We introduce a high-order cell-centered Lagrangian scheme for one dimensional elastic-plastic problems. The problem with the hypo-elastic constitutive model and the von Mises yield criterion is a classical problem and there are many numerical methods for it, including staggered Lagrangian schemes, Eulerian Godunov schemes and cell-centered Lagrangian schemes. But in these methods the wave structure of the problem is not shown in detail. Because thehypo-elastic constitutive model is a non-conservative equation, it is difficult to build approximate Riemann solvers for the governing equations with the hypo-elastic constitutive model. In this paper, we analyze the wave structure of the Riemann problem for elastic-plastic materials and then develop two-rarefaction-wave approximate Riemann solvers. Based on the developed Riemann solvers, we propose the second-order cell-centered Lagrangian scheme for one-dimensional elastic-plastic solid problems. Moreover, we use a second-order scheme to discretize the constitutive equation based on geometry conservation law in order to keep high accuracy. A number of numerical experiments are carried out, and the numerical results are compared with the ``exact'' solution and the results obtained by other authors. The comparison shows that the current scheme is convergent, stable and essentially non-oscillatory. For shock waves the current scheme is almost as good as Maire et al.'s scheme \cite{cc3} in accuracy and resolution, while for rarefactionwaves the current scheme performs better.