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This paper presents a systematic computational study to investigate the effects of crystal orientation, strain rate (impact velocity), and size (thickness) on plasticity and damage behavior of copper single crystals during the penetration process at the atomistic scale. For the penetration analysis, copper single crystals with different crystal orientations and thicknesses were impacted and penetrated by a cylindrical nickel penetrator at different initial velocities. Modified embedded atom method potentials were used to develop atomistic models, and over 250 molecular dynamics simulations were performed to fully reveal the effects of influence parameters on plasticity and damage behavior of the copper single crystals. The results show that the copper single crystal with an octal slip orientation exhibits the greatest strength and penetration resistance, while the copper crystal with the single slip orientation exhibits the lowest strength and resistance. The results further show that the strength and penetration resistance of the target increase as the thickness of the copper single crystals increases. Furthermore, as the impact velocity increases, damage and fragmentation increase. Conclusions drawn from this computational study are consistent with macroscale plasticity theories of metals and reaffirm the conclusions drawn by other researchers in previous experimental studies.