1,721,481 research outputs found
Free-energy for an elastic-plastic material in the presence of finite isothermal deformations
No full textA thermodynamic approach to hysteretic models in ferroelectrics
No full textThe purpose of the paper is to establish a constitutive model for the hysteretic properties in ferroelectrics. Both the polarization and the electric field are simultaneously independent variables so that the constitutive functions depend on both of them. This viewpoint is naturally related to the fact that an hysteresis loop is a closed curve surrounding the region of interest. For the sake of generality, the deformation of the material and the dependence on the temperature are allowed to occur. The constitutive functions are required to be consistent with the second law of thermodynamics. Among other results, the second law implies a general property on the relation between the polarization and the electric field via a differential equation. This equation shows a dependence fully characterized by the free energy and a dependence which is related to the dissipative character of the hysteresis. As a consequence, different hysteresis models may have the same free energy. Models compatible with thermodynamics are then determined by appropriate selections of the free energy and of the dissipative part. Correspondingly, major and minor hysteretic loops are plotted
On the modeling of magneto-mechanical effects in solids
Full text linkThe paper develops a thermodynamically-consistent approach to magnetostriction. This is performed by following two different approaches depending on whether a three-dimensional or a one-dimensional setting is considered. In the three-dimensional case the symmetry condition required by the balance of angular momentum results in the need of appropriate variables in the constitutive equations. These variables prove to be Euclidean invariant and comprise the so-called Lagrangian fields usually adopted in the literature. The consequences of the second law of thermodynamics are then determined for a solid described by the temperature, the deformation gradient, and the magnetic field. With this background the magnetostriction is modelled for linear or nonlinear magnetic laws. Next a one-dimensional setting is addressed mainly in connection with available experimental data. The symmetry condition becomes ineffective and hence the classical Eulerian fields are used. Based on the relations established through the thermodynamic consistency a detailed set of constitutive equations, for magnetization and strain, is established. These equations are set up so as to fit the experimental data from a one-dimensional sample under tensile stresses and magnetic fields
On the modeling of compressible viscous fluids via Burgers and Oldroyd equations
No full textThe paper develops some generalizations of the Burgers and the Oldroyd equations for the dynamics of fluids. To account for compressibility, in addition to viscosity, first the Oldroyd derivative is replaced with the Truesdell derivative. Consequently, the two equations can be given a linear form within the Lagrangian formulation. Furthermore, possible anisotropies are modeled by replacing some (scalar) coefficients with tensors. To emphasize the compressibility property, generalizations are established to allow for nonzero longitudinal viscosity. Next, the thermodynamic consistency is investigated by regarding both types of equations as rate equations, of second order and first order. The requirements on the parameters entering the two equations are derived while the linearity of the two equations allow the free energy potential be quadratic. The Oldroyd equation is found to be compatible via appropriate restrictions of the tensor coefficients, through different pairs of free energy and entropy production
Electrostriction and modelling of finitely deformable dielectrics
No full textThe paper investigates models of electrostriction by following a new approach though within the basic laws of continuum mechanics. Three general requirements are considered. Firstly, in a three-dimensional setting the balance of angular momentum implies a symmetry condition for the Cauchy stress tensor, the electric field and the electric polarization. By checking the thermodynamic consistency it is observed that constitutive equations with a separate dependence on the deformation gradient and the electric field does not satisfy the symmetry condition. Instead the symmetry is shown to hold for variables involving jointly the deformation gradient and the electric field or the polarization. This scheme in turn is found to satisfy both the thermodynamic consistency and the objectivity principle. Next electrostriction is examined by determining the deformation of an isotropic elastic solid induced by an electric field. Furthermore, it is shown that a proper dependence on the polarization or on the electric field results in elongations or contractions just as it is observed in real materials
Techniques for the Thermodynamic Consistency of Constitutive Equations
Full text linkThe paper investigates the techniques associated with the exploitation of the second law of thermodynamics as a restriction on the physically admissible processes. Though the exploitation consists of the use of the arbitrariness occurring in the Clausius-Duhem inequality, the approach emphasizes two uncommon features within the thermodynamic analysis: the representation formula, of vectors and tensors, and the entropy production. The representation is shown to be fruitful whenever more terms of the Clausius-Duhem inequality are not independent. Among the examples developed to show this feature, the paper yields the constitutive equation for hypo-elastic solids and for Maxwell-
Cattaneo-like equations of heat conduction. The entropy production is assumed to be given by a constitutive function per se and not merely the expression inherited by the other constitutive functions. This feature results in more general expressions of the representation formulae and is crucial for the compact description of hysteretic phenomena
Strain-Rate and Stress-Rate Models of Nonlinear Viscoelastic Materials
Full text linkThe paper is devoted to the modeling of nonlinear viscoelastic materials. The constitutive equations are considered in differential form via relations between strain, stress, and their derivatives in the Lagrangian description. The thermodynamic consistency is established by using the Clausius-Duhem inequality through a procedure that involves two uncommon features. Firstly, the entropy production is regarded as a positive-valued constitutive function per se. This view implies that the inequality is in fact an equation. Secondly, this statement of the second law is investigated by using an algebraic representation formula, thus arriving at quite general results for rate terms that are usually overlooked in thermodynamic analyses. Starting from strain-rate or stress-rate equations, the corresponding finite equations are derived. It then emerges that a greater generality of the constitutive equations of the classical models, such as those of Boltzmann and Maxwell, are obtained as special cases
Objective rate equations and memory properties in continuum physics
No full textThe paper deals with the modelling of material behaviours in continuum physics by means of rate equations. The research has a twofold purpose. First, to review the structure of objective time derivatives, namely invariant derivatives within the set of Euclidean transformations; known derivatives occurring in the literature are shown to be particular cases of the whole family of objective time derivatives. Second, to investigate the thermodynamic consistency of some models involving objective time derivatives. In particular, two topics are developed. One is the improvement of the constitutive equation of viscous fluids. The other topic is a possible rate equation for the stress. The thermodynamic consistency is shown in connection with the co-rotational derivative
A Thermodynamic Approach to Rate-Type Models of Elastic-Plastic Materials
No full textWithin the framework of continuum thermodynamics, a tensor-valued rate-type model of elastic-plastic materials is established. The evolution of the stress-strain relation is governed by a free-energy function and a hysteretic function proportional to the entropy production density. It is a key point of the present approach that the entropy production is given by a constitutive function consistent with the second-law inequality. The free energy depends on both stress and strain, as well as temperature. In the absence of hysteretic loops the evolution of the stress depends on the values of stress and strain and is affected by the time derivative of the strain. The hysteretic behaviour is modelled in detail in the one-dimensional case. Simple examples are established by a hysteretic function proportional to the absolute value of the strain rate. While the entropy production is formally similar to the widely used dissipation potentials the corresponding approaches are qualitatively different. The entropy production and the free energy potential are functions of the same set of physical variables and no internal variable is involved. The analysis of the second-law inequality leads to the sought constitutive relation, here in the rate-type form, for stress and strain
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