Mauro Fabrizio

Viscoelastic relaxation functions compatible with thermodynamics

Within the framework of linear viscoelasticity this paper deals with the elaboration of a procedure for testing whether a given relaxation function is compatible with thermodynamics. In order to avoid any difficulty concerning the indeterminacy of the free energy functional, the second law is stated through the Clausius property for approximately-cyclic processes. Then, by considering sinusoidal strain tensor evolutions, it is shown that the statement of the second law is equivalent to a condition which, in essence, expresses the non-negativeness of the loss modulus. This means that, in order to test for compatibility with thermodynamics, it is enough to examine what happens in sinusoidal processes.

ON EXPONENTIAL ASYMPTOTIC STABILITY IN LINEAR VISCOELASTICITY

This paper establishes results concerning the exponential decay of strong solutions of a linear hyperbolic integrodifferential equation in Hilbert space. Rather than the more commonly used assumptions that the relaxation function is non-negative, decreasing and convex, dissipation is modelled by requiring that the dynamic viscosity be a positive function. This restriction has a firm thermodynamic basis. Frequency domain methods are employed.

Ice-water and liquid-vapor phase transitions by a Ginzburg-Landau model

A model for the first order phase transitions as ice-water and liquid-vapor is proposed using the Ginzburg–Landau equation for the order parameter φ. In this model the density ρ is composed of two quantities ρ0 and ρ1 such that 1/ρ=1/ρ0+1/ρ1, where ρ1 is strictly connected to the order parameter φ. By means of this decomposition, we are able to represent the Andrew diagram without the use of the heuristic van der Waals equation.

Existence and Uniqueness

The study of differential problems related to materials with fading memory began with the work of Graffi [111, 112]. Later on, these studies were considered by many authors, and in particular, a new important description of such phenomena was given by Dafermos in [47, 46], using semigroup theory, where besides existence and uniqueness of the solution, the interesting problem of asymptotic stability was also examined.

On rigid linear heat conductors with memory

Abstract This paper studies a theory of linear rigid heat conductors with memory within the framework proposed by Gurtin and Pipkin. An approximate theory of thermodynamics is developed for this model. In such a context a new thermodynamic potential, called the free pseudoenergy, is introduced and used to define a new norm on a reduced state space. This norm is appropriate for stating a stability property and a domain of dependence inequality for solutions of the energy equation.

Internal dissipation, relaxation property, and free energy in materials with fading memory

This paper examines some features of the standard theory of materials with fading memory and emphasizes that the commonly-accepted notion of dissipation yields unexpected consequences. First, application of the Clausius-Duhem inequality to linear viscoelasticity shows that there is a free energy functional such that the so-called internal dissipation vanishes in spite of the dissipative character of the model. Second, upon the choice of a suitable function norm, the relaxation property is proved not to hold for viscoelastic solids. Finally, the particular case is considered when the relaxation function is a superposition of exponentials. Different descriptions of state are then possible which prove to be inequivalent as far as the free energy is concerned.

Damage and fatigue described by a fractional derivative model

As in 1], damage is associated with fatigue that a material undergoes. In this paper, because we work with viscoelastic solids represented by a fractional model, damage is described by the order of the fractional derivative, which represents the phase field satisfying Ginzburg-Landau equation, which describes the evolution of damage.Finally, in our model, damage is caused, not only by fatigue, but also directly by a source related to environmental factors and described by a positive time function.

Delayed Thermal Models: Stability and Thermodynamics

In this article, for a rigid medium, we investigate the constitutive equations for the heat conduction given by a single and double phase-lag patterns, suggested by Tzou. Moreover, this research is extended to some Taylor expansions. Within this work, the thermodynamic compatibility of these systems is studied. In particular, we find that the models considered by Tzou are not in agreement with the laws of thermodynamics. We have observed that this compatibility can be proved for the first-order Taylor expansions of the SPL equation. In the following, we have investigated other models obtained by different order expansions of the phase lag patterns. Also, for these new differential systems we have studied the relationships between instability and thermodynamic incompatibility. Sufficient conditions for the onset of temperature instability are given. A uniqueness result for the ill-posed problem is obtained.

Fractional rheological models for thermomechanical systems. Dissipation and free energies

Within the fractional derivative framework, we study thermomechanical models with memory and compare them with the classical Volterra theory. The fractional models involve significant differences in the type of kernels and predicts important changes in the behavior of fluids and solids. Moreover, an analysis of the thermodynamic restrictions provides compatibility conditions on the kernels and allows us to determine certain free energies, which in turn enables the definition of a topology on the history space. Finally, an analogous analysis is carried out for the phenomenon of heat propagation with memory.

Thermodynamics of non-local materials: extra fluxes and internal powers

The most usual formulation of the Laws of Thermodynamics turns out to be suitable for local or simple materials, while for non-local systems there are two different ways: either modify this usual formulation by introducing suitable extra fluxes or express the Laws of Thermodynamics in terms of internal powers directly, as we propose in this paper. The first choice is subject to the criticism that the vector fluxes must be introduced a posteriori in order to obtain the compatibility with the Laws of Thermodynamics. On the contrary, the formulation in terms of internal powers is more general, because it is a priori defined on the basis of the constitutive equations. Besides it allows to highlight, without ambiguity, the contribution of the internal powers in the variation of the thermodynamic potentials. Finally, in this paper, we consider some examples of non-local materials and derive the proper expressions of their internal powers from the power balance laws.