Equilibrium Problem for a Timoshenko Plate with a
Geometrically Nonlinear Condition of Nonpenetration for a Vertical Crack

Lazarev, N. P.; Semenova, Galina

doi:10.1134/s1990478920030126

Cited by 5 publications

(1 citation statement)

References 12 publications

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“…We refer to the paper [16], in which free boundary problems for elastic bodies with a rigid inclusion being in contact with another rigid non-deformable punch were proposed and investigated for the first time. Crack problems with unilateral non-penetration conditions between crack faces (see, for example [17][18][19][20][21][22][23][24][25][26]) are a special subclass of contact problems. Furthermore, by passages to the limits when rigidity parameters for a family of crack problems go to infinity, convergences of corresponding solutions to appropriate solutions of contact problems with the Signorini-type conditions were established in [27][28][29].…”

Section: Introductionmentioning

confidence: 99%

Signorini-Type Problems over Non-Convex Sets for Composite Bodies Contacting by Sharp Edges of Rigid Inclusions

Lazarev

Kovtunenko

2022

Mathematics

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A new type of non-classical 2D contact problem formulated over non-convex admissible sets is proposed. Specifically, we suppose that a composite body in its undeformed state touches a wedge-shaped rigid obstacle at a single contact point. Composite bodies under investigation consist of an elastic matrix and a rigid inclusion. In this case, the displacements on the set, corresponding to a rigid inclusion, have a predetermined structure that describes possible parallel shifts and rotations of the inclusion. The rigid inclusion is located on the external boundary and has the form of a wedge. The presence of the rigid inclusion imposes a new type of non-penetration condition for certain geometrical configurations of the obstacle and the body near the contact point. The sharp-shaped edges of the obstacle effect such sets of admissible displacements that may be non-convex. For the case of a thin rigid inclusion, which is described by a curve and a volume (bulk) rigid inclusion specified in a subdomain, the energy minimization problems are formulated. The solvability of the corresponding boundary value problems is proved, based on analysis of auxiliary minimization problems formulated over convex sets. Qualitative properties of the auxiliary variational problems are revealed; in particular, we have found their equivalent differential formulations. As the most important result of this study, we provide justification for a new type of mathematical model for 2D contact problems for reinforced composite bodies.

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Section: Introductionmentioning

confidence: 99%

Signorini-Type Problems over Non-Convex Sets for Composite Bodies Contacting by Sharp Edges of Rigid Inclusions

Lazarev

Kovtunenko

2022

Mathematics

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

Lazarev

Semenova

Fedotov

2023

Lobachevskii J Math

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Equilibrium Problem for a Timoshenko Plate with a Geometrically Nonlinear Condition of Nonpenetration for a Vertical Crack

Cited by 5 publications

References 12 publications

Signorini-Type Problems over Non-Convex Sets for Composite Bodies Contacting by Sharp Edges of Rigid Inclusions

Signorini-Type Problems over Non-Convex Sets for Composite Bodies Contacting by Sharp Edges of Rigid Inclusions

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

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

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