Abstract: According to the clear segmental characteristics of the stress curve of red sandstone under impact load, a constitutive model with the nonlinear body, Maxwell body and damaged body in parallel is established based on the Zhu-Wang-Tang(Z-W-T)constitutive model. According to the mechanical properties of red sandstone, the Z-W-T model is simplified, and another rate-type damage constitutive equation is derived based on the Lemaitre equivalent strain hypothesis. The applicability of two new constitutive equations is discussed and the influence of each parameter on the fitting result of the experimental data is compared and analyzed. The results show that the nonlinear viscoelastic damage constitutive equation based on Weibull distribution can well characterize the dynamic mechanical properties of pre-peak stress and the whole stress curves at room temperature and freezing temperature. However, the damage constitutive equation based on Lemaitre's hypothesis of equivalent strain can only be used to solve the pre-peak stress of red sandstone, and has some limitations in the characterization of the whole stress curves.