Continuum Mech. Thermodyn. (2000) 12: 423–433 c  Springer-Verlag 2000 Long-time behavior for the full one-dimensional Fr´ emond model for shape memory alloys Pierluigi Colli 1,∗ , Philippe Laurenc¸ot 2,∗∗ and Ulisse Stefanelli 1,∗∗∗ 1 Dipartimento di Matematica, Universit` a di Pavia, via Ferrata 1, 27100 Pavia, Italy 2 CNRS & Institut Elie Cartan, Universit´ e de Nancy I, BP 239, 54506 Vandoeuvre-les-Nancy cedex, France Received July 6, 2000 This note deals with a well-posed one-dimensional initial-boundary value problem related to the Fr´ emond thermomechanical model of structural phase transitions in shape memory materials. The long-time behavior is investigated and it is proved that the ω−limit set in a suitable topology only contains stationary states. 1 Introduction This paper is concerned with the following system of partial differential equations in terms of the unknown functions ϑ, u , χ 1 , and χ 2 , ∂ t ( c 0 ϑ − L χ 1 ) + ∂ t ( (α(ϑ) − ϑα  (ϑ)) χ 2 u x ) − h ϑ xx = α(ϑ) χ 2 u xt , (1.1) ∂ x ( u x + βα(ϑ) χ 2 ) = 0, (1.2) µ∂ t  χ 1 χ 2  − λ∂ xx  χ 1 χ 2  +  (ϑ − ϑ ∗ ) α(ϑ)u x  + ∂ I K ( χ 1 , χ 2 )   0 0  , (1.3) a.e. in Q T := (0, 1) × (0, T ), where T stands for some reference time, and c 0 , L, h , β, µ, λ, , and ϑ ∗ are positive parameters. Here, ∂ I K denotes the subdifferential of the indicator function of a non-empty, bounded, convex and closed subset K of R 2 , while α : R → R is a smooth function with suitable properties to be specified later. The nonlinear system (1.1)–(1.3) has been proposed by M. Fr´ emond [7, 9, 10, 11, 12] in order to describe the macroscopic thermomechanical evolution of a shape memory body. The latter is a metallic alloy, which exhibits this surprising behavior: it could be permanently deformed (avoiding fractures) and consequently be forced to recover its original shape just by thermal means. Indeed, in the microscopic scale, such phenomenon has been interpreted (see, e.g., [1]) as the effect of a structural phase transition (solid-solid) between two (or more) variants of the metallic lattice, due to thermomechanical treatments. In this concern, ϑ has to be regarded as the absolute temperature of the sample, while u stands for the one-dimensional longitudinal displacement and χ 1 , χ 2 are related to the pointwise proportions of the phases. Namely, the system (1.1)–(1.3) results from the coupling of the universal conservation laws for energy and momentum (the latter in a quasi-stationary form, since the term u tt is omitted) with a variational inequality describing the evolution of the phases. Let us point out that α(ϑ) represents the thermal expansion of the system and λ is a diffusion parameter related to the phase dissipation. Moreover, we refer to [7] for the physical meaning of the constants c 0 , L, h ,β,µ,, and ϑ ∗ . ∗ e-mail: pier@dragon.ian.pv.cnr.it ∗∗ e-mail: laurenco@iecn.u-nancy.fr ∗∗∗ e-mail: ulisse@dimat.unipv.it