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 models for solids that exhibit localization of deformation (in the form of shear bands) at sufficiently high levels of strain. 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 comparison of the solution to a boundary value problem using both the exact microscopic model and the corresponding approximate higher order gradient macroscopic model. In the interest of simplicity the microscopic model is a discrete periodic nonlinear elastic structure. The corresponding macroscopic model derived from it is a continuum model involving higher order gradients in the displacements. Attention is focused on the simplest such model, namely the one whose energy density involves only the second order gradient of the displacement. The discrete to continuum comparisons are done for a boundary value problem involving two different types of macroscopic material behavior. In addition the issues of stability and imperfection sensitivity of the solutions are also investigated.