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Optimal control for antiplane frictional contact problems involving nonlinearly elastic materials of hencky type. (English) Zbl 1404.74114

Summary: We consider an antiplane contact problem modeling the friction between a nonlinearly elastic body of Hencky type and a rigid foundation. We discuss the well-posedness of the model by considering two friction laws. Firstly, Tresca’s law is used to describe the friction force and leads to a variational inequality. Alternatively, a regularizing power law with a positive exponent \(r\) is considered and gives, from the mathematical point of view, a variational equation. In both contexts, we address a boundary optimal control problem by minimizing, on a nonconvex set, a cost functional with two arguments. We show the existence of at least one optimal pair for each problem. Finally, we deliver some convergence results proving that the optimal solution of the regular problem tends, when \(r\) goes to zero, to an optimal solution of the first one.

MSC:

74M05 Control, switches and devices (“smart materials”) in solid mechanics
74M15 Contact in solid mechanics
74M10 Friction in solid mechanics
74B20 Nonlinear elasticity
74G25 Global existence of solutions for equilibrium problems in solid mechanics (MSC2010)
49J40 Variational inequalities
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