A new, small-strain constitutive model, incorporating elastoplastic coupling to describe developing elastic anisotropy, and density as a state variable to capture compaction and dilation, is proposed to simulate the behaviour of granular materials, in particular sand. This developing elastic anisotropy is related to grain reorientation and is shown to be crucial to obtain localisation during strain hardening, as experiments exhibit. Post-localisation analysis is also performed under simplificative assumptions, which evinces a number of features, including softening induced by localisation, size effects and snap-back, all phenomena found in qualitative and quantitative agreement with experiments. No prior model of granular material deformation correctly captures all these behaviours. The post-localisation analysis has revealed a new form of material instability in granular materials, consisting of a saturation mechanism, in which shear bands just formed unload, permitting new bands to form. This phenomenon shares similarities with the mechanics of phase transformation in metal strips and results in a stress oscillation during increasing deformation. The investigation of this mechanism of localised deformation reveals that loose and dense sands behave in qualitatively different ways. In particular, saturation is not persistent in dense sand; rather, after several shear bands form and saturate, this process is terminated by the formation of a differently inclined shear band occurring in the material transformed by previous strain localisation. In this case, the resulting ‘global’ stress–strain curve exhibits a few stress oscillations followed by a strong softening. On the other hand, band saturation is found to be a persistent phenomenon in loose sand, yielding a continuing stress oscillation. This provides a consistent description of specific experimental results.