A model for contact of a rod with an obstacle using the damped normal compliance condition

Abstract This work models, analyses and simulates the contact process between a projectile, a linearly elastic rod and a reactive obstacle, using the ‘damped normal compliance’ contact condition. This condition explicitly takes into account the energy dissipation during the process. We establish the energy balance and study the possible steady states of the system, to which the long-time solutions converge. It is found that depending on the obstacle stiffness and damping, the rod may or may not leave completely the obstacle. The existence of a unique solution when viscosity is added is established, using the Banach fixed-point theorem, and in the limit, when the viscosity vanishes, the existence of a weak solution to the original model is obtained. The various theoretical results are depicted using computer simulations, which are based on a simple explicit finite-difference scheme. The convergence of the scheme is discussed. The simulations show that as the stiffness of the obstacle increases and the damping decreases, the penetration decreases, and the obstacle behaves as a rigid body. On the other hand, when the stiffness is low and the damping high, the system describes a rigid body that penetrates a soft material or tissue. The simulations also show the propagation of the waves generated in the rod.