# Basic Concepts in Nonlocal Approaches

13.1.1 The Early Approaches

The concept of nonlocal continuum for materials with a randomly heterogeneous microstructure was originally conceived and extensively studied for elastic materials (Eringen 1965, 1966; Kroner 1967; Levin 1971; Kunin 1968; Eringen and Edelen (1972); Eringen and Ari 1983). For such materials, the constitutive relation is considered as a relation between the average continuum stress tensor cf(x) and strain tensor є(х), which are defined as the statistical averages of the randomly scattered microstresses over a suitable representative volume of the material centered at point x (Fig. 13.1.1a – b).

Intuitively, the justification for nonlocal averaging may be explained by Fig. 13.1.1b, showing a representative volume of the material with an aggregate microstmcture. (The representative volume is, in the statistical theory, defined as the smallest volume for which the statistics of the microstructure are not changed by shifting the volume.) The formation of a crack in the center of this element obviously does not depend only on the continuum strain at the center of the crack, but on the overall deformation of this representative volume, which determines the strain energy content and thus the energy release from this volume.

The simplest way to introduce a nonlocal strain measure is to define the average strain tensor as

є(х) •■= —j-r / a{x-s)e{s)dV(s) = [ a(x, s)s[s)dV(s) (13.1.1).

Er(X) Jv Jv

in which є(х) is the usual (local) strain tensor at point x, V the volume of the structure, and a(r) is a scalar function of the distance r — – x — sj between the point at which the average is taken and the point contributing to that average; Vr is a normalizing factor introduced so that, for uniform strain, the average is also uniform and coincides with the local value. It is a simple matter to find the required relationships