Due to the significant thermal-mechanical effects during hot spot formation in polymer-bonded explosives (PBXs), a coupled mechanical-thermal constitutive relationship was developed to predict the mechanical behaviors and the thermal reactions of polymer-bonded explosives (PBXs), which includes viscoelastic responses, the Visco-SCRAM model, equations of state, and the Arrhenius first-order chemical kinetics model. The proposed model was implemented in the finite element code ABAQUS using the user subroutine VUMAT. Split Hopkinson pressure bar (SHPB) and confined Steven test simulations for PBX 9501 were performed to verify the model and understand the stress and temperature fields under different low impact velocities. The results predicted by the proposed model agreed well with the experimental results. This suggested that the model could be used to better simulate the thermal-mechanical responses of explosives. Moreover, compared with the previous analysis models and methods, the method in this paper has wider applicability because it does not require an empirical ignition criterion and hotspot model. It can provide a reference for the study of the thermal-mechanical responses of explosives and the localized ignition problem under low amplitude impact scenarios.