Three-dimensional Landau theory describing the martensitic phase transformation of shape-memory alloys

Abstract

In shape-memory alloys a first-order martensitic phase transition is responsible for pseudo-elastic and for ferro-elastic stress-strain relations. To describe this behaviour a modified Landau theory is proposed in which the free energy of the crystal depends on the temperature and on the full strain tensor. The energy is invariant with respect to the cubic point group Oh of the high-temperature phase. To predict the cubic-to-monoclinic phase transitionofb-phaseshape-memory alloysanexpansion uptosixthorder instrain isnecessary for which, for the class of alloys considered, odd terms may be neglected. For a CuAlNi alloy the expansion coefficients are determined by comparison with experimental results. In contrast to classical Landau theory of second-order phase transitions, not only a single second-order but also a fourth-order expansion coefficient depend on temperature.

Simulation of Nonlinear Thermomechanical Waves with an Empirical Low Dimensional Model

“…The dynamical behavior of the SMA rod is strongly nonlinear with coupling effects between the elastic and thermal fields, first order martensitic transformations, and hysteresis [3,11,10,19]. The simulation is performed for a Au 23 Cu 30 Zn 47 rod of length L = 1cm based on Eq.(1).…”

“…The well known Falk model has been constructed on the basis of conservation laws for linear momentum and energy, and thermo-dynamical consistency. To model the coupled thermo-mechanical wave interactions and the first order phase transitions in the shape memory alloys, we use the following 1D mathematical model [3,11,19]:…”

Well-posedness of a thermo-mechanical model for shape memory alloys under tension

“…Indeed, SMA behavior has been investigated at all scales (microscopic, mesoscopic with volume fractions, macroscopic) and by means of a full menagerie of modelling perspectives. Even restricting to the realm of macroscopic-phenomenological models (which is the focus of this paper), the different modelling options available are many and diversified and the corresponding cross-validation is still under assessment [6,18,19,21,22,27,28,[35][36][37][38][39]42,43]. On the contrary, the mathematical treatment of full thermo-mechanical problems for SMAs is less developed for the only comprehensive results in this sense refer to either the original formulations or modifications of the Frémond [19] and the Falk and Konopka [17,18] models.…”

“…Even restricting to the realm of macroscopic-phenomenological models (which is the focus of this paper), the different modelling options available are many and diversified and the corresponding cross-validation is still under assessment [6,18,19,21,22,27,28,[35][36][37][38][39]42,43]. On the contrary, the mathematical treatment of full thermo-mechanical problems for SMAs is less developed for the only comprehensive results in this sense refer to either the original formulations or modifications of the Frémond [19] and the Falk and Konopka [17,18] models. With no claim of completeness, the reader is referred to [1,2,[12][13][14]24,34,45] and the related references for a comprehensive collection of results.…”

A Dynamic Model for Phase Transformations in 3D Samples of Shape Memory Alloys

“…Our better understanding of such dynamics can be achieved with multi-physics multiscale models which assist the researchers in designing new materials and devices by harnessing the shape memory effect. Phenomenological framework based on Landau theory of martensitic transformations (e.g., [1]) has become a convenient choice for basic building blocks in the computational models. However, most of the phenomenological models are applicable at macroscopic and mesoscopic scales as discussed in [2], and the strain field in such model is often beyond the resolution of bain strain.…”