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Using molecular-dynamics simulation we simulate nanoindentation into the three principal surfaces - the (100), (110), and (111) surface - of Cu and Al. In the elastic regime, the simulation data agree fairly well with the linear elastic theory of indentation into an elastically anisotropic substrate. With increasing indentation depth, the effect of pressure hardening becomes visible. When the critical stress for dislocation nucleation is reached, even the elastically isotropic Al shows a strong dependence of the force-displacement curves on the surface orientation. After the load drop, when plasticity has set in, the influence of the surface orientation is lost, and the contact pressure (hardness) becomes independent of the surface orientation.