In this study a phenomenological constitutive model is proposed to describe the finite, nonlinear, viscoelastic behavior of glassy polymers up to the yield point. It is assumed that the deformation behavior of a glassy polymer up to the yield point is completely determined by the linear relaxation time spectrum and that the nonlinear effect of stress is to alter the intrinsic time scale of the material. A quantitative three‐dimensional constitutive equation for polycarbonate as a model polymer was obtained by approximating the linear relaxation time spectrum by eighteen Leonov modes, all exhibiting the same stress dependence. A single Leonov mode is a Maxwell model employing a relaxation time that is dependent on an equivalent stress proportional to the Von Mises stress. Furthermore, a Leonov mode separates the (elastic) hydrostatic and (viscoelastic) deviatoric stress response and accounts for the geometrical complexities associated with simultaneous elastic and plastic deformation. Using a single set of parameters, the multi‐mode Leonov model is capable of describing realistic constant strain rate experiments, including the strain rate dependent yield behavior. It is also capable of giving a quantitative description of nonlinear stress‐relaxation experiments.