Relativistic thermodynamics of viscoelastic dielectrics

Archive for Rational Mechanics and Analysis -     

A nonlinear field theory of deformable dielectrics

Two difficulties have long troubled the field theory of dielectric solids. First, when two electric charges are placed inside a dielectric solid, the force between them is not a measurable quantity. Second, when a dielectric solid deforms, the true electric field and true electric displacement are not work conjugates. These difficulties are circumvented in a new formulation of the theory in this paper. Imagine that each material particle in a dielectric is attached with a weight and a battery, and prescribe a field of virtual displacement and a field of virtual voltage. Associated with the virtual work done by the weights and inertia, define the nominal stress as the conjugate to the gradient of the virtual displacement. Associated with the virtual work done by the batteries, define the nominal electric displacement as the conjugate to the gradient of virtual voltage. The approach does not start with Newton's laws of mechanics and Maxwell-Faraday theory of electrostatics, but produces them as consequences. The definitions lead to familiar and decoupled field equations. Electromechanical coupling enters the theory through material laws. In the limiting case of a fluid dielectric, the theory recovers the Maxwell stress. The approach is developed for finite deformation, and is applicable to both elastic and inelastic dielectrics. As applications of the theory, we discuss material laws for elastic dielectrics, and study infinitesimal fields superimposed upon a given field, including phenomena such as vibration, wave propagation, and bifurcation.     

On thermoelastic dielectrics

Constitutive equations for a linear thermoelastic dielectric are derived from the energy balance equation assuming dependence of the stored energy function on the strain tensor, the polarization vector, the polarization gradient tensor and entropy. A method is indicated for constructing a hierarchy of constitutive equations for materials with arbitrary symmetry by introducing various thermodynamic potentials. Maxwell's relations are constructed for the thermodynamic potential W^L. The entropy inequality is used to obtain stability conditions for an elastic dielectric in equilibrium under prescribed boundary constraints. Frequencies are explicitly determined for a plane wave propagating along the x₁-axis in an infinite centro-symmetric isotropic thermoelastic dielectric.     

On the symmetry of deformable crystals

We prove existence, uniqueness, and regularity properties for a solution u of the Bellman-Dirichlet equation of dynamic programming: (1) $$\left\{ \begin{gathered} \max {\text{ }}\{ L^i u + f^i = 0{\text{ in }}\Omega \hfill \\ i{\text{ = 1,2 }} \hfill \\ u{\text{ = 0 on }}\partial \Omega , \hfill \\ \end{gathered} \right.$$ where L ¹ and L ² are two second order, uniformly elliptic operators. The method of proof is to rephrase (1) as a variational inequality for the operator K=L ²(L ¹)^?1 in L ²(Ω) and to invoke known existence theorems. For sufficiently nice f ¹ and f ² we prove in addition that u is in H ³(Ω)?C ^2,α(Ω) (for some 0<α<1) and hence is a classical solution of (1).     

On continuum thermodynamics

Within the scope of classical continuum thermodynamics, we elaborate on the basic concepts and adopt a different approach from usual to the formulation of conservation laws and an entropy production inequality, both for a single phase continuum and for a mixture of any number of constituents. These conservation laws and the entropy inequality can be regarded as applicable to both local and nonlocal problems. In the case of a single phase continuum and for a simple material which is homogeneous in its reference configuration, under fairly mild smoothness assumptions, we prove that all the conservation laws reduce to the usual classical ones and the entropy production inequality reduces to the Clausius-Duhem inequality. Some attention is given to possible redundancies in the basic concepts, as well as to alternative forms of the energy equation and the entropy inequality. The latter is particularly significant in regard to different but equivalent formulations of mixture theory.     

Dynamic stability and bifurcation analysis in fractional thermodynamics

In mechanics, viscoelasticity was the first field of applications in studying geomaterials. Further possibilities arise in spatial non-locality. Non-local materials were already studied in the 1960s by several authors as a part of continuum mechanics and are still in focus of interest because of the rising importance of materials with internal micro- and nano-structure. When material instability gained more interest, non-local behavior appeared in a different aspect. The problem was concerned to numerical analysis, because then instability zones exhibited singular properties for local constitutive equations. In dynamic stability analysis, mathematical aspects of non-locality were studied by using the theory of dynamic systems. There the basic set of equations describing the behavior of continua was transformed to an abstract dynamic system consisting of differential operators acting on the perturbation field variables. Such functions should satisfy homogeneous boundary conditions and act as indicators of stability of a selected state of the body under consideration. Dynamic systems approach results in conditions for cases, when the differential operators have critical eigenvalues of zero real parts (dynamic stability or instability conditions). When the critical eigenvalues have non-trivial eigenspace, the way of loss of stability is classified as a typical (or generic) bifurcation. Our experiences show that material non-locality and the generic nature of bifurcation at instability are connected, and the basic functions of the non-trivial eigenspace can be used to determine internal length quantities of non-local mechanics. Fractional calculus is already successfully used in thermo-elasticity. In the paper, non-locality is introduced via fractional strain into the constitutive relations of various conventional types. Then, by defining dynamic systems, stability and bifurcation are studied for states of thermo-mechanical solids. Stability conditions and genericity conditions are presented for constitutive relations under consideration.     

On the thermodynamics of viscoelastic continua

Summary The thermodynamic foundations of a theory of permanent deformations are considered: in the axioms, an appropriate definition of free enthalpy plays an essential rôle. Our developments are parallel (but in a certain sense dual) to those which occur in a theory of viscoelastic materials, due to Coleman and Noll.

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Sommario Si propone una trattazione dei fondamenti della teoria delle deformazioni permanenti (plastiche, dovute a scorrimento a caldo, ecc.), legata a certi postulati termomeccanici dove gioca ruolo essenziale una opportuna definizione di entalpia libera. La trattazione ha un carattere in certo senso duale di quella relativa ai continui viscoelastici di Coleman e Noll.