Thermodynamic relations for rubberlike elasticity on simple elongation

Abstract

Thermodynamic relations pertaining to rubberlike elasticity on simple elongation are derived systematically in terms of the four alternate sets of independent variables (T, V, L), (T, p, L), (T,V, α) and (T, p, λ), where λ is the elongation ratio and α is the same corrected for volume dilation. Formulae, which facilitate changes of independent variables from one set to another, are given. The volume dilation coefficient and the correction terms, required to evaluate the energetic contribution to the elastic force from measurements performed under constant pressure, are all derived as a function of the thermoelastic equation of state. Approximate expressions for estimating these quantities, which were proposed previously on the basis of either the infinitesimal theory of elasticity or the statistical theory of gaussian networks, are examined critically in light of the exact relations derived here; also the nature of the approximations involved is pointed out. Upon analysis of volume dilation data available in the literature, new approximate expressions for these quantities, which apply when the stress-strain relation obeys the Mooney-Rivlin equation, are proposed.