Existence results for a contact problem with varying friction coefficient and nonlinear forces

Abstract

We consider the rate‐independent problem of a particle moving in a three‐dimensional half space subject to a time‐dependent nonlinear restoring force having a convex potential and to Coulomb friction along the flat boundary of the half space, where the friction coefficient may vary along the boundary. Our existence result allows for solutions that may switch arbitrarily often between unconstrained motion in the interior and contact where the solutions may switch between sticking and frictional sliding. However, our existence result is local and guarantees continuous solutions only as long as the convexity of the potential is strong enough to compensate the variation of the friction coefficient times the contact force. By simple examples we show that our sufficient conditions are also necessary.Our method is based on the energetic formulation of rate‐independent systems as developed in [MTL02, MiT04]. We generalize the time‐incremental minimization procedure of [MiR06] for the present situation of a non‐associative flow rule.

Quasi‐static contact problem with finitely many degrees of freedom and dry friction

“…Hence a strong decrease of the frictional force can lead to a jump as simple examples already show, see Chap. 4 in [22] or the famous counterexample of Klarbring in [8].…”

“…In the foregoing paper [22] the case N = 1 was considered under the strong assumption α ∅ > q {1} . Due to α {1} ≥ α ∅ this strong assumption implies (N4), which is composed by α ∅ > 0 and α {1} > q {1} .…”

“…In this section we prove that the limit function z γ ∈ W 1,∞ ([0, T e ], A e ) of Theorem 4.8 represents a solution of the simplified problem. In literature this step is well known if the dissipation functional is continuous with respect to z, for example see [22]. Thus it is sufficient to show Ψ γ to be continuous on a set D, which contains all discrete solutions.…”

“…3 we present the principal novelty of this article: the introduction of a smoothened dissipation Ψ (j) γ depending on the parameter γ > 0, see (3.5) for a precise definition and a physical motivation. This simplification allows us to construct a solution z γ applying techniques similar to those found in [13,22]. The construction is carried out in Sect.…”

“…In [22] existence was established for a single particle system, i.e. N = 1, under a similar assumption α * > q * .…”