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УДК 539.4
Исследование изгибного поведения одно- и многослойных углеродных нанотрубок на упругом основании в рамках метода оптимизации
A. Timesli
Университет им. Хасана II, Касабланка, 20670, Марокко
В рамках нелокальной теории цилиндрических оболочек изучено поведение при продольном изгибе многослойных и однослойных углеродных нанотрубок, внедренных в упругую среду. Однослойная нанотрубка рассматривается как цилиндрическая оболочка, а многослойная — как несколько вложенных друг в друга однослойных нанотрубок, взаимодействующих посредством ван-дер-ваальсовых сил. Взаимодействие между матрицей и внешним слоем моделируется как основание с использованием моделей Winkler, Pasternak и Kerr. Разработана методика оптимизации для оценки нелокальной критической нагрузки одно- и многослойных углеродных нанотрубок. Предложены аналитические формулы для описания потери устойчивости однослойных нанотрубок, погруженных в упругую среду, без учета влияния нелокального параметра. Данные уравнения учитывают ван-дер-ваальсовы взаимодействия между соседними трубками и влияние членов, содержащих разницу радиусов трубок, которые обычно не учитываются в уравнениях. Численно исследовано влияние количества слоев, нелокального параметра, а также параметров упругого основания, а также стабильность углеродных на-нотрубок в зависимости от различных параметров.
Ключевые слова: поведение при изгибе, многослойные углеродные нанотрубки, ван-дер-ваальсово взаимодействие, нелокальная теория упругости, теория цилиндрических оболочек
DOI 10.24412/1683-805X-2021-6-36-40
Buckling behavior of SWCNTs and MWCNTs resting on elastic foundations using an optimization technique
A. Timesli
AICSE Laboratory, National Higher School of Arts and Crafts, Hassan II University of Casablanca,
Casablanca, 20670, Morocco
This paper aims to investigate the buckling behavior of multi walled carbon nanotubes (MWCNTs) and single walled carbon nanotubes (SWCNTs) embedded in an elastic medium using the nonlocal cylindrical shell theory. The SWCNT is treated as a cylindrical shell and MWCNT is considered as multiple SWCNTs nested inside one another, they interact with each other via van der Waals interactions. The interaction between the matrix and the outer wall is modeled as a foundation using Winkler, Pasternak, and Kerr models. An optimization technique is developed to estimate the nonlocal critical buckling load of SWCNT and MWCNT. Furthermore, analytical formulas are proposed to describe the buckling behavior of SWCNTs embedded in an elastic medium without taking into account the effects of the nonlocal parameter. In the proposed formulas, van der Waals interactions between adjacent tubes and the effect of terms involving tube radii differences are taken into account, although they are generally neglected in expressions published in the literature. The effects of the number of layers, the nonlocal parameter, and the elastic foundation parameters are investigated. Moreover, the effects of different parameters on the stability behavior of the carbon nano-tubes are also discussed.
Keywords: buckling behavior, multi-walled carbon nanotubes, van der Waals interaction, nonlocal elasticity theory, cylindrical shell theory
© Timesli A., 2021
1. Introduction
Carbon nanotubes (CNTs) were discovered by Ii-jima [1]. Since that time, many scientific research activities have been carried out on CNTs. In these scientific research works, the study of the mechanical behavior of CNTs is based on two main types of modeling: atomistic modeling [2-4] and continuum mechanic modeling [5-7]. However, atomistic modeling is very expensive, which limits the computational capacity. To give an idea on the problem of the atomistic modeling in molecular dynamics (MD) simulation [8], we refer readers to the work of Liew et al. [9] which shows that the calculation of buckling behavior for SWCNTs needs 36 hours using a single CPU SGI origin 2000 system and 2000 atoms. In addition, the computation time increases considerably with increasing number of atoms or number of layers of MWCNT. The analysis of SWCNT and MWCNT was conducted by several continuum models [1016]. One of the most important researches is the excellent mechanical resilience of CNTs embedded in a matrix, the researchers continue to study this kind of problems until now as evidenced by the following research work [5, 7, 17-28]. In the following, we present some examples of scientific research for each foundation model. For the Winkler foundation model based on a single parameter, the stability analysis of cantilevered curved microtubules in axons is presented by Shariati et al. [29], the impacts of covering MAP Tau proteins have been taken into account using the Winkler elastic medium. Chaabane et al. [30] studied the mechanic behavior of functionally graded materials (FGM) beams using a model based on the hyperbolic shear deformation theory, the authors also discussed the effect of several parameters such as the Winkler spring constant, fundamental frequency, normal and shear stresses. Here are some other recent research papers [13, 31, 32] that study the effect of the Winkler elastic medium on CNTs. Using the Pasternak foundation model based on two parameters, Chikr et al. [33] studied the buckling behavior of material sandwich plates with several boundary conditions using a refined trigonometric shear deformation theory. Refrafi et al. [34] analyzed the buckling and hygrothermal of the FGM sandwich plate embedded in an elastic foundation. Bousahla et al. [35] studied the buckling and vibration of composite beams reinforced by SWCNTs and embedded in an elastic foundation. Bellal et al. [36] studied the buckling behavior of a single layer graphene sheet seated on an visco-Pasternak medium. Tounsi et al. [37] analyzed the static effects of nonlinear hygrothermo-mechani-
cal loading on advanced FGM ceramic-metal plates. Boukhlif et al. [38] presented a dynamic investigation of FGM plates seated on an elastic foundation. Some works focus on the study of structures resting on viscoelastic foundations [39-41]. Boulefrakh et al. [39] studied the FGM plates embedded in visco-Pas-ternak foundations using a quasi three-dimensional (quasi 3D) model of hyperbolic shear deformation. Malikan et al. [40] analyzed the damped forced vibration of SWCNTs seated on a viscoelastic foundation and under a thermal environment. Malikan et al. [41] used a viscoelastic nanobeam resting on a visco-Pasternak foundation to examine the transient response of CNTs. There is still recent work on the Pasternak model such as [42, 43]. For the Kerr foundation model based on three parameters, Kaddari et al. [44] discussed the mechanic behavior of functionally graded porous plates embedded in elastic foundations. Timesli [45] analyzed the buckling of double-walled carbon nanotubes (DWCNTs) seated on Winkler, Pasternak, and Kerr elastic foundations using the nonlocal Donnell shell theory. Addou et al. [46] investigated the dynamic behavior of FGM plates with porosities using different elastic foundation models.
There are several applications of CNTs such as nanocomposite structures, some references are given in the following. Arshid et al. [47] analyzed the vibration of FGM microplates with porosities embedded in polymeric nanocomposite patches, an innovative plate theory is also used to take into account the hygrothermal effect. Bendenia et al. [48] presented studies on defections, stresses and free vibration of FGM reinforced by CNTs and embedded in an elastic foundation of the Pasternak-type model. Al-Furjan et al. [49] investigated frequency analysis of viscoelas-tic multi-phase reinforced fully symmetric systems resting on an elastic foundation. Bourada et al. [50] analyzed the dynamic and stability of SWCNT reinforced concrete beam on elastic foundation using an integral first-order shear deformation beam theory. Al-Furjan et al. [51] investigated characteristics of the propagated wave in a sandwich doubly curved nanocomposite panel, a parametric study is presented such as the effects of the weight fraction of CNTs. Timesli [6] studied the stability analysis of the concrete cylindrical shell reinforced by SWCNTs and embedded in an elastic foundation using the Donnell cylindrical shell theory.
The continuum model of CNT based on Donnell shell theory [6, 52, 53] was chosen in this work to describe the mechanical behavior of CNTs.
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Received 10.03.2021, revised 09.07.2021, accepted 09.07.2021
This is an excerpt of the article "Buckling Behavior of SWCNTs and MWCNTs Resting on Elastic Foundations Using an Optimization Technique". Full text of the paper is published in Physical Mesomechanics Journal. DOI: 10.1134/S1029959922020047
Сведения об авторе
Abdelaziz Timesli, Prof., Hassan II University of Casablanca, Morocco, [email protected], [email protected]
