Thermodynamics and the Constitutive Relations for Second Sound in Crystals

Summary

A derivation is given of the implications of the second law of thermodynamics for the constitutive equations of materials for which the heat flux vector q and the temperatureθ obey the relation

$$ T\left( \theta \right)q + q = - K\left( \theta \right)grad\theta , $$

with T(θ) and K(θ) non-singular second-order tensors that, as functions of θ, depend on the material under consideration. The relation (†), which is a natural generalization to anisotropic media of the relation of Cattaneo, has been used by Pao and Banerjee to describe second sound in dielectric crystals. It is here shown that when (†) holds the specific internal energy e depends not only on θ but also on q; that is:

$$ e = {e_0}\left( \theta \right) + q.A\left( \theta \right)q, $$

where e ₀ is the classical or “equilibrium” internal energy, and A is determined by K and T:

$$ A\left( \theta \right) = - \frac{{{\theta ^2}}}{2}\frac{d}{{d\theta }}\left( {\frac{{Z\left( \theta \right)}}{{{\theta ^2}}}} \right),Z\left( \theta \right) = K{\left( \theta \right)^{ - 1}}T\left( \theta \right). $$

. It is also shown that the second law implies that Z(θ) is a symmetric tensor and that K(θ) is positive definite. It is observed that if Z(θ) and Z(θ)^-1 A (θ) are positive definite and ∂e/∂θ is positive, a temperature-rate wave, i.e., a singular surface across which there is a jump in θ, will travel faster if it propagates opposite to, rather than parallel to the heat flux.

An Italian language version of this paper appeared in Volume 68 of the Rendiconti del Seminario Matematico della Università di Padova under the title: >Il secondo suono nei cristalli: Termodinamica ed equazioni costitutive.