Elastic equations of state and force–deformation relations

Abstract

An equation of state is an equation interrelating the various properties required to characterize a system. The elastic equation of state for a rubber network thus specifies the relationship between the applied forces, the resulting deformations, and the molecular structure of the network. It is obtained according to the thermodynamic expression τ_t = V⁻¹ λ_t (∂ΔA_el/∂λ_t)_T,V, for t = 1, 2, 3, where τ_t is the "true" stress along the tth coordinate direction, defined as the force per unit deformed area (Flory, 1961). ΔA_el is the elastic free energy. The quantity λ_t is the ratio of the final length to the reference length along that direction. The subscripts T, V indicate that the differentiation is performed at fixed temperature and volume. The index t may take values 1, 2, 3, or x, y, z. The equation may conveniently be interpreted by considering a prismatic block of a network with sides L₀₁, L₀₂, and L₀₃, and volume V₀ in the reference state that represents the state of formation of the network. In the case of network formation in solution, V₀ represents the total volume of polymer and solvent. The dimensions L₁₁, L₁₂, and L₁₃ give the network at the start of the experiment, where the volume V may be different from V₀ depending on the amount of solvent present relative to that during formation. The deformed state occurs under forces f₁, f₂, and f₃ applied along the three coordinate directions. The equality of the volumes before and after the application of the stresses is the result of assuming incompressibility: experimental data in simple tension show that the volume change in ordinary networks is negligibly small in comparison to changes in linear dimensions.