Nonlocal Mechanics in the Framework of the General Nonlocal Theory

Abstract

A continuum is stable if and only if its body and inertia forces are balanced globally with surface tractions acting on it. The global balance requirement automatically postulates a local balance if a material particle interacts with its direct neighbors. However, when the interaction of the particle is nonlocal and extended to interactions between all particles of the continuum, the local balance is violated, and the material is still stable only if the restrictive global balance is satisfied. The latter continuum behavior can be described by a nonlocal theory. The nonlocal theory developed by Eringen postulates the global balance requirement by representing the stress field depends on the properties of the whole continuum. Eringen assumed a single nonlocal kernel function for all the material coefficients. Here, we show that utilizing a single attenuation function puts limits to the nonlocal model where it would break down when applied to materials operate at high frequency ranges. We also show that Eringen’s nonlocal model is limited for slowly varying acoustic waves and low frequencies. To exceed this limit, the general nonlocal theory is presented. The general nonlocal theory outweighs Eringen’s nonlocal theory for considering different nonlocal kernels for the different material coefficients. The mechanics of particles, the mechanics of linear elastic continua, and the materials dispersion are discussed in the framework of the general nonlocal theory. In addition, the reduction of the general nonlocal theory to the strain gradient theory is interpreted. In addition, we indicate that the strain gradient theory can capture the same phenomena as the general nonlocal theory as long as various strain gradients are considered.