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

Abstract

Abstract. We present a model of the full thermo-mechanical evolution of a shape memory body undergoing a uniaxial tensile stress. The well-posedness of the related quasi-static thermo-inelastic problem is addressed by means of hysteresis operators techniques. As a by-product, details on a time-discretization of the problem are provided.Mathematics Subject Classification. 74N30, 74C05, 35K55.

Existence and time-discretization for the finite-strain Souza–Auricchio constitutive model for shape-memory alloys

“…where H stands for the classical Heaviside function. Further details on this issue are to be found in Krejci-Stefanelli09,Krejci-Stefanelli10 [27,28].…”

“…The Souza-Auricchio model has a sound mathematical foundation: it has been analyzed from the viewpoint of existence and approximation of solutions of the isothermal threedimensional quasi-static evolution problem in ams [3] and convergence rates for space-time discretizations mpps,mpps2 [38,39]. Results in the non-isothermal case have also been obtained els2,Krejci-Stefanelli09,Krejci-Stefanelli10,mpp,mpp2 [18,27,28,40,37]. Moreover, extensions of the original model to residual plasticity c3,auricchio07b,els [7,8,17], more realistic non-symmetric behaviors and transformation-dependent material parameters ars2 [9], and the ferromagnetic shape-memory effect paperalpha,gamm,bs,bks,hmm [1,2,11,12,47] are also available.…”

“…This perspective is followed in els2,mpp,mpp2 [18,40,37] in the small-strain regime. On the contrary, a fully thermomechanically coupled problem for the small-strain model is solved in one spatial dimension in Krejci-Stefanelli09,Krejci-Stefanelli10 [27,28]. Note that the smoothness on the given teperature here assumed is weaker than the corresponding one from mpp,mpp2 [40,37] exactly in the same spirit as in els2 [18].…”

Quasistatic Isothermal Evolution of Shape Memory Alloys

“…The ATL model is scalar: the actual phase structure of the material is completely encoded in the scalar internal variable ξ and the martensitic reorientation phenomenon is described in (1.1) by assuming a priori that the relevant direction of ε tr is directly given by the stress σ via F . This is of course a crude simplification with respect to the truly tensorial nature of superelasticity in polycrystals, some results in the direction of coping with a tensorial internal variable may be found in [5,6,7,8,9,31,32]. Still, the simplicity of the ATL model, its robustness with respect to discretizations [10], and its ability to capture experimental evidence are remarkably promising.…”

Existence and linearization for the Souza-Auricchio model at finite strains

“…In this situation, one can assume that the heat produced in the specimen is immediately transferred to the surrounding environment acting indeed as a time-dependent heat bath. Note that the Souza-Auricchio model with unknown temperature has been addressed in one space dimension only [38,39].…”