ON HIGHER-ORDER GRADIENT CONTINUUM-THEORIES IN 1-D NONLINEAR ELASTICITY - DERIVATION FROM AND COMPARISON TO THE CORRESPONDING DISCRETE MODELS

Higher order gradient continuum theories have often been proposed as m odels for solids that exhibit localization of deformation (in the form of shear bands) at sufficiently high levels of strain. These models i ncorporate a length scale for the localized deformation zone and are e ither postulated or justified from micromechanical considerations. Of interest here is the consistent derivation of such models from a given microstructure and the subsequent comparison of the solution to a bou ndary value problem using both the exact microscopic model and the cor responding approximate higher order gradient macroscopic model. In the interest of simplicity the microscopic model is a discrete periodic n onlinear elastic structure. The corresponding macroscopic model derive d from it is a continuum model involving higher order gradients in the displacements. Attention is focused on the simplest such model, namel y the one whose energy density involves only the second order gradient of the displacement. The discrete to continuum comparisons are done f or a boundary value problem involving two different types of macroscop ic material behavior. In addition the issues of stability and imperfec tion sensitivity of the solutions are also investigated.