Nonlinear Rubberlike Viscoelasticity—A Molecular Approach

The difference between the dynamic and equilibrium tension in a macromolecular chain is expressed as a ``functional'' of the variation of the nonlinear strain function φ(r*) developed in the kinetic theory of elasticity, with respect to time through the interval (0,t) r*≡r/rm where r is the vectorial end-to-end distance of the molecular chain and rm is the maximum separation). The ``functional'' is expanded in an integral series analogous to Taylor's series and higher terms are neglected to obtain a linear integral equation for the viscously retarded response of the network chain. The equation obtained is a generalized one-dimensional Boltzmann's superposition equation. It is then shown that the time-dependent response of the molecular chain is independent of the magnitude of the deformation and, consequently, is of the same analytical form whether the deformation is infinitesimal or finite. From this it necessarily follows that there cannot be an inconsistency at finite stress and strain which is not allowed at infinitesimal excitations. Thus the response at finite excitations can be treated generally by employing the ``generalized'' superposition equation and the same techniques which have been utilized in the linear theories. Employing the usual kinetic theory assumptions, equations are developed for the macroscopic response of a well-vulcanized rubber. Experimental data obtained in creep, stress relaxation, and dynamic stress-strain for three different elastomers are presented which support the approach outlined. Some consequences of the theory are discussed.