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The paper develops a new approach to the analysis of a solid-liquid phase diagram in terms of the chemical potentials of the single phases. The approach stems from the basic hypothesis that chemical potentials between different phases are equal at equilibrium. To make the idea operative, the mathematical dependence of the pertinent chemical potentials on the temperature and the mass fraction is determined by means of Aspen Plus; for definiteness a sodium chloride solution is examined. The phase diagram is established and the analytical properties of the curve are obtained. Moreover, the functional dependence of the mass fraction on the temperature at the freezing point is determined. The satisfactory agreement with new experimental data proves the validity of the procedure. The methodology so developed, provides an improved and rigorous mathematical overview for the analysis of phase diagrams. The approach is likely to be applicable to the investigation of solutions with several constituents or alloys. [ABSTRACT FROM AUTHOR]

The paper deals with viscoelastic solids subject to aging and fatigue effects. First a scheme is set up on the basis of physical insights, provided by the analysis of a rheological model, by letting the elastic moduli and the viscosity coefficients be functions of time. Hence the functional providing the stress in terms of the strain is established. The procedure has the advantage of showing how the dependence on the present value and that on the history of the strain are influenced by the rheological parameters. The three-dimensional version is then considered and the thermodynamic restrictions are determined. Next a second approach is set up by letting the stress be given by a constitutive equation involving a fractional derivative. The coefficient characterizing the kernel is assumed to obey an evolution equation that incorporates the fatigue effects. Compatibility with thermodynamics is shown to hold with an appropriate internal structural power. [ABSTRACT FROM AUTHOR]

The modeling of heat conduction is considered by letting the time derivative, in the Cattaneo-Maxwell equation, be replaced by a derivative of fractional order. The purpose of this new approach is to overcome some drawbacks of the Cattaneo-Maxwell equation, for instance possible fluctuations which violate the non-negativity of the absolute temperature. Consistency with thermodynamics is shown to hold for a suitable free energy potential, that is in fact a functional of the summed history of the heat flux, subject to a suitable restriction on the set of admissible histories. Compatibility with wave propagation at a finite speed is investigated in connection with temperature-rate waves. It follows that though, as expected, this is the case for the Cattaneo-Maxwell equation, the model involving the fractional derivative does not allow the propagation at a finite speed. Nevertheless, this new model provides a good description of wave-like profiles in thermal propagation phenomena, whereas Fourier's law does not. [ABSTRACT FROM AUTHOR]

Abstract: Phase transitions between two phases are modelled as space regions where an order parameter, or phase field, changes smoothly. A thermodynamic approach is developed by allowing for the nonlocal character of the continuum. The phase field is regarded as an internal variable and the kinetic or evolution equation is viewed as a constitutive equation. Along with the other constitutive equations, the unknown evolution equation is required to satisfy the second law of thermodynamics. Necessary and sufficient restrictions placed by thermodynamics are derived for the constitutive equations and, furthermore, a general form of the evolution equation for the order parameter is obtained within the schemes of a non-conserved or a conserved phase field. Based on the thermodynamic restrictions, a model for the ice–water transition is established which allows for superheating and undercooling. A model is also provided for the transition in superconducting materials. [Copyright &y& Elsevier]

Models of dielectric solids subject to large deformations are established by following a thermodynamic approach. The models are quite general in that they account for viscoelastic properties and allow electric and thermal conduction. A preliminary analysis is devoted to the selection of fields for the polarization and the electric field; the appropriate fields are required to comply with the balance of angular momentum and to enjoy the Euclidean invariance. Next, the thermodynamic restrictions for the constitutive equations are investigated using a wide set of variables allowing for the joint properties of viscoelastic solids, electric and heat conductors, dielectrics with memory, and hysteretic ferroelectrics. Particular attention is devoted to models for soft ferroelectrics, such as BTS ceramics. The advantage of this approach is that a few constitutive parameters provide a good fit of material behaviour. A dependence on the gradient of the electric field is also considered. The generality and the accuracy of the models are improved by means of two features. The entropy production is regarded as a constitutive property per se, while the consequences of the thermodynamic inequalities are made explicit by means of representation formulae.

Models of ferromagnetic hysteresis are established by following a thermodynamic approach. The class of constitutive properties is required to obey the second law, expressed by the Clausius-Duhem inequality, and the Euclidean invariance. While the second law states that the entropy production is non-negative for every admissible thermodynamic process, here the entropy production is viewed as a non-negative constitutive function. In a three-dimensional setting, the magnetic field and the magnetization are represented by invariant vectors. Next, hysteretic properties are shown to require that the entropy production is needed in an appropriate form merely to account for different behavior in the loading and the unloading portions of the loops. In the special case of a one-dimensional setting, a detailed model is determined for the magnetization function, in terms of a given susceptibility function. Starting from different initial magnetized states, hysteresis cycles are obtained by solving a nonlinear ordinary differential system. Cyclic processes with large and small amplitudes are established for materials such as soft iron.

This paper develops a phase-field approach to describe the damage within continuum mechanics. The body is associated with the standard stresses and body forces of macroscopic character. As is the case in many contexts, the phase field is a scalar variable whose time rate is governed by a constitutive equation. The generality of the approach allows the modeling of non-stationary heat conduction, mechanical hysteretic effects, and the macroscopic damage. The thermodynamic consistency is investigated through the constraint of the Clausius-Duhem inequality following the standard procedure of Rational Thermodynamics. The entropy production is considered as a constitutive function; this view was proved to be essential in the elaboration of hysteretic models. Here, the scheme is improved by viewing the entropy production as a sum of two terms, one induced by the other constitutive equations and one given by a constitutive equation of the entropy production per se.

The properties of viscoelastic solids subject to a magnetic field are modelled within two thermodynamically consistent approaches that are typical of models with a non-instantaneous response. One is based on memory functionals: the reversible changes are described by the instantaneous response, while the dissipativity is expressed by the dependence on histories. The other approach involves objective rate equations. While memory functionals lead to the difficulty of determining thermodynamically consistent free energy functionals, rate equations result in a simpler scheme. The greater simplicity allows the discovery of, in particular, models of magneto-hyperelastic materials, magneto-hypoelastic materials, and various forms of magneto-viscoelastic behaviour. The novelty of the procedure is based on two features: a representation formula, originating from the entropy inequality, and the use of the entropy production as a constitutive function. Relations with other approaches in the literature are examined in detail.

The thermodynamic consistency of linear viscoelastic models is investigated. First, the classical Boltzmann law of stress-strain is considered. The kernel (Boltzmann function) is shown to be consistent only if the half-range sine transform is negative definite. The existence of free-energy functionals is shown to place further restrictions. Next, the Boltzmann function is examined in the unbounded power law form. The consistency is found to hold if the stress functional involves the strain history, not the strain-rate history. The stress is next taken to be given by a fractional order derivative of the strain. In addition to the constitutive equations involving strain-rate histories, finding a free-energy functional, consistent with the second law, seems to be an open problem.