On evolution equations for elastic deformation and the notion of hyperelasticity

For elastic–plastic and elastic–viscoplastic materials it is possible to introduce an evolution equation for an elastic deformation measure. Also, it is possible to develop constitutive equations for which the stress and strain energy are functions of elastic deformation only, the stress is determined by a derivative of the strain energy function and the associated material response is rate-independent and non-dissipative in the absence of the rate of inelasticity. Yet, these equations do not necessarily exhibit hyperelastic response in the elastic range. The objective of this paper is to emphasize the importance of satisfying an additional condition that requires the work done between two configurations to be insensitive to the history and rate of total deformation. This work condition places restrictions on the evolution equation which ensure that the integrated elastic deformation measure is a function of total deformation only. Also, it is argued that there is no need to complicate the evolution equation for elastic deformation to accommodate alternative strain measures since the nonlinearities of these strain measures can be absorbed into the form of the strain energy function.