A Perturbation Result for Dynamical Contact Problems

Abstract

This paper is intended to be a first step towards the continuous dependence of dynamical contact problems on the initial data as v^iell as the uniqueness of a solution. Moreover, it provides the basis for a proof of the convergence of popular time integration schemes as the Newmark method. We study a frictionless dynamical contact problem betvv'een both linearly elastic and viscoelastic bodies which is formulated via the Signorini contact conditions. For viscoelastic materials fulfilling the Kelvin-Voigt constitutive law, we find a characterization of the class of problems which satisfy a perturbation result in a non-trivial mix of norms in function space. This characterization is given in the form of a stability condition on the contact stresses at the contact boundaries. Furthermore, we present perturbation results for two well-established approximations of the classical Signorini condition: The Signorini condition formulated in velocities and the model of normal compliance, both satisfying even a sharper version of our stability condition.

Consistency results on Newmark methods for dynamical contact problems

“…The model is based on Signorini's contact conditions and has been described at length in [12] for the viscoelastic case, or in [4], respectively, for the purely elastic case. For the convenience of the reader, we here merely collect the notation used therein.…”

“…[12]). Both elasticity and viscoelasticity tensors are assumed to be sufficiently smooth, symmetric, and uniformly positive definite.…”

“…In what follows, we consider three variants of the Newmark method after space discretization. These three schemes have already been described in [4] for the purely elastic case, but-in view of our perturbation results in [12]-we here give the generalization to the viscoelastic case. In order to fix notation, let the continuous time interval [0, T ] be subdivided by N + 1 discrete time points…”

“…We derive an estimate for the local discretization error in a special norm suggested by our previous perturbation results in [12]. We start with the analysis of the discretization error for the Newmark scheme (N-M/CS), which we afterwards transfer to the classical scheme (N-CL).…”

“…In fact, the problem turned out to be really hard. As a preparatory step, we recently studied the stability of dynamical contact problems under perturbation of the initial data [12]. That study gave us the idea about a non-trivial mix of norms in function space which we will exploit here.…”