A 2-D sliding model of micro-scale was established between two rough surfaces exhibiting fractal behavior, and the stress and strain was discussed by using the finite element analysis. A pair of asperities was analyzed take into account elastic-plastic material properties with material failure. The transient stress/strain distribution of the rough solid was presented. The numerical results showed that the equivalent plastic strain versus different depth which at different locations showed different laws. Failure began beneath the surface at the region of highest plastic strain. A shear stress zone exited at a certain depth from the contact surface in the rear of the contact asperity. This shear stress zone on the one hand prevented the crack from extending along the depth direction, on the other hand, prompted the crack along an extension of the parallel to the direction of the friction surface crack growth and leads to final fracture.