chebЧебышевский сборникChebyshevskii Sbornik2226-8383Tula State Lev Tolstoy Pedagogical University10.22405/2226-8383-2022-23-4-285-307cheb-1395Research ArticleИстория математики и приложенийСomputer scienceТеоретический и численный анализ локализации пластических деформаций в изотропных дилатирующих неассоциативных средах в условиях плоской деформацииTheoretical and numerical plastic strain localization analysis at plane strain of isotropic dilating non-associated media at plane strain conditionsЛевинВладимир АнатольевичLevinVladimir Anatol’evich

доктор физико-математических наук, профессор

doctor of physical and mathematical sciences, professor

v.a.levin@mail.ruКрапивинКирилл ЮрьевичKrapivinKirill Yurievichk.krapivn@gmail.comМосковский государственный университет им. М. В. ЛомоносоваРоссияLomonosov Moscow State UniversityRussian FederationООО «Фидесис»РоссияCAE FidesysRussian Federation202216012023234285307Copyright © Левин В.А., Крапивин К.Ю., 20222022Левин В.А., Крапивин К.Ю.Levin V.A., Krapivin K.Y.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://www.chebsbornik.ru/jour/article/view/1395

Статья посвящена предельному равновесию и локализации пластических деформаций вдоль сдвиговых полос в пластических дилатирующих средах. Получены уравнения характеристик систем уравнений для напряжений и скоростей в плоскодеформированном состоянии для произвольной функции поверхности текучести с зависимостью от первых двух инвариантов и неассоциативным законом течения в рамках жесткопоастического подхода. Получены уравнения для напряжений вдоль характеристик в предельном состоянии и исследована область гиперболичности. Приведена численная модель решения упругопластической задачи галеркиновскими уравнениями на спектральных элементах высокого прядка. Проведены численные эксперименты для линейной функции поверхности текучести с целью установить границы диапазона возможных наклонов сдвиговых полос и проэкзаминировать теоретические результаты.

The article is devoted to the limit plastic state and localization of plastic deformations along shear bands in dilating media with a non-associative flow rule. The equations of the characteristics of systems of equations for stresses and velocities in a plane strain state for anarbitrary function of the yield surface dependenе on the first two invariants in the rigid-plastic framework are obtained. Equations for stresses along the characteristics for the limit state and the condition of their hyperbolicity are obtained. A numerical model of the solution of the elastic-plastic problem by Galerkin equations on high-order spectral elements is presented.Numerical experiments have been carried out for the linear function of the yield surface in order to establish the boundaries of the range of possible slopes of the shear bands and to test the theoretical results.

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