PLANE PROBLEMS ABOUT THE ACTION OF OSCILLATING LOAD ON THE BOUNDARY OF AN ELASTIC ISOTROPIC LAYER IN THE PRESENCE OF SURFACE STRESSES

Author details

T.I. Kalinina
South-Russian State Polytechnic University (NPI) named after M.I. Platov, Novocherkassk, Russian Federation
A.V. Nasedkin
Southern Federal University, Rostov on Don, Russian Federation

https://doi.org/10.15593/perm.mech/2023.1.05

In this paper, symmetric and antisymmetric plane problems about the action of oscillating load on the boundary of an elastic isotropic nanothin layer are considered. The nanoscale layer thickness is considered by introducing surface stresses in accordance with the Gurtin-Murdoch theory. According to this theory, it is assumed that, in addition to external loads, surface stresses act on the layer boundaries, which are described by Hooke's “surface” law. As a result, the properties of the elastic material of the layer with nanoscale thickness become different from the properties of the material of a regular-sized body, which is typical for nanomechanics problems. A standard technique was used for the solution of formulated problems, including the application of limiting absorption principle, the Fourier transform over infinitely extended coordinate and the theory of residues for finding the inverse Fourier transform. It is shown how it is possible to obtain solutions in the form of series in natural waves, in which the wave numbers are defined as the roots of the corresponding dispersion equations. For a specific example, dispersion relations were studied and graphs of the first dispersion curves were plotted. The behavior of barrier frequencies, changes in wave numbers and zones of existence of backward waves at different nanoscale layer thicknesses are analyzed. The results of the analysis showed that for an ultrathin layer, surface effects have a significant impact on the dispersion relations, and the trends in the dispersion curves can differ significantly for different modes and layer thicknesses.

Keywords

steady-state oscillationsdispersion equationelastic layerplane problemnanothicknesssurface stressGurtin-Murdoch model

ПЛОСКИЕ ЗАДАЧИ О ДЕЙСТВИИ ОСЦИЛЛИРУЮЩЕЙ НАГРУЗКИ НА ГРАНИЦЕ УПРУГОЙ ИЗОТРОПНОЙ ПОЛОСЫ ПРИ НАЛИЧИИ ПОВЕРХНОСТНЫХ НАПРЯЖЕНИЙ

Рассматриваются симметричная и антисимметричная плоские задачи о действии осциллирующей нагрузки на границе упругой изотропной нанотонкой полосы. Наноразмерность полосы учитывается введением поверхностных напряжений по теории Гуртина – Мурдоха. Согласно данной теории, принимается, что на торцах полосы помимо внешних нагрузок действуют также поверхностные напряжения, которые описываются «поверхностным» законом Гука. В такой модели интегральные свойства упругого материала полосы с наноразмерной толщиной становятся отличными от свойств материала тела обычной размерности, что характерно для задач наномеханики. Для решения поставленных задач использовалась стандартная техника, включающая применение принципа предельного поглощения, преобразования Фурье по бесконечно протяженной координате и теории вычетов для нахождения обратного преобразования Фурье. Показано, как можно получить решения в виде рядов по собственным волнам, в которых волновые числа определяются как корни соответствующих дисперсионных уравнений. Для конкретного примера были изучены дисперсионные соотношения и построены графики первых дисперсионных кривых. Проанализировано поведение частот запирания, изменение волновых чисел и зон существования обратных волн при различных наноразмерных толщинах полосы. Результаты проведенного анализа показали, что для ультра-тонкой полосы поверхностные эффекты оказывают существенное влияние на дисперсионные соотношения, причем тенденции изменения дисперсионных кривых могут существенно отличаться для различных мод и толщинах полосы.

Ключевые слова

установившиеся колебаниядисперсионное уравнениеупругая полосаплоская задачананотолщинаповерхностное напряжениемодель Гуртина – Мурдоха

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Article information

Published: 29 May 2023
Issue date: 29 May 2023
Issue: No. 1 (2023)