Micro-to-Meso Scale Limit for Shape-Memory-Alloy Models with Thermal Coupling

Abstract

Modeling of shape-memory alloys represents a multiscale problem due to occurrence of martensite/austenite phase transformation and a microstructure in the deformation gradient typical for martensitic phase. Inspired by relaxation in static situation, a limit passage between two modeling scales, called micro-and meso-scales, is performed for the corresponding evolution variants with considering activated phase transformation and even thermodynamically consistent thermal coupling. The mesoscopic model captures possible fine spatial oscillations of the deformation gradient by means of gradient Young measures. In particular, the mesoscopic model is justified as a limit from the microscopic scale and existence of its solutions is proved.

A microscopically motivated constitutive model for shape memory alloys: Formulation, analysis and computations

“…Let us note, however, that in case of a larger specimen when the heat equation is included in the governing system, such a regularity is usually not obtained (see e.g. [65, 66]).…”

An experimentally-fitted thermodynamical constitutive model for polycrystalline shape memory alloys

“…where a AM (•) > 0, a MA (•) > 0, and a reo (•) > 0 are given coefficients depending continuously on their arguments; such a form has already been used in, e.g., [9]. Note also that r TOT (π, ξ, θ; •, •, •) is convex and non-negative, and r RI (π, ξ, θ; •, •) is positively homogeneous (of degree 1); this is why r RI is called the rate-independent (part of the) dissipation potential.…”

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

“…The complex behavior of shape-memory materials is nowadays addressed by a menagerie of models [28,44,61] focusing on different scales (atomistic, microscopic with micro-structures [9], mesoscopic with volume fractions, macroscopic [13,64,63]), emphasizing different principles (minimization of stored energy vs. maximization of dissipation, phenomenology vs. rational crystallography and Thermodynamics [62]), considering different structures (single crystals vs. polycrystalline aggregates, possibly including intragranular interaction and viscosity [58]), and having ambition for different ranges of applicability, including electromagnetic coupling [66,67]. A minimal selection of macroscopic thermomechanically coupled models can be found in [3,32,37,43,59,60,70].…”