The constitutive behaviour of amorphous polymers near the glass transition contains many features general to this class of materials. A new, physically based, three-dimensional constitutive model has been developed for simulating this wide range of features in models of polymer products and processes. In particular, the model displays glassy response at low temperatures and short time-scales, and rubber-like response at high temperatures and long time-scales, and is therefore an example of a glass-rubber constitutive model. Its basis is the assumed additivity of free energies of bond distortion and conformation perturbation. For the elastic bond distortion stress-strain law and flow model, and the conformational entropy function, the model employs linear elasticity, Eyring viscous flow and the Edwards-Vilgis entropy function, respectively. Glass structure and temperature dependence are introduced through the Vogel-Tammann-Fulcher and Arrhenius equations for viscosity, respectively. With parameters obtained for poly(ethylene terephthalate) in a companion paper, the model was solved numerically to simulate a variety of uniaxial strain sequences, and found to replicate well the characteristic patterns of behaviour of amorphous polymers in the temperature region of interest, over a wide range of experimental situations from small to large strains. The only major deficiency, resulting from the simplifying assumption of a single activation barrier height, is its being too localized in the time domain. At high temperatures and long times, applicability of the model is limited by the onset of conformational relaxation.