A diffusion theory of the visco‐elasticity of rubbery polymers in finite elastic strain

Abstract

Abstract The molecular mechanism of stress relaxation in a linear polymeric material is conceived of as a process of diffusion in phase space of molecular configuration. The analysis is an extension of the Rouse theory of linear visco‐elasticity to finite, three‐dimensional deformations. A molecule, either open end or crosslinked, is treated as a chain of submolecules, each just long enough that its net length follows approximately the Gaussian distribution law. Variables are separated by transformation to normal coordinates, and diffusion equations are obtained of the form where ρ n is the density in configuration space of the n ‐mode of molecular configuration, and D n is the diffusion constant characteristic of the mode. An exact solution exists in the form Here β n − 1 is the measure of the recoverable strain in the u n direction of the n ‐mode, the instantaneous value of the corresponding partial stress being proportional to β n 2 − 1. Any partial stress obeys the Boltzmann superposition law, but the total stress does not. The transform equations connecting such pairs of functions as creep and relaxation in the linear theory are here not valid; but, as in the linear theory, in a given case of a single imposed strain the stresses are expressed in terms of a distribution function of relaxation times. Expressions are obtained for the tangential and normal stressed in continuous simple shear at a finite rate. To explain non‐Newtonian viscosity it is necessary to postulate either that the intertangling of the molecules is reduced or that rheological flow units develop in the material which are larger than the individual molecules.