On a New Class of Models in Elasticity

Recently, Rajagopal and co-workers have shown (see Rajagopal [1], Rajagopal and Srinivasa [2],[3], Bustamante and Rajagopal[4], Rajagopal and Saccomandi [5]) that if by an elastic body one means a body that is incapable of dissipation, then the class of such bodies is far larger than either Green elastic or for that matter Cauchy elastic bodies as one could model elastic bodies using implicit constitutive relations between the Cauchy stress and the deformation gradient or implicit constitutive relations that are rate equations involving the Piola-Kirchhoff stress and the Green-St.Venant Strain (see Rajagopal and Srinivasa [2]). Such a generalized framework allows one to develop models whose linearization with regard to the smallness of the displacement gradient allows one to obtain models that have limited linearized strains even while the stresses are very large. Such a possibility has important consequences to problems which, within the context of the classical linearized theory, leads to singularities. In this short paper, we illustrate the implications of such models by considering simple problems within the context of a specific model belonging to the general class, wherein the strains remain small as the stresses tend to very large values.