Non-Linear Elastic Bodies Exhibiting Limiting Small Strain

In this paper we develop a constitutive model for describing the elastic response of solids that does not stem from either classical Cauchy or Green elasticity. In contrast to the classical theory, we show that it is possible to obtain a constitutive model wherein the linearized strain is related to the stress in a non-linear manner. The specific choice that we make allows for the stress to be arbitrarily large while the strain remains small (consistent with the assumption used in the linearization of the non-linear strain) or below some limiting value. Such models are worth investigating in detail as they have relevance to problems involving cracks as well as other problems wherein one finds strain singularities within the classical theory of linearized elasticity, and to models that exhibit limited stretch.