A finite volume cell-centered Lagrangian hydrodynamics approach, formulated in Cartesian frame, is presented for solving elasto-plastic response of solids in general unstructured grids. Because solid materials can sustain significant shear deformation, evolution equations for stress and strain fields are solved in addition to mass, momentum, and energy conservation laws. The total stress is split into deviatoric shear stress and dilatational components. The dilatational response of the material is modeled using the Mie-Grüneisen equation of state. A predicted trial elastic deviatoric stress state is evolved assuming a pure elastic deformation in accordance with the hypo-elastic stress-strain relation. The evolution equations are advanced in time by constructing vertex velocity and corner traction force vectors using multi-dimensional Riemann solutions erected at mesh vertices. Conservation of momentum and total energy along with the increase in entropy principle are invoked for computing these quantities at the vertices. Final state of deviatoric stress is effected via radial return algorithm based on the J-2 von Mises yield condition. The scheme presented in this work is second-order accurate both in space and time. The suitability of the scheme is evinced by solving one- and two-dimensional benchmark problems both in structured grids and in unstructured grids with polygonal cells. Copyright © 2013 John Wiley & Sons, Ltd.