  This paper reports a theoretical and numerical framework to model nonlinear
waves in elastic-plastic solids. Formulated in the Eulerian frame, the
governing equations employed include the continuity equation, the momentum
equation, and an elastic-plastic constitutive relation. The complete governing
equations are a set of first-order, fully coupled partial differential
equations with source terms. The primary unknowns are velocities and deviatoric
stresses. By casting the governing equations into a vector-matrix form, we
derive the eigenvalues of the Jacobian matrix to show the wave speeds. The
eigenvalues are also used to calculate the Courant number for numerical
stability. The model equations are solved using the Space-Time Conservation
Element and Solution Element (CESE) method. The approach is validated by
comparing our numerical results to an analytical solution for the special case
of longitudinal wave motion.
