Introduction

Since the mid-seventies there has been a strong and remarkably successful effort to use 3-dimensional nonlinear thermoelasticity theory for modelling the behavior of crystalline solids in the range of finite deformations. Extremely common examples of this behavior are encountered in mechanical twinning and solid state phase transitions with loss of symmetry. The contributions to this subject by a number of authors stem from some earlier works by Ericksen, in which he proposed a theory of material symmetry for nonlinearly elastic solids that can describe crystalline substances having many nontrivially related natural states. His approach to the material symmetry of crystals is, in a sense, more local than the by now standard one of Coleman and Noll (1964), also adopted by Truesdell and Noll (1965); according to them there is a reference configuration such that the related material symmetry group is one of the 32 crystallographic point groups, augmented by the central inversion. This basically extends to nonlinearly elastic crystals the assumption of classical linear elasticity that the invariance of the energy is dictated by the point group describing the geometric symmetry of the underlying crystal lattice in a natural state, that is, an unstressed stable equilibrium state, taken as reference configuration. While this view is universally adopted for determining the independent elastic moduli of crystalline solids of any given symmetry (see Love (1927)), it is too rigid when extended to nonlinear regimes, where material symmetry is not necessarily imposed by the geometric symmetry of any specific reference configuration. The approach of Ericksen (1978) thus aimed at an elastic theory capable of accounting for the common phenomena of crystal mechanics mentioned above. These could hardly be described1 by means of the earlier elastic models, even nonlinear, in which invariance is based on a given point group. Ericksen (1970), (1977), (1980b), (1987) achieved his goal by relying on molecular models to construct the global material symmetry that the constitutive equations of a crystal should exhibit. This invariance group turns out to be an infinite, discrete group in which there are nonorthogonal as well as simple shearing transformations. These notions allow elasticity theory to become flexible enough to deal with the aforemen-

tioned phenomena, which laid for long outside its range. Indeed, some of the nonorthogonal transformations contained in such a large symmetry group play an important role in the description of mechanical twins, in particular transformation twins, as pairwise homogeneous coherent natural states for crystals. This description includes properties of twins considered to be true by the mineralogists, Friedel (1926) and Buerger (1945) among others. On the basis of these ideas and of the earlier calculations of Ericksen (1981a), a twinning equation has been derived which describes the allowed twinning operations and the kinematics of twins. This equation has been studied by many authors and a great deal is now known, although a full explicit classification of natural states is still missing.