Nonlocal continuum-based modeling of mechanical characteristics of nanoscopic structures

Accepted Manuscript Nonlocal continuum-based modeling of mechanical characteristics of nanoscopic structures Hashem Rafii-Tabar, Esmaeal Ghavanloo, S. Ahmad Fazelzadeh PII: DOI: Reference:

S0370-1573(16)30102-8 http://dx.doi.org/10.1016/j.physrep.2016.05.003 PLREP 1905

To appear in:

Physics Reports

Accepted date: 25 May 2016 Please cite this article as: H. Rafii-Tabar, E. Ghavanloo, S.A. Fazelzadeh, Nonlocal continuum-based modeling of mechanical characteristics of nanoscopic structures, Physics Reports (2016), http://dx.doi.org/10.1016/j.physrep.2016.05.003 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

*Manuscript

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Nonlocal continuum-based modeling of mechanical characteristics of nanoscopic structures Hashem Rafii-Tabara,∗, Esmaeal Ghavanloob,∗, S. Ahmad Fazelzadehb a

Department of Medical Physics and Biomedical Engineering, Faculty of Medicine, Shahid Beheshti University of Medical Sciences, Tehran, Iran b School of Mechanical Engineering, Shiraz University, Shiraz 71963-16548, Iran

Abstract Insight into the mechanical characteristics of nanoscopic structures is of fundamental interest and indeed poses a great challenge to the research communities around the world. These structures are ultra fine in size and consequently performing standard experiments to measure their various properties is an extremely difficult and expensive endeavor. Hence, to predict the mechanical characteristics of the nanoscopic structures, different theoretical models, numerical modeling techniques, and computer-based simulation methods have been developed. Among several proposed approaches, the nonlocal continuum-based modeling is of particular significance because the results obtained from this modeling for different nanoscopic structures are in very good agreement with the data obtained from both experimental and atomistic-based studies. A review of the essentials of this model together with its applications is presented here. Our paper is a self contained presentation of the nonlocal elasticity theory and contains the analysis of the recent works employing this model within the field of nanoscopic structures. In this review, the concepts from both the classical (local) and the nonlocal elasticity theories are presented and their applications to static and dynamic behavior of nanoscopic structures with various morphologies are discussed. We first introduce the various nanoscopic structures, both carbon-based and non carbon-based types, and then after a brief review of the definitions and concepts from classical elasticity theory, and the basic assumptions underlying Corresponding authors Email addresses: [email protected] (Hashem Rafii-Tabar), [email protected] (Esmaeal Ghavanloo) ∗

Preprint submitted to Physics Reports

May 19, 2016

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

size-dependent continuum theories, the mathematical details of the nonlocal elasticity theory are presented. A comprehensive discussion on the nonlocal version of the beam, the plate and the shell theories that are employed in modeling of the mechanical properties and behavior of nanoscopic structures is then provided. Next, an overview of the current literature discussing the application of the nonlocal models of nanoscopic carbon allotropes is presented. We then discuss the application of the models to the investigation of the properties of nanoscopic structures from different materials and with different types of morphologies. Furthermore, we also present recent developments in the application of the nonlocal models. Finally, conclusions and discussions regarding the potentiality of these models for future research are provided.

2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Contents 1 Introduction

6

2 Nanoscopic structures 2.1 Carbon-based nanoscopic structures 2.1.1 Fullerenes . . . . . . . . . . 2.1.2 Carbon nanotubes . . . . . 2.1.3 Graphene . . . . . . . . . . 2.2 Nanoparticles . . . . . . . . . . . . 2.3 Nanowires . . . . . . . . . . . . . .

. . . . . .

12 13 14 14 16 16 17

3 Essential concepts from nonlocal continuum elasticity theory 3.1 A brief review of classical (local) continuum elasticity theory . 3.1.1 Concept of strain . . . . . . . . . . . . . . . . . . . . . 3.1.2 Concept of stress . . . . . . . . . . . . . . . . . . . . . 3.1.3 Stress-strain relation for elastic materials . . . . . . . . 3.1.4 Governing equations . . . . . . . . . . . . . . . . . . . 3.2 Size-dependent continuum theories . . . . . . . . . . . . . . . 3.3 Nonlocality and nonlocal averaging . . . . . . . . . . . . . . . 3.4 Integral-type nonlocal elasticity theory . . . . . . . . . . . . . 3.5 Laplacian-based nonlocal elasticity theory . . . . . . . . . . . 3.5.1 Laplacian of the stress tensor . . . . . . . . . . . . . . 3.5.2 Nonlocal constitutive relations . . . . . . . . . . . . . .

19 19 19 20 20 21 23 24 26 28 29 31

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

4 Computational modeling of the mechanical behavior of nanoscopic structures 34 4.1 Mechanical behavior of materials: useful definitions and concepts 34 4.2 Nonlocal beam models . . . . . . . . . . . . . . . . . . . . . . 36 4.2.1 The Euler-Bernoulli beam theory . . . . . . . . . . . . 36 4.2.2 The Timoshenko beam theory . . . . . . . . . . . . . . 40 4.3 Nonlocal cylindrical shell models . . . . . . . . . . . . . . . . 44 4.3.1 Classical shell theory . . . . . . . . . . . . . . . . . . . 44 4.3.2 First order shear deformation theory . . . . . . . . . . 50 4.4 Nonlocal plate model . . . . . . . . . . . . . . . . . . . . . . . 55 4.4.1 The Kirchhoff plate theory . . . . . . . . . . . . . . . . 55 4.4.2 The Mindlin plate theory . . . . . . . . . . . . . . . . . 58 4.5 Nonlocal spherical shell model . . . . . . . . . . . . . . . . . . 60 3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

4.6

Nonlocal elasticity theory for breathing-mode of nano-objects . 64 4.6.1 Nonlocal model for nanosphere . . . . . . . . . . . . . 64 4.6.2 Nonlocal model for elongated nanocrystals . . . . . . . 67

5 Determination of the elastic properties and the nonlocal rameter of carbon-based nanoscopic structures 5.1 Young's modulus and effective wall thickness of SWCNTs . 5.2 Anisotropic elastic constants of SWCNTs . . . . . . . . . . 5.3 Anisotropic mechanical properties of graphene sheets . . . 5.4 Nonlocal Parameter for CNTs . . . . . . . . . . . . . . . . 5.5 Evaluation of the nonlocal parameter for SWCNTs . . . .

pa. . . . .

. . . . .

69 69 71 77 79 81

6 Mechanical characteristics of high aspect ratio CNTs: nanobeams 86 6.1 Application of static analysis to CNTs . . . . . . . . . . . . . 86 6.1.1 Structural deformations of CNTs under transverse loadings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 6.1.2 Buckling of CNT . . . . . . . . . . . . . . . . . . . . . 87 6.2 Application to dynamic analysis of CNT . . . . . . . . . . . . 90 6.2.1 Linear flexural vibrations of CNTs . . . . . . . . . . . 90 6.2.2 Linear longitudinal vibrations of nanotube heterojunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 6.2.3 Nonlinear vibrations of CNTs . . . . . . . . . . . . . . 93 6.2.4 Wave propagation in CNTs . . . . . . . . . . . . . . . 94 7 Mechanical characteristics of low aspect drical nanoshells 7.1 Shell-based approach to CNT buckling . 7.1.1 Axial buckling . . . . . . . . . . . 7.1.2 Torsional buckling . . . . . . . . 7.2 Vibrational properties of CNTs . . . . . 7.2.1 The shell-like vibrations of CNTs 7.2.2 Phonon dispersion of SWCNTs . 7.3 Wave propagation in CNTs . . . . . . .

ratio CNTs: cylin. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

96 96 96 98 100 100 102 103

8 Application of nonlocal continuum theory to modeling graphene sheets 105 8.1 Buckling of graphene sheets . . . . . . . . . . . . . . . . . . . 105 8.2 Vibration of graphene sheets . . . . . . . . . . . . . . . . . . . 106 4

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

8.3

8.2.1 Single-layered graphene sheets . . . . 8.2.2 Multi-layered graphene sheets . . . . 8.2.3 Graphene-based resonant sensors . . Propagation of waves in the graphene sheets

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

106 108 110 111

9 Dynamical characteristics of spherical nanoscopic structures113 9.1 Axisymmetric vibration of spherical nanoshells . . . . . . . . . 113 9.2 Radial vibration of spherical nanoparticles . . . . . . . . . . . 115 10 Application of nonlocal models to other types of structures 10.1 Mechanical modeling of elongated nanocrystals . . 10.1.1 Bending and buckling of nanowires . . . . 10.1.2 Dynamic analysis of nanowires . . . . . . . 10.2 Nonlocal-based modeling of nanopeapods . . . . . 10.3 Analysis of micro/nanobridge test . . . . . . . . .

nanoscopic 117 . . . . . . . 117 . . . . . . . 117 . . . . . . . 118 . . . . . . . 119 . . . . . . . 119

11 Some recent developments in the application of the nonlocal models 121 11.1 A different nonlocal stress model and its stiffness enhancement effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 11.2 Self-adjointness of Eringen's nonlocal elasticity . . . . . . . . . 121 11.3 Application of nonlocal continuum theory to piezoelectric nanoscopic structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 12 Concluding remarks and future prospects

5

124

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

1. Introduction The synthesis and characterization of the nanoscopic structures with various morphologies have generated enormous interest in both basic and applied research. Various types of biological as well as non-biological nanoscopic structures with very different functionalities have been constructed and reported in the literature over the past two decades. These fundamental and exotic elementary material units possess unique physical and chemical properties when compared to aggregates of individual atoms, molecules and their bulk composites. Many key developments in the fields of nanoscale science and nanotechnology are directly related to a proper understanding of the structure and function of these nanoscopic structures. Understanding the size and shape dependence of the mechanical characteristics of the nanoscopic structures is an essential prerequisite for an accurate and efficient design, fabrication and assembly of nanostructured materials and nanodevices with a predictable behavior. Therefore, the development of efficient methods to evaluate their mechanical behavior poses a multitude of challenging research problems to the theoretical, computational and experimental nanoscience community. Mechanical properties can be predicted via various theoretical and computational methodologies. These include different quantum mechanical-based methods, two classical statistical mechanics-based techniques, namely Molecular Dynamics (MD) and Monte Carlo (MC) simulation methods [1], and also via continuum-based theories [2]. Evidently, the most accurate methods to describe the mechanics of the nanoscopic structures are the quantum mechanical-based methods, as expressed by solutions of the Schr¨odinger equation embodying the motion of the electrons and the nuclei [3]. However, quantum-mechanical calculations are computationally very expensive and time-consuming and are only practical for comparatively small systems containing a few tens to a few hundreds of atoms. In MD simulation [4], the space-time trajectories of individual atoms in an assembly of N atoms, or molecules, are computed within the framework of either a Newtonian (i.e., deterministic) dynamics or a Langevin-type (i.e., stochastic) dynamics. In both of these frameworks, the initial position coordinates and velocities of the atoms are the input parameters. A classical MD-based simulation proceeds by considering a model nanoscopic structure consisting of, say, N atoms confined in a simulation cell of volume V . The choice of N critically depends on the amount of computational power avail6

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

able. The simulation cell, also referred to as the central cell, is periodically replicated in all spatial dimensions. This leads to the generation of the periodic images of this cell, as well as the original N atoms confined in the central cell. This replication is referred to as the periodic boundary condition (PBC), and its implementation is necessary to compensate for the unwanted consequences of the artificial surfaces that emerge as a result of the finite size of the nanoscopic structure considered. The energetics and dynamics of the N atoms are obtained from prescribed two-body or many-body phenomenological interatomic potentials, from which the forces experienced by individual atoms are computed at each simulation time step. While the MD simulation is essentially a deterministic method, the MC simulation method, on the other hand, uses probabilistic concepts. Like in the MD simulation, in this method too, the model-system is composed of N interacting atoms, confined to a volume V and is given a set of initial position coordinates. The initial configuration is then allowed to evolve by successive random displacements of the atoms. The successive configurations so generated are not, however, all acceptable, and a decision must made as to whether to accept or reject a particular configuration. The decision must ensure that, asymptotically, the configuration part of the phase space is sampled according to the probability density pertinent to a particular statistical-mechanical ensemble [5]. Compared to quantum mechanical-based methods, the classical statistical mechanics-based methods can be used to study larger systems. However, these methods typically need very high-performance computational facilities. In addition, MD simulations have several limitations with respect to boundary conditions, time steps and incorporation of the temperature effects [6]. The practical applications of the atomistic modeling techniques are limited to systems containing a relatively small number of molecules or atoms and are usually confined to studies of rather short-lived phenomena, from picoseconds to nanoseconds. Alternative approaches are, therefore, highly desirable, and it is well-known that continuum mechanics-based models might offer alternative and complementary methodologies for the analysis of the nanoscopic structures including nanoparticles [7], quantum dots [8], fullerenes [9, 10], single- and multi-walled nanotubes [11, 12, 13, 14], graphene sheets [15], nanowires [16], nanocones [17] and nanotori [18]. Continuum mechanics is probably the most well-known and sophisticated branch of classical mechanics which deals with the mechanical properties and behavior of materials. In this mechanics, it is assumed that the physical or mechanical 7

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

properties of systems are distributed continuously in the spaces they occupy [19]. Continuum-based methods are much faster than the atomistic modeling methods, working with a much reduced number of degrees of freedom. These models can be useful in the study of long-range phenomena in extended atomic systems since they capture the collective behavior of the atoms. It has been shown in many computational studies that the results obtained from continuum mechanics approaches are in good agreement with the experimental and atomistic-based results. In the classical continuum theory, it is assumed that the stress at a material point is dependent on the strain, strain rate and strain history at that point. This model is referred to as the local model due to the absence of information from neighboring material points [20]. Since all nanoscopic structures consist of discrete elements (molecules and atoms), a question is naturally posed about the applicability and limitations of the classical continuum mechanics to the description of mechanical behavior of these structures [21]. Classical continuum-based simulations are scale free and cannot reflect the nanoscale physical laws of the nanoscopic structures. Furthermore, this model cannot incorporate the small-scale effects into the formulation and is therefore not adequate for the analysis of the nanoscopic structures. The applicability limitations reflect the fact that at small sizes the lattice spacing between individual atoms becomes increasingly important and the discrete nature of the material can no longer be homogenized into a continuum medium. In other words, the physical properties at the nanoscale are size-dependent and the small-scale effect should be considered for a better prediction of the mechanical behavior of the nanoscopic structures [22]. Consequently, the classical continuum theories are expected to fail when the microscopic and macroscopic length scales are comparable, as the mechanical properties at a point are affected by the neighboring points. To overcome these limitations, modified continuum models have been proposed which can look at larger systems but are also able to include nanoscale size-effects. There are different ways to modify the classical continuum theories so that their range of applicability can be extended to problems with smaller characteristic lengths, with results closer to lattice dynamics in a wider range of wave lengths. Generally, the modified continuum models which include information on the size scale of the structure are termed the nonlocal models. Different nonlocal continuum theories, such as the couplestress theory [23, 24], the Cosserat or the micropolar continuum theory [25], the nonlocal continuum field theories [26, 27, 28] and the gradient elasticity 8

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

theory [29] have been developed. The nonlocal models allow the integration of size-effects into the classical continuum models. One of the most familiar classes of modified continuum models utilizes the concepts from nonlocality. Nonlocal field theories provide efficient mechanisms for capturing small-scale effects induced by microstructures, or the discrete nature of the material at smaller scales. The main idea behind the nonlocal field theories is to obtain a relationship between macroscopic and microscopic quantities [30]. The nonlocal effect is implemented through the introduction of a nonlocal parameter which depends on the material, the molecular structure and the internal characteristic length [31]. The appearance of the nonlocal continuum field theories dates back from the 1970s when modeling elastic materials with long range forces was considered [26, 32, 33]. A systematic rational procedure for the nonlocal theories was established by Eringen and co-workers [27, 28, 34]. Many topics of the nonlocal continuum field theories have been established by Eringen[35], including nonlocal elasticity, nonlocal fluid mechanics, nonlocal electromagnetism, nonlocal thermoelasticity, nonlocal memory-dependent elasticity, and nonlocal piezoelectricity etc. The nonlocal elasticity theory states that the stress at a point in a body depends not only on the local stress at that particular point, but also on the spatial integrals, with weighted averages, of the local stress contribution from all other points in the body. The nonlocal elasticity theory differs from the classical or local one in the stress-strain constitutive relations only. The general form of the constitutive relations involves an integral over the whole body. This form of the nonlocal elasticity theory is called the strong or the integral nonlocal model [36]. The strong nonlocal theory received relatively less attention and it was investigated by a limited group of researchers [37, 38, 39, 40, 41, 42], due to the fact that it is mathematically difficult to obtain the solution to problems. To overcome the limitations of the strong nonlocal theory, Eringen [43] showed that it is possible to represent the integral constitutive relations in an equivalent differential form. Although an equivalent, but approximate, expression of the nonlocal stress in a differential form was also derived for different nonlocal moduli, this differential nonlocal stress relation seemed not to have attracted the attention for some time. Application of the nonlocal elasticity to nanoscopic structures was initially addressed by Peddieson et al. [44], when it was employed to formulate a nonlocal version of the Euler-Bernoulli beam model, and it was concluded that the nonlocal elasticity could potentially play a useful role in nanotech9

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

nology applications. That study attracted extensive attention in the nanomechanics community. Subsequently a series of research papers appeared in top international journals in mechanics and applied physics. The mechanical behavior of various nanoscopic structures including fullerenes, carbon nanotubes, graphene sheets, nanoparticles, nanowires, nanocones etc, have been investigated successfully. It has been shown in many applications that the results from the nonlocal elasticity for different nanoscopic structures are in good agreement with the results from the experimental and atomistic-based approaches. Therefore, the nonlocal elasticity could play an important role in the study of novel nanoscopic structures that have not been fully explored or even previously addressed before. In fact, the nonlocal elasticity has become of great importance in nanoscience and nanoengineering research activities. This fact is well illustrated in Fig. 1, with the data acquired from the Scopus database. Although several reviews on the application of the nonlocal continuum modeling of some nanoscopic structures have appeared so far [45, 46], however, to-date, no review that exclusively deals with the formulation of nonlocal mathematical and computational modeling has been presented. The extensive nonlocal continuum modeling of nanoscopic structures has now reached such a mature stage that an up-to-date review is now called for to summarize our current status of knowledge particularly for the benefit of the computational condensed matter physics and computational materials modeling communities. This is the motivation behind this review. It is a rather comprehensive and self-contained review, covering materials up to 2015. For the review to be self-contained, we have presented all the relevant theories, computational methods and physical models essential for grasping the existing research materials and for future research. The typical audience should be graduate students and researchers from different fields of computational condensed matter physics, computational materials science, computational nanoscience and nanotechnology. As the topics we have covered are outside the scope of conventional university courses in theoretical solid state physics, nanophysics, computational physics or materials science, we have attempted to provide an easy-to-follow description of their fundamental theories so as to make the pertinent research materials accessible to as a wide community as possible. In this review, we have surveyed the nonlocal continuumbased studies concerned with the mechanical behavior of different nanoscopic structures, covering their vibrational behavior, structural stability, buckling behavior, wave propagation in them, and their static deformations. The 10

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

nanoscopic structures we have considered different morphologies including fullerenes, carbon nanotubes, nanoparticles, nanowires, and graphene sheets. We have tried to cover most of the research materials pertinent to the nonlocal modeling of the mechanical properties and behavior of the nanoscopic structures that we have been able to locate and deemed to be relevant to this review. Obviously, our selection has been limited, and in no way makes a claim to be either exhaustive or be a judgment on the quality of those works that have somehow not been included in this study. The existing literature on the physics and mechanics of nanoscopic structures is vast and is being continuously expanded. The organization of this review is as follows. Following the Introduction Section, we introduce, in Section 2, the various nanoscopic structures, both carbon-based and non carbon-based types, the computational modeling of whose properties have been the subject of the extensive investigations that have tried to summarize in this review for the benefit of the physics and materials science communities. In Section 3, the essential concepts from both the local and nonlocal continuum elasticity theories are presented, with emphasis on the description of the concepts from the nonlocal theories. This is followed in Section 4 by a comprehensive discussion of the nonlocal version of the theories that are normally employed in continuum modeling of the mechanical properties and behavior of materials. In Sections 5 to 8, we review the applications of the nonlocal continuum modeling to the static and dynamic properties of nanoscopic carbon allotropes. Sections 9 and 10 are concerned with the application of the model to the investigation of the properties of other types of nanoscopic structures including spherical nanoparticles, elongated nanocrystals etc. In Section 11, we present some of the recent developments in the application of the nonlocal models, while Section 12 contains a summary of our review and the concluding remarks. A comment should be made about the convention that we have used to spell the words that are accompanied by the prefix ’nano’. In all cases, we have connected this prefix to the word. For example, for nano wire, we have written nanowire, or for nano structure, we have written nanostructure etc.

11

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

2. Nanoscopic structures The fundamental entities with which nanoscopic science is concerned are the physical, chemical and biological nanoscopic structures that form the building blocks of inanimate and animate matter. Nanoscopic structures operate on highly reduced time and energy scales. They are formed from a countable (finite) number of atoms and molecules, and their sizes lie in the range between individual molecules and microscopic structures. Normally, at least one of their dimensions is smaller than 100 nm. Their characteristic feature is their high surface-to-volume ratio. Nanotechnology is concerned with constructing nanoscopic devices and components from interacting assemblies of individual nanoscopic structures. Depending on the type of a nanoscopic structure involved, namely, soft or hard, nanoscience and nanotechnology have been categorized into soft (wet) and hard (dry) subfields. The bottomup approach to nanotechnology involves constructing nanoscopic structures from below the nanoscale, atom by atom and molecule by molecule, by a precise positioning of these fundamental units at specified locations. This has led, for the first time in human history, to the possibility of designing and fabricating devices, components, and materials, including biomaterials, that exhibit totally different and novel physical, chemical, and biological properties, as compared to the behavior of single molecules and bulk phase materials. As a consequence, nanoscopic science and technology allow for the purposeful manipulation and structural and functional transformation of condensed phases at their most elementary levels. Such manipulation offers the possibility of exercising a complete control over the properties and over the functioning of physical and biological matter at the atomic and molecular levels, implying that we can interrogate physical and biological matter atom-by-atom and induce predetermined property changes in them. The top-down approach to nanotechnology, however, provides a practical framework for the realization of the miniaturization program, initiated in the classic paper of Feynman [47], one of whose aims is to produce nanoscopic structure-based low-dimensional quantum-scale devices starting from microscopic scales. Whatever strategy is followed to reach the nanoscale, there is no doubt that the resulting structures and processes that unfold at this scale display remarkable and unique properties. For example, nanostructured materials (i.e., materials composed of nanosized grains, or materials containing injected nanoscopic structures, such as carbon nanotubes) have very different mechanical, thermal, electronic, and optical properties compared to their 12

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

counterparts with microsized grains. For instance, it is known that nanostructured Fe, Cu, and Ni have electrical resistances, respectively, 55%, 15%, and 35% higher than their corresponding coarse-grained polycrystalline samples [48]. Numerous different types of nanoscopic structures with different sizes, shapes and compositions have been reported in the literature with their behavior strongly dependent on their sizes, shapes, dimensionality and morphologies. The criterion which has been employed to classify various types of the nanoscopic structures is their dimensionality. These structures are generally classified as zero-dimensional (0D), one-dimensional (1D) and twodimensional (2D) [49]. In zero-dimensional nanoscopic structures, all three dimensions are restricted to a few tens of nanometers. Atomic clusters, nanoparticles, quantum dots and fullerenes are well-known examples of 0D nanoscopic structures. The nanoscopic structures with two dimensions in nanometer size fall under the category of 1D structures. In various fields of nanotechnology, one-dimensional nanoscopic structures have paved the way for numerous advances in both fundamental and applied sciences [50]. One-dimensional structures represent the smallest-dimension structures that can efficiently transport charge carriers. Nanowires, nanorods, nanofibers, nanotubes, nanobelts and nanoribbons are among the 1D nanoscopic structures. Two-dimensional structures, such as thin films, nanoplates and nanosheets, have two of their dimensions lying outside the nanometric size range [51]. 2.1. Carbon-based nanoscopic structures Carbon is one of the most interesting elements in the periodic table and plays a vital role in nature and life, and represents a very important building block in nanomaterials and nanoscopic structures. Many of the new crystalline forms of carbon have only been experimentally discovered in the last few decades [52]. The reason for this critical role is the potentiality of carbon to bond in so many different ways. The bonding can lead to various known carbon allotropes such as diamond, different types of fullerenes, several kinds of nanotubes, graphite, and graphene sheets. These carbon-based nanoscopic structures play a central role in nanomaterial science and nanotechnology due to their exotic mechanical, electronic, thermal and transport properties. A new era in carbon-base materials science and technology was commenced when in 1985 the C60 molecule, the so-called Buckminsterfullerene or the buckyball, was synthesized [53]. The discovery of this nanoscopic 13

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

structure set into motion a very active world-wide research effort that is still growing [54]. In 1991, Iijima [55] discovered another allotrope of carbon in a tubular form, called the carbon nanotube (CNT), and in 2004 the single graphene sheet was isolated [56, 57]. Carbon nanoscopic structures with various morphologies, such as spherical (C60 fullerene (Fig. 1a)), cylindrical (CNT (Fig. 1b)) and planar (graphene (Fig. 1c)), have different behavior vis-a-vis their dimensionality which ranges from 0D to 2D. 2.1.1. Fullerenes One of the most familiar types of carbon nanoscopic structures is the fullerene. Fullerenes are cage-like carbon molecules with their carbon atoms arranged on a sphere with pentagonal and hexagonal faces. The fullerenes form a family with members ranging from C60 to C4860 [58]. Based on the structural shape, there are two main types of fullerenes; spherical or nearspherical, and ellipsoidal/non-spherical. Figure 3 shows the structure and geometry of some fullerene molecules. The C60 is one of the earliest discovered fullerenes and it led to a revolution in nanotechnology [53]. Its 60 atoms are located at the vertices of a regular truncated icosahedron and every carbon site is equivalent to every other site. The average nearest neighbor C-C bond in C60 is 0.144 nm, which is almost identical to that in graphite (0.142 nm). Each atom is trigonally bonded to other atoms, the same as that in graphite, and most of the faces on the cage are hexagons. There are 20 hexagonal and 12 pentagonal faces, and the molecular diameter is ∼ 0.710 nm. There are a total of 1812 fullerene isomers with 60 carbon atoms [59] among which the icosahedral isomer is the only one with firmly established experimental properties, and is believed to be the most thermodynamically stable isomer. 2.1.2. Carbon nanotubes In 1991, the CNT was experimentally observed by Iijima using transmission electron microscopy [54]. This type of CNT was the so called multi-wall carbon nanotube (MWCNT) consisting of several single-wall carbon nanotubes (SWCNTs) of different diameters nested inside one another (Fig. 4). Two years later, the SWCNT was also experimentally obtained [60]. Nowadays, MWCNTs and SWCNTs are produced by different techniques such arc-discharge, laser-ablation, and catalytic growth [61].

14

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

An SWCNT can be visualized as a single layer of graphite (called a graphene sheet) that has been rolled into a hollow cylinder. It should be noted that this is not the practical way to make the nanotubes. However, the above description helps illustrate the nature of the SWCNT structure. In principle, an infinite number of graphene geometries can exist, because a graphene sheet can be rolled up with different angles. The rolling up of the sheet is described by the chiral vector (Fig. 5) that can be expressed as ~ h = n~a1 + m~a2 C

(1)

where the integers n and m are the number of steps along the carbon bonds of the hexagonal lattice, and the ~a1 and ~a2 are the two unit-cell basis vectors of the two-dimensional graphene sheet. An SWCNT can thus be uniquely indexed by a pair of integers (n, m) to represent its chirality or helicity [62]. If n 6= 0, and m = 0, i.e., an (n, 0) nanotube, the SWCNT is named a ‘zigzag’ SWCNT. If n = m i.e., an (n, n) nanotube, the SWCNT is called an ‘armchair’ SWCNT. In all other cases, the SWCNT is called a ‘chiral’ SWCNT. Examples of the three kinds of SWCNTs, i.e., the zigzag, the armchair, and the chiral, are shown in Fig. 6. The structures of the armchair and zigzag nanotubes have a high degree of symmetry. These terms refer to the arrangement of hexagons around the circumference. The diameter d of an SWCNT is given by aC−C p 2 3(n + nm + m2 ) (2) d= π

where aC−C is the carbon-carbon bond length (0.142 nm). There is no geometrical constraint on the diameter of SWCNTs. However, it has been shown that collapsing the SWCNT into a flattened two-layer ribbon is energetically more favorable than maintaining the tubular morphology beyond a diameter value of ≈ 2.5nm [63]. This implies that physically the SWCNTs larger than ≈ 2.5 nm are not stable. The angle between the chiral vector and the zigzag axis of the graphene sheet is called the chiral angle θ, and it is defined as √ 3m −1 ) (3) θ = tan ( m + 2n Because of the high geometrical symmetry, the zigzag and armchair nanotubes are achiral nanotubes, whereas the SWCNTs with a chiral angle of 15

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

0 < θ < π/6 are chiral nanotubes. The values of (n, m) determine the chirality of the SWCNTs and affect their optical, mechanical and electronic properties. SWCNTs with |n−m| = 3i are metallic and those with |n−m| = 3i±1 are semiconducting, where i is an integer. An MWCNT can be considered as a nested collection of concentric SWCNTs with different diameters. The length and diameter of these structures differ from those of SWCNTs and their properties are also very different. Typical dimensions of multi-walled carbon nanotubes consist of the outer diameter in the range 2-20 nm, the inner diameter in the range 1-3 nm, and the length in the range 1-100 µm. The interlayer distance in MWCNTs is close to the distance between graphite layers, i.e., approximately 0.34 nm. Among the different MWCNTs, the double-walled carbon nanotube (DWCNT) and the triple-walled carbon nanotube (TWCNT) are the most studied structures as they have some of the unique properties of SWCNTs as well as the structural stability of MWCNTs [64]. The pair of chiral indices of a DWCNT are denoted by (ni ,mi )@(no ,mo ) where (ni ,mi ) and (no ,mo ) are the chiral indices pertinent to the inner and outer nanotubes in the DWCNT. It should be noted that for MWCNTs, the chirality of the inner walls is hard to ascertain. 2.1.3. Graphene Graphene is a sheet composed of sp2 -bonded carbon atoms, which are tightly packed into a two dimensional honeycomb lattice. Graphene is considered to be the building block for graphitic materials [65]. Graphite consists entirely of layers of graphene stacked on top of each other. For a number of years the isolation of a stable graphene sheet was considered to be improbable. Many experiments were attempted to produce graphene sheets by chemical reactions, but these were not successful [66]. Finally, Novoselov and coworkers [56] were able to isolate an individual layer of graphite crystallites using transparent adhesive tape. Graphene is a promising material in different technological applications due to its unique mechanical, thermal, electrical and optical characteristics. It is also the basic building block of other carbon nanoscopic structures such as carbon nanotubes. 2.2. Nanoparticles The term nanoparticle is a very general one and mainly refers to ultrafine particles in the nanometer size range. The definition of the nanoparticles differs depending upon the materials and fields of application. In a narrow sense, 16

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

they are regarded as particles smaller than 10-20 nm but, generally speaking, all particles in the 1 nm−1µm range may still be defined as nanoparticles [67]. The principal parameters describing nanoparticles are their shapes, sizes, and their morphologies. By controlling the shape during synthesis, several nanoparticles with different geometries including spherical, octahedral, decahedra and triangular plates can be produced. Figure 7 displays the relaxed geometries of gold, silver, and platinum nanoparticles with icosahedral (ICO), Marks decahedral (DEC), and FCC truncated octahedral (TOC) morphologies [68]. Nanoparticles are present in the nature, and have been also created for thousands of years as by-products of combustion and cooking, and more recently from industrial and vehicle exhausts. Nanoparticles represent intermediate structures between bulk materials and atomic, ionic, and molecular structures. Bulk materials have constant physical properties regardless of their sizes, whereas size-dependent properties are often associated with nanoscopic structures. Methods for synthesizing nanoparticles are typically grouped into two categories: top-down and bottom-up approaches. The first approach involves the decomposition of the bulk material into nanosized structures or particles. This approach may address milling or attrition and volatilization of a solid followed by condensation of the volatilized components. The alternative approach, the bottom-up, refers to the buildup of a material atom-by-atom, molecule-by-molecule, or cluster-by-cluster in a gas phase or in solution via chemical methods. 2.3. Nanowires Nanowires are one-dimensional anisotropic nanoscopic structures with a small diameter, and a large surface to-volume ratio. Nanowires have two quantum-confined directions and one unconfined direction available for electrical conduction [69]. Thus, their physical properties are different from those of structures with different scale and dimensionality. These unique properties make these structures ideal candidates for future applications in the fields of electronics, optics, magnetic medium, thermoelectronic, sensor devices, and nanoelectromechanical systems (NEMS). The potential of nanowires in future applications has led to a significant interest in the experimental and theoretical characterization of their size-dependent properties.

17

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Different types of nanowires including metallic, semi-conducting and insulator can be synthesized. Metallic nanowires can be grown from Cu, Ag, Pt, Ni and Au. Semiconductor nanowires are emerging as a powerful class of materials that, through controlled growth and organization, are opening up novel opportunities for nanoscale electronic and photonic devices. These can be grown from silicon, or titanium dioxide. In addition, nanowires could have circular, semicircular, triangular, rectangular, pentagonal, hexagonal or other unconventional cross-sectional shapes [70].

18

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

3. Essential concepts from nonlocal continuum elasticity theory 3.1. A brief review of classical (local) continuum elasticity theory 3.1.1. Concept of strain We begin our development of the elasticity theory with the concept of strain. Strain is defined as the ratio of the change in length to the original length, so it is a dimensionless quantity. Let us consider a material point P in a body which is located originally at the coordinates X = (X1 , X2 , X3 )T . As a result of application of external loads, the material is deformed or strained, and the point P is displaced to the point P´ with coordinates x = (x1 , x2 , x3 )T . Therefore, the displacement vector is defined as u = x − X = ui ei

(4)

where u1 , u2 and u3 are the components of the displacement vector, and e1 , e2 , e3 are the unit vectors in the directions of coordinate axes. The gradient of the displacement vector is a tensor and may be written as [71] ∇u = ui,j

(5)

The displacement gradient tensor is commonly decomposed into a symmetric and a skew symmetric part as ui,j = εij + ωij

(6)

where   1 ∂ui ∂uj + 2 ∂xj ∂xi   1 ∂ui ∂uj ωij = − 2 ∂xj ∂xi εij =

(7)

in which the symmetric tensor, εij , is called the infinitesimal strain tensor, and the skew symmetric tensor, ωij , is called the rotation tensor. The diagonal elements of the strain tensor are referred to as the normal strains, whereas the off-diagonal elements are called the shear strains.

19

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

3.1.2. Concept of stress The concept of the stress was introduced into the theory of elasticity by Cauchy, and it is defined as force divided by the area on which it operates. The stress at a point A in a body is, therefore, defined as σ = lim

∆A→0

∆P ∆A

(8)

in which ∆P is the internal force acting on a small area ∆A in the body. In general, we note that each point A could be thought of as an infinitesimal cube that is reduced in size in a limiting process (Fig. 8). There are nine stress components at each point A   σ11 σ12 σ13 [σij ] =  σ21 σ22 σ23  (9) σ31 σ32 σ33 The diagonal components of the stress tensor, σij , are referred to as the normal stresses and the off-diagonal components are called the shear stresses.

3.1.3. Stress-strain relation for elastic materials The stress and strain tensors, defined above, are related to each other and this relationship involves the material 's properties. Mathematical expressions relating these quantities together are referred to as constitutive equations. The constitutive equations serve to describe the material properties of the medium when it is subjected to external loads. For small deformations and linear elastic materials, the stress tensor is linearly related to the strain tensor by the fourth-rank elasticity tensor C, and this relation is given by the general form of Hooke's law σij = Cijkl εkl

(10)

The 81 coefficients Cijkl are referred to as the elastic constants of the material. Based on the symmetry of the stress and strain tensors, it is clear that Cijkl = Cjikl = Cijlk (11) whereby the 81 coefficients are reduced to 36 independent coefficients. This makes it possible to express Eq. (10) in a contracted notation, or the Voigt matrix notation. This matrix can be obtained by mapping the pair of indices 20

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

ij to a single index χ. The general rules to contract the indices are (11)↔(1), (22)↔(2), (33)↔(3), (23)↔(4), (13)↔(5) and (12)↔(6). It should be noted that the symmetry imposed by the strain energy implies that the 6×6 C matrix is symmetric; that is, Cij =Cji , and thus only 21 independent elastic constants remain. Therefore, the stress-strain relation for anisotropic elastic materials, in the matrix form, has the structure      σ1 C11 C12 C13 C14 C15 C16 ε1  σ2   C12 C22 C23 C24 C25 C26   ε2        σ3   C13 C23 C33 C34 C35 C36   ε3   =   (12)  σ4   C14 C24 C34 C44 C45 C46   ε4        σ5   C15 C25 C35 C45 C55 C56   ε5  σ6 C16 C26 C36 C46 C56 C66 ε6

The number of independent elastic constants is further reduced when the crystal structure of the material has symmetries [72]. A material with three mutually perpendicular planes of symmetry is called orthotropic. Thus, the elasticity matrix for the orthotropic case reduces to only nine independent coefficients given by   C11 C12 C13 0 0 0  C12 C22 C23 0 0 0     C13 C23 C33 0 0 0    Cij =  (13) 0 0 C44 0 0   0   0 0 0 0 C55 0  0 0 0 0 0 C66

In case of complete symmetry, the material is referred to as isotropic wherein its mechanical properties are the same in any direction. As a result, the stress-strain relation for isotropic elastic materials is given by σij =

νE E εij + εkk δij 1+ν (1 + ν)(1 − 2ν)

(14)

where E is the Young modulus and ν is the Poisson ratio. 3.1.4. Governing equations Consider a closed sub-domain with volume Ω and exterior surface S within an elastic body. The elastic body is acted upon by a general distribution of surface tractions tn and a body force B per unit volume. The 21

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

resulting force acting on the sub-domain is, in index notation, given by Z Z Z Z σji nj dS + Bi dΩ (15) Fi = tni dS + Bi dΩ = Ω

S

S

Ω

where the components nj are the direction cosines of the unit vector n perpendicular to the surface. The surface integral of the traction forces can be transformed to the volume integral by Green-Gauss theorem Z Z σji nj dS = σji,j dΩ (16) S

Ω

Using Eq. (16), Eq. (15) is rewritten as Z Fi = (σji,j + Bi )dΩ

(17)

Ω

Next, the linear momentum is defined by Z Gi = ρu˙ i dΩ

(18)

Ω

where ρ is the mass density of the material and u˙ i is the component of the material velocity. Newton's law of motion requires that Fi = G˙ i

(19)

Substituting Eqs. (17) and (18) into Eq. (19), we obtain Z Z (σji,j + Bi )dΩ = ρ¨ ui dΩ Ω

(20)

Ω

Since the volume Ω is arbitrary, Eq. (20) reduces to Cauchy's first law of motion [73]: σji,j + Bi = ρ¨ ui (21) The stress distribution can be obtained by solving Eq. (21) with appropriate boundary and initial conditions.

22

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

3.2. Size-dependent continuum theories Experimental observations have confirmed that some nanoscopic phenomena cannot be predicted by classical continuum theories. The inability of the classical continuum theories to deal with this issue arises from the absence of a material length scale in constitutive equations [74]. Consequently, modified continuum theories incorporating the length scale are called for. Such classical models can be generalized to accommodate the underlying microscopic, or nanoscopic, structure by expanding the constitutive relations leading to what is referred to as higher order continua [75]. Higher order gradient theories are generalizations of the classical continuum theories which include the material length scales and higher order terms to account for microstructural and size-effects. In these theories, the standard constitutive equations, Eq. (10), are generalized by incorporating the higher order derivatives of strains, stresses and/or accelerations. Gradient-dependent constitutive equations can be used to model deformation processes in very small volumes to capture the size-effects [76, 77]. Consequently, the higher order gradient theories are physically motivated by the underlying mechanisms and related microstructural observations [78]. A systematic way to incorporate the strain and stress gradients in the constitutive response of materials is to utilize implicit constitutive equations. The idea of the implicit constitutive equations was first introduced by Morgan [79]. In the case of higher order gradient theories, it is convenient to employ the implicit constitutive equation of the form [80] f (σ, ε, ∇2 σ, ∇2 ε) = 0

(22)

where f is a general linear function of the stress and strain tensors and their Laplacians. In the special case of a linear isotropic tensor function, Eq. (22) is rewritten as (a1 ε + a2 σ)I + a3 ε + a4 σ + ∇2 [(a5 ε + a6 σ)I + a7 ε + a8 σ] = 0

(23)

in which the coefficients ai are constitutive coefficients and I is the identity matrix. It should be noted that various size-dependent continuum theories, such as the strain gradient theory, the stress gradient theory and the dynamically consistent gradient elasticity can be obtained by an appropriate selection of the constitutive coefficients. The following gradient models may be obtained on the basis of the implicit constitutive equation (23) σ = C : (1 − l12 ∇2 )ε 23

(24)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

(1 − l22 ∇2 )σ = C : ε

(25)

(1 − l32 ∇2 )σ = C : (1 − l42 ∇2 )ε

(26)

wherein ‘:’ denotes the double-dot product, and l1 , l2 , l3 and l4 are the material length scale parameters. The constitutive equation (24) is the simplified strain gradient theory which has been popularized since 1990s through the work of Aifantis [81]. Equation (25) is the standard constitutive equation in the stress gradient (nonlocal) theory which was derived by Eringen [43]. The model presented by Eq. (26) is the constitutive equation of the dynamically consistent gradient elasticity [77]. 3.3. Nonlocality and nonlocal averaging The domain of application of the classical continuum theories is closely coupled to the length and time scales. If Le denotes the external characteristic length (e.g., the crack length, the wavelength, the size of the sample) and Li presents the internal characteristic length (e.g., the lattice parameter, the size of a grain), for Le /Li >> 1 , the classical continuum theories can predict the results with sufficient accuracy. In the nanoscale, we usually have Le /Li ∼ 1 and so the local theories fail. Thus, we must resort to either atomistic or nonlocal theories which can account for the long-range interatomic interactions. Similarly, for the dynamical problems, there will be a scale Te /Ti , where Te is the external characteristic time (e.g., the time of application of the external loads, the period of vibration) and Ti is the internal characteristic time (e.g., the relaxation time, the time for a signal to travel between molecules). Again, classical theories fail when Te /Ti ∼ 1. Thus, the description of the physical phenomenon in space-time requires nonlocality and memory effects scaled by Le /Li and Te /Ti . There are three main types of nonlocality; spatial, temporal (memory effect) and space-time. To present the types of nonlocality, we follow the work of Povstenko [82]. A physical quantity (the effect r) at a point x and time t is locally dependent on another physical quantity (the cause p) at the same point and time via the following general form r(x1 , x2 , x3 , t) = r(p(x1 , x2 , x3 , t))

(27)

Spatial nonlocality implies that the effect R at the point x and time t is ´ at the same time. Thus, the a functional of the causes at all other points x 24

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

weighted spatial average, or nonlocal average, of a local field r(x1 , x2 , x3 , t) within the domain V is defined by Z (28) R(x1 , x2 , x3 , t) = α(|x − x0 | , τ )r(p(x01 , x02 , x03 , t))dx01 dx02 dx03 V

where α is a weight function or a scalar kernel function whose arguments are the Euclidean distance |x − x0 | and the parameter τ is proportional to a characteristic length ratio Li /Le . It should be noted that the volume V is smaller than the total volume of the body. For the case of the stress-strain constitutive relation, the cause is the strain ε, and the effect is the stress σ. According to Eq. (28), the corresponding nonlocal constitutive equation takes on the following form: Z σijnl (x1 , x2 , x3 , t) = α(|x − x0 | , τ )Cijkl εkj (x01 , x02 , x03 , t)dx01 dx02 dx03 (29) V

In the case of materials with memory, the effect R at the material point x at time t depends on the history of causes at the same x at all times prior to time t: R(x1 , x2 , x3 , t) =

Zt

β(t − t0 , η)r(p(x1 , x2 , x3 , t0 ))dt0

(30)

0

wherein β is a kernel of the integral and the parameter η is proportional to the characteristic time ratio Ti /Te . Using Eq. (30), the temporal nonlocal constitutive equation takes on the following form: σijnl (x1 , x2 , x3 , t)

=

Zt

β(t − t0 , η)Cijkl εkl (x1 , x2 , x3 , t0 )dt0

(31)

0

It is natural to expect that nonlocality in space may be accompanied with memory effects as well. Such space-time nonlocality may be expressed in the general form R(x1 , x2 , x3 , t) =

Rt 0

R

V

$(|x − x0 | , t − t0 , τ, η)r(p(x0 1 , x0 2 , x0 3 , t0 ))dx0 1 dx0 2 dx0 3 dt0 25

(32)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

in which $ is the nonlocal kernel. The nonlocal constitutive equation for a material with memory effect and space nonlocality is given by σijnl (x1 , x2 , x3 , t) =

Rt R 0

V

$(|x − x0 | , t − t0 , τ, η)Cijkl εij (x0 1 , x0 2 , x0 3 , t0 )dx0 1 dx0 2 dx0 3 dt0 (33)

Here we see that the stress at a reference point x, at time t, is a functional of strains at all other points for all past and present times. When τ → 0 the nonlocality disappears and the nonlocal theory reduces to the classical one, and when η → 0 the memory is eliminated. 3.4. Integral-type nonlocal elasticity theory With the discovery of the lattice structure of crystals, it became known that interatomic forces acting on an atom depend on the relative changes of distances between this atom and neighboring atoms. This is equivalent to the concept of nonlocal elasticity theory which states that the stress at a point x is a functional of the strains at all other points in the body. According to the theory of nonlocal elasticity theory, the basic equations for linear homogeneous elastic solids are expressed as tij,j + Bi = ρ¨ ui

tij (x1 , x2 , x3 ) =

Z

V

α(|x − x0 | , τ )Cijkl εkj (x01 , x02 , x03 )dx01 dx02 dx03

(34)

(35)

  1 ∂ui ∂uj + (36) 2 ∂xj ∂xx where tij is the nonlocal stress tensor. Any nonlocal generalization must consider some basic requirements in order to provide physically rational formulations. The nonlocal kernel α(|x − x0 | , τ ) has the following properties: (i) α(|x − x0 | , τ ) is a positive continuous function of |x − x0 |. (ii) The weight function is usually adjusted such that the nonlocal field corresponding to a constant local field remains constant even in the vicinity of the boundary. In order to guarantee this requirement, the weight function must satisfy the normalization condition Z ∀x ∈ V (37) α(|x − x0 | , τ )dx01 dx02 dx03 = 1 εij =

V

26

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

(iii) From the nonlocal averaging integral (35), it is clear that the kernel function has the dimension of (length)-3 . As a result, it must always contain at least one length-scale parameter with the dimension of length. The length-scale parameter, τ (= e0 Li /Le ), relates to the microscopic structure of the material, and it is used to transmit information from the surrounding material to a given material point. e0 is a non-dimensional nonlocal scaling parameter which has been assumed to be a constant appropriate to each material. The nonlocal parameter will be discussed in later sections. (iv) The kernel function must reach a maximum at x = x0 and must tend rapidly to zero at large distances. (v) When τ → 0, α must revert to the Dirac delta function, δ, i.e. lim α(|x − x0 | , τ ) = δ(|x − x0 |)

(38)

τ →0

and the nonlocal elasticity reduces to the classical elasticity (Fig. 9). (vi) When τ → 1, the nonlocal theory should approximate to the atomistic lattice dynamics [43]. The nonlocal kernel function α can be determined by matching the dispersion curves of plane waves with those of atomistic lattice dynamics (or experiments). Several appropriate functions have been reported [43] for one-, two- and three-dimensional problems. For one-dimensional kernel: Type I   ( |x−x0 | 1 1 − |x − x0 | < τ Le 0 τ Le τ Le (39) α(|x − x | , τ ) = 0 |x − x0 | > τ Le Type II α(|x − x0 | , τ ) = Type III 0

α(|x − x | , τ ) =

  1 |x − x0 | exp − 2τ Le τ Le

Le

1 √

|x − x0 |2 exp − L2e τ πτ

For two-dimensional kernel: Type I α(|x − x0 | , τ ) =

1 K0 2πL2e τ 2 27



|x − x0 | Le τ



!

(40)

(41)

(42)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

where K0 is the modified Bessel's function of the second kind. Type II ! 0 2 |x − x | 1 exp − α(|x − x0 | , τ ) = πL2e τ L2e τ For three-dimensional Kernel: Type I 0

α(|x − x | , τ ) = Type II α(|x − x0 | , τ ) =

1 (πL2e τ )3/2

|x − x0 |2 exp − L2e τ

!

  |x − x0 | 1 0 |x − x | exp − 4πL2e τ 2 Le τ

(43)

(44)

(45)

3.5. Laplacian-based nonlocal elasticity theory For homogeneous bodies, the linear theory leads to a set of integro-partial differential equations for the displacement field, which are generally difficult to solve. Therefore, the treatment of this problem by means of differential equations is promising. Eringen [43] developed an equivalent nonlocal elasticity theory in which the spatial integrals are replaced by differential operators. This model is called Laplacian-based nonlocal elasticity theory. To obtain the equivalent differential equations, it is assumed that α is Green's function of a linear differential operator Lα(|x − x0 | , τ ) = δ(x − x0 )

(46)

If such an operator can be found, then by applying L to Eq. (35), the following differential equation is obtained. Ltij = Cijkl εkl

(47)

Similarly, by applying L to the integro-partial differential equation (34), it is correspondingly reduced to the partial differential equation σij,j + L(fi − ρ¨ ui ) = 0

(48)

In this case, we need to solve partial differential equations, instead of integro-partial differential equations. By matching the dispersion curves with

28

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

lattice models, Eringen [83] proposed a nonlocal model with the linear differential operator L defined by L = 1 − (e0 Li )2 ∇2

(49)

where ∇2 is the Laplace operator. Therefore, the nonlocal constitutive relation may be simplified to t − (e0 Li )2 ∇2 t = C : ε

(50)

Note that ∇2 t denotes the Laplacian of the stress tensor. In Cartesian coordinates, the Laplacian of the stress tensor can be obtained by directly applying the Laplace operator to each component of the tensor [84]. However, the Laplacian of a tensor in an orthogonal curvilinear coordinate system does not simply pass through and operate on each of the individual tensor components as in the Cartesian case. In other words, the component of the Laplacian of the stress tensor, (∇2 t)ij , is not equal to the Laplacian of the stress tenser component, ∇2 (tij ). An expression of the Laplacian of the stress tensor, in the orthogonal curvilinear coordinates, is presented in the next section. 3.5.1. Laplacian of the stress tensor Here, we provide a brief account of the derivation of the Laplacian of the stress tensor in the curvilinear coordinates. Consider the general case in which three orthogonal curvilinear coordinates are denoted by ξ1 , ξ2 , ξ3 and let ek (k =1, 2, 3) be the corresponding curvilinear base vectors (Fig. 10). In the curvilinear system, we can write dt = ∇tdr ≡ µdr

(51)

where ∇t ≡ µ is the gradient of the stress tensor which is a third-order tensor. In addition, the vector dr is defined by dr =

3 X

hk dξk ek

(52)

k=1

wherein h1 , h2 , h3 are called the scale factors which are nonnegative functions of position. Substituting Eq. (52) into Eq. (51), the following relation is obtained 3 X dt = µdr = [hk dξk (µek )] (53) k=1

29

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

It should be noted that µek is a second-order tensor given by µek = µmnk em en

(54)

By using Eq. (54), relation (53) gives dt = µdr =

3 X

[hk µmnk dξk em en ]

(55)

k=1

For such a system, the stress tensor is given by t = tij ei ej

(56)

Then, the differential of the stress tensor is obtained as dt = dtij ei ej + tij dei ej + tij ei dej where

(57)

∂tij dξl , dei = Γipq dξp eq (58) ∂ξl are the Christoffel symbols. Substituting Eq. (58) into Eq. dtij =

in which Γipq (57), we have

dt =

X ∂tij X tqj Γqmi + tiq Γqmj + ∂ξm q q

!

dξm ei ej

(59)

Comparing Eq. (59) with Eq. (55), the gradient of the stress tensor is obtained as ! X 1 ∂tij X µijm = tiq Γqmj (60) tqj Γqmi + + hm ∂ξm q q Similarly, after very lengthy derivations, the gradient of the third-order tensor µ, which is a fourth-order tensor can be determined as P

ijkl

1 hl



∂µijk ∂ξl

+

P m

∇∇t = ∇µ =  P P µmjk Γmli + µimk Γmlj + µijm Γmlk ei ej ek el (61) m

m

Finally, the Laplacian of the stress tensor, which is a second-order tensor, can be obtained as ! X 1 ∂µijk X X X 2 ∇ t= + µmjk Γmki + µimk Γmkj + µijm Γmkk ei ej hk ∂ξk m m m ijk

(62)

30

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

3.5.2. Nonlocal constitutive relations In this section, we will give the nonlocal constitutive relations in the Cartesian, cylindrical and spherical coordinate systems. These are useful for modeling nanoscopic structures with different morphologies. The nonlocal stress-strain constitutive relations for an orthotropic material (e.g. a graphene sheet) in Cartesian coordinates are expressed as

where

txx − (e0 Li )2 (∇2 txx )
tyy − (e0 Li )2 ∇2 tyy
tzz − (e0 Li )2 ∇2 tzz
tyz − (e0 Li )2 ∇2 tyz
txz − (e0 Li )2 ∇2 txz
txy − (e0 Li )2 ∇2 txy ∇2 (∗) =

= = = = = =

C11 εxx + C12 εyy + C13 εzz C12 εxx + C22 εyy + C23 εzz C13 εxx + C23 εyy + C33 εzz C44 εyz C55 εxz C66 εxy

∂ 2 (∗) ∂ 2 (∗) ∂ 2 (∗) + + ∂x2 ∂y 2 ∂z 2

(63)

(64)

Carbon nanotubes may be modeled as nonlocal cylindrical shells. Thus, the nonlocal stress-strain constitutive relations for an anisotropic material in cylindrical coordinates (r, θ, z) are given by   2 4 ∂trθ 2 − 2 (trr − tθθ ) = trr − (e0 Li ) ∇c trr − 2 r ∂θ r C11 εrr + C12 εθθ + C13 εzz + C14 εθz + C15 εrz + C16 εrθ 2

  2 4 ∂trθ tθθ − (e0 Li )2 ∇2c tθθ + 2 + 2 (trr− tθθ ) = r ∂θ r C12 εrr + C22 εθθ + C23 εzz + C24 εθz + C25 εrz + C26 εrθ
tzz − (e0 Li )2 ∇2c tzz = C13 εrr + C23 εθθ + C33 εzz + C34 εθz + C35 εrz + C36 εrθ   2 ∂tzr 1 tθz − (e0 Li )2 ∇2c tθz + 2 − 2 tθz = r ∂θ r C14 εrr + C24 εθθ + C34 εzz + C44 εθz + C45 εrz + C46 εrθ 31

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

 1 2 ∂tzθ + 2 tzr = tzr − (e0 Li ) − 2 r ∂θ r C15 εrr + C25 εθθ + C35 εzz + C45 εθz + C55 εrz + C56 εrθ 2



∇2c tzr

    2 ∂trr ∂tθθ 4 − − 2 trθ = trθ − (e0 Li )2 ∇2c trθ + 2 r ∂θ ∂θ r C16 εrr + C26 εθθ + C36 εzz + C46 εθz + C46 εrz + C66 εrθ where

(65)

1 ∂ 2 (∗) ∂ 2 (∗) ∂ 2 (∗) 1 ∂(∗) + + + (66) ∂r2 r ∂r r2 ∂θ2 ∂z 2 As expected, in cylindrical coordinates, the component of the Laplacian of stress tensor, (∇2 t)ij , is not equal to the Laplacian of the stress tenser component, ∇2 (tij ). Several nanoscopic structures including spherical fullerenes and nanoparticles can be modeled as spherical bodies. Thus, the nonlocal stress-strain constitutive relations for an orthotropic material in spherical coordinates (R, θ, φ), given by the following relations, are required ∇2c (∗) =

tRR − (e0 Li )2 [∇2s tRR − −

4 ∂tφR 4 ∂tRθ − 2 R2 ∂θ R sin θ ∂φ

4 cot θ 2 tRθ − 2 (2tRR − tθθ − tφφ )] = C11 εRR + C12 εθθ + C13 εφφ 2 R R

4 cot θ ∂tθφ 2 4 ∂tRθ − 2 + 2 (tRR − tθθ ) 2 R ∂θ R sin θ ∂φ R 2cot2 θ 2tθφ + 2 (tθφ − tθθ sin θ + tφφ sin θ) + 2 ] R sin θ R sin θ = C12 εRR + C22 εθθ + C23 εφφ tθθ − (e0 Li )2 [∇2s tθθ +

4 cot θ ∂tθφ 4 ∂tφR tRR − tφφ + 2 +2 2 R sin θ ∂φ R sin θ ∂φ R2 2 4 cot θ 2tθφ cot θ + tRθ − + 2 (2tθθ sin θ − 2tφφ sin θ − 2tθφ )] 2 R R sin θ R sin θ = C13 εRR + C23 εθθ + C33 εφφ tφφ − (e0 Li )2 [∇2s tφφ +

32

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

2 ∂tRθ 2 ∂tφR + 2 tθφ − (e0 Li )2 [∇2s tθφ + 2 R ∂θ R sin θ ∂φ     2 cot θ ∂tθθ ∂tφφ 1 tθθ − tφφ + 2 − − 2 + 2tθφ + 2tφR cot θ R sin θ ∂φ ∂φ R sin θ cot2 θ − 2 (tθθ − tφφ + 4tθφ sin θ)] = C44 εθφ R sin θ   ∂tRR ∂tφφ 2 − tφR − (e0 Li )2 [∇2s tφR + 2 R sin θ ∂φ ∂φ 2 ∂tθφ 2 cot θ ∂tRθ 5tφR 4 cot θ tRθ − 2 + 2 − 2 − tθφ − 2 2 R ∂θ R sin θ ∂φ R R R sin θ cot2 θ (tRθ + tφR sin θ)] = C55 εφR − 2 R sin θ   2 ∂tRR ∂tθθ 2 ∂tθφ tRθ − (e0 Li )2 [∇2s tRθ + 2 − − 2 R ∂θ ∂θ R sin θ ∂φ 2 cot θ ∂tφR 5tRθ 2 cot θ tφR − 2 − 2 − (tθθ − tφφ ) + 2 2 R sin θ ∂φ R R R sin θ cot2 θ (tφR − tRθ sin θ)] = C66 εRθ (67) + 2 R sin θ in which ∇2s (∗) =

∂ 2 (∗) ∂ 2 (∗) 2 ∂(∗) 1 ∂ 2 (∗) cot θ ∂(∗) 1 + + 2 + 2 + 2 2 2 2 ∂R R ∂R R ∂θ R ∂θ R sin θ ∂φ2

33

(68)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

4. Computational modeling of the mechanical behavior of nanoscopic structures A relatively new area of the mechanics is the study of mechanical properties and mechanical behavior of nanoscale systems and nanoscopic structures. This area is referred to as nanomechanics. It covers a wide range of static (e.g. bending and buckling) and dynamic (e.g. vibration and wave propagation) problems. In this section, a comprehensive survey of theories, appropriate for modeling the mechanical behavior of nanoscopic structures, based on the nonlocal elasticity theory is presented. In the following, after a brief introduction to some important mechanical behaviors, such as bending, buckling, vibration and wave propagation in general, we will focus on the nonlocal modeling of different structures. 4.1. Mechanical behavior of materials: useful definitions and concepts In the present section, we will introduce some useful definitions and concepts pertinent to the mechanical behavior of the nanoscopic structures that will be studied in other sections. For a more scholarly (and comprehensive) coverage of the topics, the reader is referred to various textbooks (for example [85, 86, 87]). A physical phenomenon associated with a thin and/or slender structure deforming abruptly from its initial configuration due to an applied compression is referred to as buckling [88]. It is due to mechanical instability and is usually independent of material strength. For example, when a rod is subjected to an axial compressive force, it first shortens slightly, but at a critical load the rod bows out, and it is said that the rod has buckled. The load at which buckling occurs depends on the stiffness of a component, not upon the strength of its materials. This behavior can be mostly seen in different members, such as columns, beams, rods, arches, rings, plates and shells, under diverse loading conditions. Experimental evidence reveal the occurrence of buckling in several nanoscopic structures including nanotubes, graphene sheets and nanowires [89, 90, 91, 92, 93, 94, 95, 96]. The buckling of nanoscopic structures has special characteristics relevant to engineering and scientific applications. Because of the experimental difficulty in measuring the critical buckling load, several theoretical techniques have been developed. Another important mechanical behavior is vibration. In general terminology, the term vibration is used to describe an oscillatory motion of a body or a mechanical system about its equilibrium position. The vibration of a system 34

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

involves the transfer of its potential energy to kinetic energy, and vice versa, alternately. Vibration analysis usually involves the prediction of frequencies and the corresponding mode shapes of the system, with frequency being the number of energy transformation cycles per second. As an example, consider the vibration of a thin rod. It can manifest itself in three modes; longitudinal, torsional and lateral [97]. The longitudinal or extensional mode is defined by the periodic extension and contraction of elements of the central-line. The lateral or flexural mode is characterized by the periodic bending and straightening of portions of the central-line. The torsional mode refers to angular displacement of the cross-sections of the rod around its longitudinal axis. Vibrations play an important role in nature and in engineering applications. At the nanoscale, vibrations are inherent in most physical systems due to thermal motion of the atoms. Therefore, vibrational behavior of nanoscopic structures is of importance in many nanodevice applications, such as high frequency oscillators and sensors. In addition, the elastic properties of very small objects, like nanoscopic structures, can also be investigated in a fundamental manner by means of studying their vibrational characteristics. The first experimental studies of vibration of such structures were performed in spinel crystallites using Raman spectroscopy [98], and were further extended to other nanoscopic structures, such as spherical metal nanoparticles [99], semiconductor quantum dots [100] and carbon nanotubes [101]. It is also useful to define the concept of wave propagation. Any localized disturbance in a physical medium is transmitted to other points of the medium through the phenomenon of wave propagation. Although the propagation of disturbance in a solid takes place at a microscopic level, through the interaction of atoms of the solid, the physics of wave propagation is formulated in terms of such properties as the density and elastic constants of the solid body. The emergence and propagation of elastic waves in deformable bodies play a key role in many practical applications [86]. For example, mechanical properties such as Young's modulus can be determined by measuring the behavior of waves through the medium. Among the many nondestructive techniques, high-frequency acoustic wave technique has been regarded as a very efficient method to characterize elastic media composed of microscopic or nanoscopic structures [102]. Wave propagation studies mainly include the estimation of the wave number and the wave speed. Structures such as CNTs and graphene sheets and nanowires can act as media for propagation of waves of the order of terahertz (THz). Propagation of THz waves in nanoscopic structures has opened a new perspective for scientific and applied research. 35

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

4.2. Nonlocal beam models It is well-known that some nanoscopic structures such as CNTs and nanowires are slender tubular-like structures with high aspect ratios (lengthto-diameter ratio). Thus, they can behave like an elastic beam during motion and deformation induced by external loads (Fig. 11). Beams are structural members which offer resistance to bending prompted by applied loads. Various available beam theories, including the Euler-Bernoulli, the Timoshenko, the Reddy, and the Levinson beam theories, are often employed in modeling the mechanical response of nanotubes and nanowires. In this review, we only introduce the Euler-Bernoulli and the Timoshenko beam theories. To describe the beam theories, it is assumed that the x-coordinate is taken along the length of the beam and the z-coordinate along the thickness of the beam. In a beam theory, displacements u1 and u3 along the x- and z-directions are only functions of the x and z coordinates and time t. The following stress resultants are defined for use in the following sections: Z Z N= txx dA, M= ztxx dA (69) A

A

where N and M are the resultant inplane force and bending moment respectively. In addition, A is the cross sectional area of the beam. A detailed discussion of nonlocal beam theories for bending, buckling and vibration has been developed in Refs. [103, 104], from which we follow. 4.2.1. The Euler-Bernoulli beam theory Among the continuum-based beam models, the most extensively used is the Euler-Bernoulli beam theory, due to the simplicity of its mathematical structure. In this beam theory, the rotary inertia and the shear deformation of the beam are neglected. Moreover, it is assumed that a cross section which is perpendicular to the undeformed axis of the beam remains perpendicular to the deformed beam axis during the bending. The Euler-Bernoulli beam is described by the following displacement field: u1 = u(x, t) − z

∂wEB , ∂x

u3 = wEB (x, t)

(70)

where u(x, t) and wEB (x, t) are respectively the axial and the transverse displacements of the middle plane (i.e., z = 0) of the beam and the superscript

36

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

‘EB’ denotes the quantities in the Euler-Bernoulli beam theory. According to Eq. (7), the nonzero strain at an arbitrary point of the beam is 0 EB εEB xx = εxx + zκ

(71)

where ε0xx is the extensional strain of the middle plane and κEB is the corresponding bending strain. These are defined as ε0xx =

∂u ∂x

(72)

∂ 2 wEB ∂x2 The virtual strain energy in the Euler-Bernoulli beam theory is κEB = −

δU

EB

=

ZLb

(M EB δκEB + N EB δε0xx − P EB

∂wEB ∂δwEB )dx ∂x ∂x

(73)

(74)

0

where P EB is the applied axial compressive force. The kinetic energy of the beam is  2  EB 2 ! ZLb 1 ∂u ∂w EB T = m1 dx (75) + 2 ∂t ∂t 0

where m1 is the mass per unit length of the beam. In addition, the virtual work due to external loads is δW

EB

=

ZLb

(qδwEB + pδu)dx

(76)

0

in which p and q are the axial and transverse distributed forces (measured per unit length). The equations of motion and the boundary conditions can be derived via Hamilton's variational principle [86] that might be formulated as Zt2
δT EB − δU EB + δW EB dt = 0 (77) t1

37

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Applying the usual variational techniques to (77), the equations of motion are obtained as: ∂N EB ∂ 2u + p = m1 2 (78) ∂x ∂t   EB ∂ 2 wEB ∂ 2 M EB ∂ EB ∂w P = m + q − (79) 1 ∂x2 ∂x ∂x ∂t2 The boundary conditions involve specifying one member of each of the following three pairs at x = 0 and x = Lb : u wEB

EB

or

∂M ∂x

− P EB ∂wEB ∂x

or EB

N EB

∂w ∂x

≡ V EB

or

M EB

(80)

Here V EB denotes the equivalent shear force. Using Eq. (63), the nonlocal stress-strain relation for the one-dimensional problem can be written in the following form ∂ 2 txx txx − (e0 Li )2 = Eεxx (81) ∂x2 Therefore, the nonlocal constitutive relationships can be described as N EB − (e0 Li )2

∂ 2 N EB = EAε0xx ∂x2

(82)

∂ 2 M EB = EIκEB (83) ∂x2 where I is the cross-sectional moment of inertia. Substituting Eqs. (82)-(83) into Eqs. (78)-(79), the equations of motion in terms of axial and transverse displacements of the middle plane of the beam are obtained as  2  4 2 ∂ ∂u ∂ u 2 ∂ u 2∂ p (EA ) + p − (e0 Li ) = m1 − (e0 Li ) (84) ∂x ∂x ∂x2 ∂t2 ∂x2 ∂t2 M EB − (e0 Li )2

    EB ∂ 2 wEB ∂ EB ∂w −EI +q− P ∂x2 ∂x ∂x     2 EB ∂ ∂ ∂w ∂ 2 wEB ∂ 2 wEB 2 EB +(e0 Li ) P − q + m = m (85) 1 1 ∂x2 ∂x ∂x ∂t2 ∂t2 ∂2 ∂x2

38

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

The equations of motion of the conventional Euler-Bernoulli beam theory are obtained from Eqs. (84) and (85) by setting e0 = 0. Equations (84) and (85) have been respectively employed to determine the longitudinal and transverse motions of the beams. So far, the analysis has been general without reference to the boundary conditions. For reasons of simplicity, only the transverse motion of the beam with simply supported boundary condition is considered here, namely wEB = M EB = 0 at both ends of the beam. The displacement wEB which satisfies this boundary condition can be written as: wEB (x, t) =

∞ X

Wi sin λi x exp(jωi t)

(86)

i=1

where Wi is the displacement amplitude, λi (λi = iπ/Lb , i is the half-axial wave-number) is the wave-number along the longitudinal direction, ωi √ is the angular frequency related to the frequency f by ωi = 2πfi and j = −1. The distributed load q can also be written as ∞ X

q(x) =

Qi sin

i=1

where Qi =

2 Lb

Z

iπx Lb

Lb

q(x) sin

0

(87)

iπx dx Lb

(88)

Substitution of the expansions for wEB and q from Eqs. (86) and (87) into the equation of motion (85), leads to   ((1 + (e0 Li )2 λ2i ) P EB λ2i + m1 ωi2 − EIλ4i )Wi + (1 + (e0 Li )2 λ2i )Qi = 0 (89)

For the bending problem, by setting P EB and ωi to zero in Eq. (89), the static deflection is obtained wEB (x) =

∞ X (1 + (e0 Li )2 λ2 )Qi i

i=1

λ4i EI

sin λi x

(90)

For the buckling problem, the distributed force q and ωi are set to zero and i = 1. Thus, we obtain PcrEB =

1 π 2 EI 2 2 1 + (e0 Li ) λ1 L2b 39

(91)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

For the free vibration problem, it is assumed that the applied axial compressive force P EB and the distributed force q are equal to zero. The natural frequencies are given by ωi2 =

1 λ4i EI (1 + (e0 Li )2 λ2i )m1

(92)

To study the wave propagation in the beam, the wavenumbers are obtained by solving the following characteristic equation for λ EIλ4 − (1 + (e0 Li )2 λ2 )m1 ω 2 = 0 The wavenumbers are obtained as v q u u m (e L )2 ω 2 ± m ω 2 (4EI + m (e L )4 ω 2 ) t 1 0 i 1 1 0 i λ=± 2ELb

(93)

(94)

For a specific nonlocal scaling parameter, frequency, and other material parameters of the beam, two wavenumbers are purely real and the other two wavenumbers are purely imaginary. The real wavenumber gives rise to the propagating component while the imaginary wavenumber gives rise to the spatially damped mode. The interested reader can find additional details of the effect of boundary conditions in Refs. [105, 106]. 4.2.2. The Timoshenko beam theory For short nanotubes and nanowires, i.e., those with an aspect ratio between 10 and 50, which are projected to have extensive nanotechnology applications, the contributions of both the shear deformation and rotary inertia play a significant role and must therefore be included. It has been shown in different studies [107, 108, 109, 110] that the mechanical properties of the CNTs obtained using the Timoshenko beam theory reflect a truer representation of these properties. In the Timoshenko beam model, the plane sections of the beam also remain plane, as in the Euler-Bernoulli beam theory, but they no longer remain normal to the deformed beam axis, as a result of both the shear deformation and the rotary inertia. In the Timoshenko beam theory, the displacement fields are of the form u1 = u(x, t) + zϕ(x, t), 40

u3 = wT (x, t)

(95)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

where ϕ(x, t) denotes the rotation of the cross-section and the superscript ‘T ’ denotes the quantities in the Timoshenko beam theory. The nonzero strains in the Timoshenko beam theory are εTxx =

∂u ∂ϕ +z = ε0xx + zκT , ∂x ∂x

2εTxz =

∂wT +ϕ=γ ∂x

(96)

Here, γ is the transverse shear strain. The virtual strain energy in the Timoshenko beam theory is T

δU =

ZLb

(M T δκT + Qδγ + N T δε0xx − P T

∂wT ∂δwT )dx ∂x ∂x

(97)

0

where Q is the resultant shear force, which is defined as Z Q = σxz dA

(98)

A

The kinetic energy of the beam, which includes the rotary inertia, is 1 T = 2 T

ZL 0

m1



∂u ∂t

2

+ m1



∂wT ∂t

2

+ m2



∂ϕ ∂t

2 !

dx

(99)

in which m2 is the mass moment of inertia of the beam. Furthermore, the virtual work due to external loads is T

δW =

ZLb

(qδwT + pδu)dx

(100)

0

Using Hamilton's variational principle, the equations of motion are obtained as ∂N T ∂ 2u + p = m1 2 (101) ∂x ∂t   ∂Q ∂ ∂wT ∂ 2 wT +q− PT = m1 (102) ∂x ∂x ∂x ∂t2 ∂M T ∂ 2ϕ − Q = m2 2 ∂x ∂t 41

(103)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

The boundary conditions involve specifying one member of each of the following two pairs or NT T ∂w ≡VT Q − PT ∂x ϕT or MT u

wT

or

(104)

In the nonlocal Timoshenko beam theory, the nonlocal constitutive relations are given by ∂ 2N T N T − (e0 Li )2 = EAε0xx (105) ∂x2 Q − (e0 Li )2

∂ 2Q = Ks AGγ ∂x2

(106)

∂ 2M T = EIκT (107) ∂x2 where Ks is the shear correction coefficient [85] and G is the shear modulus of elasticity. Substituting Eqs. (105)-(107) into Eqs. (101)-(103), the equations of motion of the beam are obtained as  2  2 4 ∂u ∂ ∂ u 2 ∂ u 2∂ p (EA ) + p − (e0 Li ) = m1 − (e0 Li ) (108) ∂x ∂x ∂x2 ∂t2 ∂x2 ∂t2 M T − (e0 Li )2

∂ ∂wT ∂ ∂wT (GAKs (ϕ + )) + q − (P T ) ∂x ∂x ∂x ∂x ∂ 2 wT ∂ 4 wT ∂2 ∂ ∂wT +(e0 Li )2 2 [ (P T ) − q] = m1 ( 2 − (e0 Li )2 2 2 ) (109) ∂x ∂x ∂x ∂t ∂x ∂t ∂ ∂x



EI

∂ϕ ∂x



   2  4 ∂wT ∂ ϕ 2 ∂ ϕ − GAKs ϕ + = m2 − (e L ) (110) 0 i ∂x ∂t2 ∂x2 ∂t2

In a manner similar to the Euler-Bernoulli beam, we only consider the transverse motion of the beam with simply supported boundary conditions, namely: wT = M T = 0 at both ends of the beam. The displacement wT and the rotation ϕ which satisfy the boundary conditions can be written as: wT (x, t) =

∞ X

Wi sin λi x exp(jωi t)

i=1

42

(111)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

ϕ(x, t) =

∞ X

Φi cos λi x exp(jωi t)

(112)

i=1

By substituting the expansions for wT and ϕ from Eqs. (111) and (112) into the equations of motion (109) and (110), we obtain −GAKs λi (Φi + λi Wi ) + (1 + (e0 Li )2 λ2i )Qi +(1 + (e0 Li )2 λ2i )P T λ2i Wi + (1 + (e0 Li )2 λ2i )m1 ωi2 Wi = 0

−EIλ2i Φi − GAKs (Φi + λi Wi ) + (1 + (e0 Li )2 λ2i )m2 ωi2 Φi = 0

(113)

(114)

For a static bending, we obtain wT (x) =

∞  X

1+

i=1

ϕ(x) = −

λ2i EI GAKs



(1 + (e0 Li )2 λ2i )Qi sin λi x λ4i EI

∞ X (1 + (e0 Li )2 λ2 )Qi i

i=1

λ3i EI

cos λi x

(115)

(116)

It should be noted that the inclusion of both the transverse shear and the nonlocal constitutive equations increases the deflection. However, for the simply supported beam considered here, the inclusion of the transverse shear has no effect on the rotation ϕ. The Timoshenko critical buckling load is given by PcrT =

(1 +

GAKs (e0 Li )2 λ21 )(GAKs

π 2 EI + λ21 EI) L2b

(117)

The natural frequencies and wavenumbers of the nonlocal Timoshenko beam theory can be computed from the characteristic equation m1 m2 (1 + (e0 Li )2 λ2i )2 ωi4 GAKs     λ2 EI − m1 1 + i + m2 λ2i (1 + (e0 Li )2 λ2i )ωi2 + EIλ4i = 0 (118) GAKs

43

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

4.3. Nonlocal cylindrical shell models In addition to nonlocal beam models, CNTs with low aspect ratios have been modeled with elastic shell models since they exhibit many shell-like mechanical characteristics. Hence, nonlocal shell models have become indispensable especially when the length-to-radius ratio of CNTs is reduced. Several nonlocal shell models, based on the Donnell theory, the Fl¨ ugge theory, the Sanders theory and the Novozhilov theory, employing different sets of assumptions, levels of complexity and distinct category of approximations, have been developed in the recent years for predicting the mechanical behavior of CNTs [111, 112, 113, 114, 115]. Based on the thickness-to-radius ratio, the nonlocal shell theories are classified as nonlocal classical shell theory and nonlocal first order shear deformation theory. It is well-known that the classical shell theory is an adequate formalism for analyzing the cylindrical shells wherein the thickness-to-radius ratio is smaller than 0.05. In the classical shell theory, the transverse shear can be neglected. For the CNTs with high values of thickness-to-radius ratio, the shell model for the CNTs requires transition to the shear deformation shell theory. Thus, the first order shear deformation theory has been widely employed for the thick CNTs. In this section, we present the general nonlocal formulations for modeling the mechanical behavior of the thin and thick CNTs by using the Fl¨ ugge shell theory. The Fl¨ ugge theory is known as a highly reliable theory that can be used for most shapes regardless of the size of their cross-sectional radii. It should be noted that no model has been developed so far which is capable of accommodating the combined effects of the different design parameters. 4.3.1. Classical shell theory The atomistic structure of an SWCNT and its equivalent continuumbased model, and the corresponding appropriate coordinate system are shown in Fig. 12. The cylindrical shell is assumed to have the length LCN T , radius R and an effective wall thickness h. There have been differing suggestions concerning the choice of the wall thickness for an equivalent continuum shell model. For a detailed discussion on the issue of effective thickness, the reader can refer to Section 5 of this review. In the case of a circular cylindrical shell, we use x and θ as axial and circumferential angular coordinates, respectively, and z is the coordinate along the thickness (outward) of the shell. The displacements at an arbitrary point on the shell in the axial, circumferential, and radial directions are 44

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

denoted by u1 , u2 , and u3 , respectively. In the classical shell theory, the displacement components are assumed to be ∂w(x, θ, t) ∂x ∂w(x, θ, t) u2 (x, θ, z, t) = v(x, θ, t) − z R∂θ u3 (x, θ, t) = w(x, θ, t) u1 (x, θ, z, t) = u(x, θ, t) − z

(119)

where u, v and w are the middle surface displacements. The governing equations for the thin-walled cylindrical shell subjected to an axial compressive ¯ , a torsional load, T¯, and a uniform external pressure, p¯, are [116, 117] load, N 2 ¯ 2 1 ∂Nxθ 1 ∂Mxθ ∂Nxx ¯ ∂ u − 2T ∂ u + − −N 2 2 ∂x  R ∂θ 2R  ∂θ ∂x R ∂x∂θ 1 ∂ 2u 1 ∂w ∂ 2u −¯ pR − = ρh 2 R2 ∂θ2 R ∂x ∂t 2 1 ∂Nθθ ∂Nxθ 3 ∂Mxθ 1 ∂Mθθ ¯∂ v + + + 2 −N 2 R ∂θ ∂x 2R ∂x  ∂x  R 2 ∂θ  2 ¯ ∂ v ∂w 1 ∂ v 1 ∂w 2T ∂ 2v + + 2 − − p¯R = ρh 2 2 2 R ∂x∂θ ∂x R ∂θ R ∂θ ∂t

(120)

(121)

  ∂ 2 Mxx 2 ∂ 2 Mxθ 1 ∂ 2 Mθθ Nθθ 2T¯ ∂v ∂ 2w + + 2 − − + − ∂x2 R ∂x∂θ R ∂θ2 R R ∂x∂θ ∂x   2 2 2 ¯ ∂ w − p¯R 1 ∂ w + 1 ∂u − 1 ∂v = ρh ∂ w (122) −N ∂x2 R2 ∂θ2 R ∂x R2 ∂θ ∂t2 where ρh is the mass density per unit lateral area of the SWCNT and Nxx , Nθθ and Nxθ are the force resultants and Mxx , Mθθ , Mxθ are the moment resultants defined as h

{Nxx , Nθθ , Nxθ } =

Z2

−h 2

  z z {txx 1 + , tθθ , txθ 1 + }dz R R h

{Mxx , Mθθ , Mxθ } =

Z2

−h 2

  z z z, tθθ z, txθ 1 + z}dz (123) {txx 1 + R R 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

The strain components εxx , εθθ and εxθ at an arbitrary point on the shell are related to the middle surface strains ε0xx , ε0θθ and ε0xθ and to the changes in the curvature of the middle surface κxx , κθθ and κxθ via {εxx , εθθ , εxθ } = {ε0xx , ε0θθ , ε0xθ } + z{κxx , κθθ , κxθ }

(124)

According to the Fl¨ ugge shell theory [118], the middle surface straindisplacement relationships and changes in the curvature for the circular cylindrical shell are expressed as {ε0xx , ε0θθ , ε0xθ } = {

{κxx , κθθ , κxθ } = {−

1 ∂u ∂u 1 ∂v w ∂v , + , + } ∂x R ∂θ R ∂x R ∂θ

(125)

1 ∂ 2w w 2 ∂ 2w 1 ∂u 1 ∂v ∂ 2w , − − , − − + } (126) ∂x2 R2 ∂θ2 R2 R ∂x∂θ R2 ∂θ R ∂x

SWCNTs have been frequently modeled as isotropic elastic shells [119, 120, 121, 122, 123, 124]. However, there are obvious evidences such as chirality-dependent elastic moduli [125, 126, 127, 128], coupling of extension and twist [129, 130, 131], and chirality-dependent critical buckling strain [132], showing that SWCNTs exhibit remarkable chirality induced anisotropic elastic properties which should not be neglected in some cases. Therefore, here we propose a general anisotropic elastic shell model including chirality effect to study the mechanical behavior of SWCNTs. In addition, most researchers have used ∇2 (tij ) instead of (∇2 t)ij because of the difficulty in deriving the governing equations and their solutions. Using this assumption, the stress and strain relations (65) can be rewritten as   2 1 ∂ 2 txx 1 ∂ txx 2 + 2 = (Y11 εxx + Y12 εθθ + Y13 εxθ ) (127) txx − (e0 Li ) 2 2 ∂x R ∂θ h tθθ − (e0 Li )



∂ 2 tθθ 1 ∂ 2 tθθ + ∂x2 R2 ∂θ2



=

1 (Y12 εxx + Y22 εθθ + Y23 εxθ ) h

(128)

txθ − (e0 Li )2



∂ 2 txθ 1 ∂ 2 txθ + 2 2 ∂x R ∂θ2



=

1 (Y13 εxx + Y23 εθθ + Y33 εxθ ) h

(129)

2

46

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

The elements Yij illustrate the anisotropic surface elastic constants of an arbitrary SWCNT. These elastic constants are defined in Section 5. In the nonlocal anisotropic elastic shell model, the force and moment resultants are defined based on the stress components in Eqs. (127)-(129), and thus can be expressed as   2 ∂u Y12 ∂v 1 ∂ 2 Nxx ∂ Nxx 2 = Y11 + ( + w) + + 2 Nxx − (e0 Li ) ∂x2 R ∂θ2 ∂x R ∂θ ∂v 1 ∂u D11 ∂ 2 w D12 1 ∂ 2 w w Y13 ( + )− ( − + 2) ∂x R ∂θ R ∂x2 R R2 ∂θ2 R 2 ∂ 2w 1 ∂u 1 ∂v D13 (− − 2 + ) (130) + R R ∂x∂θ R ∂θ R ∂x 

 ∂ 2 Nθθ 1 ∂ 2 Nθθ + = ∂x2 R2 ∂θ2 ∂v 1 ∂u ∂u Y22 ∂v + ( + w) + Y23 ( + ) Y12 ∂x R ∂θ ∂x R ∂θ Nθθ − (e0 Li )2

(131)

 1 ∂ 2 Nxθ ∂ 2 Nxθ ∂u Y23 ∂v + + ( + w) = Y13 ∂x2 R2 ∂θ2 ∂x R ∂θ ∂v 1 ∂u D13 ∂ 2 w D23 1 ∂ 2 w w +Y33 ( + )− − ( + 2) + ∂x R ∂θ R ∂x2 R R2 ∂θ2 R 2 ∂ 2w 1 ∂u 1 ∂v D33 (− − + ) (132) R R ∂x∂θ R2 ∂θ R ∂x Nxθ − (e0 Li )2



 1 ∂ 2 Mxx D11 ∂u D12 ∂v ∂ 2 Mxx + 2 + ( + w) Mxx − (e0 Li ) = 2 2 ∂x R ∂θ R ∂x R ∂θ D13 ∂v 1 ∂u ∂ 2w 1 ∂ 2w w + ( + ) − D11 2 − D12 ( 2 2 + 2 ) R ∂x R ∂θ ∂x R ∂θ R 2 ∂ 2w 1 ∂u 1 ∂v +D13 (− − + ) (133) R ∂x∂θ R2 ∂θ R ∂x 2



 1 ∂ 2 Mθθ ∂ 2w ∂ 2 Mθθ + = −D Mθθ − (e0 Li ) 12 ∂x2 R2 ∂θ2 ∂x2 2 2 1 ∂ w w 2 ∂ w 1 ∂u 1 ∂v −D22 ( 2 2 + 2 ) + D23 (− − + ) R ∂θ R R ∂x∂θ R2 ∂θ R ∂x 2



47

(134)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

 ∂ 2 Mxθ D13 ∂u D23 ∂v 1 ∂ 2 Mxθ Mxθ − (e0 Li ) + 2 = + ( + w) ∂x2 R ∂θ2 R ∂x R ∂θ D33 ∂v 1 ∂u ∂ 2w 1 ∂ 2w w + ( + ) − D13 2 − D23 ( 2 2 + 2 ) R ∂x R ∂θ ∂x R ∂θ R 1 ∂u 1 ∂v 2 ∂ 2w − 2 + ) (135) +D33 (− R ∂x∂θ R ∂θ R ∂x 2



wherein the elements of the bending stiffness matrix (Dij ) can be defined as Dij =

Yij h2 12

(136)

After substitution of Eqs. (130)-(135) into Eqs. (120)-(122), the governing equations for the anisotropic thin-walled cylindrical shell in terms of the axial, circumferential and radial displacements of the mean surface, (u, v, w), are obtained as 2Y13 3 D13 ∂ 2 Y33 D33 ∂ 2 ∂2 +( − ) + ( 2 − 4 ) 2 ]u 2 3 ∂x R 2 R ∂x∂θ R R ∂θ D13 ∂ 2 Y12 + Y33 ∂ 2 Y23 D23 ∂ 2 +[(Y13 + 2 ) 2 + ( ) + ( 2 − 4 ) 2 ]v R ∂x R ∂x∂θ R R ∂θ Y12 D12 ∂ Y23 D23 ∂ D11 ∂ 3 +[( − 3) +( 2 − 4 ) − R R ∂x R R ∂θ R ∂x3 3 3 D12 + D33 ∂ 5 D13 ∂ D23 ∂ 3 −( ) − − ]w 3 2 2 2 ∂x∂θ 2 R ∂x ∂θ 2R4 ∂θ3  R2  ∂ 2 u 2T¯ ∂ 2 u 1 ∂ 2u 1 ∂w ∂ u ¯ = ρh 2 + N 2 + + p¯R − ∂t ∂x R ∂x∂θ R2 ∂θ2 R ∂x  4    4 4 ∂ u 1 ∂ u ∂ u 1 ∂ 4u 2 ¯ −(e0 Li ) {ρh + +N + ∂x2 ∂t2 R2 ∂θ2 ∂t2 ∂x4 R2 ∂θ2 ∂x2   2T¯ ∂ 4u 1 ∂ 4u + + R ∂x3 ∂θ R2 ∂x∂θ3   1 ∂ 4u 1 ∂ 3w 1 ∂ 4u 1 ∂ 3w +¯ pR − + 4 4 − 3 } (137) R2 ∂x2 ∂θ2 R ∂x3 R ∂θ R ∂x∂θ2 [Y11

[(Y13 +

3 D13 ∂ 2 Y33 + Y12 D33 ∂ 2 Y23 D23 ∂ 2 ) + ( − ) + ( − 4 ) 2 ]u 2 R2 ∂x2 R R3 ∂x∂θ R2 R ∂θ 48

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

D33 ∂ 2 Y22 ∂ 2 2Y23 5 D23 ∂ 2 ) + ( + ) + ]v R2 ∂x2 R 2 R3 ∂x∂θ R2 ∂θ2 3 Y22 D22 ∂ 5 D13 ∂ Y23 D23 ∂ − 3) +( 2 − 4 ) − +[( R R ∂x R R ∂θ 2 R ∂x3 3 9 D23 ∂ D22 ∂ 3 5D33 + D12 ∂ 3 − −( ) 2 − 4 ]w 3 2 2 2R ∂x∂θ R  ∂x ∂θ R ∂θ3  ∂w 1 ∂ 2v ∂ 2 v 2T¯ ∂ 2 v 1 ∂w ∂ 2v ¯ + + p¯R + = ρh 2 + N 2 + ∂t ∂x R ∂x∂θ ∂x R2 ∂θ2 R2 ∂θ     4 ∂ 4v ∂ v 1 ∂ 4v 1 ∂ 4v 2 ¯ +N −(e0 Li ) {ρh + + ∂x2 ∂t2 R2 ∂θ2 ∂t2 ∂x4 R2 ∂x2 ∂θ2   2T¯ ∂ 4v ∂ 3w 1 ∂ 4v 1 ∂ 3w + + + + R ∂x3 ∂θ ∂x3 R2 ∂x∂θ3 R2 ∂x∂θ2   4 1 ∂ 3w 1 ∂ 4v 1 ∂ 3w 1 ∂ v + + + +¯ pR } (138) R2 ∂x2 ∂θ2 R2 ∂x2 ∂θ R4 ∂θ4 R4 ∂θ3 +[(Y33 + 4

Y23 ∂ D11 ∂ 3 2D13 ∂ 3 D23 ∂ 3 Y12 ∂ − 2 + + − ]u R ∂x R ∂θ R ∂x3 R2 ∂x2 ∂θ R4 ∂θ3 3 3 Y23 ∂ Y22 ∂ D13 ∂ D12 + 4D33 ∂ 3D23 ∂ 3 +[− − 2 +2 + ( ) + ]v R ∂x R ∂θ R ∂x3 R2 ∂x2 ∂θ R3 ∂x∂θ2 Y22 D22 ∂ 2 D22 ∂ 4 ∂4 +[− 2 − 4 − 4 − D11 4 2 4 R R ∂θ R ∂θ ∂x 2D12 + 4D33 ∂4 4D13 ∂ 4 4D23 ∂ 4 −( ) − − ]w 3 R2 ∂x2 ∂θ2 R ∂x3 ∂θ R3 ∂x∂θ     2 2 2 2 ¯ ∂ w ¯ ∂ w + 2T − ∂ w + ∂v + p¯R 1 ∂ w + 1 ∂u − 1 ∂v = ρh 2 + N ∂t ∂x2 R ∂x∂θ ∂x R2 ∂θ2 R ∂x R2 ∂θ  4   4  4 4 ∂ w 1 ∂ w ¯ ∂ w+ 1 ∂ w −(e0 Li )2 {ρh + 2 2 2 +N 2 2 ∂x ∂t R ∂θ ∂t ∂x4 R2 ∂x2 ∂θ2   2T¯ ∂ 4w ∂ 3v 1 ∂ 4w 1 ∂ 3v + − 3 + 3− 2 + 2 3 R ∂x ∂θ ∂x R ∂x∂θ R ∂x∂θ2   1 ∂ 4w 1 ∂ 3u 1 ∂ 3v 1 ∂ 3u 1 ∂ 3v 1 ∂ 4w +¯ pR + − + + − }(139) R2 ∂x2 ∂θ2 R ∂x3 R2 ∂x2 ∂θ R4 ∂θ4 R3 ∂x∂θ2 R4 ∂θ3 [−

Simplified forms of these equations have been developed in Refs. [115, 133]. These three coupled equations with appropriate boundary conditions can be solved by different approaches. The detailed description of the numer49

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

ical methods is however beyond the scope of this review, and the interested reader is referred to appropriate textbooks (for example [134, 135, 136]). 4.3.2. First order shear deformation theory According to the first order shear deformation theory, the displacement fields are of the form u1 (x, θ, z, t) = u(x, θ, t) + zβx (x, θ, t) u2 (x, θ, z, t) = v(x, θ, t) + zβθ (x, θ, t) u3 (x, θ, t) = w(x, θ, t)

(140)

where βx and βθ are the rotations of the transverse normal about the x- and θ-axes, respectively. The equations of motion for cylindrical shells in the first order shear deformation theory are 2 ¯ 2 1 ∂Nxθ ∂Nxx ¯ ∂ u − 2T ∂ u + −N 2 ∂x  R ∂θ ∂x R ∂x∂θ  1 ∂ 2u 1 ∂w ∂ 2u −¯ pR − = ρh R2 ∂θ2 R ∂x ∂t2

  ∂Nxθ 1 ∂Nθθ Qθ ∂ 2 v 2T¯ ∂ 2 v ∂w ¯ + + −N 2 − + ∂x R ∂θ R ∂x R ∂x∂θ ∂x   2 2 1 ∂ v 1 ∂w ∂ v −¯ pR + = ρh 2 R2 ∂θ2 R2 ∂θ ∂t  2 2 ¯ 1 ∂Qθ Nθθ ∂Qx ¯ ∂ w − 2T − ∂ w + ∂v + − −N ∂x R ∂θ R ∂x2 R ∂x∂θ ∂x   2 2 1 ∂ w 1 ∂u 1 ∂v ∂ w −¯ pR + − = ρh 2 R2 ∂θ2 R ∂x R2 ∂θ ∂t ∂Mxx 1 ∂Mxθ ρh3 ∂ 2 βx + − Qx = ∂x R ∂θ 12 ∂t2 ∂Mxθ 1 ∂Mθθ ρh3 ∂ 2 βθ + − Qθ = ∂x R ∂θ 12 ∂t2 50

(141)

(142)

(143)

(144)

(145)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

in which Qx , Qθ are the transverse shear stress resultants defined as h

{Qx , Qθ } =

Z2

{txz (1 +

z ), tθz }dz R

(146)

−h 2

The strain components are defined by {εxx , εθθ , εθz , εxz , εxθ } = {ε0xx , ε0θθ , ε0θz , ε0xz , ε0xθ } + z{κxx , κθθ , 0, 0, κxθ } (147) According to the first order shear deformation theory [118], the middle surface strain-displacement relationships and changes in the curvature for a circular cylindrical shell are {ε0xx , ε0θθ , ε0θz , ε0xz , ε0xθ } = {

{κxx , κθθ , κxθ } = {

1 ∂w v ∂w ∂v 1 ∂u ∂u 1 ∂v w , + , βθ + − , βx + , + } ∂x R ∂θ R R ∂θ R ∂x ∂x R ∂θ (148)

∂βx 1 ∂βθ w 1 ∂v 1 ∂βx 1 ∂u ∂βθ , − 2− 2 , − 2 + } (149) ∂x R ∂θ R R ∂θ R ∂θ R ∂θ ∂x

For the zigzag and armchair SWCNTs, the stress-strain relationships are obtained by using Eqs. (13) and (65) as  2  ∂ txx 1 ∂ 2 txx txx − (e0 Li )2 + = C11 εxx + C12 εθθ (150) ∂x2 R2 ∂θ2  ∂ 2 tθθ 1 ∂ 2 tθθ tθθ − (e0 Li ) + 2 = C12 εxx + C22 εθθ ∂x2 R ∂θ2   2 ∂ tθz 1 ∂ 2 tθz tθz − (e0 Li )2 + = C44 εθz ∂x2 R2 ∂θ2  2  ∂ txz 1 ∂ 2 txz 2 txz − (e0 Li ) + 2 = C55 εxz ∂x2 R ∂θ2  2  ∂ txθ 1 ∂ 2 txθ 2 txθ − (e0 Li ) + 2 = C66 εxθ ∂x2 R ∂θ2 2



51

(151) (152) (153) (154)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

For the orthotropic SWCNTs, which is assumed to be in a state of plane stress, Cij are defined as νθ Ex Eθ Ex , C12 = , C22 = 1 − ν x υθ 1 − νx νθ 1 − νx νθ = Gθz , C55 = Gxz , C66 = Gxθ

C11 = C44

(155)

where Ex and Eθ are Young's moduli in the longitudinal and circumferential directions, respectively, Gxθ is in-plane shear modulus, Gxz and Gθz are the transverse shear moduli and νx is Poisson's ratio along the axis of the SWCNT (Poisson's ratio along the circumferential direction is νθ = νx (Eθ /Ex ) ). By substituting Eqs. (148) and (149) into Eqs. (150)-(154), and then substituting the resulting equations into Eqs. (123) and (146), the force and moment results can be derived as   2 1 ∂ 2 Nxx ∂ Nxx 2 + 2 Nxx − (e0 Li ) ∂x2 R ∂θ2 D11 ∂βx D12 ∂βθ w 1 ∂v ∂u A12 ∂v + ( + w) + + 2( − − )(156) = A11 ∂x R ∂θ R ∂x R ∂θ R R ∂θ Nθθ − (e0 Li )2



1 ∂ 2 Nθθ ∂ 2 Nθθ + 2 2 ∂x R ∂θ2



= A12

∂u A22 ∂v + ( + w) ∂x R ∂θ

 1 ∂ 2 Nxθ ∂ 2 Nxθ Nxθ − (e0 Li ) + 2 ∂x2 R ∂θ2 ∂v 1 ∂u D66 ∂βθ 1 ∂u 1 ∂βx = A66 ( + )+ ( − 2 + ) ∂x R ∂θ R ∂x R ∂θ R ∂θ 2

(157)



(158)

 ∂ 2 Mxx 1 ∂ 2 Mxx + ∂x2 R2 ∂θ2 D11 ∂u D12 ∂v ∂βx D12 ∂βθ w 1 ∂v = + 2( + w) + D11 + ( − − )(159) R ∂x R ∂θ ∂x R ∂θ R R ∂θ

Mxx − (e0 Li )2



 ∂ 2 Mθθ 1 ∂ 2 Mθθ Mθθ − (e0 Li ) + 2 ∂x2 R ∂θ2 ∂βx D22 ∂βθ w 1 ∂v = D12 + ( − − ) ∂x R ∂θ R R ∂θ 2



52

(160)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65



 ∂ 2 Mxθ 1 ∂ 2 Mxθ Mxθ − (e0 Li ) + 2 ∂x2 R ∂θ2 1 ∂u ∂βθ 1 ∂u 1 ∂βx D66 ∂v ( + ) + D66 ( − 2 + ) = R ∂x R ∂θ ∂x R ∂θ R ∂θ 2

Qx − (e0 Li )2 Qθ − (e0 Li )2





∂ 2 Qx 1 ∂ 2 Qx + 2 2 ∂x R ∂θ2

1 ∂ 2 Qθ ∂ 2 Qθ + ∂x2 R2 ∂θ2





= Ks A55 (βx +

= Ks A44 (βθ +

∂w ) ∂x

v 1 ∂w − ) R ∂θ R

(161)

(162)

(163)

where the extensional (Aij ) and the bending (Dij ) stiffnesses are defined as  Aij = Cij h, Dij = Cij h3 12, i, j = 1, 2, 6 A44 = C44 h, A55 = C55 h (164) Finally, substituting Eqs. (156)-(163) into Eqs. (141)-(145), the equations of motion for thick SWCNTs in terms of the longitudinal, circumferential and radial displacements (u, v, w) of the middle surface of the SWCNTs and the rotations of tangents to the middle surface (βx and βθ ) are obtained as     ∂2 A66 ∂ 2 (A12 + A66 )R2 − D12 ∂ 2 v A11 2 + 2 2 u + ∂x R ∂θ R3 ∂x∂θ       A12 R2 − D12 ∂ D11 ∂ 2 D12 ∂ 2 + w + β + βθ x R3 ∂x R ∂x2 R2 ∂x∂θ    ∂ 2u ∂ 2 u 2T¯ ∂ 2 u 1 ∂ 2u 1 ∂w ¯ = ρh 2 + N 2 + + p¯R − ∂t ∂x R ∂x∂θ R2 ∂θ2 R ∂x  4    4 4 ∂ u 1 ∂ u ∂ u 1 ∂ 4u 2 ¯ −(e0 Li ) {ρh + +N + ∂x2 ∂t2 R2 ∂θ2 ∂t2 ∂x4 R2 ∂θ2 ∂x2   2T¯ ∂ 4u 1 ∂ 4u + + R ∂x3 ∂θ R2 ∂x∂θ3   1 ∂ 4u 1 ∂ 3w 1 ∂ 4u 1 ∂ 3w +¯ pR − + 4 4 − 3 } (165) R2 ∂x2 ∂θ2 R ∂x3 R ∂θ R ∂x∂θ2 53

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65



   ∂2 (A12 + A66 )R2 − D66 ∂ 2 A22 ∂ 2 Ks A44 u + A v + − 66 R3 ∂x∂θ ∂x2 R2 ∂θ2 R2       A22 + Ks A44 ∂ D66 ∂ 2 D66 ∂ 2 Ks A44 + w + β + βθ + x R2 ∂θ R2 ∂x∂θ R ∂x2 R     2 ¯  ∂ 2v ∂ 2v ∂w 1 ∂ 2v 1 ∂w ¯ ∂ v + 2T = ρh 2 + N + + p ¯ R + ∂t ∂x2 R ∂x∂θ ∂x R2 ∂θ2 R2 ∂θ     4 4 4 ∂ v 1 ∂ 4v ¯ ∂ v+ 1 ∂ v + N −(e0 Li )2 {ρh + ∂x2 ∂t2 R2 ∂θ2 ∂t2 ∂x4 R2 ∂x2 ∂θ2   2T¯ ∂ 3w 1 ∂ 4v 1 ∂ 3w ∂ 4v + + + 2 + R ∂x3 ∂θ ∂x3 R ∂x∂θ3 R2 ∂x∂θ2   1 ∂ 3w 1 ∂ 4v 1 ∂ 3w 1 ∂ 4v + + + +¯ pR } (166) R2 ∂x2 ∂θ2 R2 ∂x2 ∂θ R4 ∂θ4 R4 ∂θ3     A12 ∂ A22 + Ks A44 ∂ − u+ − v R ∂x R2 ∂θ       A22 Ks A44 ∂ 2 ∂ Ks A44 ∂ ∂2 w + Ks A55 βx + βθ + Ks A55 2 − 2 + ∂x R R2 ∂θ2 ∂x R ∂θ      ∂ 2w ∂ 2 w 2T¯ ∂v 1 ∂u 1 ∂v ∂ 2w 1 ∂ 2w ¯ = ρh 2 + N 2 + + + − − + p¯R ∂t ∂x R ∂x∂θ ∂x R2 ∂θ2 R ∂x R2 ∂θ  4    4 4 ∂ w 1 ∂ 4w ¯ ∂ w+ 1 ∂ w + −(e0 Li )2 {ρh + N ∂x2 ∂t2 R2 ∂θ2 ∂t2 ∂x4 R2 ∂x2 ∂θ2   4 3 4 3 ¯ 2T ∂ v 1 ∂ w 1 ∂ v ∂ w + − 3 + 3− 2 + R ∂x ∂θ ∂x R ∂x∂θ3 R2 ∂x∂θ2   1 ∂ 4w 1 ∂ 3u 1 ∂ 3v 1 ∂ 4w 1 ∂ 3u 1 ∂ 3v +¯ pR + − + + 3 − }(167) R2 ∂x2 ∂θ2 R ∂x3 R2 ∂x2 ∂θ R4 ∂θ4 R ∂x∂θ2 R4 ∂θ3 

   D11 ∂ 2 D66 ∂ 2 ∂ − 3 u + −Ks A55 w R ∂x2 R ∂θ2 ∂x     ∂2 D66 ∂ 2 (D12 + D66 ) ∂ 2 + D11 2 + 2 − Ks A55 βx + βθ ∂x R ∂θ2 R ∂x∂θ    ρh3 ∂ 2 βx ∂ 4 βx 1 ∂ 4 βx 2 = − (e L ) + (168) 0 i 12 ∂t2 ∂x2 ∂t2 R2 ∂θ2 ∂t2 54

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65



   D66 ∂ 2 D22 + Ks A44 R2 ∂ D22 ∂ 2 Ks A44 v+ − w − 3 + R ∂x2 R ∂θ2 R R3 ∂θ     (D12 + D66 ) ∂ 2 ∂2 D22 ∂ 2 + βx + D66 2 + 2 − Ks A44 βθ R ∂x∂θ ∂x R ∂θ2    ρh3 ∂ 2 βθ ∂ 4 βθ 1 ∂ 4 βθ 2 (169) = − (e L ) + 0 i 12 ∂t2 ∂x2 ∂t2 R2 ∂θ2 ∂t2 From Eqs. (165)-(169), it is easily seen that the classical or local Fl¨ ugge's shell theory [117, 137] is recovered if the parameter e0 is set to zero. 4.4. Nonlocal plate model In modeling micro or nanoelectromechanical systems (MEMS or NEMS) and devices, some mechanical components such as thin film elements [138], nanosheet resonators [139], paddle-like resonators [140], two-dimensional suspended nanoscopic structures [141] and graphene sheets have to be modeled by a two-dimensional plate-like structure. Particularly, graphene sheets are modeled as continuous and homogenous plates (Fig. 13), while the lattice spacing between individual carbon atoms in the nanomaterials is ignored, implying that the carbon atoms are uniformly distributed over the surface of each nanoscopic structure with a constant surface density. Generally, two well-known plate theories are utilized in the mechanical modeling of the graphene sheets. These are the Kirchhoff plate theory and the Mindlin plate theory. The Kirchhoff plate theory applies to thin plates and neglects the effect of transverse shear deformation, whereas the Mindlin plate theory is a first order shear deformable plate theory and incorporates this effect which becomes significant in thick plates and shear deformable plates [142]. A detailed discussion on nonlocal plate theories has been presented in Ref. [142], from which we follow. 4.4.1. The Kirchhoff plate theory In this section, we consider the classical thin plate theory, which is based on the Kirchhoff following assumptions: (a) Straight lines perpendicular to the mid-surface (i.e., the transverse normals) before deformation remain straight after deformation. (b) The transverse normals do not experience elongation (i.e., they are inextensible). 55

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

(c) The transverse normals rotate such that they remain perpendicular to the mid-surface after deformation. The implications of Kirchhoff's assumptions imply that the transverse strains (εxz , εyz , εzz ) are zero, and consequently, the transverse stresses (σxz , σyz , σzz ) do not enter the theory [85]. Consider a graphene sheet of length L in the x direction, with width W in the y direction and thickness h. The displacements of an arbitrary point on the sheet are related to the displacements of middle surface by the following relations ∂w(x, y, t) ∂x ∂w(x, y, t) u2 (x, y, z, t) = v(x, y, t) − z ∂y u3 (x, y, t) = w(x, y, t) u1 (x, y, z, t) = u(x, y, t) − z

(170)

where u, v and w denote displacements of the point (x, y,0) along x, y and z directions, respectively. According to Eq. (7), the strains are expressed as ∂u ∂ 2w −z 2 ∂x ∂x ∂ 2w ∂v −z 2 = ∂y ∂y 1 ∂u ∂v ∂ 2w = ( + − 2z ) 2 ∂y ∂x ∂y∂x

εxx = εyy εxy

(171)

It is well-known that the principal of virtual work is independent of constitutive relations. So, this can be utilized to derive the governing equations of the graphene sheets. Using the principle of virtual displacements, the following governing equations can be obtained [143]: 2 ∂ 2 Mxx ∂ 2 Myy ∂ 2 Mxy ¯xx ∂ w + + 2 − N ∂x2 ∂y 2 ∂x∂y ∂x2 2 2 2 ¯xy ∂ w − N ¯yy ∂ w = ρh ∂ w −2N ∂x∂y ∂y 2 ∂t2

(172)

where ρh is the mass density per unit lateral area of the graphene sheet and ¯xx , N ¯yy and N ¯xy are the applied in-plane compressive and shear forces N

56

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

measured per unit length. In addition, Mxx , Myy and Mxy are the stress couple resultants which are defined as h

{Mxx , Myy , Mxy } =

Z2

{txx z, tyy z, txy z}dz

(173)

−h 2

Using the strain-displacement relationships (171), the stress-strain relationships (63) and the definition of stress resultants (173), we can express the stress resultants in terms of displacements as follows   2 ∂ 2w ∂ 2w ∂ Mxx ∂ 2 Mxx 2 + − D = −D Mxx − (e0 Li ) 12 11 ∂x2 ∂y 2 ∂x2 ∂y 2  2  2 2 ∂ Myy ∂ Myy ∂ w ∂ 2w Myy − (e0 Li )2 + − D = −D 22 12 ∂x2 ∂y 2 ∂y 2 ∂x2   2 2 2 ∂ w ∂ Mxy ∂ Mxy + (174) Mxy − (e0 Li )2 = −2D66 2 2 ∂x ∂y ∂x∂y where D11 , D22 , D66 and D12 denote the bending rigidity of the orthotropic graphene sheet and are defined by E1 h3 , 12(1 − ν12 ν21 ) E2 h3 = , 12(1 − ν12 ν21 )

ν12 E2 h3 12(1 − ν12 ν21 ) G12 h3 = 12

D11 =

D12 =

D22

D66

(175)

Here E1 and E2 are the Young moduli in directions 1 and 2, respectively, G12 is the shear modulus and ν12 and ν21 denote Poisson's ratios. Finally, using Eqs. (172) and (174), we obtain the governing differential equation of motion in terms of the transverse displacement ∂ 4w ∂ 4w ∂ 4w D11 4 + 2(D12 + 2D66 ) 2 2 + D22 4 ∂x ∂x ∂y ∂y    2 ∂2 ∂ 2 + + 1 − (e0 Li ) ∂x2 ∂y 2 2 2 2 ∂ 2w ¯xx ∂ w + 2N ¯xy ∂ w + N ¯yy ∂ w ) = 0 ×(ρh 2 + N ∂t ∂x2 ∂x∂y ∂y 2 57

(176)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

4.4.2. The Mindlin plate theory The simplest shear deformation plate theory is the first-order theory, also referred to as the Mindlin plate theory [144], and it is based on the displacement fields u1 (x, y, z, t) = u(x, y, t) + zβx (x, y, t) u2 (x, y, z, t) = v(x, y, t) + zβy (x, y, t) u3 (x, y, t) = w(x, y, t)

(177)

where βx and βy denote rotations about the y- and x-axes, respectively. Based on Eqs. (7) and (177), the strains are expressed as ∂βx ∂v ∂βy 1 ∂w ∂u +z , εyy = +z , εyz = (βy + ) ∂x ∂x ∂y ∂y 2 ∂y   1 ∂u ∂v ∂βx ∂βy ∂w 1 = ( + +z + ) (178) ), εxz = (βx + 2 ∂y ∂x ∂y ∂x 2 ∂y

εxx = εxy

The equilibrium equations of the first order plate theory are given by   2 2 2 2 ∂Qx ∂Qy ¯xx ∂ w + 2N ¯xy ∂ w + N ¯yy ∂ w = ρh ∂ w + − N (179) 2 2 ∂x ∂y ∂x ∂x∂y ∂y ∂t2 ∂Mxx ∂Mxy ρh3 ∂ 2 βx + − Qx = ∂x ∂y 12 ∂t2 ∂Mxy ∂Myy ρh3 ∂ 2 βy + − Qy = ∂x ∂y 12 ∂t2 where Qx , Qy are the transverse shear stress resultants, defined by

(180) (181)

h

{Qx , Qy } =

Z2

{txz , tyz }dz

(182)

−h 2

In the first order shear deformation theory, a shear correction factor (Ks ) is introduced to correct for the discrepancy between the actual transverse shear force distributions and those computed using the kinematic relations of the first order shear deformation theory. The shear correction factors depend not only on the geometric parameters but also on the loading and boundary conditions of the plate. 58

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Using the strain-displacement relationship (178), the stress-strain relationships (63) and the stress resultants definition (173) and (182), the stress resultants in terms of displacements are obtained as follows  2  ∂ Mxx ∂ 2 Mxx ∂βx ∂βy Mxx − (e0 Li )2 = D11 + D12 + ∂x2 ∂y 2 ∂x ∂y  2  2 ∂ M ∂β ∂β ∂ M yy x y yy Myy − (e0 Li )2 = D12 + D22 + ∂x2 ∂y 2 ∂x ∂y  2    ∂ Mxy ∂ 2 Mxy ∂βx ∂βy Mxy − (e0 Li )2 + + = D 66 ∂x2 ∂y 2 ∂y ∂x  2  2 Q ∂ ∂ Q ∂w x x Qx − (e0 Li )2 + ) = Ks A55 (βx + ∂x2 ∂y 2 ∂x  2  ∂ Qy ∂ 2 Qy ∂w Qy − (e0 Li )2 + ) (183) = Ks A44 (βy + 2 2 ∂x ∂y ∂y wherein A44 and A55 are defined as

A44 = G23 h,

A55 = G12 h

(184)

in which G23 is the transverse shear modulus. Substituting Eqs. (183) into Eqs. (179)-(181), the nonlocal governing differential equations of motion for graphene sheets based on the Mindlin plate theory are derived as ∂βy ∂ 2 w ∂βx ∂ 2 w + ) + Ks A44 ( + ) 2 ∂x ∂x ∂y ∂y 2 ∂2 ∂2 = (1 − (e0 Li )2 ( 2 + 2 )) ∂x ∂y 2 2 2 ∂ w ∂ 2w ¯xx ¯xy ∂ w + N ¯yy ∂ w ) + 2 N ×(ρh 2 + N ∂t ∂x2 ∂x∂y ∂y 2

Ks A55 (

(185)

 2  ∂ 2 βx ∂ 2 βy ∂ 2 βy ∂ βx ∂w + D + D + − Ks A55 (βx + ) 12 66 ∂x2 ∂x∂y ∂y 2 ∂x∂y ∂x    4 ρh3 ∂ 2 βx ∂ βx ∂ 4 βx 2 = − (e L ) + (186) 0 i 12 ∂t2 ∂x2 ∂t2 ∂y 2 ∂t2 D11



 ∂ 2 βx ∂ 2 βx ∂ 2 βy ∂ 2 βy ∂w D66 + + D + D − Ks A44 (βy + ) 12 22 2 2 ∂x∂y ∂x ∂x∂y ∂y ∂y    ρh3 ∂ 2 βy ∂ 4 βy ∂ 4 βy 2 = − (e L ) + (187) 0 i 12 ∂t2 ∂x2 ∂t2 ∂y 2 ∂t2 59

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

In order to obtain the mechanical behavior of the nanoplates, these three equations must be solved simultaneously. Note that higher order models for analyzing the graphene sheets have not, so far, been found to be necessary for the analysis of nanoplates [46]. 4.5. Nonlocal spherical shell model Progress in microtechnology and nanotechnology requires the production of various types of spherical micro/nanostructures such as different types of fullerenes, bimetallic spherical particles, metal nanoshells and spherical shell resonators. Moreover, in various modern biomedical and biological applications, some organisms, such as biological cells and spherical viruses can be modeled as spherical membrane-like structures. Therefore, the static and dynamic analyses of micro/nanosized spherical structures can form interesting research problems. In this section the equations governing the dynamics of the spherical membrane shells are formulated using the nonlocal differential constitutive relations of Eringen. The Legendre polynomials and the associated Legendre polynomials are employed to obtain an analytical solution for the axisymmetric vibration of the spherical shell-like nanoscopic structures. Based on the present formulation, two types of spherical shell-like structures including a spherical fullerene (Fig. 14a) and a hollow spherical virus (Fig. 14b), can be studied. It should be noted that the realistic analysis of the fullerenes and viruses can be very complicated. Consequently, idealized models have been proposed and used to study their behavior. Fullerenes may be considered to have a wall sufficiently thin that the stresses and strains do not vary across the shell wall. This approximation is prompted by the membrane theory [145]. Here, we only present the nonlocal formulations for axisymmetric vibration modes because they are the low order modes which are expected to be of most interest in computing electron-phonon interactions. Discussion of the axisymmetric vibration modes of fullerenes has been only considered in [146], from which we follow. Deformation of a spherical membrane shell can be analyzed in terms of the deformation of its middle surface. The symbols u, v and w are used to denote the meridional (θ), the circumferential (φ) and the radial (r) displacements, respectively (Fig. 15). We consider torsionless axisymmetric vibrations of a closed spherical shell of thickness h and radius R measured from the undeformed middle surface.

60

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

For torsionless axisymmetric deformations, we have ∂(∗) = 0, ∂φ

v=0

(188)

where (∗) denotes any function of the dependent variable. The remaining displacements are now expressible as u = u(φ, t),

w = w(φ, t)

(189)

The kinematic relations for the normal strains εφφ and εθθ are given by   1 ∂u +w εθθ = R ∂θ 1 εφφ = (u cot θ + w) (190) R In a similar manner to the case of the cylindrical shell, we approximate (∇2 t)ij by ∇2 (tij ). The nonlocal constitutive equations are described by    ∂tθθ E 1 ∂ 2 tθθ 2 + cot θ (εθθ + νεφφ ) (191) tθθ − (e0 Li ) = R2 ∂θ2 ∂θ 1 − ν2 2

tφφ − (e0 Li )



1 R2



∂ 2 tφφ ∂tφφ + cot θ ∂θ2 ∂θ



=

E (εφφ + νεθθ ) 1 − ν2

(192)

where E is the elastic modulus of the spherical structure and ν denotes Poisson's ratio. In the nonlocal elastic shell model, the stress resultants are defined in terms of the stress components in Eqs. (191) and (192), and thus can be expressed as follows by referencing the kinematic relations   (e0 Li )2 ∂ 2 Nθθ ∂Nθθ + cot θ Nθθ − R2 ∂θ2 ∂θ   Eh ∂u = + w + ν (u cot θ + w) (193) R(1 − ν 2 ) ∂θ   ∂Nφφ (e0 Li )2 ∂ 2 Nφφ Nφφ − + cot θ R2 ∂θ2 ∂θ    Eh ∂u = u cot θ + w + ν +w R(1 − ν 2 ) ∂θ 61

(194)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

The dynamic equilibrium equations of the stress resultants are given by [147] ∂ 2u ∂Nθθ + (Nθθ − Nφφ ) cot θ = Rρh 2 (195) ∂θ ∂t ∂ 2w (196) ∂t2 in which ρh is the mass density per unit lateral area of the spherical shell. Substituting Eqs. (193) and (194) into Eqs. (195) and (196), the equations of motion for the spherical shell-like nanoscopic structures in terms of meridional and radial displacements of the mean surface of spherical shell, (u, w), are obtained as −(Nφφ + Nθθ ) = Rρh

∂w ∂ 2 u ∂u + cot θ − (ν + cot2 θ)u + (1 + ν) 2 ∂θ ∂θ ∂φθ ρ(1 − ν 2 )R2 ∂ 2 u (e0 Li )2 [ 2 − = ∂t R2  4E  3 ∂ u ∂ u ∂ 2u 2 × + cot θ − (1 + cot θ) ] ∂t2 ∂θ2 ∂t2 ∂θ ∂t2 ∂u ρ(1 − ν)R2 ∂ 2 w (e0 Li )2 + u cot θ + 2w = − [ 2 − ∂θ E ∂t R2 3 4 ∂ w ∂ w ×( 2 2 + 2 cot θ)] ∂t ∂θ ∂t ∂θ

(197)

(198)

The solutions of Eqs. (197) and (198) are of the types u(θ, t) = An Pn1 (cos θ) exp(jωt)

(199)

w(θ, t) = Bn Pn (cos θ) exp(jωt)

(200)

where ω is the natural frequency, An and Bn are constant coefficients, Pn (cos θ) are the Legendre polynomials of the first kind and Pn1 (cos θ) are the associated Legendre polynomials of the first kind and first order [148]. Substitution of Eqs. (199) and (200) into Eqs. (197) and (198) leads to two sets of homogeneous algebraic equations for An and Bn , i.e., [−n(n + 1) + (1 − ν) + Ω2 (1 − ν 2 ) 62

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

(e0 Li )2 n(n + 1))]An − (1 + ν)Bn = 0 R2 ( ) (e0 Li )2 2 n(n + 1)An + [2 − Ω (1 − ν) 1 + n(n + 1) ]Bn = 0 (201) R2

×(1 +

where

ρR2 ω 2 (202) E It should be noted that the higher order derivatives of Pn (cos θ) and Pn1 (cos θ) have been eliminated by the recursive use of the differential equations which they satisfy. For the non-trivial solution, the determinant of this set of equations must be zero. Hence, the frequency equation is obtained as follows )2 ( ) ( 2 2 (e L ) (e L ) 0 i 0 i n(n + 1) − Ω2 1 + n(n + 1) Ω4 (1 − ν 2 ) 1 + R2 R2 Ω2 =

× [1 + 3ν + n(n + 1)] − [2 − n(n + 1)] = 0

(203)

It is obvious that the dimensionless frequency parameters (the roots of Eq. (203)) can be expressed as q [1 + 3ν + n(n + 1)] + [1 + 3ν + n(n + 1)]2 − 4(1 − ν 2 ) [n(n + 1) − 2] h i a2n = 2 2(1 − ν 2 ) 1 + (e0RL2i ) n(n + 1) (204) q [1 + 3ν + n(n + 1)]2 − 4(1 − ν 2 ) [n(n + 1) − 2] h i b2n = 2 2(1 − ν 2 ) 1 + (e0RL2i ) n(n + 1) (205) We have an upper branch (an ) and a lower branch (bn ) of the frequency spectrum. It should be noted that, for n = 0, the frequency corresponding to the lower branch is imaginary and is, therefore, a spurious frequency. The upper branch for n = 0 gives the dimensionless natural frequency r 2 a0 = (206) 1−ν [1 + 3ν + n(n + 1)] −

63

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Although this lower frequency of the upper branch is higher than all of the frequencies of the lower branch, it is referred to as the fundamental frequency because it involves a purely radial motion and it is the frequency of the single mode which occurs in the symmetrical free vibration (u = 0 and w 6= 0) of the closed spherical membrane shell. This mode is also known as the breathing-mode and is of particular significance. Moreover, it can be seen from Eq. (205) that b1 has a null frequency which defines a rigid-body motion. 4.6. Nonlocal elasticity theory for breathing-mode of nano-objects Breathing-mode is a low frequency mode displayed by nano-objects of different shapes and is observed via Raman spectroscopy. The breathingmode corresponds to the symmetric synchronous in-phase vibration of all atoms in the radial direction. This mode is popular and is routinely accessible to optical mode for identifying the diameter and elastic properties of the nano-objects. Thus, determining the frequency of the breathing-mode has become a challenging research topic over the past decades [149]. 4.6.1. Nonlocal model for nanosphere Let us first consider the breathing-mode of nanoparticles. Since nanoparticles have spherical or quasi-spherical shapes, their morphology can be approximated as solid nanospheres (Fig. 16). Consider a solid sphere having an outer radius Ro . In a spherical coordinate system (R, θ, φ), the nonzero component of displacement can be written as w = w(R, t), where w is the radial displacement, a function of the radial coordinate R. Therefore, the strain components in terms of the radial displacement are ∂w w (207) εθθ = εφφ = , εRR = R ∂R Generally, the distortion of the spherical symmetry in nanoparticles results in the introduction of anisotropy in their properties [150]. Moreover, recent advances have permitted the creation of high quality elastically anisotropic nanoparticles [151]. Based on Eq. (67), for anisotropic nanoparticles, the stress-strain relation is expressed by  2  ∂ tθθ 2 ∂tθθ 2tθθ 2tRR 2 tθθ − (e0 Li ) + − 2 + 2 = c11 εθθ + c12 εφφ + c13 εRR ∂R2 R ∂R R R (208) 64

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65



∂ 2 tφφ 2 ∂tφφ 2tφφ 2tRR tφφ − (e0 Li ) − 2 + 2 + 2 ∂R R ∂R R R = c12 εθθ + c22 εφφ + c23 εRR 2





∂ 2 tRR 2 ∂tRR 4tRR 2(tθθ + tφφ ) − 2 + + 2 ∂R R ∂R R R2 = c13 εθθ + c23 εφφ + c33 εRR tRR − (e0 Li )2

(209) 

(210)

where cij are the components of elastic constants. In the absence of a body force, the equation of motion is expressed as ∂ 2w ∂tRR 2tRR − tθθ − tφφ + =ρ 2 ∂R R ∂t

(211)

in which ρ is the mass density of the sphere. Using Eqs. (207) to (211), the nonlocal equation of motion for the radial vibration of the anisotropic nanoparticles in term of radial displacement is obtained as   2   4 2 ∂ 3w 2 ∂ 2w 2 ∂w ∂ w ∂ w ρ(e0 Li )2 + − + c + 33 ∂R2 ∂t2 R ∂R∂t2 R2 ∂t2 ∂R2 R ∂R 2 ∂ w w + 2 (c13 + c23 − c11 − c22 − 2c12 ) − ρ 2 = 0 (212) R ∂t It is easily seen from the above equation that the classical, or local, theory is recovered when the parameter e0 is set identically to zero. Moreover, the order of equation (212) with respect to the parameter R is 2 and it is the same as that for the local model. For the free vibration analysis, it is assumed that the displacement w varies harmonically with respect to the time variable t as follows w(R, t) = W (R) exp(jωt) (213) where ω is the angular frequency. Substituting Eq. (213) into Eq. (212), we obtain the following equation   d2 W 2 dW B1 2 + + + B2 W = 0 (214) dR2 R dR R2 65

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

where B1 =

c13 + c23 − c11 − c22 − 2c12 + 2ρ(e0 Li )2 ω 2 c33 − ρ(e0 Li )2 ω 2 B22 =

ρω 2 c33 − ρ(e0 Li )2 ω 2

(215) (216)

Equation (214) is a Bessel-type equation and the general solution of the Bessel equation corresponding to Eq. (214) is [148] 1 W (R) = √ [A1 Jζ (B2 R) + A2 Yζ (B2 R)] (217) R √ where A1 and A2 are unknown constants, ζ = 0.5 1 − 4B1 , Jζ and Yζ are Bessel functions of first and second kind of order ζ, respectively. As the displacement must remain finite at the center of the nanoparticle, we must .√ R when R = 0. The set A2 = 0 to remove the infinite value of Yζ (B2 R) resulting equation is Jζ (B2 R) W (R) = A1 √ (218) R For the stress-free boundary condition, tRR = 0 at the external radius Ro and thus c13 + c23 W (Ro ) dW =− (219) dR R=Ro c33 Ro Substituting Eq. (218) into the boundary condition (219), the frequency equation is obtained as   1 c13 + c23 − (B2 Ro )Jζ+1 (B2 Ro ) = 0 (220) Jζ (B2 Ro ) ζ − + 2 c33

By solving equation (220), we can obtain the natural frequencies of the nanoparticle with cubic, tetragonal, trigonal and hexagonal symmetries. It should be noted that the lowest frequency is the breathing-mode which is critical to the characterization of the nanoparticles.

66

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

4.6.2. Nonlocal model for elongated nanocrystals Elongated nanocrystals (i.e., rod-shaped), also known as nanowires, are probably the most studied nanocrystal systems after the spherical ones. The typical lengths of nanowires span from tens of nanometers to tens of micrometers, while their diameters are generally in the range of a few nm. Nanowires can have circular, semicircular, triangular, rectangular, pentagonal, hexagonal or any unconventional cross-sectional shapes. Here, we model the nanowires as infinitely long cylinders with circular cross sections and radii Rnw . For predicting the radial breathing-mode in a cylindrical coordinate system (r, θ, z), the nonzero component of displacement can be denoted as w = w(r, t) where w is the radial displacement. The nonzero strain components in terms of the radial displacement are given by εrr =

∂w , ∂r

εθθ =

w r

Using Eq. (65), the stress-strain relation is expressed as   2 2 ∂w w ∂ trr 1 ∂trr + − (t − t ) = C11 + C12 trr − (e0 Li )2 rr θθ ∂r2 r ∂r r2 ∂r r tθθ − (e0 Li )

2



 ∂ 2 tθθ 1 ∂tθθ 2 ∂u w + + 2 (trr − tθθ ) = C12 w + C11 ∂r2 r ∂r r ∂r r

(221)

(222)

(223)

In the absence of a body force, the radial governing equation of the long cylinder is expressed as ∂trr trr − tθθ ∂ 2w + =ρ 2 ∂r r ∂t

(224)

wherein ρ is the mass density of the nanowire. Using Eqs. (221)-(224), the nonlocal equation of motion is obtained  4  ∂ w 1 ∂ 3w ∂ 4w 2 ρ(e0 Li ) + − ∂r2 ∂t2 r ∂r∂t2 ∂r2 ∂t2  2  ∂ w 1 ∂w ∂ 2w w +C11 + − ρ =0 (225) − ∂r2 r ∂r r2 ∂t2 In the free vibration analysis, it is assumed that the displacement w varies harmonically with respect to the time variable t as follows w(r, t) = W (r) exp(jωt) 67

(226)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

where ω is the angular frequency. Substituting Eq. (226) into Eq. (225), we obtain the following equation   d2 W 1 dW r 2 + + B2r − W =0 (227) dr dr r where B=

ξ p , Rnw 1 − e2 ξ 2

ξ=

ρω 2 Rnw , C11

e=

e0 Li Rnw

(228)

The general solution of Eq. (227) is given by W (r) = A1 J1 (Br) + A2 Y1 (Br)

(229)

where A1 and A2 are unknown constants, J1 and Y1 are the Bessel functions of first- and second-kind, of order 1 respectively. As the displacement must remain finite at the center of the nanowire, we must set A2 = 0 to remove the infinite value of Y1 (Br) when r = 0. The resulting equation becomes ! ξr p W (r) = A1 J1 (230) Rnw 1 − e2 ξ 2

For the stress-free boundary condition, trr = 0 at the external radius Rnw and thus dW C12 C11 + W (Rnw ) =0 (231) dr Rnw r=Rnw

Substituting Eq. (230) into the boundary condition (231), the frequency equation is obtained as ! !  ζ ζ C12 ζ p J2 p − J1 p 1+ = 0 (232) C11 1 − e2 ζ 2 1 − e2 ζ 2 1 − e2 ζ 2

It is obvious that the frequency of the breathing-mode can be determined by finding the lowest root of Eq. (232).

68

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

5. Determination of the elastic properties and the nonlocal parameter of carbon-based nanoscopic structures In spite of the computational efficiency, as well as the reasonable accuracy associated with the nonlocal continuum theory, it suffers from a substantial deficiency related to an appropriate definition of such mechanical properties as elastic constants and effective thickness. Furthermore, there is no agreement in the research community about the choice of the nonlocal parameter. There has been no consensus on the exact values of the mechanical properties and the nonlocal parameter, and considerable inconsistencies have emerged in the literature. To overcome these obstacles in describing the mechanical properties of the carbon-based nanoscopic structures, our aim has been to gain an indepth insight into the issues underlying the inconsistent results ubiquitous in the literature, and shed some lights on the long standing problem related to the controversial notion of the effective thickness of the nanoscopic structures. In this connection, we offer a discussion of the so-called Yakobson's paradox which refers to the dispersion in the results pertinent to the mechanical properties of carbon-based structures and attributable to the uncertainty concerning the value of the thickness of the nanoscopic structures [152]. In addition, we present an accurate molecular mechanics-based model for predicting the anisotropic elastic constants and the nonlocal parameters of the SWCNTs having an arbitrary chirality. 5.1. Young's modulus and effective wall thickness of SWCNTs The definition of elastic moduli for a solid implies a spatial uniformity of the material, at least in a statistically average sense. However, it is wellknown that a CNT has a discrete atomistic structure and does not have a continuous spatial distribution. Nevertheless, to apply continuum-based theories to CNTs, the basic mechanical quantities such as Young's and shear moduli, Poisson's ratio and wall thickness have to be defined appropriately. The response of a cylinder of length l and cross-sectional area A, to an extensional force, F , is described by its Young's modulus, E, and the relationship between these physical quantities is given by Hooke's law ∆l F =E (233) A l where ∆l is the small deformation change and the relative extension, ∆l/l, is the unidirectional strain along the direction of the applied force. Techniques 69

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

are now available that allow for the application of forces with magnitude in the pico- to micro-Newton range, so as observations can made of deformations in the Angstroms-to-micrometer range [153]. Hence, to estimate the Young modulus of the CNTs, the cross section area A must be chosen accurately. For the CNTs, the cross section area A can be defined as 2πRh in which R is the average radius of the nanotube and h is its effective wall thickness. The value of the radius is reasonably defined by Eq. (2) whereas the effective thickness h is an ambiguous parameter since a nanotube does not have a continuous wall. Unfortunately, considerable inconsistencies have developed in the literature, wherein the reported obtained values for the effective thickness of SWCNTs have varied from 0.0617 nm to 0.69 nm [154]. This large variations generate values of Young's modulus of SWCNTs ranging from 1 TPa to 5.5 TPa. For a more comprehensive discussion of the wall thickness and Young's modulus of SWCNTs, the reader is referred to the review by Wang and Zhang [155]. Table 1 lists a few representative values of the effective wall thickness and the corresponding Young's modulus of the SWCNTs which have been reported [12, 156, 157, 158, 159, 160, 161, 162, 163]. Table 1: The effective wall thickness and Young's modulus of SWCNTs Author

Wall thickness h (nm) Yakobson et al. [12] 0.066 Lu [156] 0.34 Krishnan et al. [157] 0.34 Zhou et al. [158] 0.074 Belytschko et al. [159] 0.34 Li and Chou [160] 0.34 Vodenitcharova and Zhang [161] 0.0617 Pantano et al. [162] 0.075 Cai et al. [163] 0.32

Young's modulus E (TPa) 5.5 0.974 1.25 5.1 0.94 1.01

Eh (Jm-2 )

4.88 4.84 1.06

301 363 340

363 332 425 377 320 343

As is seen from this Table, the majority of the previous results are distributed in the vicinity of Eh = 360 Jm-2 , with the lowest Eh around 300 Jm-2 and the highest around 425 Jm-2 . The parameter Eh is known as the in-plane stiffness, which can be determined without knowing E and h separately. Specifically, without defining the effective thickness of the SWCNTs, the in-plane stiffness can be directly calculated via various techniques. 70

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

One of the most important studies about the wall thickness of the SWCNTs was carried out by Vodenitcharova and Zhang [161]. Based on the consideration of force equilibrium and equivalence, they proposed a necessary condition (but not sufficient) for justifying an effective thickness of SWCNTs that should be smaller than the theoretical diameter of a carbon atom (v0.142 nm). This is known as the Vodenitcharova-Zhang criterion. Their argument is that the cross-section of a nanotube contains only a number of atoms and that the forces in the nanotube are transmitted through these atoms. However, in a continuum mechanics-based model the same forces are transmitted through a continuous wall. Hence, the effective wall thickness cannot be greater than or equal to the theoretical diameter of a carbon atom, otherwise, the nanotube's equilibrium cannot be maintained [164]. Based on this criterion, the assumed value of 0.34 nm, as well as those larger than 0.142 nm, e.g., 0.69 and 0.147 nm, should be excluded in future investigations. 5.2. Anisotropic elastic constants of SWCNTs The above mentioned studies concerned mainly two major elastic properties, namely, the longitudinal elastic modulus and the Poisson's ratio, i.e., the SWCNTs were considered as isotropic elastic materials with two independent elastic constants. However, according to the arrangements of the carbon atoms in the SWCNTs, it is obvious that the strengths of the bonds in the longitudinal and circumferential directions are not equal. It should be noted that the armchair and zigzag SWCNTs are symmetric about the longitudinal direction and thus they exhibit transverse-isotropic or orthotropic behavior in the plane normal to the longitudinal axis [165, 166]. In the case of chiral nanotubes, however, the carbon bonds are not symmetric about the longitudinal axis and so their mechanical properties are fully anisotropic. Generally, the chirality of the SWCNTs results in the introduction of anisotropy in their elastic properties. Therefore, nanotubes cannot be modeled by isotropic models and there have been some studies concerned with the anisotropic mechanical properties and behavior of SWCNTs. Wu et al. [167] developed a finite-deformation shell theory directly from the interatomic potential for an SWCNT by using the modified Born rule. They showed that the SWCNTs with diameter smaller than 1 nm have anisotropic mechanical properties. Based on the theory developed by Wu et al. [167], Peng et al. [168] argued that SWCNTs cannot be represented by conventional isotropic shell models because their constitutive relation involves the coupling between tension and curvature and between bending and 71

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

strain. Using an orthotropic in-plane stress-strain relation for the zigzag and armchair SWCNTs, Ru [169] developed an anisotropic elastic shell model, which can explain the chirality-dependent mechanical behavior of small-radii nanotubes. To facilitate the study of the axial strain-induced torsion of chiral SWCNTs with actual sizes, a two-dimensional continuum model has been proposed which can be used to describe the anisotropic properties due to the nanotube's chirality [131]. A molecular based anisotropic shell model has been developed for predicting the mechanical behavior of SWCNTs [170]. It includes the coupling of axial, circumferential, and torsional strains in SWCNTs, their radial breathing-mode frequency, and their longitudinal and torsional wave speeds. Here, our aim is to obtain the elastic properties of SWCNTs with arbitrary chirality on the basis of molecular mechanics. To proceed, we follow the derivation of these properties as given by [170]. To investigate the anisotropic elastic properties of SWCNTs, the atomic interactions between the carbon atoms must be accurately modeled, and molecular mechanics is employed as one of the most common approaches to describe these interactions. According to molecular mechanics, the total potential energy UT of a molecular system can be expressed as a sum of bonded and non-bonded interactions [171] UT = Uρ + Uθ + Uω + Uτ + UvdW + Ues

(234)

The bonded interactions consist of the bond stretching, Uρ , and the angular distortions including the bond angle bending, Uθ , the bond inversion, Uω , and the angle torsion, Uτ . The non-bonded interactions consist of the van der Waals (vdW), UvdW and the electrostatic, Ues terms. For an SWCNT subjected to an axial loading at small strains, it is assumed that the two energy terms associated with the bond stretching and the angle variation are only significant in the total molecular potential energy, and the other terms may be negligible. Furthermore, due to the small deformations and the atomic interactions near the equilibrium structure, the total molecular potential energy UT of the SWCNT is expressable as UT = Uρ + Uθ =

1X 1X KC−C (dri )2 + CC−C (dθj )2 2 i 2 j

(235)

where dri is the elongation of bond i and dθj is the variation of bond angle j (Fig. 17). 72

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

In addition, KC−C is an elastic stiffness to model the force-stretch relationship of the C-C bonds and CC−C is referred to as the effective stiffness associated with the angular distortion of bonds. For a representative atom, there are three chemical bonds r1 , r2 , r3 and three bond angles θ1 , θ2 , θ3 (Fig. 18). The application of external loads will result in three bond elongations dr1 , dr2 , dr3 and three bond angle variations dθ1 , dθ2 , dθ3 . The relationship between external loads and variations in bond lengths and bond angles can be determined by equilibrium and the geometry of the structure. Force equilibrium in bond extension leads to      p1   dr1  p2 dr2 = KC−C (236)     p3 dr3 where p1 , p2 and p3 are the internal forces along the C-C bonds. From the moment equilibrium of the three bonds, we have      M11 M12 M13  dφ1   q1  2C C−C  M21 M22 M23  dφ2 q2 (237) =     aC−C M31 M32 M33 dφ3 q3 in which q1 , q2 and q3 are the internal forces perpendicular to the C-C bonds and the matrix [M] is defined by Mij = −Nik Njk , where Nij =

(

i, j, k = 1, 2, 3 (sum over k)

cos φi sin φk cos ψj −sin φj cos φk sin θj

0

i 6= j 6= k i=j

(238)

(239)

The structural parameters θi , φi and ψi are illustrated in Fig. 18 and can be defined as functions of chiral indices (n, m):   2n + m 4π 2π √ φ1 = arccos , φ2 = + φ1 , φ3 = + φ1 (240) 2 2 3 3 2 n + nm + m π cos φ1 + nm + m2 π π ψ2 = √ cos( + φ1 ) 2 2 3 n + nm + m π π ψ3 = √ cos( − φ1 ) 3 n2 + nm + m2 ψ1 = √

n2

73

(241)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

θi = arccos(sin φj sin φk cos ψi + cos φj cos φk ) i 6= j 6= k

(242)

The axial, circumferential and the shear strains can be calculated by the geometrical deformation of the SWCNT as follows [127]: εxx =

d[(2n + m)r1 cos φ1 − (n − m)r2 cos φ2 − (2m + n)r3 cos φ3 ] √ 3aC−C n2 + mn + m2 εθθ =

εxθ =

d[mr1 sin φ1 − (n + m)r2 sin φ2 + nr3 sin φ3 ] √ √ 3aC−C n2 + mn + m2

d[(2n + m)r1 sin φ1 − (n − m)r2 sin φ2 − (2m + n)r3 sin φ3 ] √ 3aC−C n2 + mn + m2

(243)

(244)

(245)

Equations (243)-(245) can be rewritten in a matrix form as {ε} = where {ε} =



1 [B] {p} + [J] {dφ} KC−C aC−C

εxx εθθ εxθ

T

(246)

and [B] and [J] are given by

1 [B] = √ 2 3 n + mn + m2   (2n √+ m) cos φ1 −(n √ − m) cos φ2 −(2m √ + n) cos φ3 (247) × 3m sin φ1 − 3(n + m) sin φ2 3n sin φ3 (2n + m) sin φ1 −(n − m) sin φ2 −(2m + n) sin φ3 1 [J] = √ 3 n2 + mn + m2   −(2n (n − m) sin φ2 (2m √ + m) sin φ1 √ √ + n) sin φ3  × (248) 3m cos φ1 − 3(n + m) cos φ2 3n cos φ3 (2n + m) cos φ1 −(n − m) cos φ2 −(2m + n) cos φ3

Local equilibrium of the representative atom requires that

(p1 sin φ1 + p2 sin φ2 + p3 sin φ3 ) − (q1 cos φ1 + q2 cos φ2 + q3 cos φ3 ) = 0 (249) 74

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

(p1 cos φ1 + p2 cos φ2 + p3 cos φ3 ) + (q1 sin φ1 + q2 sin φ2 + q3 sin φ3 ) = 0 (250) The compatibility equations for a stress-free SWCNT are expressed as d[mr1 cos φ1 − (n + m)r2 cos φ2 + nr3 cos φ3 ] = 0

(251)

Rewriting Eqs. (249)-(251) in the matrix form, the following relation is obtained [U] {p} + [V] {q} + KC−C aC−C [W] {dφ} = {0} (252) where {0} = {0, 0, 0}T and [U], [V], and [W] are given by   sin φ1 sin φ2 sin φ3 cos φ2 cos φ3  [U] =  cos φ1 m cos φ1 −(n + m) cos φ2 n cos φ3 

 − cos φ1 − cos φ2 − cos φ3 sin φ2 sin φ3  [V] =  sin φ1 0 0 0   0 0 0  0 0 0 [W] =  −m sin φ1 (n + m) sin φ2 −n sin φ3

(253)

(254)

(255)

Substitution of Eqs. (236) and (246) into Eq. (252) leads to {dφ} =

 [U] [B]−1 [J] −

2CC−C [V] [M] − [W] KC−C a2C−C

≡ [F] {ε}

−1

[U] [B]−1 {ε} (256)

The internal forces can be obtained as a function of the strains by substituting Eq. (256) into Eq. (246), i.e., {p} = KC−C aC−C [B]−1 ( [I] − [J] [F] ) {ε} ≡ KC−C aC−C [G] {ε}

(257)

in which [I] is the identity matrix. From Eqs. (236) and (257), it is known that bond elongations can be obtained by {dr} = aC−C [G] {ε} 75

(258)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

In addition, the bond angle variations {dθ} can be expressed by {dθ} = [Q] {dφ}

(259)

Qij = −Nji

(260)

where By substituting Eq. (256), we have {dθ} = [Q] [F] {ε} ≡ [R] {ε}

(261)

The surface density of the strain energy, i.e., the strain energy per unit area on the cylindrical surface of the SWCNT, can be obtained by the strain energy per atom divided by the area of an atom, i.e. CC−C 1 {ε}T [R]T [R] {ε}) Π0 = √ ({ε}T [G]T [G] {ε} + 2 2 K a 3 3 C−C C−C

(262)

The expression for the anisotropic surface elastic constants is obtained as follows ∂ 2 Π0 Yij = (263) ∂εi ∂εj According to Eqs. (262) and (263), the closed-form expression of anisotropic surface elastic constants of an SWCNT is developed as 2 CC−C Yij = √ (KC−C Gli Glj + 2 2 Rli Rlj ) aC−C 3 3

(264)

In particular, for isotropic case, it is readily shown that Y11 = Y22 = Y33 =

Eh , 1 − ν2

Eh , 2(1 + ν)

Y12 = Y21 =

Ehν 1 − ν2

Y13 = Y31 = Y23 = Y32 = 0

(265)

To determine the numerical results, the basic quantities which have to be defined appropriately are the elastic constants associated with the bond stretching and the angular distortion of the bonds. These constants can be determined from the experimental and/or numerical data on the elastic constant of graphene sheets. The values of these force constants are KC−C =742 N/m and CC−C =1.42 nN nm according to [125]. 76

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Figure 19 shows the surface elastic constants as a function of the diameter of the SWCNTs with different chiralities. It is observed that the surface Young's modulus Y11 (=Y22 ) and surface shear modulus (Y33 ) of SWCNTs increase and Y12 (=Y21 ) decreases with an increase in the nantube's diameter, especially for small radii nanotubes. In addition, the surface Young's modulus for zigzag nanotubes is much more sensitive to the nanotube diameter than that for armchair nanotubes. In contrast, the surface shear modulus for armchair nanotubes is much more sensitive to the nanotube diameter than that for zigzag nanotubes. On the other hand, with an increase in nanotube's chiral angle, Y11 (=Y22 ) increases while Y33 and Y12 (=Y21 ) decrease. It is also seen that Y13 (=Y31 = −Y23 = −Y32 ) is indeed very small (two orders of magnitude smaller than others). As can be expected, the chirality- and size-dependence may be ignored when the nanotube diameter is larger than 2.0 nm, and the elastic properties approach the limiting values for graphene sheets. 5.3. Anisotropic mechanical properties of graphene sheets Many investigations have described the mechanical behavior of the graphene sheet employing an isotropic assumption, and so two elastic parameters (e.g. Young's modulus and Poisson's ratio) have been used. Similar to SWCNTs, a large variation of the Young modulus and Poisson's ratio, as well as the effective thickness have been obtained and reported in the literature [126, 152, 172, 173, 174, 175, 176, 177] (see Table 2). A large scatter of the Young modulus values was brought into attention for the first time in [173]. Table 2: Mechanical properties of the graphene sheets

77

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Author Kudin et al. [172] Arroy and Belytschko [173] Wang et al. [126] Huang et al. [152] Reddy et al. [174] Hemmasizadeh et al. [175] Reddy et al. [176] Armchair sheets Zigzag sheets Sakhaee-Pour [177] Armchair sheets Zigzag sheets Chiral sheets

Wall Young's Poisson's thickness h (nm) modulus E (TPa) ratio ν 0.335 1.029 0.149 0.34 0.694 0.142 0.0665 5.07 0.158 0.0874-0.0618 2.69-3.81 0.142 0.0811-0.0574 2.99-4.23 0.397 0.34 0.659-0.682 0.367-0.416 0.34 1.01 343 0.34 0.34

1.095-1.125 1.106-1.201

0.445-0.498 0.442-0.465

0.34 0.34 0.34

1.042 1.040 0.992

1.285 1.441 1.129

Poot and van der Zant [178] have, however, pointed out that a graphene flake is stiffer along the principle direction than the other directions. Furthermore, anisotropic mechanical properties are observed for graphene sheets along different load directions [179, 180, 181]. The anisotropic mechanical properties are attributed to the hexagonal structure of the unit cell of the graphene. Thus, some questions arise as to whether the graphene sheet is isotropic and how much its mechanical properties differ along different directions? On the basis of MD simulations, the anisotropic elastic properties of single-layered graphene sheets with an armchair and a zigzag helicity were predicted by Shen et al. [182]. In their MD simulations, the large-scale atomic/molecular massively parallel simulator (LAMMPS) software [183] was employed and the adaptive intermolecular reactive empirical bond order potential (AIREBO) [184] was used. Two types of single-layered graphene sheets with different values of aspect ratios were considered, i.e., an armchair sheet with aspect ratios AR = 1.97, 1.44 and 1.01, and a zigzag sheet with aspect ratios AR = 1.95, 1.45 and 0.99. Four independent material parameters were needed to completely describe the elastic behavior of the graphene sheets, which are denoted by E1 , E2 , G12 and ν12 , respectively. In order to determine these properties, three MD simulation tests that included 78

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

uniaxial tension, shear load and transverse uniform pressure tests were carried out in the temperature range from 300 K to 700 K. In addition, by directly computing the bending deflections in the MD simulation, the effective thickness of single-layered graphene sheets was uniquely determined. The results including the values of Young's moduli along the principle axes, and the shear modulus and Poisson's ratio are listed in Table 3, from [182] at room temperature. Table 3: Material properties and geometrical parameters of single-layered graphene sheets at room temperature Armchair Armchair Armchair Sheet 1 Sheet 2 Sheet 3 Properties E1 (TPa) E2 (TPa) G12 (TPa) ν12 Geometrical parameters Lx (nm) Ly (nm) h (nm)

Zigzag Sheet 1

Zigzag Sheet 2

Zigzag Sheet 3

2.434 2.473 1.039 0.197

2.154 2.168 0.923 0.202

1.949 1.962 0.846 0.201

2.145 2.097 0.938 0.223

2.067 2.040 0.913 0.204

1.987 1.974 0.857 0.205

9.519 4.844 0.129

6.995 4.847 0.143

4.888 4.855 0.156

9.496 4.877 0.145

7.065 4.887 0.149

4.855 4.888 0.154

5.4. Nonlocal Parameter for CNTs Compared with the strong revival of interest in improving the nonlocal structural models for CNTs, the estimation of the length scale coefficient or the nonlocal parameter, e0 , has not received the full attention it deserves. One of the most important steps in the application of nonlocal elasticity theory to nanoscopic structures is the determination of the nonlocal parameter. Owing to the complexity of the experiments, different approaches including Lattice Dynamics (LD) [43, 185], MD simulations [186, 187] and molecular structural mechanics [188, 189] have been used to provide an estimate of this parameter in various nonlocal models. Initially, Eringen [43] proposed the value to be e0 ≈ 0.39, based on the matching of the dispersion curves via nonlocal theory for plane wave and Born-Karman model of lattice dynamics applied at the Brillouin zone boundary. He also proposed the value e0 ≈ 0.31 in his investigation of Rayleigh 79

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

surface wave using the nonlocal continuum mechanics and lattice dynamics. Based on the MD simulations √ of the SWCNTs, Wang and Hu [190] recommended the value e0 = 1/ 12 ≈ 0.288 for the wave propagation in the nonlocal Euler and Timoshenko beam models. Zhang et al. [191] evaluated the value of 0.82 for this parameter through the comparison of theoretical buckling strain obtained via the nonlocal thin shell model [192] with those obtained by molecular mechanics simulations, given by Sears and Batra [193]. Based on the wave propagation analysis, Wang [194] estimated e0 aC−C < 2.0 nm for frequencies greater than 10 THz. He found that the adopted value depends on the crystal structure in the lattice dynamics. The values of e0 for different chiral angles were predicted [132] by curve fitting the results obtained via the MD simulations and those obtained by the nonlocal Donell shell model for the buckling behavior of SWCNTs. The values obtained varied from a minimum of 0.546 for a (15, 4) chiral shell to a maximum of 1.043 for a (11, 9) chiral shell. To analyze the radial shell buckling of the MWCNTs, and by adopting the average stress in the equilibrium equations, Xie et al. [195] suggested the value of e0 ≈ 0.456. Based on the comparison between the nonlocal Timoshenko beam theory and the MD simulation results of free vibration, Duan et al. [196] found that the values of e0 varied between 0 and 19 depending on the aspect ratio, the boundary conditions and the mode shapes of the nanoscopic structures. Furthermore, they suggested a physical explanation, namely, that e0 allocates part of the kinetic energy to the strain energy. Hu and coworkers [197] evaluated the values of e0 for SWCNTs and DWCNTs by comparing the wave dispersion obtained from nonlocal the Fl¨ ugge shell theory and MD simulation. The estimated value of e0 for transverse wave in a (10,10)@(15,15) DWCNT to be 0.6. Furthermore, the values e0 = 0.2 and 0.23 were predicted for torsional wave in a (10,10) SWCNT and a (15,15) SWCNT, respectively. Based on the MD simulation results, Zhang et al. [198] adopted e0 as 0.3 and 3 in the nonlocal shell and beam models, respectively, to predict the critical strain values of axially compressed SWCNTs. Ansari et al. [199] matched the results of the MD simulations with those of the nonlocal shell model and extracted the nonlocal parameter for vibrations of DWCNTs with different boundary conditions. They concluded that the chirality does not have a considerable influence on the calibrated values of e0 . However, this conclusion was contradicted by further theoretical studies [31, 187, 188]. Utilizing molecular structural mechanics, the explicit expressions for e0 of zigzag and armchair SWCNTs were computed [188]. The value of e0 was respectively found to be 80

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

0.3221 and 0.3458 for zigzag and armchair SWCNTs. In another study, the following relations for e0 of the zigzag and armchair CNTs were obtained on the basis of the bending characteristics of the CNTs [187], i.e., For zigzag CNTs: R + 0.2024 (266) e0 = 0.0297 aC−C and for armchair CNTs: e0 = 0.052

R + 0.1528 aC−C

(267)

The detailed procedure for obtaining these relations can be found in Ref. [187]. Recently, simple analytical expressions of e0 related to geometrical properties and vibration modes were obtained using the Pad´e approximantion and several beam theories [200]. √ It has √ been found that the e0 for the vibration of CNTs ranges from 1/ 6 to1/ 12 depending on the geometry and vibration modes. The first methodical attempt to develop a new formulation for predicting the e0 of the SWCNTs with an arbitrary chirality was presented by Ghavanloo and Fazelzadeh [31]. They concluded that instead of adopting a fixed value for the e0 , its predicted values depend on the nanotube's chirality, the nanotube's diameter, the aspect ratio, and the wave mode shapes. In the next section, we follow the derivation of e0 for SWCNTs with an arbitrary chirality as given in [31]. Based on the above review, it is clear that the magnitudes of the e0 are scattered due to the effects of various factors such as the nanotube's diameter, the chirality, the aspect ratio (length/diameter), and the boundary conditions of the nanotube. Moreover, e0 is different for diverse physical problems and there is no rigorous study made on estimating it so far, with the extension of the previous methods to different cases being quite difficult. Thus, more studies are required to determine e0 more accurately for the SWCNTs. Finally, it should be noted that the research on the prediction of the e0 for different carbon-based nanoscopic structures is still in its primary stages [201, 202, 203] and it is hoped that it will expand in the near future. 5.5. Evaluation of the nonlocal parameter for SWCNTs Let us return to Eq. (65) and rewrite the constitutive relations for an anisotropic SWCNT, i.e.,   2 1 ∂ 2 txx 1 ∗ ∂ txx 2 ∗ ∗ + 2 = (B11 εxx + B12 εθθ + B13 εxθ ) (268) txx − (e0 Li ) 2 2 ∂x R ∂θ h 81

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

2

tθθ − (e0 Li )

2

txθ − (e0 Li )



∂ 2 tθθ 1 ∂ 2 tθθ 2 + − 2 tθθ 2 2 2 ∂x R ∂θ R



∂ 2 txθ 1 ∂ 2 txθ 1 + − 2 txθ ∂x2 R2 ∂θ2 R



=



=

1 ∗ ∗ ∗ εxθ ) εθθ + B23 (B εxx + B22 h 12 (269) 1 ∗ ∗ ∗ (B εxx + B23 εθθ + B33 εxθ ) h 13 (270)

where the constants Bij∗ are given by  ∗   H1 H2 0 H3 0 B11 ∗   H1 −H2  B22 0 H 0 3  ∗    B12   H4 0 0 −H 0 3  ∗ =  B33   H5 0 0 −H 0 3  ∗    B13   0 0 − 21 H2 0 −H3 ∗ B23 0 H3 0 0 − 21 H2



  1    sin 2ϑ      cos 2ϑ      cos 4ϑ   sin 4ϑ

(271)

and all constants Hi (i= 1,2 5) are related to four in-plane elastic constants ∗ ∗ ∗ ∗ (C11 , C12 , C22 and C33 ) as follows: 1 ∗ ∗ ∗ ∗ H1 = (3C11 + 3C22 + 2C12 + 4C33 ) 8

(272)

1 ∗ ∗ − C22 ) H2 = (C11 2

(273)

1 ∗ ∗ ∗ ∗ + C22 − 2C12 − 4C33 ) H3 = (C11 8

(274)

1 ∗ ∗ ∗ ∗ H4 = (C11 + C22 + 6C12 − 4C33 ) 8

(275)

1 ∗ ∗ ∗ ∗ H5 = (C11 + C22 − 2C12 + 4C33 ) 8 Here, the four independent elastic constants are specified by ∗ ∗ C11 = C22 =

∗ C33

Eh 1 − ν2

Ehν 1 − ν2 Eh = 2(1 + ν)

∗ C12 =

82

(276)

(277)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

It should be noted that any chiral SWCNT has a chiral angle difference of less than π/12 with respect to either a zigzag or an armchair SWCNT. Therefore, if ϑ is the rotation angle between a chiral SWCNT and either a zigzag or an armchair SWCNT, then (−π/12 ≤ ϑ ≤ π/12 ), in either the clockwise or the counterclockwise direction. Thus, according this definition and Eq. (3), the rotation angle is defined by   2n + m π −1 √ (278) ϑ = cos − 2 2 12 2 n + mn + m Equations (268)-(270) are rewritten in the matrix form as 1 ∗ [B ] {ε} h where [L∗ ] is a diagonal matrix and its diagonal elements are  2  ∂ 1 ∂2 2 ∗ L11 = 1 − (e0 Li ) + ∂x2 R2 ∂θ2  2  ∂ 1 ∂2 2 2 ∗ L22 = 1 − (e0 Li ) + − ∂x2 R2 ∂θ2 R2  2  ∂ 1 ∂2 1 2 ∗ L33 = 1 − (e0 Li ) + − ∂x2 R2 ∂θ2 R2 [L∗ ] {t} =

(279)

(280)

From a different point of view, and based on Eq. (264), we get the equivalent-continuum constitutive relation for the SWCNT as 1 [Y] {ε} (281) h Equations (279) and (281) are combined and the atomistic-continuum relation is obtained as [L∗ ] [Y] {ε} = [B∗ ] {ε} (282) {t} =

By substituting Eq. (125) into Eq. (282) and simplifying the resulting equation, the following relation is derived, i.e.,  ∗   ∗  ∗ L11 Y11 L∗11 Y12 L∗11 Y13 E11 0 E13 ∗ ∗  E23 ( L∗22 Y12 L∗22 Y22 L∗22 Y23  −  0 E22 ) ∗ ∗ ∗ ∗ ∗ ∗ L33 Y13 L33 Y23 L33 Y33 E31 E32 E33     ∗  ∗ ∗ B11 B12 B13  u   0  ∗ ∗ ∗  B22 B23 v 0 ×  B12 = (283)     ∗ ∗ ∗ B13 B23 B33 w 0 83

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

wherein ∂2 ∂ ∗ , E13 = −z 2 ∂x ∂x  2  1 ∂ 1 1 ∂ ∗ = , E23 = − z 2 − 1 R ∂θ R R ∂θ2 ∂ 1 ∂ z ∂ z ∂ ∗ = = − 2 , E32 + R ∂θ R ∂θ ∂x R ∂x 2z ∂ 2 w =− R ∂x∂θ

∗ E11 = ∗ E22 ∗ E31 ∗ E33

(284)

For a harmonic wave field in the SWCNT, the displacement field can be written in a complex form as      u   U¯  v V¯ exp(jλξ x + jsθ + jωt) (285) =    ¯  W w

¯ are the displacement amplitudes in the x, θ, z direcwhere U¯ , V¯ and W tions, respectively, λξ (λξ = ξπ/L CN T , ξ is the half-axial wave number and LCN T is the length of the SWCNT) is the wavenumber along the longitudinal direction, s is the wavenumber in the circumferential direction and ω is the angular frequency. By substituting the displacement field in Eq.(283) and after making some simplifications, we obtain   ∗ ∗ ∗ ∆1 Y13 − B13 ∆1 Y11 − B11 ∆1 Y12 − B12 ∗ ∗ ∗   ∆2 Y12 − B12 ∆2 Y22 − B22 ∆2 Y23 − B23 ∗ ∗ ∗ ∆3 Y13 − B13 ∆3 Y23 − B23 ∆3 Y33 − B33      U¯ 0 jλξ 0 zλξ 2 1 1 1 2 ¯     V 0  0 js − z (−s − 1) × = (286) R R R2 1 z z 2z ¯ W 0 js jλ + js − jλ λ s ξ ξ R R2 R R ξ

where

  2 ∆1 = 1 + (e0 Li ) λξ 2 +   2 ∆2 = 1 + (e0 Li ) λξ 2 +   2 ∆3 = 1 + (e0 Li ) λξ 2 + 84

 s2 R2  (s2 + 2) R2  2 (s + 1) R2

(287)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

For the non-trivial solution, the determinant of this set of equations must be zero, i.e.,   ∗ ∗ ∗ ∆1 Y11 − B11 ∆1 Y12 − B12 ∆1 Y13 − B13 ∗ ∗ ∗  ∆2 Y22 − B22 ∆2 Y23 − B23 = 0, if λξ 6= 0 (288) det  ∆2 Y12 − B12 ∗ ∗ ∗ ∆3 Y13 − B13 ∆3 Y23 − B23 ∆3 Y33 − B33

θθ There are three eigenvalues for the system which are exx 0 Li , e0 Li and , and these are functions of chirality (n, m) of the SWCNT and the longitudinal and circumferential wavenumbers. In order to evaluate the e0 of the SWCNTs with different geometrical characteristics, the following parameters are used: Eh = 360 TPa nm, ν = 0.16, KC−C =742 N/m and CC−C =1.42 nN nm. Figures 20(a)-(c) demonstrate the predicted e0 of the SWCNTs for the case of first longitudinal mode of wave propagation (ξ = 1, s=0). Similar results are obtained and are shown in Figs. 20(d)-(f) for the case of first coupled longitudinal-torsional mode (ξ = 1, s=1). To elucidate the effect of the aspect ratio (LCN T /2R), the predicted e0 for the first longitudinal mode and the first coupled longitudinaltorsional mode versus the aspect ratio is plotted in Fig. 21 for two armchair SWCNTs. It is seen from Fig. 21(a) that, for the case of first coupled longitudinal torsional mode, the e0 in the longitudinal direction (about 0.43) is not sensitive to the aspect ratio. However, for the case of longitudinal mode, the aspect ratio has an important effect on the longitudinal nonlocal parameter. The longitudinal nonlocal parameter increases linearly as the aspect ratio increases. Furthermore, as shown in Fig. 21(b) and 21(c), the other xθ nonlocal parameters (eθθ 0 and e0 ) are not sensitive to the aspect ratio.

exθ 0 Li

85

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

6. Mechanical characteristics of high aspect ratio CNTs: nanobeams 6.1. Application of static analysis to CNTs The cylindrical geometry of a CNT ensures that it offers a significant resistance to externally applied loads and bending moments. When CNTs are subjected to static loads, the performance of these structures is heavily influenced. Hence, understanding the static behavior of CNTs, including their bending and buckling properties, is of importance for their future applications. Bending is one of the most important modes displayed during structural deformations of CNTs (Fig. 22), and several experimental and numerical investigations have focused on this issue vis-a-vis SWCNTs and MWCNTs. Iijima and coworkers observed the bending of SWCNTs and MWCNTs in the course of high resolution electron microscopy (Fig. 23) [204]. The bending behavior of the CNTs seems fully reversible and no fracture was observed during the load/unload phases [205]. Since CNTs are hollow cylindrical structures with high aspect ratios, they are highly sensitive to buckling or structural instability when subjected to axial, torsional and radial compressive loads (Fig. 24) [206]. Hence, many theoretical and experimental studies have been performed to investigate the buckling behavior and mode shapes of CNTs subject to various types of applied loads [207]. 6.1.1. Structural deformations of CNTs under transverse loadings The first attempt to investigate the static behavior of CNTs employing the nonlocal elasticity theory was carried out by Wang and Liew [105]. The bending deformations of CNTs subject to two types of transverse loads i.e., concentrated and distributed forces, were studied on the basis of the EulerBernoulli beam theory, given in Eq. (85), and the Timoshenko beam theory, given in Eqs. (109) and (110). Three different experimentally varied boundary conditions, such as cantilevered, simply supported and fixed end-fixed end were considered. Figure 25, from [105], displays the variations of non-dimensional deformation, EIw/P Lb 3 , of the cantilever nanotube subjected to a concentrated force applied at the middle of the nanotube with the non-dimensional length, x/Lb at e0 aC−C /Lb = 0.2 and d/Lb = 0.2. The local and nonlocal Euler-Bernoulli beam theories and the local and nonlocal Timoshenko beam theories are denoted by LE, NE, LT, and NT respectively. It is observed that the deformations are not very sensitive to the nonlocal modeling at any locations 86

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

on the left-side of the point force, and that the small scale effect is initiated immediately after the location of the point force. Similar results are displayed in Fig. 26, also taken from [105] for the fixed end-fixed end nanotube subjected to a concentrated force applied at the middle of the nanotube. In contrast to the cantilever nanotube, the small scale effect is significant along the whole of the nanotube and a sudden change in the variation of the deformation appears at the force location. It was also reported that the small scale effect is negligible when the length of the nanotube is above 20 nm. This implies that physically the nonlocal effect would not manifest itself in micro-systems. In summary, the results from this study show that the nonlocal effect is important in modeling the CNTs in their static responses which have important consequences for tribological, electronic and mechanical applications of nanotubes. Based on this conclusion, several studies have been performed in this field to describe the bending of nanobeams and CNTs [103, 104, 208, 209, 210, 211]. 6.1.2. Buckling of CNT In recent years, several studies on buckling of CNTs (both SWCNTs and MWCNTs) have been presented based on the nonlocal elasticity beam theories. The first study, using the nonlocal elasticity theory, was conducted by Sudak [212], predicting the buckling of MWCNTs subjected to an axial compressive load. In that study, the nonlocal Euler-Bernoulli beam model, given in Eq. (85), was employed and it was demonstrated that the smallscale effects significantly contribute to the buckling solution of MWCNTs. An MWCNT is composed of concentric elastic thin cylindrical shells, which interact through vdW forces acting between two adjacent layers. Based on the nonlocal Euler-Bernoulli beam theory, Wang et al. [213] derived the general and explicit solutions to analyze the buckling of CNTs with various boundary conditions as P EB =

1 1 + (e0 a)2



β1 π Lb

2

β1 2 π 2 EI L2b

(289)

where β1 takes on different values for different modes and boundary conditions and a = Li is the C-C bond length. For the first three buckling modes, the values of β1 are 1, 2 and 3 for a simply supported CNT; 0.5, 1.5 and 2.5 for a cantilevered CNT; and β1 = 2, 4 and 6 for a fixed end-fixed end CNT. 87

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

From relation (289), it is easily seen that Eq. (91) is recovered if β1 is set to 1. The small scale effect on buckling loads of CNTs is shown in Fig. 27, taken from [213]. In this way, the buckling load ratio is defined as Load ratio =

Buckling load from nonlocal theory Buckling load from local theory

(290)

In Fig. 27, the load ratio for the first buckling mode of a simply supported CNT is depicted for different nonlocal scale coefficients, namely e0 a = 0.5 nm, e0 a = 1 nm, and e0 a = 2 nm. It can be seen that the small scale effect is more significant for CNTs having a shorter length. The length-dependent scale effect is to be expected since the nonlocal phenomena are definitely more pronounced within short distances in substances. The modified Euler-Bernoulli beam model was developed based on both the stress gradient (nonlocal elasticity) and the strain gradient approaches for predicting the buckling of CNTs [214]. The critical buckling loads predicted by these approaches for a simply supported boundary condition are shown in Fig. 28 [214]. As expected, the two approaches give the same predictions of the load if the nonlocal scale coefficient e0 a is very low (that is e0 a/L < 0.05). It is observed that the stress gradient approach predicts a higher load than the strain gradient approach. Thus, the strain gradient model provides a lower bound on the buckling load of CNTs modeled in the manner described here, and the stress gradient model provides the upper bound on the load for nanotubes. A new type of nanoscopic structure, called the complex-nanobeam-system, has been reported in the literature. One of the most complex-nanobeamsystems is the double-nanobeam-system (DNBS) because of the important role it plays in the field of nano-optomechanical systems (Fig. 29) [215]. The DNBS is a complex system consisting of two nanobeams coupled together by vdW forces, elastic medium, etc. (Fig. 30) [216]. As can be seen in Fig. 30, the coupling mechanism is modeled by a distributed array of vertical parallel springs. Since gaining an insight into the buckling behavior of the DNBS is important in order to ascertain the structural integrity of these nanoscopic systems, different buckling modes of the DNBS, namely the out-of-phase mode (Fig. 31a), the in-phase mode (Fig. 31b), and when one of the nanobeams is fixed (Fig. 31c), have been discussed by Murmu and Adhikari [217] based on the nonlocal Euler-Bernoulli beam theory. In the computational modeling, the properties of the nanobeams were considered for a (5, 5) armchair SWCNT 88

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

with a length 20 nm and radius of 0.34 nm. Furthermore, the Young modulus was taken to be 0.971 TPa, the nanotube density was taken as 2300 kg/m3 and the non-dimensional stiffness parameter of the coupling springs between the SWCNTs was assumed to be K = 30. To show the influence of small-scale effect on the critical buckling load of the coupled-carbon-nanotube-systems, a buckling load reduction percent (BLRP) parameter was introduced as Buckling load from local theory - Buckling load from nonlocal theory ×100 Buckling load from local theory (291) Figure 32, taken from [217], shows the variations of BLRP with the scale coefficient (e0 a/L) for different modes. It can be observed that the buckling load decreases with an increase in the scale coefficient. The reduction in the buckling load is due to the nonlocal effect which reduces the stiffness of the material. As can also be seen from this figure, the scale coefficient significantly reduces the in-phase buckling load (thus higher BLRP) compared to the other cases considered. The robust mechanical properties, high aspect ratio and small diameter of CNTs make them ideal for use as tips in scanning probe microscopes, especially in the atomic force microscope (AFM) (Fig. 33) [218]. Prediction of the buckling load of CNTs is a fundamental design issue in the use of CNTs as AFM probes [219]. To investigate the buckling of CNT as AFM probes, a model of CNTs clamped in an elastic medium (Fig. 34) was proposed and the nonlocal Euler-Bernoulli beam model was used to investigate its behavior [220]. In the numerical simulation, the aspect ratio, the Young modulus and the diameter of SWCNTs, were taken to be 30, 1.0 TPa and 1.0 nm, respectively. As expected, the buckling instability of SWCNTs can occur in different mode shapes. Figure 35, taken from [220], shows the first four buckling mode shapes of SWCNTs with L2 /L1 = 2 and the local condition (e0 = 0). It can be seen from this figure that the nanotubes clamped in an elastic medium undergo a slight deflection compared to the exposed parts. This physically implies that the buckling instability of SWCNTs can increase due to deflection when they are clamped embedded within flexible elastic bodies. As expected, the maximum buckling instability occurs in the first mode with a critical buckling strain of 0.0068. To elucidate the nonlocal effect on the buckling of SWCNTs used as AFM probes, the dispersion curves of the critical buckling stress as a function of BLRP =

89

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

buckling mode are shown in Fig. 36. In the first four buckling modes, the critical buckling stresses decrease with an increase in e0 a from 0 to 2.0 nm. In other words, the buckling stress has an inverse relationship with the nonlocal scale coefficient e0 a. For the fundamental mode 1, the critical buckling stresses of the SWCNT probe are hardly affected by the nonlocal scale coefficient, which are 0.68, 0.67, and 0.63 GPa for e0 a = 0, 1.0, and 2.0 nm, respectively. In summary, it is found that the nonlocal beam theories are useful for the prediction of long and slender nanoscopic structures in various applications such as nanotube-polymer composites and AFM probs. 6.2. Application to dynamic analysis of CNT In this section, we will consider the applications of the nonlocal continuum elasticity to the analysis of dynamics of CNTs. The study of vibration and wave propagation in CNTs is of great interest in the nanotechnology applications owing also to their relevance in the study of electronic and optical properties of CNTs. Furthermore, the vibrational characteristics of CNTs are important in such highly complex and sensitive devices as MEMS, NEMS, high-frequency resonators and sensors. All systems have a set of associated natural frequencies and corresponding mode shapes at which they are prompted to vibrate when exited. Natural frequencies form essential properties of nanoscopic structures and they, in turn, depend on the elastic and geometrical properties. Hence, finding the natural frequencies is a fundamental step to the prediction of the dynamical response of a system. 6.2.1. Linear flexural vibrations of CNTs In 2005, Zhang et al. [191] first reported the application of a nonlocal beam model to study the transverse vibrations of DWCNTs. They found that the inclusion of nonlocality has significant effects on the form and magnitudes of natural frequencies and amplitude ratios. On the basis of the nonlocal Euler-Bernoulli and Timoshenko beam theories, the vibrations of both SWCNTs and DWCNTs were studied by Wang and Varadan [221]. The small-scale effects on the natural frequencies of the simply supported CNTs were explicitly derived and the numerical results were calculated visa-vis the nonlocal scale coefficient e0 a. To illustrate the small-scale effect

90

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

and using Eq. (92), the following frequency ratio was introduced as Natural frequency from nonlocal theory 1 =q 2 2 Natural frequency from local theory 1 + (e0 a)2 iLπ2 (292) The variations of the first three frequency ratios of the SWCNTs at length (L = 10 nm) with the parameter e0 a are plotted in Fig. 37, taken from [221]. It was found that the frequency ratio became smaller when the smallscale effect was taken into account, implying that the classical continuumbased model could overestimate the frequency of SWCNTs. Furthermore, the small-scale effect was more pronounced in higher modes. To elucidate the effect of the nanotube's length, SWCNTs with L = 10, 15 and 20 nm were considered. Figure 38, also taken from [221], depicts the results of this simulation. As can be seen, the lower frequency ratio is observed for shorter CNTs. For example, about 3% difference is seen between the local and nonlocal results for L = 10 nm, whereas only 0.005% difference is observed for L = 20 nm at e0 a ≈ 0.8. These results are in good qualitative agreement with the previous experimental results, reported by Krishnan et al [157]. In summary, it can be safely concluded that the classical continuumbased models are quite applicable and valid, and convenient, for predicting the natural frequencies of long CNTs, especially at lower vibration modes. Thermal vibrations of CNTs are closely linked to their resonance properties and so they are likely to be key issues in their application in highfrequency mechanical resonators [222]. Due to the importance of thermal vibrations, the general equation for transverse vibrations of SWCNTs was formulated on the basis of the nonlocal Timoshenko beam theory and thermal elasticity [109]. In numerical simulation, a simply supported SWCNT was considered and the parameters that were used consisted of the Young modulus (E = 1 TPa), the effective thickness (h = 0.35 nm), the mass density (ρ = 2.3 g cm-3 ) and the nanotube's diameter (d = 0.7 nm). Moreover, the coefficients of thermal expansion for CNTs were assumed to be −1.6 × 10−6 K-1 and 1.1 × 10−6 K-1 for low temperature and high temperature regimes, respectively. To investigate the effect of temperature change, the frequency ratio was given by Frequency ratio =

χth =

Nonlocal natural frequency with temperature change Nonlocal natural frequency without temperature change

(293)

The effects of temperature change on the second mode natural frequencies 91

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

of an SWCNT with the aspect ratio of L/d = 40, and the nonlocal scale coefficient e0 a = 2 nm are shown in Figs. 39(a) and 39(b), taken from [109], for the low and high temperature regimes respectively. As can be seen, in both cases, the thermal effect on the natural frequencies becomes more significant with the increase in the aspect ratio. For the case of low temperature, it is also observed from Fig. 39(a) that the frequency ratios are greater than unity. This means that physically the values of the frequency when the thermal effects are taken into consideration are larger than those when the influence of temperature change is ignored. In the high temperature regime, in contrast to the low temperature, the values of frequency, when considering the thermal effect, are smaller than those when the influence of temperature change is excluded. In another study [223], the transverse vibration of an elastically-connected carbon nanotube system carrying a moving nanoparticle was studied on the basis of the nonlocal Euler-Bernoulli beam theory. The two nanotubes were assumed to have same properties and connected to each other continuously by elastic springs. It was shown that the dynamic behavior of the doublecarbon nanotube system was really affected by the small scale parameter, the velocity of the nanoparticle and the stiffness of the elastic layer. Due to the magnetic properties of the CNTs and their mechanical behavior in a magnetic field, the investigation on their dynamic characteristics under an applied magnetic field has also attracted a considerable attention within the scientific and engineering communities [224]. Murmu et al. [225] developed an analytical approach, based on the nonlocal Euler-Bernoulli beam theory, to study the effect of an applied longitudinal magnetic field on the transverse vibration of a DWCNT. In this way, the effect of the longitudinal magnetic field was modeled by the Lorentz magnetic force obtained from Maxwell's equation. Furthermore, while the nonlocal effects were considered in the longitudinal direction, they were ignored in the circumferential direction of the CNTs. Both the same phase (SP) and anti-phase (AP) vibration modes of the nanotubes of DWCNTs were reported. Their results showed that the applied longitudinal magnetic field increased the natural frequency of the DWCNT (Fig. 40) [225]. This behavior was considered to be a direct evidence for the coupling of vibrating nanotubes and the longitudinal magnetic field. In another nonlocal modeling [226], the resonance frequencies and stability of a fixed end-fixed end CNT carrying an electric current, which was subjected to an applied longitudinal magnetic field and an axial compres92

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

sive force due to thermal stress, were investigated on the basis of a three dimensional nonlocal beam model (Fig. 41). The most relevant results from the numerical simulations are shown in Fig. 42, taken from [226]. Fig. 42a and 42b depict the variations of the dimensionless fundamental natural frequency and the compressive axial force with the dimensionless magnetic field respectively for several values of the small scale parameter. It was shown that the fundamental resonance frequency of the CNT decreased when the magnetic field and the compressive axial force were increased. This meant physically that both the magnetic field and the compressive force reduce the stiffness of the CNT. Furthermore, the area under each curve, which represents the stable region, diminishes when the small scale effect is considerable. 6.2.2. Linear longitudinal vibrations of nanotube heterojunctions The appearance of topological defects in the hexagonal lattice network of CNTs can lead to a change of their chirality. Two CNTs with different chiralities can be connected to form a nanotube heterojunction simply by the introduction of a heptagon and pentagon pair [227, 228] (see Fig. 43 taken from [228]). For example, one metallic SWCNT can be connected to a semiconducting SWCNT, displaying a range of interesting functions, different from the constituent SWCNTs. Thus, nanotube heterojunctions may provide new design opportunities. The application of the nonlocal continuum-based model for predicting the longitudinal vibration of nanotube heterojunctions was considered by Filiz and Aydogdu [229]. They presented general equations of the longitudinal motion of n-segmented carbon nanotubes and then calculated the natural frequencies of CNTs with two segments. In numerical simulations, the fixed end-fixed end and fixed end-free end boundary conditions were considered. They found that the longitudinal vibration frequencies of CNT heterojunctions were highly over estimated within the classical model due to neglecting the effect of small length scale. 6.2.3. Nonlinear vibrations of CNTs Several investigations [230, 231, 232, 233] have shown that the deformation of CNTs is a nonlinear phenomneon when they are subjected to large external loads and that their vibration amplitude exceeds that associated with small perturbations. Generally, the nonlinear behavior has two typical characteristics: 1) the response of a nonlinear system is not linearly related 93

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

to the input force or displacement and 2) nonlinear systems do not satisfy the principle of superposition. A nonlocal Timoshenko beam model was reformulated by using von Karman geometric nonlinearity to study the nonlinear free vibration of SWCNTs [234]. It was reported that the nonlocal parameter had a significant effect on the linear and nonlinear frequencies while the nonlinear mode shapes were not sensitive to nonlocality. Furthermore, as expected, the linear and nonlinear frequencies decreased with an increase in the nonlocal parameter. Similar results have also been reported elsewhere [235]. In recent years, the use of CNTs as novel atomic transport channels, has received some attention [236, 237]. Transportation of atoms inside the SWCNTs would induce longitudinal and flexural vibrations. Since the efficiency of the atomic transportation is dependent on the interaction forces between the moving atoms and the inner surface of the CNT, understanding of the nonlinear vibrations of an SWCNT for the translocation of atoms is an important task. Kiani [238] studied the nonlinear vibrations of SWCNTs as nanoparticle delivery nanodevices employing the nonlocal Rayleigh beam theory. The vdW forces between the moving nanoparticle and the SWCNT atoms and the frictional force were taken into account in the proposed nonlocal beam model. In that work, it was shown that the maximum longitudinal displacement, as well as the nonlocal axial force within the SWCNT, would increase as the vdW force between the moving nanoparticle and the nanotube increased. It was also presented that the nonlocal continuum-based modeling was an efficient technique for investigating this problem. 6.2.4. Wave propagation in CNTs There have been theoretical attempts to understand wave propagation in CNTs through the nonlocal beam theories. Wang [194] first suggested the nonlocal Euler-Bernoulli and Timoshenko beam models in order to predict the wave propagation in CNTs. This study revealed the significance of the nonlocal effect and the limitation of the applicability of the classical continuum-based models in the prediction of wave propagation in CNTs. From the numerical simulations, small scale effect was clearly observed on CNTs wave characteristics at lower wavelength and smaller diameters. This finding was qualitatively in agreement with the published experimental report [157]. Wang and Hu [190] studied the flexural wave propagation in armchair (5, 5) and (10, 10) SWCNTs by using the nonlocal Euler-Bernoulli and Timoshenko beam models and MD simulations based on the Terroff94

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Brenner potential. Their results indicated that the nonlocal Timoshenko beam model was able to offer a much better prediction than the classical and nonlocal Euler beam model for the flexural wave dispersion modeled by using MD simulations (see Fig. 44). Based on the comparison between the results of the nonlocal Timoshenko beam model and MD simulations, they proposed the value of e0 = 0.288 for a wide range of wavelengths. In another work [239], the consistent equations of motion for the nonlocal Euler-Bernoulli and Timoshenko beam models were derived and some issues on the nonlocal beam theories were discussed. The Timoshenko beam model was utilized to predict the wave propagation in SWCNTs and DWCNTs. Narendar and Gopalakrishnan [240] examined the nonlocal effect on the wave propagation in MWCNTs based on the Timoshenko beam model. They reported that the nonlocal parameter introduced certain band gap region in both flexural and shear wave modes of SWCNTs and MWCNTs when the wavenumber tended to infinity and the group speed tended to zero (see Fig. 45). The frequency at which this phenomenon occurred was called the escape frequency and beyond this frequency there was no wave propagation. Furthermore, it was seen that the escape frequencies of the flexural and shear wave modes of MWCNTs were inversely proportional to the nonlocal scale coefficient. Yang et al. [241] established a new analytical nonlocal Timoshenko beam model for the analysis of the wave propagation in DWCNTs. Comparison with the previous publications showed that their proposed model lead to an increased stiffness effect. Recently, the longitudinal wave propagation in MWCNTs was investigated using the nonlocal elasticity theory [242]. The effect of the vdW force and the possibility of relative displacement between layers were considered.

95

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

7. Mechanical characteristics of low aspect ratio CNTs: cylindrical nanoshells For low aspect ratio CNTs, both global and local deformations are present. Hence, the CNT should be considered as an elastic cylindrical shell rather than an elastic beam [243]. Several elastic shell models have been effectively used to study the mechanical characteristics of the CNTs especially during bending [244], buckling under compression loads [121, 245, 246], vibrations [14, 120, 247] and wave propagations [248, 249]. In this section, a comprehensive analysis of the mechanical characteristics of the CNTs is provided on the basis of the nonlocal continuum-based shell theories discussed in Section 4.3. 7.1. Shell-based approach to CNT buckling Under compressive and torsional or bending stresses, a CNT suddenly undergoes a large deformation and it is said to have buckled. It should be noted that compressive stresses may result in both global, or beam, buckling (see Section 6.2.1) and local, or shell, buckling within the CNT. In the case of shell buckling, the CNT deforms in a wavy fashion to lower the compressive energy, while the axis of CNT remains straight just as in a thin-shell buckling case (Fig. 46 taken from [250]). Usually CNTs with small aspect ratios undergo shell buckling, whereas those with large aspect ratios undergo beam buckling [206]. The buckling behavior of the SWCNTs with a large aspect ratio (L/d > 10) can be predicted by the local Euler-Bernoulli beam model, and the nonlocal Euler-Bernoulli beam model does not provide a better model for predicting the buckling in these types of SWCNTs [198]. However, when SWCNTs have moderate aspect ratios (8 < L/d < 10), it is necessary to apply the nonlocal beam theories for a better prediction of the buckling behavior. For short SWCNTs with large diameters (L/d < 8), it has also been shown that the nonlocal shell model with an appropriate nonlocal parameter could predict the critical buckling strains in good agreement with the results obtained from MD simulations. The above discussion is summarized in Fig. 47. The main results reported in the literature concerning the study of the shell buckling of CNTs based on the nonlocal shell model are reviewed in the next section. 7.1.1. Axial buckling The first study of nonlocal shell buckling of CNTs under axial loading was reported in 2006 by Wang et al. [213]. They modeled the CNT as an 96

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

isotropic cylindrical shell and used the Fl¨ ugge shell theory. It was concluded that the nonlocal effect leads to a reduction in the critical buckling strains. Buckling and postbuckling analyses of axially compressed DWCNTs have also been studied [251]. Since CNTs have low values of radius-to-thickness ratio, in that study the DWCNTs were modeled as nonlocal shear deformable cylindrical shells. Furthermore, MD simulations were carried out to obtain the size-dependent and temperature-dependent material properties of CNTs, and to estimate the nonlocal scale coefficient by matching the buckling loads of CNTs obtained from the MD simulations and that obtained from the nonlocal shear deformable shell model. Table 4, taken from [251], lists a set of selected values of the buckling loads and the corresponding critical strains for a (9,9) SWCNT and a (9,9)@(14,14) DWCNT with L=5.32 nm at room temperature, calculated by MD simulation and the nonlocal shell model. The last column of this table is the calibrated value of the nonlocal scale coefficient e0 a. As can be seen, by using the calibrated values, the buckling loads and the corresponding critical strains obtained from the nonlocal shear deformable shell model match fairly closely the results from the MD simulations. Table 4: Computed buckling loads and the corresponding critical strains for a (9,9) SWCNT and a (9,9)@(14,14) DWCNT CNTs

MD simulation

(9,9)

0.0528 79.430 0.0499 113.994

Critical strain Buckling load (nN) (9,9)@(14,14) Critical strain Buckling load (nN)

Nonlocal shell model 0.0543 79.427 0.0502 113.999

e0 a (nm) 0.0495 0.0198

Ansari and coworkers [114, 252] performed a set of nonlocal Donnell shell modeling and MD simulations of the buckling behavior of SWCNTs subjected to axial compressive loads. Initially the authors developed a nonlocal shell model on the basis of the first order shear deformation theory and thermal elasticity. Then, the MD simulator “NanoHive” [253] was utilized to predict the buckling behavior of SWCNTs for different boundary conditions. In these works, the SWCNTs were subject to four boundary conditions, namely the simply supported-simply supported, the fixed end-fixed end, the fixed end-simply supported and the fixed end-free end, were considered. In MD 97

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

simulations, to simulate the simply supported and fixed end boundary conditions, one layer and four layers of carbon atoms at the end of SWCNTs were spatially fixed respectively. The schematic representation of the four sets of boundary conditions is shown in Fig. 48a, taken from [252]. The results from the simulations are displayed in Figs. 48b-48e. To find the appropriate value of e0 a, the results obtained from MD simulations were matched with those from the nonlocal shell model through a nonlinear least square fitting procedure. It was observed that the nonlocal continuum-based shell model with its proper values of e0 a was capable of accurately predicting the critical axial buckling loads of the SWCNTs. A further study later published by the same group [254] confirmed these findings on the basis of a new nonlocal atomistic-based shell theory. Their model had a unique feature because it was independent of the widely scattered values of Young's modulus and the effective thickness of SWCNTs. The authors presented the three dimensional buckling mode shapes of an SWCNT (see Fig. 49, taken from [254]). Summing up the results from these simulations, it was found that the nonlocal continuum-based shell models were better than their local counterparts for predicting the buckling behavior of CNTs with low values of aspect ratio. 7.1.2. Torsional buckling Following the observation of twisted SWCNTs by STM [255], it is known that SWCNTs are prone to the distortion of their sp2 -bond structure under a torsional load. It is also proved that the electronic properties of SWCNTs are affected by twisting deformations [256, 257]. Hence, several research groups were able to use SWCNTs and MWCNTs as torsional spring elements in the design of electromechanical switches, and paddle oscillators [258, 259, 260, 261]. The understanding of a CNT's torsional properties, especially its torsional buckling load, is crucial for the design of new nanodevices. Due to this fact, several investigations were carried out to study the torsional buckling behavior of CNTs on the basis of the nonlocal shell modes. The torsional response of SWCNTs was investigated by developing a nonlocal continuum shell model and also by MD simulations [113]. In this connection, Eringen's constitutive relations were incorporated into the Timoshenko and the Donnell shell models. The nonlocal parameter and the effective wall thickness of the SWCNTs were also obtained from MD simulations through a nonlinear optimization process. In brief, from simulations it was found that 98

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

the buckling mode-shape obtained from the nonlocal continuum and MD models agreed closely (see Fig. 50, taken from [113]). From a quantitative point of view, the buckling wave number calculated from the nonlocal shell model was 3.7 while the number of longitudinal waves in the buckled shape of SWCNT from MD simulations was 3.6. On the basis of minimizing the Euclidean norm of the difference between the buckling torques obtained from the MD simulations and those from the nonlocal elasticity shell model, the effective wall thickness of an armchair SWCNT and a zigzag SWCNT was determined to be 0.85 ˚ A .Furthermore, the values of e0 = 0.8 and 0.6 were recommended for the armchair and zigzag nanotubes respectively. It was also shown that the nonlocal continuum shell models provided a much better match with the MD results than the classical models, over a range of diameters (Fig. 51). In summary, it was found that the size-effects in torsional buckling of SWCNTs becomes significant when the SWCNT diameter is small, i.e., for diameters smaller than 1.5 nm there is more than 15 % difference between the results from the classical shell model and the MD simulation. In another combined MD simulation and the nonlocal continuum-based modeling study [112], the buckling and postbuckling of DWCNTs subjected to an external torque were determined at temperatures T = 300, 700 and 1200 K. In the continuum-based modeling part, the higher order shear deformation shell theory with a von Karman-Donnell-type of kinematic nonlinearity and Eringen's nonlocal theory, as given in Eqs. (150)-(154), was utilized. Furthermore, the MD simulations employed the velocity-Verlet algorithm [262] to integrate the equations of motion, and the Brenner “second generation” REBO potential [263] was used to describe the short-range covalent C-C interactions and the Lennard-Jones 12-6 potential [264] to model the long-range vdW interaction. Table 5, taken from [112], lists a set of selected values of the buckling torque for a (9,9) SWCNT and a (9,9)@(14,14) DWCNT with L=5.32 and 8.29 nm respectively at room temperature, calculated by MD simulation and the nonlocal shell model. Table 5: Computed the buckling torque for a (9,9) SWCNT and a (9,9)@(14,14) DWCNT

99

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

CNTs L = 5.32 nm (9,9) (9,9)@(14,14) L = 8.29 nm (9,9) (9,9)@(14,14)

MD simulation Nonlocal shell model

e0 a (nm)

18.33 59.85

18.36 59.85

0.554 0.495

14.66 43.56

14.66 43.61

0.761 0.717

The postbuckling torque-rotation curves for a (9, 9)@(14,14) DWCNT are shown in Fig. 52, taken from [112]. In this figure, the results from MD simulations and the nonlocal shell model at room temperature are compared and it is seen that the two approaches are in a reasonable agreement when e0 a = 0.495 nm. The small scale effect on the torsional buckling of MWCNTs coupled with a temperature change was investigated by Hao et al. [265] on the basis of the Donnell shell model. The cylindrical layers of the carbon were assumed to be concentrically nested, and were coupled through vdW forces between two adjacent layers. The results of the simulation are illustrated in Fig. 53, from [265]. These results show that the buckling loads as predicted by the classical shell model is larger than those predicted by the nonlocal shell model. It can also be seen that the buckling shear forces are related to the axial wavenumber, ma , and the circumferential wave-number na , where each pair (ma , na ) corresponds to only one minimum buckling force. 7.2. Vibrational properties of CNTs 7.2.1. The shell-like vibrations of CNTs Arash and Ansari [266] developed a nonlocal shell model, on the basis of the Donnell shell theory and isotropic assumption, to analyze the vibrational characteristics of SWCNTs under different boundary conditions. They also evaluated the nonlocal scale coefficient for a wide range of aspect ratios via MD simulations. The computed values for e0 a associated with SWCNTs for fixed end-fixed end and fixed end-free end boundary conditions were found to be 1.7 and 2 nm, respectively. Furthermore, to calibrate the values of e0 a for armchair and zigzag DWCNTs, Ansari and coworkers [199] investigated the free vibration of a (5,5)@(10,10) DWCNT by the nonlocal Donnell shell theory and MD simulations. The computed frequencies by the nonlocal shell model and MD simulations for different boundary conditions are displayed 100

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

in Figs. 54a-54c, taken from [199]. It can be observed that the results from the nonlocal shell model and MD simulations agree quite well. Wang et al. [186] used the nonlocal Timoshenko beam and the nonlocal Fl¨ ugge shell theories to study the transverse vibrations of slender SWCNTs. A (5,5) SWCNT with fixed end boundary condition was considered. The effective data that were used consisted of the bending stiffness (D = 2 eV), the in-plane stiffness (K = 360 J/m2 ), the mass density per unit lateral area (ρh = (2.27 g/cm3 )×0.34nm) and the nonlocal scale coefficient (e0 a = 0.21 nm). These data correspond to the effective thickness h = 0.1 nm and the equivalent Young's modulus E = 3.5 TPa adopted for SWCNTs modeled as Timoshenko beams. The local and nonlocal beam-like vibration frequencies (n = 1) predicted by the shell models are displayed in Fig. 55, taken from [186], compared to the results from the local and nonlocal beam models and MD simulation results given by Ref. [196]. The results from the nonlocal shell model have been shown to be in good agreement with the MD simulation results. Physically this may be due to the circumferential nonlocal effect arising from the atom-atom interaction in the circumferential direction. Furthermore, a comparison of the results from the nonlocal beam and shell models shows that different physical explanations are required for the way the nonlocal effect influences the transverse vibrations of SWCNTs. Consequently, the discrepancy between these models cannot be resolved by just choosing different values of e0 . Das et al. [267] derived the frequency expressions for inextensional vibration modes of zigzag SWCNTs using two nonlocal shell theories, namely, the stress and the strain gradient shell theories, and molecular mechanics simulations employing the MM3 potential. Two types of the inextensional Rayleigh and Love modes for a (15, 15) SWCNT are illustrated in Fig. 56, taken from [268]. In cases of the inextensional deformations, the midsurface of the SWCNT deforms without stretching. By comparing the results from the nonlocal shell models with the results from molecular mechanics simulations, the value of the nonlocal parameter, e0 , was also computed. For (q,0) √ SWCNT, the value of e0 was found to be 0.45 for an even q value, and 2q/π(q + 1) for an odd q value. Since CNTs are recently used as reinforcements in new composites [269, 270], understanding the effect of the surrounding elastic medium on their dynamic behavior is important. Motivated by this idea, Mikhasev [271] studied the axisymmetric vibrations of a long SWCNT embedded in a nonhomogeneous elastic matrix using the nonlocal Fl¨ ugge cylindrical shell theory. It was 101

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

shown that including the nanoscale effect into the continuum model allowed new eigenmodes, localized in a vicinity of the weakest curve which is assumed to be far from the nanotube edges, to be revealed. 7.2.2. Phonon dispersion of SWCNTs The carbon atoms in an SWCNT are constantly vibrating about their mean positions. These vibrations are known as phonons which are particlewaves and propagate through the SWCNTs. Generally, the phonons are characterized by the frequency-wavevector dispersion. The phonon dispersion in SWCNTs is complicated and mathematically involved [52]. Unfortunately, the full phonon dispersion of SWCNTs cannot be experimentally determined. The phonon dispersion of SWCNTs can be obtained within a force-constant model [61]. The calculated phonon dispersion of a (10,10) SWCNT using the force-constant model and the continuum shell is shown in Fig. 57. Recently, it was shown that the phonon dispersion of SWCNTs could be calculated by the continuum-based models [14, 247, 272, 273]. The first calculation of the phonon dispersion of SWCNTs by using the nonlocal shell model was reported in 2012 by Fazelzadeh and Ghavanloo [115]. They developed a nonlocal anisotropic elastic shell model on the basis of the Fl¨ ugge shell theory, discussed in section 4.3.1, the nonlocal constitutive relations and the molecular mechanics formulation, given in section 5.2. Figure 58, taken from [115], shows the effect of the nonlocal scale coefficient, e0 a, on the axisymmetric phonon-dispersion of an armchair (15,15) SWCNT. The longitudinal, radial and torsional modes are denoted by L, R and T respectively. The results from Fig. 55 show that the frequency of the T-mode is decoupled from the longitudinal and radial modes. Furthermore, the L-mode (i.e., L1 and L2) and the R-mode (i.e., R1 and R2) are strongly coupled with each other in a transition zone of the two modes. It can also be seen that the effect of the nonlocal scale coefficient on the frequencies of the SWCNT is dependent on the axial wavelength. This effect increases with decreasing axial wavelength (increasing λq ). Summing up the results from this simulation, it is seen that the nonlocal anisotropic shell theory predicts smaller frequency values as compared with the local continuum results and that the frequencies decrease when the nonlocal scale coefficient increases. Actually, the nonlocal theory provides a more flexible model. Furthermore, the results show that the effect of the nonlocal parameter on the frequency is significant for SWCNTs with lower aspect ratios and lower diameters. 102

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

7.3. Wave propagation in CNTs Application of the nonlocal elastic shell theory to the study of wave propagation in CNTs was explored by Wang and Varadan [111]. Different wavenumbers along both the longitudinal and circumferential directions were considered in both theoretical analyses and numerical simulations. It was found that the proposed nonlocal the Fl¨ ugge shell theory was indispensable for predicting the CNT phonon dispersion relations. However, they could not obtain the wave properties of the SWCNT at all frequencies and also no MD simulations were available for the validation of their shell models. These shortcomings were improved by Hu et al. [197]. They used the same nonlocal shell model and provided its verification by using MD simulations. The simulations were performed for a (15,15) armchair SWCNT, a (20,0) zigzag SWCNT, and a (10,10)@(15,15) DWCNT. The results of their calculations of the transverse wave are illustrated in Fig. 59, taken from [197]. The results show that the nonlocal shell model could predict the MD results better than the local shell model. The value e0 = 0.6 was used for the transverse wave for all these CNTs. The good agreement between the results from the nonlocal elastic cylindrical shell model and the MD simulations verified the effectiveness of the nonlocal shell model. Selim [274] studied the propagation of dilatation wave in SWCNTs on the basis of the nonlocal Fl¨ ugge shell theory. The dispersion characteristics were discussed by the use of group velocity, phase velocity and their ratio. The results revealed that an increase in the nonlocal parameter leads to a reduction in both velocities. Similar phenomena are also reported in [111, 275, 276]. The unexpected results were calculated by Yang and Lim [124] when proposing a new nonlocal cylindrical shell model for axisymmetric wave propagation in CNTs. They have shown that the stiffness of the SWCNTs increased, rather than decreased, and so the phase velocity of the wave in the SWCNTs was enhanced when the small scale parameter was increased. The effect of the nonlocal parameter on the coupled wave propagation in SWCNTs was studied by Narendar and Gopalakrishnan [277] on the basis of the first order shear deformation theory. They indicated that there were four modes of wave propagation, namely, axial, flexural, shear and contractional (see Fig. 60 taken from [277]). It was reported that all the wave modes tended to infinity at an escape frequency (see section 6.2.4). It should be noted that the wavenumber before the escape frequency is real and after that it is purely imaginary. The variations of escape frequencies of the flexural and the shear wave modes with e0 a are shown in Fig. 61, taken from [277]. 103

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

As expected, the escape frequency decreases with the increase in e0 a, for all the wave modes. Khademolhosseini et al. [278] have studied the torsional vibration and wave propagation in SWCNTs. Both the nonlocal cylindrical shell model and MD simulation were employed. The natural frequency was used to find the group and phase velocities of torsional waves propagating in SWCNTs. Through the comparison of the results of MD simulations and the nonlocal dispersion relations, it was found that the nonlocal parameter, e0 , was around 0.18 for the torsional waves propagating in (6,6) and (10,10) nanotubes. Using this value, the nonlocal shell model could predict a reliable result with a maximum error of 0.5% (see Fig. 62, taken from [278]). In summary, the superiority of the nonlocal shell model over the classical one in predicting the real dispersion behavior of torsional waves in SWCNTs has been established.

104

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

8. Application of nonlocal continuum theory to modeling graphene sheets The mechanical characteristics of graphene sheets are determined by the nature of their carbon bonds. Each carbon atom has four free electrons which can form bonds with other atoms. In the formation process of the graphene sheet, each carbon atom forms strong planar σ-bonds with its three nearest neighbors. In graphite, this plane is rather very weakly bonded to other planes via vertical π-bonds [52]. Although discovery of the graphene sheet has a very short history, it is very significant that the mechanical characteristics of single and multi-layered graphene sheets have been studied and reported by an unusually large number of researchers employing the nonlocal continuum theory. In this section, a selective review of the pertinent studies on the mechanical behavior of the graphene sheets is presented. 8.1. Buckling of graphene sheets Under compressive loading, the in-plane deformation of the graphene sheet is very small before structural instability or buckling sets in (Fig. 63) [279, 280]. Different types of buckling are possible and are observed in graphene sheets, due to their relatively small bending rigidity, see Fig. 64 taken from [281, 282]. The buckling analysis of single-layered graphene sheets under uniaxial and biaxial compressions was studied by Pradhan and coworkers [283, 284, 285, 286] using the nonlocal continuum mechanics. Numerical results showed that the buckling load obtained from the nonlocal models was smaller compared to the predictions of the local elasticity theory. This reduction was also reported in the nonlocal modeling of the buckling of circular and quadrilateral graphene sheets [287, 288]. In a combined MD simulation and nonlocal continuum-based modeling study [289], the biaxial buckling characteristics of single-layered graphene sheets were investigated. In the continuum-based modeling part, different types of the nonlocal plate theory, namely the Kirchhoff plate theory, the Mindlin plate theory, and the higher order shear deformation theory, were utilized. Furthermore, quasi-static MD simulations, wherein the deformation is independent of the strain rate, were performed on the armchair and zigzag single-layered graphene sheets. The buckling mode shape of simplysupported graphene sheet under biaxial compressive strain is shown in Fig. 65, taken from [289]. 105

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

To evaluate the correct values of e0 a corresponding to each type of the nonlocal plate model, the results of MD simulations were fitted with those computed by the nonlocal elasticity plate models through a least-square fitting procedure. The estimated values for e0 a corresponding to different types of the nonlocal plate model and chirality are listed in Table 6. It can be seen that the chirality does not play an important role in the biaxial buckling of single-layered graphene sheets. Table 6: Computed values for e0 a corresponding to different types of the nonlocal plate model and chirality Nonlocal plate model e0 a (nm) Armchair single-layered graphene sheets Classical or Kirchhoff plate theory 1.85 First order shear deformation or Mindlin plate theory 1.81 Higher order shear deformation theory 1.78 Zigzag single-layered graphene sheets Classical or Kirchhoff plate theory 1.85 First order shear deformation or Mindlin plate theory 1.81 Higher order shear deformation theory 1.78 Summing up the results from this computation, one very interesting conclusion emerges, namely the nonlocal plate models with their appropriate values of the nonlocal parameter have an excellent potential to predict the biaxial buckling of single-layered graphene sheets. In the aforementioned studies, it was assumed that the material properties of the graphene sheet have no uncertainties whereas it is well-known that the material properties may have some degree of ambiguity associated with them. Recently, the biaxial buckling of a rectangular orthotropic graphene sheet with the material properties displaying uncertain-but-bounded variations around their nominal values was investigated [290] on the basis of the nonlocal Kirchhoff plate theory. To include the uncertainty in material properties, the convex model [291] was employed. It was found that the effect of this uncertainty played a significant role in estimating the buckling of single-layered graphene sheets. 8.2. Vibration of graphene sheets 8.2.1. Single-layered graphene sheets On the basis of the nonlocal constitutive relations of Eringen, the authors Pradhan and kumar [292, 293] reformulated the classical plate theory 106

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

and the first order shear deformation theory to study the free vibrations of single-layered graphene sheets. However, no results from MD simulations were available to validate their plate models. This weakness was addressed by Ansari et al. [202]. They used the nonlocal Mindlin plate theory, discussed in Section 4.4.2, and verified it by using MD simulations. In Table 7, we report some computed values from the nonlocal model and MD simulations. Comparison of the results indicates that there is an excellent agreement between the two sets of numerical solutions. Another similar table of data can be found in [202]. Table 7: Comparison between the fundamental frequencies (THz) of the nonlocal model and those computed from MD simulation for armchair graphene sheets Length of square graphene sheet (nm) 10 15 20 25 30 35 40 45 50

MD simulations 0.0595014 0.0277928 0.0158141 0.0099975 0.0070712 0.0052993 0.0041017 0.0032614 0.0026197

Nonlocal plate model (e0 a = 1.34 nm) 0.0592359 0.0284945 0.0165309 0.0107393 0.0075201 0.0055531 0.0042657 0.0033782 0.0027408

In a further combined MD simulations and the nonlocal-based plate modeling [294], the linear and nonlinear vibrations of single-layered graphene sheets in thermal environments were investigated. The single-layered graphene sheet was modeled as an orthotropic thin plate with a von Karman-type of kinematic nonlinearity. Furthermore, the MD simulations were performed with the LAMMPS software. For numerical analysis, six types of armchair and zigzag single-layered graphene sheets were considered. The material properties of these graphene sheets are listed in Table 3. The natural frequencies for these graphene sheets at room temperature are listed in Table 8, taken from [294]. The results show that the nonlocal plate model with properly selected nonlocal scale coefficients and material properties can provide a remarkably accurate prediction of the graphene sheet behavior. 107

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Table 8: Computed natural frequency (GHz) of linear vibration for six types of graphene sheets Graphene sheet Armchair Sheet 1 Armchair Sheet 2 Armchair Sheet 3 Zigzag Sheet 1 Zigzag Sheet 2 Zigzag Sheet 3

MD simulation 57.2 76.3 114.4 66.8 81.1 114.4

Nonlocal plate model e0 a (nm) 57.0 0.67 76.4 0.47 114.4 0.27 66.3 0.47 81.2 0.32 114.2 0.22

In spite of the extensive use of the nonlocal plate models, the physical implication and some fundamental issues of the nonlocal theory have not been clarified and discussed. Wang et al. [295] explained the aforementioned issues for the transverse vibration of graphene sheets. They showed that the nonlocal interactions between the atoms induce a surface compression. The surface compression and curvature change result in a distributed transverse load on the graphene sheets, which reduces their equivalent structural rigidity and downshifts their vibration frequency. A similar explanation has been pointed to by Eringen [43]. In most of previous studies, the graphene sheets were modeled as elastic plates notwithstanding the fact that they display viscoelastic structural damping similar to many materials. Recently, viscoelastic properties of a graphene oxide sheet were reported experimentally by Su et al. [296]. The vibration analysis of the viscoelastic orthotropic graphene sheets was investigated on the basis of the nonlocal Kirchhoff plate theory and the KelvinVoigt model [297]. The results revealed that the frequency was significantly influenced by the structural damping of the graphene sheet, the damping coefficient of the viscoelastic foundation and the nonlocal parameter. Also, it was found that the frequency increased when the values of the structural and external damping was decreased, see Fig. 66, taken from [297]. 8.2.2. Multi-layered graphene sheets It is known that the bending stiffness of the single-layered graphene sheet is low. Consequently, to increase it, multi-layered sheets can be considered. The multi-layered graphene sheets are multiple parallel sheets which are coupled to each other with the vdW interatomic forces acting between adjacent layers. These vdW forces induce considerable changes in the physical properties of the multi-layered graphene sheets. 108

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Pradhan and Phadikar [298] employed the nonlocal plate theory for the vibration analysis of multi-layered graphene sheets having orthotropic properties and embedded in a polymer matrix. Navier solutions were obtained for the natural frequencies for simply supported boundary conditions and the effects of the nonlocal parameter, length and elastic modulus were also examined. They showed that the nonlocal effect is quite important and needs to be included in the modeling of the multi-layered graphene sheets. Ansari and coworkers [299, 300, 301] extended the work of Pradhan and Phadikar [298] for different boundary conditions employing the nonlocal Mindlin plate theory. For numerical calculations, they used the finite element method and the generalized differential quadrature method. In another theoretical study using the nonlocal plate theory [302], the nonlinear vibrations of multi-layered graphene sheets were studied by considering the von Karman hypothesis. They found that the small scale effect does not have a significant influence on the nonlinear frequency ratio of multi-layered graphene sheets. However, the underlying physics of these results was not discussed at all. Lin [303, 304] investigated the vibration characteristics of free and embedded multi-layered graphene sheets using the nonlocal continuum-based modeling and the generalized differential quadrature method. The results showed that the vibration modes can generally be classified into three types; the lower classical synchronized modes which are independent of vdW forces (Fig. 67a), the middle vdW enhanced modes which are largely determined by the presence of vdW interactions (Fig. 67b), and the higher mixed modes which are the combinations of the classical synchronized modes and the vdW enhanced modes (Fig. 67c). Unfortunately, the above mentioned papers as well as some other similar works [305, 306, 307] do not verify their results against MD simulations and other atomistic-based approaches. Furthermore, although the details of these investigations are not the same, their approaches, formulations and results are very similar. Recently, the vibration characteristics of multi-layered graphene sheets were investigated via MD simulations and the nonlocal plate elasticity [308, 309]. The results revealed that the small scale effect plays an important role in the vibration of the multi-layered graphene sheets. Similar to CNTs, good agreement was obtained between the results from the nonlocal continuum theory and MD simulations.

109

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

8.2.3. Graphene-based resonant sensors Among various available chemical and physical sensing methods for nanoobjects, resonance-based sensors form novel devices in the field of nanometrology [310]. Their detection criterion is established on the basis of measuring the resonant frequency shift of the sensor which is sensitive to the size and position of the attached object. Although the resonant sensing application of graphene sheets have not been shown experimentally up to now, however, many researchers have raised the possibility of using the graphene sheets as a nanoscale label-free mass sensor. The first contribution to the vibration analysis of graphene-based mass sensors was made by Shen et al. [311] on the basis of the nonlocal Kirchhoff plate theory. The effects of the mass and position of the attached nanoparticle on the frequency shift of the graphene sheet were discussed. Their numerical results illustrated that the sensitivity of graphene-based sensors was of the order of zeptogram (10-21 g), which is of the same order as mentioned in Refs. [312, 313]. To ensure the realistic applications of the graphene sheet as a nanomechanical sensor, a mathematical framework according to the nonlocal plate theory was developed and compared with a molecular mechanics approach based on the universal force field (UFF) model [314]. Acceptable agreements between the nonlocal continuum based-model and the molecular mechanics simulations were observed (see Fig. 68, taken from [314]). Consequently, it was concluded that the nonlocal model was applicable for graphene-based mass sensors. However, this work was limited to linear distribution of the attached masses. In the framework of the nonlocal plate theory, Lee et al. [315] analyzed the mass detection using a graphene-based nanoresonator. They discussed the influence of the small scale effect, the size and the aspect ratio of a single-layered graphene sheet on the sensitivity of the sensor. The possibility of double-layered graphene sheets as resonant mass sensors was first explored by Natsuki et al. [316]. In most of the previous researches, the effect of temperature change on the frequency shift of the graphene-based mass sensors was not taken into account, even though thermal gradients have enormous effect on the performance and sensitivity of sensors. Motivated by this fact, Fazelzadeh and Ghavanloo [317] investigated the vibration characteristics of graphene-based mass sensors with multiple attached nanoparticles in thermal environments. In that study, the mass sensor was modeled as a simply supported rectangu-

110

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

lar orthotropic nanoplate with multiple attached nanoparticles at different positions (Fig. 69). Since the physical dimensions of the nanoparticles were assumed to be very small compared to the in-plane dimensions of the singlelayered graphene sheet, they were modeled as point particles. To sum up the main results of that investigation, it can be stated that the sensitivity of the mass sensor increases with increasing the temperature difference and the nonlocal effect diminishes the frequency and increases the relative frequency shift (see Fig. 70, taken from [317]). Recently, application of single-layered graphene sheets as resonant sensors in the detection of ultra-fine nanoparticles was addressed by Jalali and coworkers [318, 319] using combined MD simulations and the nonlocal elasticity approach. A nonlinear nonlocal plate model carrying an attached mass-spring system was developed to study the geometric nonlinearity and the atomistic interactions between the single-layered graphene sheet and nanoparticles. In MD-based simulations, the interactions between carbon-carbon, metalmetal and metal-carbon atoms were modeled respectively via AIREBO potential, embedded atom method (EAM), and the Lennard-Jones potential. By matching the frequency shifts obtained by the nonlocal and MD simulation approaches with same vibration amplitude, a proper value of nonlocal scale coefficient was computed. A typical comparison between the MD simulation and the nonlocal continuum results for distributed nanoparticles is shown in Fig. 71, taken from [319]. In this figure, µ ¯ is the non-dimensional nonlocal scale coefficient and w¯max denotes the non-dimensional maximum vibration amplitude. Furthermore, they elucidated the influence of nonlinearity, nonlocality and the distribution of attached nanoparticles on the frequency shifts of the sensor. 8.3. Propagation of waves in the graphene sheets Ultrasonic wave propagation in the free and embedded graphene sheets was studied by Narendar and coworkers [320, 321] on the basis of the nonlocal elasticity theory incorporating the small scale effects. In these studies, the graphene sheet was modeled as an isotropic plate of one-atom thick. Similar to CNTs, it was shown that the nonlocal model introduces certain band gap region in the in-plane and flexural wave modes where no wave propagation occurs. This is manifested as the region where the phase velocity tends to zero. Similar results were also reported in Ref. [322]. However, only nonlocal continuum mechanics was employed in these studies and no calibration of 111

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

the small scale parameters was performed in these investigations. Hence, the applicability of the nonlocal theories for prediction of the propagation of waves in the graphene sheets could not be justified. To provide a comprehensive study on the wave propagations in singlelayered graphene sheets and the justification for the nonlocal model, Arash et al. [323] developed a finite element model from the weak-form of the nonlocal elastic plate model and verified it by the MD simulations. They showed that the nonlocal plate model is indispensable in predicting graphene phonon dispersion relations, especially at wavelengths of less than 1 nm. Moreover, the value of the nonlocal scale coefficient e0 a = 0.18 nm was recommended through the comparison between the results of the MD simulations and the nonlocal plate model (Fig. 72). Figure 73, taken from [323], illustrates a snapshot of a wave propagating in a graphene sheet predicted by the MD simulation and the nonlocal continuum model. Finally, they concluded that the nonlocal plate model with a proper nonlocal scale coefficient was adequate to analyze the propagation of waves in the graphene sheets.

112

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

9. Dynamical characteristics of spherical nanoscopic structures The applicability of the nonlocal elasticity theory to model spherical nanoscopic structures were not addressed up to 2012. The first attempts to capture the small scale effects of the nanoscopic structures by using the continuum equations of the nonlocal elasticity theory was carried out by Ghavanloo and Fazelzadeh [146, 324, 325]. However, the mathematical modeling of the spherical nanoscopic structures via the nonlocal elasticity is still in its nascent stage, and further extensive study is required. In this section, we review the studies performed in this field on the basis of the theoretical models described in Sections 4.5 and 4.6.1. 9.1. Axisymmetric vibration of spherical nanoshells Axisymmetric vibrational behavior of spherical fullerenes and empty spherical viruses has been examined on the basis of the nonlocal elastic theory [146]. To justify the reliability and robustness of the nonlocal model for application to spherical shell-like nanoscopic structures, the computed frequencies of the C60 (cm−1 ) obtained from the nonlocal model were compared with some existing theoretical and experimental results. The effective data that were used consisted of the Young modulus (E = 1.02 TPa), the Poisson's ratio (ν = 0.145), the mass density (ρ = 2.27 g/cm3 ) and the radius (R = 0.35 nm). Table 9, taken from [146], lists the computed values of the vibrational frequencies of the C60 and those obtained by the Raman spectroscopy [326], together with some existing theoretical results [145, 327, 328, 329], and molecular mechanics results [330]. As can be inferred from this table, the predicted numerical results from the nonlocal model are in reasonable agreement with the results reported by the experimental and atomistic-based studies. Table 9: Comparison between the results obtained from the nonlocal model and the existing theoretical and experimental results

113

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Vibration Mode a0 a1 a2 a3 a4 b2 Nonlocal continuum model [146](e0 a/R = 0.15) 491.416 588.758 798.620 1025.36 1223.36 230.220 Raman spectroscopy [326] 496 576 774 1183 1469 273 MNDO approximation [327] 610 719 924 1353 1667 263 Force-constant model [328] 492 589 788 1208 1468 269 Local continuum model [329] 496 607 833 1109 1400 273 Local continuum model [145] 503 594 894 1218 — 259 Molecular mechanics [330] 490 587 850 1187 — 269 Ten spherical fullerenes, i.e., C60 , C180 , C240 , C540 , C960 , C1500 , C2160 , C2940 , C3840 , and C4860 , were considered and their frequency spectra were computed (Fig. 74). The frequencies of the lower branch, given in Eq. (205), and the upper branch, given in Eq. (204), are plotted in Figs. 74a and 74b, respectively. It should be noted that the mode numbers n are discrete, i.e., only those points corresponding to the integer values of n are physically meaningful. Figure 75, taken from [331], shows the modal shapes for n values ranging from 0 to 3. Their numerical results revealed that all frequencies, except the breathing-mode, a0 , decreased when the nonlocal scale coefficient was increased. Similar results were observed by Zaera et al. [331]. Most spherical nanoshells usually operate in complex environments composed of an inner or outer fluid. It is well-known that the dynamic behavior of a fluid-filled spherical nanoshell is different from empty ones. Furthermore, the fluid-filled spherical nanoshell is technologically important in some modern industrial, biomedical, biological and many other nanotechnological applications [332, 333]. For this purpose, Fazelzadeh and Ghavanloo [324] studied the effect of small scale parameter on the fluid-structure interactions by investigating the axisymmetric vibrations of a fluid-filled spherical membrane nanoshell. They assumed that the nanoshell was completely filled with a compressible and inviscid fluid whose motion was governed by the wave equation. The frequency equation of the coupled axisymmetric vibration of the fluid-filled spherical membrane nanoshell was obtained by using the Legendre polynomials, the associated Legendre polynomials and the spherical Bessel functions. To sum up, the results from this study show that there exist an infinite number of frequencies for each mode and for the lower frequency branches all frequencies decrease with an increase in the nonlocal scale coefficient, and that the 114

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

rate of decrease is not identical for all branches (see Fig. 76, taken from [324]). In the aforementioned works, the in-plane forces were taken into account and the bending and twisting moments, the shear effects and the radial normal stresses were neglected. To extend the previous researches, Vila et al. [334] investigated the axisymmetric free vibration of closed thin spherical nanoshells by including the bending effects. They showed that the upper vibrational branch was slightly changed by the bending effect while the lower branch was strongly responsive to slight changes in the bending effect. 9.2. Radial vibration of spherical nanoparticles A good knowledge of the vibrational properties of the nanoparticles is of fundamental interest because it can be used as a way to characterize nanometer-scale objects and is also required for an efficient design of devices. The Raman spectroscopy technique, and the Brillouin scattering and time-resolved pump-probe experiments have been the main tools to study the vibrations of the nanoparticles over the years. The breathing-mode, or the pure radial mode of the nanoparticles (Fig. 77), is identified as the excitation of A1g mode with the in-phase radial displacement of atoms in the nanoparticles. To describe this vibrational mode for rather large nanoparticles with a radius R >2 nm, the classical continuum elastic model, which was proposed by Lamb in 1882 [335], has usually been used. The applicability of the classical model for description of nanoparticles has been confirmed using atomisticbased calculations for those larger than 2-3 nm [336, 337, 338]. However, recent studies have shown that the prediction of the vibration characteristics by the Lamb formulations may fail for ultra-small nanoparticles. Motivated by this point, Ghavanloo and Fazelzadeh [339] developed a nonlocal continuum model to study the radial vibration of anisotropic nanoparticles, for the first time. The breathing-mode frequencies of several isolated nanoparticles having cubic, hexagonal, trigonal and tetragonal morphologies were computed. Their suggested model was justified by a good agreement between the computed results from the nonlocal continuum-based model and the experimental results for several nanoparticles including gold, silver, germanium, and cadmium selenide. A further study later published by the same authors [340], confirmed the previously described validation. Figure 78, taken from [340], shows the predicted radial breathing-mode period for free Au and Pt nanoparticles in the size range of up to 4 nm along with the experimental

115

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

data obtained from time-resolved pump-probe spectroscopy and atomistic simulation [341]. They showed that the inverse relationship between the radial frequency and the nanoparticle radius was bound to fail when the nanoparticle radius was smaller than 1.5 nm. Furthermore, it was reported that the effect of the nonlocal parameter on the frequency was significant for the nanoparticles with small radii (see Fig. 79, taken from [339]). After verifying the accuracy and reliability of the nonlocal model, in another study [342], the radial vibrations of spherical nanoparticles immersed in a fluid medium were investigated. In some new applications of nanotechnology, the investigation of dynamical characteristics of nanoscopic structures subject to an external magnetic field has been addressed. Therefore, in another study, the effects of a circumferential magnetic field on the radial breathing-mode frequency of a magnetically sensitive nanoparticle were examined [340] on the basis of the nonlocal continuum model. To sum up the results from these works, it was noticed that the findings point directly to the possibility of employing the nonlocal continuum modeling as an effective way in simulating the nanoparticles.

116

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

10. Application of nonlocal models to other types of nanoscopic structures 10.1. Mechanical modeling of elongated nanocrystals It has been experimentally established that the mechanical properties of the nanocrystals are size-dependent. Experimental investigations on this topic appear to have been started in the 1990s. Wong et al. [343] successfully measured the elastic bending of tiny beams of SiC. The bending strength measurements of Si and SiO2 nanobeams were carried out by Sundararajan and Bhushan [344] using atomic force microscopy (AFM). They found that the bending strength of a specimen showed a size dependence (Fig. 80). In another investigations [345, 346], it has been shown that the elastic moduli of ZnO nanowires, which are experimentally obtained from the measurement of the resonant frequencies of the nanowires, are also sizedependent. It has been found that the elastic modulus of ZnO nanowires increases with a decrease in the nanowire diameter. The size-dependent mechanical properties was also observed in both the tensile and bending experiments of ZnO nanowires, where the bending modulus and the tensile modulus increased with a decrease in the nanowire diameter from 80 nm to 20 nm [347]. In addition, the size-dependent vibrational properties of nanocrystals have also been reported in combined experimental and computational studies [348, 349]. In the following, we will concentrate on the application of the nonlocal continuum models to study the size-dependent mechanical characteristics of elongated nanocrystals. We restrict our review to some selective works. 10.1.1. Bending and buckling of nanowires Nanowires have a huge surface-to-bulk ratio and surface energy. To understand the effect of surface energy on the mechanical responses of the nanowires, a surface elasticity model provided by Gurtin and Murdoch [350, 351] has been widely used during the past few years to study the surface effect on the bending, buckling, and vibration of the nanowires. In this model, the surface layer is modeled as a zero-thickness film that overlays the bulk without slipping. Excellent reviews have appeared on the application of surface elasticity to nanoscopic structures [352, 353, 354], and the reader is referred to those works for more in-depth coverage.

117

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

The bending deformation due to a uniformly distributed load and the buckling load of the nanowires with various boundary conditions were investigated by incorporating simultaneously both the nonlocal effect in the bulk of the nanowire and the surface forces [355]. The results from Ref. [355] show that the buckling load increases when the surface stress effect is taken into account, while the nonlocal elasticity computes the buckling load with a slight decrease (Fig. 81). The reason for this behavior is that the nonlocal effect promotes a reduction in the internal interaction force which directly results in the reduction of the buckling load. In practical applications, such as nanotweezers and AFM tips, nanowires may exhibit a nonlinear elastic large deformation curve which is known as the elastica (Fig. 82, taken from [356]). Several efforts toward elastica type buckling analysis of the nanowires and nanorods have been made on the basis of the nonlocal elasticity theory [357, 358, 359, 360]. It is seen that the nonlocal effect reduces the postbuckling load and causes more deflection in the nanorod (Fig. 83). In another study, the buckling of the nanowires was investigated by Hu et al. [361] on the basis of a modified core-shell model. The modified model was utilized to predict the size effect of the Young modulus of the nanowires and was verified with the reported experimental data [346]. They found that the surface layer thickness should be taken into account for an exact characterization of the buckling behavior of the nanowires. 10.1.2. Dynamic analysis of nanowires The dynamic response of an embedded conducting nanowire subject to an axial magnetic shock was investigated within the nonlocal continuum theory and Maxwell's and Cauchy's equations [362]. In this study, the effects of the small scale parameter, the stiffness of the surrounding matrix, and the duration of the applied magnetic shock on the dynamic response of the nanowire were examined. The time development plots of the non-dimensional radial displacement of a silver nanowire are shown in Fig. 84, taken from [362]. The nonlocal formulations of the breathing-mode of the circular nanowire were obtained in detail in Section 4.6.2. Complementary formulations by incorporating both the surface elastic properties and the surface inertia effect were developed by Ghavanloo et al. [363]. The breathing frequencies of Si nanowires predicted by the nonlocal model for three cases are displayed in Fig. 85 and compared with the results from the ab initio calculations 118

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

of Trejo et al. [364]. The results from the nonlocal calculations, which agreed well with the ab initio results, show that the combined surface and the nonlocal model is suitable for the prediction of the breathing-mode frequency of nanowires with diameters smaller than 1.5 nm. 10.2. Nonlocal-based modeling of nanopeapods C60 fullerenes encapsulated in SWCNTs form the so-called nanopeapods (Fig. 86). These hybrid molecules were first observed by Smith et al. [365, 366] using high resolution transmission electron microscopy during the purification and annealing treatments of SWCNTs. The encapsulated C60 fullerenes may change the physical, electrical and mechanical properties of the SWCNT [367, 368]. Therefore, in recent years, theoretical and experimental studies have been carried out to investigate the characteristics of SWCNTs encapsulating fullerenes [379, 370, 371]. The nonlocal Donnell shell model has been employed to predict the structural instability of the nanopeapods under two different loading conditions, namely radial external pressure [372] and torsional moments [373]. A pristine (10,10) SWCNT and a C60 @(10,10) nanopeapod were considered. The results for the (10,10) SWCNT were computed and compared with the available MD simulation results. Then, the critical loads for the C60 @(10,10) nanopeapod was predicted. The simulation results for the critical torsional moment of the C60 @(10,10) nanopeapod and (10,10) SWCNT are shown in Fig. 87, taken from [373]. The numerical results demonstrate that the stability resistance increases by more than 100% due to the presence of C60 fullerenes in the nanotube. This is in good agreement with some MD simulation results [374, 375]. Finally, it was stated that employing the nonlocal continuumbased theory can significantly diminish the gap between the results of the classical continuum theory and the MD simulations. 10.3. Analysis of micro/nanobridge test Among the mechanical characterization techniques for low dimensional materials, the bending test is one of the simplest and most important experimental methods. There are two main types of bending test: the cantilever beam bending test and the micro/nanobridge test. In the micro/nanobridge test, both ends of a micro/nanobeam are bonded onto a substrate (see Fig. 88, taken from [376]). Since the size-dependent properties of materials have been observed in tests on micro/nanometer sized samples [377, 378], Fan and coworkers [376] 119

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

analyzed the micro/nanobridge test on the basis of the nonlocal Timoshenko beam theory. Furthermore, the energy release rate and the phase angle for the delamination test were predicted by the nonlocal elasticity approach.

120

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

11. Some recent developments in the application of the nonlocal models So far, we have given rather comprehensive general formulations of the standard nonlocal continuum-based models and have analyzed their applications in the study of the mechanical characteristics and functioning of different nanoscopic structures. Recently, some new ideas about the nonlocal continuum approach have emerged in the specialized literature, which are different from the works reported in this review. To complement the discussion about the nonlocal theory, we include some important ideas from these latest studies in the following section. 11.1. A different nonlocal stress model and its stiffness enhancement effects As reviewed in previous sections, the mechanical behavior of various nanoscopic structures with different morphologies was modeled on the basis of application of the nonlocal continuum theory. A majority of these studies have led to one important conclusion, namely an increase in the nonlocal parameter results in a reduced structural stiffness of the nanoscopic structures. Consequently, larger bending deflection, lower vibration frequency, lower buckling load and lower wave propagation velocity are obtained from the nonlocal models in comparison with the local versions. Recently, Lim [209] has pointed out that the above conclusion should be re-examined. In contrast to previous conclusions, Lim and coworkers have reported that the stiffness was consistently enhanced with an increase in the nonlocal parameter [124, 379]. As a consequence of an enhanced structural stiffness, the bending deflection and the tensile extension were reduced, while the vibration frequency, the critical buckling load and the wave propagation velocity were increased. Furthermore, their proposed governing equations and the corresponding boundary conditions involve essential higher order differential terms which have opposite signs to terms in the previous studies. However, to our knowledge, their conclusions have not been confirmed by other researchers. Judgments about these unexpected findings, must await reliable experiments and/or atomistic-based calculations. 11.2. Self-adjointness of Eringen's nonlocal elasticity One issue that is debated about the applicability of Eringen's nonlocal constitutive relation, Eq. (50), is whether the nonlocal models are selfadjoint, and the governing equations can be derived via a single-energy functional. In a series of important papers, Reddy [103, 380] has raised the 121

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

possibility of the nonself -adjointness of Eringen's model and the impossibility of constructing the underlying quadratic energy functional for such nonlocal beam models. Later, Adali [381, 382, 383], showed that the governing equations of Eringen's type nonlocal elasticity that is applied to the beam and plate models can be derived from a single-energy functional. As has been recently shown [384], it is possible to build an energy function for the bending, vibration, and buckling problems using the nonlocal EulerBernoulli beams, as well as the higher order beam models. More recently, Challamel and coworkers [385] found that Eringen's nonlocal elasticity may be self-adjoint for some problems having specific boundary conditions. The following conclusions may be drawn from their study: 1. The buckling problem studied within the nonlocal beam model is self-adjoint for all boundary conditions. 2. Under all circumstances the nonlocal Euler-Bernoulli beam model can be obtained from a single energy function for the case of simply supported boundary conditions. 3. For the vibration analysis of the fixed end-free end nanobeams, the nonlocal beam mode forms a nonself-adjoint problem. This implies that it is not possible to build an associated functional energy, and the problem is clearly nonconservative. Furthermore, a surprising and paradoxical result emerges in this case, wherein the small length scale effect tends to stiffen the local system, in contrast to the softening phenomenon obtained for all the other boundary conditions. Finally, it should be noted that the research on the self-adjointness of the nonlocal models is still in its initial stages and it is hoped that it will generalize and expand in the near future. 11.3. Application of nonlocal continuum theory to piezoelectric nanoscopic structures Recently, the application of the nonlocal elasticity theory was extended to piezoelectric materials. Wang and Wang [386] studied the bending and electromechanical coupling behavior of piezoelectric nanowires on the basis of the combined nonlocal piezoelectric theory and the surface elasticity theory. Based on the nonlocal piezoelectric cylindrical shell theory, Arani and coworkers [387, 388] studied the electro-thermo-mechanical buckling and vibration of double-walled boron nitride nanotubes embedded in a bundle of CNTs. The buckling and postbuckling behavior of piezoelectric nanobeams

122

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

were investigated by Liu et al. [389], using the piezoelectric nonlocal Timoshenko beam theory. Further studies were carried out by Ke and coworkers [390, 391] on the basis of the nonlocal piezoelectric Timoshenko beam theory to investigate the thermoelectric-mechanical vibration of the piezoelectric nanobeams. They found that the natural frequency of the piezoelectric nanobeam was quite sensitive to the thermoelectric-mechanical loadings. In a further study [392], the analytical solution for natural frequencies of the piezoelectric nanoplate was developed by the same group. In summary, studies are lacking on the static and dynamic behavior of the magneto-electroelastic nanoscopic structures via the nonlocal continuum models. Hence, future studies are called for to evaluate the application of the nonlocal theory in the simulation of the static and dynamic behaviors of these structures.

123

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

12. Concluding remarks and future prospects Nanotechnology is likely to form the pivotal technology in the 21st century. It is predicted that nanoscince and nanotechnology will promote revolutionary changes in multitude of fields. Among the diverse branches of nanoscience and nanotechnology, the synthesis and characterization of nanoscopic structures form an interdisciplinary and cross disciplinary area in both theoretical and applied research. This subfield of nanotechnology has witnessed an explosive growth worldwide over the past few years. In this extensive and rather comprehensive review, we have attempted to survey the field of the nonlocal continuum-based theories in order to provide an alternative strategy for computational modeling of the mechanical characteristics of nanoscopic structures. In this endeavor we have completely reviewed the research that has been based on the use of this modeling approach. Unlike the atomistic-based computational modeling, the nonlocal continuum-based modeling is not computationally expensive and does not demand extensive hardware resources, and so it is very suitable for application to more complex many-body systems without time or length scale limitations. We should reiterate that the discrete nature of the nanoscopic structures is captured in this type of modeling. Let us very briefly highlight some of the key results that can be harvested from this review. We have seen that the mechanical behavior, i.e., the bending, the buckling, the vibration and the wave propagation of high aspect ratio CNTs can be consistently predicted, and verified with the MD simulation results, by using the nonlocal beam theories. It has been found that to correctly describe the behavior of the high aspect ratio CNTs, the magnitude of the nonlocal parameter had to be determined appropriately. In other words, the determination of the magnitude of the nonlocal parameter is the key issue in a successful application of the nonlocal beam models. In this regard, more studies are needed to fully evaluate the nonlocal parameter for CNTs. Furthermore, it is observed that the surface stresses as well as the interactions with the surrounding media of the nanotubes affect their static and dynamic behavior, and the correct modeling of these stresses and interactions forms a set of challenging research problems. Next, we considered the application of the nonlocal cylindrical shell models to the prediction of the mechanical characteristics of low aspect ratio CNTs. In the case of buckling analysis, it was found that the nonlocal continuum-based shell models were more appropriate than their local coun124

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

terpart for the prediction of the axial and tortional buckling of SWCNTs having small diameters i.e., for diameters smaller than 1.5 nm. Although the application of the nonlocal shell models to the investigation of CNT buckling is successful, nevertheless, only few results are available in the literature related to the nonlocal modeling of buckling of CNTs that are subject to a combination of loads. This is a significant problem, and further research is called for using the nonlocal shell models. The investigation of the vibrational properties of CNTs formed another field of application of the nonlocal shell models. The computations of frequencies pertinent to various vibrational modes, i.e., the longitudinal, the radial and the torsional modes, the phonon dispersion and the group and phase velocities of waves in CNTs have been the focal points in modeling research studies. It has been found that the nonlocal shell model could predict the MD results better than the local shell model. The good agreement between the results from the nonlocal elastic cylindrical shell model and the MD simulations is a further testimony to the effectiveness of the nonlocal shell model. Furthermore, it is seen that the nonlocal shell models predict smaller frequency values and wave velocities as compared with the local continuum models and that the frequencies decrease when the nonlocal scale coefficient increases. Actually, the nonlocal theory provides a more flexible model. This finding is valid for SWCNTs as well as MWCNTs. Transportation of encapsulated atoms and nanoparticles in the SWCNTs has been identified and so a dynamic simulation of this phenomenon on the basis of the nonlocal continuum models can be an interesting research area. Furthermore, investigation of the vibrational properties of the SWCNTs with structural defects is another valuable field for future research. Generally, defects may degrade the stiffness and strength of the nanoscopic structures. The effect of defects can be modeled by using the nonlocal continuum theories. Single- and multi-layered graphene sheets have been among the most investigated materials in research related to nanoscopic structures, and have found applications in many fields, such as optics, sensing and biosensing etc. To discuss the relevance of the nonlocal plate theories to the prediction of mechanical behavior of graphene sheets, we have examined some selective papers in this field. It is seen that the nonlocal plate models with their appropriate values of the nonlocal parameter have an excellent prospect to predict the buckling, the vibration and the wave propagation in single- and multi-layered graphene sheets. Future research along these lines, i.e., employing the nonlocal continuum-based models, could help reduce the number of 125

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

very time-consuming and costly MD simulations that must be implemented to obtain estimates of mechanical and elastic properties of graphene sheets. The modeling of dynamical characteristics of spherical nanoscopic structures is another topic dealt with in this review. We have seen that the modeling based on the use of the nonlocal continuum-based elasticity theories has been implemented and has been quite successful in providing deep insights into the dynamics of spherical fullerenes, metal nanoparticles and quantum dots, as well as providing quantitative estimates of the pertinent variables. Efforts made so far to study the properties of spherical nanoscopic structures by the nonlocal theories have been rather meager. To complete the discussion about the nonlocal continuum-based models, we have also included its application to other types of nanoscopic structures including nanowires, nanorods and nanopeapods. Furthermore, some new ideas and challenges in this field have been mentioned in Section 11. In this review our main aim has been to show, as far as possible, that the results obtained from the nonlocal models are in very good agreement, both qualitatively and quantitatively, with the data obtained from both experimental and atomistic-based studies where they are available. Notwithstanding the undoubted successes and inherent potential power of the nonlocal continuum-based models, it should also be remarked that they also suffer from some limitations. Realistic modeling of the nanoscopic structures would require very accurate definition of the mechanical properties such as elastic constants and effective thickness. Another question of vital importance to a simulation is the correct choice of the nonlocal parameter. Data obtained by different authors on the magnitude of the nonlocal parameter are not yet unitary and unambiguous. All of these data reveal, however, that the value of the nonlocal parameter depends crucially on the state of motion, the boundary condition and the mode shapes. In this connection, future research should help clarify this question. Furthermore, modeling the mechanical characteristics of biological nanoscopic structures, with various morphologies, in the fields of medical nanoscience and nanotechnology is a challenge for a long time to come, and can be performed by using the nonlocal continuum-based models in the future. Notwithstanding the fact that the nonlocal continuum-based models provide powerful tools for characterizing the mechanical behavior of the nanoscopic structures, they lack the ability to address the thermal characteristics of these structures, while MD simulations have definite advantage for application in this area. 126

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

It should be noted that the higher order nonlocal continuum-based models involve complicated equations that are not easy to handle. Furthermore, there does not seem to be any strong reason suggesting the necessity of employing the higher order continuum-based models for the analysis of the nanoscopic structures. In summary, the nonlocal Euler-Bernoulli and Timoshenko beam theories are appropriate theories for application to the modeling of long CNTs and the elastic shell models, based on the classical shell theory and the first order shear deformation theory, are adequate for modeling the static and dynamic behavior of short CNTs. Furthermore, the nonlocal Kirchhoff plate theory and the Mindlin plate theory are adequate to analyze the graphene sheets. The preparation of this review was made possible owing to the recently obtained experimental, theoretical and computational results obtained from the works of several research groups involved in the field of modeling the physical and structural properties of nanoscopic structures. We have tried to provide a comprehensive review. However, there is little doubt that some papers and reports have not come to our attention and have been missed out in this review. This, in no way is a judgment on the quality of those works that have not been included in our review, and we welcome suggestions from researchers in this field.

127

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

References [1] J.M. Haile, Molecular Dynamics Simulation: Elementary Methods, J. Wiley& Sons Inc, New York, 1992. [2] I.A. Guz, A.A. Rodger, A.N. Guz, J.J. Rushchitsky, Developing the mechanical models for nanomaterials, Compos. Part A 38(4) (2007) 1234-1250. [3] D.D. Vvedensky, Multiscale modelling of nanostructures, J. Phys.: Condens. Matter 16 (2004) R1537-R1576. [4] H. Rafii-Tabar, Computational physics of carbon nanotubes, Cambridge University Press, Cambridge, 2008. [5] J.P. Hansen, I.R. McDonald, Theory of simple liquids, Academic Press, New York, 1986. [6] W.K. Liu, E.G. Karpov, S. Zhang, H.S. Park, An introduction to computational nanomechanics and materials, Comput. Methods Appl. Mech. Engg. 193 (2004) 1529-1578. [7] A. Crut, P. Maioli, N. Del Fatti, F. Valle, Acoustic vibrations of metal nano-objects: time-domain investigations, Phys. Rep. 549 (2015) 1-43. [8] N. Combe, J.R. Huntzinger, A. Mlayah, Vibrations of quantum dots and light scattering properties: Atomistic versus continuous models, Phys. Rev. B 76 (2007) 205425. [9] D. Baowan, N. Thamwattana, J.M. Hill, Continuum modelling of spherical and spheroidal carbon onions, Eur. Phys. J. D 44 (2007) 117123. [10] E. Ghavanloo, S.A. Fazelzadeh, Continuum modeling of breathing-like modes of spherical carbon onions, Phys. Lett. A 379 (2015) 1600-1606. [11] X. Yao, Q. Han, A continuum mechanics nonlinear postbuckling analysis for single-walled carbon nanotubes under torque, Eur. J. Mech. A/Solids 27 (2008) 796-807.

128

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

[12] B.I. Yakobson, C.J. Brabec, J. Bernholc, Nanomechanics of carbon tubes: instability beyond linear response, Phys. Rev. Lett. 76 (1996) 2511-2514. [13] X.Q. He, S. Kitipornchai, K.M. Liew, Buckling analysis of multi-walled carbon nanotubes: a continuum model accounting for van der Waals interaction, J. Mech. Phys. Solids 53 (2005) 303-326. [14] E. Ghavanloo, S.A. Fazelzadeh, Vibration characteristics of singlewalled carbon nanotubes based on an anisotropic elastic shell model including chirality effect, Appl. Math. Model. 36 (2012) 4988-5000. [15] A. Assadi, B. Farshi, Vibration characteristics of circular nanoplates, J. Appl. Phys. 108 (2010) 074312. [16] M. Hu, X. Wang, G.V. Hartland, P. Mulvaney, J.P. Juste, J.E. Sader, Vibrational response of nanorods to ultrafast laser induced heating: theoretical and experimental analysis, J. Am. Chem. Soc. 125 (2003) 14925-14933. [17] J.W. Yan, K.M. Liew, L.H. He, Predicting mechanical properties of single-walled carbon nanocones using a higher-order gradient continuum computational framework, Compos. Struct. 94 (2012) 3271-3277. [18] F. Alisafaei, R. Ansari, A. Alipour, A semi-analytical approach for the interaction of carbon nanotori, Physica E 58 (2014) 63-66. [19] W.Q. Chen, The renaissance of continuum mechanics, J. Zhejiang Univ. Sc. A 15(4) (2014) 231-240. [20] J. Fatemi, F. Vankeulen, P.R. Onck, Generalized continuum theories: application to stress analysis in bone, Meccanica 37 (2002) 385-396. [21] I.A. Guz, Continuum solid mechanics at nano-scale: how small can it go?, J. Nanomater. Mol. Nanotechnol. 1 (2012) 1. [22] T.T. Zhu, A.J. Bushby, D.J. Dunstan, Materials mechanical size effects: a review, Mater. Technol. 23(4) (2008) 193-209. [23] R. Mindlin, H. Tiersten, Effects of couple-stresses in linear elasticity, Arch. Ration. Mech. Anal. 11 (1962) 415-448. 129

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

[24] R.A. Toupin, Theories of elasticity with couple stress, Arch. Ration. Mech. Anal. 17 (1964) 85-112. [25] Y. Yoshiyuki, An intrinsic theory of a Cosserat continuum, Int. J. Solids Struct. 4 (1968) 1013-1023. [26] D.G.B. Edelen, A.E. Green, N. Laws, Nonlocal continuum mechanics, Arch. Rat. Mech. Anal. 43 (1971) 36-44. [27] A.C. Eringen, Nonlocal polar elastic continua, Int. J. Eng. Sci. 10 (1972) 1-16. [28] A.C. Eringen, Linear theory of nonlocal elasticity and dispersion of plane waves, Int. J. Eng. Sci. 10 (1972) 425-435. [29] E.C. Aifantis, On the role of gradients in the localization of deformation and fracture. Int. J. Eng. Sci. 30 (1992) 1279-1299. [30] Z.G. Zhou, B. Wang, Nonlocal theory solution of two collinear cracks in the functionally graded materials, Int. J. Solids Struct. 43 (2006) 887-898. [31] E. Ghavanloo, S.A. Fazelzadeh, Evaluation of nonlocal parameter for single-walled carbon nanotubes with arbitrary chirality, Meccanica. In Press (2016). [32] E. Krner, Elasticity theory of materials with long range cohesive forces, Int. J. Solids Struct. 3 (1967) 731-742. [33] D.G.B. Edelen, N. Laws, On the thermodynamics of systems with nonlocality, Arch. Rat. Mech. Anal. 43 (1971) 24-35. [34] A.C. Eringen, C.G. Speziale, B.S. Kim, Crack-tip problem in non-local elasticity, J. Mech. Phys. Solids 25 (1977) 339-355. [35] A.C. Eringen, Nonlocal Continuum Field Theories, Springer-Verlag, NewYork, 2002. [36] C. Polizzotto, Nonlocal elasticity and related variational principles, Int. J. Solids Struct. 38 (2001) 7359-7380.

130

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

[37] A.A. Pisano, P. Fuschi, Closed form solution for a nonlocal elastic bar in tension, Int. J. Solids Struct. 40 (2003) 13-23. [38] C. Polizzotto, P. Fuschi, A.A. Pisano, A nonhomogeneous nonlocal elasticity model, Eur. J. Mech. A/Solids 25 (2006) 308-333. [39] A.A. Pisano, A. Sofi, P. Fuschi, Finite element solutions for nonhomogeneous nonlocal elastic problems, Mech. Res. Commun. 36 (2009) 755-761. [40] T.M. Atanackovic,B. Stankovic, Generalized wave equation in nonlocal elasticity, Acta Mech. 208 (2009) 1-10. [41] M. Di Paola, G. Failla, M. Zingales, Physically-based approach to the mechanics of strong non-local linear elasticity theory, J. Elast. 97 (2009) 103-130. [42] N. Di Paola, G. Failla, A. Pirrotta, A. Sofi, M. Zingales, The mechanically based non-local elasticity: an overview of main results and future challenges, Phil. Trans. R. Soc. A 371 (2013) 20120433. [43] A.C. Eringen, On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, J. Appl. Phys. 54 (1983) 4703-4710. [44] J. Peddieson, G.R. Buchanan, R.P. McNitt, Application of nonlocal continuum models to nanotechnology, Int. J. Eng. Sci. 41 (2003) 305312. [45] B. Arash, Q. Wang, A review on the application of nonlocal elastic models in modeling of carbon nanotubes and graphenes, Comp. Mater. Sci. 51 (2012) 303-313. [46] Y.Z. Wang, F.M. Li, Dynamical properties of nanotubes with nonlocal continuum theory: a review, Sci. China Phys. Mech. 55 (2012) 12101224. [47] R.P. Feynman, There's plenty of room at the bottom, Eng. Sci. 23 (1960) 22-36. [48] K. Pekala, M. Pekala, Low Temperature Transport Properties of Nanocrystalline Cu, Fe and Ni, Nanostruct. Mater. 6 (1995) 819-822. 131

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

[49] G. Gao, Nanostructures and Nanomaterials: Synthesis, Properties, and Applications, Imperial College Press, Singapore, 2004. [50] S.V.N.T. Kuchibhatla, A.S. Karakoti, D.Bera, S. Seal, One dimensional nanostructured materials, Prog. Mater. Sci. 52 (2007) 699-913. [51] J.N. Tiwari, R.N. Tiwari, K.S. Kim, Zero-dimensional, onedimensional, two-dimensional and three-dimensional nanostructured materials for advanced electrochemical energy devices, Prog. Mater. Sci. 57 (2012) 724-803. [52] H.S.P. Wong, D. Akinwande, Carbon Nanotube and Graphene Device Physics, Cambridge University Press, Cambridge, 2011. [53] H. Kroto, J. Heath, S. Obrien, R. Curl, R. Smalley, C60 Buckminsterfullerene, Nature 318 (1985) 162-163. [54] O.A. Shenderova, V.V. Zhirnov, D.W. Brenner, Carbon nanostructures, Crit. Rev. Solid State 27(3/4) (2002) 227-356. [55] S. Iijima, Helical microtubules of graphitic carbon, Nature 354 (1991) 56-58. [56] K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, Y. Zhang, S.V. Dubonos, I.V. Grigorieva, A.A. Firsov, Electric field effect in atomically thin carbon films, Science 306 (2004) 666-669. [57] K.S. Novoselov, D. Jiang, F. Schedin, T.J. Booth, V.V. Khotkevich, S.V. Morozov, A.K. Geim, Two dimensional atomic crystals, Proc. Natl. Acad. Sci. USA 102 (2005) 10451-10453. [58] B.C. Wang, H.W. Wang, J.C. Chang, H.C. Tso, Y.M. Chou, More spherical large fullerenes and multi-layer fullerene cages, J. Mol. Struc. 540 (2001) 171-176. [59] H. Zhang, D. Ye, Y. Liu, A combination of Clar number and Kekul count as an indicator of relative stability of fullerene isomers of C60, J. Math. Chem. 48 (2010) 733-740. [60] S. Iijima, T. Ichihashi, Single-shell carbon nanotubes of 1-nm diameter, Nature 363 (1993) 603-605. 132

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

[61] V.N. Popov, Carbon nanotubes: properties and application, Mat. Sci. Eng. R 43 (2004) 61-102. [62] H. Rafii-Tabar, Computational modelling of thermo-mechanical and transport properties of carbon nanotubes, Phys. Rep. 390 (2004) 235452. [63] J. Tersoff, R.S. Ruoff, Structural properties of a carbon-nanotube crystal, Phys. Rev. Lett. 73 (1994) 676-679. [64] S. Kawasaki, Y. Kanamori, Y. Iwai, F. Okino, H. Touhara, H. Muramatsu, T. Hayashi, Y.A. Kim, M. Endo, Structural properties of pristine and fluorinated double-walled carbon nanotubes under high pressure, J. Phys. Chem. Solids 69(5-6) (2008) 1203-1205. [65] C. Srinivasan, Graphene- mother of all graphitic materials, Curr. Sci. 92 (2007) 1338-1339. [66] M.I. Katsnelson, Graphene: carbon in two dimensions, Mater. Today 10 (2007) 20-27. [67] M. Hosokawa, K. Nogi, M. Naito, T. Yokoyama, Nanoparticle Technology Handbook, Elsevier, Oxford, 2007. [68] H.E. Sauceda, D. Mongin, P. Maioli, A. Crut, M. Pellarin, N. Del Fatti, F. Vallee, I.L. Garzon, Vibrational properties of metal nanoparticles: atomistic simulation and comparison with time-resolved investigation, J. Phys. Chem. C 116 (2012) 25147-25156. [69] J. Sarkar, G.G. Khan, A. Basumallick, Nanowires: properties, applications and synthesis via porous anodic aluminium oxide template, Bull. Mater. Sci., 30 (2007) 271-290. [70] A. Paul, M. Luisier, G. Klimeck, Influence of cross-section geometry and wire orientation on the phonon shifts in ultra-scaled Si nanowires, J. Appl. Phys. 110 (2011) 094308. [71] M.H. Sadd, Elasticity: Theory, Applications, and Numerics, Elsevier, New York, 2009. [72] E. Schreiber, O.L. Anderson, N. Soga, Elastic Constants and their Measurement, McGraw-Hill, New York, 1973. 133

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

[73] T.J. Chung, Applied Continuum Mechanics, Cambridge University Press, Cambridge, 1996. [74] H. Askes, E.C. Aifantis, Gradient elasticity in statics and dynamics: an overview of formulations, length scale identification procedures, finite element implementations and new results, Int. J. Solids Struct. 48 (2011) 1962-1990. [75] H. Askes, A.V. Metrikine, Higher-order continua derived from discrete media: continualisation aspects and boundary conditions, Int. J. Solids Struct. 42 (2005) 187-202. [76] E.C. Aifantis, Update on a class of gradient theories, Mech. Mater. 35 (2003) 259-280. [77] H. Askes, E.C. Aifantis, Gradient elasticity and flexural wave dispersion in carbon nanotubes, Phys. Rev. B 80 (2009) 195412. [78] S. Forest, E.C. Aifantis, Some links between recent gradient thermoelastoplasticity theories and the thermomechanics of generalized continua, Int. J. Solids Struct. 47 (2010) 3367-3376. [79] A.J.A. Morgan, Some properties of media defined by constitutive equations in implicit form, Int. J. Eng. Sci. 4 (1966) 155-178. [80] E.C. Aifantis, On the gradient approach-Relation to Eringen's nonlocal theory, Int. J. Eng. Sci. 49 (2011) 1367-1377. [81] E.C. Aifantis, On the role of gradients in the localization of deformation and fracture, Int. J. Eng. Sci. 30 (1992) 1279-1299. [82] Y.Z. Povstenko, The nonlocal theory of elasticity and its applications to the description of defects in solid bodies, J. Math. Sci. 97 (1999) 3840-3845. [83] A.C. Eringen, Non-local Polar Field Models, Academic Press, New York, 1996. [84] S. Forest, K. Sab, Stress gradient continuum theory, Mech. Res. Commun. 40 (2012) 16-25.

134

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

[85] C.M. Wang, C.Y. Wang, J.N. Reddy, Exact Solutions for Buckling of Structural Members, CRC Press, Florida, 2005. [86] S.S. Rao, Vibration of Continuous Systems, J. Wiley& Sons Inc, New Jersey, 2007. [87] E. Ventsel, T. Krauthammer, Thin Plates and Shells: Theory, Analysis, and Applications, CRC press, New York, 2001. [88] C. H. Yoo, S. C. Lee, Stability of Structures: Principles and Applications, Elsevier, Amsterdam, 2011. [89] J.F. Waters, P.R. Guduru, M. Jouzi, J.M. Xu, T. Hanlon, S. Suresh, Shell buckling of individual multiwalled carbon nanotubes using nanoindentation, Appl. Phys. Lett. 87 (2005) 103109. [90] S.C. Hung, Y.K. Su, T.H. Fang, S.J. Chang, L.W. Ji, Buckling instabilities in GaN nanotubes under uniaxial compression, Nanotechnology 16 (2005) 2203-2208. [91] P.R. Guduru, Z. Xia, Shell buckling of imperfect multiwalled carbon nanotubes-experiments and analysis, Exp. Mech. 47 (2007) 153-161. [92] H.W. Yap, R.S. Lakes, R.W. Carpick, Mechanical instabilities of individual multiwalled carbon nanotubes under cyclic axial compression, Nano Lett. 7 (2007) 1149-1154. [93] S.J. Young, L.W. Ji, S.J. Chang, T.H. Fang, T.J. Hsueh, Nanoindentation of vertical ZnO nanowires, Physica E 39 (2007) 240-243. [94] S.Y. Ryu, J. Xiao, W.I. Park, K.S. Son, Y.Y. Huang, U. Paik, J.A. Rogers, Lateral buckling mechanics in silicon nanowires on elastomeric substrates, Nano Lett. 9 (2009) 3214-3219. [95] J. Zhao, M.R. He, S. Dai, J.Q. Huang, F. Wei, J. Zhu, TEM observations of buckling and fracture modes for compressed thick multiwall carbon nanotubes, Carbon 49 (2011) 206-213. [96] Y. Mao, W.L. Wang, D. Wei, E. Kaxiras, J.G. Sodroski, Graphene structures at an extreme degree of buckling, ACS Nano 5 (2011) 13951400. 135

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

[97] A.E.H. Love, A Treatise on the Mathematical Theory of Elasticity, Cambridge University Press, Cambridge, 1927. [98] E. Duval, A. Boukenter, B. Champagnon, Vibration eigenmodes and size of microcrystallites in glass: observation by very-low-frequency Raman scattering, Phys. Rev. Lett. 56 (1986) 2052-2055. [99] M. Fujii, T. Nagareda, S. Hayashi, K. Yamamoto, Low-frequency Raman scattering from small silver particles embedded in SiO2 thin films, Phys. Rev. B 44 (1991) 6243-6248. [100] A. Tanaka, S. Onari, T. Arai, Low-frequency Raman scattering from CdS microcrystals embedded in a germanium dioxide glass matrix, Phys. Rev. B 47 (1993) 1237-1243. [101] A.M. Rao, E. Richter, S. Bandow, B. Chase, P.C. Eklund, K.A. Williams, et al., Diameter-selective Raman scattering from vibrational modes in carbon nanotubes, Science 275 (1997) 187-191. [102] G.L. Huang, C.T. Sun, F. Song, Wave propagation in elastic media with micro/nano-structures, in: A. Petrin (Ed.), Wave Propagation in Materials for Modern Applications, INTECH, Croatia, 2010, pp. 201224. [103] J.N. Reddy, Nonlocal theories for bending, buckling and vibration of beams, Int. J. Eng. Sci. 45 (2007) 288-307. [104] M. Aydogdu, A general nonlocal beam theory: its application to nanobeam bending, buckling and vibration, Physica E 41 (2009) 16511655. [105] Q. Wang, K.M. Liew, Application of nonlocal continuum mechanics to static analysis of micro- and nano-structures, Phys. Lett. A 363 (2007) 236-242. [106] R. Ansari, S. Sahmani, Small scale effect on vibrational response of single-walled carbon nanotubes with different boundary conditions based on nonlocal beam models, Commun. Nonlinear Sci. Numer. Simulat. 17 (2012) 1965-1979.

136

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

[107] C.M. Wang, Y.Y. Zhang, X.Q. He, Vibration of nonlocal Timoshenko beams, Nanotechnology 18 (2007) 105401. [108] N. Khosravian, H. Rafii-Tabar, Computational modelling of a nonviscous fluid flow in a multi-walled carbon nanotube modelled as a Timoshenko beam, Nanotechnology 19 (2008) 275703. [109] A. Benzair, A. Tounsi, A. Besseghier, H. Heireche, N. Moulay, L. Boumia, The thermal effect on vibration of single-walled carbon nanotubes using nonlocal Timoshenko beam theory, J. Phys. D Appl. Phys. 41 (2008) 225404-225413. [110] E. Ghavanloo, S.A. Fazelzadeh, Flow-thermoelastic vibration and instability analysis of viscoelastic carbon nanotubes embedded in viscous fluid, Physica E 44 (2011) 17-24. [111] Q. Wang, V.K. Varadan, Application of nonlocal elastic shell theory in wave propagation analysis of carbon nanotubes, Smart Mater. Struct. 16 (2007) 178-190. [112] H.S. Shen, C.L. Zhang, Torsional buckling and postbuckling of doublewalled carbon nanotubes by nonlocal shear deformable shell model, Compos. Struct. 92 (2010) 1073-1084. [113] F. Khademolhosseini, R.K.N.D. Rajapakse, A. Nojeh, Torsional buckling of carbon nanotubes based on nonlocal elasticity shell models, Comp. Mater. Sci. 48 (2010) 736-742. [114] R. Ansari, S. Sahmani, H. Rouhi, Axial buckling analysis of singlewalled carbon nanotubes in thermal environments via the RayleighRitz technique, Comp. Mater. Sci. 50 (2011) 3050-3055. [115] S.A. Fazelzadeh, E. Ghavanloo, Nonlocal anisotropic elastic shell model for vibrations of single-walled carbon nanotubes with arbitrary chirality, Compos. Struct. 94 (2012) 1016-1022. [116] W. Fl¨ ugge, Stresses in Shells, Springer, Berlin, 1960. [117] B. Gu, Y.W. Mai, C.Q. Ru, Mechanics of microtubules modeled as orthotropic elastic shells with transverse shearing, Acta Mech. 207 (2009) 195-209. 137

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

[118] M. Amabili, Nonlinear Vibrations and Stability of Shells and Plates, Cambridge University Press, New York, 2008. [119] C.Y. Wang, C.Q. Ru, A. Mioduchowski, Applicability and limitations of simplified elastic shell equations for carbon nanotubes, J. Appl. Mech. 71 (2004) 622-631. [120] C.Y. Wang, C.F. Li, S. Adhikari, Axisymmetric vibration of singlewalled carbon nanotubes in water, Phys. Lett. A 374 (2010) 2467-2474. [121] N. Silvestre, C.M. Wang, Y.Y. Zhang, Y. Xiang, Sanders shell model for buckling of single-walled carbon nanotubes with small aspect ratio, Compos. Struct. 93 (2011) 1683-1691. [122] Y. Yan, W.Q. Wang, L.X. Zhang, Applied multiscale method to analysis of nonlinear vibration for double-walled carbon nanotubes, Appl. Math. Model. 35 (2011) 2279-2289. [123] M.S. Hoseinzadeh, S.E. Khadem, Thermoelastic vibration and damping analysis of double-walled carbon nanotubes based on shell theory, Physica E 43 (2011) 1146-1154. [124] Y. Yang, C.W. Lim, A new nonlocal cylindrical shell model for axisymmetric wave propagation in carbon nanotubes, Adv. Sci. Lett. 4 (2011) 121-131. [125] T. Chang, H. Gao, Size dependent elastic properties of a single-walled carbon nanotube via a molecular mechanics model, J. Mech. Phys. Solids 51 (2003) 1059-1074. [126] L.F. Wang, Q.S. Zheng, J.Z. Liu, Q. Jiang, Size dependence of the thinwall models for carbon nanotubes, Phys. Rev. Lett. 95 (2005) 105501. [127] T. Chang, J. Geng, X. Guo, Prediction of chirality- and size-dependent elastic properties of single-walled carbon nanotubes via a molecular mechanics model, Proc. R. Soc. A 462 (2006) 2523-2540. [128] G. Cao, X. Chen, The effects of chirality and boundary conditions on the mechanical properties of single-walled carbon nanotubes, Int. J. Solids Struct. 44 (2007) 5447-5465.

138

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

[129] K. Chandraseker, S. Mukherjee, Coupling of extension and twist in single-walled carbon nanotubes, J. Appl. Mech. 73 (2006) 315-326. [130] H.W. Zhang, L. Wang, J.B. Wang, Z.Q. Zhang, Y.G. Zhang, Torsion induced by axial strain of double-walled carbon nanotubes, Phys. Lett. A 372 (2008) 3488-3492. [131] W. Mu, M. Li, W. Wang, Z.C.O. Yang, Study of axial strain-induced torsion of single-wall carbon nanotubes using the 2D continuum anharmonic anisotropic elastic model, New J. Phys. 11 (2009) 113049. [132] Y.Y. Zhang, V.B.C. Tan, C.M. Wang, Effect of chirality on buckling behavior of single-walled carbon nanotubes, J. Appl. Phys. 100 (2006) 074304. [133] Y. Yan, W. Wang, Axisymmetric vibration of SWCNTs in water with arbitrary chirality based on nonlocal anisotropic shell model, Appl. Math. Model. 39 (2015) 3016-3023. [134] L.J. Segerlind, Applied Finite Element Analysis, J. Wiley& Sons Inc, New York, 1984. [135] C. Shu, Differential quadrature and its application in engineering, Springer, Berlin, 2000. [136] G.R. Liu, Mesh Free Methods: Moving Beyond the Finite Element Method, Taylor & Francis, 2009. [137] F. Daneshmand, E. Ghavanloo, M. Amabili, Wave propagation in protein microtubules modeled as orthotropic elastic shells including transverse shear deformations, J. Biomech. 44 (2011) 1960-1966. [138] L.B. Freund, S. Suresh, Thin Film Materials, Cambridge University Press, Cambridge, 2003. [139] J.S. Bunch, A.M. van der Zande, S.S. Verbridge, I.W. Frank, D.M. Tanenbsum, J.M. Parpia, H.G. Craighead, P.L. McEuen, Electromechanical resonators from graphene sheets, Science 315 (2007) 490-493. [140] N. Lobontiu, B. Ilic, E. Garcia, T. Reissman, H.G. Craighead, Modeling of nanofabricated paddle bridges for resonant mass sensing, Rev. Sci. Instrum. 77 (2006) 073301. 139

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

[141] M.K. Zalalutdinov, J.W. Baldwin, M.H. Marcus, R.B. Reichenbach, J.M. Parpia, B.H. Houston, Two-dimensional array of coupled nanomechanical resonators, Appl. Phys. Lett. 88 (2006) 143504. [142] P. Lu, P.Q. Zhang, H.P. Lee, C.M. Wang, J.N. Reddy, Non-local elastic plate theories, Proc. R. Soc. A 463 (2007) 3225-3240. [143] J.N. Reddy, Mechanics of Laminated Composite Plates: Theory and Analysis, CRC Press, New York, 2004. [144] R.D. Mindlin, Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates, J. Appl. Mech. 18 (1951) 31-38. [145] D. Kahn, K.W. Kim, M.A. Stroscio, Quantized vibrational modes of nanospheres and nanotubes in the elastic continuum model, J. Appl. Phys. 89 (2001) 5107. [146] E. Ghavanloo, S.A. Fazelzadeh, Nonlocal shell model for predicting axisymmetric vibration of spherical shell-like nanostructures, Mech. Adv. Mater. Struct. 22 (2015) 597-603. [147] W. Soedel, Vibrations of Shells and Plates, Marcel Dekker, Inc: New York, 2004. [148] P.M. Morse, H. Feshbach, Methods of Theoretical Physics Part II, McGraw-Hill, New York, 1953. [149] E. Ghavanloo, S.A. Fazelzadeh, H. Rafii-Tabar, Analysis of radial breathing-mode of nanostructures with various morphologies: a critical review, Int. Mater. Rev. 60 (2015) 312-329. [150] M. Sastry, A. Swami, S. Mandal, P.R. Selvakannan, New approaches to the synthesis of anisotropic, coreshell and hollow metal nanostructures, J. Mater. Chem. 15 (2005) 3161-3174. [151] H. Portals, N. Goubet, L. Saviot, S. Adichtchev, D.B. Murray, A. Mermet, E. Duval, M.P. Pilni, Probing atomic ordering and multiple twinning in metal nanocrystals through their vibrations, Proc. Natl. Acad. Sci. 105 (2008) 14784-14789. [152] Y. Huang, J. Wu, K.C. Hwang, Thickness of graphene and single-wall carbon nanotubes, Phys. Rev. B 74 (2006) 245413. 140

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

[153] S. Kasas, G. Dietler, Techniques for measuring microtubule stiffness, Curr. Nanosci. 3 (2007) 79-96. [154] G. M. Odegard, T. S. Gates, L. M. Nicholson, K. E. Wise, Equivalentcontinuum modeling of nano-structured materials, Compos. Sci. Technol. 62 (2002) 1869-1880. [155] C.Y. Wang, L.C. Zhang, A critical assessment of the elastic properties and effective wall thickness of single-walled carbon nanotubes, Nanotechnology 19 (2008) 75705. [156] J.P. Lu, Elastic properties of carbon nanotubes and nanoropes, Phys. Rev. Lett. 79 (1997) 1297-1300. [157] A. Krishnan, E. Dujardin, T.W. Ebbesen, P.N. Yianilos, M.M.J. Treacy, Young's modulus of single-walled nanotubes, Phys. Rev. B 58 (1998) 14013-14019. [158] X. Zhou, J. Zhou, Z.C. Ou-Yang, Strain energy and Young's modulus of singlewall carbon nanotubes calculated fromelectronic energy-band theory, Phys. Rev. B 62 (2000) 13692-13696. [159] T. Belytschko, S.P. Xiao, G.C. Schatz, R.S. Ruoff, Atomistic simulations of nanotube fracture, Phys. Rev. B 65 (2002) 235430. [160] C. Li, T.W. Chou, A structural mechanics approach for analysis of carbon nanotubes, Int. J. Sol. Struct. 40 (2003) 2487-2499. [161] T. Vodenitcharova, L.C. Zhang, Effective wall thickness of a singlewalled carbon nanotube, Phys. Rev. B 68 (2003) 165401. [162] A. Pantano, D.M. Parks, M.C. Boyce, Mechanics of deformation of single- and multi-wall carbon nanotubes, J. Mech. Phys. Solids 52 (2004) 789-821. [163] J. Cai, C. Y. Wang, T. Yu, S. Yu, Wall thickness of single-walled carbon nanotubes and its Young's modulus, Phys. Scr. 79 (2009) 025702. [164] L.C. Zhang, On the mechanics of single-walled carbon nanotubes, J. Mater. Process. Tech. 209 (2009) 4223-4228.

141

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

[165] L. Shen, J. Li, Transversely isotropic elastic properties of single-walled carbon nanotubes, Phys. Rev. B 69 (2004) 045414. [166] A. Montazeri, M. Sadeghi, R. Naghdabadi, H. Rafii-Tabar, Multiscale modeling of the effect of carbon nanotube orientation on the shear deformation properties of reinforced polymer-based composites, Phys. Lett. A 375 (2011) 1588-1597. [167] J. Wu, K.C. Hwang, Y. Huang, An atomistic-based finite-deformation shell theory for single-wall carbon nanotubes, J. Mech. Phys. Solids 56 (2008) 279-292. [168] J. Peng, J. Wu, K.C. Hwang, J. Song, Y. Huang, Can a single-wall carbon nanotube be modeled as a thin shell?, J. Mech. Phys. Solids 56 (2008) 2213-2224. [169] C.Q. Ru, Chirality-dependent mechanical behavior of carbon nanotubes based on an anisotropic elastic shell model, Math. Mech. Solids 14 (2009) 88-101. [170] T. Chang, A molecular based anisotropic shell model for single-walled carbon nanotubes, J. Mech. Phys. Solids 58 (2010) 1422-1433. [171] A. Hinchliffe, Molecular Modelling for Beginners, J. Wiley& Sons Inc, New York, 2008. [172] K.N. Kudin, G.E. Scuseria, I.B. Yakobson, C2F, BN, and C nanoshell elasticity from ab initio computations, Phys. Rev. B 64 (2001) 235406. [173] M. Arroyo, T. Belytschko, Finite crystal elasticity of carbon nanotubes based on the exponential CauchyBorn rule, Phys. Rev. B 69 (2004) 115415. [174] C.D. Reddy, S. Rajendran, K.M. Liew, Equilibrium configuration and continuum elastic properties of finite sized graphene, Nanotechnology 17 (2006) 864-870. [175] A. Hemmasizadeh, M. Mahzoon, E. Hadi, R. Khandan, A method for developing the equivalent continuum model of a single layer graphene sheet, Thin Solid Films 516 (2008) 7636-7640.

142

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

[176] C.D. Reddy, S. Rajendran, K.M. Liew, Equilibrium continuum modeling of graphene sheets, Int. J. Nanosci. 4 (2005) 631-636. [177] A. Sakhaee-Pour, Elastic properties of single-layered graphene sheet, Solid State Commun. 149 (2009) 91-95. [178] M. Poot, H.S.J. van der Zant, Nanomechanical properties of few-layer graphene membranes, Appl. Phys. Lett. 92 (2008) 063111. [179] F. Scarpa, S. Adhikari, A.S. Phani, Effective elastic mechanical properties of single layer graphene sheets, Nanotechnology 20 (2009) 065709. [180] Z. Ni, H. Bu, M. Zou, H. Yi, K. Bi, Y. Chen, Anisotropic mechanical properties of graphene sheets from molecular dynamics, Physica B 405 (2010) 1301-1306. [181] E. Mareni; A. Ibrahimbegovic, J. Sori, P.A. Guidault, Homogenized elastic properties of graphene for small deformations, Materials 6 (2013) 3764-3782. [182] L. Shen, H.S. Shen, C.L. Zhang, Temperature-dependent elastic properties of single layer graphene sheets, Mater. Design 31 (2010) 44454449. [183] P. Steve, P. Crozier, A. Thompson, LAMMPS-large-scale atomic/molecular massively parallel simulator, Sandia National Laboratories 18 (2007). [184] S.J. Stuart, A. B. Tutein, J. A. Harrison, A reactive potential for hydrocarbons with intermolecular interactions, J. Chem. Phys. 112 (2000): 6472-6486. [185] M. Aydogdu, Longitudinal wave propagation in nanorods using a general nonlocal unimodal rod theory and calibration of nonlocal parameter with lattice dynamics, Int. J. Eng. Sci. 56 (2012) 17-28. [186] C.Y. Wang, J. Zhang, Y.Q. Fei, T. Murmu, Circumferential nonlocal effect on vibrating nanotubules, Int. J. Mech. Sci. 58 (2012) 86-90. [187] Y.J. Liang, Q. Han, Prediction of nonlocal scale parameter for carbon nanotubes, Sci. China Phys. Mech. 55(9) (2012) 1670-1678. 143

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

[188] S. Narendar, D.R. Mahapatra, S. Gopalakrishnan, Prediction of nonlocal scaling parameter for armchair and zigzag single-walled carbon nanotubes based on molecular structural mechanics, nonlocal elasticity and wave propagation, Int. J. Eng. Sci. 49 (2011) 509-522. [189] M. Malag, E. Benvenuti, A. Simone, One-dimensional nonlocal elasticity for tensile single-walled carbon nanotubes: A molecular structural mechanics characterization, Eur. J. Mech. A/Solids 54 (2015): 160-170. [190] L.F. Wang, H.Y. Hu, Flexural wave propagation in single-walled carbon nanotubes, Phys. Rev. B 71 (2005) 195412. [191] Y.Q. Zhang, G.R. Liu, X.Y. Xie, Free transverse vibrations of doublewalled carbon nanotubes using a theory of nonlocal elasticity, Phys. Rev. B 71 (2005) 195404. [192] Y.Y.Q. Zhang, G.R. Liu, J.S. Wang, Small-scale effects on buckling of multiwalled carbon nanotubes under axial compression, Phys. Rev. Lett. 70 (2004) 205430. [193] A. Sears, R.C. Batra, Macroscopic properties of carbon nanotubes from molecular-mechanics simulations, Phys. Rev. B 69 (2004) 235406. [194] Q. Wang, Wave propagation in carbon nanotubes via nonlocal continuum mechanics, J. Appl. Phys. 98 (2005) 124301. [195] G.Q. Xie, X. Han, G.R. Liu, S.Y. Long, Effect of small size-scale on the radial buckling pressure of a simply supported multiwalled carbon nanotube, Smart Mater. Struct. 15 (2006) 1143-1149. [196] W.H. Duan, C.M. Wang, Y.Y. Zhang, Calibration of nonlocal scaling effect parameter for free vibration of carbon nanotubes by molecular dynamics, J. Appl. Phys. 101 (2007) 024305. [197] Y.G. Hu, K.M. Liew, Q. Wang, X.Q. He, B.I. Yakobson, Nonlocal shell model for elastic wave propagation in single- and double-walled carbon nanotubes, J. Mech. Phys. Solids 56 (2008) 3475-3485. [198] Y.Y. Zhang, C.M. Wang, W.H. Duan, Y. Xiang, Z. Zong, Assessment of continuum mechanics models in predicting buckling strains of singlewalled carbon nanotubes, Nanotechnology 20 (2009) 395707. 144

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

[199] R. Ansari, H. Rouhi, S. Sahmani, Calibration of the analytical nonlocal shell model for vibrations of double-walled carbon nanotubes with arbitrary boundary conditions using molecular dynamics, Int. J. Mech. Sci. 53 (2011) 786-792. [200] W.H. Duan, N. Challamel, C.M. Wang, Z. Ding, Development of analytical vibration solutions for microstructured beam model to calibrate length scale coefficient in nonlocal Timoshenko beams, J. Appl. Phys. 114 (2013) 104312. [201] X. Zhang,K. Jiao,P. Sharma,B.I. Yakobson, An atomistic and nonclassical continuum field theoretic perspective of elastic interactions between defects (force dipoles) of various symmetries and application to graphene, J. Mech. Phys. Solids 54(11) (2006) 2304-2329. [202] R. Ansari, S. Sahmani, B. Arash, Nonlocal plate model for free vibrations of single-layered graphene sheets, Phys. Lett. A 375(1) (2010) 53-62. [203] L.Y. Huang, Q. Han, Y.J. Liang, Calibration of nonlocal scale effect parameter for bending single-layered graphene sheet under molecular dynamics, Nano 7(5) (2012) 1250033. [204] S. Iijima, C. Brabec, A. Maiti, J. Bernholc, Structural exibility of carbon nanotubes, J. Chem. Phys. 104(5) (1996) 2089-2092. [205] S.Y. Wu, Reversible electromechanical characteristics of carbon nanotubes under local-probe manipulation, Nature 405 (2000) 769-772. [206] B. Motevalli, A. Montazeri, J.Z. Liu, H. Rafii-Tabar, Comparison of continuum-based and atomistic-based modeling of axial buckling of carbon nanotubes subject to hydrostatic pressure, Comp. Mater. Sci. 79 (2013) 619-626. [207] B. Motevalli, A. Montazeri, R. Tavakoli-Darestani, H. Rafii-Tabar, Modeling the buckling behavior of carbon nanotubes under simultaneous combination of compressive and torsional loads, Physica E 46 (2012) 139-148. References

145

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

[208] C. M. Wang, S. Kitipornchai, C. W. Lim, M. Eisenberger, Beam bending solutions based on nonlocal Timoshenko beam theory, J. Eng. Mech. 134 (2008) 475-481. [209] C. W. Lim, On the truth of nanoscale for nanobeams based on nonlocal elastic stress field theory: equilibrium, governing equation and static deflection, Appl. Math. Mech. 31(1) (2010) 37-54. [210] C. W. Lim, Q. Yang, Nonlocal thermal-elasticity for nanobeam deformation: exact solutions with stiffness enhancement effects, J. Appl. Phys. 110 (2011) 013514. [211] C. Li, L. Yao, W. Chen, S. Li, Comments on nonlocal effects in nanocantilever beams, Int. J. Eng. Sci. 87 (2015) 47-57. [212] L.J. Sudak, Column buckling of multiwalled carbon nanotubes using nonlocal continuum mechanics, J. Appl. Phys. 94(11) (2003) 7281-7287. [213] Q. Wang, V.K. Varadan, S.T. Quek, Small scale effect on elastic buckling of carbon nanotubes with nonlocal continuum models, Phys. Lett. A 357 (2006) 130-135. [214] D. Kumar, C. Heinrich, A.M. Waas, Buckling analysis of carbon nanotubes modeled using nonlocal continuum theories, J. Appl. Phys. 103 (2008) 073521. [215] I.W. Frank, P.B. Deotare, M.W. McCutcheon, M. Loncar, Programmable photonic crystal nanobeam cavities, Opt. Express 18 (2010) 8705-8712. [216] T. Murmu, S. Adhikari, Nonlocal transverse vibration of doublenanobeam-systems, J. Appl. Phys. 108 (2010) 083514. [217] T. Murmu, S. Adhikari, Axial instability of double-nanobeam-systems, Phys. Lett. A 375 (2011) 601-608. [218] S.S. Wong, E. Joselevich, A.T. Woolley, C.L. Cheung, C.M. Lieber, Covalently functionalized nanotubes as nanometre-sized probes in chemistry and biology, Nature 394 (1998) 52-55.

146

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

[219] T. Natsuki, Q.Q. Ni, M. Endo, Stability analysis of double-walled carbon nanotubes as AFM probes based on a continuum model, Carbon 49 (2011) 2532-2537. [220] J.X. Shi, T. Natsuki, X.W. Lei, Q.Q. Ni, Buckling instability of carbon nanotube atomic force microscope probe clamped in an elastic medium, J. Nanotech. Eng. Med. 3 (2012) 020903. [221] Q. Wang, V.K. Varadan, Vibration of carbon nanotubes studied using nonlocal continuum mechanics, Smart Mater. Struct. 15 (2006) 659666. [222] E.H. Feng, R.E. Jones, Equilibrium thermal vibrations of carbon nanotubes, Phys. Rev. B 81 (2010) 125436. [223] M. imek, Nonlocal effects in the forced vibration of an elastically connected double-carbon nanotube system under a moving nanoparticle, Comp. Mater. Sci. 50 (2011) 2112-2123. [224] H. Wang, K. Dong, F. Men, Y.J. Yan, X. Wang, Influences of longitudinal magnetic field on wave propagation in carbon nanotubes embedded in elastic matrix, Appl. Math. Model. 34 (2010) 878-889. [225] T. Murmu, M.A. McCarthy, S. Adhikari, Vibration response of doublewalled carbon nanotubes subjected to an externally applied longitudinal magnetic field: A nonlocal elasticity approach, J. Sound Vib. 331 (2012) 5069-5086. [226] R.D. Firouz-Abadi, A.R. Hosseinian, Resonance frequencies and stability of a current-carrying suspended nanobeam in a longitudinal magnetic field, Theor. Appl. Mech. Lett. 2 (2012) 031012. [227] L. Chico, V.H. Crespi, L.X. Benedict, S.G. Louie, M.L. Cohen, Pure carbon nanoscale devices: nanotube heterojunctions, Phys. Rev. Lett. 76 (1996) 971-974. [228] Z. Qin, Q.H. Qin, X.Q. Feng, Mechanical property of carbon nanotubes with intramolecular junctions: molecular dynamics simulations, Phys. Lett. A 372 (2008) 6661-6666.

147

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

[229] S. Filiz, M. Aydogdu, Axial vibration of carbon nanotube heterojunctions using nonlocal elasticity, Comp. Mater. Sci. 49 (2010) 619-627. [230] Y.M. Fu, J.W. Hong, X.Q. Wang, Analysis of nonlinear vibration for embedded carbon nanotubes, J. Sound Vib. 296 (2006) 746-756. [231] M.A. Hawwa, H.M. Al-Qahtani, Nonlinear oscillations of a doublewalled carbon nanotube, Comp. Mater. Sci. 48 (2010) 140-143. [232] R. Ansari, M. Hemmatnezhad, Nonlinear vibrations of embedded multi-walled carbon nanotubes using a variational approach, Math. Comput. Model. 53 (2011) 927-938. [233] A. Hajnayeb, S.E. Khadem, Nonlinear vibration and stability analysis of a double-walled carbon nanotube under electrostatic actuation, J. Sound Vib. 331 (2012) 2443-2456. [234] J. Yang, L.L. Ke, S. Kitipornchai, Nonlinear free vibration of singlewalled carbon nanotubes using nonlocal Timoshenko beam theory, Physica E 42 (2010) 1727-1735. [235] H.S. Shen, C.L. Zhang, Nonlocal beam model for nonlinear analysis of carbon nanotubes on elastomeric substrates, Comp. Mater. Sci. 50 (2011) 1022-1029. [236] Q. Wang, Atomic transportation via carbon nanotubes, Nano Lett. 9 (2009) 245-249. [237] B.D. Chen, C.L. Yang, J.S. Yang, M.S. Wang, X.G. Ma, The translocation of DNA oligonucleotide with the water stream in carbon nanotubes, Comput. Theor. Chem. 991 (2012) 93-97. [238] K. Kiani, Nonlinear vibrations of a single-walled carbon nanotube for delivering of nanoparticles, Nonlinear Dyn. 76 (2014) 1885-1903. [239] P. Lu, H.P. Lee, C. Lu, P.Q. Zhang, Application of nonlocal beam models for carbon nanotubes, Int. J. Solids Struct. 44 (2007) 52895300. [240] S. Narendar, S. Gopalakrishnan, Nonlocal scale effects on wave propagation in multi-walled carbon nanotubes, Comp. Mater. Sci. 47 (2009) 526-538. 148

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

[241] Y. Yang, L. Zhang, C.W. Lim, Wave propagation in double-walled carbon nanotubes on a novel analytically nonlocal Timoshenko-beam model, J. Sound Vib. 330 (2011) 1704-1717. [242] M. Aydogdu, Longitudinal wave propagation in multiwalled carbon nanotubes, Compos. Struct. 107 (2014) 578-584. [243] C.Q. Ru, Elastic models for carbon nanotubes, in: H. S. Nalwa (Ed.), Encyclopedia of Nanoscience and Nanotechnology, American Scientific, Stevenson Ranch, CA, Vol. 2, 2004, pp. 731-744. [244] M. Shaban, A. Alibeigloo, Three dimensional vibration and bending analysis of carbon nanotubes embedded in elastic medium based on theory of elasticity, Lat. Am. J. Solids Stru. 11 (2014) 2122-2140. [245] X. Q. He, C. Qu, Q. H. Qin, C. M. Wang, Buckling and postbuckling analysis of multi-walled carbon nanotubes based on the continuum shell model, Int. J. Struct. Stab. Dy. 7(4) (2007) 629-645. [246] A. Ghorbanpour Arani, R. Rahmani, A. Arefmanesh, Elastic buckling analysis of single-walled carbon nanotube under combined loading by using the ANSYS software, Physica E 40 (2008) 2390-2395. [247] C.Y. Wang, L.C. Zhang, An elastic shell model for characterizing single-walled carbon nanotubes, Nanotechnology 19 (2008) 195704. [248] L. Wang, W. Guo, H. Hu, Group velocity of wave propagation in carbon nanotubes, Proc. R. Soc. A 464 (2008) 1423-1438. [249] T. Usuki, K. Yogo, Beam equations for multi-walled carbon nanotubes derived from Fl¨ ugge shell theory, Proc. R. Soc. A 465 (2009) 1199-1226. [250] F. Khademolhosseini, Nonlocal continuum shell models for torsion of single-walled carbon nanotubes (M.Sc. Thesis) University of British Columbia, Vancouver, 2009. [251] H.S. Shen, C.L. Zhang, Nonlocal shear deformable shell model for postbuckling of axially compressed double-walled carbon nanotubes embedded in an elastic matrix, J. Appl. Mech. 77 (2010) 041006.

149

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

[252] R. Ansari, S. Sahmani, H. Rouhi, RayleighRitz axial buckling analysis of single-walled carbon nanotubes with different boundary conditions, Phys. Lett. A 375 (2011) 1255-1263. [253] Nanorex Inc. NanoHive-1 v.1.2.0-b1, http://www.nanoengineer-1.com. [254] R. Ansari, A. Shahabodini, H. Rouhi, A thickness-independent nonlocal shell model for describing the stability behavior of carbon nanotubes under compression, Compos. Struct. 100 (2013) 323-331. [255] W. Clauss, D.J. Bergeron, A.T. Johnson, Atomic resolution STM imaging of a twisted single-wall carbon nanotube, Phys. Rev. B 59(8) (1998) R4266. [256] C.E. Giusca, Y. Tison, S.R.P. Silva, Atomic and electronic structure in collapsed carbon nanotubes evidenced by scanning tunneling microscopy, Phys. Rev. B 76 (2007) 035429. [257] C.E. Giusca, Y. Tison, S.R.P. Silva, Evidence for metal-semiconductor transitions in twisted and collapsed double-walled carbon nanotubes by scanning tunneling microscopy, Nano Lett. 8(10) (2008) 3350-3356. [258] S.J. Papadakis, A.R. Hall, P.A. Williams, L. Vicci, M.R. Falvo, R. Superfine, S. Washburn, Resonant oscillators with carbon-nanotube torsion springs, Phys. Rev. Lett. 93 (2004) 146101-146101. [259] A.R. Hall, L. An, J. Liu, L. Vicci, M.R. Falvo, R. Superfine, S. Washburn, Experimental measurement of single-wall carbon nanotube torsional properties, Phys. Rev. Lett. 96(25) (2006) 256102. [260] A.R. Hall, M.R. Falvo, R. Superfine, S. Washburn, Electromechanical response of single-walled carbon nanotubes to torsional strain in a selfcontained device, Nat. Nanotechnol. 2 (2007) 413-416. [261] A.R. Hall, M.R. Falvo, R. Superfine, S. Washburn, A self-sensing nanomechanical resonator built on a single-walled carbon nanotube, Nano Lett. 8(11) (2008) 3746-3749. [262] N.S. Martys, R.D. Mountain, Velocity Verlet algorithm for dissipativeparticle-dynamics-based models of suspensions, Phys. Rev. E 59 (1999) 3733. 150

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

[263] D.W. Brenner, O.A. Shenderova, J.A. Harrison, S.J. Stuart, B. Ni, S.B. Sinnott, A second generation reactive empirical bond order (REBO) potential energy expression for hydrocarbons, J. Phys. Condens. Matter. 14 (2002) 783-802. [264] L.A. Girifalco, M. Hodak, R.S. Lee, Carbon nanotubes, buckyballs, ropes, and a universal graphitic potential, Phys. Rev. B 62 (2000) 13104-13110. [265] M.J. Hao, X.M. Guo, Q. Wang, Small-scale effect on torsional buckling of multi-walled carbon nanotubes, Eur. J. Mech. A/Solids 29 (2010) 49-55. [266] B. Arash, R. Ansari, Evaluation of nonlocal parameter in the vibrations of single-walled carbon nanotubes with initial strain, Physica E 42 (2010) 2058-2064. [267] S.L. Das, T. Mandal, S.S. Gupta, Inextensional vibration of zig-zag single-walled carbon nanotubes using nonlocal elasticity theories, Int. J. Solids Struct. 50 (2013) 2792-2797. [268] S.S. Gupta, F.G. Bosco, R.C. Batra, Wall thickness and elastic moduli of single-walled carbon nanotubes from frequencies of axial, torsional and inextensional modes of vibration, Comput. Mater. Sci. 47 (2010) 1049-1059. [269] M. Cadek, J.N. Coleman, V. Barron, K. Hedicke, W.J. Blau, Morphological and mechanical properties of carbon-nanotube-reinforced semicrystalline and amorphous polymer composites, Appl. Phys. Lett. 81 (2002) 5123-5125. [270] S.A. Fazelzadeh, S. Pouresmaeeli, E. Ghavanloo, Aeroelastic characteristics of functionally graded carbon nanotube-reinforced composite plates under a supersonic flow, Comput. Methods Appl. Mech. Engrg. 285 (2015) 714-729. [271] G. Mikhasev, On localized modes of free vibrations of single-walled carbon nanotubes embedded in nonhomogeneous elastic medium, ZAMMZ. Angew. Math. Mech. 94 (2014) 130-141.

151

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

[272] M. Mitra, S. Gopalakrishnan, Vibrational characteristics of singlewalled carbon-nanotube: time and frequency domain analysis, J. Appl. Phys. 101 (2007) 114320. [273] V. Sundararaghavan, A. Waas, Non-local continuum modeling of carbon nanotubes: physical interpretation of non-local kernels using atomistic simulations, J. Mech. Phys. Solids 59 (2011) 1191-1203. [274] M. M. Selim, Dispersion of dilatation wave propagation in single-wall carbon nanotubes using nonlocal scale effects, J. Nanopart. Res. 13 (2011) 1229-1235. [275] L. Wang, H. Hu, W. Guo, Validation of the non-local elastic shell model for studing longitudinal waves in single-walled carbon nanotubes, Nanotechnology 17 (2006) 1408-1415. [276] Z.M. Islam, P. Jia, C.W. Lim, Torsional wave propagation and vibration of circular nanostructures based on nonlocal elasticity theory, Int. J. Appl. Mech. 6 (2014) 1450011. [277] S. Narendar, S. Gopalakrishnan, Nonlocal continuum mechanics formulation for axial, flexural, shear and contraction coupled wave propagation in single walled carbon nanotubes, Lat. Am. J. Solids Stru. 9 (2012) 497-513. [278] F. Khademolhosseini, A.S. Phani, A. Nojeh, N. Rajapakse, Nonlocal continuum modeling and molecular dynamics simulation of torsional vibration of carbon nanotubes, IEEE T. Nanotechnol. 11 (2012) 34-43. [279] Y. Gao, P. Hao, Mechanical properties of monolayer graphene under tensile and compressive loading, Physica E 41 (2009) 1561-1566. [280] Q. Lu, R. Huang, Nonlinear mechanics of single-atomic-layer graphene sheets, Int. J. Appl. Mech. 1 (2009) 443-467. [281] C. Androulidakis, E.N. Koukaras, O. Frank, G. Tsoukleri, D. Sfyris, J. Parthenios, N. Pugno, K. Papagelis, K.S. Novoselov, C. Galiotis, Failure processes in embedded monolayer graphene under axial compression, Sci Rep. 4 (2014) 5271.

152

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

[282] W.H. Duan, K. Gong, Q. Wang, Controlling the formation of wrinkles in a single layer graphene sheet subjected to in-plane shear, Carbon 49 (2011) 3107-3112. [283] S.C. Pradhan, Buckling of single layer graphene sheet based on nonlocal elasticity and higher order shear deformation theory, Phys. Lett. A 373 (2009) 4182-4188. [284] S.C. Pradhan, T. Murmu, Small scale effect on the buckling of singlelayered graphene sheets under biaxial compression via nonlocal continuum mechanics, Comput. Mater. Sci. 47 (2009) 268-274. [285] S.C. Pradhan, T. Murmu, Small scale effect on the buckling analysis of single-layered graphene sheet embedded in an elastic medium based on nonlocal plate theory, Physica E 42 (2010) 1293-1301. [286] S.C. Pradhan, A. Kumar, Buckling analysis of single layered graphene sheet under biaxial compression using nonlocal elasticity theory and DQ method, J. Comput. Theor. Nanos. 8 (2011) 1325-1334. [287] A. Farajpour, M. Mohammadi, A.R. Shahidi, M. Mahzoon, Axisymmetric buckling of the circular graphene sheets with the nonlocal continuum plate model, Physica E 43 (2011) 1820-1825. [288] H. Babaei, A. R. Shahidi, Small-scale effects on the buckling of quadrilateral nanoplates based on nonlocal elasticity theory using the Galerkin method, Arch. Appl. Mech. 81 (2011) 1051-1062. [289] R. Ansari, S. Sahmani, Prediction of biaxial buckling behavior of singlelayered graphene sheets based on nonlocal plate models and molecular dynamics simulations, Appl. Math. Model. 37 (2013) 7338-7351. [290] I.S. Radebe, S. Adali, Buckling and sensitivity analysis of nonlocal orthotropic nanoplates with uncertain material properties, Compos. Part B 56 (2014) 840-846. [291] Y. Ben-Haim, I. Elishakoff, Convex Models of Uncertainty in Applied Mechanics, Elsevier, Amsterdam, 1990. [292] S.C. Pradhan, A. Kumar, Vibration analysis of orthotropic graphene sheets embedded in Pasternak elastic medium using nonlocal elasticity 153

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

theory and differential quadrature method, Comput. Mater. Sci. 50 (2010) 239-245. [293] S.C. Pradhan, A. Kumar, Vibration analysis of orthotropic graphene sheets using nonlocal elasticity theory and differential quadrature method, Compos. Struct. 93 (2011) 774-779. [294] L. Shen, H.S. Shen, C.L. Zhang, Nonlocal plate model for nonlinear vibration of single layer graphene sheets in thermal environments, Comput. Mater. Sci. 48 (2010) 680-685. [295] C.Y. Wang, T. Murmu, S. Adhikari, Mechanisms of nonlocal effect on the vibration of nanoplates, Appl. Phys. Lett. 98 (2011) 153101. [296] Y. Su, H. Wei, R. Gao, Z. Yang, J. Zhang, Z. Zhong, Y. Zhang, Exceptional negative thermal expansion and viscoelastic properties of graphene oxide paper, Carbon 50 (2012) 2804-2809. [297] S. Pouresmaeeli, E. Ghavanloo, S.A. Fazelzadeh, Vibration analysis of viscoelastic orthotropic nanoplates resting on viscoelastic medium, Compos. Struct. 96 (2013) 405-410. [298] S.C. Pradhan, J.K. Phadikar, Small scale effect on vibration of embedded multilayered graphene sheets based on nonlocal continuum models, Phys. Lett. A 373 (2009) 1062-1069. [299] R. Ansari, R. Rajabiehfard, B. Arash, Nonlocal finite element model for vibrations of embedded multi-layered graphene sheets, Comput. Mater. Sci. 49 (2010) 831-838. [300] R. Ansari, B. Arash, H. Rouhi, Nanoscale vibration analysis of embedded multi-layered graphene sheets under various boundary conditions, Comput. Mater. Sci. 50 (2011) 3091-3100. [301] R. Ansari, B. Arash, H. Rouhi, Vibration characteristics of embedded multi-layered graphene sheets with different boundary conditions via nonlocal elasticity, Compos. Struct. 93 (2011) 2419-2429. [302] E. Jomehzadeh, A.R. Saidi, A study on large amplitude vibration of multilayered graphene sheets, Comput. Mater. Sci. 50 (2011) 10431051. 154

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

[303] R.M. Lin, Nanoscale vibration characterization of multi-layered graphene sheets embedded in an elastic medium, Comput. Mater. Sci. 53 (2012) 44-52. [304] R.M. Lin, Nanoscale vibration characteristics of multi-layered graphene sheets, Mech. Syst. Signal Pr. 29 (2012) 251-261. [305] H. Babaei, A.R. Shahidi, Vibration of quadrilateral embedded multilayered graphene sheets based on nonlocal continuum models using the Galerkin method, Acta Mech. Sin. 27 (2011) 967-976. [306] T. Murmu, S. Adhikari, Nonlocal vibration of bonded doublenanoplate-systems, Compos. Part B 42 (2011) 1901-1911. [307] S. Pouresmaeeli, S.A. Fazelzadeh, E. Ghavanloo, Exact solution for nonlocal vibration of double-orthotropic nanoplates embedded in elastic medium, Compos. Part B 43 (2012) 3384-3390. [308] H.S. Shen, Y.M. Xu, C.L. Zhang, Prediction of nonlinear vibration of bilayer graphene sheets in thermal environments via molecular dynamics simulations and nonlocal elasticity, Comput. Methods Appl. Mech. Engrg. 267 (2013) 458-470. [309] R. Nazemnezhad, S. Hosseini-Hashemi, Free vibration analysis of multilayer graphene nanoribbons incorporating interlayer shear effect via molecular dynamics simulations and nonlocal elasticity, Phys. Lett. A 378 (2014) 3225-3232. [310] J.J. Li, K.D Zhu, All-optical mass sensing with coupled mechanical resonator systems, Phys. Rep. 525 (2013) 223-254. [311] Z.B. Shen, H.L. Tang, D.K. Li, G.J. Tang, Vibration of single-layered graphene sheet-based nanomechanical sensor via nonlocal Kirchhoff plate theory, Comput. Mater. Sci. 61 (2012) 200-205. [312] A. Sakhaee-Pour, M.T. Ahmadian, A. Vafai, Applications of singlelayered graphene sheets as mass sensors and atomistic dust detectors, Solid State Commun. 145 (2008) 168-172. [313] S. Adhikari, R.Chowdhury, Zeptogram sensing from gigahertz vibration: graphene based nanosensor, Physica E 44 (2012) 1528-1534. 155

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

[314] T. Murmu, S. Adhikari, Nonlocal mass nanosensors based on vibrating monolayer graphene sheets, Sensor. Actuat. B-Chem. 188 (2013) 13191327. [315] H.L. Lee, Y.C. Yang, W.J. Chang, Mass detection using a graphenebased nanomechanical resonator, Jpn. J. Appl. Phys. 52 (2013) 025101. [316] T. Natsuki, J.X. Shi, Q.Q. Ni, Vibration analysis of nanomechanical mass sensor using double-layered graphene sheets resonators, J. Appl. Phys. 114 (2013) 094307. [317] S.A. Fazelzadeh, E. Ghavanloo, Nanoscale mass sensing based on vibration of single-layered graphene sheet in thermal environments, Acta Mech. Sin. 30 (2014) 84-91. [318] S.K. Jalali, M.H. Naei, N.M. Pugno, A mixed approach for studying size effects and connecting interactions of planar nano structures as resonant mass sensors, Microsyst. Technol. 21 (2015) 2375-2386. [319] S.K. Jalali, M.H. Naei, N.M. Pugno, Graphene-based resonant sensors for detection of ultra-fine nanoparticles: molecular dynamics and nonlocal elasticity investigations, Nano 10 (2015) 1550024. [320] S. Narendar, D.R. Mahapatra, S. Gopalakrishnan, Investigation of the effect of nonlocal scale on ultrasonic wave dispersion characteristics of a monolayer graphene, Comput. Mater. Sci. 49 (2010) 734-742. [321] S. Narendar, S. Gopalakrishnan, Nonlocal continuum mechanics based ultrasonic flexural wave dispersion characteristics of a monolayer graphene embedded in polymer matrix, Compos. Part B 43 (2012) 3096-3103. [322] H. Liu, J.L.Yang, Elastic wave propagation in a single-layered graphene sheet on two-parameter elastic foundation via nonlocal elasticity, Physica E 44 (2012) 1236-1240. [323] B. Arash, Q. Wang, K.M. Liew, Wave propagation in graphene sheets with nonlocal elastic theory via finite element formulation, Comput. Methods Appl. Mech. Engrg. 223-224 (2012) 1-9.

156

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

[324] S.A. Fazelzadeh, E. Ghavanloo, Coupled axisymmetric vibration of nonlocal fluid-filled closed spherical membrane shell, Acta Mech. 223 (2012) 2011-2020. [325] E. Ghavanloo, S.A. Fazelzadeh, Nonlocal elasticity theory for radial vibration of nanoscale spherical shells, Eur. J. Mech. A/Solids 41 (2013) 37-42. [326] A. Vassallo, L. Pang, P. Coleclarke, M.Wilson, Emission FTIR study of C60 thermal stability and oxidation, J. Am. Chem. Soc. 113 (1991) 7820-7821. [327] R.E. Stanton, M.D. Newton, Normal vibrational modes of Buckminsterfullerene, J. Phys. Chem. 92 (1988) 2141-2145. [328] R.A. Jishi, R.M. Mirie, M.S. Dresselhaus, Force-constant model for the vibrational modes in C60, Phys. Rev. B 45 (1992) 13685-13689. [329] L.T. Chadderton, Axisymmetric vibrational modes of fullerene C60, J. Phys. Chem. Solids 54 (1993) 1027-1033. [330] S. Adhikari, R. Chowdhury, Vibration spectra of fullerene family, Phys. Lett. A 375 (2011) 2166-2170. [331] R. Zaera, J. Fernndez-Sez, J.A. Loya, Axisymmetric free vibration of closed thin spherical nano-shell, Compos. Struct. 104 (2013) 154-161. [332] K. Tachibana, S. Tachibana, Application of ultrasound energy as a new drug delivery system, Jpn. J. Appl. Phys. 38 (1999) 3014-3019. [333] J. Hu, Z. Qiu, T.C. Su, Axisymmetric vibrations of a viscous-fluid-filled piezoelectric spherical shell and the associated radiation of sound, J. Sound Vib. 330 (2011) 5982-6005. [334] J. Vila, R. Zaera, J. Fernndez-Sez, Axisymmetric free vibration of closed thin spherical nanoshells with bending effects, J. Vib. Control (2015) doi: 10.1177/1077546314565808. [335] H. Lamb, On the Vibrations of a Spherical Shell, Proc. London Math. Soc. s1-14(1) (1882) 50-56.

157

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

[336] D.B. Murray, L. Saviot, Acoustic vibrations of embedded spherical nanoparticles, Physica E 26 (2005) 417-421. [337] S.K. Gupta, S. Sahoo, P.K. Jha, A.K. Arora, Y.M. Azhniuk, Observation of torsional mode in CdS1xSex nanoparticles in a borosilicate glass, J. Appl. Phys. 106 (2009) 024307. [338] V. Mankad, S.K. Gupta, P.K. Jha, N.N. Ovsyuk, G.A. Kachurin, Lowfrequency Raman scattering from Si/Ge nanocrystals in different matrixes caused by acoustic phonon quantization, J. Appl. Phys. 112 (2012) 054318. [339] E. Ghavanloo, S.A. Fazelzadeh, Radial vibration of free anisotropic nanoparticles based on nonlocal continuum mechanics, Nanotechnology 24 (2013) 075702. [340] E. Ghavanloo, S.A. Fazelzadeh, T. Murmu, S. Adhikari, Radial breathing-mode frequency of elastically confined spherical nanoparticles subjected to circumferential magnetic field, Physica E 66 (2015) 228-233. [341] H.E. Sauceda, D. Mongin, P. Maioli, A. Crut, M. Pellarin, N. Del Fatti, F. Valle, I.L. Garzon, Vibrational properties of metal nanoparticles: Atomistic simulation and comparison with time-resolved investigation, J. Phys. Chem. C 116 (2012) 25147-25156. [342] S.A. Fazelzadeh, E. Ghavanloo, Radial vibration characteristics of spherical nanoparticles immersed in fluid medium, Mod. Phys. Lett. B 27 (3013) 1350186. [343] E. Wong, P.E. Sheehan, C.M. Lieber, Nanobeam mechanics: elasticity, strength, and toughness of nanorods and nanotubes, Science 277 (1997) 1971-1975. [344] S. Sundararajan, B. Bhushan, Development of AFM-based techniques to measure mechanical properties of nanoscale structures, Sensor. Actuat. A 101(2002) 338-351. [345] C.Q. Chen, Y. Shi, Y.S. Zhang, J. Zhu, Y.J. Yan, Size-dependence of Young's modulus in ZnO nanowires, Phys. Rev. Lett. 96 (2006) 075505. 158

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

[346] R. Agrawal, B. Peng, E.E. Gdoutos, H.D. Espinosa, Elasticity size effects in ZnO nanowires a combined experimental-computational approach, Nano Lett. 8 (2008) 3668-3674. [347] F. Xu, Q. Qin, A. Mishra, Y. Gu, Y. Zhu, Mechanical properties of ZnO nanowires under different loading modes, Nano Res. 3 (2010) 271-280. [348] J.J. Shiang, R.H. Wolters, J.R. Heath, Theory of size-dependent resonance Raman intensities in InP nanocrystals, J. Chem. Phys. 106(22) (1997) 8981-8994. [349] G. Cerullo, S. De Silvestri, Size-dependent dynamics of coherent acoustic phonons in nanocrystal quantum dots, Phys. Rev. B 60(3) (1999) 1928-1932. [350] M.E. Gurtin, A.I. Murdoch, A continuum theory of elasticmaterial surfaces, Arch. Ration. Mech. Anal 57 (1975) 291-323. [351] M.E. Gurtin, A.I. Murdoch, Surface stress in solids, Int. J. Solids Struct. 14 (1978) 431-440. [352] H.L. Duan, J. Wang, B.L. Karihaloo, Theory of elasticity at the nanoscale, Adv. Appl. Mech. 42 (2008) 1-68. [353] J.C. Hamilton, W.G. Wolfer, Theories of surface elasticity for nanoscale objects, Surf. Sci. 603 (2009) 1284-1291. [354] K.F. Wang, B.L. Wang, T. Kitamura, A review on the application of modified continuum models in modeling and simulation of nanostructures, Acta Mech. Sin. In Press (2016). [355] C. Juntarasaid, T. Pulngern, S. Chucheepsakul, Bending and buckling of nanowires including the effects of surface stress and nonlocal elasticity, Physica E 46 (2012) 68-76. [356] V. Cimalla, C.C. Rhlig, J. Pezoldt, M. Niebelsch¨ utz, O. Ambacher, K. Br¨ uchner, M. Hein, J. Weber, S. Milenkovic, A. J. Smith, A.W. Hassel, Nanomechanics of single crystalline tungsten nanowires, J. Nanomater. 2008 (2008) 638947.

159

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

[357] C.M. Wang, Y. Xiang, S. Kitipornchai, Postbuckling of nano rods/tubes based on nonlocal beam theory, Int. J. Appl. Mech. 1 (2009) 259-266. [358] V.B. Glavardanov, D.T. Spasic, T.M. Atanackovic, Stability and optimal shape of Pfl¨ uger micro/nano beam, Int. J. Solids Struct. 49 (2012) 2559-2567. [359] S.P. Xu, C.M. Wang, M.R. Xu, Buckling analysis of shear deformable nanorods within the framework of nonlocal elasticity theory, Physica E 44 (2012) 1380-1385. [360] N. Challamel, A. Kocsis, C.M. Wang, Discrete and non-local elastica, Int. J. Nonlin. Mech. 77 (2015) 128-140. [361] K.M. Hu, W.M. Zhang, Z.Y. Zhong, Z.K. Peng, G. Meng, Effect of surface layer thickness on buckling and vibration of nonlocal nanowires, Phys. Lett. A 378 (2014) 650-654. [362] K. Kiani, Magneto-elasto-dynamic analysis of an elastically confined conducting nanowire due to an axial magnetic shock, Phys. Lett. A 376 (2012) 1679-1685. [363] E. Ghavanloo, S. A. Fazelzadeh, H. Rafii-Tabar, Nonlocal continuumbased modeling of breathing mode of nanowires including surface stress and surface inertia effects, Physica B 440 (2014) 43-47. [364] A. Trejo, R. Vazquez-Medina, G.I. Duchen, M. Cruz-Irisson, Anisotropic effects on the radial breathing mode of silicon nanowires: an ab initio study, Physica E 51 (2013) 10-14. [365] B.W. Smith, M. Monthioux, D.E. Luzzi, Encapsulated C60 in carbon nanotubes, Nature 396 (1998) 323-324. [366] B.W. Smith, M. Monthioux, D.E. Luzzi, Carbon nanotube encapsulated fullerenes: a unique class of hybrid materials, Chem. Phys. Lett. 315 (1999) 31-36. [367] E.G. Noya, D. Srivastava, L.A. Chernozatonskii, M. Menon, Thermal conductivity of carbon nanotube peapods, Phys. Rev. B 70 (2004) 115416. 160

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

[368] R. Ansari, F. Sadeghi, B. Motevalli, A comprehensive study on the oscillation frequency of spherical fullerenes in carbon nanotubes under different system parameters, Commun. Nonlinear Sci. Numer. Simulat. 18 (2013) 769-784. [369] A.N. Sohi, R. Naghdabadi, Stability of C60-peapods under hydrostatic pressure, Acta Mater. 55 (2007) 5483-5488. [370] B.J. Cox, N. Thamwattana, J.M. Hill, Mechanics of atoms and fullerenes in single-walled carbon nanotubes: I. acceptance and suction energies, Proc. R. Soc. A 463 (2007) 461-476. [371] S.K. Joung, T. Okazaki, S. Okada, S. Iijima, Weak response of metallic single-walled carbon nanotubes to C60 encapsulation studied by resonance Raman spectroscopy, J. Phys. Chem. C 116 (2012) 23844-23850. [372] M. Asghari, R. Naghdabadi, J. Rafati-Heravi, Small scale effects on the stability of carbon nano-peapods under radial pressure, Physica E 43 (2011) 1050-1055. [373] M. Asghari, J. Rafati, R. Naghdabadi, Torsional instability of carbon nano-peapods based on the nonlocal elastic shell theory, Physica E 47 (2013) 316-323. [374] B.W. Jeong, J.K. Lim, S.B. Sinnott, Elastic torsional responses of carbon nanotube systems, J. Appl. Phys. 101 (2007) 084309. [375] Q. Wang, Torsional instability of carbon nanotubes encapsulating C60 fullerenes, Carbon 47 (2009) 507-512. [376] C. Fan, M. Zhao, Y. Zhu, H. Liu, T. Zhang, Analysis of micro/nanobridge test based on nonlocal elasticity, Int. J. Solids Struct. 49 (2012) 2168-2176. [377] C.Y. Nam, P. Jaroenapibal, D. Tham, D.E. Luzzi, S. Evoy, J.E. Fischer, Diameter-dependent electromechanical properties of GaN nanowires, Nano Lett. 6 (2006) 153-158. [378] S.R. Puri, A. Acharya, D. Dimiduk, Modeling dislocation sources and size effects at initial yield in continuum plasticity, J. Mech. Mater. Struct. 4 (2009) 1603-1618. 161

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

[379] Y. Yang, C.W. Lim, Non-classical stiffness strengthening size effects for free vibration of a nonlocal nanostructure, Int. J. Mech. Sci. 54 (2012) 57-68. [380] J.N. Reddy, Nonlocal nonlinear formulations for bending of classical and shear deformation theories of beams and plates, Int. J. Eng. Sci. 48 (2010) 1507-1518. [381] S. Adali, Variational principles for multi-walled carbon nanotubes undergoing buckling based on nonlocal elasticity theory, Phys. Lett. A 372 (2008) 5701-5705. [382] S. Adali, Variational principles for transversely vibrating multi-walled carbon nanotubes based on nonlocal Euler-Bernoulli beam models, Nano Lett. 9 (2009) 1737-1741. [383] S. Adali, Variational principles for nonlocal continuum model of orthotropic graphene sheets embedded in an elastic medium, Acta Math. Sci. 32B(1) (2012) 325-338. [384] N. Challamel, Variational formulation of gradient or/and nonlocal higher-order shear elasticity beams, Compos. Struct. 105 (2013) 351368. [385] N. Challamel, Z. Zhang, C.M. Wang, J.N. Reddy, Q. Wang, T. Michelitsch, B. Collet, On nonconservativeness of Eringen's nonlocal elasticity in beam mechanics: correction from a discrete-based approach, Arch. Appl. Mech. 84 (2014) 1275-1292. [386] K.F. Wang, B.L. Wang, The electromechanical coupling behavior of piezoelectric nanowires: Surface and small-scale effects, Europhys. Lett. 97 (2012) 66005. [387] A.G. Arani, S. Amir, A.R. Shajari, M.R. Mozdianfard, Electro-thermomechanical buckling of DWBNNTs embedded in bundle of CNTs using nonlocal piezoelasticity cylindrical shell theory, Compos. Part B 43 (2012) 195-203. [388] A.G. Arani, S. Amir, A.R. Shajari, M.R. Mozdianfard, Z. Khoddami Maraghi, M, Mohammadimehr, Electro-thermal nonlocal vibration analysis of embedded DWBNNTs, P. I. Mech. Eng. C-J. Mec. 226 (2012) 1410-1422. 162

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

[389] C. Liu, L.L. Ke, Y.S. Wang, Buckling and postbuckling of sizedependent piezoelectric Timoshenko nanobeams subject to thermoelectro-mechanical loadings, Int. J. Struct. Stab. Dy. 14 (2014) 1350067. [390] L.L. Ke, Y.S. Wang, Thermoelectric-mechanical vibration of piezoelectric nanobeams based on the nonlocal theory, Smart Mater. Struct. 21 (2012) 025018. [391] L.L. Ke, Y.S. Wang, Z.D. Wang, Nonlinear vibration of the piezoelectric nanobeams based on the nonlocal theory, Comput. Struct. 94 (2012) 2038-2047. [392] C. Liu, L.L. Ke, Y.S. Wang, J. Yang, S. Kitipornchai, Thermo-electromechanical vibration of piezoelectric nanoplates based on the nonlocal theory, Compos. Struct. 106 (2013) 167-174.

163

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Figure captions

Fig. 1 Number of papers published in journals included in the database of Scopus. Fig. 2. Nanoscopic carbon structures with various dimensions. (a) C60 fullerene. (b) Single-walled carbon nanotube. (c) Graphene sheet. Fig. 3. Geometry of some fullerenes; (a) C60 , (b)C70 and (c) C80 . Fig. 4. Structural configuration of an MWCNT. Fig. 5. Unrolled graphene sheet with definitions of basis unit vectors and chiral vector. Fig. 6. Cross section of different types of carbon nanotubes. (a) (9,0) zigzag SWCNT. (b) (5,5) armchair SWCNT. (c) (7,3) chiral SWCNT. Fig. 7. Morphology of the metal nanoparticles with size (number of atoms), and diameter (maximum interatomic distance): TOC (left column), DEC (middle column), and ICO (right column) [68]. Fig. 8. Components of the stress. Fig. 9. Schematic comparison between local and nonlocal kernel functions Fig. 10. Orthogonal curvilinear coordinates Fig. 11. Molecular structure of CNT and its equivalent beam model Fig. 12. Schematic illustration of an equivalent continuum shell structure and the coordinate system of an SWCNT. Fig. 13. Atomistic and equivalent-continuum structures of a graphene sheet Fig. 14. Spherical shell-like nanoscopic structures; (a) Schematic of a C240 fullerene and (b) Structure of a cowpea chlorotic mottle virus (CCMV). Fig. 15. Coordinate system definitions for a spherical shell. Fig. 16. Atomistic and equivalent-continuum structure of idealized models of silica nanoparticle. Fig. 17. Representation of interatomic interaction in molecular mechanics theory (a) elongation of bond and (b) variation of bond angle Fig. 18. Schematic illustration of a chiral SWCNT (a) Global structure, (b) side view and (c) top view of the local structure. Fig. 19. Chirality dependent anisotropic elastic properties of SWCNTs. (a) Y11 (= Y22 ), (b) Y12 (= Y21 ), (c) Y13 (= Y31 = −Y23 = −Y32 ), and (d) Y33 [170].

164

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Fig. 20. Variations of nonlocal parameter with diameter for different types of SWCNTs. (a)-(c) first longitudinal mode (ξ = 1, s=0) and (d)-(f) first coupled longitudinal-torsional mode ( ξ = 1, s=1) Fig. 21. Variations of nonlocal parameters with aspect ratio; (a) exx 0 , xθ (b) eθθ 0 and (c)e0 . Fig. 22. Deformed and undeformed configurations of an SWCNT. Fig. 23. High-resolution TEM image of a bent SWCNT [204] Fig. 24. Buckling deformation of a (5,5) armchair SWCNT subject to axial compression load and under hydrostatic pressure: (a) pure compression, (b) 1 GPa, (c) 2 GPa [206]. Fig. 25. (a) Schematic illustration of a cantilever CNT subjected to concentrated force. (b) Static deformation [105]. Fig. 26. (a) Schematic illustration of a fixed end-fixed end CNT subjected to concentrated force. (b) Static deformation [105]. Fig. 27. Small scale effect on buckling load ratio of a simply supported CNT [213]. Fig. 28. Buckling loads of simply supported CNTs based on the stress gradient and strain gradient approaches [214]. Fig. 29. A SEM Image of a double-nanobeam-system [215]. Fig. 30. Schematic configuration of a simply supported DNBS [216]. Fig. 31. Different buckling modes of the DNBS [217]. Fig. 32. Effect of scale-coefficient on BLRP in coupled-SWCNT-systems [217]. Fig. 33. A CNT connected to the end of a conventional silicon probe used in SEM [218]. Fig. 34. (a) Schematic SWCNT used as an AFM probe (b) the continuum model for the clamped SWCNT [220]. Fig. 35. The first four buckling mode shapes of clamped SWCNT used as AFM probes [220]. Fig. 36. Nonlocal effect on the buckling of clamped SWCNTs [220]. Fig. 37. Effect of the nonlocal scale parameter on the frequency ratio of an SWCNT [221]. Fig. 38. Variations of the frequency ratio with scale parameter for three different values of tubes length [221]. Fig. 39. Thermal effects on vibrational frequencies of an SWCNT (a) Low temperature (b) High temperature [109]. Fig. 40. (a) Molecular structure of an armchair DWCNT subjected to an axial magnetic field (b) Variations of first frequency ratios for DWCNT 165

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

considering same- and anti-phase vibration with and without magnetic field [225]. Fig. 41. Schematic representation of a current-carrying suspended CNT subjected to an applied longitudinal magnetic field [226]. Fig. 42. (a) Variations of the fundamental resonance frequency of the CNT with the dimensionless applied magnetic field parameter (b) Variations of the critical compressive axial force with the dimensionless applied magnetic field parameter [226]. Fig. 43. Molecular structure of CNT heterojunctions (a) (5, 5)-(9, 9) heterojunction (b) (9, 0)-(14, 0) heterojunction [228]. Fig. 44. Variations of phase velocity of flexural wave with wave length (a) for a (5,5) SWCNT (b) for a (10,10) SWCNT [190]. Fig. 45. (a) Spectrum curves for DWCNT and (b) dispersion curves for DWCNTs for various nonlocal scaling parameters [240]. Fig. 46. Axial buckling shape of a (10, 10) armchair nanotube [250]. Fig. 47. Comparison between the critical strains obtained from MD simulations and continuum mechanics models (a) (5, 5) SWCNT and (b) (7, 7) SWCNT [198]. Fig. 48. (a) Schematic representation of the boundary conditions of an SWCNT; Comparison of the critical axial buckling loads obtained by local and nonlocal shell models with those from MD simulations corresponding to (b) Simply supported-simply supported boundary condition; (c) Fixed endfixed end boundary condition; (c) Fixed end -simply supported boundary condition; (d) Fixed end- free boundary condition [252]. Fig. 49. Buckling mode shapes of an SWCNT in the fifth circumferential mode number (R = 8.5 nm, L/R = 5): (a) first axial mode, (b) second axial mode, (c) third axial mode, and (d) fourth axial mode [254]. Fig. 50. Buckling mode-shape of a (8, 0) zigzag SWCNT (a) Nonlocal continuum shell model and (b) MD simulations [113]. Fig. 51. Comparison of buckling torques from classical and nonlocal shell models with MD results (a), (b) for a zigzag SWCNT and (c), (d) for an armchair SWCNT [113]. Fig. 52. Comparisons of postbuckling torquerotation curves for a DWCNT [112]. Fig. 53. Relationship of critical shear stress of fifth-layered MWCNTs and wave-number (m, n) [265]. Fig. 54. Computed results from both nonlocal shell model and MD simulations for (a) simply supported simply supported, (b) fixed end-fixed 166

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

end and (c) fixed end-simply supported boundary conditions [199]. Fig. 55. Variations of fundamental frequency of a (5,5) SWCNT based on classical beam (fBeam ) and shell (fShell ) models, nonlocal beam (fN Beam ) and shell (fN Shell ) models, and MD simulations [186]. Fig. 56. (a) The lowest inextensional Rayleigh mode and (b) the lowest inextensional Love mode [268]. Fig. 57. (a) The calculated phonon dispersion of a (10,10) SWCNT obtained using the force-constant model [61] (b) Axisymmetric modes corresponding to the torsional, transverse and longitudinal vibrations of the SWCNT calculated by the continuum shell model. Fig. 58. Comparisons of frequency curves of a (15,15) SWCNT with different values of the small scale parameter for the axisymmetric case [115]. Fig. 59. Dispersion relation of transverse wave for (a) an armchair SWCNT (15, 15), (b) a zigzag SWCNT (20, 0) and (c) a (10,10)@(15,15) DWCNT [197]. Fig. 60. Prediction of four modes of wave propagation in an SWCNT obtained from local and nonlocal shell models [277]. Fig. 61. Effect of the small scale parameter on the escape frequencies of axial, flexural, shear and contractional waves [277]. Fig. 62. Comparison of the classical and nonlocal shell models for the normalized group velocity versus wavenumber with MD simulation results for (a) a (6,6) SWCNT and (b) a (10,10) SWCNT [278]. Fig. 63. Variations of buckling amplitude of a graphene sheet with compressive strain [280]. Fig. 64. Different types of the buckling; (a) rippling mode [281], (b) wrinkles [282]. Fig. 65. Computed buckling mode shape of graphene sheet under biaxial compressive strain [289]. Fig. 66. Variations of eigenfrequency with nonlocal scale coefficient (µ = e0 a); (a) imaginary part and (b) real part of eigenfrequency for different damping coefficient of the matrix, (c) imaginary part and (d) real part of eigenfrequency for different values of structural damping [297]. Fig. 67. Typical mode shapes of a triple-layered graphene sheet; (a) lower classical synchronized mode (b) middle vdW enhanced mode (c) higher mixed mode [304]. Fig. 68. Identified attached mass from the frequency-shift of a cantilevered single-layered graphene sheet resonator for two possible arrangements of attached objects [314]. 167

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Fig. 69. Single-layered graphene sheet with multiple attached nanoparticles; (a) molecular representation and (b) equivalent continuum model [317]. Fig. 70. Coupled effects of the nonlocal scale coefficient (µ = e0 a), temperature difference (θ) and mass of the nanoparticle on the relative frequency shift [317]. Fig. 71. Comparison between MD simulation and nonlocal continuum results for the distributed nanoparticles [319]. Fig. 72. Comparison between the phase velocities in a graphene sheet with a size of 3.62 nm × 15.03 nm computed from MD simulations, the local plate model, and the nonlocal model [323]. Fig. 73. Snapshots of wave propagation in a graphene sheet computed from (a) the nonlocal finite element model and (b) MD simulations [323]. Fig. 74. Frequency spectra for ten spherical fullerenes with e0 aR = 0.15;(a) lower branch and (b) upper branch [146]. Fig. 75. Modal shapes for: (a) n = 0, (b) n = 1, (c) n = 2, and (d) n = 3 [331]. Fig. 76. Comparisons of the nondimensional frequency spectra with different values of the nonlocal scale coefficient (µ = e0 a) [324]. Fig. 77. Breathing-mode of a nanoparticle. Fig. 78. Radial-breathing mode periods for (a) free Au and (b) free Pt nanoparticles [340]. Fig. 79. Variations of the breathing-mode frequencies for (a) gold and (b) platinum nanoparticles as a function of 1/R [339]. Fig. 80. Bending strength of Si and SiO2 nanobeams obtained from bending experiments [344]. Fig. 81. Variations of non-dimensional buckling load of fixed end-fixed end nanowires by varying the diameter as well as the aspect ratio [355]. Fig. 82. Series of optical images showing the nonlinear elastic large deformation of tungsten nanowires [356]. Fig. 83. Buckled shapes of the nanorod corresponding to various small length scale parameters [357]. Fig. 84. The time development plots of the non-dimensional radial displacement of a silver nanowire for (a) e0 a = 0, (b) e0 a = 0.1, (c) e0 a = 0.2 [362]. Fig. 85. Comparison between the results from the nonlocal model and those computed from the ab initio calculations [363]. Fig. 86. Molecular structure of a nanopeapod.

168

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Fig. 87. Variations of the computed critical torsional moment of the C60@(10,10) nanopeapod and (10,10)SWCNT with L = 12.62 nm with the scale coefficient η(= e0 a) [373]. Fig. 88. Schematics of the bridge test with a cylindrical indenter [376].

169

Figure 1

Figure 2a

Figure 2b

Figure 2c

Figure 3a

Figure 3b

Figure 3c

Figure 4

Figure 5

Figure 6a

Figure 6b

Figure 6c

Figure 7

Figure 8

Figure 9

Figure 10

Figure 11

Figure 12

Figure 13

Figure 14a

Figure 14b

Figure 15

Figure 16

Figure 17

Figure 18

Figure 19

Figure 20

Figure 21

Figure 22

Figure 23

Figure 24

Figure 25a

Figure 25b

Figure 26a

Figure 26b

Figure 27

Figure 28

Figure 29

Figure 30

Figure 31a

Figure 31b

Figure 31c

Figure 32

Figure 33

Figure 34a

Figure 34b

Figure 35

Figure 36

Figure 37

Figure 38

Figure 39a

Figure 39b

Figure 40a

Figure 40b

Figure 41

Figure 42a

Figure 42b

Figure 43a

Figure 43b

Figure 44a

Figure 44b

Figure 45a

Figure 45b

Figure 46

Figure 47

Figure 48a

Figure 48b

Figure 48c

Figure 48d

Figure 48e

Figure 49

Figure 50

Figure 51

Figure 52

Figure 53

Figure 54a

Figure 54b

Figure 54c

Figure 55

Figure 56

Figure 57a

Figure 57b

Figure 58

Figure 59a

Figure 59b

Figure 59c

Figure 60

Figure 61

Figure 62a

Figure 62b

Figure 63

Figure 64a

Figure 64b

Figure 65

Figure 66a

Figure 66b

Figure 66c

Figure 66d

Figure 67a

Figure 67b

Figure 67c

Figure 68a

Figure 68b

Figure 69a

Figure 69b

Figure 70

Figure 71

Figure 72

Figure 73

Figure 74a

Figure 74b

Figure 75

Figure 76

Figure 77

Figure 78

Figure 79

Figure 80

Figure 81

Figure 82

Figure 83

Figure 84

Figure 85

Figure 86

Figure 87

Figure 88