In this paper studied is the crises in the Duffing vibro-impact oscillator with non-viscously damping by the composite cell coordinate system method. It is assumed that the non-viscously damping depends on the past history of the velocities other than the instantaneous generalized velocities. The energy dissipation behaviors of real structural materials can be preferably represented in the non-viscously damping models. Numerical simulations show that as the damping coefficient or the relaxation parameter or the recovery coefficient is varied, there appear two kinds of crises: one is the interior crisis, which results from the collision between a chaotic attractor and a chaotic saddle on the basin boundary, and the other is the regular/chaotic boundary crisis, which is due to the collision of a chaotic attractor with a periodic/chaotic saddle on its basin boundary. All the crises result in a sudden change in size and shape of the attractor.