On the nonlinear elastic response of bodies in the small strain range

The classical linearized approximation to describe the elastic response of solids is the most widely used model in solid mechanics. This approximate model is arrived at by assuming that the norm of the displacement gradient is sufficiently small so that one can neglect the square of the norm in terms of the norm. Recent experimental results on Titanium and Gum metal alloys, among other alloys, indicate with unmistakable clarity a nonlinear relationship between the strain and the stress in the range of strain wherein one would have to use the classical linearized theory of elasticity, namely wherein the square of the norm of the strain can be ignored with regard to the value of the strain, leading to a dilemma concerning the modeling of the response, as the classical nonlinear Cauchy elastic model would collapse to the linearized elastic model in this range. A novel and important generalization of the theory of elastic materials has been suggested by Rajagopal in Appl Math 48: 279-319, 2003 and Zeit Angew Math Phys 58: 309-317, 2007 that allows for an approximation wherein the linearized strain can be a nonlinear function of the stress. In this paper, we show how this new theory can be used to describe the new experiments on Titanium and Gum metal alloys and also clarify several issues concerning the domain of application of the classical linearized theory.