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Existence of solutions to discrete semicoercive frictional contact problems. (English) Zbl 0961.74046

The purpose of this important paper is to study the existence of solutions to discrete semicoercive frictional contact problems. The discrete incremental friction problem is investigated for positive semidefinite stiffness matrix (i.e. the semicoercive case is treated). Physically, the positive semidefiniteness describes a situation where the structure or body is not anchored (when certain directions for external loads are not permissible), so that it is free to perform rigid body displacements on top of local deformations. The authors derive governing equations of the discrete incremental version of the problem. The problem is considered as an incremental instance in a sequence of problems, which means that an extra load term appears as data as compared to the original Duvaut-Lions problem.
The first of two existence results is obtained for a large class of friction laws (including nonlinear laws such as classical and anisotropic versions of Coulomb’s law and laws where the friction cone is piecewise linear). The second existence result is valid for piecewise linear laws. The authors give precise proofs of lemmas and theorems, and establish new sufficient conditions for the existence of solution to discrete (three-dimensional, deformable body, small displacement) frictional contact problem with a positive semidefinite stiffness matrix. The approach is based on a quasi-variational inequality formulation, and makes extensive use of known results form linear complementarity theory.

MSC:

74M15 Contact in solid mechanics
74M10 Friction in solid mechanics
49J40 Variational inequalities

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