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. I. INTRO~XJCTI~N A FEATURE SHARED BY MANY ductile solids when sufficiently strained, is the transition of their deformation field from a smoothly varying pattern into a highly localized deformation pattern in the form of a “slzenr hand”. This instability phenomenon is local, i.e. it appears at any point whose stresses reach a critical level, and it is modeled in continuum mechanics as a loss ofellipticity in the incremental equilibrium equations of the solid. The characteristic surfaces of the governing equations indicate the position of the localized deformation zones. This approach has been proposed in the context of elasticity by HADAMARD (1903) and subsequently for rate independent elastoplastic solids by THOMAS (196 I), HILL (I 962) and MANDEL (I 966). Numerous works have subsequently concentrated on the study of the localization of deformation’s dependence on the assumed constitutive model. For further information on this subject, the interested reader is referred to KNOWLES and STERNBERC (1977) for elastic materials and RICE (1976) for inelastic ones.