Entropy principle, non-regular processes, and generalized exploitation procedures

Abstract

The classical Coleman-Noll approach to the exploitation of the entropy principle regards the classical balances of mass, linear and angular momentum and energy as differential constraints for the entropy inequality, and presupposes that the second law of thermodynamics is a restriction on the constitutive equations describing the material properties [B. D. Coleman and W. Noll, “The thermodynamics of elastic materials with heat conduction and viscosity,” Arch. Rational Mech. Anal. 13, 167–178 (1963)10.1007/BF01262690]. In 1996, Muschik and Ehrentraut proved that this presupposition may be confirmed by a rigorous proof, provided that an amendment to the classical second law of thermodynamics, which asserts that, except in equilibria, reversible process directions in state space do not exist, is postulated [“An amendment to the second law,” J. Non-Equilib. Thermodyn. 21, 175–192 (1996)10.1515/jnet.1996.21.2.175]. In their paper, the authors considered regular processes only. In a recent article [V. Triani and V. A. Cimmelli, “Interpretation of second law of thermodynamics in the presence of interfaces,” Continuum. Mech. Thermodyn. 24, 165–174 (2012)10.1007/s00161-011-0231-8], we proved that the result above remains valid in the presence of interfaces across which the unknown fields suffer jump discontinuities. Here, we show that the same conclusions achieved by Muschik and Ehrentraut and Triani and Cimmelli hold when the classical Coleman-Noll and Liu [“Method of Lagrange multipliers for exploitation of the entropy principle,” Arch. Rational Mech. Anal. 46, 131–148 (1972)10.1007/BF00250688] procedures for the exploitation of the second law, are generalized by considering also the gradients of the fundamental balance equations as constraints for the entropy inequality.