Taking the strain rate as an environmental factor, an incremental stress–strain–strain rate relation of concrete is established based on the proposed multi-physical modeling method. By incorporating strain rate into the strength parameters, a dynamic closed yield function is proposed, which can reasonably determine the elastic domain boundaries under different strain rates. The strain rate is treated as an independent variable when a consistency condition is performed on the yield function to determine the plastic multiplier. Therefore, the plastic deformation caused by the independent change of strain rate can be captured. The non-orthogonal gradient of the yield function defined the direction of the plastic flow. Additionally, the corresponding loading/unloading criterion is established to judge the conditions of triggering plastic deformation of concrete materials under the joint drive of strain and strain rate. By contrasting model simulations with experimental findings obtained by the author and published papers, including tests with various stress states and strain rate ranges, the model's competence is assessed. In particular, the results for a design experiment with a rapidly changing strain rate demonstrate the model’s ability to capture stress state changes caused by strain rate variations, ensuring calculations consistently satisfy the consistency condition—a capability beyond conventional dynamic stress-strain models. The established model will show good performance in predicting the mechanical properties of concrete subjected to explosion or impact with a sharp change in strain rate.