Nonlinear waves in thermoelastic dielectrics

This paper is addressed to the analysis of wave propagation in electroelastic materials. First the balance equations are reviewed and the entropy inequality is established. Next the constitutive equations are considered for a deformable and heat-conducting dielectric. To allow for discontinuity wave propagation, an appropriate objective rate equation of the heat flux is considered. The thermodynamic consistency of the whole set of constitutive equations is established. Next the nonlinear evolution equations so determined are tested in relation to wave propagation properties. Waves are investigated in the form of weak discontinuities and the whole system of equations for the jumps is obtained. As a particular simple case the propagation into an unperturbed region is examined. Both the classical electromagnetic waves and the thermal waves are found to occur. In both cases the mechanical term is found to be induced by the electrical or the thermal wave discontinuity.


1.
Introduction. The modelling of electromechanical interactions within nonlinear continuum physics is a subject of interest in many respects. As with continuum mechanics, the Eulerian and Lagrangian descriptions of continuum physics differ for the appropriate functions representing forces and stresses, in addition to the difference about the use of the current configuration or the reference configuration. The two descriptions of the electromagnetic fields provide the magnetic field H, the magnetic induction B, the electric field E and the electric displacement D D D in the Lagrangian description while the corresponding description of the polarization (density) P and the magnetization M are not uniquely defined. This is so because the balance equations of electromagnetism in integral form involve H and E through line integrals and B and D D D through surface integrals.
The electromechanical interactions are described by the balance equations. Yet the literature shows differences among various representations of force, couple, and power in an electromagnetic continuum [14,6,1,5]. In addition, contributions to the electromechanical interactions arise from the selection of the independent variables in the constitutive equations. In an electroelastic material we might think that E is the natural variable of electric character. However, by the principle of objectivity the effect of E on any scalar function should be the same as that supplied by the vector QE, Q being any time-dependent rotation tensor. A natural hint is that we assume the dependence on E via the invariant scalar E · E. Yet we might observe that there are also invariant vectors namely F T E, F −1 E, and JF T E, JF −1 E, where F is the deformation gradient, the superposed T means transpose, and J = det F. Notation. We consider a body occupying the time dependent region R t ⊂ E 3 . The motion is described by means of the function χ(X, t) providing the position vector x ∈ R t in terms of the position vector X, in a reference configuration R, and the time t, so that R t = χ(R, t). The deformation is described by means of the deformation gradient F = ∂ X χ, F iK = ∂ X K χ i . The symbol ∇ denotes the gradient in the current configuration Ω, ε the internal energy density (per unit mass), T the Cauchy stress, L the velocity gradient, q the heat flux vector, r the (external) heat supply, ρ the mass density. Owing to the polar character of polarizable media, the stress T need not be symmetric.
2. Second law inequality. Since the electric field E and the polarization P are dependent on the frame of reference, for definiteness we let E and P be the fields at the local frame of reference at rest with the material.
Relative to general models of electromagnetic solids we assume the magnetization and the electric current are zero. Moreover, no magnetic field is applied to the body. The balance equations of linear momentum, angular momentum, and energy can be written in the form 1 (see, e.g., [14,12]) where π = P/ρ is the polarization per unit mass, b is the mechanical body force density (per unit mass), and f P is the body force density on the electric dipoles.
Particle-like arguments about the action of the field on single dipoles justify the assumption [12] ρf B being possibly induced by E via Ampère's law. For definiteness and simplicity here we let this being the dominant term for weak magnetic fields B. The entropy per unit mass η is assumed to satisfy the inequality where k is the extra-entropy flux. Substitution of ∇·q−ρr from the energy equation (2) gives Upon some rearrangements we can write the inequality in the form We state the second law of thermodynamics by saying that inequality (4) has to hold for all possible processes compatible with the balance equations.
In [3] and [5] the total stress tensor is considered as the sum of the elastic stress tensor, here T, and the Maxwell stress tensor, The stress τ m is used systematically to express the boundary conditions and the use is allowed by the identity The proof of (5) is based on the assumptions the vanishing of ∇ × × E being a consequence of the assumed time independence of the pertinent fields. For, since ∇ · D = 0 and D = 0 E + P then Since ∇ × × E = 0 then (E · ∇)E = 1 2 ∇(E · E) and hence (5) follows. This in turn shows that the use of τ m is less general than the use of (3) as the body force. In addition, as remarked also in [5], there are different definitions of the Maxwell stress. That is why we prefer to employ the body force (P · ∇)E rather than the stress τ m .
3. Constitutive assumptions and objectivity. To describe the evolution of a thermoelastic dielectric we let the response functions of the material be determined by the deformation gradient F, the temperature θ, and the electric field E. The dependence on the gradient ∇E is allowed so that nonlocal properties can be described. Moreover the dependence on the time derivativeĖ is allowed so that appropriate evolution features can be modelled. Heat conduction is also allowed and this is thought to be modelled via the dependence both on ∇θ and on the heat flux q. Hence we might say that the constitutive equations have the form and the like for the other constitutive functions. Yet the constitutive functions are required to satisfy the principle of objectivity.
Let F, F * be two frames of reference. The vector positions x and x * of a point, relative to F and F * , are related by ( [16], §17) where c(t) is an arbitrary vector-valued function while Q(t) is a proper orthogonal tensor function, det Q = 1. We denote by The material time derivative of an objective tensor is not objective in thaṫ The analogue holds for an objective vector. Instead, a tensor function A(x, t), x ∈ Ω, is invariant if, under the change of frame F → F * , satisfies the invariance property The deformation gradient F is an objective vector (see [7]), Hence the Cauchy-Green tensor Likewise, for any objective vector u the vector F T u is invariant, Hence, letting E and q be objective we can say that the vectors are invariant; incidentally, Ξ is just the field E 1 of [4]. This in turn implies thaṫ Let f be any invariant scalar or vector. The gradient ∇f , in the current configuration, satisfies the relation and hence ∇f is objective (vector). As a consequence F T ∇f is invariant, Indeed, Based on these properties, next we investigate the model of nonlinear deformable dielectrics by letting be the set of independent variables.
It is worth remarking that objectivity and invariance need not single out a unique set of independent variables. First we observe that Indeed, JF −1 u is the material vector provided by Nanson's formula [9,10]. Moreover, JF −1 P is the vector variable used in [15] to describe elastic dielectrics. The appropriate choice of physical variables is related to the constitutive properties under consideration.
The modelling of heat conduction hinges on the by-now standard view that Fourier's law is not compatible with wave propagation and that, instead, equations of the Maxwell-Cattaneo type should be considered to obtain models allowing for finite wave speed. In this regard we observe that, sinceḞ = LF, Accordingly, the three derivatives are, repectively, F T , JF T , and JF −1 times appropriate objective derivatives. Indeed, are the Cotter-Rivlin derivative and the Truesdell derivative [11] of q.
4. Thermodynamic restrictions. We assume φ, η, T,Q Q Q, and k are functions of Γ. Indeed, we suppose φ is differentiable while the remaining functions are continuous. Evaluation ofφ and substitution in (4) gives The arbitrariness and the linearity ofΞ,∇ R θ, andθ imply Hence we have Some identities allow a better understanding of the structure of (8). SinceḞ = LF then Further, J∇ · k = ∇ R · k R , k R := JF −1 k. As a consequence, upon multiplication by J/θ, inequality (8) This inequality suggests that we let The arbitrariness and the linearity of D, W,Ė imply that Upon multiplying by θ/J the remaining inequality we find Equations (7), (10)-(13) are sufficient for the validity of the second law inequality. Further conclusions follow if the functionQ Q Q is given an explicit form. For definiteness we letQ τ and κ being possibly dependent on J, θ. The parameter τ is a positive valued function, τ 2 ≥ τ ≥ τ 1 > 0. We can then solve equation (14) on the interval [t 0 , t] and find that For any bounded value Q Q Q(t 0 ), the limit as t 0 → −∞ gives We now examine the consistency between equation (14) and inequality (13). Upon substitution we have Since we can write inequality (16) in the form The arbitrariness of ∇ R θ implies that the inequality holds if and only if A direct integration of (17) gives The positive definiteness of C −1 implies that inequality (20) holds if and only if κ > 0. Accordingly, as with the classical Fourier's law, the second law provides the positive valuedness of the heat conductivity.
It is worth pointing out that the rate equation (14) is invariant under a change of reference configuration. Let R 1 and R 2 be two reference configurations and let F 0 be the (constant) deformation gradient associated with the deformation R 1 → R 2 . Let F 1 and F 2 be the deformation gradients relative to R 1 and R 2 . Hence at any X ∈ R 1 . Now, if F is the deformation gradient then Relative to R 1 , eq. (14) becomes Replacing for any pair of deformation gradients connected by F 2 = F 1 F 0 . The same invariance property follows from the solution (15).
5. Evolution equations. The equations governing the evolution of the body are now established by means of the balance equations and the constitutive properties.
To get a more tractable model we neglect the dependence on ∇ R Ξ and let q = 0. We also neglect the body force ρb and the heat supply ρr. The equation of motion (1) reduces to ρv = ∇ · T + (P · ∇)E.
(24) To evaluateη it is convenient to consider q as a variable, rather than Q Q Q. For definiteness, based on (19) we assume φ takes the form Hence observe Q Q Q · C −1 Q Q Q = q 2 . As a consequence the entropy per unit mass η can be written At q = 0, it is natural to assume thatη is positive ifθ is positive or rather η is an increasing function of θ. This is the case for an ideal gas ( [18], §9-2). Accordingly we let Φ < 0. The valueq is determined by the rate equation (14). Sincė Q Q Q =Ḟ T q + F Tq then equation (14) becomes As a consequence,

Substitution in (24) results in
The mass density ρ is subject to the continuity equatioṅ The occurrence of ∇E in the equation of motion (22) requires that we consider also Maxwell's equations. Since we are dealing with non-magnetic materials then we let B = µ 0 H. In addition we let the electric current be zero. Since to fix ideas, we let the electric displacement D D D be a nonlinear function of E by assumingφ (C, Ξ) = −h(|E|). Hence, by (12) we find As a consequence Hence we can write Maxwell's equations in the form (29) Finally, in suffix notation, the equation of motion (22) becomes Equations (25)-(30) constitute the system of governing equations in the unknown functions ρ, χ, q, θ, H, E. Some properties of the system (25)-(30) are now investigated in connection with weak discontinuity waves.
6. Weak discontinuity waves. A propagating surface σ(t) is a weak discontinuity wave if the fields ρ, v, F, θ, q, H, E, with v =χ, F = ∇ R χ, have the following properties. 1) ρ, v, F, q, θ, H, E are continuous functions of x and t jointly for all x ∈ R t , t ∈ R; 2)ρ, ∇ρ,v,Ḟ, ∇ R F,q,θ,Ḣ, ∇H,Ė, ∇E and all higher order derivatives suffer, at most, jump discontinuities across σ but are continuous in x and t jointly everywhere else.
As an immediate consequence it follows that Let Ω + and Ω − be the subregions of R t ahead of and behind σ, R t = Ω − ∪σ∪Ω + . Also let n be the unit normal to σ pointing into Ω + . Let [[g]] denote the discontinuity of g across σ. By definition, [[g]] = g − − g + , where g − and g + are the limit values of g(x, t) as x approaches the pertinent point on σ(t) while remaining within Ω + and Ω − , respectively.
At a weak discontinuity the jumps associated with the system (25)-(30) satisfy The jumps are subject to the geometrical and kinematical conditions of compatibility ( [17], chapter C). Denote by ∂ n g = n · ∇g the normal derivative of a field g. By the geometrical conditions of compatibility if the vector field w is continuous across σ then U being referred to as the local speed of propagation of σ.
We now investigate the system (31)   The nonlinear model so established is investigated in connection with the propagation of weak discontinuity waves. In this regard some new compatibility conditions are derived that are associated with the simultaneous occurrence of material and spatial derivatives such asv, F and L, ∇θ. After deriving the whole system of equation in the unknown discontinuities, the particular case of propgation in an unperturbed region is examined. It follows that two types of waves are allowed: electromagnetic waves and thermal waves with local speed of propagation given by (51)