The constitutive models at high loading rates are usually investigated by introducing additional and empirical rate-dependent components into classical models initially developed for quasi-static cases. In addition, the dynamic stress–strain curves of many materials are non-monotonic. The yield drop phenomenon can be observed in series of experiments and often neglected in the simulation of stress–strain diagrams. The structural-temporal approach proposed by Petrov and Morozov provides a new viewpoint for understanding the dynamic response of materials. They indicate that in the limit case of low-rate loading it transfers into the classical quasi-static models by neglecting certain insignificant components. This work aims to derive a constitutive model from the structural-temporal approach rather than using a direct empirical fitting approach. In this paper, the incremental version of the relaxation plasticity model (IRP model) is introduced within the framework of the structural-temporal approach. Possible model scenarios are discussed and compared with the well-known Johnson–Cook model. Theoretical results under uniaxial loading conditions are obtained and compared with experimental data for the aluminium alloy 6082-T6, API 5L X70 steel, titanium alloy Ti–6Al–4V, and aluminium alloy 2519A. It is shown that the IRP model describes the non-monotonic behaviour of the stress–strain relationship and performs better than the Johnson–Cook model for mentioned materials.