Based on the linear hardening plastic constitutive model, the theoretical solution of the elastic-plastic spherical stress wave field under impact load is established. Firstly, the influence of the impact load unloading rate on the propagation of the spherical stress wave is analyzed, and three different types of propagation images are obtained. On this basis, the method of theoretically calculating the spherical wave equation in elastic stage, plastic loading stage and unloading stage is established separately, and the calculation scheme of particle displacement, particle velocity, stress and strain is given. Compared with the existing theoretical methods, this method takes into account the different propagation patterns of the stress waves under different unloading rates, and shows how to calculate the stress wave parameters in unloading stage, which is more applicable. This method is used to calculate the elastic-plastic spherical stress wave field under constant shock load and exponential attenuation shock load. The calculated results are in good agreement with those from the existing theoretical method and numerical simulation results in the elastic stage and also in the plastic loading stage. In the unloading stage, the existing theoretical method is no longer applicable, while the results obtained in this paper are in good agreement with the numerical simulation results, which verifies the correctness of the theoretical method.