DERIVATION OF HIGHER-ORDER GRADIENT CONTINUUM-THEORIES IN 2,3-D NONLINEAR ELASTICITY FROM PERIODIC LATTICE MODELS

SOLIDS THAT EXHIBIT localization of deformation (in the form of shear bands) at sufficiently high levels of strain. are frequently modeled b y gradient type non-local constitutive laws. i.e. continuum theories t hat include higher order deformation gradients. These models incorpora te a length scale for the localized deformation zone and are either po stulated or justified from micromechanical considerations. Of interest here is the consistent derivation of such models from a given microst ructure and the subsequent investigation of their localization and sta bility behavior under finite strains. In the interest of simplicity. t he microscopic model is a discrete. periodic. non-linear elastic latti ce structure in two or three dimensions, The corresponding microscopic model is a continuum constitutive law involving displacement gradient s of all orders. Attention is focused on the simplest such model. name ly the one whose energy density includes gradients of the displacement s only up to the second order. The relation between the ellipticity of the resulting first (local) and second (non-local) order gradient mod els at finite strains. the stability of uniform strain solutions and t he possibility of localized deformation zones is discussed. The invest igations of the resulting continuum are done for two different microst ructures. the second one of which approximates the behavior of perfect monatomic crystals in plane strain. Localized strain solutions based on the continuum approximation are possible with the first microstruct ure but not with the second. Implications for the stability of three-d imensional crystals using realistic interaction potentials arc also di scussed.