-
[show abstract]
[hide abstract]
ABSTRACT: First-order phase transitions are modelled by a non-homogeneous, time-dependent scalar-valued order parameter or phase field. The time dependence of the order parameter is viewed as arising from a balance law of the structure order. The gross motion is disregarded and hence the body is regarded merely as a heat conductor. Compatibility of the constitutive functions with thermodynamics is exploited by expressing the second law through the classical Clausius–Duhem inequality. First, a model for conductors without memory is set up and the order parameter is shown to satisfy a maximum theorem. Next, heat conductors with memory are considered. Different evolution problems are established through a system of differential equations whose form is related to the manner in which the memory property is represented. Copyright © 2007 John Wiley & Sons, Ltd.
Mathematical Methods in the Applied Sciences 03/2008; 31(6):627 - 653. · 0.74 Impact Factor
-
[show abstract]
[hide abstract]
ABSTRACT: The modelling of materials with memory is characterized by the process and the initial state. Because of the equivalence between histories, the state is identified with the response generated by the trivial process on the given initial strain. The process is the strain since the initial time. The occurrence of histories up to past infinity is then avoided. Within this setting, the inversion of the constitutive equation and a variational formulation are examined. Moreover a virtual‐work type formulation provides a simpler weak form of the initial‐value problem. Uniqueness of the solution to the initial‐value problem is then established. Copyright © 2004 John Wiley & Sons, Ltd.
Mathematical Methods in the Applied Sciences 09/2004; 28(2):233 - 251. · 0.74 Impact Factor
-
[show abstract]
[hide abstract]
ABSTRACT: 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.
Journal of Elasticity 07/1995; 40(2):107-122. · 1.11 Impact Factor
-
[show abstract]
[hide abstract]
ABSTRACT: A linear viscoelastic solid is considered along with the complete set of thermodynamic restrictions on the relaxation function. It is shown that such reslrictions imply the validity of a dissipativity condition, so far regarded as unrelated to the second law. Next it is remarked that the thermodynamic restrictions imply the convexity of a commonly used bilinear functional and the stationarity only if the class of displacement field is appropriate. Then it is proved that the Laplace transform of the solution to the mixed problem gives the strict minimum of a bilinear functional and vice versa. Finally, a bilinear functional with a weight function is considered and it is shown that the solution to the mixed problem gives the strict minimum and vice versa.
Continuum Mechanics and Thermodynamics 08/1989; 1(3):197-211. · 1.31 Impact Factor
-
[show abstract]
[hide abstract]
ABSTRACT: 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.
Journal of Elasticity 12/1987; 19(1):63-75. · 1.11 Impact Factor
-
[show abstract]
[hide abstract]
ABSTRACT: A thermodynamic framework is provided for the modelling of smart materials acted upon by electromagnetic fields. The description is based on the notion of state and process which, jointly, determine the response of the material. The view is taken that only the balance equations which are compatible with thermodynamics represent physically-admissible behaviours. Concerning simple materials, linear, anisotropic materials with memory are considered in detail by disregarding the effects of motion. Necessary and sufficient conditions are determined for the relaxation functions to be compatible with the second law. Next, ferromagnetic and ferroelectric materials are considered. Liquid crystals are described both as materials with microstructure of a micropolar form and as materials within a generalized form of the director model. The modelling of nonsimple materials requires that the second law is considered in a global form. This is exemplified by considering dielectrics with quadruples and spatially-dispersive crystals.
Mathematical and Computer Modelling.
-
[show abstract]
[hide abstract]
ABSTRACT: The paper provides a modelling of the magnetization curve and of the ferromagnetic–paramagnetic transition within a continuum thermodynamic setting. The general model of the nonlinear, time dependent behaviour of ferromagnetic materials is accomplished by regarding the magnetization vector as an internal variable, namely as a vector field whose time evolution is a constitutive equation subject to the requirements of the second law of thermodynamics. The exchange interaction of the magnetization is modelled through a dependence of the free energy on the magnetization gradient. Consistent with the non-simple character of the material, the second law allows for a non-zero extra-entropy flux. A general three-dimensional scheme is elaborated which seems to be new in the literature. The three-dimensional setting is then established for stationary and homogeneous fields thus finding the collinearity and the corresponding form of the magnetic susceptibility. The whole evolution problem for the temperature and the magnetization is provided so that temperature-induced transition processes are allowed. The model accounts also for the dependence of the saturation magnetization on the temperature. Also for the sake of comparison with the existing literature, the evolution equations for the direction and the intensity of magnetization are derived. Known models, such as those of Landau–Lifshitz and Gilbert, are recovered as particular cases of saturated bodies. Next, the model is made more specific so as to account in detail for the saturation, the residual or spontaneous magnetization and the coercive field. First, the classical potential, which traces back to Ginzburg, and the Weiss model are revisited. The corresponding lack of the saturation effect or the description via implicit relations are emphasized. Hence, a new potential, with a logarithmic dependence on the magnetization, is investigated which provides the residual magnetization and the coercive field in an explicit way and satisfies expected properties of the residual magnetization as a function of the temperature.
International Journal of Engineering Science.
