The study presents the methodology and results of investigation into the viscoelasticity of polymethyl methacrylate (PMMA) and polyamide PA-6 in tensile and compressive testing. The experiments enabled the separation of the volumetric and deviatoric components of the strain tensor. A significant dependence of the shear modulus and a weak dependence of the bulk modulus on hydrostatic stress were revealed. The features of PMMA viscoelastic deformation were studied over a wide range of operating temperatures. It was found that the ratio of the elastic and viscous strain intensities is weakly dependent on both hydrostatic stress and temperature of the medium. Constitutive equations are presented for viscoelasticity under a volumetric stress state at various combinations of elastic and viscous strain rates in the stress range from the creep limit to the forced elasticity limit, as well as during recovery after complete unloading. The equations do not explicitly include time and accumulated viscous strain, so they are applicable to processes with arbitrary strain or stress growth patterns. Comparison with experiment confirmed the high modeling accuracy of viscous PMMA deformation for prescribed law of elastic strain change. Several successive stages of viscoelastic deformation during loading and subsequent unloading were revealed. At the first stage, viscous strain does not increase, at the second stage, it increases only with increasing load, at the third stage, it also increases under constant load. Upon load reversal, the viscous strain rate decreases to zero, then changes sign and increases, approaching the elastic strain rate. The developed mathematical apparatus is intended for modeling the cyclic alternating loading of a spherical shell with internal and external pressure under operating conditions of a manned submersible vehicle.