Theoretical analysis of discrete contact problems with Coulomb friction. (English) Zbl 1265.74069

Ligurský, Tomáš

Appl. Math., Praha 57, No. 3, 263-295 (2012).

The discrete static two-dimensional Signorini problem with Coulomb friction of linear elasticity is analyzed. The coefficient of friction \(\mathcal {F}\) is space-dependent. First, it is shown that the problem is always solvable and that its solution is unique if the magnitude of \(\mathcal {F}\) is below a certain bound. For \(\mathcal {F}\) varrying, under any bound smaller than the previous one, the Lipschitz continuity of the solution in dependence on \(\mathcal {F}\) as well as on the load is proved. Two different reformulations of the problem, one based on generalized equations, another on non-smooth equations, are presented in order to investigate locally the potentially non-unique solutions. With the help of the first reformulation, it is shown that the study of the local behaviour of the solution with respect to \(\mathcal {F}\) can be equivalently replaced by the study of its dependence on the load. The second one helps to show that the local uniqueness with respect to the load holds provided particular Jacobian matrices dependent on the contact status of the solutions have the same non-vanishing determinant sign. The benefit of the proposed approach is illustrated in detail on a simple example whose solutions are exactly calculated.
The paper is very well written. The presented deep analysis may be of a remarkable help to specialists working in computation of static frictional contact problems, despite its limitation to the 2D situation.

Reviewer: Jiří Jarušek (Praha)

MSC:

74M10	Friction in solid mechanics
74G20	Local existence of solutions (near a given solution) for equilibrium problems in solid mechanics (MSC2010)
74G55	Qualitative behavior of solutions of equilibrium problems in solid mechanics

Keywords:

unilateral contact; Coulomb friction; local uniqueness; qualitative behaviour

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