Effective properties of thermoperistatic random structure composites: Some background principles

The basic feature of the peridynamic model considered here is a continuum description of a material’s behavior as the integrated nonlocal force interactions between infinitesimal particles. In contrast to 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. A theory of thermoelastic composite materials (CMs) with nonlocal thermoperistatic properties of multiphase constituents of arbitrary geometry is analyzed for statistically homogeneous CMs subjected to homogeneous loading. A generalization of the Hill’s equality to peristatic composites is proved. The classical representations of effective elastic moduli through the mechanical influence functions for elastic CMs are generalized to the case of peristatics, and the energetic definition of effective elastic moduli is proposed. The general results establishing the links between the effective properties (effective elastic moduli, effective thermal expansion) and the corresponding mechanical and transformation influence functions are obtained by the use of the decomposition of local fields into load and residual fields. Effective properties of thermoperistatic CM are expressed through the introduced local stress polarization tensor averaged over the extended inclusion phase. A detected similarity of results for both the peristatic and locally elastic composites is explained fundamentally, as the methods used for obtaining the results widely exploit the Hill’s condition and the self-adjointness of the stress operator. However, the representation of effective properties for composites with both the local thermoelastic and nonlocal thermoperistatic properties do not always coincide. Therefore, the representation of effective eigenfields through mechanical influence functions generalizing Levin’s representation does not in general hold for thermoperistatic CMs; this is demonstrated for a one-dimensional numerical example.