matmessМатематические заметки СВФУMathematical notes of NEFU2411-93262587-876XСеверо-Восточный федеральный университет имени М.К. Аммосова10.25587/SVFU.2023.88.57.004matmess-103Research ArticleМАТЕМАТИКАЗадача о T-образном сопряжении двух тонких включений Тимошенко в двумерном упругом телеThe problem of T-shaped junction of two thin Timoshenko inclusions in a two-dimensional elastic bodyПоповаТ. С.PopovaT. S.

Попова Татьяна Семеновна

ул. Кулаковского, 48, Якутск 677000

Tatiana S. Popova

48 Kulakovsky Street, Yakutsk, 677000 Russia

ptsokt@mail.ruСеверо-Восточный федеральный университет имени М.К. АммосоваРоссияAmmosov North-Eastern Federal University, Institute of Mathematics and InformaticsRussian Federation2023300620233024055Copyright © Попова Т.С., 20232023Попова Т.С.Popova T.S.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://matmess.elpub.ru/jour/article/view/103

Исследуется задача о равновесии двумерного упругого тела, содержащего два контактирующих тонких включения прямолинейной формы. Включения являются упругими и моделируются в рамках теории балок Тимошенко. Включения пересекаются под прямым углом, и одно из включений отслаивается от упругой матрицы, образуя трещину. Задача ставится как вариационная, при этом получена полная дифференциальная формулировка в виде краевой задачи, в том числе в общей точке включений выписаны условия сопряжения. На берегах разреза задаются граничные условия вида неравенств. Доказана эквивалентность вариационной и дифференциальной постановок задачи при условии достаточной гладкости решений. Обоснован предельный переход по параметру жесткости одного из включений.

We consider the equilibrium problem for a two-dimensional elastic body containing two contacting thin inclusions of a rectilinear shape. The inclusions are elastic and are modeled within the framework of the theory of Timoshenko beams. The inclusions intersect at a right angle, and one of the inclusions delaminates from the elastic matrix, forming a crack. The problem is posed as a variational one and a complete differential formulation is obtained in the form of a boundary value problem, including junction conditions at a common point of inclusions. On the edges of the cut, boundary conditions of the form of inequalities are specified. The equivalence of the variational and differential formulations of the problem is proved under the condition of sufficient smoothness of the solutions. The passage to the limit with respect to the stiffness parameter of one of the inclusions is substantiated.

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The authors declare that there are no conflicts of interest present.