Derivation of Higher Order Gradient Continuum Theories in 2,3-d Non-linear Elasticity from Periodic Lattice Models

SOLIDS THAT I:XHIUIT localization of deformation (in the form of shear bands) at sufficiently high levels of strain, are ftrcquently modeled by gradient type non-local constitutive laws. i.e. continuum theories that include higher order deformation gradients. 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 consistent derivation of such models from a given microstructure and the subsequent invcstigation of their localization and stability behavior under finite strains. In the interest of simplicity. the microscopic model is a discrete, periodic, non-linear elastic lattice structure in two or three dimensions. The corresponding macroscopic model is a continuum constitutivc law involving displacement gradients of all orders. Attention is focused on the simplest such model. namely the one whose energy density includes gradients of the displacements only up to the second order. The relation between the ellipticity of the resulting first (local) and second (non-local) order gradient models at finite strains, the stability of uniform strain solutions and the possibility of localized deformation zones is discussed. The investigations of the resulting continuum are done for two different microstructures, 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 tirst microstructure but not with the scc<>nd. Implications for the stability of three-dimensional crystals using realistic interaction potentials are also discussed.