On a class of non-dissipative materials that are not hyperelastic

The purpose of this brief note is to develop fully Eulerian, implicit constitutive equations for the mechanical response of a class of materials that do not dissipate mechanical work in any process. We show that such materials can be modelled by obtaining a form for the Helmholtz potential as a function of the current mass density, the Cauchy stress and certain other parameters that capture anisotropic response. The resulting constitutive equations are of the form , where and are functions of the state variables of the system. The class of materials that can be obtained from such a constitutive relation is considerably more general than conventional Green-elastic hyperelastic materials. Such response functions may be suitable for the modelling of biological tissue where, due to the constant remodelling that takes place, there may be no physical meaning to a ‘reference configuration’.