Temperature, entropy and chemical potential

Characteristics of state, corresponding to temperature, entropy, and chemical potential are obtained [1 - 4] from pure mechanical reasons, by means of special mathematical formulation of the energy balance equation (2001), obtained by separation of the stress tensors in elastic and dissipative components. The second law of thermodynamics gives additional limitations for the introduced characteristics, and this completes their formal definition. The reduced equation of energy balance is obtained in the terms of free energy. The main purpose of this equation is to determine the arguments on which the free energy depends. It is shown that to define first the internal energy, and then the entropy and chemical potential, is impossible. All these quantities should be introduced simultaneously. To set the relations between the internal energy, entropy, chemical potential, pressure, etc., the reduced equation of energy balance is used. It is shown that the free energy is a function of temperature, density of particles, and strain measures, where all these arguments are independent. The Cauchy-Green's relations relating entropy, chemical potential and tensors of elastic stresses with temperature, density of particles and measures of deformation are obtained. Hence the concrete definition of the constitutive equations requires the setting of the free energy only.

The equations characterizing role of entropy and chemical potential in formation of internal energy are obtained. Constitutive equations for the vector of energy flux [3] are offered. In a particular case these equations give the analogue of the Fourier-Stocks law.

  1. Zhilin P.A. Basic equations of the theory of non-elastic media // Proc. of the XXVIII Summer School “Actual Problems in Mechanics”. St. Petersburg. 2001. P. 14-58. (In Russian.)
  2. Zhilin P.A. Phase Transitions and General Theory of Elasto-Plastic Bodies // Proceedings of XXIX Summer School - Conference “Advanced Problems in Mechanics”. St. Petersburg, Russia, 2002. P. 36-48.
  3. Zhilin P.A. Mathematical theory of non-elastic media // Uspehi mechaniki (Advances in mechanics). Vol. 2. N 4. 2003. P. 3-36. (In Russian.)
  4. Zhilin P.A. On the general theory of non-elastic media // Mechanics of materials and strength of constructions. Proc. of St. Petersburg State Polytechnical University. N 489. 2004. P. 8-27. (In Russian.)