MODELING OF PRESTRESSED PLATES WITH MATERIAL INHOMOGENEITY, PERFORATIONS AND INCLUSIONS

Author details

Ivan V. Bogachev
Southern Federal University, Rostov-on-Don, Russian Federation
Rostislav D. Nedin
Southern Federal University, Rostov-on-Don, Russian Federation

https://doi.org/10.15593/perm.mech/2023.3.02

In the present article, we propose the model of in-plane oscillations of inhomogeneous prestressed plates, both solid ones and those containing a set of holes and inclusions made of different materials. We treat the plates' mechanical properties and the prestress tensor components in the considered 2D problem statement as functions of two coordinates. In order to formulate the boundary value problems of steady-state in-plane vibrations of plates, we employ the general linearized formulation for an elastic body under conditions of an initial stress-strain state. The developed vibration model makes it possible to specify an arbitrary type of prestress state in the plate in the form of analytical dependences, as well as numerically, by solving the corresponding static problem, in which prestresses arise as a result of applying some initial load. To implement the finite element (FE) approach to solving the problems, we formulated the weak problem statement by projecting the original governing equations on the field of test displacements satisfying the essential boundary conditions. To increase the accuracy of calculations for plates with holes and inclusions, the local refinement of FE meshes are used. The proposed approach to calculating plate vibrations is implemented as a software package via FreeFem++. A method for assessing the effect of prestress on dynamic plates' characteristics under various types of loads is described; a comprehensive analysis is carried out to identify the probing modes, frequency ranges and response pickup areas, most sensitive to the prestress changes, for each of the plates. We systematize and generalize the results obtained during the analysis, give a few practical recommendations on the choice of probing modes for each type of the plates considered, allowing to perform the most efficient schemes for identifying the prestress components.

Keywords

prestresselastic platesinhomogeneityperforationinclusionssensitivity analysisacoustic methodfinite element methodFreeFem++.

МОДЕЛИРОВАНИЕ СПЛОШНЫХ И ИМЕЮЩИХ ОТВЕРСТИЯ И ВКЛЮЧЕНИЯ НЕОДНОРОДНЫХ ПРЕДВАРИТЕЛЬНО НАПРЯЖЕННЫХ ПЛАСТИН

Предложена модель планарных колебаний неоднородных предварительно напряженных пластин, как сплошных, так и имеющих набор отверстий и включений из других материалов. Свойства пластин и компоненты тензора предварительных напряжений (ПН) в рассмотренной плоской постановке считались функциями двух координат. Для формулировки краевых задач об установившихся планарных колебаниях пластин использована общая линеаризованная постановка задачи о колебаниях тела в условиях предварительного напряженно-деформированного состояния. Разработанная модель колебаний дает возможность задания произвольного типа предварительного состояния в пластине: как в виде аналитических зависимостей, так и численно – с помощью решения соответствующей задачи статики, в которой предварительные напряжения возникают в результате приложения некоторой начальной нагрузки. Для реализации конечно-элементного (КЭ) подхода к решению задач сформулирована слабая постановка задачи на основе проектирования исходных уравнений на поле возможных перемещений, удовлетворяющее главным граничным условиям. Для увеличения точности расчетов для пластин с отверстиями и включениями в этих областях использовалось локальное сгущение КЭ-сеток. Разработанный подход для расчета колебаний пластин реализован в виде программного комплекса в КЭ- пакете FreeFem++. Предложена методика оценки влияния ПН на динамические характеристики при различных видах нагрузок, с использованием которой был проведен комплексный анализ по выявлению наиболее чувствительных к изменению ПН-режимов зондирования, частотных диапазонов и областей считывания отклика для каждой из пластин. Полученные в процессе анализа результаты были систематизированы и обобщены, и был дан ряд практических рекомендаций по выбору режимов зондирования для каждого вида пластин, с помощью которых могут быть построены наиболее эффективные схемы идентификации трех компонент ПН.

Ключевые слова

предварительные напряженияупругие пластинынеоднородностьперфорациявключенияанализ чувствительностиакустический методметод конечных элементовFreeFem++.

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Article information

Published: 3 July 2023
Issue date: 3 July 2023
Issue: No. 3 (2023)