Nonlocal elasticity approach for free longitudinal vibration of circular truncated nanocones and method of determining the range of nonlocal small scale

Li, C.,Sui, S.H.,Chen, L.,Yao, L.Q. Techno-Press 2018 Smart Structures and Systems, An International Jou Vol.21 No.3

The free longitudinal vibration of a circular truncated nanocone is investigated based on the nonlocal elasticity theory. Exact analytical formulations for tapered nanostructures are derived and the nonlinear differential governing equation of motion is developed. The nonlocal small scale effect unavailable in classical continuum theory is addressed to reveal the long-range interaction of atoms implicated in nonlocal constitutive relation. Unlike most previous studies applying the truncation method to the infinite higher-order differential equation, this paper aims to consider all higher-order terms to show the overall nonlocality. The explicit solution of nonlocal stress for longitudinal deformation is determined and it is an infinite series incorporating the classical stress derived in classical mechanics of materials and the infinite higher-order derivative of longitudinal displacement. Subsequently, the first three modes natural frequencies are calculated numerically and the significant effects of nonlocal small scale and vertex angle on natural frequencies are examined. The coupling phenomenon of natural frequency is observed and it is induced by the combined effects of nonlocal small scale and vertex angle. The critical value of nonlocal small scale is defined, and after that a new proposal for determining the range of nonlocal small scale is put forward since the principle of choosing the nonlocal small scale is still unclear at present. Additionally, two different types of nonlocal effects, namely the nonlocal stiffness weakening and strengthening, reversed phenomena existing in nanostructures are observed and verified. Hence the opposite nonlocal effects are resolved again clearly. The nano-engineers dealing with a circular truncated nanocone-based sensors and oscillators may benefit from the present work.

Incompatibility Between Quantum Mechanics and Leggett-Type Nonlocal Hidden Variable Theory for a Singlet State

이강호,강기천 한국물리학회 2009 THE JOURNAL OF THE KOREAN PHYSICAL SOCIETY Vol.55 No.6

Local hidden variable theory has been abandoned because it violates the Bell inequality. However, the Bell inequality does not tell us which assumption, either locality or reality or even both, should be discarded in local hidden variable theory. A recently proposed incompatibility theorem for Leggett-type nonlocal hidden variable theory provides some progress in addressing this question. We propose a very simple incompatibility theorem by inspecting the properties of the singlet state within Leggett-type nonlocal hidden variable theory and discuss the characteristics of the incompatibility. Local hidden variable theory has been abandoned because it violates the Bell inequality. However, the Bell inequality does not tell us which assumption, either locality or reality or even both, should be discarded in local hidden variable theory. A recently proposed incompatibility theorem for Leggett-type nonlocal hidden variable theory provides some progress in addressing this question. We propose a very simple incompatibility theorem by inspecting the properties of the singlet state within Leggett-type nonlocal hidden variable theory and discuss the characteristics of the incompatibility.

Nonlocal elasticity approach for free longitudinal vibration of circular truncated nanocones and method of determining the range of nonlocal small scale

C. Li,S.H. Sui,L. Chen,L.Q. Yao 국제구조공학회 2018 Smart Structures and Systems, An International Jou Vol.21 No.3

The free longitudinal vibration of a circular truncated nanocone is investigated based on the nonlocal elasticity theory. Exact analytical formulations for tapered nanostructures are derived and the nonlinear differential governing equation of motion is developed. The nonlocal small scale effect unavailable in classical continuum theory is addressed to reveal the long-range interaction of atoms implicated in nonlocal constitutive relation. Unlike most previous studies applying the truncation method to the infinite higher-order differential equation, this paper aims to consider all higher-order terms to show the overall nonlocality. The explicit solution of nonlocal stress for longitudinal deformation is determined and it is an infinite series incorporating the classical stress derived in classical mechanics of materials and the infinite higher-order derivative of longitudinal displacement. Subsequently, the first three modes natural frequencies are calculated numerically and the significant effects of nonlocal small scale and vertex angle on natural frequencies are examined. The coupling phenomenon of natural frequency is observed and it is induced by the combined effects of nonlocal small scale and vertex angle. The critical value of nonlocal small scale is defined, and after that a new proposal for determining the range of nonlocal small scale is put forward since the principle of choosing the nonlocal small scale is still unclear at present. Additionally, two different types of nonlocal effects, namely the nonlocal stiffness weakening and strengthening, reversed phenomena existing in nanostructures are observed and verified. Hence the opposite nonlocal effects are resolved again clearly. The nano-engineers dealing with a circular truncated nanocone-based sensors and oscillators may benefit from the present work.

Static analysis of 2D-FG nonlocal porous tube using gradient strain theory and based on the first and higher-order beam theory

Xiaozhong Zhang,Jian-Feng Li,Yan Cui,Mostafa Habibi,H. Elhosiny Ali,Ibrahim AlBaijan,Tayebeh Mahmoudi 국제구조공학회 2023 Steel and Composite Structures, An International J Vol.49 No.3

This article focuses on the study of the buckling behavior of two-dimensional functionally graded (2D-FG) nanosize tubes, including porosity, based on the first shear deformation and higher-order theory of the tube. The nano-scale tube is simulated using the nonlocal gradient strain theory, and the general equations and boundary conditions are derived using Hamilton’s principle for the Zhang-Fu’s tube model (as a higher-order theory) and Timoshenko beam theory. Finally, the derived equations are solved using a numerical method for both simply-supported and clamped boundary conditions. A parametric study is performed to investigate the effects of different parameters, such as axial and radial FG power indices, porosity parameter, and nonlocal gradient strain parameters, on the buckling behavior of the bi-dimensional functionally graded porous tube. Keywords: Nonlocal strain gradient theory; buckling; Zhang-Fu’s tube model; Timoshenko theory; Two-dimensional functionally graded materials; Nanotubes; Higher-order theory.

A mechanical response of functionally graded nanoscale beam: an assessment of a refined nonlocal shear deformation theory beam theory

Amine Zemri,Mohammed Sid Ahmed Houari,Abdelmoumen Anis Bousahla,Abdelouahed Tounsi 국제구조공학회 2015 Structural Engineering and Mechanics, An Int'l Jou Vol.54 No.4

This paper presents a nonlocal shear deformation beam theory for bending, buckling, and vibration of functionally graded (FG) nanobeams using the nonlocal differential constitutive relations of Eringen. The developed theory account for higher-order variation of transverse shear strain through the depth of the nanobeam, and satisfy the stress-free boundary conditions on the top and bottom surfaces of the nanobeam. A shear correction factor, therefore, is not required. In addition, this nonlocal nanobeam model incorporates the length scale parameter which can capture the small scale effect and it has strong similarities with Euler–Bernoulli beam model in some aspects such as equations of motion, boundary conditions, and stress resultant expressions. The material properties of the FG nanobeam are assumed to vary in the thickness direction. The equations of motion are derived from Hamilton’s principle. Analytical solutions are presented for a simply supported FG nanobeam, and the obtained results compare well with those predicted by the nonlocal Timoshenko beam theory.

A nonlocal Layerwise theory for free vibration analysis of nanobeams with various boundary conditions on Winkler-Pasternak foundation

Mahsa Najafi,Isa Ahmadi 국제구조공학회 2021 Steel and Composite Structures, An International J Vol.40 No.1

In this study, a nonlocal Layerwise theory is presented for free vibration analysis of nanobeams resting on an elastic foundation. Eringen’s nonlocal elasticity theory is used to consider the small-scale effect on behavior of nanobeam. The governing equations are obtained by employing Hamilton’s principle and Layerwise theory of beams and Eringen’s nonlocal constitutive equation. The presented theory takes into account the in-plane and transverse normal and shear strain in the modeling of the nanobeam and can predict more accurate results. The governing equations of the beam are solved by Navier's method for Simple-Simple boundary conditions and semi-analytical methods to obtain the natural frequency for various boundary conditions including Clamped-Simple (C-S), Clamped-Clamped (C-C) and Free-Free (F-F) boundary conditions. Predictions of the present theory are compared with benchmark results in the literature. Effects of nonlocal parameter, Pasternak shear coefficient, Winkler spring coefficient, boundary conditions, and the aspect ratio on the free vibration of nanobeams are studied. The flexural mode and thickness mode natural frequencies of the nanobeam are predicted. It is shown that the predictions of present method are more accurate than the equivalent single layer theories. The theoretical developments and formulation presented herein should also be served to analyze the mechanical behavior of various nanostructures with various loading and boundary conditions.

A mechanical response of functionally graded nanoscale beam: an assessment of a refined nonlocal shear deformation theory beam theory

Zemri, Amine,Houari, Mohammed Sid Ahmed,Bousahla, Abdelmoumen Anis,Tounsi, Abdelouahed Techno-Press 2015 Structural Engineering and Mechanics, An Int'l Jou Vol.54 No.4

This paper presents a nonlocal shear deformation beam theory for bending, buckling, and vibration of functionally graded (FG) nanobeams using the nonlocal differential constitutive relations of Eringen. The developed theory account for higher-order variation of transverse shear strain through the depth of the nanobeam, and satisfy the stress-free boundary conditions on the top and bottom surfaces of the nanobeam. A shear correction factor, therefore, is not required. In addition, this nonlocal nanobeam model incorporates the length scale parameter which can capture the small scale effect and it has strong similarities with Euler-Bernoulli beam model in some aspects such as equations of motion, boundary conditions, and stress resultant expressions. The material properties of the FG nanobeam are assumed to vary in the thickness direction. The equations of motion are derived from Hamilton's principle. Analytical solutions are presented for a simply supported FG nanobeam, and the obtained results compare well with those predicted by the nonlocal Timoshenko beam theory.

A quasi-3D nonlocal theory for free vibration analysis of functionally graded sandwich nanobeams on elastic foundations

Mofareh Hassan Ghazwani,Ali Alnujaie,Pham Van Vinh,Abdelouahed Tounsi Techno-Press 2024 Advances in nano research Vol.16 No.3

The main aims of this study are to develop a new nonlocal quasi-3D theory for the free vibration behaviors of the functionally graded sandwich nanobeams. The sandwich beams consist of a ceramic core and two functionally graded material layers resting on elastic foundations. The two layers, linear spring stiffness and shear layer, are used to model the effects of the elastic foundations. The size-effect is considered using nonlocal elasticity theory. The governing equations of the motion of the functionally graded sandwich nanobeams are obtained via Hamilton's principle in combination with nonlocal elasticity theory. Then the Navier's solution technique is used to solve the governing equations of the motion to achieve the nonlocal free vibration behaviors of the nanobeams. A deep parametric study is also provided to demonstrate the effects of some parameters, such as length-to-height ratio, power-law index, nonlocal parameter, and two parameters of the elastic foundation, on the free vibration behaviors of the functionally graded sandwich nanobeams.

Dynamic instability region analysis of sandwich piezoelectric nano-beam with FG-CNTRCs face-sheets based on various high-order shear deformation and nonlocal strain gradient theory

Mohammad Arefi,Mahmoud Pourjamshidian,Ali Ghorbanpour Arani 국제구조공학회 2019 Steel and Composite Structures, An International J Vol.32 No.2

In this research, the dynamic instability region (DIR) of the sandwich nano-beams are investigated based on nonlocal strain gradient elasticity theory (NSGET) and various higher order shear deformation beam theories (HSDBTs). The sandwich piezoelectric nano-beam is including a homogenous core and face-sheets reinforced with functionally graded (FG) carbon nanotubes (CNTs). In present study, three patterns of CNTs are employed in order to reinforce the top and bottom face-sheets of the beam. In addition, different higher-order shear deformation beam theories such as trigonometric shear deformation beam theory (TSDBT), exponential shear deformation beam theory (ESDBT), hyperbolic shear deformation beam theory (HSDBT), and Aydogdu shear deformation beam theory (ASDBT) are considered to extract the governing equations for different boundary conditions. The beam is subjected to thermal and electrical loads while is resting on Visco-Pasternak foundation. Hamilton principle is used to derive the governing equations of motion based on various shear deformation theories. In order to analysis of the dynamic instability behaviors, the linear governing equations of motion are solved using differential quadrature method (DQM). After verification with validated reference, comprehensive numerical results are presented to investigate the influence of important parameters such as various shear deformation theories, nonlocal parameter, strain gradient parameter, the volume fraction of the CNTs, various distributions of the CNTs, different boundary conditions, dimensionless geometric parameters, Visco-Pasternak foundation parameters, applied voltage and temperature change on the dynamic instability characteristics of sandwich piezoelectric nano-beam.

Propagation of waves with nonlocal effects for vibration response of armchair double-walled CNTs

Ali, Zainab,Khadimallah, Mohamed Amine,Hussain, Muzamal,Asghar, Sehar,Al-Thobiani, Faisal,Elbahar, Mohamed,Elimame, Elaloui,Tounsi, Abdelouahed Techno-Press 2021 Advances in nano research Vol.11 No.2

In this paper, vibration characteristics of double-walled carbon nanotubes (CNTs) are studied based upon nonlocal elastic shell theory. The significance of small scale is being perceived by developing nonlocal Love shell model. The wave propagation approach has been utilized to frame the governing equations as eigen value system. The influence of nonlocal parameter subjected to diverse end supports has been overtly analyzed. An appropriate selection of material properties and nonlocal parameter has been considered. The influence of changing mechanical parameter Poisson's ratio has been investigated in detail. It is found that the frequencies decrease as nonlocal parameter increases and for the certain values of nonlocal parameter against range of Poisson ratio rise slowly with length double-walled CNTs. The dominance of boundary conditions via nonlocal parameter is shown graphically. The results generated furnish the evidence regarding applicability of nonlocal shell model and also verified by earlier published literature.