On the local form of the second law of thermodynamics in continuum mechanics
The Clausius-Duhem inequality a(r, t) > 0, a widely adopted axiom in continuum mechanics, leads to the conclusion that for many materials the entropy s cannot depend on gradients like the temperature gradient g and the velocity gradient e. But this is at variance with the received view (since Gibbs) that entropy is a function of thermodynamic state, however detailed that state description may be. Gradients, and even higher derivatives of macroscopic variables, may be included as state variables (although only on macroscopic time scales shorter than or comparable with their natural relaxation times), and the fundamental property of entropy is its convexity—the more detailed the specification of state, the smaller is the corresponding value of entropy. The entropy of a perfect monatomic gas is evaluated via the maximum principle, on the assumption that g and e are state coordinates, and it is found that s does depend on g • g and e : e, contrary to statements appearing in many textbooks on continuum mechanics. The source of the error in these works is shown to lie in applying <r > 0 to relations involving second derivatives. The correct form of the Clausius-Duhem inequality contains only first-order derivatives; that is, it must be confined to linear constitutive relations. In this form the inequality is a consequence of the convexity of s, which is a somewhat more general manifestation of the second law.