On the Necessity of Positive Semi-Definite Conductivity and Onsager Reciprocity in Modeling Heat Conduction in Anisotropic Media

Abstract

It is demonstrated by a concise standard derivation, motivated by principles of rational continuum mechanics and irreversible thermodynamics augmented by novel detailed examples, that for heat conduction in linearly anisotropic solids: (1) common restrictions placed on the form of the thermal conductivity tensor are insufficient to guarantee satisfaction of the second law of thermodynamics, and (2) satisfaction of the first and second laws of thermodynamics alone is still insufficient to insure agreement between heat flow predictions and observation. An additional constraint beyond that given in many standard studies, namely that all three principal invariants of the conductivity tensor be positive semi-definite, is imposed in order to guarantee satisfaction of the entropy inequality. Thus constrained, such a theory remains under-restricted and can admit purely cyclic heat fluxes, which are not observed in nature. Imposition of the conjectures of Duhamel and Stokes, which are in fact earlier specific incarnations of Onsager’s reciprocity theory, on the constitutive model relating heat flux to temperature gradient is a sufficient remedy.