A thermodynamically consistent model of shape memory alloys in three dimensions is studied. The thermoelasticity system, based on the strain tensor, its gradient and the absolute temperature, generalizes the well-known one-dimensional Falk model. Under simplifying structural assumptions we prove global in time existence and uniqueness of the solution.
A justification of the two-dimensional nonlinear “membrane” equations for a plate made of a Saint Venant-Kirchhoff material has been given by Fox et al. [9] by means of the method of formal asymptotic expansions applied to the three-dimensional equations of nonlinear elasticity. This model, which retains the material-frame indifference of the original three dimensional problem in the sense that its energy density is invariant under the rotations of , is equivalent to finding the critical points...
The constitutive equations of rate type for a class of thermo-hypo-elastic materials are derived within the framework of the extended irreversible thermodynamics.
Bhattacharya and Kohn have used small-strain (geometrically linear) elasticity to analyze the recoverable strains of shape-memory polycrystals. The adequacy of small-strain theory is open to question, however, since some shape-memory materials recover as much as 10 percent strain. This paper provides the first progress toward an analogous geometrically nonlinear theory. We consider a model problem, involving polycrystals made from a two-variant elastic material in two space dimensions. The linear theory...
We prove the existence of global in time weak solutions to a three-dimensional system of equations arising in a simple version of the Fried-Gurtin model for the isothermal phase transition in solids. In this model the phase is characterized by an order parameter. The problem considered here has the form of a coupled system of three-dimensional elasticity and parabolic equations. The system is studied with the help of the Faedo-Galerkin method using energy estimates.