{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,8]],"date-time":"2025-10-08T10:40:04Z","timestamp":1759920004583,"version":"build-2065373602"},"reference-count":31,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2025,1,10]],"date-time":"2025-01-10T00:00:00Z","timestamp":1736467200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This paper is concerned with an equilibrium problem for an elastic structure consisting of a plate and an elastic beam connected to each other at a given point. We consider two cases: In the first one, the elastic beam is connected to a rigid part of the elastic plate; in the second case, contact occurs between two elastic bodies. The elastic plate may contain a thin rigid delaminated inclusion. Neumann-type boundary conditions are considered at the external boundary of the plate. The existence of a solution to the considered problems is proven. A sufficient and necessary condition imposed onto the external forces for the solvability of the problems is found. Passages to the limit with respect to the rigidity parameter of the elastic beam are justified. For all problems, we analyze variational statements as well as differential ones.<\/jats:p>","DOI":"10.3390\/axioms14010049","type":"journal-article","created":{"date-parts":[[2025,1,10]],"date-time":"2025-01-10T03:48:11Z","timestamp":1736480891000},"page":"49","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["On Equilibrium Problem for T-Shape Elastic Structure"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9540-4053","authenticated-orcid":false,"given":"Alexander","family":"Khludnev","sequence":"first","affiliation":[{"name":"Lavrentyev Institute of Hydrodynamics of SB RAS, Novosibirsk 630090, Russia"}]}],"member":"1968","published-online":{"date-parts":[[2025,1,10]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"3743","DOI":"10.1177\/1081286519850608","article-title":"Optimal location of a rigid inclusion in equilibrium problems for inhomogeneous Kirchhoff-Love plates with a crack","volume":"24","author":"Lazarev","year":"2019","journal-title":"Math. Mech. Solids"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"103505","DOI":"10.1016\/j.nonrwa.2021.103505","article-title":"Shape derivatives of energy and regularity of minimizers for shallow elastic shells with cohesive cracks","volume":"65","author":"Shcherbakov","year":"2022","journal-title":"Nonlinear Anal. Real World Appl."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"530","DOI":"10.1007\/s10958-019-04179-z","article-title":"On contact between a thin obstacle and a plate containing a thin inclusion","volume":"237","author":"Furtsev","year":"2019","journal-title":"J. Math. Sci."},{"key":"ref_4","first-page":"1256","article-title":"Thin inclusions in elastic bodies crossing an external boundary","volume":"95","author":"Khludnev","year":"2015","journal-title":"J. Appl. Math. Mech."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"113540","DOI":"10.1016\/j.cam.2021.113540","article-title":"T-shape inclusion in elastic body with a damage parameter","volume":"393","author":"Khludnev","year":"2021","journal-title":"J. Comp. Appl. Math."},{"doi-asserted-by":"crossref","unstructured":"Khludnev, A.M. (2023). Elasticity problem with a cusp between thin inclusion and boundary. Axioms, 12.","key":"ref_6","DOI":"10.3390\/axioms12121081"},{"key":"ref_7","first-page":"39","article-title":"An uncoupled limit model for a high-contrast problem in a thin multi-structure","volume":"33","author":"Gaudiello","year":"2022","journal-title":"Rend. Lincei Mat. Appl."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1016\/j.anihpc.2007.06.002","article-title":"Asymptotic analysis, in a thin multidomain, of minimizing maps with values in S2","volume":"26","author":"Gaudiello","year":"2009","journal-title":"Ann. Inst. Henri Poincar\u00e8 Anal. Non Lin\u00e8aire"},{"key":"ref_9","first-page":"51","article-title":"Numerical solution of the equilibrium problem fora two-dimensional elastic body with a thin semirigid inclusion","volume":"28","author":"Popova","year":"2021","journal-title":"Math. Notes NEFU"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"4143","DOI":"10.1134\/S1995080223100311","article-title":"On numerical solving of junction problem for the thin rigid and elastic inclusions in elastic body","volume":"44","author":"Popova","year":"2023","journal-title":"Lobachevskii J. Math."},{"key":"ref_11","first-page":"E201700192","article-title":"The asymptotic analysis of a junction of two elastic beams","volume":"99","author":"Leugering","year":"2019","journal-title":"J. Appl. Math. Mech."},{"key":"ref_12","first-page":"E201900227","article-title":"Asymptotic analysis of a bit brace shaped junction of thin rods","volume":"100","author":"Leugering","year":"2020","journal-title":"J. Appl. Math. Mech."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"e202000063","DOI":"10.1002\/zamm.202000063","article-title":"On junction problem with damage parameter for Timoshenko and rigid inclusions inside elastic body","volume":"100","author":"Khludnev","year":"2020","journal-title":"Z. Angew. Math. Mech."},{"doi-asserted-by":"crossref","unstructured":"Rudoy, E. (2020). Asymptotic justification of models of plates containing inside hard thin inclusions. Technologies, 8.","key":"ref_14","DOI":"10.20944\/preprints202010.0038.v1"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"562","DOI":"10.1016\/j.ijsolstr.2020.06.044","article-title":"Variational approach to modeling soft and stiff interfaces in the Kirchhoff-Love theory of plates","volume":"202","author":"Furtsev","year":"2020","journal-title":"Int. J. Solids Struct."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1407","DOI":"10.1007\/s11012-016-0481-4","article-title":"Asymptotic interface models in magneto-electro-thermoelastic composites","volume":"52","author":"Serpilli","year":"2017","journal-title":"Meccanica"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1016\/j.ijsolstr.2019.07.014","article-title":"An asymptotic derivation of a general imperfect interface law for linear multiphysics composites","volume":"180","author":"Serpilli","year":"2019","journal-title":"Int. J. Solids Struct."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1016\/j.jde.2021.08.032","article-title":"Limit models for thin heterogeneous structures with high contrast","volume":"302","author":"Gaudiello","year":"2021","journal-title":"J. Diff. Equat."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1007\/s10958-012-0971-4","article-title":"Plane problem of elasticity theory for anisotropic bodies with thin elastic inclusions","volume":"186","author":"Pasternak","year":"2012","journal-title":"J. Math. Sci."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1016\/j.ijsolstr.2020.11.006","article-title":"Dispersion of rigid line inclusions as stiffeners and shear band instability triggers","volume":"210","author":"Goudarzi","year":"2021","journal-title":"Int. J. Solids Struct."},{"key":"ref_21","first-page":"1965","article-title":"Homogenization of an elastic material reinforced with thin rigid von Karman ribbons","volume":"24","year":"2018","journal-title":"Math. Mech. Solids."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1016\/0167-6636(94)90034-5","article-title":"On crack, rigid-line fiber, and interface interactions","volume":"19","author":"Hu","year":"1994","journal-title":"Mech. Mater."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"2267","DOI":"10.1016\/j.compscitech.2008.04.015","article-title":"Properties of rigid-line inclusions as building blocks of naturally occurring composites","volume":"68","author":"Pingle","year":"2008","journal-title":"Compos. Sci. Technol."},{"doi-asserted-by":"crossref","unstructured":"Kozlov, V.A., Ma\u2019zya, V.G., and Movchan, A.B. (1999). Asymptotic Analysis of Fields in a Multi-Structure, Oxford University Press.","key":"ref_24","DOI":"10.1093\/oso\/9780198514954.001.0001"},{"unstructured":"Panasenko, G. (2005). Multi-Scale Modelling for Structures and Composites, Springer.","key":"ref_25"},{"unstructured":"Khludnev, A.M. (2010). Elasticity Problems in Non-Smooth Domains, Fizmatlit.","key":"ref_26"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"134997","DOI":"10.1016\/j.conbuildmat.2024.134997","article-title":"A new elastic T-shape joint of longitudinally connected slab ballastless track structure for releasing thermal stress caused by non-uniform temperature loads","volume":"416","author":"He","year":"2024","journal-title":"Constr. Build. Mater."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"231","DOI":"10.1080\/03601218308907443","article-title":"Shape Design Sensitivity Analysis of Elastic Structures","volume":"11","author":"Choi","year":"2007","journal-title":"J. Struct. Mech."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"377","DOI":"10.1016\/S0997-7538(00)00175-3","article-title":"Junction of thin plates","volume":"19","author":"Titeux","year":"2000","journal-title":"Eur. J. Mech. A Solids"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"717","DOI":"10.1016\/S1631-073X(02)02543-8","article-title":"On the junction of elastic plates and beams","volume":"335","author":"Gaudiello","year":"2002","journal-title":"Comptes Rendus Math."},{"key":"ref_31","first-page":"179","article-title":"Modeling of the junction between a plate and a rod in nonlinear elasticity","volume":"7","author":"Gruais","year":"1993","journal-title":"Asymptot. Anal."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/1\/49\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,8]],"date-time":"2025-10-08T10:26:20Z","timestamp":1759919180000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/1\/49"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,1,10]]},"references-count":31,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2025,1]]}},"alternative-id":["axioms14010049"],"URL":"https:\/\/doi.org\/10.3390\/axioms14010049","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,1,10]]}}}