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Abstract
We study the contact problems for elastic plates with a rigid inclusion. We consider the case of frictionless contact between the rigid part of the plate and a rigid obstacle. The contact is modeled with the Signorini-type nonpenetration condition. The deformation of the transversely isotropic elastic part of the plate is described by the Timoshenko model. We analyze the dependence of solutions to the contact problems on the size of rigid inclusion. The existence of a solution to the optimal control problem is proved. For that problem, the cost functional characterizes the deviation of the displacement vector from a given function, whereas the size parameter of rigid inclusion is chosen as the control function.
Highlights
Applications of composite materials are growing vastly along with the development of research interests concerning material behavior
It is of interest to investigate the contact problems for plates that are reinforced by rigid inclusions
A detailed proof of a similar result for Timoshenko plate containing a crack along a thin rigid inclusion can be found in [ ]
Summary
Applications of composite materials are growing vastly along with the development of research interests concerning material behavior. \ωt ) , χ ∈ H ( \ωt) , with the constant ct > independent of χ These properties of the energy functional (χ ), bilinear form B( , ·, ·), and set Kt allow us to establish the existence of a unique solution ξt = (Ut, ut, φt) ∈ Kt for problem ( ) (see [ ]). The symmetry and continuity of the bilinear form B( , ·, ·) and the properties of the set Kt provide (see [ ]) the equivalence of problem ( ) to the variational inequality ξt ∈ Kt, B( , ξt, χ – ξt) ≥ F(χ – ξt) d ∀χ = (W , w, ψ) ∈ Kt. In parallel with the contact problem for a plate with a volume rigid inclusion, we consider the contact problem for an elastic plate with a thin inclusion. A detailed proof of a similar result for Timoshenko plate containing a crack along a thin rigid inclusion can be found in [ ]
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