Phase-field models for transition phenomena in materials with hysteresis
Claudio Giorgi
Non-isothermal phase-field models of transition phenomena in materials with hysteresis are considered within the framework of the Ginzburg-Landau theory. Our attempt is to capture the relation between phase-transition and hysteresis (either mechanical or magnetic). All models are required to be compatible with thermodynamics and to fit well the shape of the major hysteresis loop. Focusing on uniform cyclic processes, numerical simulations at different temperatures are performed.
keywords: austenite-martensite transition free energy potentials. Ginzburg-Landau theory First and second order phase transitions ferromagnetic transition hysteresis loops
Claudio Giorgi Davide Guidetti Maria Grazia Naso
1.Mauro Fabrizio. This volume entitled ``Mathematical Models and Analytical Problems in Modern Continuum Thermomechanics" is dedicated to Mauro Fabrizio on the occasion of his retirement.

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Phase transition and separation in compressible Cahn-Hilliard fluids
Mauro Fabrizio Claudio Giorgi Angelo Morro
The paper provides a scheme for phase separation and transition by accounting for diffusion, dynamic equations and consistency with thermodynamics. The constituents are compressible fluids thus improving the model of a previous approach. Moreover a possible saturation effect for the concentration of a constituent is made explicit. The mass densities of the constituents are independent of temperature. The evolution of concentration is described by the standard equation for mixtures but the balance of energy and entropy of the mixture are stated as for a single constituent. However, due to the non-simple character of the mixture, an extra-energy flux is allowed to occur. Also motion and diffusion effects are considered by letting the stress in the mixture have additive viscous terms and, remarkably, the chemical potential contains a quadratic term in the stretching tensor. As a result a whole set of evolution equations is set up for the concentration, the velocity, and the temperature. Shear-induced mixing and demixing are examined. A maximum theorem is proved which implies that the concentration of the mixture has values from 0 to 1 as is required from the physical standpoint.
keywords: stretching-induced mixing. evolution equations Cahn-Hilliard fluids Phase separation phase transition
Isotropic-nematic phase transitions in liquid crystals
Mauro Fabrizio Claudio Giorgi Angelo Morro
The paper derives the evolution equations for a nematic liquid crystal, under the action of an electromagnetic field, and characterizes the transition between the isotropic and the nematic state. The non-simple character of the continuum is described by means of the director, of the degree of orientation and their space and time derivatives. Both the degree of orientation and the director are regarded as internal variables and their evolution is established by requiring compatibility with the second law of thermodynamics. As a result, admissible forms of the evolution equations are found in terms of appropriate terms arising from a free-enthalpy potential. For definiteness a free-enthalpy is then considered which provides directly the dielectric and magnetic anisotropies. A characterization is given of thermally-induced transitions with the degree of orientation as a phase parameter.
keywords: Dynamic phase transitions liquid crystals thermodynamics of continua nonsimple materials.
Sandra Carillo Claudio Giorgi Maurizio Grasselli
This Special Issue is dedicated to celebrate Mauro Fabrizio's 70th Birthday.
    It is a pleasure and an honour for us to devote it to Mauro with deep appreciation and friendship for the scientist as well as for the man.
    Mauro's wide and intense research activity touched many branches of Mathematical Physics. This fact is testified by the variety of subjects studied in the contributions collected here.
    Mauro was born on December 17, 1940. He graduated in Bologna in 1965. Dario Graffi, a renowned italian mathematical physicist, was his advisor. He has been full professor in the Universities of Salerno and Ferrara before returning to his Alma Mater. Since 1967 to present he published over 160 papers and 5 books.
    For over 45 years, he has been greatly influential through his research contributions in a several areas of Mechanics and Thermodynamics. In particular, the development of the mathematical modeling of Complex Systems.
    In these areas, starting from Dario Graffi's ideas, Mauro obtained a number of important results in mathematical modeling in continuous thermomechanics, materials with fading memory and hereditary system, electromagnetism of continuous media, first and second order phase transition models.
    Mauro has always been a stimulating and open minded Colleague as well as a reliable mentor to many young scientists within the mathematical community. His deep questions and sharp remarks are well known among all the people who had the chance to have him in the audience.
    Among the several Mauro's recognitions, we only recall that, on June 22, 2012, the prestigious ``Premio Linceo per la Meccanica e applicazioni e Matematica" was bestowed upon him by Giorgio Napolitano, the President of the Italian Republic.
    The study of complex systems is a multi-faceted area where many different mathematical tools come into play. From functional analysis to calculus of variations, from geometric analysis to semigroup theory and, of course, numerical methods.
    The present volume collects 31 peer reviewed contributions of a number of leading scholars in the analysis of mathematical models. It aims to present an overview of some challenging research lines and to stimulate further investigations.
    We are grateful to all the authors. They did a great job.
    May 23, 2014
Mathematical modeling of phase transition and separation in fluids: A unified approach
Alessia Berti Claudio Giorgi Angelo Morro
A unified phase-field continuum theory is developed for transition and separation phenomena. A nonlocal formulation of the second law which involves an extra-entropy flux gives the basis of the thermodynamic approach. The phase-field is regarded as an additional variable related to some phase concentration, and its evolution is ruled by a balance equation, where flux and source terms are (unknown) constitutive functions. This evolution equation reduces to an equation of the rate-type when the flux is negligible, and it takes the form of a diffusion equation when the source term is disregarded. On this background, a general model for first-order transition and separation processes in a compressible fluid or fluid mixture is developed. Upon some simplifications, we apply it to the liquid-vapor phase change induced either by temperature or by pressure and we derive the expression of the vapor pressure curve. Taking into account the flux term, the sign of the diffusivity is discusssed.
keywords: phase-field theories Ginzburg-Landau equation Cahn-Hilliard equation. phase separation Continuum thermodynamics phase transition
On the Green-Naghdi Type III heat conduction model
Claudio Giorgi Diego Grandi Vittorino Pata
In this work, we compare different constitutive models of heat flux in a rigid heat conductor. In particular, we investigate the relation between the solutions of the Green-Naghdi type III equation and those of the classical Fourier heat equation. The latter is often referred to as a limit case of the former one, as (formally) obtained by letting certain small positive parameter $\epsilon$ vanish. In presence of steady heat sources, we prove that the type III equation may be considered as a perturbation of the Fourier one only if the solutions are compared on a finite time interval of order $1/\epsilon$, whereas significant differences occur in the longterm. Moreover, for a bar with finite length and prescribed heat flux at its ends, the solutions to the type III equation do not converge asymptotically in time to the steady solutions to the corresponding Fourier model. This suggests that the Green-Naghdi type III theory is not to be viewed as comprehensive of the Fourier theory, at least when either asymptotic or stationary phenomena are involved.
keywords: singular limit. Fourier law Heat conduction Green-Naghdi theory
Free energies and pseudo-elastic transitions for shape memory alloys
Alessia Berti Claudio Giorgi Elena Vuk
A one-dimensional model for a shape memory alloy is proposed. It provides a simplified description of the pseudo-elastic regime, where stress-induced transitions from austenitic to oriented martensitic phases occurs. The stress-strain evolution is ruled by a bilinear rate-independent o.d.e. which also accounts for the fine structure of minor hysteresis loops and applies to the case of single crystals only. The temperature enters the model as a parameter through the yield limit $y$.Above the critical temperature $\theta_A^*$, the austenite-martensite phase transformations are described by a Ginzburg-Landau theory involving an order parameter $φ$, which is related to the anelastic deformation. As usual, the basic ingredient is the Gibbs free energy, $\zeta$, which is a function of the order parameter, the stress and the temperature. Unlike other approaches, the expression of this thermodynamic potential is derived rather then assumed, here. The explicit expressions of the minimum and maximum free energies are obtained by exploiting the Clausius-Duhem inequality, which ensures the compatibility with thermodynamics, and the complete controllability of the system. This allows us to highlight the role of the Ginzburg-Landau equation when phase transitions in materials with hysteresis are involved.
keywords: hysteresis loops Shape memory alloys stress-rate materials austenite-martensite transition Ginzburg-Landau theory. pseudo-elasticity free energy potentials
On the viscoelastic coupled suspension bridge
Ivana Bochicchio Claudio Giorgi Elena Vuk
In this paper we discuss the asymptotic behavior of a doubly nonlinear problem describing the vibrations of a coupled suspension bridge. The single-span road-bed is modeled as an extensible viscoelastic beam which is simply supported at the ends. The main cable is modeled by a viscoelastic string and is connected to the road-bed by a distributed system of one-sided elastic springs. A constant axial force $p$ is applied at one end of the deck, and time-independent vertical loads are allowed to act both on the road-bed and on the suspension cable. For this general model we obtain original results, including the existence of a regular global attractor for all $p\in\mathbb{R}$.
keywords: viscoelastic beam Suspension bridge system viscoelastic string nonlinear oscillations global attractor.
Solidification and separation in saline water
Mauro Fabrizio Claudio Giorgi Angelo Morro
Motivated by the formation of brine channels, this paper is devoted to a continuum model for salt separation and phase transition in saline water. The mass density and the concentrations of salt and ice are the pertinent variables describing saline water. Hence the balance of mass is considered for the single constituents (salt, water, ice). To keep the model as simple as possible, the balance of momentum and energy are considered for the mixture as a whole. However, due to the internal structure of the mixture, an extra-energy flux is allowed to occur in addition to the heat flux. Also, the mixture is allowed to be viscous. The constitutive equations involve the dependence on the temperature, the mass density of the mixture, the salt concentration and the ice concentration, in addition to the stretching tensor, and the gradient of temperature and concentrations. The balance of mass for the single constituents eventually result in the evolution equations for the concentrations. A whole set of constitutive equations compatible with thermodynamics are established. A free energy function is given which allows for capturing the main feature which occurs during the freezing of the salted water. That is, the salt entrapment in small regions (brine channels) where the cryoscopic effect forbids complete ice formation.
keywords: evolution equations Ginzburg-Landau theory. freezing of saline water solid-liquid phase transition phase separation Thermodynamics of solutions

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