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**Example text**

Proof. Ni ) = 0, ∀ i ∈ IC , ˙ = u1 . u(0) = u0 , u(0) u ¯ u ˜ . (7) The second equation together with the contact condition uniquely deﬁne u ˜ as soon as u ¯ is given. Furthermore, u ˜ depends Lipschitz continuously on u ¯. The ﬁrst equation is a lipschitz ordinary diﬀerential equation. On the discretization of contact problems in elastodynamics 35 Fig. 1. A disc before and during the ﬁrst contact. Theorem 3. Problem (1) with the equivalent mass matrix is energy conserving. We refer to [1] for a completely proof.

N ˙ i ) = 0. N (2) The expression (2) in terms of velocity is very close to the one introduced in [7] and corresponds to the one introduced in [5]. The velocity is to be understood as a right derivative and the second condition implies in fact the non-interpenetration. We discretize the elastodynamic part in Problem (1) by a midpoint scheme as follows: On the discretization of contact problems in elastodynamics 0 u et v 0 donn´es, 1 u = u0 + ∆t v 0 + ∆t z(∆t )/ lim z(∆t ) = 0, ∆t −→0 un+1 − 2un + un−1 un+1 + 2un + un−1 M +K 2 4 ∆t i,n λ N , ∀ n ≥ 1, = f + i N 33 (3) i∈IC where ∆t is the time parameter.

Thus, 3D contact which can be seen as an interaction between two surfaces is reduced to an interaction between two 24 A. Konyukhov and K. Schweizerhof Fig. 1. Two dimensional contact as a special case of three dimensional contact boundary curves in the 2D case. One of both boundary curves resp. surfaces is chosen as the master contact curve resp. surface. A coordinate system is considered on the boundary, either for a surface in 3D or for a curve in 2D. On the plane we deﬁne a curvilinear coordinate system associated with the curve by introducing two principal vectors as a basis: the tangent vector ∂ρ and the unit normal vector ν.