A Numerical Remark on the Time Discretization of Contact Problems in Nonlinear Elasticity

Abstract

The time discretization of contact-problems in elasticity is a difficult task, since the non-penetration condition at the contact interface can lead to instabilities in displacements, stresses, and energy. For the case of linear elasticity, in (Deuflhard et al., Int J Numer Methods Eng 73(9):1274–1290, 2008), a contact stabilized Newmark scheme has been proposed, which employs a discrete L 2-projection at the contact boundary for stabilization and which can shown to be energy dissipative. Here, we combine this contact-stabilization with an approach presented on (Gonzalez, Comput Methods Appl Mech Eng 190(13–14):1763–1783, 2000) for the time discretization of unconstrained problems in nonlinear mechanics. We apply the resulting combined scheme to contact problems with non-linear non-penetration constraints and non-linear material laws and numerically investigate its behavior. Although our combined scheme is not proven to be energy dissipative, it does not show any decrease in energy and the resulting displacements and forces at the contact boundary show a highly stable behaviour.