Thermodynamic formulation of the constitutive equations for solids and fluids

Abstract

A general thermodynamic approach to material constitutive equations is presented and analyzed. The discussion is developed on piezoelectric and magnetostrictive materials as they pose modeling complexities that are suitable for the development of a comprehensive theory. The contributions of the work comprise: a general treatment of the problem, which includes kinetic energy, thermal effects and material internal dissipation in the energy balance; a simple matrix approach to clearly describe the thermoelastic, piezoelectric, pyroelectric, magnetostrictive and pyromagnetic relations; an analysis of the possible material properties involved in the definition of the constitutive equations; the introduction of a modified expression for the complementary strain–energy density, which allows the state equations commonly used in fluid dynamics to be derived from the constitutive equations obtained for piezoelectric crystals.Eight major options have been identified for the selection of the independent variables in piezoelectric non-linear crystals (the same holds for magnetostrictive materials), consequently, eight distinct thermodynamic potentials can generally be identified, resulting in eight different sets of constitutive equations. However, when linear materials are considered and thermal effects are disregarded, the thermodynamic potentials that can be used to derive the constitutive relations can always be reduced to a free-energy type function: this decreases the number of definable constitutive properties to a great extent.The developed concepts have been applied to evaluate the differences that can occur in the physical properties of solids and liquids as the operation mode changes from isothermal to adiabatic or from isochoric to isobaric. Significant differences have generally been observed between isothermal and isentropic values in the piezoelectric properties of materials that exhibit strong pyroelectric and thermal expansion effects. The thermal effects are generally unimportant for Young’s modulus, whereas they are remarkable for bulk modulus of elasticity, even though the latter is a mechanical parameter of interest only for fluids. Finally, noteworthy differences have been found between the principal specific heats: the isobaric to isochoric specific heat ratio does not depend on the phase (whether liquid or solid), and should be evaluated case-by-case.