LETTERE A:L NIJOVO CIMENTO VOL. 16, N. 15 7 Agosto 1976 Some Remarks on the Local Form of the Second Principle of Thermodynamics. A. PAGLIETTI Mathematical Institute, University o] Ox]ord - 24-29, St. Giles, Ox]ord 0X1 3LB, England (ricevuto il !2 Maggio 1976) Consider a continuous body made up of a set of particles X. Let ~, ~1, 0, s and h denote the mass density, the specific entropy, the absolute temperature, the specific heating supply and the heating flux vector. These quantities depend, in general, on the particle X and on time t. Assume that s is positive when it represents an amount of heat absorbed by the body in the unit time and that, moreover, h is outward directed with respect to the body surface whenever it rcprescnts an amount of heat which goes into the body in the unit time. The following local form of the so-called Clausius- Duhem inequality s 1 divh+ 1 h.gradO~O (1) ~l o - e o oo ~ is postulated as the second principle of thermodynamics in a number of works on continuum mechanics (see the articles quoted in 0) for a bibliography on the argument). Whereas it is sometimes discussed whether inequality (1) represents the most general form of the second principle, nobody seems to have observed up to now that the above inequality cannot be the expression of that principle at all, at least until we agree that the second principle is a physical postulate which forbids the occur- rence of perpetual motions of the second kind. Since we shall show that relation (1) does not exclude the possibility of such motions, it follows that it cannot be regarded as the mathematical formulation of the second principle. Let T be the symmetric Cauchy stress tensor and let F be the deformation gradient. The stress working per unit volume is given by (2) w = tr (TFF -~) . The local form of the first principle of thermodynanfies can then be expressed in the well-known form (3) ~i=w+divh+ Os, (1) ~V. NOLI,: The role o] the second law o/ thermodynamics in classical continuum physics, in ~llodern Developments in Thermody~amics, edited by B. GAL-OIr (Ncw York, N.Y., 1974), p. 117. 475