Higher-order gradient continuum models for nonlinear solids with periodic microstructure and their application to failure by localized deformation.

Abstract

Many solids, when sufficiently strained, exhibit a change in deformation pattern from one which is relatively smooth and slowly varying to one which is concentrated in a narrow zone, i.e., localized deformation. This occurs when it is energetically more favorable to intensely deform a small region of the solid than to deform the entire solid more uniformly. The attributes of localized deformation, in particular the zone size, depend strongly on the properties in the small region which is intensely deformed, i.e., the properties of the microstructure. Higher order gradient continuum theories have often been proposed as models for solids that exhibit localization of deformation. These models incorporate a length scale for the localized deformation zone and are either postulated or justified from micromechanical considerations. Of interest here is the derivation of higher order gradient continuum models from the properties of the microstructure and the use of these models to determine the effect of the microstructure on the solid's stability and model localization of deformation. Here only periodic solids are considered. The first part of the study considers models which are discrete, nonlinear elastic, lattice structures. The higher order gradient model is an elastic energy involving second gradients of the displacements. For one dimensional lattices, discrete to continuum comparisons are performed for a boundary value problem involving two different types of macroscopic material behavior. Stability and imperfection sensitivity of the solutions are investigated. In higher dimensions the relation between the ellipticity of the classical and the higher order gradient models at finite strains, the stability of uniform strain solutions, and the possibility of localized deformation is discussed. Implications on the stability of perfect monatomic crystals in plane strain are presented. The second part of the study considers periodic continua. Of interest is the determination of the first instability and its dependence on the underlying structure. Homogenization theory is used to incorporate the microscale size in the stability analysis. The effect of the microstructure on the solid's stability indicates whether the microstructure is stabilizing or destabilizing.