Microscopic constitutive equation theory for the nonlinear mechanical response of polymer glasses

Building on recent statistical mechanical theories of segmental activated barrier hopping and elasticity in quiescent and stressed polymeric materials, we construct a constitutive equation description at a generalized Maxwell model level for the nonlinear mechanical response of polymer glasses. The key physics is contained in a deformation-dependent elastic modulus and alpha relaxation time. Calculations of the temperature and strain rate dependence of the stress-strain curves, yield stress, and yield strain are presented for atactic poly(methyl methacrylate) (PMMA). The strain rate dependences of these properties are roughly logarithmic with upward deviations at very high rates. Yield stresses grow approximately linearly as temperature is lowered. Stress-strain curves at different temperatures and strain rates can be collapsed onto a master curve based on nondimensionalizing stress by the plateau yield stress and strain by an effective critical strain. Quantitative comparisons of the theory with several experimental measurements of the elastic modulus and stress-strain response of PMMA glasses reveal good agreement. Predictions are also made for how deformation reduces the segmental relaxation time. In the plastic flow regime a "shear thinning" type of dependence of the alpha time on strain rate occurs corresponding to an effective power law decay with strain rate. Proper nondimensionalization results in a master curve that collapses the temperature dependence. In the dynamic yielding regime the segmental relaxation time is predicted to be (much) shorter than the inverse strain rate. Hence, our results provide theoretical support for the qualitative idea that stress-induced local plastic flow can be thought of as a "devitrification transition". The implication of our results for the question of beyond what length scale a polymer glass deforms affinely is briefly addressed.