The Souza-Auricchio model for shape-memory alloys

Diego Grandi, Ulisse Stefanelli
2014 Discrete and Continuous Dynamical Systems. Series S  
Shape-memory alloys are active materials, their amazing thermoelectromechanical behavior is at the basis of a variety of innovative applications. Many models have been set forth in order to describe this complex behavior. Among these the so-called Souza-Auricchio model appears as remarkably simple in terms of mechanical assumptions yet accurate in the description of threedimensional experiments and robust with respect to approximations. Our aim is to survey here the current literature on the
more » ... iterature on the Souza-Auricchio model, with a specific focus on modeling. 2010 Mathematics Subject Classification. 74C05, 35Q74, 49J40, 80A17, 74F15. 723 724 DIEGO GRANDI AND ULISSE STEFANELLI ε ε σ σ θ ε h Figure 1. Schematic illustration of the super-elastic (left), shapememory (center), and magnetic effects (right). The amazing behavior of SMAs, originally observed in the '60, has attracted enormous attention ever since. The motivation for such an interest is the unprecedented applicative possibilities offered by SMAs. In particular, these are nowadays exploited in a variety of innovative devices including sensors, actuators, MEMS and in a number of different fields from Biomechanics and Medical Engineering, to Seismic and Aerospace Engineering. The paramount importance of SMAs in applications has triggered an extremely active research activity in the last decades and a whole menagerie of models has been proposed by addressing different alloys (NiTi, CuAlNi, Ni 2 MnGa, among the most important) at different scales (atomistic, microscopic with microstructures, mesoscopic with volume fractions, macroscopic) and emphasizing different principles (minimization of stored energy vs. maximization of dissipation, phenomenology vs. rational crystallography and Thermodynamics) and different structures (single crystals vs. polycrystals and structures) [114] . These models have of course ambitions for different ranges of applicability (from lab single-crystal experiments to commercially exploitable tools) and different abilities to fit particular experiments and to explain microstructures, stress/strain relations, or hysteresis. It is beyond our purposes to even attempt a review of the huge literature on these themes. By restricting to the case of macroscopic thermomechanically coupled systems, which are the most relevant for our purposes, we shall however minimally mention the modeling propositions in [10, 11, 56, 59, 76, 108, 109, 110, 111, 128] . From the more mathematical perspective, a distinguished role is played by the Frémond model [49, 50, 51] and the Falk-Konopka model [46, 47] . These have received extended consideration from the point of view of existence, approximation, and qualitative behavior of solutions. The reader is referred to [27, 31, 32, 52] and [28, 101, 136] and the references therein for a collection of results. This survey is aimed at reporting on a specific macroscopic model originally proposed by Souza, Mamiya, & Zouain [124] and then combined with finite elements by Auricchio & Petrini [13, 14, 15] . We refer to this as the Souza-Auricchio model (SA) in the following. Our interest in the SA model is motivated from one hand by its capability of reproducing the macroscopic behavior of SMAs within a simple variational frame and from the other hand on its amenability to a complete mathematical and numerical discussion. Our survey is intended to highlight the indeed remarkable features of the SA model, to describe its extensions to thermal ferromagnetic and plastic couplings, and to record the available corresponding mathematical results. By reflecting the progressive development of research of the SA model, this survey is divided into two parts. Sections 2-4 are devoted to the introduction and the mathematical settlement of the SA model in its original isothermal setting.
doi:10.3934/dcdss.2015.8.723 fatcat:ntxyxexflzbfbhudc6uz2lvoie