By T.A. Laursen (auth.), Peter Wriggers Professor Dr., Udo Nackenhorst Professor Dr. (eds.)

Contact mechanics was once and is a crucial department in mechanics which covers a large box of theoretical, numerical and experimental investigations. during this conscientiously edited ebook the reader will receive a state of the art evaluation on formula, mathematical research and numerical resolution systems of touch difficulties. The contributions gathered during this quantity summarize the lectures awarded in the course of the 4^{th} touch Mechanics foreign Symposium (CMIS) held in Hannover, Germany, 2005, through top scientists within the quarter of touch mechanics.

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**Sample text**

For both approaches, adding a Coulomb friction condition is not a diﬃculty from the stability view point. However, this condition could be badly approximated for the ﬁrst approach because it depends on the normal stress. References 1. B. Khenous, PhD thesis, INSA of Toulouse, in preparation. 2. B. Khenous, Y. Renard & J. Pommier, Hybrid discretization of the Signorini problem with Coulomb friction. Theoretical aspects and comparison of some numerical solvers, to appear. 3. U. Kim, A boundary thin obstacle problem for a wave equation, Com.

We assume a vanishing initial gap between the structure and the rigid foundation and we denote u0 , u1 and T the given initial displacement, initial velocity and time simulation respectively. The space semi-discretization of the elastodynamic contact problem with nodal contact condition is deﬁned for a Lagrange ﬁnite element method as follows. Ni ) = 0, ∀ i ∈ IC , λ i N ˙ = u1 , u(0) = u0 , u(0) where d is the number of degrees of freedom for the displacement u, the notations M, K, f stend for the mass matrix, the stiﬀness matrix and the given force densities respectively, IC is the set of the contact boundary indices.

For the iterative scheme the choice τ in 2D is done by of the next sets Aτk+1 and Ik+1 uτ )p (λτ )p + cτ |(λτ )p,s | − gp > 0 , Aτk+1 := p ∈ S : (ˆ τ Ik+1 := p ∈ S : (ˆ uτ )p (λτ )p + cτ |(λτ )p,s | − gp ≤ 0 , cτ > 0. e. the friction bound gp is updated in each step. For details, we refer to [7]. As examples we consider two axisymmetric problems in the 3D case with a linearized St. Venant–Kirchhoﬀ material. Due to the symmetry of the problem setting, we can use cylinder coordinates and end up with a 2D formulation.