Analytical solutions are obtained for stress, velocity, and strain at the wave front in a suddenly loaded semi-infinite rod of a material with a linear instantaneous response and nonlinear inelastic response. The material properties are assumed to depend on position directly or through a dependence on a prescribed nonuniform temperature field. Detailed solutions are obtained for two examples—a nonlinear viscoelastic material with temperature dependent parameters and a rate-sensitive plastic material which may have temperature dependent parameters and yield point.