An Equilibrium Problem for a Kirchhoff–Love Plate, Contacting an Obstacle by Top and Bottom Edges

Lazarev, N. P.; Semenova, Galina; Fedotov, E. D.

doi:10.1134/s1995080223020245

Cited by 3 publications

(1 citation statement)

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“…The oblique interaction phenomena in Kirchhoff-Love plates were considered earlier in [24] for inclined cracks, in [25,26] for contact with sharp edges of rigid inclusions and in [27] for the plate that is in contact by its side edges with inclined obstacle surfaces. The solvability of the corresponding variational inequalities has been established.…”

Section: Introductionmentioning

confidence: 99%

Variational inequality for a Timoshenko plate contacting at the boundary with an inclined obstacle

Kovtunenko,

Lazarev

2024

Phil. Trans. R. Soc. A.

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A class of variational inequalities describing the equilibrium of elastic Timoshenko plates whose boundary is in contact with the side surface of an inclined obstacle is considered. At the plate boundary, mixed conditions of Dirichlet type and a non-penetration condition of inequality type are imposed on displacements in the mid-plane. The novelty consists of modelling oblique interaction with the inclined obstacle which takes into account shear deformation and rotation of transverse cross-sections in the plate. For proposed problems of equilibrium of the plate contacting the inclined obstacle, the unique solvability of the corresponding variational inequality is proved. Under the assumption that the variational solution is smooth enough, optimality conditions are obtained in the form of equilibrium equations and relations revealing the mechanical properties of integrated stresses, moments and generalized displacements on the contact part of the boundary. Accounting for complementarity type conditions owing to the contact of the plate with the inclined obstacle, a primal-dual variational formulation of the obstacle problem is derived. A semi-smooth Newton method based on a generalized gradient is constructed and performed as a primal-dual active-set algorithm. It is advantageous for efficient numerical solution of the problem, provided by a super-linear estimate for the corresponding iterates in function spaces. This article is part of the theme issue ‘Non-smooth variational problems with applications in mechanics’.

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