Valeria Berti

Hysteresis and phase transitions for one-dimensional and three-dimensional models in shape memory alloys

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 (2010)] 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 φ and the Ginzburg–Landau equation, ruling the evolution of φ, allows us to prove a maximum principle, ensuring the boundedness of φ itself.

WELL POSEDNESS FOR A PHASE TRANSITION MODEL IN SUPERCONDUCTIVITY WITH VELOCITY AND MAGNETIC CRITICAL FIELDS

In this paper we present a Ginzburg–Landau model to describe the phenomenon of superconductivity as a second-order phase transition. The model proposed, which also includes thermal effects, allows to explain the existence of threshold values, both of the magnetic field and of the superconducting current, beyond which superconductivity vanishes. This is achieved by introducing a constitutive equation for the magnetic induction where the magnetic permeability depends on the complex order parameter. The model is proved to be consistent with thermodynamical principles. The resulting differential system is studied under the assumption that the temperature is a fixed parameter and its well-posedness is proved.

Existence and uniqueness for a non-isothermal dynamical Ginzburg-Landau model of superconductivity

In this paper, we present a time-dependent Ginzburg-Landau model which describes the phenomenon of superconductivity taking into account thermal effects. We modify the classical time-dependent Ginzburg-Landau equations by including temperature dependence. We prove that this model is compatible with the laws of thermodynamics. Moreover it allows us to express the critical magnetic field, which distinguishes the superconductive phase from the normal state, as a function of the absolute temperature. The theoretical temperature dependence of the threshold magnetic field agrees with the experimental results. Finally, we prove the existence and the uniqueness of the solutions of the non-isothemal Ginzburg-Landau equations.

Well-posedness of an isothermal diffusive model for binary mixtures of incompressible fluids

We consider a model describing the behaviour of a mixture of two incompressible fluids with the same density under isothermal conditions. The model consists of three balance equations: a continuity equation, a Navier?Stokes equation for the mean velocity of the mixture and a diffusion equation (Cahn?Hilliard equation). We assume that the chemical potential depends on the velocity of the mixture in such a way that an increase in the velocity improves the miscibility of the mixture. We examine the thermodynamic consistence of the model which leads to the introduction of an additional constitutive force in the motion equation. Then, we prove the existence and uniqueness of the solution of the resulting differential problem.

The Minimum Free Energy For Isothermal Dielectrics With Memory

Abstract The minimum free energy for a rigid dielectric with linear memory is found under isothermal conditions. The resulting expression is given in the frequency domain, i.e., in terms of the Fourier transform of the variables. By means of such a thermodynamic potential and of the Clausius-Duhem inequality an explicit formula of the dissipation is obtained. Thus the minimum free energy restores the customary relation between the dissipativity and the Clausius-Duhem inequality, that had been proved to fail when memory effects occur. Finally, by virtue of some of its properties, the minimum free energy is viewed as the square of a norm in a suitable space of the variables.

Parabolic-hyperbolic time-dependent Ginzburg-Landau-Maxwell equations

This article is devoted to the long-term dynamics of a parabolic-hyperbolic system arising in superconductivity. In the literature, the existence and uniqueness of the solution have been investigated but, to our knowledge, no asymptotic result is available. For the bidimensional model we prove that the system generates a dissipative semigroup in a proper phase-space where it possesses a (regular) global attractor. Then, we show the existence of an exponential attractor whose basin of attraction coincides with the whole phase-space. Thus, in particular, this exponential attractor contains the global attractor which, as a consequence, is of finite fractal dimension.

A thermodynamically consistent Ginzburg–Landau model for superfluid transition in liquid helium

In this paper, we propose a thermodynamically consistent model for superfluid-normal phase transition in liquid helium, accounting for variations of temperature and density. The phase transition is described by means of an order parameter, according to the Ginzburg–Landau theory, emphasizing the analogies between superfluidity and superconductivity. The normal component of the velocity is assumed to be compressible, and the usual phase diagram of liquid helium is recovered. Moreover, the continuity equation leads to a dependence between density and temperature in agreement with the experimental data.

Global and exponential attractors for a Ginzburg-Landau model of superfluidity

The long-time behavior of the solutions for a non-isothermal model in superfluidity is investigated. The model describes the transition between the normal and the superfluid phase in liquid 4 He by means of a non-linear differential system, where the concentration of the superfluid phase satisfies a non-isothermal Ginzburg-Landau equation. This system, which turns out to be consistent with thermodynamical principles and whose well-posedness has been recently proved, has been shown to admit a Lyapunov functional. This allows to prove existence of the global attractor which consists of the unstable manifold of the stationary solutions. Finally, by exploiting recent techinques of semigroups theory, we prove the existence of an exponential attractor of finite fractal dimension which contains the global attractor.

Some Existence and Uniqueness Results for a Stationary Ginzburg-Landau Model of Superconductivity

In this article we study the steady problem for gauge invariant Ginzburg-Landau equations obtained from a Gibbs free energy functional expressed in terms of observable variables and prove some existence and uniqueness results.

Internal structures, phase field and balance laws in continuum mechanics

A brief review of some models of phase transitions within the continuum mechanics is presented. A special study is directed to the phenomena of superconductivity, ferromagnetism, shape memory alloys and the behavior of a binary mixture using Cahn- Hilliard equation.