SINGLE-PHASE ENERGY MINIMIZERS FOR MATERIALS WITH NONLOCAL SPATIAL DEPENDENCE

We consider a one-dimensional body and assume that the total stored en ergy functional depends not only on the local strain field but also on the spatial average of the strain held over the body, weighted with a n influence kernel. We investigate the problem of minimizing the total stored energy subject to given end displacements, The general existen ce theory for this problem is reviewed. Then, we narrow our study and concentrate on certain fundamental aspects of nonlocal spatial depende nce by restricting our consideration to the case of a convex local ene rgy and an exponential-type influence function for the nonlocal part. We find explicit solutions and show their characteristic properties as a function of the parameter that measures the extent of influence in the nonlocal kernel. We then study in detail the behavior that results when the total stored energy functional loses its coercivity. In this case, issues concerning the local and global stability of extremal fi elds are considered.