Equilibrium problem for elastic plate with thin rigid inclusion crossing an external boundary

Khludnev, A. M.; Fankina, I. V.

doi:10.1007/s00033-021-01553-3

Cited by 12 publications

(11 citation statements)

References 27 publications

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“…Nonlinear boundary conditions in the form of a system of equalities and inequalities, specified on the crack faces, ensure their mutual nonpenetration. A large number of results on the existence of a solution, the existence and formulas of derivatives with respect to the shape of a domain, optimal control problems, contact problems, inverse problems can be found, for example, in [1][2][3][4][5][6][7][8][9][10][11][12][13]. The study of equilibrium problems for elastic bodies with Euler-Bernoulli inclusions and cracks was carried out in [14][15][16][17][18].…”

Section: Introductionmentioning

confidence: 99%

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Noncoercive problems for elastic bodies with thin elastic inclusions

Khludnev

Fankina

2023

Math Methods in App Sciences

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The paper concerns boundary value problems for an elastic body with a thin elastic inclusion for a noncoercive case. The inclusion is assumed to be delaminated thus forming a crack between the inclusion and the surrounding elastic body. To provide a mutual nonpenetration between crack faces, we consider inequality type boundary conditions with unknown set of a contact. In the paper, a solution existence of the equilibrium problems is proved for the cases when one or two given points of the inclusion are fixed as well as and for the case without these restrictions.

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Section: Introductionmentioning

confidence: 99%

“…Some results on noncoercive problems can be found in [23][24][25][26][27]. In [28], the noncoercive equilibrium problem for a Kirchhoff-Love plate with a thin rigid inclusion was considered.…”

Section: Introductionmentioning

confidence: 99%

Noncoercive problems for elastic bodies with thin elastic inclusions

Khludnev

Fankina

2023

Math Methods in App Sciences

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“…Our particular methods of non-smooth analysis stem from the variational approach to nonpenetrating cracks in solids developed by Khludnev & Kovtunenko [10] and co-authors (e.g. [11][12][13] and other works related to asymptotic analysis [14][15][16] and numerical techniques [17]).…”

Section: Introductionmentioning

confidence: 99%

Quasi-variational inequality for the nonlinear indentation problem: a power-law hardening model

Kovtunenko

2022

Phil. Trans. R. Soc. A.

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The Boussinesq problem, which describes quasi-static indentation of a rigid punch into a deformable body, is studied within the context of nonlinear constitutive equations. By this, the material response expresses the linearized strain in terms of the stress and cannot be inverted in general. A contact area between the punch and the body is unknown a priori , whereas the total contact force is prescribed and yields a non-local integral condition. Consequently, the unilateral indentation problem is stated as a quasi-variational inequality for unknown variables of displacement, stress and indentation depth. The Lagrange multiplier approach is applied in order to establish well-posedness to the underlying physically and geometrically nonlinear problem based on augmented penalty regularization and applying the minimax theorem of Ekeland and Témam. A sufficient solvability condition implies response functions that are bounded, hemi-continuous, coercive and obey a convex potential. A typical example is power-law hardening models for titanium alloys, Norton–Hoff and Ramberg–Osgood materials. This article is part of the theme issue ‘Non-smooth variational problems and applications’.

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“…Among this type of nonlinear mathematical models, a wide range of various problems for Kirchhoff-Love plates in the framework of elastic constitutive relations has been studied [4,6,[13][14][15][16]. Problems for elastic plates with rigid inclusions are investigated in [17][18][19][20][21][22][23]. Thermoelastic models of plates with cracks have been studied, e.g.…”

Section: Introductionmentioning

confidence: 99%

Equilibrium problem for a thermoelastic Kirchhoff–Love plate with a delaminated flat rigid inclusion

Lazarev

2022

Phil. Trans. R. Soc. A.

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A new mathematical model describing an equilibrium of a thermoelastic heterogeneous Kirchhoff–Love plate is considered. A corresponding nonlinear variational problem is formulated with respect to a two-dimensional domain with a cut. This cut corresponds to an interfacial crack located on a given part of the boundary of a flat rigid inclusion. The flat inclusion is described by a cylindrical surface. Due to the presence of the flat rigid inclusion in the plate, restrictions of the functions describing displacements to the corresponding curves satisfy special constraints having a linear form. Displacement boundary conditions of an inequality type are set on the crack faces that ensure a mutual non-penetration of opposite crack faces. Solvability of the problem is proved. Under the assumption that the solution of the variational problem is smooth enough, an equivalent differential formulation is found. This article is part of the theme issue ‘Non-smooth variational problems and applications’.

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Equilibrium problem for elastic plate with thin rigid inclusion crossing an external boundary

Cited by 12 publications

References 27 publications

Noncoercive problems for elastic bodies with thin elastic inclusions

Noncoercive problems for elastic bodies with thin elastic inclusions

Quasi-variational inequality for the nonlinear indentation problem: a power-law hardening model

Equilibrium problem for a thermoelastic Kirchhoff–Love plate with a delaminated flat rigid inclusion

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