Elastic-Plastic Bending of Rectangular Plates in a 3D Setting

Anton E. Makseev, Pavel V. Dunchenkin, Tatiana V. Yakovleva, Maksim V. Zhigalov, Anton V. Krysko, Vadim A. Krys’ko

Author details

Anton E. Makseev
Yuri Gagarin State Technical University of Saratov, Saratov, Russian Federation
Pavel V. Dunchenkin
Yuri Gagarin State Technical University of Saratov, Saratov, Russian Federation
Tatiana V. Yakovleva
Yuri Gagarin State Technical University of Saratov, Saratov, Russian Federation
Maksim V. Zhigalov
Yuri Gagarin State Technical University of Saratov, Saratov, Russian Federation
Anton V. Krysko
Yuri Gagarin State Technical University of Saratov, Saratov, Russian Federation
Vadim A. Krys’ko
Yuri Gagarin State Technical University of Saratov, Saratov, Russian Federation

https://doi.org/10.15593/perm.mech/2025.4.09

This paper presents mathematical models of physically nonlinear plates in a three-dimensional setting. The mathematical model is constructed using three-dimensional Cauchy equilibrium equations. The equations of state are obtained in displacements. The theory of small elastic-plastic deformations is employed. The von Mises plasticity criterion is used to determine plastic deformations. Birger's method of variable elastic parameters is employed to solve the resulting nonlinear problem. This method reduces the solution of nonlinear equations to the sequential solution of linear equations at each step of the iterative procedure of the method. The numerical study is based on the finite element method using the Comsol Multiphysics package. The convergence of the solutions depending on the number of finite elements is studied. This paper analyzes the stress-strain state of a square plate, rigidly fixed at the edges, depending on the ratio of the linear size to the thickness. For each of the cases considered, the load-deflection relationships, the distribution of plastic zones by the plate volume, and the distribution of displacement values w by the plate thickness are obtained.

Keywords

plastic deformationsfinite element methodBirger's method of variable elasticity parametersdeformation theory of plasticitythree-dimensional problemstress-strain state

УПРУГОПЛАСТИЧЕСКИЙ ИЗГИБ ПРЯМОУГОЛЬНЫХ В ПЛАНЕ ПЛАСТИН В 3D-ПОСТАНОВКЕ

Построены математические модели физически нелинейных пластин в трехмерной постановке. Математическая модель строится на основании трехмерных уравнений равновесия Коши. Уравнения состояния получены в перемещениях. Используется теория малых упругопластических деформаций. При определении пластических деформаций используется критерий пластичности Мизеса. Для решения полученной нелинейной задачи используется метод переменных параметров упругости И.А. Биргера, сводящий решение нелинейных уравнений к последовательному решению линейных уравнений на каждом шаге итерационной процедуры метода. Численное исследование проводится на основе метода конечных элементов с использованием пакета Comsol Multiphysics. Исследовалась сходимость решений в зависимости от количества конечных элементов. В работе анализируется напряженно-деформированное состояние квадратной пластинки, жестко заделанной по краям, в зависимости от отношения линейного размера к толщине. Для каждого из рассмотренных случаев получены зависимости «нагрузка–прогиб», распределение пластических зон по объему пластинки и значений перемещения в то толщине пластинки.

Ключевые слова

пластические деформацииметод конечных элементовметод переменных параметров упругости И.А. Биргерадеформационная теория пластичноститрехмерная задачанапряженно-деформированное состояние

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Article information

Published: 8 December 2025
Issue date: 8 December 2025
Issue: No. 4 (2025)