1,721,127 research outputs found
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
Modelling of Electro-Viscoelastic Materials through Rate Equations
Full text linkModels 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
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
Generalities on constitutive models
No full textThe balance equations of a body form an under-determined differential system, insufficient to yield specific results unless further relations are supplied. The balance equations for a continuum (free from internal structures) are the continuity equation, the equation of motion, and the balance of energy, namely five equations, for the fourteen unknowns (mass density, velocity, symmetric stress, energy density, heat flux) in the pertinent space-time domain. The insufficiency of the balance equations to solve a dynamic problem is conceptual and is consistent with the fact that different material properties are expected to provide different responses. Mathematically the material properties of the body are expressed by constitutive equations, or constitutive assumptions, which provide a model of the material behaviour. Constitutive equations are not a mere mathematical model. They have to be physically admissible, and this is ascertained through the compatibility with the objectivity principle and the second law of thermodynamics
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
A 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
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