matmessМатематические заметки СВФУMathematical notes of NEFU2411-93262587-876XСеверо-Восточный федеральный университет имени М.К. Аммосова10.25587/2411-9326-2024-3-93-120matmess-73Research ArticleМАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕЧисленное решение задачи о Т-образном сопряжении двух тонких включений Тимошенко в двумерном упругом телеNumerical solution of the problem of T-shaped junction of two thin Timoshenko inclusions in a two-dimentional elastic bodyПоповаТ. С.PopovaT. S.

Попова Татьяна Семеновна

ул. Кулаковского, 48, Якутск 677000

Tatiana S. Popova

48 Kulakovsky Street, Yakutsk 677000

ptsokt@mail.ruСеверо-Восточный федеральный университет имени М. К. АммосоваРоссияAmmosov North-Eastern Federal University, Institute of Mathematics and InformaticsRussian Federation20243009202431393120Copyright © Попова Т.С., 20242024Попова 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/73

Разработан алгоритм численного решения задачи о равновесии двумерного упругого тела, содержащего два тонких упругих включения. Включения моделируются в рамках теории балок Тимошенко и пересекаются под прямым углом во внутренней точке одного из них, образуя Т-образную конструкцию в упругом теле. Одно из включений отслаивается от упругой матрицы, образуя трещину. На берегах трещины как на части границы области задаются граничные условия вида неравенств. Наличие данного вида краевых условий приводит к нелинейности задачи и постановке в виде вариационного неравенства. Для разработки алгоритма численного решения поставленной задачи формулируется приближенная задача о поиске седловой точки лагранжиана. Доказана сходимость по прямой переменной решений приближенной задачи к решению исходной задачи. Построен итерационный алгоритм типа Удзавы и показана его сходимость. Приведены примеры численной реализации

An algorithm for the numerical solution of the equilibrium problem of a two-dimensional elastic body containing two thin elastic inclusions is developed. The inclusions are modeled within the framework of the theory of Timoshenko beams and intersect at right angle at an internal point of one of them, forming a T-shaped structure in an elastic body. One of the inclusions delaminates from the elastic matrix, forming a crack. On the crack faces, as part of the domain boundary, boundary conditions of the inequality form are specified. The presence of this type of boundary conditions leads to nonlinearity of the problem and formulation in the form of a variational inequality. To develop an algorithm for the numerical solution of the problem, an approximate problem of finding the saddle point of the Lagrangian is formulated. The convergence of solutions of the approximate problem to the solution of the original problem is proven. An iterative Uzawa-type algorithm is constructed and its convergence is shown. Examples of numerical implementation are given.

вариационное неравенствовключение Тимошенкотонкое включениетрещинаусловия непрониканиянелинейные граничные условиязадача сопряженияалгоритм Удзавыметод конечных элементов.variational inequalityTimoshenko inclusionthin elastic inclusioncracknon-penetration conditionsnonlinear boundary conditionsjunction problemUzawa algorithmfinite element method.ReferencesItou H., Khludnev A. M. On delaminated thin Timoshenko inclusions inside elastic bodies // Math. Meth. Appl. Sci. 2016. V. 39. P. 4980–4993.Itou H. and Khludnev A. M., “On delaminated thin Timoshenko inclusions inside elastic bodies,” Math. Meth. Appl. Sci., 39, 4980–4993 (2016).Khludnev A. M., Leugering G. R. On Timoshenko thin elastic inclusions inside elastic bodies // Math. Mech. Solids. 2015. V. 20, N 5. P. 495–511.Khludnev A. M. and Leugering G. R., “On Timoshenko thin elastic inclusions inside elastic bodies,” Math. Mech. Solids, 20, No. 5, 495–511 (2015).Khludnev A. M., Popova T. S. 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The authors declare that there are no conflicts of interest present.