A note on the linearization of the constitutive relations of non-linear elastic bodies

Abstract

Within the context of the non-linear theory of Cauchy elastic bodies (hence Green elastic bodies which are a sub-set of Cauchy elastic bodies wherein the stress is derivable from a potential), linearization with regard to the gradient of displacement, in the sense that the squares of the norms of the gradient of displacement can be neglected in comparison tothe norm of the gradient of displacement, leads inexorably to the classical linearized elastic model. It is however common, especially in work related to inelastic bodies, to see expressions for the Cauchy stress as a nonlinear function of the linearized strain. Even though such models are outside the purview of purely elastic response, the nonlinear relationship between the stress and the linearized strain is also often assumed to hold in the elastic range also. While the linearized strain being a nonlinear function of the stress has no basis within the context of the classical Cauchy elasticity theory, we show that a proper justification can be provided for such models within the context of the new class of constitutive relations that have been developed to describe the response of elastic bodies by Rajagopal [19], and these models can be generalized to also describe the inelastic response in the small strain regime.