Basic Representations of New Background of Analytical Micromechanics

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

“…So, the finite element analysis (FEA) in the truncation method is used for the modeling of the infinite medium by an increased size sample. Immediate considerations of the infinite medium were performed by either the boundary integral equation methods [4,5] or the volume integral equation ones [6].…”

“…For statistically homogeneous thermoperidynamic media subjected to homogeneous volumetric macro boundary loading, one proposed the background principles in Buryachenko [14,15] in the framework of the bond-based approach considered. It was proved that the effective behavior of this media is governed by conventional effective constitutive equation as in the local thermoelasticity theory [6]. The general results establishing the links between the effective properties (effective elastic moduli, effective thermal expansion) and the corresponding mechanical and transformation influence functions (do not miss with the influence functions in peridynamics) are obtained using both the decomposition of local fields into the load and residual fields as well as extraction from the material properties a constituent of the matrix properties.…”

“…Wide expansion of the methods of locally elastic micromechanics into nonlocal phenomena [14,15] was supported by a critical generalization of micromechanics. Namely, Buryachenko [20,21] proposed the general integral equation (GIE) of microinhomogeneous media forming the second background of micromechanics (the first one is based on the effective field hypothesis (EFH) proposed by Faraday, Poisson, Mossotti, Clausius, Lorenz, and Maxwell (1830–1880), see for Buryachenko [22]). The critical features of the new GIEs are the absent of a direct dependence of GIE on both the Green function and the constitutive law (either local or nonlocal) without restrictions of the conventional micromechanics (such as acceptance of the EFH and ellipsoidal symmetry of microtopology) that offers opportunities for a fundamental jump in multiscale and multiphysics researches with drastically improved accuracy of local field estimations (even to the point of correction of a sign [22]).…”

“…It was made by the exploitation of the most popular tools and concepts used in conventional elasticity of CMs and adapted to peridynamics. The basic hypotheses of locally elastic micromechanics (in the version of the second background of micromechanics [22]) are generalized to their peridynamic counterparts in Buryachenko [23]. One proposes a generalization of a dilute approximation method to their peridynamic counterpart in the sense that the volume fraction of the particles is small and their mutual interaction can be neglected.…”

“…In such a case, the effective properties of both the peridynamic composites and locally elastic ones are described by a constant tensor of local effective moduli. However, even for locally elastic composites subjected to inhomogeneous loading, the effective deformations are described by a nonlocal (either the differential or integral) operator [6,22,25] relating a statistical average of stresses in the point being considered with a statistical average of strains in the vicinity of this point. However, to the best authors’ knowledge, there are no academic publications analyzing the mentioned effective deformations of locally elastic composites due to prescribed inhomogeneous body forces although the practical importance of this problem is well known, for example, for laser heating of CMs actively used in both the additive manufacturing [26] and military applications [27].…”