We present a constitutive model for particle-binder composites that accounts for finite-deformation kinematics, nonlinear elastoplasticity without apparent yield, cyclic hysteresis, and progressive stress-softening before the attainment of stable cyclic response. The model is based on deformation mechanisms experimentally observed during quasi-static monotonic and cyclic compression of mock plastic-bonded explosives (PBX) at large strain. An additive decomposition of strain energy into elastic and inelastic parts is assumed, where the elastic response is modeled using Ogden hyperelasticity while the inelastic response is described using yield-surface-free endochronic plasticity based on the concepts of internal variables and of evolution or rate equations. Stress-softening is modeled using two approaches; a discontinuous isotropic damage model to appropriately describe the softening in the overall loading–unloading response, and a material scale function to describe the progressive cyclic softening until cyclic stabilization. A nonlinear multivariate optimization procedure is developed to estimate the elastoplastic model parameters from nominal stress–strain experimental compression data. Finally, a correlation between model parameters and the unique stress–strain response of mock PBX specimens with differing concentrations of aluminum is identified, thus establishing a relationship between model parameters and material composition.