A contact problem for an elastic plate with a thin rigid inclusion

Fankina, I. V.

doi:10.1134/s1990478916030030

Cited by 7 publications

(12 citation statements)

References 4 publications

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“…Furthermore, as was proved in [1], the two-dimensional contact problem for a rigid body with a Signorini type boundary condition can be considered as a limiting case for a family of equilibrium problems modelling two-dimensional bodies with cracks. Analogous results for the Kirchhoff-Love plates were established in [6,9].…”

Section: Introductionsupporting

confidence: 52%

“…Now we are going to make explicit the relationship between the problems (9) and (19). It turns out that the solutions ξ λ converge to a limiting functionξ as λ → 0.…”

Section: Fig 3 Geometry Of the Extended Domainmentioning

confidence: 99%

“…It turns out that the solutions ξ λ converge to a limiting functionξ as λ → 0. Moreover, the restriction ofξ onto Ω 1 is a solution to the variational inequality (9).…”

Section: Fig 3 Geometry Of the Extended Domainmentioning

confidence: 99%

“…The fictitious domain method for nonlinear models, describing equilibrium of n-dimensional (n = 2, 3) elastic bodies with nonlinear boundary conditions in the form of inequalities was elaborated in [1][2][3][4][5][6][7][8][9][10]. Particularly, non-linear models for plates are studied in [6][7][8][9]. Furthermore, as was proved in [1], the two-dimensional contact problem for a rigid body with a Signorini type boundary condition can be considered as a limiting case for a family of equilibrium problems modelling two-dimensional bodies with cracks.…”

Section: Introductionmentioning

confidence: 99%

“…The method of the fictitious domain has proven useful in establishing the solvability of problems that describe equilibrium of bodies with cracks crossing the external boundary at zero angles [2,4,8,9]. In the last years, within the framework of crack models subject to non-penetration boundary conditions, numerous works have been published, see, for example, [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25].…”

Section: Introductionmentioning

confidence: 99%

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Fictitious Domain Method for Equilibrium Problems of the Kirchhoff-Love Plates with Nonpenetration Conditions for Known Configurations of Plate Edges

Lazarev

Everstov

Романова

et al. 2019

Journal of Siberian Federal University. Mathematics &Amp; Physics

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New models are investigated in this paper, that describe equilibrium states of plates with Signorini type nonpenetration conditions. In these models, it is assumed that under appropriate loading, plates have special deformations with already known configurations of edges. For this case, we deal with new non-penetration conditions that allow us to describe more precisely the possibility of contact interaction of plate edges. Using the method of fictitious domains, it is proved that an original contact problem for a plate can be obtained by passing to the limit when a rigidity parameter tends to infinity from a family of auxiliary problems formulated in a wider domain. The mentioned family of problems model an equilibrium state of plates with a crack and depend on the positive rigidity parameter. For these problems, to prevent a mutual penetration of the opposite crack faces boundary conditions of inequality type are imposed on the inner boundary corresponding to the crack. For the problem, describing a plate with a crack that intersects the external boundary at zero angle (a case of a boundary having one cusp), the unique solvability is proved.

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Section: Introductionsupporting

confidence: 52%

“…Now we are going to make explicit the relationship between the problems (9) and (19). It turns out that the solutions ξ λ converge to a limiting functionξ as λ → 0.…”

Section: Fig 3 Geometry Of the Extended Domainmentioning

confidence: 99%

“…It turns out that the solutions ξ λ converge to a limiting functionξ as λ → 0. Moreover, the restriction ofξ onto Ω 1 is a solution to the variational inequality (9).…”

Section: Fig 3 Geometry Of the Extended Domainmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Fictitious Domain Method for Equilibrium Problems of the Kirchhoff-Love Plates with Nonpenetration Conditions for Known Configurations of Plate Edges

Lazarev

Everstov

Романова

et al. 2019

Journal of Siberian Federal University. Mathematics &Amp; Physics

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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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On hyperelastic solid with thin rigid inclusion and crack subjected to global injectivity condition

Furtsev,

Rudoy,

Sazhenkov

2024

Phil. Trans. R. Soc. A.

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The paper investigates a problem concerning the equilibrium of a solid body containing a thin rigid inclusion and a crack. It is assumed that the body is hyperelastic, therefore, it is described within the framework of finite strain theory. One of the peculiarities of this problem is a global injectivity constraint, which prevents the body, the crack faces and the inclusion from both mutual and self penetration. First, the paper deals with the differential formulation of the problem. Next, we consider energy minimization, showing that the latter provides the weak formulation of the former. Finally, the existence of the weak solution is demonstrated through the use of the variational technique. This article is part of the theme issue ‘Non-smooth variational problems with applications in mechanics’.

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A contact problem for an elastic plate with a thin rigid inclusion

Cited by 7 publications

References 4 publications

Fictitious Domain Method for Equilibrium Problems of the Kirchhoff-Love Plates with Nonpenetration Conditions for Known Configurations of Plate Edges

Fictitious Domain Method for Equilibrium Problems of the Kirchhoff-Love Plates with Nonpenetration Conditions for Known Configurations of Plate Edges

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

On hyperelastic solid with thin rigid inclusion and crack subjected to global injectivity condition

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