The study of the inelastic deformation behavior of six amorphous and semicrystalline polymers was performed to develop and verify the capabilities of a constitutive material model. The test conditions consisted of piecewise constant strain rates for loading and unloading. Immediate control mode switching capability permitted using load control for creep and recovery tests. Positive, nonlinear rate sensitivity was observed in all cases for monotonic loading and the prior loading rate was found to have a strong influence on creep, relaxation and strain recovery (emulating creep at zero stress) tests. In particular, a fast prior rate engenders a larger change in the output variable: strain in conditions of creep and stress drop in relaxation. Based on the absence of any distinctive deformation traits, the preponderance of data collected in the experimentation program suggests that both categories of polymers can be modeled using the same phenomenological approach. Modeling of the experimental data is introduced with a uniaxial form of the Viscoplasticity Theory Based on Overstress for Polymers (VBOP). Simulations and model predictions are provided for various loading histories. Additional modifications necessary to extend the theory to finite deformation and inelastic compressibility are then presented. An objective formulation is obtained in the Eulerian framework together with the recently proposed logarithmic spin by Xiao [Xiao, H., Bruhns, O., and Meyers, A., 1997, “Hypoelesticity Model Based Upon the Logarithmic Stress Rate,” J. Elast., 47, pp. 51–68].