ntvНаучно-технический вестник информационных технологий, механики и оптикиScientific and Technical Journal of Information Technologies, Mechanics and Optics2226-14942500-0373Университет ИТМО10.17586/2226-1494-2024-24-4-629-636ntv-343Research ArticleМАТЕМАТИЧЕСКОЕ И КОМПЬЮТЕРНОЕ МОДЕЛИРОВАНИЕMODELING AND SIMULATIONПотеря устойчивости защемленной по контуру прямоугольной нанопластиныInstability of a rectangular CCCC-nanoplatehttps://orcid.org/0000-0002-8295-7089СухотеринМ. В.SukhoterinM. V.

Сухотерин Михаил Васильевич — доктор технических наук, доцент, заведующий кафедрой

Санкт-Петербург, 198035

Mikhail V. Sukhoterin — D.Sc., Associate Professor, Head of Department

Saint Petersburg, 198035

sukhoterinmv@gumrf.ruhttps://orcid.org/0009-0000-6476-6783СосновскаяА. А.SosnovskayaA. A.

Сосновская Анна Анатольевна — старший преподаватель

Санкт-Петербург, 198035

Anna A. Sosnovskaya — Senior Lecturer

Saint Petersburg, 198035

sosnovskayaaa@gumrf.ruГосударственный университет морского и речного флота имени адмирала С.О. МакароваРоссияAdmiral Makarov State University of Maritime and Inland ShippingRussian Federation202411122024244629636Copyright © Сухотерин М.В., Сосновская А.А., 20242024Сухотерин М.В., Сосновская А.А.Sukhoterin M.V., Sosnovskaya A.A.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://ntv.elpub.ru/jour/article/view/343

Введение. Изучен спектр критических нагрузок и форм равновесия CCCC-нанопластины (С — clamped edge, защемленный край) при различных значениях нелокального нанопараметра. Метод. Симметричные формы представлены двумя гиперболо-тригонометрическими рядами по двум координатам, которые подчинялись основному дифференциальному уравнению физического состояния А.Д. Эрингена. Граничные условия отсутствия прогибов и углов поворота защемленных граней были удовлетворены полностью. В результате получена однородная бесконечная система линейных алгебраических уравнений относительно неизвестных коэффициентов гиперболо-тригонометрических рядов, содержащая в качестве основного параметра относительную сжимающую нагрузку. Преобразованная система включает только одну последовательность коэффициентов. Построен итерационный процесс поиска нетривиального решения в сочетании с методом перебора величины нагрузки. Основные результаты. Для каждого значения нелокального параметра найдены первые четыре критические нагрузки для симметричных форм закритического равновесия и получены их 3D-изображения. Установлено, что критические нагрузки убывали с ростом нелокального параметра. Исследовано влияние на точность результатов количества членов, удерживаемых в рядах, и числа итераций. Обсуждение. Полученные результаты могут быть использованы при проектировании различных наноразмерных smart-конструкций.

The spectrum of critical loads and equilibrium forms of a CCCC-nanoplate (C — clamped edge) at various values of a non–local nanoparameter has been studied. The symmetric solution is represented by two hyperbolo-trigonometric series in two coordinates which obeyed the basic differential equation of the physical state of Eringen. The boundary conditions for the absence of deflections and angles of rotation of the pinched faces were precisely satisfied. As a result, a homogeneous infinite system of linear algebraic equations with respect to unknown coefficients of these series is obtained containing a relative compressive load as the main parameter. After the conversion, the system began to contain only one sequence of coefficients. An iterative process of searching for a non-trivial solution in combination with the method of iterating over the load value is constructed. For each value of the nonlocal parameter, the first four critical loads for symmetric forms of supercritical equilibrium are found and their 3D images are obtained. It was found that critical loads decreased with an increase in the nonlocal parameter. The influence of the number of members held in rows and the number of iterations on the accuracy of the results is investigated. The results obtained can be used in the design of various nanoscale smart structures.

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