Abstract The paper presents a phenomenological material model for a superimposed elastic–viscoelastic–plastoelastic stress response with damage at large strains and considers details of its numerical implementation. The formulation is suitable for the simulation of carbon-black filled rubbers in monotonic and cyclic deformation processes under isothermal conditions. The underlying key approach is an experimentally motivated a priori decomposition of the local stress response into three constitutive branches which act in parallel: a rubber–elastic ground–stress response, a rate-dependent viscoelastic overstress response and a rate-independent plastoelastic overstress response. The damage is assumed to act isotropically on all three branches. These three branches are represented in a completely analogous format within separate eigenvalue spaces, where we apply a recently proposed compact setting of finite inelasticity based on developing reference metric tensors. On the numerical side, we propose a time integration scheme which exploits intrinsically the modular structure of the proposed constitutive model. This is achieved on the basis of a convenient operator split of the local evolution system, which we decouple into a stress evolution problem and a parameter evolution problem. The constitutive functions involved in the proposed model are specified for a particular filled rubber on the basis of a parameter identification process. The paper concludes with some numerical examples which demonstrate the overall response of the proposed model by means of a representative set of numerical examples.