In this paper we develop a description for the mechanical response of non-dissipative solids. Unlike the classical approach, a relative deformation gradient tensor is introduced and, based on the second law of thermodynamics, the Cauchy stress is derived from a potential function for the relative deformation tensor and the preferred directions associated with the material symmetry possessed by a non-dissipative solid. The requirement of Galilean invariance and the symmetry of the Cauchy stress are then employed to restrict the form of the potential, with the help of invariance theory. Finally, the implications of the formulation are explored by considering special cases of symmetry such as isotropy, transverse isotropy and orthotropy. The difference between the present approach and the standard treatment of elasticity are discussed. © 2006 Elsevier B.V. All rights reserved.