G.A. Kluitenberg

On linear dynamical equations of state for isotropic media I: General formalism☆

In this paper we investigate the relation (dynamical equation of state) among the (mechanical hydrostatic) pressure P, the volume ν and the temperature T of an isotropic medium with an arbitrary number, say n, of scalar internal degrees of freedom. It is assumed that the irreversible processes in the medium are due to volume viscosity and to changes in the internal variables and that these processes can be described with the aid of non-equilibrium thermodynamics. It is shown that an explicit form for the dynamical equation of state may be obtained if in the neighbourhood of some state of thermodynamic equilibrium it is permissible to consider the equilibrium pressure and the thermodynamic affinities (conjugate to the internal degrees of freedom) as linear functions of ν, T and the internal variables and if, moreover, the phenomenological coefficients, which occur in the equations for the irreversible processes, may be considered as constants. This dynamical equation of state has the form of a linear relation among P, ν, T, the first n derivatives with respect to time of P and of T and the first n+1 derivatives with respect to time of ν.

On linear dynamical equations of state for isotropic media II: Some cases of special interest

In a previous paper we have investigated the relation (dynamical equation of state) among the hydrostatic pressure P, the volume v and the temperature T of an isotropic medium with an arbitrary number, say n, of scalar internal degress of freedom. It has been shown that linearization of the theory leads to a dynamical equation of state which has the form of a linear relation among P, v, T, the first n derivatives with respect to time of P and T and the first n+1 derivatives with respect to time of v. In this paper we give a more detailed investigation of the coefficients of P, v and T in the linear dynamical equation of state. Furthermore, we consider the case of media without volume viscosity. It is shown that for these media the derivative with respect to time of order n+1 of the volume does not occur in the dynamical equation of state. Finally, we pay special attention to media with one and with two scalar internal variables.

On the propagation of linear transverse acoustic waves in isotropic media with mechanical relaxation phenomena due to viscosity and a tensorial internal variable

In this paper we consider the propagation of linear transverse acoustic waves in isotropic media in which mechanical relaxation phenomena occur. It is assumed that the irreversible mechanical processes in the medium are due to viscosity and to changes in a tensorial internal variable and that these processes can be described with the aid of non-equilibrium thermodynamics. The viscous flow phenomenon is analogous to the viscous flow of ordinary fluids. In particular we investigate the velocity and attenuation of the waves and we consider the limiting cases of waves high and low frequencies. For high frequencies we obtain expressions for the phase velocity and attenuation which are analogous to those for viscous fluids and it is seen that the tensorial internal variables does not influence the propagation of waves. If the frequencies are sufficiently low, the expression for the phase velocity is analogous to the expression for the phase velocity in purely elastic media, while the damping is influenced by both irreversible phenomena and the imaginary part of the complex wave number is proportional to the square of the frequency.

On the propagation of linear transverse acoustic waves in isotropic media with mechanical relaxation phenomena due to viscosity and a tensorial internal variable. II. Some cases of special interest (Poynting-Thomson, Jeffreys, Maxwell, Kelvin-Voigt, Hooke and Newton media)

The propagation of linear transverse acoustic waves in isotropic media in which mechanical relaxation phenomena occur was considered in a previous paper. In particular expressions for the velocity and attenuation of the waves were obtained and the limiting cases of waves with high and low frequencies were discussed. In the present paper we investigate the propagation of linear transverse acoustic waves in Poynting-Thomson, Jeffreys, Maxwell, Kelvin-Voigt, Hooke and Newton media. We show that the dispersion relations for these waves may be considered as degeneracies of the dispersion relation which we derived in the general case of a viscoanelastic medium with memory. In particular we investigate the explicit dependence of the dispersion relations on the thermodynamic parameters and the phenomenological coefficients.

On generalizations of the Debye equation for dielectric relaxation

In some previous papers one of us (G.A.K.) discussed dielectric relaxation phenomena with the aid of non-equilibrium thermodynamics. In particular the Debye equation for dielectric relaxation in polar liquids was derived. It was also noted that generalizations of the Debye equation may be derived if one assumes that several microscopic phenomena occur which give rise to dielectric relaxation and that the contributions of these microscopic phenomena to the macroscopic polarization may be introduced as vectorial internal degrees of freedom in the entropy. If it is assumed that there are n vectorial internal degrees of freedom an explicit from for the relaxation equation may be derived, provided the developed formalism may be linearized. This relaxation equation has the form of a linear relation among the electric field E, the first n derivatives with respect to time of this field, the polarization vector P and the first n + 1 derivatives with respect to time of P. It is the purpose of the present paper to give full details of the derivations of the above mentioned results. It is also shown in this paper that if a part of the total polarization P is reversible (i.e. if this part does not contribute to the entropy production) the coefficient of the time derivative of order n + 1 of P in the relaxation equation is zero.

On the propagation of linear longitudinal acoustic waves in isotropic media with shear and volume viscosity and a tensorial internal variable: I. General formalism☆

In this paper we consider the propagation of linear longitudinal acoustic waves in isotropic media with viscosity and a tensorial internal variable. We use a thermodynamic theory for mechanical relaxation phenomena proposed by one of us in some previous papers. In particular we investigate the velocity and attenuation of the waves and we consider the limiting cases of high and low frequencies. We show that at high frequencies the tensorial internal degree of freedom does not influence the wave phenomena. If the frequencies are sufficiently low, the phase velocity depends only on the coefficients which occur in the equations of state, while the attenuation of the waves also depends on the coefficients which occur in the phenomenological equations which describe the irreversible processes of viscous flow and changes in the tensorial internal variable.

On the propagation of linear longitudinal acoustic waves in isotropic media with shear and volume viscosity and a tensorial internal variable. II: Some cases of special interest (Poynting-Thomson, Jeffreys, Maxwell, Kelvin-Voigt, Hooke and Newton media)

In a previous paper the propagation of linear longitudinal acoustic waves in isotropic media with shear and volume viscosity and a tensorial internal variable was considered and the expressions for the velocity and attenuation of the waves were obtained. In the present paper we investigate the propagation of linear longitudinal acoustic waves in Poynting-Thomson, Jeffreys, Maxwell, Kelvin-Voigt, Hooke and Newton media. We show that the dispersion relations for these waves may be considered as degeneracies of the dispersion relation which we derived in the general case of a viscoanelastic medium with memory. In particular we investigate the explicit dependence of the dispersion relations on the thermodynamic parameters and the phenomenological coefficients and the limiting cases of waves with high and low frequencies are discussed.

On the heat dissipation function for dielectric relaxation phenomena in anisotropic media

The heat dissipation function for polarizable anisotropic media in which phenomena of dielectric relaxation occcur is derived as a generalization of the heat dissipation function studied in the case that the media are isotropic. The methods of non-equilibrium thermodynamics are used. It is seen that the linearization of the theory leads to a dielectric relaxation equation for anisotropic polarizable media which has the form of a linear relation among the temperature, the components of the electric field vector and of the total polarization vector, the first derivatives with respect to time of the components of these vectors and of the temperature and the components of the second derivative with respect to time of the total polarization vector. It is shown that the heat dissipation function is due to irreversible phenomena of viscous flow, electric conduction and dielectric relaxation. Finally, the obtained results are applied to the particular cases of Debye media and De Groot-Mazur media.

On transformations of internal vectorial variables in the thermodynamic theory of dielectric relaxation and the invariance of Onsager's reciprocal relations

In some previous papers one of us developed a theory of dielectric and magnetic relaxation phenomena based on non-equilibrium thermodynamics. In one of these papers possible cross effects between dielectric (or magnetic) relaxation and heat conduction were discussed. It was shown in particular that Onsagers relations for these cross effects are invariant under a transformation of the vectorial internal variable. (This variable was introduced in the description of dielectric relaxation). In the present paper we shall extend the discussion to a broader class of irreversible phenomena. (Two types of dielectric or magnetic relaxation, which give rise to two relaxation times, heat conduction, electrical conduction and viscous flow.)