We propose a nodal high-order discontinuous Galerkin method for 1D wave propagation in nonlinear media. We solve the elastodynamic equations written in the velocity-strain formulation and apply an upwind flux adapted to heterogeneous media with nonlinear constitutive behavior coupling stress and strain. Accuracy, convergence and stability of the method are studied through several numerical applications. Hysteresis loops distinguishing loading and unloading-reloading paths are also taken into account. We investigate several effects of nonlinearity in wave propagation, such as the generation of high frequencies and the frequency shift of resonant peaks to lower frequencies. Finally, we compare the results for both nonlinear models, with and without hysteresis, and highlight the effects of the former on the stabilization of the numerical scheme.