Identifiability of the Landau–Ginzburg Potential in a Mathematical Model of Shape Memory Alloys (original) (raw)
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On the macroscopic free energy functions for shape memory alloys
Journal of the Mechanics and Physics of Solids, 2001
Most of the models for the macroscopic behaviour of shape memory alloys (SMA) rely upon the assumption of equal material properties of the phases while, on the contrary, experiments show signiÿcant di erences. On the basis of the variational formulation of the problem governing the behaviour of a linear elastic heterogeneous material with prescribed eigenstrains, macroscopic free energies for SMA are deÿned taking into account the phase heterogeneity. The general structure and the dependence on the macroscopic state variables of such functions are discussed and formal expressions in terms of proper concentration tensors given. In the case of an underlying two-phase microstructure exact connections between the quantities that determine the free energies (macroscopic transformation strain, interaction energy, e ective thermal expansion tensor) and the e ective elastic compliance are derived. Estimates of the SMA macroscopic free energies based on Reuss, Voigt and Hashin-Shtrikman bounds for the e ective elastic moduli are explicitly calculated and compared in the speciÿc case of a NiTi alloy.
We consider a strongly nonlinear PDE system describing solid-solid phase transitions in shape memory alloys. The system accounts for the evolution of an order parameter (related to different symmetries of the crystal lattice in the phase configurations), of the stress (and the displacement), and of the absolute temperature. The resulting equations present several technical difficulties to be tackled: in particular, we emphasize the presence of nonlinear coupling terms, higher order dissipative contributions, possibly multivalued operators. As for the evolution of temperature, a highly nonlinear parabolic equation has to be solved for a right hand side that is controlled only in L^1. We prove the existence of a solution for a regularized version, by use of a time discretization technique. Then, we perform suitable a priori estimates which allow us pass to the limit and find a weak global-in-time solution to the system.
Inverse Problems, 1998
The nonlinear partial differential equations considered here arise from the conservation laws of linear momentum and energy, and describe structural phase transitions (martensitic transformations) in one-dimensional Shape Memory Alloys (SMA) with non-convex Landau-Ginzburg free energy potentials. This system is formally written as a nonlinear abstract Cauchy problem in an appropriate Hilbert Space. A quasilinearization-based algorithm for parameter identification in this kind of Cauchy problems is proposed. Sufficient conditions for the convergence of the algorithm are derived in terms of the regularity of the solutions with respect to the parameters. Numerical examples are presented in which the algorithm is applied to recover the non-physical parameters describing the free energy potential in SMA, both from exact and noisy data.
A non isothermal Ginzburg-Landau model for phase transitions in shape memory alloys
Meccanica, 2010
A thermodynamical model for martensitic phase transitions in shape memory alloys is formulated in this paper in the framework of the Ginzburg-Landau approach to phase transitions. A single order parameter is chosen to represent the austenite parent phase and two mirror related martensite variants. A free energy previously proposed in the literature (Levitas et al. in Phys. Rev. B 66:134206, 2002; Phys. Rev. B 66:134207, 2002; Phys. Rev. B 68:134201, 2003) is employed, in its simplest form, as the main constitutive content of the model. In this paper we treat timedependent Ginzburg-Landau equation as a balance law on the structure order and we couple it to a energy balance equation, thus allowing to account of heat transfer processes. We obtain a coupled thermo-mechanical problem whose consistency with the Second Law is verified.
A shape memory alloy model by a second order phase transition
Mechanics Research Communications, 2016
Motivated by the experimental results of the paper [1] and unlike the general theories of shape memory alloys (SMAs), in this paper we suggest for such materials a phase field model by a second order phase transition. So that, with this new system we obtain a simulation of phase dynamics very convenient to describe the natural behavior of these materials. The differential system is governed by the motion equation, the heat equation and the Ginzburg-Landau (GL) equation and by a constitutive law between the phase field, the temperature, the strain and the stress. The use of this new model is characterized by new potentials of the GL equation and by a new dependence on the temperature in the constitutive equation. Using this new model, we obtain simulations in better agreement with experimental data and respect to previous work [2].
Variational principle for shape memory alloys
arXiv: Materials Science, 2018
The quasistatic problem of shape memory alloys is reviewed within the phenomenological mechanics of solids without microphysics analysis. The assumption is that the temperature variation rate is small. Reissner's type of generalized variational principle is presented, and its mathematical justification is given for three-dimensional bodies made of shape memory materials.
Journal of Mathematical Physics, 2010
By means of the Ginzburg-Landau theory of phase transitions, we study a nonisothermal model to characterize the austenite-martensite transition in shape memory alloys. In the first part of this paper, the one-dimensional model proposed by Berti et al. ͓"Phase transitions in shape memory alloys: A non-isothermal Ginzburg-Landau model," Physica D 239, 95 ͑2010͔͒ is modified by varying the expression of the free energy. In this way, the description of the phenomenon of hysteresis, typical of these materials, is improved and the related stress-strain curves are recovered. Then, a generalization of this model to the three-dimensional case is proposed and its consistency with the principles of thermodynamics is proven. Unlike other three-dimensional models, the transition is characterized by a scalar valued order parameter and the Ginzburg-Landau equation, ruling the evolution of , allows us to prove a maximum principle, ensuring the boundedness of itself.
A three-dimensional non-isothermal Ginzburg–Landau phase-field model for shape memory alloys
Modelling and Simulation in Materials Science and Engineering, 2014
In this paper, a macroscopic three dimensional non-isothermal model is proposed to describe hysteresis phenomena and phase transformations in shape memory alloys (SMAs). The model is of phase-field type and is based on the Ginzburg-Landau theory. The hysteresis and phase transformations are governed by the kinetic phase evolution equation using the scalar order parameter, conservation laws of momentum and energy, and a non-linear coupling between stress, strain, and the order parameter in a differential form. One of the important features of the model is that the phase transformation is governed by the stress tensor as opposed to the transformation strain tensor typically used in the literature. The model takes into account different properties of austenite and martensite phases based on the compliance tensor as a function of the order parameter and stress. Representative numerical simulations on a SMA specimen reproduce hysteretic behaviors observed experimentally in the literature.