The nature of physical ageing is illustrated by an overview of the literature and by explanatory experiments with amorphous polyurethane samples. Thereby, the idea is motivated to use the history of the volumetric strain to describe physical ageing. This concept is related to former ideas out of the literature. On the basis of a deviatoric–volumetric split of the Helmholtz free energy, the volumetric strain is introduced in the constitutive model approach. The whole set of constitutive equations is derived from the free energy in combination with the Clausius–Duhem inequality. To predict the temporal development of the volumetric strain, i.e. the freezing and unfreezing during passing the glass transition and the long-term relaxation below the glass transition, a linear thermo-visco-elastic model is used which is enhanced with thermal strain elements. The evolution equations for the deviatoric viscous strains in dependence on the current and the equilibrium volumetric strain are able to model physical ageing. The properties of the developed model are shown by means of numerical examples.