Molecular dynamics simulations are used to investigate the micro-mechanisms of nano-indentation for tip to substrate contact. The method combines a many-body interatomic potential derived from the nearest-neighbor EAM and brownian dynamics (BD) approach to simulate a rigid tip indenting Cu (001) surface. Elastic contact and plastic instability of the crystal are investigated through the loading-unloading cycle, the variations of the system potential energy versus the tip approach, the atomic stress distributions and the portraits of atomic trajectories and configurations. For elastic indentation, we find that atomistic stress distributions resembling roughly to those of the continuum Hertzian fields, except for a jump-to-contact phenomenon in the initial contact stage. When the tip approach is beyond some critical value, plastic instability of the substrate occurs, and both the contact load and potential energy decrease dramatically. Detailed calculations reveal that material yield at the atomic level is still governed by the von Mises shear strain-energy criterion, while atomistic trajectories show that the displacements in (010) plane of atoms near the contact region is similar to that in Johnson＇s cavity model, accompanied by atomic cross-layer movements in [010] direction to release the strain energy. The crystal defects after plastic indentation include subsurface cavities, surface atomic steps and plastic indent.