Abstract: Using the similarity between the transformation of viscoelastic materials constitutive relation and elastic materials, the Laplace transform solutions of elastic modulus and Poisson’s ratio of fractional Kelvin viscoelastic model were obtained. The solid propellant grain was regarded as viscoelastic medium, and the stress-strain relationship of solid propellant grain was described by the fractional derivative Kelvin constitutive model. The Laplace solutions of the stress of solid propellant grain described by fractional derivative viscoelastic model under uniform internal pressure were obtained by using the elastic-viscoelastic correspondence principle based on the elastic solution of stress, and the solutions in time domain were also got by the inverse Laplace transform. The results indicated that the radial stress is always compressive stress and the circumferential stress is always pulling stress, and the solution of fractional derivative Kelvin viscoelastic model can be degenerated to the solution of classic Kelvin viscoelastic, and the absolute value of stress will be greater if the order of fractional derivative is greater