Analysis and Simulation of Contact Problems (Lecture Notes by Peter Wriggers, Udo Nackenhorst

By Peter Wriggers, Udo Nackenhorst

This rigorously edited e-book deals a cutting-edge evaluate on formula, mathematical research and numerical answer tactics of touch difficulties. The contributions accumulated during this quantity summarize the lectures offered by means of top scientists within the quarter of touch mechanics, through the 4th touch Mechanics foreign Symposium (CMIS) held in Hannover, Germany, 2005.

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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.

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