The viscoelastic Poisson′s ration master curve of solid propellant was established on the basis of the time-temperature equivalence principle and WLF equation, and then the viscoelastic constitutive model of solid propellant was proposed. The viscoelastic constitutive model in incremental form was deduced on the basis of the incremental finite element method and by adopting the fully explicit integration method. The constitutive model was realized in finite element software by using the consistent tangent stiffness acquired according to the programming rules of user subroutine in MSC.Marc. The stress-strain mechanical response of solid propellant grains was analyzed by using the viscoelastic and the elastic Poisson′s ration under solidification process and ignition pressurization process. Furthermore, the effects of different kind of Poisson′s rations on the stress-strain field were compared. The method and result can provide a reference for the structural integrity analysis of engine grain.