A 2D (two-dimensional) random cellular solid model was built using FEM (finite element method) based on a modified Voronoi tessellation technique. A sequence of permanent biaxial compression deformations was applied on the model to obtain a series of re-entrant random cellular solid structures with different area compression ratios. The Poisson's ratio and energy absorption capacity of cellular solid models with different initial relative densities (0.032 and 0.039) were studied at different area compression ratios. The results showed that the Poisson's ratio first decreased and then increased with increasing compression strain. A minimum Poisson's ratio of approximately -0.38 was achieved with an appropriate compression strain. An empirical fitting rule was established which can best fit the 2D simulation to 3D experimental results for foams. The cellular solids with minimum negative Poisson's ratio can exhibit the highest energy absorption capacity. Furthermore, mechanical properties of the random cellular solid model were compared with 2D regular honeycomb models with both concave- and convex-shaped cells. Results showed that the energy absorption capacity of the three models increased with an increasing dynamic compression velocity. The random foam model exhibited the highest increase rate in energy absorption capacity with the increasing compression velocity.