support and related prospective problems

Abstract

We consider a static problem for statistically homogeneous matrix linear bond-based peridynamic composite materials (CMs) subjected to body force with compact support. Estimation of the effective displacements is performed by the exploitation of the most popular tools and concepts used in conventional local elasticity of composite materials (CMs) with their adaptation to peridynamics. The method is based on estimation of a perturbator introduced by one inclusion inside the infinite peristatic matrix subjected to the body force. The statistical averages of both the displacements and stresses are estimated by summation of these perturbators for all possible locations of inclusions. It leads to obtaining of a new general integral equation (GIE) where a renormalizing procedure is not required for the involved absolutely convergent integrals. It allows us to estimate not only the effective nonlocal constitutive equation but also to evaluate the statistical field averages (inhomogeneous and nonlocal) inside the phases at the fine scale that is critically important for advanced modeling (e.g. for any nonlinear phenomena potentially considered in the future). In particular, in the generalized effective field method (EFM) proposed, the effective field is evaluated from self-consistent estimations using closing of a corresponding integral equation in the framework of the quasi-crystalline approximation. In so doing, the classical effective field hypothesis is relaxed, and the hypothesis of the ellipsoidal symmetry of the random structure of CMs is not used. The method proposed is a promising ingredient at the construction of provably robust data-driven surrogate equations describing both the effective behavior and effective displacement concentration operator inside inclusions. Some opportunities for generalizations of the method proposed are considered. Numerical results for the estimation of effective deformations are obtained for one-dimensional (1D) statistically homogeneous peridynamic composite bar with the prescribed self-equilibrated body forces.

Second Moment of Displacement State and Effective Energy-Based Criteria in Peridynamic Micromechanics of Random Structure Composites

“…The mentioned unique scheme is applicable for CMs of either statistically homogeneous or inhomogeneous (functionally graded) structures, with the phases described by a wide class of constitutive laws (see [22]) such as the local models (elasticity, conductivity, coupled physical phenomena), weakly nonlocal theories (covering higher-grade models, higher-order models, and micromorphic models), and strongly nonlocal models (strain-type and displacement type, peridynamics) and subjected to either homogeneous or inhomogeneous loading (such as e.g. inhomogeneous body forces, see [22,24,25]). In particular, a formally slight modification (which is simultaneously a conceptual one in essence) of the scheme [15] in the present paper is described at the end of Subsection 6.2.…”

“…Equations ( 22)- (24) were presented for pure mechanical loading (at β ≡ 0) resulting the structural displacements, u(x). The basic equations of thermoelasticity can be obtained from the peridynamic counterpart of Eq.…”