Effective properties of thermoperistatic random structure composites: Some background principles

Abstract

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.

Modeling of One Inclusion in the Infinite Peristatic Matrix Subjected to Homogeneous Remote Loading

“…were S stands for the unit sphere and d m denotes a differential solid angle on S in the direction of any unite vector m. Lehoucq and Silling [32] proved (26) for 3D case while the case d = 2 [13] can be justified in a similar manner. Equation 26 at d = 1 can also be reduced to the representations [58,62] by the variable exchange x−z → r, x + y → s. Indeed, the origin-centered unit 1D "sphere" is the set {-1 , 1}, which has a measure of 2.…”

“…Peridynamics is a continuum theory where each infinitesimal volume interacts with an infinite numbers of other volumes within the horizon. However, in numerical implementation described by Silling and Askari [54], the region w is discretized into a set of nodes p, each with a finite known volume (called full volume)V p = h d defined by the size h. Taken together, the nodes form a grid x p with the total number p ∈ [1, N max ] of nodes covering the total macrovolume w. The spatially discretized form of the equilibrium (6) and (13) replaces the integral by the finite sum for each node p…”

“…The simulation is stopped when the tolerance reaches 0.1%. We control the variations of u ρ only over the domain v l := v ∪ v because according to [10,13] the effective moduli depend on the strain distribution u(x) only in the domain…”

“…Buryachenko [10,13] established similarity of the general integral equations for both locally elastic CMs and peristatic ones that opens the opportunities to straightforward generalization of their solutions for locally elastic CMs to the peristatic ones. These models contain as a necessary ingredient the solution for one inclusion in the infinite matrix subjected to the homogeneous effective field (called basic problem).…”

“…Immediate consideration of the infinite medium can be performed by the Green function (see [61,63]) technique which is reduced to the volume integral equation method (VIEM) well developed in the theory of locally elastic CMs (see for references [8]). Introduction of the micropolarization tensor by Buryachenko [10,13] (generalizing the notion of polarization tensor in local elasticity, see, e.g., [8]) provides a great scope for generalization of the VIEM to its peristatic counterpart that can be easily performed as a straightforward exploitation of the method [9] proposed for solution of a corresponding basic problem with nonlocally elastic (in the sense of Eringen [23]) properties of constituents. However, this version of the VIEM is not considered in the current publication in more details.…”