Fibre-reinforced polymer composites are widely used in structural applications due to their high specific stiffness and strength. In some applications the response of dynamically loaded composite components must be analysed. For example, in crash analyses of structural components, where very high loading rates occurs, the composite behaviour is not fully understood. For this, we present a novel transversely isotropic viscoelasticviscoplastic constitutive model for a unidirectional carbon-epoxy composite. The model is micromechanically motivated so that the matrix and fibre materials of the composite are treated as micromechanical constituents at the ply scale. Based on the Hill-Mandel condition, the phases are homogenized via the macroscopic and fluctuating strain fields. To arrive at a simple but still representative model, a simplistic ansatz is applied to the structure of the fluctuating strains leading to a non-standard homogenized response of the composite. The model is applied to the non-linear rate dependent anisotropic ply behaviour under quasi-static and dynamic loading at different off-axis angles. For a simple viscoelastic-viscoplastic prototype for the rate dependent matrix response, there is a good correlation between measured and model response of the IM7-8552 material system in compression and tension.