Computational Continuum Mechanics of Nanoscopic Structures: Nonlocal Elasticity Approaches [1st ed.] 978-3-030-11649-1, 978-3-030-11650-7

Table of contents :
Front Matter ....Pages i-xvii
Introduction (Esmaeal Ghavanloo, Hashem Rafii-Tabar, Seyed Ahmad Fazelzadeh)....Pages 1-7
Front Matter ....Pages 9-9
Fundamental Tenets of Nanomechanics (Esmaeal Ghavanloo, Hashem Rafii-Tabar, Seyed Ahmad Fazelzadeh)....Pages 11-39
Fundamental Notions from Classical Continuum Mechanics (Esmaeal Ghavanloo, Hashem Rafii-Tabar, Seyed Ahmad Fazelzadeh)....Pages 41-60
Essential Concepts from Nonlocal Elasticity Theory (Esmaeal Ghavanloo, Hashem Rafii-Tabar, Seyed Ahmad Fazelzadeh)....Pages 61-86
Nonlocal Modelling of Nanoscopic Structures (Esmaeal Ghavanloo, Hashem Rafii-Tabar, Seyed Ahmad Fazelzadeh)....Pages 87-113
Elastic Properties of Carbon-Based Nanoscopic Structures (Esmaeal Ghavanloo, Hashem Rafii-Tabar, Seyed Ahmad Fazelzadeh)....Pages 115-139
Front Matter ....Pages 141-141
Computational Modelling of the Vibrational Characteristics of Zero-Dimensional Nanoscopic Structures (Esmaeal Ghavanloo, Hashem Rafii-Tabar, Seyed Ahmad Fazelzadeh)....Pages 143-159
Modelling the Mechanical Characteristics of One-Dimensional Nanoscopic Structures (Esmaeal Ghavanloo, Hashem Rafii-Tabar, Seyed Ahmad Fazelzadeh)....Pages 161-185
Modelling the Mechanical Characteristics of Carbon Nanotubes: A Nonlocal Differential Approach (Esmaeal Ghavanloo, Hashem Rafii-Tabar, Seyed Ahmad Fazelzadeh)....Pages 187-217
Application of Nonlocal Elasticity Theory to Modelling of Two-Dimensional Structures (Esmaeal Ghavanloo, Hashem Rafii-Tabar, Seyed Ahmad Fazelzadeh)....Pages 219-239
Nonlocal Elasticity Models for Mechanics of Complex Nanoscopic Structures (Esmaeal Ghavanloo, Hashem Rafii-Tabar, Seyed Ahmad Fazelzadeh)....Pages 241-260
Recent Developments and Future Challenges in the Application of Nonlocal Elasticity Theory (Esmaeal Ghavanloo, Hashem Rafii-Tabar, Seyed Ahmad Fazelzadeh)....Pages 261-275

Citation preview

Springer Tracts in Mechanical Engineering

Esmaeal Ghavanloo Hashem Rafii-Tabar Seyed Ahmad Fazelzadeh

Computational Continuum Mechanics of Nanoscopic Structures Nonlocal Elasticity Approaches

Springer Tracts in Mechanical Engineering Series Editors Seung-Bok Choi, Inha University, Incheon, South Korea Haibin Duan, Beijing University of Aeronautics and Astronautics, Beijing, P.R. China Yili Fu, Harbin Institute of Technology, Harbin, P.R. China Carlos Guardiola, Universitat Politècnica de València, València, Spain Jian-Qiao Sun, University of California, Merced, USA Young W. Kwon, Naval Postgraduate School, Monterey, CA, USA

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Esmaeal Ghavanloo Hashem Rafii-Tabar Seyed Ahmad Fazelzadeh •

•

Computational Continuum Mechanics of Nanoscopic Structures Nonlocal Elasticity Approaches

123

Esmaeal Ghavanloo School of Mechanical Engineering Shiraz University Shiraz, Iran Seyed Ahmad Fazelzadeh School of Mechanical Engineering Shiraz University Shiraz, Iran

Hashem Rafii-Tabar Department of Medical Physics and Biomedical Engineering Shahid Beheshti University of Medical Sciences Tehran, Iran The Physics Branch Iran Academy of Sciences Tehran, Iran

ISSN 2195-9862 ISSN 2195-9870 (electronic) Springer Tracts in Mechanical Engineering ISBN 978-3-030-11649-1 ISBN 978-3-030-11650-7 (eBook) https://doi.org/10.1007/978-3-030-11650-7 Library of Congress Control Number: 2018967439 © Springer Nature Switzerland AG 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To my wife, Nazanin, for all her encouragement, patience and understanding and my Parents for their unconditional support Esmaeal Ghavanloo To my mentor Richard Feynman who was the first to raise the possibility of manipulating single atoms, an idea that eventually led to the emergence of the fields of nanoscience and nanotechnology Hashem Rafii-Tabar To my father, wife and children for their unwavering support and to the memory of my mother for her love and encouragement Seyed Ahmad Fazelzadeh

Preface

Following the enlightening talk by Richard Feynman in 1959 suggesting that the manipulation of single atoms is not contradicted by the laws of physics, a new perspective was ushered in scientific research in 1970s, namely, the transition from micro- and macro-scale science and technology to nanoscale science and technology, since many exotic phenomena both in non-living and living systems emerge and manifest themselves at this scale. The mechanical properties of materials at the nanoscale are very different from the properties of the same materials at the microand macro-scale. A deep understanding of the mechanical characteristics of nanoscopic structures is of great importance from both pure and applied perspectives. Owing to the complexity of performing accurate experimental measurements at nanoscopic scales, and also the high computational costs associated with quantum-mechanical-based and atomistic-based computer modelling and simulations, an alternative and very accurate methodology, namely, computational continuum mechanics-based modelling of the nanoscopic structures, has attracted a significant amount of attention worldwide. Over the past decade, computational modelling based on the use of nonlocal continuum theories has emerged as a very promising approach whereby, among other things, size dependency of the material properties is accommodated into the standard local continuum theory. Therefore, the nonlocal elasticity models play an important role in the study of the properties of nanoscopic structures that have not been fully explored or even addressed in the past. In fact, the nonlocal elasticity theory has become a great investigative tool in both basic and applied research in nanoscience and nanoengineering. The nonlocal elasticity theory-based models are currently being developed rapidly and are continuously being updated and modified. Notwithstanding its importance, there are only a few books dealing with this subject and its applications, particularly to nanoscience, that are simple enough for a wide range of audiences to follow, who would like to learn the nonlocal elasticity models for analysis of low-dimensional structures.

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This book provides a comprehensive treatment of the nonlocal elasticity theory as applied to the prediction of the mechanical characteristics of various types of nanoscopic structures having different morphologies and functional behaviours. In writing the book, we pursued three main objectives: (1) To introduce the fundamental notions and advanced concepts of the nonlocal elasticity approaches. (2) To present the necessary theoretical knowledge and practical tools to apply the nonlocal elasticity theory to the study of the mechanical behaviour of nanoscopic structures. (3) To provide in-depth discussions of the vast and rapidly expanding research works pertinent to the nonlocal modelling of the nanoscopic structures. The topics covered in this book are such that we have assumed that the reader has a background in continuum mechanics and university-level mathematics. However, we have attempted to provide an easy-to-follow description of the fundamental theories so as to make the pertinent research materials accessible to as a wide community as possible. This book is mainly written for practising scientists. The typical audience of this book could include scientists in the fields of computational condensed matter physics, computational materials science, computational nanoscience and nanotechnology, and nanomechanics. By providing enough details, it can also be useful for senior undergraduate, postgraduate and Ph.D. students, as well as practicing engineers. We hope that all the readers will gain a good understanding of the subject as they explore the pages of this book. We believe that there still exist a wide variety of new, unexplored areas of research within the nonlocal modelling of nanoscopic structures. Finally, it should be noted that the completion of this book would not have been possible without the patient and support of our families and friends. Their understanding and sacrifices are greatly appreciated. Shiraz, Iran Tehran, Iran Shiraz, Iran November 2018

Esmaeal Ghavanloo Hashem Rafii-Tabar Seyed Ahmad Fazelzadeh

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Part I 2

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Fundamental Tenets of Nanomechanics . . . . . . . . . . . . . . . . . 2.1 Nanoscopic Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Carbon-Based Nanoscopic Structures . . . . . . . . . . 2.1.2 Nanoparticles . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Nanowires . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.4 Biological Nanoscopic Structures . . . . . . . . . . . . 2.2 Techniques for Modelling Mechanical Characteristics of Nanoscopic Structures . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Atomistic-Based Modelling Methods . . . . . . . . . . 2.2.2 Continuum Modelling Methods . . . . . . . . . . . . . . 2.2.3 Multiscale Simulation Methods . . . . . . . . . . . . . . 2.3 Mechanical Behaviour of Nanoscopic Structures: Pertinent Definitions and Concepts . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Bending Behaviour of One-Dimensional Nanoscopic Structures . . . . . . . . . . . . . . . . . . . . 2.3.2 Buckling Behaviour . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Vibrational Behaviour . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fundamental Notions from Classical Continuum Mechanics 3.1 Displacement and Strain . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Strain Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Principal Strains . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Spherical and Deviatoric Strains . . . . . . . . . . . .

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3.1.4 Compatibility Conditions . . . . . . . . 3.1.5 Plane Strain Condition . . . . . . . . . . 3.2 Stress Tensor . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Concept of Stress . . . . . . . . . . . . . . 3.2.2 Stress on an Arbitrary Surface . . . . . 3.2.3 Principal Axes of Stress . . . . . . . . . 3.2.4 Hydrostatic and Deviatoric Stresses . 3.3 Constitutive Equations . . . . . . . . . . . . . . . . 3.3.1 Generalized Hooke’s Law . . . . . . . . 3.3.2 Material Symmetry . . . . . . . . . . . . . 3.4 Governing Equations . . . . . . . . . . . . . . . . . . 3.4.1 Equations of Motion . . . . . . . . . . . . 3.4.2 Navier Equations of Motion . . . . . . 3.5 Elastic Energy . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

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Essential Concepts from Nonlocal Elasticity Theory . . 4.1 Nonlocality and Nonlocal Averaging . . . . . . . . . . 4.2 Spatial Nonlocal Kernels . . . . . . . . . . . . . . . . . . . 4.2.1 Properties of Nonlocal Kernels . . . . . . . . 4.2.2 Different Types of Nonlocal Kernels . . . . 4.2.3 Two-Phase Kernels . . . . . . . . . . . . . . . . . 4.3 Gradient-Based Nonlocal Elasticity Theory . . . . . . 4.3.1 Green’s Function Kernels . . . . . . . . . . . . 4.3.2 Alternative Derivation of Gradient-Based Nonlocal Elasticity Theory . . . . . . . . . . . 4.4 Laplacian of the Stress Tensor . . . . . . . . . . . . . . . 4.4.1 Cylindrical Coordinates . . . . . . . . . . . . . 4.4.2 Spherical Coordinates . . . . . . . . . . . . . . . 4.5 Nonlocal Anisotropic Elasticity . . . . . . . . . . . . . . 4.6 Nonlocal Kernels in a Finite Domain . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Nonlocal Modelling of Nanoscopic Structures . . . . . . . . . . . . 5.1 Nonlocal Beam Models . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Nonlocal Euler-Bernoulli Beam Theory . . . . . . . 5.1.2 Nonlocal Timoshenko Beam Theory . . . . . . . . . 5.2 Nonlocal Cylindrical Shell Models . . . . . . . . . . . . . . . . . 5.2.1 Governing Equations for Circular Cylindrical Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Nonlocal Classical Shell Model . . . . . . . . . . . . . 5.2.3 Nonlocal First-Order Shear Deformation Theory 5.3 Nonlocal Plate Models . . . . . . . . . . . . . . . . . . . . . . . . .

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5.3.1 The Kirchhoff Plate Theory . . . . . . . . . 5.3.2 The Reissner-Mindlin Plate Theory . . . 5.3.3 Circular Nanoplate . . . . . . . . . . . . . . . 5.4 Nonlocal Membrane Theory for Spherical Shell 5.5 Nonlocal Models for Nanocrystals . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

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Elastic Properties of Carbon-Based Nanoscopic Structures . 6.1 Young’s Modulus and Effective Wall Thickness of SWCNTs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Anisotropic Elastic Constants of SWCNTs . . . . . . . . . . 6.3 Nonlocal Parameter for Carbon Nanotubes . . . . . . . . . . 6.4 Evaluation of the Nonlocal Parameter for SWCNTs . . . 6.5 Anisotropic Mechanical Properties of Graphene Sheets . 6.6 Effective Young’s Modulus of Fullerene Molecules . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Part II 7

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Computational Modelling of the Vibrational Characteristics of Zero-Dimensional Nanoscopic Structures . . . . . . . . . . . . . 7.1 Historical Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Axisymmetric Vibration of Spherical Nanoshells 7.1.2 Breathing-Mode of Spherical Nanoparticles . . . . 7.2 Axisymmetric Vibrational Modes of Spherical Fullerene Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Vibration of Spherical Nanoparticles . . . . . . . . . . . . . . . 7.3.1 Radial Vibration of Isolated Nanoparticles . . . . . 7.3.2 Vibration Under Radial Body Force . . . . . . . . . 7.3.3 Frequency Analysis of Nanoparticle Subject to Surface Periodic Excitation . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Modelling the Mechanical Characteristics of One-Dimensional Nanoscopic Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Measurement of Mechanical Characteristics of One-Dimensional Nanoscopic Structures . . . . . . . . . . . . 8.1.1 Nanoscale Tensile Test . . . . . . . . . . . . . . . . . . . . . 8.1.2 Beam Bending Techniques . . . . . . . . . . . . . . . . . . 8.1.3 Mechanical Resonance Technique . . . . . . . . . . . . . 8.1.4 Nanoindentation Technique . . . . . . . . . . . . . . . . . . 8.1.5 Challenges in the Mechanical Characterisation of ODNSs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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8.2

Vibration of Nanorods . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Extensional Vibration . . . . . . . . . . . . . . . . . . . 8.2.2 Breathing Vibration . . . . . . . . . . . . . . . . . . . . 8.3 Static Deformation of Nanorods Under Axial Loadings . 8.3.1 Nonlocal Differential Approach . . . . . . . . . . . . 8.3.2 Two-Phase Nonlocal Integral Approach . . . . . . 8.3.3 Nanorod in Tension . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Modelling the Mechanical Characteristics of Carbon Nanotubes: A Nonlocal Differential Approach . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Classification of Carbon Nanotubes . . . . . . . . . . . . . . . . . . . 9.2 Large Aspect Ratio SWCNTs: Nanobeams . . . . . . . . . . . . . . 9.2.1 Bending Deformations of SWCNTs Subject to Transverse Loading . . . . . . . . . . . . . . . . . . . . . . 9.2.2 Buckling of SWCNTs . . . . . . . . . . . . . . . . . . . . . . . 9.2.3 Postbuckling Behaviour of Highly Deformable Nanobeams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.4 Linear Flexural Vibrations of Carbon Nanotubes . . . 9.3 Low Aspect Ratio SWCNTs: Cylindrical Nanoshells . . . . . . 9.3.1 Shell-Like Buckling of SWCNTs . . . . . . . . . . . . . . 9.3.2 Vibrational Properties of SWCNTs . . . . . . . . . . . . . 9.3.3 Radial Breathing Mode . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10 Application of Nonlocal Elasticity Theory to Modelling of Two-Dimensional Structures . . . . . . . . . . . . . . . . . . . 10.1 Bending Deformation of Graphene Sheets Under a Transverse Load . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Buckling of Single-Layered Graphene Sheets . . . . . 10.2.1 Rectangular Shape . . . . . . . . . . . . . . . . . . 10.2.2 Circular Shape . . . . . . . . . . . . . . . . . . . . . 10.3 Deformation of Geometrically Imperfect Circular Graphene Sheets . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Transverse Vibration of Graphene Sheets . . . . . . . . 10.4.1 Transverse Free Vibration of Rectangular Graphene Sheet . . . . . . . . . . . . . . . . . . . . 10.4.2 Viscoelastic Graphene Sheets Embedded in a Viscoelastic Medium . . . . . . . . . . . . . 10.5 Application of the Nonlocal Anisotropic Elasticity to Modelling Graphene Sheets . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Contents

11 Nonlocal Elasticity Models for Mechanics of Complex Nanoscopic Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Application of the Nonlocal Models to Nanoscale Devices 11.1.1 Resonant Sensors . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Nanoturbine . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Mechanical Characteristics of Biological Nanoscopic Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.1 Mechanical Behaviour of Protein Microtubules . . 11.2.2 Analysis of Lipid Tubules . . . . . . . . . . . . . . . . . . 11.2.3 Low Frequency Vibrational Modes of Spherical Viruses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.4 Persistence Length of Collagen Molecules . . . . . . 11.3 Other Types of Nanoscopic Structures . . . . . . . . . . . . . . . 11.3.1 Nanotube-Based Heterojunctions . . . . . . . . . . . . . 11.3.2 Nanopeapods . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.3 Y-Junction Carbon Nanotubes . . . . . . . . . . . . . . . 11.3.4 Piezoelectric Nanoscopic Structures . . . . . . . . . . . 11.3.5 Carbon Honeycombs . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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12 Recent Developments and Future Challenges in the Application of Nonlocal Elasticity Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Nonlocal Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Self-adjointness of Eringen’s Nonlocal Elasticity . . . . . . . . . 12.3 Nonlocal Constitutive Boundary Conditions . . . . . . . . . . . . . 12.4 Nonlocal Fully Intrinsic Equations . . . . . . . . . . . . . . . . . . . . 12.5 Microstructure-Based Nonlocal Model . . . . . . . . . . . . . . . . . 12.6 Stress-Driven Nonlocal Integral Model . . . . . . . . . . . . . . . . . 12.7 Concluding Remarks and Future Prospects . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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261 261 263 263 265 268 273 273 274

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About the Authors

Esmaeal Ghavanloo is currently Assistant Professor of Mechanical Engineering at Shiraz University, Iran. He received B.Sc., M.Sc. and Ph.D. degrees from Shiraz University in 2007, 2009 and 2013, respectively, all in Mechanical Engineering. He is author of more than 60 peer-reviewed articles in the areas of mechanics of nanostructures, nanocomposite material, cellular biomechanics and nonlocal elasticity theory. He is also an active reviewer of several scientific and international journals. He has made important contributions to the application of the nonlocal elasticity theory to the nanoscopic structures. He has received “Dr. Kazem Ashtiani Award” in the year 2013 from Iran’s National Elites Foundation. Hashem Rafii-Tabar is currently the Distinguished Professor of Theoretical and Computational Nanoscience and Nanotechnology at the Department of Medical Physics and Biomedical Engineering, Shahid Beheshti University of Medical Sciences in Iran. He was educated in England, including a Research Fellowship at the University of Oxford. He was a Visiting Research Professor at the Institute of Materials Research at the University of Tohoku (Japan). He was the founder of the nanotechnology field in Iran and was the Head of the Nanotechnology Committee of the Ministry of Science, Research and Technology in Iran. He is a member of the Iran’s Academy of Sciences and was elected as an Eminent Science Personality (Chehreh Mandegar) in 2006. His tens of publications cover several distinct areas of theoretical physics, and nanoscience, including nano-neuroscience. He shared the Elegant Work Prize of the Institute of Materials (London) in 1994 for his investigations in the field of computational nanoscience. His book “Computational Physics of Carbon Nanotubes” published by the Cambridge University Press in 2008 was the first to cover this field and has been republished several times. Seyed Ahmad Fazelzadeh is currently Professor of Mechanical Engineering at Shiraz University in Iran. He was a Visiting Research Professor at the Zienkiewicz Centre for Computational Engineering, Swansea University, UK. He obtained the B.Sc. degree (Honors) in Mechanical Engineering in 1991 from the Shiraz University. He received the M.Sc. degree in Applied Mechanics and Ph.D. degree xv

xvi

About the Authors

in Aerospace Engineering from Sharif University of Technology in 1994 and 2002, respectively. His research and teaching span over 25 years and his expertise and leadership in the fields of structural dynamics and stability, aeroelasticity, nanocomposite, nanoscopics and smart structures. To his credit thus far he has six book chapters, one book, over 110 technical papers in prestigious journals, 125 papers in the international conference proceedings and one national Patent. He serves on the Editorial Boards of Applied Mathematical Modelling, Journal of Mechanics of Advanced Composite Structures and Sharif Journal of Mechanical Engineering. He is an active researcher and dedicated teacher and has received numerous awards and recognitions for excellence in teaching and research. He is a member of the Iranian Society of Mechanical Engineering and was elected as distinguished professor in 2018.

Abbreviations

AIREBO CCMV CGMD DFT DWCNT FCC FRF HRTEM IPR LAMMPS LD MC MD MEMS MWCNT NEMS ODNS RBM SEM SLGS STM SWCNT YJCNT

Adaptive intermolecular reactive empirical bond order potential Cowpea chlorotic mottle virus Coarse-grained molecular dynamics Density functional theory Double-walled carbon nanotube Face-centered cubic Frequency response function High-resolution transmission electron microscopy Isolated pentagon rule Large-scale atomic/molecular massively parallel simulator Lattice Dynamics Monte Carlo Molecular Dynamics Microelectromechanical systems Multi-walled carbon nanotube Nanoelectromechanical systems One-dimensional nanoscopic structure Radial breathing mode Scanning electron microscope Single-layer graphene sheet Scanning tunnelling microscope Single-walled carbon nanotube Y-junction carbon nanotube

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Chapter 1

Introduction

Small structures formed from a countable number of particles (atoms or molecules) and whose physical and chemical properties are functions of their sizes, are called nanoscopic structures. These structures are located between individual atoms (the size of an atom being about 0.1 nm) and large clusters consisting of up to 108 atoms or molecules. It is a standard practice in the nanoscience community that a structure is termed as a nanoscopic structure if at least one of its dimensions lies in the range 1–100 nm. Novel properties and functions are associated with nanoscopic structures due to reduction in their dimensionalities. These small structures have promoted a revolution in science and technology, and they are utilized in making new devices as well as leading to new technologies. Furthermore, Nature has already employed nanoscopic structures to perform vital biological functions. Hence, these exotic structures have occupied the centre stage in pure and applied research due to their widespread applications in various fields of nanoscience, nanoengineering, nanotechnology, materials science and technology, medical physics, biomedical engineering, biotechnology, molecular biology and genetics, information technology, and neural science and neural engineering. The invention of the scanning tunnelling microscope (STM) has opened a new perspective for the characterization, measurement and manipulation of nanoscopic structures. Following this invention, various types of biological and non-biological nanoscopic structures with various morphologies and functionalities have been discovered, synthesized and reported during the past four decades. Phenomenological study of nanoscopic structures, either as single isolated structures or as components in nanoscale machines and systems, forms an active research area in applied science and engineering. The investigation into their mechanical characteristics, in particular, forms one of the most intense fields of research. Understanding the mechanical properties and behaviour of a material system at nanoscale level is a necessary requirement for an efficient and accurate design, fabrication and assembly of nanoscale systems, nanomaterials, nanodevices and nanodrugs. Accordingly, the development of well-understood theoretical, computational and experimental frameworks for characterizing the mechanical structure and behaviour of the nanoscopic © Springer Nature Switzerland AG 2019 E. Ghavanloo et al., Computational Continuum Mechanics of Nanoscopic Structures, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-3-030-11650-7_1

1

2

1 Introduction

structures is very desirable, and it poses a huge number of challenges for researchers in this field. Since the measurement of the mechanical behaviour of nanoscopic structures is a time consuming and rather expensive endeavour, significant efforts have been invested to explore their properties and behaviour through theoretical and mathematical approaches. These approaches can generally be classified into four categories; atomistic-based simulation methods, multi-scale (coarse-grained) modelling methods, continuum-based methods, and modified continuum approaches. Molecular Dynamics (MD) and Monte Carlo (MC) simulation approaches are the most well-known types of the atomistic-based simulation approaches. In the MD simulation, the dynamic behaviour of constituent particles is predicted by numerically solving the differential equations of their motion, and so this method may be appropriate for modelling both their equilibrium and non-equilibrium properties. In contrast to the MD simulation, the MC simulation methods do not involve the use of equations of motion and consequently they are appropriate for studying the phenomena in thermodynamic equilibrium [1]. Although the atomistic-based simulation methods have meet with outstanding success in computational physics, their applications are limited to simple systems with a relatively small number of molecules or atoms because the use of these methods requires access to very high-performance computational facilities. Furthermore, there are other limitations related to time steps, constraints, boundary conditions and temperature effects [2]. To overcome these limitations, it is necessary to decrease the number of degrees of freedom by grouping atoms into larger particles or beads, and replacing the atomic interactions by bead-bead interactions. This process is known as coarse-graining [3]. The concept of coarse-graining has been widely used in creating a macroscopic theory of materials from their microscopic degrees of freedom [4]. Generally, there is no systematic approach for coarse-graining that is applicable to all materials, but several procedures have been proposed for coarse-graining of particular systems [5]. One of the most familiar classes of coarse-grained methods is coarse-grained molecular dynamics (CGMD) that has been developed by Rudd and Broughtony [6]. The CGMD is a natural bridging technique which provides a coupling between the various scales, allowing the results of calculations at lower length and time scales to be used as input parameters in the calculations at higher scales. Important atomistic effects can be captured by using the CGMD without the computational cost of standard atomistic-based simulation methods. Although the coarse-grained multiscale methods have proven to be reliable techniques for the simulation of material behaviour, the implementation of the model varies from system to system. Furthermore, it is confirmed that continuum-based models provide efficient alternative methods to investigate the mechanical behaviour of nanoscopic structures. The main assumption underlying the continuum-based models is that the material properties are distributed continuously throughout the spaces occupied by them and that the empty spaces between the atoms or molecules are ignored. From the computational point of view, these methods are very effective and highly efficient because they can work with reduced degrees of freedom. The main advantage of the continuumbased methods over the molecular simulation methods is their capability for performing computations without limitation in time and size scales. In these models,

1 Introduction

3

the deformation of a continuous medium is related to the applied external forces. In the classical continuum theory, the basic assumption is that the stress at a material point is only dependent on the strain, strain rate and strain history at that particular point. In other word, the strain outside an arbitrary neighbourhood of the material point is disregarded. Since information from neighbouring material points is not used, these models are referred to as local models [7]. Although the classical continuum theories can be used as very useful tools for modelling the mechanical properties of nanoscopic structures, however, several questions are naturally raised about the applicability and limitations of these theories at very small-scales. One of the most important limitations concerns the fact that the discrete nature of the nanoscopic structures cannot be accurately homogenized into a continuum background medium. Furthermore, it should be noted that the material properties at the very small-scales are size-and geometry-dependent and the small-length scale effect becomes significant in the mechanical characteristics of the nanoscopic structures. In addition, the continuum-based computational methods are scale independent and cannot reflect the nanoscale physical laws of the nanoscopic structures. Consequently, computational models based on the classical continuum theories fail when the size of the structure becomes comparable with the internal characteristic length of the material under consideration. To overcome these limitations of the classical continuum theories, a series of modified continuum approaches have been proposed for accommodating the nanoscale size-effects. There are different ways to modify the classical continuum theories to extend their applicability and reduce their limitations. Modified continuum-based methods possess both the advantages of atomistic-based simulation methods and continuum-based methods. Generally, the modified continuum-based models accommodate nanoscale size-effects and are referred to as the nonlocal models [8]. To incorporate the small-scale effects into the theoretical modelling of the nanoscopic structures, different modified continuum theories, namely the Cosserat continuum theory [9], the micromorphic continuum theory [10], the rotation gradient or couple-stress theories [11], the higher-order strain-gradient theories [12–14], the nonlocal continuum field theories [15] and the atomistic-continuum theories [16] have been proposed. In modified models, the atomistic information and size-effects are integrated into the classical continuum models. The most widely used theory for analysing the nanoscopic structures is the nonlocal elasticity theory. In the context of the nonlocal elasticity, it is assumed that the stress field at a given reference point requires the knowledge of the strain field at every other point in the body. Based on this conceptual idea, the long-range interatomic interactions between material points can be taken into account. In the original nonlocal model developed by Eringen [15], the stress is obtained from the integral of the nonlocal averaging strain over the whole body. Therefore, the stress-strain constitutive relations are only the difference between the nonlocal and classical theories, whereas the equilibrium and compatibility equations remain unaltered. The nonlocal effect is introduced via a nonlocal parameter which depends on several parameters including the molecular structure, the internal characteristic length and the material type.

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1 Introduction

The appearance of the nonlocal elasticity theory dates from the 1970s in studies concerned with the modelling of the elastic wave dispersion in crystals. A formal thermomechanical- and variational-based derivation of the constitutive equations of this theory is given by Eringen and co-workers [17–19]. Rogula [20] has collected his works on the mathematical structure of the nonlocal elasticity theory and the different types of nonlocal stress-strain relations in a book, studying various problems in continuum mechanics. Many sub-fields of the nonlocal field theories have been established by Eringen [15], including nonlocal pure elastic continua, nonlocal fluid mechanics, nonlocal electromagnetism, nonlocal thermoelasticity, nonlocal memory-dependent elasticity, nonlocal piezoelectricity etc. However, the nonlocal elasticity theory has received relatively less attention and has only been employed by a limited number of research groups. An exact, or approximate, solution to the nonlocal integral function can be determined under some very special circumstances using the Green function approach, and hence its use is rather limited. Although an equivalent formulation of the nonlocal elasticity in a differential form has also been derived, this differential nonlocal stress relation seems not to have attracted the attention of the research community for some time. Over the past decade, nonlocal elasticity theory has emerged as a promising size-dependant continuum theory. Significant progress has been made in fundamental and applied computational research in this area. Application of nonlocal continuum-based modelling to nanoscopic structures has attracted significant attention in the nanomechanics community. The mechanical properties of different nanoscopic structures such as carbon allotropes, nanowires, nanoparticles, etc., have been successfully investigated. The results obtained from the nonlocal elasticity models for various nanostructures are in excellent agreement with those obtained from experimental measurements and atomistic simulations. Although there exist few books and review papers covering the subject of the nonlocal continuum mechanics, however, to our knowledge, no book that systematically addresses all aspects of the nonlocal continuum mechanics appropriate for nanoscopic structures, and provides the underlying computational methods, and mathematical theories, and considers the applications of these methods and theories to the investigation of their mechanical characteristics has been published. The book by Eringen [15] is a significant book in this field. That book explains the fundamental formulation of the nonlocal field theory. However, it is not easy to follow for a wide range of researchers in different fields of computational science who are interested in the analysis of the mechanical characteristics of low-dimensional structures. The book by Gopalakrishnan and Narendar [21] describes the fundamental and advanced concepts of wave propagation in the nanoscopic structures. That book is based on the nonlocal elasticity theory and it predominantly addresses the wave behaviour in carbon nanotubes and graphene sheets. The book is useful but is more specialized because it focuses only on the wave propagation. A recent book by Karlicic and co-workers [22] provides an introduction to the nonlocal elasticity theory for static, dynamic and stability analyses of structural elements such as rods, beams and plates. In writing that book, it was assumed that the readers were familiar with the basic engineering mechanics and having a graduate-level mechanics background. The authors collected and classified their previous published papers, while the valuable results by other researchers were not considered.

1 Introduction

5

The extensive nonlocal continuum-based modelling of nanoscopic structures has reached such a mature stage that an up-to-date book is now called for to collect the current status of our knowledge, particularly for those who would benefit from learning the theoretical and computational methods of the nonlocal elasticity approach for investigating the mechanical characteristics of low-dimensional structures. This is the motivation behind the writing of the present book. Our book aims to systematically present and discuss the past and the current developments, recent attempts, challenges, and results addressing the mechanical characteristics of the nanoscopic structures within the framework of nonlocal elasticity theory. There are many novel and unique features to this book. It is a rather comprehensive and self-contained book that utilizes the research materials and publications produced up to the year 2019. One of the main features of this book is to give all the relevant preliminaries, mathematical techniques, theoretical assumptions, physical methods and possible limitations of the nonlocal approach. To make the self-learning an enjoyable journey for our readers, our motto is to spell out all the details even if they may seem elementary. Furthermore, to help with the comprehension of the topics, some practical applications are also presented. Although this book is primarily concerned with the theoretical and computational concepts, some experimental results are also discussed. Another feature of this book is its in-depth discussions of the vast and rapidly expanding research works pertinent to the nonlocal modelling of the nanoscopic structures. The integration of teaching and research is another key feature. This book is largely written for practicing research scientists but, by providing enough details, we hope that it is also useful for senior undergraduate and postgraduate students, and post-doctoral research scientists. Researchers and practicing engineers working in the field of nanoscopic structures can also find this book useful. Our purpose is to help the interested reader to follow the current widely used nonlocal models. In this book we present the nonlocal continuum-based mechanics methods concerned with the mechanical behaviour of different biological as well as non-biological nanoscopic structures, covering their static deformations, structural stability, buckling behaviour and vibrational behaviour. We have endeavoured to cover most of the research materials pertinent to the nonlocal-based modelling of the nanoscopic structures and their vital fundamental applications and relevant references of nonlocal elasticity theory. The book is divided into two parts. The first part consists of Chaps. 2–6, covering the relevant preliminaries, theoretical frameworks, mathematical and physical methods and possible limitations of the nonlocal approach required to understand the mechanical characteristics of nanoscopic structures. The second part consists of Chaps. 7–12, which concerns the application of the theoretical considerations and models described in Part I to the study of static and dynamic behaviour of nanoscopic structures with various morphologies. Chapter 2 provides the essential background in nanomechanical engineering techniques including the modelling techniques that are employed for investigating the mechanical properties and behaviour of nanoscopic structures. The various types of biological as well as non-biological nanoscopic structures, whose mechanical properties have been the subject of extensive investigations, are also presented. Chapter 3 addresses the theoretical background and summarizes

6

1 Introduction

the most important notions in continuum mechanics including the concepts of strain and compatibility, stress and equilibrium, elasticity and constitutive equations. In Chap. 4, we begin to introduce the notion of nonlocality and discuss the necessity of nonlocal modelling for nanoscopic structures. Furthermore, different types of nonlocal relations are discussed rather comprehensively and various nonlocal kernels are presented. To our knowledge, this is the first time that these kernels have been listed, and detailed descriptions of their properties given. Chapter 5 provides the computational nonlocal models for beams, plates, shells and nanocrystals, describing the static and dynamic behaviour of these nanoscopic structures. These models are employed in such areas of nanomechanics as vibration, structural stability and static deformations. Estimates of the elastic properties of nanoscopic structures play a vital role in the application of continuum-based modelling. A comprehensive discussion pertinent to the elastic properties of the nanoscopic structures is presented in Chap. 6. In Chaps. 7–10, we survey the applications of the nonlocal continuum modelling to the static and dynamic properties of zero-, one- and two-dimensional nanoscopic structures. Chapter 11 describes the application of nonlocal models to simulate the mechanical behaviour of complicated nanoscopic structures including nanoscale devices, biological nanoscopic structures and etc. In recent years, some new developments and challenges in the applications of the nonlocal elasticity theory (e.g. nonlocal paradox, nonlocal constitutive boundary conditions, stress-driven nonlocal integral model, nonlocal fully intrinsic equations, modified nonlocal models and structures etc.) have been presented in the nanomechanics community. To cover these aspects of the nonlocal elasticity theory, these new topics are covered as well in Chap. 12. The material presented in Part II is obtained from key research papers that we have selected from among thousands of publications in this field. Obviously, our selection could not have been exhaustive and some significant papers and reports may not have come to our attention when writing this book. We, therefore, welcome constructive suggestions from the research community in this regard.

References 1. A. Satoh, Introduction to Practice of Molecular Simulation (Elsevier, London, 2011) 2. W.K. Liu, E.G. Karpov, S. Zhang, H.S. Park, An introduction to computational nanomechanics and materials. Comput. Methods Appl. Mech. Eng. 193, 1529–1578 (2004) 3. H. Weiss, P. Deglmann, P.J. in’t Veld, M. Cetinkaya, E. Schreiner, Multiscale materials modeling in an industrial environment, Ann. Rev. Chem. Biomol. 7, 65–86 (2016) 4. N. Ramakrishnan, P.B. Sunil Kumar, R. Radhakrishnan, Mesoscale computational studies of membrane bilayer remodeling by curvature-inducing proteins. Phys. Rep. 543, 1–60 (2014) 5. T.S. Gates, G.M. Odegard, S.J.V. Frankland, T.C. Clancy, Computational materials: multi-scale modeling and simulation of nanostructured materials. Compos. Sci. Technol. 65, 2416–2434 (2005) 6. R.E. Rudd, J.Q. Broughtony, Coarse grained molecular dynamics and the atomic limit of finite elements. Phys. Rev. B 58, R5893–R5896 (1998)

References

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7. J. Fatemi, F. Vankeulen, P.R. Onck, Generalized continuum theories: application to stress analysis in bone. Meccanica 37, 385–396 (2002) 8. H. Rafii-Tabar, E. Ghavanloo, S.A. Fazelzadeh, Nonlocal continuum-based modeling of mechanical characteristics of nanoscopic structures. Phys. Rep. 638, 1–97 (2016) 9. Y. Yoshiyuki, An intrinsic theory of a Cosserat continuum. Int. J. Solids Struct. 4, 1013–1023 (1968) 10. A.C. Eringen, E.S. Suhubi, Nonlinear theory of simple micro-elastic solids. Int. J. Eng. Sci. 2, 189–203 (1964) 11. A.R. Hadjesfandiari, G.F. Dargush, Couple stress theory for solids. Int. J. Solids Struct. 48, 2496–2510 (2011) 12. R.D. Mindlin, N.N. Eshel, On first strain-gradient theories in linear elasticity. Int. J. Solids Struct. 4, 109–124 (1968) 13. 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, 1962–1990 (2011) 14. C.W. Lim, G. Zhang, J.N. Reddy, A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation. J. Mech. Phys. Solids 78, 298–313 (2015) 15. A.C. Eringen, Nonlocal Continuum Field Theories (Springer, NewYork, 2002) 16. V.B. Shenoy, R. Miller, E.B. Tadmor, D. Rodney, R. Phillips, M. Ortiz, An adaptive finite element approach to atomic-scale mechanics-the quasi continuum method. J. Mech. Phys. Solids 47, 611–642 (1999) 17. A.C. Eringen, Linear theory of nonlocal elasticity and dispersion of plane waves. Int. J. Eng. Sci. 10, 425–435 (1972) 18. A.C. Eringen, D.G.B. Edelen, On nonlocal elasticity. Int. J. Eng. Sci. 10, 233–248 (1972) 19. A.C. Eringen, C.G. Speziale, B.S. Kim, Crack-tip problem in non-local elasticity. J. Mech. Phys. Solids 25, 339–355 (1977) 20. D. Rogula, Nonlocal Theory of Material Media (Springer, Berlin, 1982) 21. S. Gopalakrishnan, S. Narendar, Wave Propagation in Nanostructures: Nonlocal Continuum Mechanics Formulations (Springer, Switzerland, 2013) 22. D. Karlicic, T. Murmu, S. Adhikari, M. McCarthy, Nonlocal Structural Mechanics (WileyISTE, London, 2016)

Part I

Chapter 2

Fundamental Tenets of Nanomechanics

Nanoscience is concerned with the study and understanding of phenomena that manifest themselves at the nanoscale, and nanotechnology refers to the synthesis, control, engineering, and manipulation of atomistic and molecular systems and structures. The appearance of nanoscience and nanotechnology has prompted fundamental and radical changes in varieties of scientific and engineering disciplines. Among the diverse branches of the nanoscience, the study and characterization of mechanical behaviour of physical systems at nanoscale form an interdisciplinary and cross disciplinary area referred to as nanomechanics. This branch of nanoscience has emerged from the interplay of classical mechanics, quantum mechanics, solid-state physics, and materials physics and chemistry. Nanomechanics differs from macroscopic mechanics in various aspects, and contains many subfields such as mechanics of nanocomposites, molecular dynamics, molecular mechanics, design of microelectromechanical and nanoelectromechanical systems (MEMS and NEMS), and cellular biomechanics [1]. Nanoscale mechanics provides the scientific foundation and the infrastructure for nanotechnology and nanoengineering. This subfield of nanoscience has witnessed an explosive growth worldwide over the past decades. In this chapter, we present a brief introduction to essential concepts underlying nanomechanics. The three sections in this chapter are organized as follows. An overview of different nonbiological as well as biological nanoscopic structures, whose mechanical properties have been the subject of the extensive investigations, is given in Sect. 2.1. In Sect. 2.2, the techniques for modelling the mechanical properties of nanoscopic structures will be briefly reviewed. Finally, in Sect. 2.3, we will introduce some useful definitions and concepts pertinent to this mechanical behaviour that will be further used in other chapters.

© Springer Nature Switzerland AG 2019 E. Ghavanloo et al., Computational Continuum Mechanics of Nanoscopic Structures, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-3-030-11650-7_2

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2 Fundamental Tenets of Nanomechanics

2.1 Nanoscopic Structures Nanoscopic structures are the entry points into a new realm in modern physical and biological sciences. They characterise the building blocks of inanimate and animate matter. These structures are composed of a countable number of elementary units (such as atoms) packed very closely together. Nanoscopic structures have significantly large surface-to-volume ratio, and operate on highly reduced time and energy scales. The miniaturization of materials from macroscale to the nanoscale leads to radical changes in their physical properties. As a result, they exhibit physical properties that are distinctively different from the properties of bulk materials composed of the same atoms. For example, the melting temperatures of various nanowires have been found to be lower than their corresponding bulk forms [2]. Furthermore, the strength of nanowires can reach up to 100 times that in corresponding bulk materials [3]. Nanotechnology is concerned with designing and constructing devices, components and systems from assemblies of isolated nanoscopic structures. Depending on the type of nanoscopic structure involved, namely, soft biological or organic nanoscopic structures and hard inorganic ones, nanotechnology has been classified into soft and hard nanotechnologies. There are two approaches to nanotechnology for the synthesis and the fabrication of nanoscopic structures; the bottom-up and top-down approaches. The bottom-up approach is based on the self-assembly phenomenon, i.e., the process of constructing complex structures from simpler ones. This approach involves molecular manipulation and molecular engineering to construct nanoscopic structures, molecular machines and molecular devices by a precise positioning of atoms, molecules or clusters at prespecified locations (Fig. 2.1). Three main bottom-up technologies for producing the nanoscopic structures are (1) the supramolecular and molecular chemistry, (2) the scanning probe microscopy, and (3) biotechnology [4]. Consequently, these techniques in nanotechnology allow for the structural transformation and the purposeful manipulation of condensed phases at their most elementary levels. Furthermore, we can interrogate organic and inorganic matter atom-by-atom and induce predetermined property changes in them. In contrast, the top-down approach, initiated in the classic paper of Feynman [5], is based on the sequential decrease of the objects size by means of chemical and mechanical methods to reach the nanoscopic structures (Fig. 2.2). The milling or attrition, repeated quenching and lithography are among the most well-known top-down methods. In these methods, the nanoscopic structures are produced from microscopic scales without control at the atomic level [6]. In the past four decades, numerous novel nanoscopic structures with different morphologies, sizes, and compositions have been discovered, synthesized and reported in the literature. It is well-known that their behaviour and functionalities strongly depend on their sizes, compositions, dimensionality and morphologies. In order to properly describe the diversity of nanoscopic structures, some form of classification scheme is required. Currently, the nanoscopic structures are typically classified on the basis of their dimensions. These structures can be categorized as

2.1 Nanoscopic Structures

13

Fig. 2.1 Schematic illustration of the bottom-up approach

Fig. 2.2 Schematic illustration of the top-down approach

zero-dimensional, one-dimensional and two-dimensional [7]. The first attempt to classify the nanoscopic structures was made by Gleiter [8], however, some important nanoscopic structures including fullerenes and nanotubes were not included in that classification. Therefore, Pokropivny and Skorokhod [9] presented a modified classification scheme. The classification expressed here follows the work of Pokropivny and Skorokhod [9]. Zero-dimensional nanoscopic structures are materials that have all their three dimensions at the nanoscale. In these structures, the electrons are fully confined and so no electron delocalization occurs. Nanoparticles, quantum dots, atomic clusters, fullerenes and onions can be considered as zero-dimensional nanoscopic structures. In one-dimensional nanostructures, two of their dimensions are restricted to a few tens of nanometers while the third dimension can be several hundred nanometers. The electron confinement occurs in two dimensions and delocalization takes place along the long axis of the strictures. Therefore, these needle-like structures represent the smallest structures that can efficiently transport charge carriers. Nanorods, nanowires, nanotubes, nanobelts and nanoribbons are well-known examples of one-dimensional nanostructures. Two-dimensional nanoscopic structures, such as nanoplates, thin films, nanosheets, nanodisks and nanowalls, are materials with two of their dimensions not restricted to the nanoscale. Therefore, two-dimensional nanoscopic structures show plate-like shapes. These certain geometries exhibit unique and novel characteristics and properties.

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2 Fundamental Tenets of Nanomechanics

2.1.1 Carbon-Based Nanoscopic Structures Carbon, the sixth element in the periodic table, is one of the most important and useful elements on earth. It has been known and utilized for centuries, going back to antiquity, and it plays a critical role in Nature and life. The reason for this vital role is its tendency to combine with other elements to form compounds. Carbon atoms can bond to other carbon atoms in so many different ways. The bonding can lead to numerous known carbon allotropes. Prior to the discovery of fullerenes in 1985, diamond and graphite were the only known crystalline forms of carbon. The starting point of a new era in carbon materials is the synthesis of fullerene or the buckyball [10]. The discovery of this nanoscopic structure has generated considerable worldwide research effort that is still growing. Another allotrope of carbon in a tubular form, called the carbon nanotube, was discovered in 1991 by Iijima [11] using an arcdischarge evaporation method. This discovery has opened a fascinating new chapter in the science and technology at nanoscales. In 2004, graphene, an isolated layer of graphite, was discovered by Novoselov and co-workers [12]. Beyond these more familiar carbon-based nanoscopic structures, several other exotic forms of carbon material, such as nanohorns, nanopeapods, bucky-shuttle, carbon schwarzite and carbon nanofoam, have also emerged [13]. The carbon-based nanoscopic structures have accelerated the basic and applied research in nanomaterial science and nanotechnology. Furthermore, they are promising materials for applications in different technological and biomedical fields. Among the numerous carbon-based nanoscopic structures, we highlight the fullerene molecules, carbon nanotubes, and graphene sheets. In the following subsections, we discuss each of these structures. 2.1.1.1

Fullerenes

Fullerene molecules are one of the earliest discovered carbon-based nanoscopic structures and they have promoted a revolution in science and technology. They are closedcage molecules, and can be considered as zero-dimensional carbon-based nanoscopic structures. In these molecules, each carbon atom is bonded to three nearest neighbour carbon atoms via sp 2 hybridized bonds. In general, there are two different carbon bonds in this molecule; the double bonds with length 0.14 nm, and single bonds with length 0.146 nm. This class of nanoscopic structures is usually denoted by C N , where N is the number of carbon atoms. Note that, for a given number of carbon atoms, there are different ways to arrange the carbon atoms into a fullerene, and each possibility is termed an isomer. For example, a C60 fullerene has 1812 different isomers [14]. From a geometric point of view, the structure of fullerenes follows the Euler polyhedral theorem, which relates to the topological structures of the molecule. The Euler theorem can be expressed as follows [15]: nv + n f − ne = 2

(2.1)

2.1 Nanoscopic Structures

15

where n v , n f and n e are respectively the number of vertices, faces and edges of a polyhedron. Since every edge forms two faces, and every vertex is connected to three edges, we have (2.2) n f = n p + nh nv =

5n p + 6n h 3

(2.3)

ne =

5n p + 6n h 2

(2.4)

in which n p and n h are the number of pentagonal and hexagonal faces, respectively. Substituting Eqs. (2.2)–(2.4) into Eq. (2.1) gives n p = 12. This means that fullerenes composed of an arbitrary number of hexagonal faces and exactly 12 pentagonal faces. Furthermore, comparison of Eqs. (2.3) and (2.4) gives 3n v = 2n e , and hence the number of vertices or atoms must be even [16]. It should also be noted that a fullerene molecule is energetically stable when all its 12 pentagons are separated by hexagons. This rule is called the isolated pentagon rule (IPR). According to the Euler theorem, the smallest fullerene is C20 , which is formed by 12 pentagons and no hexagons. However, this fullerene does not satisfy the IPR, and so exhibits a very short lifetime. The smallest fullerene which satisfies the IPR is C60 . The fullerene molecules with less than 60 carbon atoms cannot have isolated pentagons and so they are highly unstable [17]. The C60 is one of the earliest observed fullerenes and it led to a revolution in carbon-based nanotechnology. It was named buckminsterfullerene, because of its similarity to geodesic dome structures designed and built by Richard Buckminster Fuller (1895-1983). Furthermore, the C60 fullerene is known as the buckyball. These names are specifically used for the C60 molecule. The most striking characteristics of the buckminsterfullerene is its high symmetry. Based on the structural configuration, there are two main types of fullerenes; spherical or near-spherical, and ellipsoidal/non-spherical [18]. For example C60 , C80 , C180 , C240 , C260 , C320 , C540 , C720 can be considered as the spherical or near-spherical fullerenes, and the ellipsoidal (or non-spherical) fullerenes include C70 , C90 , C100 etc. Figure 2.3 shows the structure and geometry of some fullerene molecules. Interestingly, the structure of fullerenes is similar to the materials that could be found in Nature. For example, spherical fullerenes have a similar structure to icosahedral viruses.

2.1.1.2

Carbon Nanotubes

Carbon nanotube is another allotrope of carbon which was experimentally synthesized by Iijima [11] in 1991. This is often considered to be the first observation of carbon nanotubes, but there are several reports indicating that the production of filamentous carbon was observed much earlier by Radushkevich and Lukyanovich [19]. This type of carbon nanotube was later called a multi-walled carbon nanotube

16

2 Fundamental Tenets of Nanomechanics

Fig. 2.3 Geometry of fullerenes C60 , C100 and C540 Fig. 2.4 Structural configuration of an MWCNT

(MWCNT) consisting of several seamless hollow cylinders nested inside one another (Fig. 2.4). In 1993, single-walled carbon nanotube (SWCNT) was synthesized by Iijima and Ichihashi [20] and Bethune et al. [21] separately. After these works, nanotubes became an object of study in various fields of nanoscience and nanotechnology. Nowadays, SWCNTs and MWCNTs are synthesized by diverse methods such catalytic growth, arc-discharge and laser-ablation [22]. An SWCNT can be imagined as a single layer of graphene sheet that has been rolled into a seamless hollow cylinder with a constant radius. Although this is not the practical technique to synthesis the nanotubes, the above description helps illustrate the formation of the SWCNT structure [23]. Theoretically, an infinite number of SWCNTs can exist, because a graphene sheet can be rolled up with different angles. The geometry and structure of an SWCNT can be specified by a chiral vector which defines how the graphene sheet is rolled up (Fig. 2.5). The chiral vector is expressed as

2.1 Nanoscopic Structures

17

Fig. 2.5 An unrolled graphene sheet with definitions of unit vectors, and the chiral vector

Ch = na1 + ma2 , n ≥ m ≥ 0

(2.5)

where n and m are integers which indicate the number of steps along the hexagonal lattice, and a1 and a2 are the unit vectors of the hexagonal lattice of the graphene sheet. The indices (n, m) are commonly referred to as the chiral indices which represent the nanotubes chirality or helicity [22]. It should be noted that the nanotube axis is perpendicular to its chiral vector. Furthermore, the length of the chiral vector is the perimeter of the SWCNT and so the diameter of an SWCNT is given by d=

|a1 |  2 |Ch | aC−C  2 = n + nm + m 2 = 3(n + nm + m 2 ) π π π

(2.6)

where aC−C is the carbon-carbon bond length (0.142 nm). From a geometrical point of view, there is no limitation regarding the diameter of SWCNTs. However, it has been shown that collapsing the SWCNT into a flattened two-layer ribbon is energetically more favourable than keeping the circular morphology beyond a diameter value of 2.5 nm. This implies that physically the SWCNTs with diameter larger than 2.5 nm are not stable [24]. It should be noted that different SWCNTs with the same diameter but with different chiralities can also be produced. For example, a (19, 0) SWCNT has an identical diameter to a (16, 5) SWCNT. Consequently, the diameter alone is insufficient to characterize SWCNTs. For a unique characterization of SWCNTs, the other parameter required is the chiral angle which is the angle between the chiral vector and the unit vector a1 (or the zigzag axis). This angle can be obtained from [16]

18

2 Fundamental Tenets of Nanomechanics

θ0 = cos

−1



2n + m

√ 2 n 2 + mn + m 2



 = sin

−1

√ 3m

√ 2 n 2 + mn + m 2

 (2.7)

With the help of the chiral angle, SWCNTs are classified as zigzag (θ0 = 0), armchair (θ0 = π/6) and chiral (0 < θ0 < π/6). From Eq. (2.7), we see that θ0 = 0 corresponds to m = 0, and hence a zigzag SWCNT is defined by (n, 0). Again from Eq. (2.7), it is found that θ0 = π/6 corresponds to m = n, and so an armchair SWCNT is defined by (n, n). Since the structures of the zigzag and the armchair nanotubes have a high degree of symmetry, they are generally known as achiral nanotubes. In all other cases, the SWCNT is referred to as a general chiral SWCNT. Figure 2.6 displays the schematic representations of the three kinds of SWCNTs, i.e., the zigzag, the armchair, and the chiral. Depending on the chirality, SWCNTs can be metallic or semiconducting. Theoretical results indicate that SWCNTs are metallic when (n − m) = 3i (i is an integer), otherwise they are semiconducting. For example, all armchair nanotubes are metallic, while a zigzag nanotube is metallic only if n is a multiple of 3. According to a tight-binding MD simulation study [25], an interesting object is obtainable by the possibility that ultrathin SWCNTs may coalesce, without suffering atomic defects, to form new SWCNTs with different dimensions. It was shown that the chiral index of the coalesced SWCNT is a vector sum of the indices of the original SWCNTs. The chiral index sum rule is [25]

Fig. 2.6 Schematic illustrations of three kinds of SWCNTs

2.1 Nanoscopic Structures

19

(n 1 , m 1 ) + (n 2 , m 2 ) = (n 1 + n 2 , m 1 + m 2 )

(2.8)

where (n 1 , m 1 ) and (n 2 , m 2 ) are the indices of the original SWCNTs. From Eq. (2.8) we see that the chirality of coalesced SWCNT may be different from that of the original SWCNTs. An MWCNT consists of two or more rolled layers of graphene sheets. There are two possible forms to describe the structures of an MWCNT: a scroll-like structure and a Russian-doll like structure [26]. In the scroll-like structure, a single-layer graphene sheet is rolled up several times around an axis with uniform chirality [27]. In the second structure, two or more SWCNTs with different diameters are concentrically nested inside each other, very much like the structure of a Russian doll. The Russian-doll like structure is the thermodynamically stable form while the scrolllike structure is a metastable form. The length and diameter of the MWCNTs differ from SWCNTs and their properties are also very different. For example, the MWCNT structure is stiffer than that of the SWCNT, especially during compression. The inner diameter of MWCNTs typically varies in the range 1–3 nm, but the outer diameter can vary between 2–20 nm. The interlayer spacing in MWCNTS is very close to the distance between the layers in natural graphite, approximately 0.34 nm. Double-walled carbon nanotubes (DWCNT) are the simplest manifestation of the MWCNTs. These structures have some of the unique properties of the SWCNTs as well as the structural stability of MWCNTs [24]. A DWCNT is represented as (n i , m i )@(n o , m o ) where (n i , m i ) are the chiral indices of the inner SWCNT and (n o , m o ) are those pertinent to the outer SWCNT. The inner and outer indices are related to each other via (n o 2 + n o m o + m o 2 ) −

  2πr g 2 1  =0 3(n i 2 + n i m i + m i 2 ) + 3 aC−C

(2.9)

where r g is the interlayer spacing between the two shells. For a given (n i , m i ) set, the chiral vector of the outer SWCNT can be determined by choosing a value for n o and then solving Eq. (2.9) for m o to the nearest integer number. For example, for a (15, 10) inner SWCNT, and n o = 27, we find m o = 6.05, which rounds to m o = 6, and, therefore, the (15, 10) SWCNT can be nested inside the (27, 6) SWCNT.

2.1.1.3

Graphene Sheets

The planar carbon allotrope, which is called graphene, is a sheet with single atomic thickness consisting of a tessellation of hexagonal arrangement of carbon atoms on a single plane (Fig. 2.7). The term graphene was first used in 1987 to describe a single layer of carbon atoms [28]. For a number of years before its discovery in 2004, graphene was actually believed to be a theoretical object because the isolation of stable two-dimensional nanoscopic structures was considered to be improbable. Graphite consists entirely of graphene layers stacked on top of each other by weak van der Waals forces. Many experiments were attempted to produce graphene sheets

20

2 Fundamental Tenets of Nanomechanics

Fig. 2.7 A schematic model of graphene sheet composed of a two-dimensional hexagonal lattice

by chemical reactions, but these were not successful [29]. Finally, Novoselov and co-workers [12] succeeded in producing real samples of graphene using transparent adhesive tape. Graphene is the thinnest material known to humankind and can be regarded as the building block for graphitic materials [30]. The discovery and synthesis of the last allotrope of carbon was one of the major developments in the carbon nanoscience and nanotechnology, following the discoveries of the fullerenes and the carbon nanotubes. The lattice structure of graphene is made from unit cells containing two carbon atoms, indicated as A and B in Fig 2.8. The cell can be considered a rhombus with √ side a = 3aC−C . As shown in Fig. 2.8, the lattice vectors in the x1 , x2 coordinates can be written as √ √   a 3a a 3a , ,− a1 = , a2 = (2.10) 2 2 2 2

Fig. 2.8 Lattice structure of a graphene sheet

2.1 Nanoscopic Structures

21

Furthermore, the vectors directed from a type A atom to the three nearest neighbours type B atoms are expressed as:  R1 =

     a a a a a , R2 = − √ , √ , 0 , R2 = − √ , − 2 3 2 3 2 3 2

(2.11)

Graphene is a promising material for different technological applications due to its highly desirable electrical, optical, mechanical and thermal characteristics. This material is undoubtedly a very smart material in our generation and so it has attracted enormous attentions from the scientific world.

2.1.2 Nanoparticles The term nanoparticle is a very general one and refers mostly to ultrafine particles. In the literature, this term is used to describe various nanoscopic structures, and so in some cases can result in confusion. In this book, we use the term nanoparticle as a generic description of either spherical semiconductor or metal particles within the nanometer size range. In a narrower sense, the nanoparticles are regarded as particles in the range 10–20 nm but, generally speaking, all particles in the 1 nm–1 µm range may still be roughly designated as nanoparticles [31]. Nanoparticles have a rich history in science, where they have been manufactured and used for thousands of years. For example, the Romans employed gold and silver nanoparticles to produce colourful stained glasses in the 4th century. In the physics parlance, nanoparticles are considered as zero-dimensional nanoscopic structures. The nanoparticle structure is generally determined by the number of atoms it contains, and the character of the chemical interaction between these atoms. Several studies [32, 33] have indicated that nanoparticles containing only a certain number of atoms have optimum stable configurations. The corresponding numbers are usually called the structural magic numbers because they arise from maximum density, minimum volume and approximately a spherical configuration. The first structural magic number for face-centered cubic (FCC) structure is 13, which corresponds to the nanoparticle with one core atom and 12 surface atoms. In this case, the nanoparticle has an icosahedron form. If other layers of atoms are added to the nanoparticle such that its crystal structure is retained, a sequence of structural magic numbers can emerge. The well-known formula for magic numbers is [4] NnB =

1 (10n l3 − 15n l2 + 11n l − 3), n l ≥ 2 3

(2.12)

where NnB is the total number of atoms and n l is the number of layers. Furthermore, the number of surface atoms, NnS , can be calculated from NnS = 10n l2 − 20n l + 12, n l ≥ 2

(2.13)

22

2 Fundamental Tenets of Nanomechanics

Table 2.1 Structural magic numbers, surface atoms and surface to volume ratios for icosahedron FCC nanoparticles nl Magic number (NnB ) Surface atoms (NnS ) Surface to volume ratio (NnS /NnB ) 2 3 4 5 6 7 8

13 55 147 309 561 923 1415

12 42 92 162 252 362 492

0.92 0.76 0.62 0.52 0.45 0.39 0.35

The number of surface atoms and the magic number for each value of n l as well as the ratio of the surface-to-volume atoms are calculated using Eqs. (2.12) and (2.13), and are listed in Table 2.1. It should be noted that the structural magic numbers for body-centered cubic (BCC) nanoparticles differ from the FCC magic numbers. The magic number for BCC nanoparticles is calculated from [34] NnB = 4n l3 + 6n l2 + 4n l + 1, n l ≥ 1

(2.14)

For a spherical nanoparticle composed of NnB atoms, the volume of nanoparticle, Vn , is 4π 3 (2.15) Vn ≈ R = V0 NnB 3 n where Rn is the equivalent radius of the nanoparticle and V0 represents the volume of an individual atom. It is assumed that the volume V0 can be expressed by V0 =

4π 3 r 3 A

(2.16)

where r A is the atomic radius. From Eqs. (2.15) and (2.16), the equivalent radius is estimated according to the relation Rn = r A (NnB )1/3 .

(2.17)

In addition, the surface area, Sn , is a very important characteristic of a nanoparticle which can be estimated from Sn ≈ 4π Rn2 = 4πr A2 (NnB )2/3

(2.18)

Finally, we note that icosahedrons are not the only possible forms of nanoparticles. In some cases, nanoparticles have different structural from such as decahedron and cuboctahedron. Figure 2.9 shows a nanoparticle with magic number 147 in three different types of packing structures, that is, icosahedron, decahedron and cuboctahedron.

2.1 Nanoscopic Structures

23

Fig. 2.9 Schematic representation of an icosahedron, a decahedron and a cuboctahedron nanoparticle composed of 147 atoms

2.1.3 Nanowires Nanowires are wire-like structures having diameters in the nanometer size range. These nanoscopic structures have one unconfined direction available for electrical conduction and two quantum-confined directions [35]. This unique property makes these structures ideal candidates for applications wherein electrical conduction, rather than tunnelling transport, is required. Furthermore, nanowires are very sensitive to their ambient environment and so are suitable for sensing applications. There is a close relationship between their functionality and their structural and geometrical characteristics. Therefore, characterizing the geometrical properties of nanowires is especially important. Even nanowires made of the same material may possess different characteristics due to differences in their crystalline size, aspect ratios and surface conditions. Different types of nanowires, including semiconducting, metallic and insulator, can be synthesized via many approaches, such as thermal processes, physical/chemical vapour deposition and wet chemical processes [36]. Metallic nanowires can be grown from copper, silver, platinum, nickel and gold. Semiconductor nanowires are emerging as a powerful class of materials that are expected to play a critical role in future nanoscale electronic and optoelectronic devices. These can be grown from silicon, germanium and titanium dioxide. Furthermore, nanowires could have triangular, rectangular, pentagonal, hexagonal, semicircular, circular or other unconventional cross-sectional shapes [37].

2.1.4 Biological Nanoscopic Structures Biological nanoscopic structures are a fundamental part of all biological simple and complex systems. Research has focused on understanding the dramatic and exotic electronic, optical, and mechanical properties of an ever increasing variety of biological nanoscopic structures, especially those that are present in cellular systems, such microtubules and molecular protein motors, such as kinesins for cellular transport, and in neurons. Microtubules are one of the filamentous structures found in the cytoplasm of eukaryotic cells. They play an essential role in the cellular mechanical stability and the preservation of the cellular morphology. They also play an essential role in many fundamental biological processes including cell division, cell

24

2 Fundamental Tenets of Nanomechanics

Fig. 2.10 Kinesin motor protein performing a stepping motion on a microtubule

motility, and intracellular transport [38]. Kinesin is an essential anterograde microtubule motor protein that transports vesicles along microtubules from the cell interior to the nucleus. Figure 2.10 shows a kinesin protein in a stepping motion on a microtubule. There are more than 30000 biological nanoscopic structures of various types in humans. Currently there is deep interest in computational modelling and experimental investigation of the structure and function of biological nanoscopic structures, and their natural and man-made design, for applications in bionanoscience, nanoneuroscience, medical physics, nanomedicine and medical nanotechnology. Another area of deep interest is the investigation of the behaviour of biological nanoscopic structures, especially the neural nanoscopic structures, in external physical fields, such the radio frequency and microwave electromagnetic fields emitted from mobile communication and power line systems. Understanding the basic science of biological nanoscopic structures involves the prediction and manipulation of their physical properties, such as their natural and man-made self-assembly from seemingly independent units that associate with each other to form organic many-body conformational states with local and nonlocal properties. The elemental units can be constructed from synthetic materials or obtained from natural functional biological macromolecules. Biological building blocks with known structures provide an extremely versatile and useful resource for designing biological nanoscopic structures. For example in designing various types of proteins from the basic amino acids, the structural databases available in large libraries of protein molecule data banks provide the appropriate amino acid chains together with a range of geometrical conformation, surfaces, and chemical properties. These data bases have been extremely useful in computational modelling studies at nanoscopic and microscopic scales. The research involving engineered peptides can expand our knowledge of the available chemical space and enhance our understanding of those properties that are desired for particular biomedical uses. The underlying trend of the past several years in the research involving biological nanoscopic structures has been to downsize the electronic, optoelectronic, and

2.1 Nanoscopic Structures

25

mechanical biological structures and bio-devices from microscopic scales to those with features in nanoscopic regime. This downsizing has led to a dramatic appearance of the difference between the microscopic phenomena describable in terms of classical physical theories and nanoscopic phenomena that normally require the use of many-body interatomic potentials, various types of force fields, or the use of manybody quantum mechanical-based approaches, such as the density functional theory (DFT). Furthermore, the synthesis of biological nanoscopic structures via colloidal chemistry has also appeared as a viable approach in recent years. These approaches, especially the computational modelling approaches, are now recognized as offering powerful tools in our endeavour to further advance our basic understanding of biological systems as well as in biomedical applications of nanotechnology.

2.2 Techniques for Modelling Mechanical Characteristics of Nanoscopic Structures An awareness and knowledge of the mechanical properties of nanoscopic structures is an essential prerequisite for an efficient designing of realistic nanodevices. Therefore, the development of effective methods to compute their mechanical behaviour poses a multitude of challenging research problems. The mechanical properties and behaviour of nanoscopic structures can be predicted by using various theoretical and computational methodologies. These techniques include quantum mechanical-based methods, atomistic modelling methods, continuum modelling methods and multiscale and multiphysics simulation methods. Obviously, the quantum mechanicalbased methods, as expressed by solutions of the many-body Schrödinger equation involving the dynamics of the electrons and the nuclei [39], are often the most accurate methods to obtain the behaviour of nanoscopic structures, but they are computationally very expensive, time-consuming and limited to the study of very small systems containing a few tens to a few hundreds of atoms [40]. The atomistic modelling techniques, based on the use of prescribed interatomic potentials and empirical force fields, are another important numerical simulation methods for the investigation of the physical, chemical, electronic, thermal, transport and mechanical properties of nanoscopic structures. Compared to the quantum mechanical-based methods, the atomistic-based methods can be used to study much larger systems. However, typical atomistic-based modelling methods are still constrained to relatively small systems containing up to several million atoms. Furthermore, these methods are usually pertinent to studies of phenomena on very short time scales, from picoseconds to nanoseconds [41]. In addition, the atomistic-based methods are computationally prohibitive because they need the use of very highperformance facilities. Consequently, the use of alternative approaches are highly desirable and it is well-known that the continuum-based modelling methods are perhaps the most computationally efficient methodologies for the analysis of the nanoscopic structures. The continuum-based methods are generally based on the use

26

2 Fundamental Tenets of Nanomechanics

of traditional mechanical structures such as rods, beams, plates and shells. These methods are much faster than the atomistic-based modelling methods and can be useful in the study of long-range phenomena in extended atomic systems. In recent years, a series of multiscale simulation methods have been developed which combine the atomistic-based and continuum-based methods. In the following subsections, we introduce three important approaches including the atomistic-based, continuumbased and multiscale-based methods.

2.2.1 Atomistic-Based Modelling Methods Atomistic-based modelling methods serve as important tools to provide fundamental descriptions of the behaviour of nanoscopic structures. These methods have been widely adopted to complement experimental mechanical testing, and to provide the required input data for continuum models. Furthermore, they can be employed to directly predict the mechanical properties and behaviour of nanoscopic structures. Using atomistic-based simulations, we can design different virtual tests, such as torsion tests, which are currently not experimentally feasible. However, it should be noted that the atomistic-based modelling techniques provide only estimates and they employ numerous simplifications and assumptions. The most popular atomisticbased methods are the MD and MC simulation methods which consider the interaction between the molecules or atoms in nanoscopic structures and nanoscale systems. Let us now discuss briefly the MD simulation method. The MD simulation method is a computer-based computational modelling technique that allows for the prediction of the physical properties of atomic, molecular, granular systems etc. In the MD simulation, the space-time trajectories of individual atoms in an assembly of N interacting atoms, or molecules, are calculated within the framework of either a Newtonian (i.e., deterministic) dynamics or a Langevin-type (i.e., stochastic) dynamics. A classical MD-based simulation proceeds by considering a model nanoscopic structure consisting of N atoms in any morphology or crystal structure confined to a simulation cell of a finite volume V representing a finite part of an infinite system. The choice of the number of atoms depends on the computational power available. The total interaction potential energy, HI , of the N atoms can be expressed in terms of the position-dependent interaction potentials of its constituent atoms. The general form of the total interaction potential energy can be expressed as [42]: H I (r1 , r2 , ..., r N ) =

N  i

V1 (ri ) +

N  N  i

j=i

V2 (ri , r j ) +

N  N  N  i

V3 (ri , r j , rk ) + ...

j=i k=i, j

(2.19) where ri is the position vector of the ith atom and Vn is called the n-body interatomic potential function. In Eq. (2.19), V1 describes the external applied force field (e.g. gravitational and electrostatic fields) and is only a function of the position

2.2 Techniques for Modelling Mechanical Characteristics of Nanoscopic Structures

27

Fig. 2.11 Pair-wise interaction of two atoms i and j

vector. Furthermore, V2 is the two-body, or pairwise, potential between atoms i and j (Fig. 2.11), and V3 denotes the three-body potential involving atoms i, j and k. All potentials, except V1 , are expressed in terms of interatomic vectors, as ri j = ri − r j . At each simulation time step, the resultant forces experienced by individual atoms are computed from ∂ HI (2.20) Fi = −∇ri HI = − ∂ri where ∇ri denotes the spatial gradient. Once the resultant forces felt by individual atoms are calculated, the instantaneous location and velocity of each atom are determined by Newton’s equations of motion Fi d 2 ri = dt 2 mi

(2.21)

where m i is the mass of the ith atom. Therefore, the motion of the interacting N atoms is described by 3N simultaneous coupled second order differential equations. These equations are solved numerically within the computational cell by various numerical integration schemes, all based on the finite difference methods. In this connection, the coupled equations of motion are discretized in time domain. Knowledge of the position vector ri (t), the velocity vector vi (t) and the force Fi (t) at simulation time t, the integration scheme allows for the computation of the same quantities at a later time step (t + dt). Among the several integration schemes available, one very simple and popular algorithm is the velocity Verlet algorithm [43]. According to this algorithm, the position and velocity vectors of the atoms are updated after each simulation time-step dt by

28

2 Fundamental Tenets of Nanomechanics

Fi (t) 1 ri (t + dt) = ri (t) + vi (t)dt + dt 2 2 mi

(2.22)

1 1 Fi (t) vi (t + dt) = vi (t) + dt 2 2 mi

(2.23)

1 Fi (t + dt) 1 vi (t + dt) = vi (t + dt) + dt 2 2 mi

(2.24)

The choice of the time step depends on several factors, such as the temperature, density and size of the atoms involved, and the nature of the force law [16]. Furthermore, the initial position and velocity vectors of the atoms are the input parameters. Therefore, the main procedure for computing the positions and velocities of atoms using the velocity Verlet algorithm can be summarized as follows. (a) (b) (c) (d) (e)

Specify the initial positions and velocities of all atoms. Choose the appropriate time step. Calculate the resultant forces experienced by individual atoms at time t. Calculate the positions of all atoms at the next time step t + dt from Eq. (2.22). Calculate the velocities of all atoms at the half time step t + dt/2 from Eq. (2.23). (f) Calculate the resultant forces experienced by individual atoms at time t + dt. (g) Update the velocities of all atoms at the next time step t + dt from Eq. (2.24). (h) Repeat the above procedures for the next time step. After the calculation of velocities at each time step, the instantaneous kinetic energy K ins , and hence the instantaneous temperature Tins and the instantaneous pressure Pins , can be computed from the equipartition and virial theorems respectively according to N 1 m i vi .vi (2.25) K ins = 2 i Tins = Pins

1 = 3V



N 1  m i vi .vi 3N k B i N  i

m i vi .vi +

N 

(2.26)  ri .Fi

(2.27)

i

where k B is the Boltzmann constant. In the absence of external fields and for a system with pairwise interactions, the instantaneous pressure can be written as

Pins

⎞ ⎛ N N N   1 ⎝ = m i vi .vi + ri j .Fi j ⎠ 3V i i j=i

in which Fi j is the force experienced by atom i due to atom j.

(2.28)

2.2 Techniques for Modelling Mechanical Characteristics of Nanoscopic Structures

29

The MD simulations can be carried out within the framework of an appropriate statistical-mechanical ensemble, such as the micro-canonical, or constant-(NVE), ensemble, the canonical, or constant-(NVT), ensemble, and the isothermal-isobaric, or constant-(NPT), ensemble depending upon which thermodynamic state variables are held fixed. Historically, the first generation of MD simulations were performed on the constant-NVE ensembles in which the number of atoms N , the volume V , and the total energy of the system E are held fixed throughout the simulation. This ensemble represents an isolated physical system wherein there is no exchange of particle and energy with the surrounding media. However, many physical phenomena cannot be modelled by this ensemble because the constraints do not correspond to those of real experimental setups. In the constant-NVT ensemble, the fixed variables are the number of atoms N , the volume V and the temperature T . In this ensemble, the temperature acts as a control parameter. The constant-NPT ensemble is very similar to the canonical ensemble, except that the pressure P, rather than V , is kept constant. MD simulations on a statistical - mechanical ensemble at constant temperature is usually performed via a thermostat algorithm [44]. Several constant temperature MD methods have been proposed. Among various approaches, the method that generates the canonical ensemble distribution in both the configuration and momentum parts of the phase space is of particular significance. The idea is referred to the Extended System Method and was proposed and mathematically formulated by Nosé [45]. Modified forms of equations of motion for this method was later proposed simultaneously by Nosé [46] and Hoover [47]. Therefore, this algorithm is generally referred to as the Nosé-Hoover thermostat. On the basis of the Nosé-Hoover thermostat, the modified equations of motion are Fi d 2 ri = − ηb vi dt 2 mi

(2.29)

where ηb is time-dependent thermodynamic friction coefficient of the heat bath, which together with the system embedded in it form the extended system. This coefficient is not a constant value and can switch between positive and negative values. It is negative if energy has to be added into the system, and positive if energy has to be removed from the system. The friction coefficient is described by the following ordinary differential equation   N 1  dηb = m i vi .vi − gk B T (2.30) dt Q i where Q is the fictitious heat bath mass and determines the time-scale of the temperature fluctuation. Furthermore, T denotes the temperature that is to be kept constant and g is the number of degrees of freedom. Note that g should be set as (3N + 1) (3N coordinates of the atoms, plus the extra degree of freedom ηb ) in virtual time sampling, and g must be set as 3N in real time sampling [46]. A velocity Verlet formulation of the modified equations of motion using this thermostat formalism is given by [48]:

30

2 Fundamental Tenets of Nanomechanics



Fi (t) 1 ri (t + dt) = ri (t) + vi (t)dt + dt 2 − ηb (t)vi (t) 2 mi

(2.31)

Fi (t) 1 1 vi (t + dt) = vi (t) + dt − ηb (t)vi (t) 2 2 mi

(2.32)

  N  1 1 ηb (t + dt) = ηb (t) + m i vi (t).vi (t) − gk B T dt 2 2Q i

(2.33)

 N   1 1 1 1 dt m i vi (t + dt).vi (t + dt) − gk B T ηb (t + dt) = ηb (t + dt) + 2 2Q 2 2 i (2.34)

1 Fi (t + dt) 2 1 vi (t + dt) = (2.35) vi (t + dt) + dt 2 + ηb (t + dt)dt 2 2 mi It should be noted that Eq. (2.35) is a much more complicated version of the standard velocity Verlet algorithm, Eq. (2.24) for updating the velocities of the atoms. The interested reader can find additional details of the MD simulation in Refs. [16, 42].

2.2.2 Continuum Modelling Methods Historically continuum mechanics is the most well-known and sophisticated branch of classical mechanics dealing with the mechanical properties and behaviour of systems modelled as continuous matter rather than as combination of discrete atoms. In this mechanics, it is assumed that the physical properties of the matter are distributed continuously in the spaces that they occupy. Furthermore, continuum theory regards matter as infinitely divisible and there is no empty space between atoms. In other word, regardless of how small is the volume elements that the matter is subdivided into, every element will contain matter [49]. To derive differential equations describing the behaviour of matter within the continuum mechanics framework, fundamental physical principles, such as the conservation of mass, the conservation of linear and angular momenta, and the conservation of energy, are assumed. Conservation of a quantity means that the quantity cannot change over time, unless an external agent intervenes. Furthermore, in continuum mechanics, different material equations or constitutive equations are used to describe the special macroscopic properties of the matter. The most important concepts of continuum mechanics including the fundamental physical principles as well as different constitutive equations will be reviewed in Chap. 3. Classical continuum mechanics is not only applicable to a majority of the problems at macroscale, but also offer an effective way to handle nanoscopic structures and

2.2 Techniques for Modelling Mechanical Characteristics of Nanoscopic Structures

31

nanoscale systems composed of millions of atoms or molecules involving only very minimal numerical computations. As a result, various continuum-based approaches have been developed to model, compute and analyse the mechanical characteristics of nanoscopic structures. Although these approaches have been widely used for modelling the mechanical properties of nanoscopic structures, however, several challenges are naturally encountered when they are applied to systems at much reduced spatial and time scales. For example, the modelling of matter via this mechanics ignores the fact that matter is made of atoms and so the discrete nature of the nanoscopic structures is dissolved into the background. Furthermore, the classical continuum modelling methods cannot incorporate the small-scale effects, such as the long-range bond forces, electric forces, magnetic forces, van der Waals interactions, hydrogen bonds, and the interaction between crystal cells, into their formulation. In addition, the methods are scale free and are not adequate for the analysis of the nanoscopic structures. Therefore, the classical continuum models fail when the microscopic and macroscopic length scales of material become comparable. To overcome the limitations of the classical continuum methods, modified continuum mechanics theories incorporating the length scale have been developed. Such classical models can be modified so as to accommodate the underlying microscopic, or nanoscopic, structure by expanding the constitutive relations leading to what is referred to as higher-order continua [50]. Higher-order continuum methods, or size-dependent continuum modelling methods, are modifications of the standard classical continuum modelling methods which now include the material length scales. Furthermore, these modified methods allow the integration of size-effects into the classical continuum models. In these theories, the standard constitutive equations are generalized by incorporating the higher-order derivatives of strains, stresses and/or accelerations. Generally, the size-dependent continuum modelling methods can be classified into three different classes, namely; the strain gradient family, the microcontinuum field theories and the nonlocal elasticity theories [51]. The strain gradient family consists of the couple stress theory [52, 53], the modified couple stress theory [54], the first and second strain gradient theories [55, 56] and the modified strain gradient theory [57]. Several microcontinuum field theories, including the micromorphic elasticity, the microstretch elasticity, the micropolar elasticity and the dilatation elasticity, have been formulated [58]. The third family of size-dependent continuum modelling methods is the nonlocal elasticity theories, which forms the main focus of this book. The nonlocal continuum field theories were originally proposed by Kröner [59] and then systematically extended by Eringen and co-workers [60–62]. In recent years, another class of higher-order continuum methods, called the nonlocal strain gradient theory, has been proposed based on a combination of the nonlocal elasticity theory and the strain gradient theory [63, 64]. A systematic way to represent the constitutive equations of materials is to utilize implicit constitutive equations. The idea of the implicit constitutive equations was first introduced by Morgan [65]. The implicit constitutive equation is given by [63] f(σ, ε, ∇ 2 σ, ∇ 2 ε) = 0

(2.36)

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2 Fundamental Tenets of Nanomechanics

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. (2.36) is rewritten as trace(a1 ε + a2 σ )I + a3 ε + a4 σ + ∇ 2 [trace(a5 ε + a6 σ )I + a7 ε + a8 σ ] = 0 (2.37) in which the coefficients ai are the constitutive coefficients and I is the identity matrix. 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 these constitutive coefficients.

2.2.3 Multiscale Simulation Methods Another important approach to modelling the nanoscopic structures and systems composed of nanoscale components is the multiscale simulation method. There are several motivations for using the multiscale simulation method. Firstly, it is not always necessary to calculate the full atomistic information in the whole simulation domain. Secondly, the limitations of both atomistic and continuum modelling methods have motivated the development of the multiscale simulation approach that couples the atomistic and continuum modelling techniques together. Generally in the multiscale simulation method, a domain is divided into several subdomains. These subdomains with fine features which capture the individual atomistic dynamics are accurately modelled using atomistic-based approaches such as MD simulations. In all other, so-called coarse grain subdomains, where accuracy is of secondary importance, the continuum modelling methods are used. The key topic in multiscale modelling is establishing the overlap between the fine and coarse grain subdomains. On the basis of the method of information transfer between different subdomains, the multiscale method is conceptually classified into two main categories, namely, the hierarchical and the concurrent branches [66]. In the hierarchical multiscale method, the simulations are performed at the fine subdomains and the information from fine-scale is extracted using a bridging methodology and then passed to the coarse grain subdomains. In this approach, the fine-scale model is embedded into the coarse grain model and the transfer of information occurs through the interface. The hierarchical multiscale method is computationally more efficient than the concurrent method. Since the multiscale simulation methods are beyond the scope of this book, the interested reader is referred to appropriate texts (for example [42, 67]).

2.3 Mechanical Behaviour of Nanoscopic Structures: Pertinent Definitions and Concepts In this section, some useful concepts and definitions pertinent to the mechanical behaviour of the nanoscopic structures covering their vibrational behaviour, buckling behaviour, and their static deformations are briefly introduced. These concepts are

2.3 Mechanical Behaviour of Nanoscopic Structures: Pertinent Definitions and Concepts

33

crucial for understanding the functionality of structures in use, and thus form the focus of many investigations. For a more comprehensive and scholarly coverage of these topics, the reader is referred to various texts (for example [68–70]).

2.3.1 Bending Behaviour of One-Dimensional Nanoscopic Structures The geometry of a one-dimensional nanoscopic structure ensures that it offers a significant resistance to externally applied loads and bending moments. When nanoscopic structures are subjected to static loads, the performance of these structures is deeply affected. Therefore, understanding the static behaviour of nanoscopic structures, including their axial deformation, bending and buckling, is of importance for their future applications. Bending is one of the most important modes of mechanical deformations observed during structural changes. Figure 2.12 shows both deformed and undeformed configurations of an SWCNT under the application of a bending moment. The first experimental observation of the bending behaviour of SWCNTs and MWCNTs was reported by Iijima and co-workers [71]. Image of high-resolution transmission electron microscopy (HRTEM) of a bent SWCNT is shown in Fig. 2.13, taken from Ref. [71].

2.3.2 Buckling Behaviour Buckling or structural instability is a physical phenomenon occurring when the deformed configuration of a thin and/or slender structure, under a compressive load

Fig. 2.12 Deformed and undeformed configurations of an SWCNT under the application of a bending moment

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2 Fundamental Tenets of Nanomechanics

Fig. 2.13 HRTEM image of a bent SWCNT as reported by Iijima et al. [71]. Reprinted with permission from publisher

undergoes a relatively sudden deformation in shape, typically with large or relatively large increase in displacements and strains. This instability is characterized by the loss of structural stiffness. It is a very important phenomenon because it often occurs without any prior warning, and can have catastrophic effects. For example, when a column is subjected to an axial compressive force, it first shortens slightly but at a critical load the column bows out, and it is said that the column has buckled. The buckled patterns strongly depend on geometric and material parameters. The load at which buckling occurs depends on the stiffness of a component, and not upon the strength of its materials. This behaviour can be mostly seen in different members, such as rods, beams, rings, arches, plates and shells, under diverse loading conditions such as axial compression, pressure, bending, torsion, and their combinations. Experimental evidence indicates the occurrence of buckling in several nanoscopic structures including SWCNTs and MWCNTs, nanowires and graphene sheets, and fullerene molecules [72–79]. Figure 2.14, taken from Ref. [78], shows the buckling of the MWCNT during a mechanical compressive test. The buckling of nanoscopic structures has special characteristics relevant to engineering and scientific applications. For example, the axial buckling of carbon nanotubes is very important for the design of the AFM probes.

2.3.3 Vibrational Behaviour Another important phenomenon is mechanical vibration. In general parlance, the term vibration refers to an oscillatory motion executed by a body or a mechanical system about its equilibrium position. In the course of the vibration of a system its kinetic energy is transferred to its potential energy, and vice versa, alternately. Vibration analysis is usually concerned with the prediction of frequencies and the corresponding mode shapes of the system, with frequency representing the number of energy transformation cycles per second. Vibrations play an essential role in Nature and in various engineering applications. At nanoscopic scales, vibrations are inherently present in most physical systems due to thermal drift of atoms. This implies that, the consideration of vibrational characteristics of nanoscopic structures is essential

2.3 Mechanical Behaviour of Nanoscopic Structures: Pertinent Definitions and Concepts

35

Fig. 2.14 Image of an MWCNT during a mechanical compression test as reported by Zhao et al. [78]. Reprinted with permission from publisher

Fig. 2.15 SEM images of the first and second vibrational modes of a cantilevered boron nanowire as reported by Ding et al. [84]. Reprinted with permission from publisher

in many nanodevices, such as high frequency oscillators and sensors. Furthermore, an insight into the elastic properties of objects highly reduced in spatial dimensions, like nanoscopic structures, can also be obtained in a fundamental manner by means of investigating their vibrational characteristics. The first experimental studies of vibrational properties of such structures were performed in spinel crystallites using Raman spectroscopy [80]. This was later extended to other nanoscopic structures, such as spherical metal nanoparticles [81], semiconductor quantum dots [82] and carbon nanotubes [83]. The scanning electron microscope (SEM) image of the first and second vibrational modes of a cantilevered boron nanowire is shown in Fig. 2.15, taken from Ref. [84].

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References 1. V. Harik (ed.), Trends in Nanoscale Mechanics (Springer, New York, 2014) 2. G. Gao, Nanostructures and Nanomaterials: Synthesis, Properties, and Applications (Imperial College Press, Singapore, 2004) 3. B. Wu, A. Heidelberg, J.J. Boland, Mechanical properties of ultrahigh-strength gold nanowires. Nat. Mater. 4, 525–529 (2005) 4. M. Ashby, P. Ferreira, D. Schodek, Nanomaterials, Nanotechnologies and Design (Elsevier, Oxford, 2009) 5. R.P. Feynman, There’s plenty of room at the bottom. Eng. Sci. 23, 22–36 (1960) 6. Y. Gogodtsi (ed.), Nanomaterials Handbook (Taylor & Francis-CRC Press, Philadelphia, 2006) 7. J.N. Tiwari, R.N. Tiwari, K.S. Kim, Zero-dimensional, one-dimensional, two-dimensional and three-dimensional nanostructured materials for advanced electrochemical energy devices. Prog. Mater. Sci. 57, 724–803 (2012) 8. H. Gleiter, Nanostructured materials: basic concepts and microstructure. Acta Mater. 48, 1–29 (2000) 9. V.V. Pokropivny, V.V. Skorokhod, Classification of nanostructures by dimensionality and concept of surface forms engineering in nanomaterial science. Mater. Sci. Eng. C 27, 990–993 (2007) 10. H. Kroto, J. Heath, S. Obrien, R. Curl, R. Smalley, C60 Buckminsterfullerene. Nature 318, 162163 (1985) 11. S. Iijima, Helical microtubules of graphitic carbon. Nature 354, 56–58 (1991) 12. 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, 666–669 (2004) 13. O.A. Shenderova, D.M. Gruen (eds.), Ultrananocrystalline Diamond: Synthesis, Properties, and Applications (William Andrew Publishing, New York, 2006) 14. H. Zhang, D. Ye, Y. Liu, A combination of Clar number and Kekulé count as an indicator of relative stability of fullerene isomers of C60 . J. Math. Chem. 48, 733–740 (2010) 15. M.S. Dresselhaus, G. Dresselhaus, P.C. Eklund, Science of Fullerenes and Carbon Nanotubes (Academic Press, San Diego, 1996) 16. H. Rafii-Tabar, Computational Physics of Carbon Nanotubes (Cambridge University Press, Cambridge, 2008) 17. H. Terrones, M. Terrones, Curved nanostructured materials. New J. Phys. 5, 126.1–126.37 (2003) 18. S. Adhikari, R. Chowdhury, Vibration spectra of fullerene family. Phys. Lett. A 375, 2166–2170 (2011) 19. L.V. Radushkevich, V.M. Lukyanovich, O structure ugleroda, obrazujucegosja pri termiceskom razlozenii okisi ugleroda na zeleznom konarte (The structure of carbon forming in thermal decomposition of carbon monoxide on an iron catalyst). Zurn. Fisc. Chim. 26, 88–95 (1952) 20. S. Iijima, T. Ichihashi, Single-shell carbon nanotubes of 1-nm diameter. Nature 363, 603–605 (1993) 21. D.S. Bethune, C.H. Kiang, M.S. de Vries, G. Gorman, R. Savoy, J. Vazquez, R. Beyers, Cobaltcatalysed growth of carbon nanotubes with single-atomic-layer walls. Nature 363, 605–607 (1993) 22. V.N. Popov, Carbon nanotubes: properties and application. Mat. Sci. Eng. R 43, 61–102 (2004) 23. H. Rafii-Tabar, E. Ghavanloo, S.A. Fazelzadeh, Nonlocal continuum-based modeling of mechanical characteristics of nanoscopic structures. Phys. Rep. 638, 1–97 (2016) 24. J. Tersoff, R.S. Ruoff, Structural properties of a carbon-nanotube crystal. Phys. Rev. Lett. 73, 676–679 (1994) 25. T. Kawai, Y. Miyamoto, O. Sugino, Y. Koga, General sum rule for chiral index of coalescing ultrathin nanotubes. Phys. Rev. Lett. 89, 085901 (2002) 26. P.J.F. Harris, Carbon Nanotube Science: Synthesis, Properties and Applications (Cambridge University Press, Cambridge, 2009)

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27. W. Ruland, A.K. Schaper, H. Hou, A. Greiner, Multi-wall carbon nanotubes with uniform chirality: evidence for scroll structures. Carbon 41, 423–427 (2003) 28. P. Delhaes, Graphite and Precursors (CRC Press, Amsterdam, 2001) 29. M.I. Katsnelson, Graphene: carbon in two dimensions. Mater. Today 10, 20–27 (2007) 30. C. Srinivasan, Graphene-mother of all graphitic materials. Curr. Sci. 92, 1338–1339 (2007) 31. M. Hosokawa, K. Nogi, M. Naito, T. Yokoyama, Nanoparticle Technology Handbook (Elsevier, Oxford, 2007) 32. T. Ikeshoji, B. Hafskjold, Y. Hashi, Y. Kawazoe, Molecular dynamics simulation for the cluster formation process of lennard-jones particles: magic numbers and characteristic features. J. Chem. Phys. 105, 5126–5137 (1996) 33. V.Y. Shevchenko, A.E. Madison, Structure of nanoparticles: I. generalized crystallography of nanoparticles and magic numbers. Glass Phys. Chem. 28, 40–43 (2002) 34. R.L. Johnston, Atomic and Molecular Clusters (CRC Press, London, 2002) 35. J. Sarkar, G.G. Khan, A. Basumallick, Nanowires: properties, applications and synthesis via porous anodic aluminium oxide template. Bull. Mater. Sci. 30, 271–290 (2007) 36. U. Cvelbar, Towards large-scale plasma-assisted synthesis of nanowires. J. Phys. D: Appl. Phys. 44, 174014 (2011) 37. 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, 094308 (2011) 38. F. Daneshmand, E. Ghavanloo, M. Amabili, Wave propagation in protein microtubules modeled as orthotropic elastic shells including transverse shear deformations. J. Biomech. 44, 1960– 1966 (2011) 39. D.D. Vvedensky, Multiscale modelling of nanostructures. J. Phys.: Condens. Matter 16, R1537– R1576 (2004) 40. A. Bartók-Pirtay, The Gaussian Approximation Potential: An Interatomic Potential Derived from First Principles Quantum Mechanics (Springer, London, 2010) 41. W.K. Liu, E.G. Karpov, S. Zhang, H.S. Park, An introduction to computational nanomechanics and materials. Comput. Methods Appl. Mech. Eng. 193, 1529–1578 (2004) 42. W.K. Liu, E.G. Karpov, H.S. Park, Nano Mechanics and Materials: Theory, Multiscale Methods and Applications (John Wiley & Sons Ltd., New Jersey, 2006) 43. M.P. Allen, D.J. Tildesley, Computer Simulation of Liquids (Clarendon Press, Oxford, 1987) 44. P.H. Hünenberger, Thermostat algorithms for molecular dynamics simulations. Adv. Polym. Sci. 173, 105–149 (2005) 45. S. Nosé, A molecular dynamics method for simulations in the canonical ensemble. Mol. Phys. 52, 255–268 (1984) 46. S. Nosé, A unified formulation of the constant temperature molecular dynamics methods. J. Chem. Phys. 81, 511–519 (1984) 47. W.G. Hoover, Canonical dynamics: equilibrium phase-space distributions. Phys. Rev. A 31, 1695–1697 (1985) 48. H. Rafii-Tabar, Modelling the nano-scale phenomena in condensed matter physics via computer-based numerical simulations. Phys. Rep. 325, 239–310 (2000) 49. F. Irgens, Continuum Mechanics (Springer, Berlin, 2008) 50. H. Askes, A.V. Metrikine, Higher-order continua derived from discrete media: continualisation aspects and boundary conditions. Int. J. Solids Struct. 42, 187–202 (2005) 51. H.T. Thai, T.P. Vo, T.K. Nguyen, S.E. Kim, A review of continuum mechanics models for size-dependent analysis of beams and plates. Comput. Struct. 177, 196–219 (2017) 52. R.D. Mindlin, H. Tiersten, Effects of couple-stresses in linear elasticity. Arch. Ration. Mech. Anal. 11, 415–448 (1962) 53. R.A. Toupin, Theories of elasticity with couple stress. Arch. Ration. Mech. Anal. 17, 85–112 (1964) 54. F. Yang, A.C.M. Chong, D.C.C. Lam, P. Tong, Couple stress based strain gradient theory for elasticity. Int. J. Solids Struct. 39, 2731–2743 (2002) 55. R.D. Mindlin, Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16, 51–78 (1964)

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56. R.D. Mindlin, Second gradient of strain and surface-tension in linear elasticity. Int. J. Solids Struct. 1, 417–438 (1965) 57. D.C.C. Lam, F. Yang, A.C.M. Chong, J. Wang, P. Tong, Experiments and theory in strain gradient elasticity. J. Mech. Phys. Solids 51, 1477–1508 (2003) 58. A.C. Eringen, Microcontinuum Field Theories: I. Foundations and Solids (Springer, New York, 1999) 59. E. Kröner, Elasticity theory of materials with long range cohesive forces. Int. J. Solids Struct. 3, 731–742 (1967) 60. A.C. Eringen, Linear theory of nonlocal elasticity and dispersion of plane waves. Int. J. Eng. Sci. 10, 425–435 (1972) 61. A.C. Eringen, D.G.B. Edelen, On nonlocal elasticity. Int. J. Eng. Sci. 10, 233–248 (1972) 62. A.C. Eringen, On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J. Appl. Phys. 54, 4703–4710 (1983) 63. E.C. Aifantis, On the gradient approach-relation to Eringens nonlocal theory. Int. J. Eng. Sci. 49, 1367–1377 (2011) 64. C. Lim, G. Zhang, J.N. Reddy, A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation. J. Mech. Phys. Solids 78, 298–313 (2015) 65. A.J.A. Morgan, Some properties of media defined by constitutive equations in implicit form. Int. J. Eng. Sci. 4, 155–178 (1966) 66. M. Tak, D. Park, T. Park, Computational coupled method for multiscale and phase analysis. J. Eng. Mater. Technol. 135, 021013 (2013) 67. J. Fish (ed.), Multiscale Methods: Bridging the Scales in Science and Engineering (Oxford University Press, Oxford, 2009) 68. C.M. Wang, C.Y. Wang, J.N. Reddy, Exact Solutions for Buckling of Structural Members (CRC Press, Florida, 2005) 69. S.S. Rao, Vibration of Continuous Systems (John Wiley & Sons Inc., New Jersey, 2007) 70. E. Ventsel, T. Krauthammer, Thin Plates and Shells: Theory, Analysis, and Applications (CRC Press, New York, 2001) 71. S. Iijima, C. Brabec, A. Maiti, J. Bernholc, Structural exibility of carbon nanotubes. J. Chem. Phys. 104, 2089–2092 (1996) 72. 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, 103109 (2005) 73. 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, 2203–2208 (2005) 74. P.R. Guduru, Z. Xia, Shell buckling of imperfect multiwalled carbon nanotubes-experiments and analysis. Exp. Mech. 47, 153–161 (2007) 75. H.W. Yap, R.S. Lakes, R.W. Carpick, Mechanical instabilities of individual multiwalled carbon nanotubes under cyclic axial compression. Nano Lett. 7, 1149–1154 (2007) 76. S.J. Young, L.W. Ji, S.J. Chang, T.H. Fang, T.J. Hsueh, Nanoindentation of vertical ZnO nanowires. Physica E 39, 240–243 (2007) 77. 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, 3214–3219 (2009) 78. 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, 206–213 (2011) 79. Y. Mao, W.L. Wang, D. Wei, E. Kaxiras, J.G. Sodroski, Graphene structures at an extreme degree of buckling. ACS Nano 5, 1395–1400 (2011) 80. 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, 2052–2055 (1986) 81. 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, 6243–6248 (1991) 82. A. Tanaka, S. Onari, T. Arai, Low-frequency Raman scattering from CdS microcrystals embedded in a germanium dioxide glass matrix. Phys. Rev. B 47, 1237–1243 (1993)

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Chapter 3

Fundamental Notions from Classical Continuum Mechanics

Understanding the basic concepts of continuum mechanics is very important when attempting to describe the materials behaviour that completely fill the occupied space. With the rapid development of science and technology, continuum mechanics has found significant applications in almost all cutting-edge research activities. In this chapter, we provide an introduction to fundamental concepts of continuum mechanics. The deformation of a continuum can be described by three types of mathematical relationships, i.e., kinematic, kinetic and constitutive equations. We will discuss these types of equations separately. First, a brief review of the kinematic equations, strain tensor and its properties is given. Next, the concept of stress in a deformable body undergoing a finite motion is introduced. We then discuss the constitutive equations that describe a material’s response under load, and stress. Finally, the general governing equations of motion are expressed in terms of the stress tensor and also the displacement field.

3.1 Displacement and Strain 3.1.1 Strain Tensor Any motion of a small region of a continuum can be considered as a combination of translation, rotation and deformation. The change in the shape of the continuum is known as a deformation. Let us consider two arbitrary neighbouring material points P and Q in the body that initially have coordinates X and X + dX respectively (Fig. 3.1). When the continuum undergoes a deformation, the material points are displaced to

© Springer Nature Switzerland AG 2019 E. Ghavanloo et al., Computational Continuum Mechanics of Nanoscopic Structures, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-3-030-11650-7_3

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3 Fundamental Notions from Classical Continuum Mechanics

Fig. 3.1 Undeformed and deformed configurations of a continuum body

the points P  and Q  with coordinates x and x + dx respectively. Therefore, the displacement vector of the point P can be written as u(X, t) = x − X = u i ei

(3.1)

where e1 , e2 , e3 are unit vectors, and u 1 , u 2 and u 3 are the components of the displacement vector. According to Fig. 3.1, one can write the following relation: dx = dX + u(X + dX, t) − u(X, t)

(3.2)

Since the P and Q are neighbouring points, we can use a Taylor series expansion around the point P to express the displacement vector u(X + dX, t) as u(X + dX, t) = u(X, t) + ∇X u(X, t)dX + O(|dX|2 )

(3.3)

where ∇X u is the displacement gradient tensor, which may be written as ⎛

∂u 1 ∂ X1

⎜ ∂u 2 ∇X u = ⎜ ⎝ ∂ X1 ∂u 3 ∂ X1

∂u 1 ∂ X2

∂u 1 ∂ X3

∂u 2 ∂ X2

∂u 2 ∂ X3

∂u 3 ∂ X2

∂u 3 ∂ X3

⎞ ⎟ ⎟ ⎠

(3.4)

3.1 Displacement and Strain

43

Fig. 3.2 The Cartesian, cylindrical and spherical coordinate systems

where subscripts 1, 2 and 3 denote the x, y and z in the Cartesian coordinate system, r , θ and z in the cylindrical coordinate system, and R, φ and θ in the spherical coordinate system. The three coordinate systems are displayed in Fig. 3.2. The displacement gradient tensor is uniquely decomposed into symmetric and antisymmetric parts as (3.5) ∇X u = ε + ω where 1 ∇X u + (∇X u)T 2 1 ω= ∇X u − (∇X u)T 2 ε=

(3.6)

in which the symmetric tensor, ε, is called the infinitesimal strain tensor, and the antisymmetric tensor, ω, is called the rotation tensor. These tensors can be written out in component forms as

 ∂u j 1 ∂u i + 2 ∂Xj ∂ Xi 

∂u j 1 ∂u i ωi j = − 2 ∂Xj ∂ Xi εi j =

(3.7)

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3 Fundamental Notions from Classical Continuum Mechanics

The diagonal elements of the strain tensor are referred to as the normal strains, whereas the off-diagonal elements are called the shear strains. It should be noted that both ε and ω behave as second-order tensors under coordinate transformations. For infinitesimal deformations of a continuum body, all components of the strain and rotation tensors are small compared to unity, i.e., εi j  1, ωi j  1

(3.8)

The strain components in the Cartesian coordinate system are expressed as ∂u y ∂u x ∂u z , ε yy = , εzz = , εx x = ∂x ∂y ∂z





 ∂u y ∂u z ∂u z 1 ∂u x 1 ∂u x 1 ∂u y + , εx z = + , ε yz = + εx y = (3.9) 2 ∂y ∂x 2 ∂z ∂x 2 ∂z ∂y

where u x , u y and u z are the displacements along the x, y and z directions respectively. Furthermore, in terms of the displacement components in the cylindrical coordinate system, the strain components can be expressed by [1]

 ur ∂u θ uθ ∂u r 1 ∂u θ 1 1 ∂u r , εθθ = + , εr θ = + − , ∂r r ∂θ r 2 r ∂θ ∂r r



 ∂u z 1 ∂u θ 1 ∂u r 1 ∂u z ∂u z , εr z = (3.10) εzz = , εθ z = + + ∂z 2 ∂z r ∂θ 2 ∂z ∂r εrr =

where u r , u θ and u z denote the displacements along the r , θ and z directions respectively. Finally, the strain-displacement relations in the spherical coordinate system are given by [1]

 uR ∂u θ ∂u R 1 ∂u φ 1 , εφφ = + , εθθ = + sin φu R + cos φu φ , εR R = ∂R R ∂φ R R sin φ ∂θ



 ∂u φ uφ 1 ∂u R ∂u θ uθ 1 1 ∂u R 1 + − , ε Rθ = + − , ε Rφ = 2 R ∂φ ∂R R 2 R sin φ ∂θ ∂R R

 1 ∂u φ 1 ∂u θ cot φu θ 1 + − (3.11) εφθ = 2 R sin φ ∂θ R ∂φ R wherein u R , u φ and u θ are the displacements along the R, φ and θ directions respectively.

3.1.2 Principal Strains Since the strain tensor is a symmetric and real tensor, there exist three orthogonal principal directions of strain and corresponding principal values at each point in the body. These can be determined by solving the equation

3.1 Displacement and Strain

45

εn p = ε p n p

(3.12)

where ε p is the principle strain and n p is the corresponding eigenvector which is the principal strain direction. For a given ε, the three principal strains are the roots of the characteristic equation, which can be expressed as [2] ε3p − Σ1 ε2p + Σ2 ε p − Σ3 = 0

(3.13)

where Σ1 , Σ2 and Σ3 are three strain invariants, given by Σ1 = ε11 + ε22 + ε33

(3.14)

2 2 2 − ε23 − ε13 Σ2 = ε11 ε22 + ε22 ε33 + ε11 ε33 − ε12

(3.15)

2 2 2 − ε22 ε13 − ε33 ε12 Σ3 = ε11 ε22 ε33 + 2ε12 ε13 ε23 − ε11 ε23

(3.16)

The roots of Eq. (3.13) are denoted by ε p1 , ε p2 and ε p3 . The principal directions n p1 , n p2 and n p3 corresponding to the principal strains are determined by substituting ε p1 , ε p2 and ε p3 , each in turn, into Eq. (3.12), and then solving the resulting equations simultaneously. If the principal strains are distinct, these three eigenvectors are mutually orthogonal and can be used to form a principal coordinate system (n p1 , n p2 , n p3 ). Notice that the strain tensor is purely diagonal, and there are no shear strain components when the tensor is written in the principal coordinate system.

3.1.3 Spherical and Deviatoric Strains In some particular applications, it is appropriate to decompose the strain tensor into a part responsible for the volume change, called volumetric or spherical strain tensor, and a part responsible for the shape change, called the deviatoric strain. The spherical strain tensor, εv , is given by εv =

1 1 trace(ε)I = (ε11 + ε22 + ε33 ) I 3 3

(3.17)

where I is the identity matrix. The deviatoric strain tensor, εD , is defined by 1 εD = ε − εv = ε − trace(ε)I 3

(3.18)

46

3 Fundamental Notions from Classical Continuum Mechanics

It should be noted that the trace of the deviatoric strain tensor is zero and therefore there is no volume change. It can be shown that the principal directions of the deviatoric strain coincide with the principal directions of the strain tensor. Furthermore, if D D D , ε p2 and ε p3 are the principal deviatoric strains, they are related to the principal ε p1 strains 1 D = ε p1 − (ε p1 + ε p2 + ε p3 ) ε p1 3 1 D ε p2 = ε p2 − (ε p1 + ε p2 + ε p3 ) 3 1 D ε p3 = ε p3 − (ε p1 + ε p2 + ε p3 ) 3

(3.19)

3.1.4 Compatibility Conditions Equations (3.9), (3.10) and (3.11), or in the index notation form Eq. (3.7), represent six equations for the six independent strain components in terms of three independent displacements. If these displacements are known, the strain components can be determined uniquely. Nevertheless, in many practical applications, the strain components are known from numerical analysis or experimental measurements. In these cases, the displacements can be obtained from the strain components by integration. However, for an arbitrary choice of strain components, there may not generally exist single-valued continuous solutions for displacements. This is not surprising because we are trying to solve six equations for only three unknowns. Therefore, in order to ensure single-valued continuous displacements, the strain components must satisfy a set of constraints knows as integrability or compatibility conditions. In index notation, the compatibility conditions are [3] ∂ 2 ε jk ∂ 2 εi j ∂ 2 εkl ∂ 2 εil + − − =0 ∂ Xk ∂ Xl ∂ Xi ∂ X j ∂ X j ∂ Xk ∂ Xi ∂ Xl

(3.20)

The details of the mathematical derivation for the compatibility conditions can be found in Ref. [1]. Although Eq. (3.20) would lead to 81 individual equations, due to the symmetry with respect to the pairs of indices (i and j) and (k and l), these equations are reduced to only six equations, i.e.,

3.1 Displacement and Strain

2

∂ 2 ε12 ∂ 2 ε11 ∂ 2 ε22 = + 2 ∂ X 1∂ X 2 ∂ X2 ∂ X 12

2

∂ 2 ε23 ∂ 2 ε22 ∂ 2 ε33 = + ∂ X 2∂ X 3 ∂ X 32 ∂ X 22

∂ 2 ε13 ∂ 2 ε11 ∂ 2 ε33 = + 2 ∂ X 1∂ X 3 ∂ X3 ∂ X 12

 ∂ε23 ∂ 2 ε11 ∂ ∂ε13 ∂ε12 − = + + ∂ X 2∂ X 3 ∂ X1 ∂ X1 ∂ X2 ∂ X3

 2 ∂ε23 ∂ ε22 ∂ ∂ε13 ∂ε12 = − + ∂ X 1∂ X 3 ∂ X2 ∂ X1 ∂ X2 ∂ X3

 2 ∂ε23 ∂ ε33 ∂ ∂ε13 ∂ε12 = + − ∂ X 1∂ X 2 ∂ X3 ∂ X1 ∂ X2 ∂ X3

47

2

(3.21)

It should be noted that these six equations are also not all independent and the six equations may be reduced to three independent fourth-order relations. The three independent fourth-order equations of strain compatibility conditions in the Cartesian coordinate system can be expressed as

 ∂εx y ∂ε yz ∂εzx ∂ 4 εx x ∂3 + − = ∂ y 2 ∂z 2 ∂ x∂ y∂z ∂ y ∂z ∂x

 4 3 ∂ε yz ∂εx y ∂ ε yy ∂εzx ∂ − + = ∂ x 2 ∂z 2 ∂ x∂ y∂z ∂ x ∂y ∂z

 ∂εx y ∂ε yz ∂εzx ∂ 4 εzz ∂3 − + + = ∂ x 2∂ y2 ∂ x∂ y∂z ∂z ∂x ∂y

(3.22)

Finally, we note that it is usually more appropriate to use the six second-order equations given by Eq. (3.21).

3.1.5 Plane Strain Condition An object is said to be in a state of plane deformation or a state of plane strain, where its geometry, its body forces and its surface tractions do not vary significantly along one of its spatial dimensions (for example along its X 3 direction). Under such a circumstance, the displacement u 3 is zero at every cross-section, and the displacements u 1 and u 2 are only functions of the X 1 and X 2 coordinates. Therefore, the strains become

48

3 Fundamental Notions from Classical Continuum Mechanics

ε13 = ε23 = ε33 = 0 ε11 = ε11 (X 1 , X 2 ) ε22 = ε22 (X 1 , X 2 ) ε12 = ε12 (X 1 , X 2 )

(3.23)

Under the plane strain condition, the three-dimensional problem is reduced to a two-dimensional one.

3.2 Stress Tensor 3.2.1 Concept of Stress The concept of stress was introduced by the French mathematician, Cauchy, in 1823. The stress is a measure of the internal forces intensity induced in a body when it is subjected to external loads. Generally, induced stresses in the body vary from point to point. In order to understand this concept, consider a general body subjected to external forces, as depicted in Fig. 3.3. To determine the traction or the stress vector at an arbitrary point P, it is necessary to pass a plane through the point P having a ˜ A small area ΔA is considered on this plane and the resultant unit normal vector n. surface force acting on it is defined by ΔP (Fig. 3.4). The stress vector at the point P is, therefore, defined as ΔP (3.24) Tn˜ = lim ΔA→0 ΔA It should be noted that the stress vector depends on the location of point P as well as the unit normal vector of the cutting plane. The component of the stress vector parallel to the unit normal vector is called the normal stress and is expressed as σn˜ = Tn˜ · n˜

Fig. 3.3 General body under external loadings

(3.25)

3.2 Stress Tensor

49

Fig. 3.4 Sectioned body

Furthermore, the component of the stress vector perpendicular to the unit vector n˜ is known as the shear stress and is defined as

(3.26) σs = Tn˜ · Tn˜ − σn˜2 In principle, an infinite number of cutting planes can pass through the point P to obtain the stress vectors at that point. Generally, the state of stress at the point P is defined by all possible stress vectors at the point. However, according to Cauchy’s stress theorem, the state of stress can be completely and uniquely described by the stress vectors on three mutually orthogonal planes. The Cauchy stress theorem states that the stress vector on a surface through the point P is uniquely determined by a ˜ Therefore, second-order stress tensor σ , which is independent of the unit vector n. the stress vector can be written as Tn˜ = σ T n˜

(3.27)

The proof of the Cauchy stress theorem can be found in Ref. [4]. Now, to specify the state of stress at an arbitrary point, we consider an infinitesimal cubic element composed of planes perpendicular to three orthogonal axes (Fig. 3.5). According to Eq. (3.27), the stress vectors Tei on the three coordinate planes are related to the stress tensor by Te1 = σ T e1 = σ11 e1 + σ12 e2 + σ13 e3 Te2 = σ T e2 = σ21 e1 + σ22 e2 + σ23 e3 Te3 = σ T e3 = σ31 e1 + σ32 e2 + σ33 e3

(3.28)

where the nine quantities σ11 , σ12 , σ13 , σ21 , σ22 , σ23 , σ31 , σ32 , σ33 are called the stress components, which completely describe the state of stress at the point. The stress tensor is normally written in matrix form as ⎛

⎞ σ11 σ12 σ13 σ = ⎝ σ21 σ22 σ23 ⎠ σ31 σ32 σ33

(3.29)

50

3 Fundamental Notions from Classical Continuum Mechanics

Fig. 3.5 Components of the stress tensor

The diagonal components are called the normal stresses and the off-diagonal components are called the shear stresses. Conceptually, it is convenient to view the stress tensor as a vector-like quantity having a magnitude and an associated direction(s) specified by unit vectors. In this connection, the stress tensor can be expressed in terms of the dyadic products of unit vectors. This is known as dyadic representation of the stress. The stress tensor or stress dyadic can be written as σ = σi j ei ⊗ ej = σ11 e1 ⊗ e1 + σ12 e1 ⊗ e2 + σ13 e1 ⊗ e3 + σ21 e2 ⊗ e1 + σ22 e2 ⊗ e2 + σ23 e2 ⊗ e3 + σ31 e3 ⊗ e1 + σ32 e3 ⊗ e2 + σ33 e3 ⊗ e3

(3.30)

where the symbol ⊗ denotes the dyadic product. Finally, it should be noted that some authors use the convention Tn˜ = σ n˜ instead of Eq. (3.27). Irrespective of the convention adopted, the results will be the same because the Cauchy stress tensor is symmetric tensor.

3.2.2 Stress on an Arbitrary Surface In this subsection, we evaluate the stress vector on an arbitrary surface. In this connection, we consider a differential tetrahedron as shown in Fig. 3.6. As can be seen from this figure, the stress vector Tn˜ acts on a plane identified by the unit normal n˜ which is defined by (3.31) n˜ = n˜ 1 e1 + n˜ 2 e2 + n˜ 3 e3 where n˜ 1 , n˜ 2 and n˜ 3 are the direction cosines of the unit vector with respect to the coordinate system. Force equilibrium for the tetrahedron gives Tn˜ d A − Te1 d A1 − Te2 d A2 − Te3 d A3 = 0

(3.32)

3.2 Stress Tensor

51

Fig. 3.6 Differential tetrahedron

where d A, d A1 , d A2 and d A3 are the areas of the planes BC D, OC D, O B D and O BC respectively. The areas d A1 , d A2 and d A3 are related to d A by d A1 = n˜ 1 d A, d A2 = n˜ 2 d A, d A3 = n˜ 3 d A

(3.33)

Furthermore, the stress vector can be written in terms of its three components as Tn˜ = T1 e1 + T2 e2 + T3 e3

(3.34)

where T1 , T2 and T3 are the components of the stress vector. Substituting from Eqs. (3.28), (3.33) and (3.34) into Eq. (3.32), these components are obtained as T1 = σ11 n˜ 1 + σ21 n˜ 2 + σ31 n˜ 3 T2 = σ12 n˜ 1 + σ22 n˜ 2 + σ32 n˜ 3 T2 = σ13 n˜ 1 + σ23 n˜ 2 + σ33 n˜ 3

(3.35)

As a result, using Eqs. (3.25), (3.31) and (3.35), the normal stress on the plane is given by σn˜ = σ11 n˜ 21 + σ22 n˜ 22 + σ33 n˜ 23 + (σ12 + σ21 )n˜ 1 n˜ 2 (3.36) + (σ13 + σ31 )n˜ 1 n˜ 3 + (σ23 + σ32 )n˜ 2 n˜ 3

52

3 Fundamental Notions from Classical Continuum Mechanics

3.2.3 Principal Axes of Stress Let us now assume that the stress tensor at a specific point in the body is known. The main purpose of this subsection is to find the orientation of a plane such that the normal stress on this plane is an extremum (maximum or minimum) and the shear stresses are zero. This plane is called the principal stress plane, and the unit normal vector to this plane is known as a principle stress direction, and the normal stress on this plane is called the principal stress. In order to extremize the normal stress subject to the constraint that n˜ · n˜ = 1, the Lagrange-multiplier method can be used. In this way, the Lagrangean function, L˜ , is defined as L˜ = σ11 n˜ 21 + σ22 n˜ 22 + σ33 n˜ 23 + (σ12 + σ21 )n˜ 1 n˜ 2 + (σ13 + σ31 )n˜ 1 n˜ 3 + (σ23 + σ32 )n˜ 2 n˜ 3 − λ∗ (n˜ 21 + n˜ 22 + n˜ 23 − 1)

(3.37)

where λ∗ is the Lagrange multiplier. The extremum of the Lagrangian is determined from ∂ L˜ ∂ L˜ ∂ L˜ ∂ L˜ = 0, = 0, = 0, =0 (3.38) ∂ n˜ 1 ∂ n˜ 2 ∂ n˜ 3 ∂λ∗ Substitution of Eq. (3.37) into Eq. (3.38), gives σ11 n˜ 1 + σ12 n˜ 2 + σ13 n˜ 3 − λ∗ n˜ 1 = 0

(3.39)

σ12 n˜ 1 + σ22 n˜ 2 + σ23 n˜ 3 − λ∗ n˜ 2 = 0

(3.40)

σ13 n˜ 1 + σ23 n˜ 2 + σ33 n˜ 3 − λ∗ n˜ 3 = 0

(3.41)

n˜ 21 + n˜ 22 + n˜ 23 − 1 = 0

(3.42)

It is noted that to obtain Eqs. (3.39)–(3.41), we use the fact that the Cauchy stress tensor is a symmetric tensor. Rewriting Eqs. (3.39)–(3.41) in the matrix form, the following relation is obtained ⎛

⎞⎛ ⎞ ⎛ ⎞ σ11 − λ∗ n˜ 1 0 σ12 σ13 ⎝ σ12 σ22 − λ∗ σ23 ⎠ ⎝ n˜ 2 ⎠ = ⎝ 0 ⎠ 0 σ13 σ23 σ33 − λ∗ n˜ 3

(3.43)

For the non-trivial solution, the determinant of this set of equations must be zero, i.e., ⎞ ⎛ σ11 − λ∗ σ12 σ13 σ22 − λ∗ σ23 ⎠ = 0 (3.44) det ⎝ σ12 σ13 σ23 σ33 − λ∗

3.2 Stress Tensor

53

Equation (3.44) is called the characteristic equation of the stress tensor. Expanding Eq. (3.44) leads to a cubic equation which has three roots σ p1 , σ p2 and σ p3 . Generally, the roots of the characteristic equation are the principal stresses. Since the stress tensor is symmetric, all principle stresses are real and the corresponding eigenvectors or principle directions are orthogonal. These directions are determined by substituting σ p1 , σ p2 and σ p3 , each in turn, into Eq. (3.43), and then solving the resulting equations together with Eq. (3.42) simultaneously.

3.2.4 Hydrostatic and Deviatoric Stresses In a similar manner to the case of the strain tensor, the stress tensor is uniquely decomposed into the hydrostatic and deviatoric stress tensors. Analogous to Eqs. (3.17) and (3.18), the hydrostatic stress is defined by ⎛

σH where σM =

⎞ σM 0 0 = ⎝ 0 σM 0 ⎠ 0 0 σM

1 1 (σ11 + σ22 + σ33 ) = (σ p 1 + σ p 2 + σ p 3 ) 3 3

Furthermore, the deviatoric stress becomes ⎛ ⎞ σ11 − σ M σ12 σ13 σ22 − σ M σ23 ⎠ σ D = ⎝ σ12 σ13 σ23 σ33 − σ M

(3.45)

(3.46)

(3.47)

Physically the hydrostatic stress tensor is associated with volumetric changes in the body, while the deviatoric stress tensor causes a distortion in the body. It can be shown that the principal directions of the deviatoric stress tensor coincide with the principal directions of the stress tensor. Furthermore, the deviatoric stress tensor is trace-free. The principal values of the deviatoric stress tensor, σ pD , are the roots of the characteristic equation [5]: (σ pD )3 − J1 (σ pD )2 − J2 (σ pD ) − J3 = 0

(3.48)

where J1 , J2 and J3 are called the invariants of the deviatoric stress tensor and are given by (3.49) J1 = 0 J2 =

 1 2 2 2 (σ11 − σ22 )2 + (σ22 − σ33 )2 + (σ33 − σ11 )2 + 6(σ12 + σ23 + σ13 ) 6 (3.50)

54

3 Fundamental Notions from Classical Continuum Mechanics

J3 = det(σ D )

(3.51)

Among these invariants, the most important one is J2 because it is proportional to the stored strain energy. It is also related to the von Mises stress, σvonMises , by σvonMises =

 3J2

(3.52)

3.3 Constitutive Equations 3.3.1 Generalized Hooke’s Law The strain and stress 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 loadings. For linear elastic materials and small deformations, the stress tensor is linearly related to the strain tensor by the fourth-rank elasticity tensor E, and this relation is given by the generalized Hooke’s law σi j = E i jkl εkl

(3.53)

Equation (3.53) says that the stress at a material point of a body is a linear function of the strain at that point. The 81 coefficients E i jkl are called the elastic constants of the material. However, these constants are not independent. Since both the strain and stress tensors are symmetric tensors, it is clear that E i jkl = E jikl = E i jlk

(3.54)

where by the 81 coefficients are reduced to 36 independent elastic constants. Furthermore, the symmetry imposed by the strain energy implies that the elasticity tensor is symmetric with respect to indices i j and kl, i.e., E i jkl = E kli j

(3.55)

Hence, only 21 independent elastic constants for the material remain. This makes it possible to express Eq. (3.53) in a contracted notation or the Voigt matrix notation. This matrix can be obtained by mapping the pair of indices to a single index. The general rules to contract the indices are (11)↔(1), (22)↔(2), (33)↔(3), (23)↔(4), (13)↔(5) and (12)↔(6). As a consequence, the constitutive equation for anisotropic elastic materials, in the matrix form, has the structure

3.3 Constitutive Equations

⎞ ⎛ E 11 σ1 ⎜ σ2 ⎟ ⎜ E 12 ⎜ ⎟ ⎜ ⎜ σ3 ⎟ ⎜ E 13 ⎜ ⎟=⎜ ⎜ σ4 ⎟ ⎜ E 14 ⎜ ⎟ ⎜ ⎝ σ5 ⎠ ⎝ E 15 σ6 E 16 ⎛

55

E 12 E 22 E 23 E 24 E 25 E 26

E 13 E 23 E 33 E 34 E 35 E 36

E 14 E 24 E 34 E 44 E 45 E 46

E 15 E 25 E 35 E 45 E 55 E 56

⎞⎛ ⎞ ε1 E 16 ⎜ ε2 ⎟ E 26 ⎟ ⎟⎜ ⎟ ⎜ ⎟ E 36 ⎟ ⎟ ⎜ ε3 ⎟ ⎟ ⎟ E 46 ⎟ ⎜ ⎜ ε4 ⎟ ⎝ ⎠ E 56 ε5 ⎠ E 66 ε6

(3.56)

Fortunately, most engineering materials possess properties of symmetry about one or more of the planes or axes, which allow the number of independent elastic constants to be further reduced [6].

3.3.2 Material Symmetry In this subsection, we explore how the symmetries of the material can be used to simplify the number of independent elastic constants. In this way, the constitutive equations for some practical cases including monoclinic, orthotropic, transversely isotropic and isotropic materials are presented. We first present the constitutive equation of a monoclinic material which has only one plane of symmetry. For the case of symmetry with respect to the 12-plane, Hooke’s laws simplify to [7] ⎞ ⎛ E 11 σ1 ⎜ σ2 ⎟ ⎜ E 12 ⎜ ⎟ ⎜ ⎜ σ3 ⎟ ⎜ E 13 ⎜ ⎟=⎜ ⎜ σ4 ⎟ ⎜ 0 ⎜ ⎟ ⎜ ⎝ σ5 ⎠ ⎝ 0 σ6 E 16 ⎛

E 12 E 22 E 23 0 0 E 26

E 13 0 0 E 23 0 0 E 33 0 0 0 E 44 E 45 0 E 45 E 55 E 36 0 0

⎞⎛ ⎞ ε1 E 16 ⎜ ε2 ⎟ E 26 ⎟ ⎟⎜ ⎟ ⎜ ⎟ E 36 ⎟ ⎟ ⎜ ε3 ⎟ ⎜ ⎟ 0 ⎟ ⎟ ⎜ ε4 ⎟ 0 ⎠ ⎝ ε5 ⎠ E 66 ε6

(3.57)

It is obvious that 13 independent elastic constants are needed to characterize the monoclinic materials. A material with three mutually perpendicular planes of symmetry is called orthotropic. Thus, the elasticity matrix for the orthotropic case reduces to only 9 independent coefficients given by ⎛

⎞ ⎛ ⎞⎛ ⎞ σ1 E 11 E 12 E 13 0 ε1 0 0 ⎜ σ2 ⎟ ⎜ E 12 E 22 E 23 0 ⎟ ⎜ ε2 ⎟ 0 0 ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ σ3 ⎟ ⎜ E 13 E 23 E 33 0 ⎜ ⎟ 0 0 ⎟ ⎜ ⎟=⎜ ⎟ ⎜ ε3 ⎟ ⎜ σ4 ⎟ ⎜ 0 ⎟ ⎟ 0 ⎟⎜ 0 0 E 44 0 ⎜ ⎟ ⎜ ⎜ ε4 ⎟ ⎝ σ5 ⎠ ⎝ 0 ⎝ ⎠ ε5 ⎠ 0 0 0 E 55 0 σ6 ε6 0 0 0 0 0 E 66

(3.58)

Another common form of material symmetry is with respect to rotations about an axis that is normal to a plane of isotropy. Within this plane, the material properties are the same in all directions. Such materials are called transversely isotropic. In this

56

3 Fundamental Notions from Classical Continuum Mechanics

case, only five independent elastic constants exist and the constitutive equation is ⎞ ⎛ E 11 E 12 E 13 0 0 σ1 ⎜ σ2 ⎟ ⎜ E 12 E 11 E 13 0 0 ⎜ ⎟ ⎜ ⎜ σ3 ⎟ ⎜ E 13 E 13 E 33 0 0 ⎜ ⎟=⎜ ⎜ σ4 ⎟ ⎜ 0 0 0 0 E 44 ⎜ ⎟ ⎜ ⎝ σ5 ⎠ ⎝ 0 0 0 0 E 55 σ6 0 0 0 0 0 ⎛

0 0 0 0 0

⎞⎛

E 11 −E 12 2

⎞ ε1 ⎟ ⎜ ε2 ⎟ ⎟⎜ ⎟ ⎟ ⎜ ε3 ⎟ ⎟⎜ ⎟ ⎟ ⎜ ε4 ⎟ ⎟⎜ ⎟ ⎠ ⎝ ε5 ⎠ ε6

(3.59)

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 constitutive equation for isotropic elastic materials is given by ⎛

⎞ ⎛ E 11 σ1 ⎜ σ2 ⎟ ⎜ E 12 ⎜ ⎟ ⎜ ⎜ σ3 ⎟ ⎜ E 12 ⎜ ⎟=⎜ ⎜ σ4 ⎟ ⎜ 0 ⎜ ⎟ ⎜ ⎝ σ5 ⎠ ⎝ 0 σ6 0

E 12 E 11 E 12 0 0 0

E 12 E 12 E 11 0 0 0

0 0 0

E 11 −E 12 2

0 0

0 0 0 0

E 11 −E 12 2

0

0 0 0 0 0

E 11 −E 12 2

⎞⎛

⎞ ε1 ⎟ ⎜ ε2 ⎟ ⎟⎜ ⎟ ⎟ ⎜ ε3 ⎟ ⎟⎜ ⎟ ⎟ ⎜ ε4 ⎟ ⎟⎜ ⎟ ⎠ ⎝ ε5 ⎠ ε6

(3.60)

It is obvious that only two independent elastic constants are needed to characterize the isotropic materials. For these materials, we have E 11 =

E(1 − ν) Eν , E 12 = (1 + ν)(1 − 2ν) (1 + ν)(1 − 2ν)

(3.61)

where E is the Young modulus and ν is the Poisson ratio. In addition, for the isotropic materials, Eq. (3.53) can be rewritten as follows: σi j =

E Eν εi j + (ε11 + ε22 + ε33 )δi j 1+ν (1 + ν)(1 − 2ν)

(3.62)

where δi j is the Kronecker delta.

3.4 Governing Equations 3.4.1 Equations of Motion Consider a closed sub-domain with volume Ω  and an exterior surface Γ  within an elastic body (Fig. 3.7). The elastic body is acted upon by a general distribution of surface tractions Tn˜ and a body force bf per unit volume. The resulting force acting on the sub-domain is

3.4 Governing Equations

57

Fig. 3.7 A closed sub-domain with volume Ω  and an exterior surface Γ 

 F=

Tn˜ dΓ  +

Γ



bf dΩ  =

Ω



 ˜ σ T ndΓ +

Γ



bf dΩ 

(3.63)

Ω

The surface integral of the traction forces can be transformed to the volume integral by the Green-Gauss theorem 

 ˜ σ T ndΓ =

Γ



Div(σ T )dΩ 

(3.64)

Ω

where Div is the divergence operator. Note that, for the case of an infinitesimal deformation, the undeformed and deformed configurations of the body can be assumed to be identical. This means that mathematically ∂/∂ X i = ∂/∂ xi . Therefore, the divergence operator can be expressed as (Div(σ T ))i =

∂σ ji ∂σ ji = ∂Xj ∂x j

(3.65)

Using Eqs. (3.64) and (3.63) is rewritten as  F=

(Div(σ T ) + bf )dΩ 

(3.66)

Ω

Next, the linear momentum is defined by  G=

ρ Ω

∂u dΩ  ∂t

(3.67)

58

3 Fundamental Notions from Classical Continuum Mechanics

where ρ is the mass density of the material. Newton’s law of motion requires that F=

∂G ∂t

(3.68)

Substituting Eqs. (3.66) and (3.67) into Eq. (3.68), we obtain 





(Div(σ ) + bf )dΩ = T

Ω

ρ Ω

∂ 2u dΩ  ∂t 2

(3.69)

Since the volume Ω  is arbitrary, Eq. (3.69) reduces to Cauchy’s first law of motion [8]: ∂ 2u (3.70) Div(σ T ) + bf = ρ 2 ∂t In component form, the equations of motion can be expressed as ∂σ ji ∂ 2ui + (b f )i = ρ 2 ∂x j ∂t

(3.71)

The stress distribution can be obtained by solving Eq. (3.70) or Eq. (3.71) with appropriate boundary and initial conditions. For time-independent problems, the equations of motion reduce to the equilibrium equations, Div(σ T ) + bf = 0

(3.72)

3.4.2 Navier Equations of Motion In this subsection, we will present the equations of motion for isotropic materials in terms of the displacement components only. In the first step, substituting Eq. (3.7) into Eq. (3.62) yields σi j =

E 2(1 + ν)



∂u j ∂u i + ∂x j ∂ xi

 +

Eν (1 + ν)(1 − 2ν)



∂u 1 ∂u 2 ∂u 3 + + ∂ x1 ∂ x2 ∂ x3

 δi j

(3.73) Next, the stress is substituted into the equations of motion to obtain the equations in terms of the components of the displacement field as, E 2(1 + ν)



∂ 2ui ∂ x 2j



∂ 1 + (1 − 2ν) ∂ xi



∂u 1 ∂u 2 ∂u 3 + + ∂ x1 ∂ x2 ∂ x3

 + (b f )i = ρ

These equations are known as the Navier equations of motion.

∂ 2ui ∂t 2 (3.74)

3.5 Elastic Energy

59

3.5 Elastic Energy Let us consider an elastic solid with volume Ω and an exterior surfaces subjected to the external body forces bf and surface tractions Tn˜ . The first law of thermodynamics, which describes the complete energy balance for any deformation process, has the following form: δU = δ Q + δW (3.75) where δU is the change in the internal energy, δ Q is the heat flowing in the solid and δW denotes the virtual work done by the surface tractions and body forces and it can be written as   δW = F · δu = (Tn˜ · δu)d S + (bf · δu)dΩ Ω S   (3.76) = Ti δu i d S + (b f )i δu i dΩ Ω

S

where δu is the virtual displacement. Using Eqs. (3.64) and (3.76) and applying the Green-Gauss theorem, the first law of thermodynamics in an adiabatic isolated system (i.e., δ Q = 0) may be rewritten as 

δU = Ω

  ∂σ ji ∂δu i + (b f )i δu i dΩ + σi j dΩ ∂x j ∂x j

(3.77)

Ω

Substituting Eq. (3.71) into Eq. (3.77), we have  δU = Ω

∂ 2ui ρ 2 δu i dΩ + ∂t

 σi j Ω

∂δu i dΩ ∂x j

(3.78)

The first term on the right-hand side of Eq. (3.78) is the variation of the total kinetic energy and the second one is the variation of elastic strain energy, δΠΩ , stored during the elastic deformation. Using Eq. (3.5), the variation of elastic strain energy is expressed as  δΠΩ =

 σi j (δεi j + δωi j )dΩ =

Ω

σi j δεi j dΩ

(3.79)

Ω

where we have used the fact that σ ji δωi j = 0. Furthermore, the variation of elastic strain energy per volume (or strain energy density) is defined as δΠ = σi j δεi j

(3.80)

60

3 Fundamental Notions from Classical Continuum Mechanics

Therefore, if the stresses are written in terms of the strains, the strain energy density can be written as a function of the strains only. Therefore, the variation of the strain energy density is obtained as δΠ =

∂Π δεi j ∂εi j

(3.81)

Comparing Eqs. (3.80) and (3.81), we find σi j =

∂Π ∂εi j

(3.82)

Similarly, the variation of complimentary strain energy density is defied by δΠ c = εi j δσi j

(3.83)

For linear elastic material, Π = Π c and so the strain energy density is Π=

1 σi j εi j 2

(3.84)

References 1. M.H. Sadd, Elasticity: Theory, Applications, and Numerics (Elsevier, New York, 2009) 2. A.N. Cleland, Foundations of Nanomechanics: From Solid-State Theory to Device Applications (Springer, New York, 2003) 3. F. Irgens, Continuum Mechanics (Springer, Berlin, 2008) 4. W.M. Lai, E. Krempl, D. Rubin, Introduction to Continuum Mechanics (Elsevier, Oxford, 2010) 5. J.G.M. van Mier, Fracture Processes of Concrete (CRC Press, London, 2017) 6. E. Schreiber, O.L. Anderson, N. Soga, Elastic Constants and Their Measurement (McGrawHill, New York, 1973) 7. J.N. Reddy, Mechanics of Laminated Composite Plates and Shells: Theory and Analysis (CRC Press, London, 2004) 8. T.J. Chung, Applied Continuum Mechanics (Cambridge University Press, Cambridge, 1996)

Chapter 4

Essential Concepts from Nonlocal Elasticity Theory

Nonlocal elasticity theory is a convenient methodology for considering the smallscale effects that are exhibited by nanoscopic structures. In recent decades, the use of nonlocal elasticity theory in mechanical modelling of these structures has seen an inflationary development. According to this highly complex, but mathematically elegant, theory, the state of stress at a given point in a material would be determined not only by the state of strain at that point, but also by the state of strain at other points. This accords with the concept of many body interatomic forces acting on a particular atom in a material which depend not only on the local interaction between this particular atom and its immediate neighbours, but also nonlocally on the interaction among all other atoms not located in the neighbour list of this particular atom. Therefore, we require a knowledge of the independent variables over all material points to obtain the mechanical state of the body at particular points. Based on this hypothesis, the long-range interatomic interactions between material points are naturally taken into account and so the results that are obtained are dependent on the size of the material body. The nonlocal elasticity theory has the great potential to predict the mechanical behaviour of both small and relatively larger nanoscopic structures without solving a large number of highly complex equations. This theory differs from the local classical elasticity theory only in the constitutive equations describing the material medium. In this chapter, the fundamental concepts and definitions of the nonlocal theory are presented at a level that is comprehensible to the non-specialist. More precisely, we focus on these essential constituents of the theory: nonlocal averaging, nonlocal kernels gradient-based nonlocal elasticity theory and nonlocal anisotropic elasticity.

4.1 Nonlocality and Nonlocal Averaging The applicability of the classical continuum theories is indeed closely coupled with the length and time scales of the problem under study. If the external characteristic length is much larger than the internal one, then the classical continuum theories © Springer Nature Switzerland AG 2019 E. Ghavanloo et al., Computational Continuum Mechanics of Nanoscopic Structures, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-3-030-11650-7_4

61

62

4 Essential Concepts from Nonlocal Elasticity Theory

can predict sufficiently accurate results. Here, the external characteristic length, L e , may refer to the crack length, the wavelength, the size of the sample, etc. while the internal characteristic length, L i , can refer to the lattice parameter, the size of a grain, the diameter of the molecular constituents of polymers, etc. In the nanoscale, the external characteristic length is of the same order of magnitude as the internal one (i.e., L e /L i ∼ 1). Consequently, the local theories fail. Under such circumstances, either atomistic or nonlocal theories must be employed to model the long-range interatomic interactions. In a similar manner, for the dynamical problems, there is a pertinent ratio, namely Te /Ti , where Ti denotes the internal characteristic time (e.g., the relaxation time, the time for a signal to travel between molecules) and Te denotes the external characteristic time (e.g., the time of application of the external loads, the period of vibration). We note that, for Te /Ti ∼ 1, the classical theories fail to provide reasonable response for many physical phenomena. Thus, the description of the physical phenomenon in space-time requires the consideration of nonlocal effects [1]. Generally, nonlocality can be of three types: spatial, temporal and mixed spatialtemporal. To define the nonlocality types, we follow the work of Povstenko [2]. Generally, a physical quantity (an effect ℘) at a point x and time t is locally dependent on another physical quantity (a cause ℵ) at the same point and time via the following form (4.1) ℘ (x1 , x2 , x3 , t) = ℘ (ℵ(x1 , x2 , x3 , t)) The spatial nonlocality implies that the effect ℘¯ at point x and time t depends on the causes from all other points x at the same time. Thus, the spatial nonlocality or nonlocal average of a local field ℘ (x1 , x2 , x3 , t) within the domain V at time t is defined by the following weighted average:  ℘(x ¯ 1 , x2 , x3 , t) =

α(x, x , λ)℘ (ℵ(x  1 , x  2 , x  3 , t))d x  1 d x  2 d x  3

(4.2)

V

where α is a positive continuous function of the so-called averaging kernel or the spatial nonlocal kernel, and the parameter λ is proportional to a characteristic length ratio L i /L e . For the case of the constitutive equation, the effect is the stress and the cause is the strain. According to Eq. (4.2), the nonlocal constitutive equation can be written as:  nl σi j (x1 , x2 , x3 , t) = α(x, x , λ)E i jkl εkl (x1 , x2 , x3 , t)d x1 d x2 d x3 (4.3) V

where σinlj is the nonlocal stress component. The spatial nonlocality deals with the long-range interaction. For an elastic material with a time-dependent memory, the

4.1 Nonlocality and Nonlocal Averaging

63

effect ℘ at the material point x at time t depends on the history of causes at the same point at all times prior to the time t. This is defined mathematically as t ℘(x ¯ 1 , x2 , x3 , t) =

β(t − t  , η)℘ (ℵ(x1 , x2 , x3 , t  ))dt 

(4.4)

0

wherein β is the time-nonlocality kernel and the parameter η is proportional to the characteristic time ratio Ti /Te . Using Eq. (4.4), the time-dependent nonlocal constitutive equation is expressed as t σinlj (x1 , x2 , x3 , t)

=

β(t − t  , η)E i jkl εkl (x1 , x2 , x3 , t  )dt 

(4.5)

0

This constitutive equation implies that the material remembers its past deformations. Such materials fall into the category of viscoelastic. When the spatial nonlocality is accompanied by time-dependence, the effect ℘ at the material point x at time t depends on the causes at all other points x and at all the times prior to t, i.e., t  ℘(x ¯ 1 , x2 , x3 , t) = 0

γ (x, x , t − t  , λ, η)℘ (ℵ(x  1 , x  2 , x  3 , t  ))d x  1 d x  2 d x  3 dt 

V

(4.6) in which γ is the space-time-nonlocality kernel. Accordingly, the nonlocal constitutive equation for a material with both space and memory effects is given by t  σinlj (x1 , x2 , x3 , t) =

γ (x, x , t − t  , λ, η)E i jkl εkl (x  1 , x  2 , x  3 , t  )d x  1 d x  2 d x  3 dt 

(4.7)

0 V

It is obvious that the spatial nonlocality and the memory effect disappear when λ → 0 and η → 0.

4.2 Spatial Nonlocal Kernels The nonlocal kernels substantially affect the spatial distribution of the variables and they play an important role in elasticity problems. Therefore, it is necessary to discuss the basic characteristics of the nonlocal kernels and their effect on the nonlocal theory. In this section, we will discuss the basic features of the spatial nonlocal kernels.

64

4 Essential Concepts from Nonlocal Elasticity Theory

4.2.1 Properties of Nonlocal Kernels From Eq. (4.3), it is clear that the spatial nonlocal kernel for N -dimensional problems (N = 1, 2 and 3) has the dimension of (length)−N . As a result, it must always contain a length scale parameter. Furthermore, for homogeneous media, the nonlocal kernel depends on the Euclidean distance and we can express in a more appropriate form as    e0 L i (4.8) α(x, x , λ) = α x − x  , λ , λ = Le where e0 is a non-dimensional nonlocal scaling parameter which has been assumed to be a constant appropriate for each material. The nonlocal kernel has the following essential properties [3]: (i) Correctly reflect the long-range interatomic interactions between material points. The maximum value of the kernel function must emerge at x = x and the function is required to decrease rapidly and monotonically to zero at large distances. This means that nonlocal interactions are only effective in a finite small neighbourhood of the point.   The nonlocal kernel is symmetric with respect to x, i.e., α(x − x  , λ) = (ii)  α(x − x , λ). (iii) In the limiting case of λ → 0, the kernel function should converge to the Dirac delta function, δ, and the nonlocal elasticity reduces to the classical elasticity (Fig. 4.1). This is known as a limit impulsivity condition [4] and it is mathematically expressed as      (4.9) lim+ α x − x  , λ = δ x − x  λ→0

(iv) Since it is expected that a uniform local ℘ results in a uniform nonlocal ℘, ¯ the following normalization condition must be satisfied:

Fig. 4.1 Schematic comparison between the local and nonlocal kernel functions

4.2 Spatial Nonlocal Kernels

65



   α x − x   , λ d V = 1

(4.10)

V

Consequently a uniform strain field would produce a uniform nonlocal stress field. In order to accommodate the nonlocal effects correctly, it is necessary that the above conditions be satisfied.

4.2.2 Different Types of Nonlocal Kernels Over the recent decades, several types of nonlocal kernels have been proposed in the literature [5, 6] related to one-, two- and three-dimensional problems. Among the many kernels that are available, the most practical ones are the cone-shaped function, the bell-shaped function, the Gaussian function and the exponential function. The cone-shaped function is expressed by ⎧

⎨ c |x−x | N    α 1 − N (λL e ) N α x − x   , λ = ⎩ 0

  x − x  ≤ λL e   x − x  > λL e

(4.11)

where N is the spatial dimension of the problem (N = 1, 2, 3). The constant α cN is obtained by using the normalization of α on an infinite domain, i.e.,  α(x, λ)d V = 1

(4.12)

V∞

where V∞ is the infinite domain wherein V is embedded within it. As a consequence, using Eq. (4.12), the constants α1c , α2c and α3c corresponding to one-, two-, or threedimensional kernels are obtained as [1] 1 λL e 2 α2c = π (λL e )2 3 α3c = 2π (λL e )3 α1c =

(4.13)

Another important kernel is the bell-shaped function defined by [7] ⎧

2 ⎨ b |x−x |2    α 1 − (λL )2 α x − x  , λ = e ⎩ 0

  x − x  ≤ λL e   x − x  > λL e

(4.14)

66

4 Essential Concepts from Nonlocal Elasticity Theory

where α b is an unknown constant which can be evaluated from the normalization condition (4.12) and is equal to 15/16λL e for one-dimension, 3/π (λL e )2 for twodimensions, and 105/32π (λL e )3 for three-dimensions. It should be noted that both the cone-shaped and bell-shaped functions have a bounded support of length 2λL e . Another possible choice is the Gaussian function [1]:    x − x  2    g  α x − x  , λ = α N exp − λL 2e

(4.15)

g

where α N is a constant and the normalization condition as given by Eq. (4.12) yields α N = (π λ)−N / 2 L −N e g

(4.16)

Clearly the Gaussian function, which is a well-known radial basis function, is a delta sequence since when λ → 0 it gives a Dirac-delta distribution. An alternative radial basis function kerenl is the exponential kernel defined by [5]:    x − x       e α x − x  , λ = α N exp − λL e

(4.17)

where 1 2λL e 1 e α2 = 2π (λL e )2 1 α3e = 8π (λL e )3 α1e =

(4.18)

Note that the Gaussian and the exponential kernel functions have an infinite support, which means that the nonlocal interaction takes place between any two points, no matter how far apart they are. However, due to computational limitations in practical calculations, these functions are truncated at the distance where their values become negligible. In order to compare the four usual types of the nonlocal kernels, the normalized kernels are shown in Fig. 4.2. To complement the discussion about the different types of nonlocal kernels, the atomistically-based kernel is also presented and discussed. In 2002, Picu [8] developed a nonlocal kernel from atomistic simulations instead of the classical Gaussian kernel. This kernel has the following form:       x − x  2 x − x  2    1 − λ1 exp − α N D x − x  , λ1 , λ2 = α atom. N λ2 L 2e (λ2 L e )2

(4.19)

4.2 Spatial Nonlocal Kernels

67

Fig. 4.2 Comparison between four usual types of the nonlocal kernel functions for λ = 0.25

where α atom. is an unknown constant which can be evaluated from the normalization N conditions on infinite domains, i.e., R α1D (r )dr = 1

2 lim

R→∞

(4.20)

0

2π R α2D (r )r dr dθ = 1

lim

R→∞ 0

(4.21)

0

2π π R α3D (r )r 2 dr sin φdφdθ = 1

lim

R→∞ 0

0

(4.22)

0

where r = |x|. Substituting Eq. (4.19) into Eqs. (4.20)–(4.22), we obtain α1atom.

√ 2 λ2 =√ , 2λ2 − λ1 > 0 π L e (2λ2 − λ1 )

α2atom. =

1 π L 2e (λ2

− λ1 )

, λ2 − λ1 > 0

(4.23)

(4.24)

68

4 Essential Concepts from Nonlocal Elasticity Theory

Fig. 4.3 Comparison between the atomistically-based and the Gaussian kernels

α3atom. =

2 . 2λ2 − 3λ1 > 0 √ √ π π L 3e λ2 (2λ2 − 3λ1 )

(4.25)

Figure 4.3 shows the atomistically-based kernel for λ1 = 0.15 and λ2 = 0.25, along with the Gaussian kernel evaluated for λ2 = 0.25 (and λ1 = 0). As the figure shows, the atomistically-based kernel becomes negative at some distance, while the nonlocal kernel must be continuous, positive and a monotonically decreasing function.

4.2.3 Two-Phase Kernels Kroner [9] and, Eringen and Kim [10] have defined an alternative nonlocal constitutive relation by a convex combination of local and nonlocal phases. A local-nonlocal constitutive relation can be expressed as [11] ti j (x1 , x2 , x3 , t) = η1 E i jkl εkl (x1 , x2 , x3 , t)  + η2 α(x, x , λ)E i jkl εkl (x1 , x2 , x3 , t)d x1 d x2 d x3 (4.26) V

where η1 and η2 are non-negative phase parameters so that η1 + η2 = 1

(4.27)

4.2 Spatial Nonlocal Kernels

69

Equation (4.26) can be viewed as a constitutive relation of a two-phase elastic material, in which phase 1 with volume fraction η1 has local characteristics and phase 2 with volume fraction η2 displays nonlocal characteristics. Note that the fully nonlocal constitutive relation is recovered by setting η1 = 0, and the local model corresponds to η2 = 0. Therefore, the two-phase kernel or the local-nonlocal kernel can be proposed to be       α˜ x − x  = η1 δ x − x  + (1 − η1 )α x − x 

(4.28)

  It should be noted that the kernel α(x − x ) can be selected from all kernels presented in Sect. 4.2.2 and other kernels will be presented in the next section.

4.3 Gradient-Based Nonlocal Elasticity Theory The solution of nonlocal elasticity problems by directly using the above kernels is mathematically difficult. As a result, by using Green’s function approach, Eringen [3] proposed an equivalent nonlocal elasticity theory in which the spatial integrals are replaced by differential operators. According to the integral-type nonlocal elasticity theory [12], the basic equations for linear homogeneous elastic solids are expressed as ∂ti j ∂ 2ui + (b f )i = ρ 2 (4.29) ∂x j ∂t  ti j (x1 , x2 , x3 , t) =

α(x, x , λ)E i jkl εkl (x1 , x2 , x3 , t)d x1 d x2 d x3

(4.30)

V

εi j =

1 2



∂u j ∂u i + ∂x j ∂ xx

(4.31)

where ti j is the nonlocal stress tensor. To obtain the associated differential form of Eq. (4.3), the kernel is assumed so that α is Green’s function of a linear differential operator L :    (4.32) L α x − x  , λ = δ(x − x ) where δ is the Dirac unit impulse. Note that the differential operator L can be an operator with constant or variable coefficients of any order. Applying the differential operator L to Eq. (4.30), the following differential equation is obtained: L ti j (x1 , x2 , x3 , t) = E i jkl εkl (x1 , x2 , x3 , t)

(4.33)

70

4 Essential Concepts from Nonlocal Elasticity Theory

Therefore, the nonlocal constitutive relation may be expressed in tensor notation as: Lt = E : ε

(4.34)

If L is a differential operator with constant coefficients, we have ∂ (L ti j ) = L ∂x j



∂ti j ∂x j

(4.35)

Using Eq. (4.35), Eq. (4.29) is reduced to ∂ (E i jkl εkl ) + L ∂x j

∂ 2ui (b f )i − ρ 2 = 0 ∂t

(4.36)

As a result, the partial differential equations must be solved instead of the integropartial differential equations. One of the most important operators is the Helmholtz operator, which is a linear second-order operator with constant coefficients. It is given by (4.37) L H = 1 − (λL e )2 ∇ 2 where ∇ 2 is the Laplace operator. Using Eq. (4.37), the nonlocal differential constitutive relation may be simplified to t − (λL e )2 ∇ 2 t = E : ε

(4.38)

wherein the symbol : denotes the double-dot product. 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 [13]. However, the Laplacian of a tensor in the orthogonal curvilinear coordinates does not simply pass through and operate on each of the individual tensor components as in the Cartesian case [12]. An expression of the Laplacian of the stress tensor, in the orthogonal curvilinear coordinates, is presented in Sect. 4.4.

4.3.1 Green’s Function Kernels Eringen [1] proposed several differential operators and found their corresponding kernels. Using the differential operator (4.37), Eq. (4.32) takes on the following form (1 − (λL e )2 ∇ 2 )α(x) = δ(x)

(4.39)

4.3 Gradient-Based Nonlocal Elasticity Theory

71

Taking the Fourier transform of Eq. (4.39), we have α(ζ ˜ )=

1 1 + (λL e )2 |ζ |2

(4.40)



where α(ζ ˜ ) = F n {α(x)} =

α(x) exp( jx · ζ )dx

(4.41)

Rn

in which Rn is the real coordinate space of n-dimensions, ζ ∈ Rn and j = Using the definition of the inverse Fourier transform, we obtain α(x) = Fn−1 {α(ζ )} =

1 (2π )n

√

−1.

 α(ζ ˜ ) exp(− jx · ζ )dζ

(4.42)

Rn

Substituting Eq. (4.40) into Eq. (4.42), the Green function kernels for one-, twoand three-dimensional problems are determined as [14]: (a) For one-dimension:    α1D x − x  , λ =

   x − x   1 exp − 2λL e λL e

(4.43)

(b) For two-dimensions:    α2D x − x  , λ =

1 K0 2π (λL e )2

  x − x   λL e

(4.44)

(c) For three-dimensions:    α3D x − x  , λ =

   x − x   1 exp − λL e 4π (λL e )2 |x − x |

(4.45)

Note that K 0 is the modified Bessel’s function of the second kind of order zero. It can easily be shown that the Green function kernels satisfy the normalization conditions. Furthermore, it should be noted that the two- and three-dimensional kernels have one singularity at x = x . To remove this singularity, Lazar and coworkers [14] proposed the nonlocal kernels which are the Green function of the bi-Helmholtz equation. The bi-Helmholtz operator is a linear elliptic fourth-order operator and it is given by LbH = 1 − ε02 L i2 ∇ 2 + γ04 L i4 ∇ 4

(4.46)

72

4 Essential Concepts from Nonlocal Elasticity Theory

where ε0 and γ0 are non-negative parameters. The bi-Helmholtz operator can be rewritten as LbH = 1 − ε02 L i2 ∇ 2 + γ04 L i4 ∇ 4 = (1 − L 2e λ21 ∇ 2 )(1 − L 2e λ22 ∇ 2 )

(4.47)

where λ1 and λ2 are nonlocal parameters defined by

 21 L i ε0 γ04 λ1 = 1+ 1−4 4 2L e ε0

 21 γ04 L i ε0 λ2 = 1− 1−4 4 2L e ε0

(4.48)

where ε04 ≥ 4γ04

(4.49)

Using Eq. (4.47), the bi-Helmholtz equation is obtained as (1 − L 2e λ21 ∇ 2 )(1 − L 2e λ22 ∇ 2 )α(x) = δ(x)

(4.50)

Therefore, the nonlocal kernels are given by [14] (a) For one-dimension: 

 Le 1 r r λ − λ , exp − exp − 1 2 2 λ21 L 2e − λ22 L 2e λ1 L e λ2 L e 1 1 α1D (0, λ1 , λ2 ) = (4.51) 2L e λ1 + λ2

α1D (r, λ1 , λ2 ) =

(b) For two-dimensions: 



 r r 1 1 K0 − K0 , α2D (r, λ1 , λ2 ) = 2π λ21 L 2e − λ22 L 2e λ1 L e λ2 L e 1 1 λ1 α2D (0, λ1 , λ2 ) = ln (4.52) 2π λ21 L 2e − λ22 L 2e λ2 (c) For three-dimensions: 



 r r 1 1 exp − − exp − , 4πr λ21 L 2e − λ22 L 2e λ1 L e λ2 L e 1 1 α3D (0, λ1 , λ2 ) = (4.53) 4π L 3e λ1 λ2 (λ1 + λ2 )

α3D (r, λ1 , λ2 ) =

4.3 Gradient-Based Nonlocal Elasticity Theory

73

  where r = x − x . From Eqs. (4.51)–(4.53), it is easily seen that the nonlocal kernels of Helmholtz type are recovered if the parameter λ2 is set to zero. Furthermore, for the limiting case λ2 → λ1 = λ, the bi-Helmholtz kernels are obtained as (a) For one-dimension: 

 1 r (λL , + r ) exp − e λL e 4(λL e )2 1 α1D (0, λ) = 4L e λ

α1D (r, λ) =

(4.54)

(b) For two-dimensions: r α2D (r, λ) = K1 4π (λL e )3 1 α2D (0, λ) = 4π (λL e )2



r λL e

, (4.55)

(c) For three-dimensions:

1 r α3D (r, λ) = , exp − λL e 8π (λL e )3 1 α3D (0, λ) = 8π (λL e )3

(4.56)

It is important to note that all kernels (4.51)–(4.56) are nonsingular. To compare the Helmholtz and bi-Helmholtz kernels, the one-dimensional kernels and their derivatives are respectively plotted in Figs. 4.4 and 4.5. It is observed that the Helmholtz kernel is continuous but its derivative has a jump discontinuity at x = 0 while the bi-Helmholtz kernel and its derivative are continuous.

4.3.2 Alternative Derivation of Gradient-Based Nonlocal Elasticity Theory In this subsection, the nonlocal integral constitutive relations in one- and two- dimensional cases are converted into the differential relations using the the Green function of the Helmholtz equation. Let us first consider the one-dimensional constitutive relation. For an infinite domain, Eq. (4.30) can be expressed as 1 t11 (x1 , t) = 2λL e

∞ −∞

   x1 − x1   exp − σ11 (x1  , t)d x1  λL e

(4.57)

74

4 Essential Concepts from Nonlocal Elasticity Theory

Fig. 4.4 Comparison between the Helmholtz and the bi-Helmholtz kernels for λ = 0.15

Fig. 4.5 Comparison between derivatives of the Helmholtz and the bi-Helmholtz kernels for λ = 0.15

where t11 and σ11 are respectively the nonlocal and local stresses. Equation (4.57) can be rewritten as 1 t11 (x1 , t) = 2λL e +

1 2λL e



x exp −∞ ∞

exp

x

x1  − x1 σ11 (x1  , t)d x1  λL e

x1 − x1  σ11 (x1  , t)d x1  λL e

(4.58)

4.3 Gradient-Based Nonlocal Elasticity Theory

75

Taking the first and second derivatives of Eq. (4.58) with respect to x1 yields 1 ∂t11 (x1 , t) = ∂ x1 2(λL e )2 1 − 2(λL e )2

2(λL e )2

∞ x



x1 − x1  exp σ11 (x1  , t)d x1  λL e

x1  − x1 σ11 (x1  , t)d x1  exp λL e

x −∞

∂ 2 t11 (x1 , t) 1 = −2σ11 (x1 , t) + λL e ∂ x12 +

1 λL e



∞ exp x



x exp −∞

(4.59)

x1  − x1 σ11 (x1  , t)d x1  λL e

x1 − x1  σ11 (x1  , t)d x1  λL e

(4.60)

Substituting Eq. (4.58) into Eq. (4.60), the resultant equation is t11 (x1 , t) − (λL e )2

∂ 2 t11 (x1 , t) = σ11 (x1 , t) ∂ x12

(4.61)

It should be noted that the nonlocal elasticity in an infinite domain is free of boundary conditions. For two-dimensional constitutive relations, we follow the work of Altan and Aifantis [15]. According to Eqs. (4.30) and (4.44), the two-dimensional nonlocal integral constitutive relation in an infinite domain is ti j (x1 , x2 , t) =

1 2π (λL e )2

∞ ∞ −∞ −∞

⎛ ⎞ (x1 − x  1 )2 + (x2 − x  2 )2 ⎝ ⎠σi j (x  , x  , t)d x  d x  K0 1 2 1 2 λL e

(4.62)

Note that Eq. (4.62) is the Fredholm integral equation of the first kind [16]. It is assumed that the expansion σi j (x1 , x2 , t) =

∞  ∞  ∂ m+n σi j (x  1 , x  2 , t) (x  1 − x1 )m (x  2 − x2 )n ∂ x1m ∂ x2n m! n! m=0 n=0

(4.63)

is valid for every pair of x1 and x2 throughout the entire domain. It is convenient to define the following new variable:

76

4 Essential Concepts from Nonlocal Elasticity Theory



∞ ∞ Imn =

K0 −∞ −∞

(x1 − x  1 )2 + (x2 − x  2 )2 m (x  1 − x1 ) (x2 − x2 )n d x1 d x2 λL e (4.64)

which can be rewritten in an equivalent form

2π ∞ Imn =

K0 0

where

0

r λL e

r m+n+1 cosm θ sinn θ dr dθ

(x1 − x1 ) = r cos θ, (x2 − x2 ) = r sin θ

(4.65)

(4.66)

It can be easily shown that Imn = 0 for all odd m and n. Therefore, Eq. (4.65) is expressed as

I2k,2l



π / 2 ∞ r 2k 2l = 4 (cos θ sin θ )dθ r 2k+2l+1 K 0 dr where l and k = 0, 1, 2, ... λL e 0

0

(4.67) Using the following formulae [17] π / 2 (cos2k θ sin2l θ )dθ = 0



∞ r

2k+2l+1

K0

0

r λL e

(2k)!(2l)! π 1 22(k+l) (k)!(l)!(k + l)! 2

(4.68)

dr = 22(k+l) (λL e )2(k+l+1) [(k + l)!]2

(4.69)

the nonlocal constitutive equation becomes ti j =

∞ ∞  

(λL e )2(k+l)

k=0 l=0

(k + l)! ∂ 2k+2l σi j (k)!(l)! ∂ x12k ∂ x22l

(4.70)

Additional simplifications can be made by taking into account the following identities p ∞  ∞  ∞   Akl = A p−q,q (4.71) k=0 l=0

p=0 q=0

k + l = p, l = q

(4.72)

4.3 Gradient-Based Nonlocal Elasticity Theory

77

The application of Eqs. (4.71) and (4.72) leads to ti j =

p ∞  

(λL e )2 p

p=0 q=0

∂ 2 p σi j ( p)! 2( ( p − q)!(q)! ∂ x1 p−q) ∂ x22q

(4.73)

Furthermore, we use the following identity which provides a further simplification. ∇ 2 p σi j =

i  j=0

∂ 2 p σi j ( p)! ( p − q)!(q)! ∂ x12( p−q) ∂ x22q

(4.74)

where ∇ 2 p = ∇ 2 ∇ 2 ...∇ 2 ( p times)

(4.75)

By using Eq. (4.74), Eq. (4.73) is rewritten a ti j =

∞ 

(λL e )2 p ∇ 2 p σi j = σi j + (λL e )2 ∇ 2 σi j + (λL e )4 ∇ 4 σi j + ...

(4.76)

p=0

Taking the Laplacian of both sides of Eq. (4.76), the following relation can be derived: (λL e )2 ∇ 2 ti j =

∞ 

(λL e )2( p+1) ∇ 2( p+1) σi j

p=0

(4.77)

= (λL e )2 ∇ 2 σi j + (λL e )4 ∇ 4 σi j + ... = ti j − σi j Finally, we obtain ti j − (λL e )2 ∇ 2 ti j = σi j

(4.78)

Note that the expression (4.78) is equivalent to Eq. (4.62).

4.4 Laplacian of the Stress Tensor In this section, we provide the derivation of the Laplacian of the stress tensor in the curvilinear coordinates. Let ζi (i = 1, 2, 3) represent the three orthogonal curvilinear coordinates, with the corresponding orthonormal basis vectors ei (Fig. 4.6). In the orthogonal curvilinear coordinates, we can write dt = ∇tdx ≡ μdx

(4.79)

78

4 Essential Concepts from Nonlocal Elasticity Theory

Fig. 4.6 Orthogonal curvilinear coordinates

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

3 

h k dζk ek

(4.80)

k=1

in which h 1 , h 2 , h 3 are called the scale factors which are nonnegative functions of position. Substituting Eq. (4.80) into Eq. (4.79), the following relation is obtained dt = μdx =

3 

h k dζk (μek )

(4.81)

k=1

Note that μek is a second-order tensor which can be expressed in the dyadic notation as (4.82) μek = μi jk ei ⊗ e j Substituting Eq. (4.82), Eq. (4.81) gives dt =

3 

h k μi jk dζk ei ⊗ e j

(4.83)

k=1

On the other hand, the nonlocal stress tensor can be expressed as t = ti j ei ⊗ e j

(4.84)

dt = dti j ei ⊗ e j + ti j dei ⊗ e j + ti j ei ⊗ de j

(4.85)

Therefore, we have

4.4 Laplacian of the Stress Tensor

where dti j =

79

∂ti j dζl , dei = Γi pq dζ p eq ∂ζl

(4.86)

with the values of Γi pq depending on the orthogonal curvilinear coordinate system. Substituting Eq. (4.86) into Eq. (4.85), we have ⎛

⎞ 3 3   ∂t i j dt = ⎝ + tq j Γqmi + tiq Γqm j ⎠ dζm ei ⊗ e j ∂ζm q=1 q=1

(4.87)

Comparing Eq. (4.87) with (4.83), the gradient of the stress tensor is obtained as ⎛ μi jm =

3 

3 

⎞

1 ⎝ ∂ti j + tq j Γqmi + tiq Γqm j ⎠ h m ∂ξm q=1 q=1

(4.88)

By using a similar procedure and very lengthy derivations, the gradient of the third-order tensor μ, which is a fourth-order tensor, can be expressed as ⎛ ⎜ ⎜ 3  3  3 3   1 ⎜ ⎜ ∇∇t = ∇μ = h ⎜ i=1 j=1 k=1 l=1 l ⎜ ⎝

⎞ μm jk Γmli ⎟ m=1 ⎟ 3 ⎟  + μimk Γml j ⎟ ⎟ ei ⊗ e j ⊗ ek ⊗ el m=1 ⎟ ⎠ 3  + μi jm Γmlk

∂μi jk ∂ζl

+

3 

m=1

(4.89) Consequently, the Laplacian of the stress tensor in orthogonal curvilinear coordinates, which is a second order tensor, can be obtained as ⎛ ∇2t =

3  3  3  i=1 j=1 k=1

1 ⎜ ⎜ hk ⎝

∂μi jk ∂ζk

+

3  m=1

+

3 

⎞ μm jk Γmki

m=1

μimk Γmk j +

3 

μi jm Γmkk

⎟ ⎟ ei ⊗ e j ⎠

(4.90)

m=1

In the following subsections, Eq. (4.90) is used to obtain the Laplacian of the stress tensor for cylindrical and spherical coordinates.

4.4.1 Cylindrical Coordinates For the case of the cylindrical coordinates (r , θ , z), we have [18]

80

4 Essential Concepts from Nonlocal Elasticity Theory

ζ1 = r, ζ2 = θ, ζ3 = z e1 = er , e2 = eθ , e3 = ez h 1 = h r = 1, h 2 = h θ = r, h 3 = h z = 1 Γ122 = Γr θθ = 1, Γ221 = Γθθr = −1

(4.91)

It should also be noted that the values of other Γi pq are zero. As a result, the Laplacian of the stress tensor in cylindrical coordinates is obtained as (∇ 2 t)rr = ∇c2 trr −

2 4 ∂tr θ − 2 (trr − tθθ ) r 2 ∂θ r

(4.92)

4 ∂tr θ 2 + 2 (trr − tθθ ) 2 r ∂θ r

(4.93)

(∇ 2 t)θθ = ∇c2 tθθ +

(∇ 2 t)zz = ∇c2 tzz (∇ 2 t)θ z = ∇c2 tθ z +

1 2 ∂tzr − 2 tθ z r 2 ∂θ r

1 2 ∂tzθ − 2 tzr 2 r ∂θ r

2 ∂trr ∂tθθ 4 2 = ∇c tr θ + 2 − − 2 tr θ r ∂θ ∂θ r

(∇ 2 t)zr = ∇c2 tzr − (∇ 2 t)r θ where ∇c2 (∗) =

1 ∂ 2 (∗) ∂ 2 (∗) ∂ 2 (∗) 1 ∂(∗) + + + ∂r 2 r ∂r r 2 ∂θ 2 ∂z 2

(4.94) (4.95) (4.96)

(4.97)

(4.98)

The same equations were given by Povstenko [19] and Povstenko and Matkovskii [20] without derivations. In addition, these equations were developed in Appendix A of the book by Eringen [1] for the cylindrical coordinates. However, the derivation is very complicated.

4.4.2 Spherical Coordinates In the spherical coordinates, the scale factors are h 1 = h R = 1, h 2 = h φ = R, h 3 = h θ = R sin φ and the nonzero components of Γi jk are Γ122 = Γ Rφφ = 1, Γ133 = Γ Rθθ = sin φ, Γ221 = Γφφ R = −1, Γ331 = Γθθ R = − sin φ, Γ233 = Γφθθ = cos φ and Γ332 = Γθθφ = − cos φ [18]. In conjunction with Eq. (4.90), the Laplacian of the stress tensor for the spherical coordinates are obtained as

4.4 Laplacian of the Stress Tensor

 2  tθθ + tφφ − 2t R R 2 R ∂ ∂t Rθ 4 4 − 2 (sin φt Rφ ) − 2 R sin φ ∂φ R sin φ ∂θ

81

(∇ 2 t) R R = ∇s2 t R R +

2 2cot 2 φ (t R R − tφφ ) − (tφφ − tθθ ) 2 R R2 4 cos φ ∂tφθ 4 ∂t Rφ − 2 2 + 2 R ∂φ R sin φ ∂θ

(4.99)

(∇ 2 t)φφ = ∇s2 tφφ +

2 2cot 2 φ (t − t ) + (tφφ − tθθ ) R R θθ R2 R2 ∂t Rθ 4 cos φ ∂tφθ 4 4 cot φ + 2 2 t Rφ + 2 + R2 R sin φ ∂θ R sin φ ∂θ

(4.100)

(∇ 2 t)θθ = ∇s2 tθθ +

2 cot φ 4t Rφ t Rφ − 2 2 − (tφφ − tθθ ) R2 R2 R sin φ 2 ∂tφθ 2 ∂ 2 cos φ ∂t Rθ (t R R − tφφ ) − 2 2 − 2 + 2 R ∂φ R sin φ ∂θ R sin φ ∂θ

(4.101)

(∇ 2 t) Rφ = ∇s2 t Rφ −

4 cot φ 4t Rθ t Rθ − 2 2 − tφθ R2 R2 R sin φ 2 ∂ 2 ∂tφθ 2 cos φ ∂t Rφ + 2 (t R R − tθθ ) + 2 2 − 2 R ∂φ R sin φ ∂θ R sin φ ∂θ

(4.102)

(∇ 2 t) Rθ = ∇s2 t Rθ −

2tφθ 4cot 2 φ 2 cot φ − tφθ − t Rθ R2 R2 R2 2 cos φ ∂ ∂t Rφ 2 ∂t Rθ 2 + 2 2 (tφφ − tθθ ) + 2 + 2 R ∂φ ∂θ R sin φ ∂θ R sin φ

(4.103)

(∇ 2 t)φθ = ∇s2 tφθ −

(4.104)

where ∇s2 (∗) =

1 ∂ 2 (∗) cot φ ∂(∗) 1 ∂ 2 (∗) ∂ 2 (∗) 2 ∂(∗) + + + + ∂ R2 R ∂R R 2 ∂φ 2 R 2 ∂φ R 2 sin2 φ ∂θ 2

(4.105)

Note that the same equations have been presented, without derivations, by Povstenko [21]. For a centro-symmetric problem where there are only normal stresses (t R R , tφφ , tθθ ) which are function of R only, the formulations derived in this section

82

4 Essential Concepts from Nonlocal Elasticity Theory

can be easily simplified to those obtained by Gao et al. [22]. Finally, it should be noted that most researchers have used ∇ 2 (ti j ) instead of (∇ 2 t)i j because of the difficulty in deriving the governing equations and their solutions.

4.5 Nonlocal Anisotropic Elasticity At this point, it is necessary to mention that two types of anisotropy might be considered for materials. The first one is the inherent material anisotropy. The second is related to anisotropic length scale effect due to the nonlocality [23]. In this section, a linear differential operator and the corresponding nonlocal kernel function for the nonlocal anisotropic elasticity are presented. A suitable choice of the differential operator to describe the effect of anisotropic nonlocality can be written as follows [23] 3 3   ∂2 Li j (4.106) L anis. = 1 − ∂ xi ∂ x j i=1 j=1 where L i j is the symmetric second-order tensor which incorporates anisotropic length scales. Using Eqs. (4.106), (4.30) and (4.32) take on the following forms respectively ⎛ ⎝1 −

i=1

⎛ ⎝tkl −

⎞ ∂2 ⎠ Li j α(x) = δ(x) ∂ xi ∂ x j j=1

3  3 

⎞ ∂ 2 tkl ⎠ Li j = σkl = E klpq ε pq ∂ xi ∂ x j j=1

3  3  i=1

(4.107)

(4.108)

Equation (4.108) represents the constitutive equations for nonlocal anisotropic elasticity. If the tensor L i j is positive definite, Eq. (4.108) is an elliptic partial differential equation of second order. The necessary and sufficient conditions for tensor L i j to be positive definite are L 11 > 0 L 11 det L 12 ⎛ L 11 det ⎝ L 12 L 13

L 12 L 22 L 12 L 22 L 23

>0 ⎞ L 13 L 23 ⎠ > 0 L 33

Note that Eq. (4.38) is recovered by setting L i j = (λL e )2 δi j where

(4.109)

4.5 Nonlocal Anisotropic Elasticity

83

 δi j =

i= j i = j

1 0

(4.110)

For two-dimensional problems, such as plane strain and anti-plane strain problems, Eq. (4.109) reduces to ti j − L 11

∂ 2 ti j ∂ 2 ti j ∂ 2 ti j − L − L = σi j 12 22 ∂ x1 ∂ x2 ∂ x12 ∂ x22

(4.111)

In some special cases such as for hexagonal and orthotropic materials, Eq. (4.111) reduces to a simpler equation, i.e., ti j − L 11

∂ 2 ti j ∂ 2 ti j − L = σi j 22 ∂ x12 ∂ x22

(4.112)

In addition, the corresponding two-dimensional nonlocal kernel must satisfy the following equation:

∂2 ∂2 ∂2 1 − L 11 2 − L 12 − L 22 2 α(x1 , x2 ) = δ(x1 )δ(x2 ) ∂ x1 ∂ x2 ∂ x1 ∂ x2

(4.113)

In order to obtain the normal form of Eq. (4.113), the following coordinate transformation is introduced.  L 11 L 22 − L 212 L 12 xˆ1 = x1 − x2 , xˆ2 = x2 (4.114) L 22 L 22 Substituting Eq. (4.114) into Eq. (4.113), we have 

L 11 L 22 − 1− L 22

L 212



∂ ∂ + 2 ˆ ∂ xˆ1 ∂ x22 2

2



 α(xˆ1 , xˆ2 ) =

L 11 L 22 − L 212 L 22

δ(xˆ1 )δ(xˆ2 ) (4.115)

Using Eq. (4.44), the solution of Eq. (4.115) is obtained as α(xˆ1 , xˆ2 ) =



1

 K0 2π L 11 L 22 − L 212

L 22 (xˆ12 + xˆ22 ) L 11 L 22 − L 212

(4.116)

Therefore, in the original coordinates, the nonlocal kernel of the nonlocal anisotropic elasticity for two-dimensional problems is α(x1 , x2 ) =

1

 K0 2π L 11 L 22 − L 212



(L 22 x12 − 2L 12 x1 x2 + L 11 x22 ) L 11 L 22 − L 212

(4.117)

84

4 Essential Concepts from Nonlocal Elasticity Theory

4.6 Nonlocal Kernels in a Finite Domain From a physical point of view, a uniform strain field must produce a uniform nonlocal stress identified with the local one everywhere in the domain. This fact was reflected in Eq. (4.10) as the normalization condition. Since all the above mentioned kernels are normalized over the infinite domain, they do not satisfy the normalization condition for all points of a finite body [24, 25]. Therefore, Eq. (4.10) is smaller than 1 for the finite domain. To overcome this limitation, two approaches have been proposed. In the first approach, a nonstandard weighting function is defined as [7] α mod (x, x ) = 

α(x, x ) α(x, x )d V (x )

(4.118)

V

where α(x, x ) is the kernel corresponding to an infinite domain. Although Eq. (4.118) satisfy the normality condition in a finite domain, this modified kernel breaks the symmetry with respect to the arguments x and x . Therefore, another modified kernel function has been proposed by Borino et al. [26]. This modified kernel is expressed as ⎡ ⎤    α mod (x, x ) = ⎣1 − α(x, x )d V (x )⎦ δ x − x  + α(x, x ) (4.119) V

The modified kernel preserves both the symmetry and the normalization conditions at each point of a finite domain. The kernel has received scant attention and has only been employed by Koutsoumaris et al. [24] and Eptaimeros et al. [27]. However, it should be noted that the infinite kernel functions can be applied accurately to finite bodies under some special conditions. In this way, and by using the Green function kernels, we define the following integrals for one-, two- and three-dimensional problems:

R R 1 r dr (4.120) exp − S1 = 2 α1D (r )dr = e0 L i e0 L i 0

2π R S2 = 0

0

1 α2D (r )r dr dθ = 2π (e0 L i )2

2ππ R α3D (r )r 2 dr sin φdφdθ =

S3 = 0 0 0

0



2π R K0 0

1 4π (e0 L i )2

r r dr dθ e0 L i

(4.121)

0

2ππ R

exp

0 0 0

−

r r dr sin φdφdθ e0 L i

(4.122)

4.6 Nonlocal Kernels in a Finite Domain

85

Table 4.1 Computed values of integrals S1 , S2 and S3 for different values of R R/e0 L i S1 S2 S3 1 2 4 6 8

0.6321 0.8647 0.9817 0.9975 0.9997

0.3981 0.7203 0.9501 0.9919 0.9988

0.2642 0.5940 0.9084 0.9826 0.9970

Table 4.1 presents the values of integrals for various radii R. As expected, the values of these integrals are generally smaller than 1 and approach 1 when R is increased. According to this table, it can be seen that one-, two- and three-dimensional Green function kernels approximately satisfy the normalization conditions for R > 6e0 L i . It can be concluded that if the smallest dimension of the finite domain is larger than 6e0 L i or 6λL e , the kernels corresponding to an infinite domain approximately satisfy the normalization condition. Therefore, the constitutive relation (4.38) will be acceptable for modelling the finite domain.

References 1. A.C. Eringen, Nonlocal Continuum Field Theories (Springer, New York, 2002) 2. Y.Z. Povstenko, The nonlocal theory of elasticity and its applications to the description of defects in solid bodies. J. Math. Sci. 97, 3840–3845 (1999) 3. A.C. Eringen, On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J. Appl. Phys. 54, 4703–4710 (1983) 4. G. Romano, R. Barretta, Nonlocal elasticity in nanobeams: the stress-driven integral model. Int. J. Eng. Sci. 115, 14–27 (2017) 5. C. Polizzotto, Nonlocal elasticity and related variational principles. Int. J. Solids Struct. 38, 7359–7380 (2001) 6. S. Ghosh, V. Sundararaghavan, A.M. Waas, Construction of multi-dimensional isotropic kernels for nonlocal elasticity based on phonon dispersion data. Int. J. Solids Struct. 51, 392–401 (2014) 7. Z.P. Ba˜zant, M. Jirasek, Nonlocal integral formulations of plasticity and damage: survey of progress. J. Eng. Mech. 128, 1119–1149 (2002) 8. R.C. Picu, The Peierls stress in non-local elasticity. J. Mech. Phys. Solids 50, 717–735 (2002) 9. E. Kroner, Elasticity theory of materials with long range cohesive forces. Int. J. Solids Struct. 3, 731–742 (1967) 10. A.C. Eringen, B.S. Kim, Relation between nonlocal elasticity and lattice dynamics. Cryst. Latt. Def. 7, 51–57 (1977) 11. S.B. Altan, Uniqueness of initial-boundary value problems in nonlocal elasticity. Int. J. Solids Struct. 25, 1271–1278 (1989) 12. H. Rafii-Tabar, E. Ghavanloo, S.A. Fazelzadeh, Nonlocal continuum-based modeling of mechanical characteristics of nanoscopic structures. Phys. Rep. 638, 1–97 (2016) 13. E. Benvenuti, A. Simone, One-dimensional nonlocal and gradient elasticity: closed-form solution and size effect. Mech. Res. Commun. 48, 46–51 (2013) 14. M. Lazar, G.A. Maugin, E.C. Aifantis, On a theory of nonlocal elasticity of bi-Helmholtz type and some applications. Int. J. Solids Struct. 43, 1404–1421 (2006)

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4 Essential Concepts from Nonlocal Elasticity Theory

15. B.S. Altan, E.C. Aifantis, On some aspects in the special theory of gradient elasticity. J. Mech. Behav. Mater. 8, 231–282 (1997) 16. B.S. Altan, Airy’s functions in nonlocal elasticity. J. Comput. Theor. Nanosci. 8, 2381–2388 (2011) 17. I.S. Gradshteyn, I.W. Ryzhik, Table of Integrals, Series and Products (Academic Press, New York, 1980) 18. W.M. Lai, E. Krempl, D. Rubin, Introduction to Continuum Mechanics (Elsevier, Oxford, 2010) 19. Y.Z. Povstenko, Straight disclinations in nonlocal elasticity. Int. J. Eng. Sci. 33, 575–582 (1995) 20. Y.Z. Povstenko, O.A. Matkovskii, Boundary dislocation in a nonlocally-elastic medium with moment stresses. J. Math. Sci. 96, 2883–2886 (1999) 21. Y.Z. Povstenko, Point defect in a nonlocal elastic medium. J. Math. Sci. 104, 1501–1505 (2001) 22. X.L. Gao, S.K. Park, H.M. Ma, Analytical solution for a pressurized thick-walled spherical shell based on a simplified strain gradient elasticity theory. Math. Mech. Solids 14, 747–758 (2009) 23. M. Lazar, E. Agiasofitou, Screw dislocation in nonlocal anisotropic elasticity. Int. J. Eng. Sci. 49, 1404–1414 (2011) 24. C.C. Koutsoumaris, K.G. Eptaimeros, G.J. Tsamasphyros, A different approach to Eringen’s nonlocal integral stress model with applications for beams. Int. J. Solids Struct. 112, 222–238 (2017) 25. G. Romano, R. Luciano, R. Barretta, M. Diaco, Nonlocal integral elasticity in nanostructures, mixtures, boundary effects and limit behaviours. Continuum Mech. Thermodyn. 30, 641–655 (2018) 26. G. Borino, B. Failla, F. Parrinello, A symmetric nonlocal damage theory. Int. J. Solids Struct. 40, 3621–3645 (2003) 27. K.G. Eptaimeros, C.C. Koutsoumaris, I.T. Dernikas, T. Zisis, Dynamical response of an embedded nanobeam by using nonlocal integral stress models. Compos. Part B 150, 255–268 (2018)

Chapter 5

Nonlocal Modelling of Nanoscopic Structures

Various types of nonlocal models have been proposed and extensively utilized to study the mechanical characteristics of nanoscopic structures with varied morphologies. These models are appropriate for describing the behaviour of ultra-small structures, as well as components embedded in nanoscale systems. Furthermore, they can accommodate the discrete nature of nanoscopic structures. The results obtained from the nonlocal models have been successfully compared with those obtained from both experimental and atomistic-based simulation studies. The nonlocal models that have been utilized include the nonlocal beam models, the nonlocal plate and shell models, and the nonlocal models for nanocrystals. The beam models are employed for computational modelling of one-dimensional nanoscopic structures such as nanowires, nanorods, nanofibers and nanotubes. The nonlocal plate models have been applied to study the mechanical characteristics of two-dimensional nanoscopic structures including thin films, nanoplates and nanosheets. The nonlocal shell model has been used to predict the behaviour of shell-like nanoscopic structures. In this chapter, we shall present the basic equations of all these highly complex models, and provide enough details in order to follow the current research materials and modelling future research problems.

5.1 Nonlocal Beam Models Nanoscale beam-like structures are highly efficient components and they have extensive practical applications in various fields of nanoscale science and nanoscale technology. Furthermore, numerous one-dimensional nanoscopic structures in biomedical science and engineering (e.g. DNA, microtubule and primary cilia) and nanoengineering (e.g. carbon nanotubes and nanowires) can be modelled accurately as beam-like components. Therefore, the nonlocal beam theories have found extensive applications in modelling the static and dynamic behaviour of one-dimensional nanoscopic structures. There are several nonlocal beam theories. © Springer Nature Switzerland AG 2019 E. Ghavanloo et al., Computational Continuum Mechanics of Nanoscopic Structures, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-3-030-11650-7_5

87

88

5 Nonlocal Modelling of Nanoscopic Structures

Fig. 5.1 Schematic illustration of a beam and the coordinate system

These are developed to describe the static and dynamic response of nanorods and nanobeams. Here we present the mathematical formulations of the nonlocal EulerBernoulli, and Timoshenko beam theories. Both of these nonlocal theories have found extensive applications in modelling the mechanical behaviour of one-dimensional nanoscopic structures. Let us consider a straight homogeneous isotropic nanobeam of length l, Young’s modulus E, cross-section A, inertia I and mass density ρ. To describe the beam theories, a right-handed coordinate system (x, y, z) is introduced so that the x-axis is chosen along the nanobeam’s length, the y and z coordinates are the out-of-plane and transverse coordinates, respectively (Fig. 5.1). Since the deformations of the nanobeam take place in the x z plane, the displacement field is given by Ux (x, z, t) = u(x, t) − z

  ∂w(x, t) ∂w(x, t) + f (z) + ψ(x, t) ∂x ∂x

U y (x, z, t) = 0 Uz (x, z, t) = w(x, t)

(5.1)

where Ux , U y and Uz are the displacement components along the x-, y- and z-axes, respectively. Furthermore, u, w and ψ denote respectively the axial and the transverse displacements of the reference line of the nanobeam (i.e., z = 0), and the rotation of cross-section. In addition, f (z) is the shape function and, for different beam theories, it must be defined. According to Eq. (3.9), the non-zero strains are εx x

  2 ∂ 2w ∂ w ∂ψ ∂u − z 2 + f (z) + = ∂x ∂x ∂x2 ∂x

(5.2)

5.1 Nonlocal Beam Models

89

2εx z =

  ∂ f (z) ∂w +ψ ∂z ∂x

(5.3)

Furthermore, for use in the future subsections, the stress resultants can be defined by the cross-sectional integral of the nonlocal stresses, i.e., 





tx x d A, M =

N= A

ztx x d A, Q = A

tx z d A

(5.4)

A

where tx x and tx z are the nonlocal normal and shear stresses, respectively. As mentioned in Chap. 4, the nonlocal constitutive relations can be expressed by the original integral constitutive relations as well as by the differential constitutive relations. Therefore, the stress resultants can be obtained on the basis of both nonlocal constitutive forms. Using the nonlocal integral constitutive relations (Eq. (4.30)) and Eqs. (5.2)–(5.4), the stress resultants can be expressed in terms of the displacements and the rotation, i.e., ⎞ ⎛ l    2 ∂ w(ξ, t) ∂u(ξ, t) dξ ⎠ d A −z E ⎝ α(x − ξ ) N (x, t) = ∂ξ ∂ξ 2 A 0 ⎞ ⎛ l   2   w(ξ, t) ∂ ∂ψ(ξ, t) dξ ⎠ d A (5.5) + E ⎝ α(x − ξ ) f (z) + ∂ξ 2 ∂ξ A 

0

⎞ ⎛ l     2 ∂ w(ξ, t) ∂u(ξ, t) dξ ⎠ d A −z M(x, t) = E z ⎝ α(x − ξ ) ∂ξ ∂ξ 2 A 0 ⎞ ⎛ l   2   w(ξ, t) ∂ ∂ψ(ξ, t) dξ ⎠ d A (5.6) + E z ⎝ α(x − ξ ) f (z) + ∂ξ 2 ∂ξ A ⎛



G⎝

Q(x, t) = A

0

l 0

⎞   ∂ f (z) ∂w(ξ, t) + ψ(ξ, t) dξ ⎠ d A (5.7) α(x − ξ ) ∂z ∂ξ 

where G is the shear modulus. In a similar manner, the stress resultants are obtained as follows by referencing the nonlocal differential constitutive relations (Eq. (4.38)) N (x, t) − l 2 λ2

∂ 2 N (x, t) ∂u(x, t) = EA ∂x2 ∂x   2 ∂ w(x, t) ∂ψ(x, t) + f (z)d A +E ∂x2 ∂x A

(5.8)

90

5 Nonlocal Modelling of Nanoscopic Structures

∂ 2 M(x, t) ∂ 2 w(x, t) = − E I ∂x2 ∂x2   2 ∂ w(x, t) ∂ψ(x, t) +E + z f (z)d A (5.9) ∂x2 ∂x A   2 ∂ f (z) ∂w(x, t) 2 2 ∂ Q (x, t) + ψ(x, t) d A (5.10) Q(x, t) − l λ =G ∂x2 ∂x A ∂z

M(x, t) − l 2 λ2

5.1.1 Nonlocal Euler-Bernoulli Beam Theory Among the different beam theories, the simplest and also the most commonly used is the Euler-Bernoulli beam theory. In this beam theory, it is assumed that the effects of both rotary inertia and shear deformation are negligible implying that the EulerBernoulli beam is applicable to long beams. Furthermore, the plane sections of the beam remain plane and orthogonal to the beam reference line during the deformation. This assumption leads to f (z) = 0 (5.11) The governing equations and the corresponding boundary conditions of the EulerBernoulli are obtained by using the extended Hamilton’s principle [1] t2 (T − π + W )dt = 0

δ

(5.12)

t1

The kinetic energy, T , of the Euler-Bernoulli beam can be expressed as 1 T = ρA 2

 l  0

∂u ∂t

2

 +

∂w ∂t

2  dx

(5.13)

In addition, the virtual strain energy, δπ , is l  δπ = 0

 ∂ 2 δw ∂δu −M dx +N ∂x2 ∂x

(5.14)

Furthermore, the total work, W , due to the axial external compressive force, P, the axial distributed force, p(x, t), and the transverse distributed force, q(x, t), can be expressed as  2  l  1 ∂w W = P + pu + qw d x (5.15) 2 ∂x 0

5.1 Nonlocal Beam Models

91

Substituting Eqs. (5.13)–(5.15) into Eq. (5.12) and applying the usual variational techniques, the Euler-Lagrange equations are obtained as ∂N ∂ 2u + p = ρA 2 ∂x ∂t

(5.16)

∂ 2w ∂ 2w ∂2 M + q − P = ρ A ∂x2 ∂x2 ∂t 2

(5.17)

In addition, the boundary conditions at x = 0 and x = l, are as follows u = 0 or N = 0 ∂w ∂M −P =0 w = 0 or ∂x ∂x ∂w = 0 or M = 0 ∂x

(5.18)

Substituting Eqs. (5.5)–(5.7) into Eqs. (5.16) and (5.17), the governing equations for the nonlocal integral model in terms of axial and transverse displacements are given by ⎡ l ⎤  ∂ ⎣ ∂u(ξ, t) ⎦ ∂ 2 u(x, t) EA α(x − ξ ) (5.19) dξ + p(x, t) = ρ A ∂x ∂ξ ∂t 2 0

⎡ ∂2 − EI 2 ⎣ ∂x

l 0

⎤ ∂ 2 w(ξ, t) ⎦ α(x − ξ ) dξ + q(x, t) ∂ξ 2 −P

∂ 2 w(x, t) ∂ 2 w(x, t) = ρ A ∂x2 ∂t 2 (5.20)

Similarly, the governing equations according to the nonlocal stress resultants (5.8)–(5.10) are obtained as   2 4 ∂2 p ∂ u ∂ 2u 2 2 ∂ u − l λ E A 2 + p − l 2 λ2 2 = ρ A ∂x ∂x ∂t 2 ∂ x 2 ∂t 2 (5.21) − EI

∂ 4w ∂ 2q + q − (l 2 λ2 ) 2 4 ∂x ∂x  2   2 4  4 ∂ w ∂ w 2 2∂ w 2 2 ∂ w = ρA −P −l λ −l λ ∂x2 ∂x4 ∂t 2 ∂ x 2 ∂t 2 (5.22)

92

5 Nonlocal Modelling of Nanoscopic Structures

It should be noted that the axial deformation problems of one-dimensional nanoscopic structures are defined by Eq. (5.19) or Eq. (5.21). Furthermore, Eq. (5.20) or Eq. (5.22) can be used for modelling the transverse deformation of nanobeams. As we shall see later on, in general, the nonlocal differential model is not equivalent to the nonlocal integral model.

5.1.2 Nonlocal Timoshenko Beam Theory For short one-dimensional nanoscopic structures, the rotary inertia and the shear deformation exert significant effects and so must be included. To incorporate the effects of the rotary inertia and the shear deformation, Timoshenko [2] proposed a more refined beam theory, known as the Timoshenko beam theory or the first-order shear deformation theory. This theory relaxes the normality restriction that requires the plane sections to remain orthogonal to deformed beam reference line during the deformation. Therefore, we have f (z) = z (5.23) The kinetic energy of the Timoshenko beam including the rotary inertia is given by 1 T = ρA 2

 l  0

∂u ∂t

2

 +

∂w ∂t

2   l  1 ∂ψ 2 dx + ρI dx 2 ∂t

(5.24)

0

The strain energy of the beam is determined from 1 π= 2

l  M 0

  ∂ψ ∂u ∂w +N +Q + ψ dx ∂x ∂x ∂x

(5.25)

Substituting Eqs. (5.15), (5.24) and (5.25) into the extended Hamilton’s principle and applying the DuBois-Reymond Lemma [3] leads to the following set of the Euler-Lagrange equations ∂N ∂ 2u + p = ρA 2 (5.26) ∂x ∂t ∂ 2w ∂ 2w ∂Q + q − P 2 = ρA 2 ∂x ∂x ∂t

(5.27)

∂ 2φ ∂M − Q = ρI 2 ∂x ∂t

(5.28)

5.1 Nonlocal Beam Models

93

Furthermore, the appropriate boundary conditions are u = 0 or N = 0 w = 0 or Q − P ψ = 0 or M = 0

∂w =0 ∂x (5.29)

Using Eqs. (5.5)–(5.7), the governing equations of the nonlocal integral Timoshenko beam model can be found as ⎤ ⎡ l  ∂ 2 u(x, t) ∂u(ξ, t) ⎦ ∂ ⎣ dξ + p(x, t) = ρ A α(x − ξ ) (5.30) EA ∂x ∂ξ ∂t 2 ⎡ ∂ ⎣ Ks G A ∂x

0

l 0

⎤  ∂ 2 w(x, t) ∂w(ξ, t) + ψ(ξ, t) dξ ⎦ + q(x, t) − P α(x − ξ ) ∂ξ ∂x2 

= ρA

∂ 2 w(x, t) ∂t 2

(5.31)

⎤ ⎡ l    l ∂ ⎣ ∂ψ(ξ, t) ⎦ ∂w(ξ, t) EI dξ − K s G A α(x − ξ ) + ψ(ξ, t) dξ α(x − ξ ) ∂x ∂ξ ∂ξ 0

0

∂ 2 ψ(x, t) = ρI ∂t 2

(5.32)

where K s is the shear correction coefficient defined as the ratio of the shear stress at the middle to the average shear stress [4]. The equations of motion of the classical Timoshenko beam theory are obtained from Eqs. (5.30)–(5.32) by setting α(x − ξ ) = δ(x − ξ ) where δ is the Dirac delta function. On the other hand, the governing equations of the nonlocal differential beam model are obtained as   2 2 4 ∂ u ∂ 2u 2 2∂ p 2 2 ∂ u + p − l λ = ρ A − l λ ∂x2 ∂x2 ∂t 2 ∂ x 2 ∂t 2  2   2 2 4  ∂ w ∂ψ ∂ w 2 2 ∂ q 2 2∂ w G AK s + q − (l λ ) 2 − P + −l λ ∂x2 ∂x ∂x ∂x2 ∂x4   2 4 ∂ w ∂ w = ρA − l 2 λ2 2 2 2 ∂t ∂ x ∂t     4 ∂ 2ψ ∂w ∂ 2ψ 2 2 ∂ ψ E I 2 − G AK s + ψ = ρI − l λ ∂x ∂x ∂t 2 ∂ x 2 ∂t 2 EA

(5.33)

(5.34)

(5.35)

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5 Nonlocal Modelling of Nanoscopic Structures

The equations of motion of the classical Timoshenko beam theory are obtained from Eqs. (5.33)–(5.35) by setting λ = 0.

5.2 Nonlocal Cylindrical Shell Models With decreasing aspect ratios of the structures, the one-dimensional nanoscopic structures exhibit shell-like behaviour during their motion and deformation. Furthermore, it has been shown that the shell-like nature of tubular nanoscopic structures should be considered when studying their mechanical behaviour. Therefore, several available shell theories, including the Donnell theory [5], the Sanders theory [6], the Flügge theory [7] and the Novozhilov theory [8], are all reformulated using the nonlocal differential constitutive relations to investigate the mechanical characteristics of nanotubes. These nonlocal shell models have been developed on the basis of different sets of assumptions and distinct approximations. In this section, we shall explain the mathematical formulations of the nonlocal circular cylindrical shell models which will be used in analyses throughout the book.

5.2.1 Governing Equations for Circular Cylindrical Shells Let us consider a circular cylindrical shell with wall thickness h, middle-surface radius R, length l and mass density ρ. To establish the mathematical formulations, the cylindrical coordinate system (x, θ , z) is used where x, θ and z define the axial, the circumferential and the transverse directions, respectively (Fig. 5.2). It should be noted that the governing equations are independent of the constitutive relations. Therefore, the local governing equations can be applied to formulate the

Fig. 5.2 Schematic illustration of a circular cylindrical shell and the coordinate system

5.2 Nonlocal Cylindrical Shell Models

95

nonlocal shells. The governing equations for the circular cylindrical shell subjected to an axial compressive load, N¯ , are [9, 10] 2 ∂ Nx x 1 ∂ N xθ C1 ∂ Mxθ ∂ 2u ¯∂ u + N + − = ρh ∂x R ∂θ 2R 2 ∂θ ∂t 2 ∂x2

(5.36)

∂ N xθ C1 ∂ Mxθ ∂ 2v 1 ∂ Nθθ C1 ∂ 2v + + Qθ + = ρh 2 + N¯ 2 ∂x R ∂θ R R ∂x ∂t ∂x

(5.37)

∂ Qx ∂ 2w 1 ∂ Qθ Nθθ ∂ 2w + − = ρh 2 + N¯ 2 ∂x R 1∂θ R ∂t ∂x

(5.38)

1 ∂ Mxθ ∂ Mx x ρh 3 ∂ 2 βx + − Qx = ∂x R ∂θ 12 ∂t 2

(5.39)

∂ Mxθ ρh 3 ∂ 2 βθ 1 ∂ Mθθ + − Qθ = ∂x R ∂θ 12 ∂t 2

(5.40)

where u, v and w refer to the mid-surface displacements along the x, θ and z coordinates respectively, and βx and βθ are the rotations of the transverse normal about the x- and θ -axes, respectively. Furthermore, N x x , Nθθ and N xθ are the in-face stress resultants, Q x and Q θ are the transverse shear stress resultants, and Mx x , Mθθ and Mxθ are the stress couple resultants. These stress resultants are defined by h

   2   C1 z C1 z {N x x , Nθθ , N xθ } = tx x 1 + , tθθ , txθ 1 + dz R R

(5.41)

− h2 h

   2   C1 z C1 z {Mx x , Mθθ , Mxθ } = tx x 1 + , tθθ , txθ 1 + zdz R R

(5.42)

− h2 h

  2   C1 z {Q x , Q θ } = tx z 1 + , tθ z dz R

(5.43)

− h2

where tx x and tθθ are the nonlocal in-plane stresses, txθ , tx z and tθ z are the nonlocal shear stresses. Using Eqs. (4.38), (5.41), (5.42) and (5.43), the nonlocal constitutive equations of a specially orthotropic material are

96

5 Nonlocal Modelling of Nanoscopic Structures

⎛

⎞ ⎛ ⎞⎛ 0 ⎞ Nx x A11 A12 0 εx x 0 ⎠ L H C ⎝ Nθθ ⎠ = ⎝ A12 A22 0 ⎠ ⎝ εθθ 0 N xθ γxθ 0 0 A66 ⎞⎛ ⎞ ⎛ κx x D D 0 C1 ⎝ 11 12 0 0 0 ⎠ ⎝ κθθ ⎠ + R κ 0 0 D 66

⎛

(5.44)

xθ

⎞ ⎞⎛ 0 ⎞ ⎛ Mx x εx x D11 D12 0 C 1 0 ⎠ ⎝ 0 0 0 ⎠ ⎝ εθθ L H C ⎝ Mθθ ⎠ = R 0 Mxθ γxθ 0 0 D66 ⎛ ⎞⎛ ⎞ D11 D12 0 κx x + ⎝ D12 D22 0 ⎠ ⎝ κθθ ⎠ κxθ 0 0 D66     0  Qθ S44 0 γθ z = LH C Qx γx0z 0 S55

(5.45)

(5.46)

where Ai j , Sii , and Di j are respectively the extensional, the shear and the bending stiffnesses. They are defined as h

2 {Ai j , Di j } =

{1, z 2 }Ci j dz

(i, j = 1, 2, 6)

(5.47)

− h2 h

2 Sii = K s

Cii dz

(i = 4, 5)

(5.48)

− h2

in which K s is the shear correction coefficient and Ci j are the reduced elastic stiffnesses and defined as Ex νθ E x Eθ , C12 = , C22 = , 1 − νx νθ 1 − νx νθ 1 − νx νθ = G θ z , C55 = G x z , C66 = G xθ

C11 = C44

(5.49)

where E x and E θ are the longitudinal and circumferential Young’s moduli, respectively, G xθ is the in-plane shear modulus, G x z and G zθ are the transverse shear moduli, and νx and νθ denote Poisson’s ratios. Furthermore, L H C is the Helmholtz operator in the polar coordinate system and it is given by L H C = 1 − l 2 λ2 (

∂2 1 ∂2 + 2 2) 2 ∂x R ∂θ

(5.50)

5.2 Nonlocal Cylindrical Shell Models

97

0 0 In Eqs. (5.44)–(5.46), εx0 x , εθθ , γxθ , γθ0z , and γx0z are the mid-surface strains and κx x , κθθ and κxθ are the curvature and twist changes of the mid-surface. The strain and curvature components are given by [9]

εx0 x γx0z κθθ

  1 ∂u ∂u 1 ∂v ∂v 0 0 , εθθ = + w , γxθ + , = = ∂x R ∂θ ∂x R ∂θ ∂w 1 ∂w C1 ∂βx , γθ0z = βθ + − v, κx x = , = βx + ∂x R ∂θ R  ∂x  ∂βθ C1 ∂v 1 ∂u 1 ∂βθ 1 ∂βx , κxθ = + + − = R ∂θ R ∂θ ∂x R ∂x R ∂θ

(5.51)

The weighting constant C1 is introduced in order to allow for reducing the foregoing expressions to those corresponding to the Donnell shell theory or to the Sanders shell theory. If the coefficient C1 = 0, Eqs. (5.36)–(5.40) are reduced to a shear deformable version of the Donnell shell theory [5]. For the case C1 = 1, Eqs. (5.36)– (5.40) correspond to the generalization of the Sanders theory [6]. By neglecting the rotary inertia term (ρh 3 /12), and eliminating the transverse shear stress resultants, the five governing differential equations (5.36)–(5.40) can be reduced to the following system of three governing equations, i.e., 2 ∂ Nx x 1 ∂ N xθ C1 ∂ Mxθ ∂ 2u ¯∂ u + N + − = ρh ∂x R ∂θ 2R 2 ∂θ ∂t 2 ∂x2

(5.52)

1 ∂ Nθθ C1 ∂ Mθθ 3C1 ∂ Mxθ ∂ 2v ∂ 2v ∂ N xθ + + 2 + = ρh 2 + N¯ 2 ∂x R ∂θ R ∂θ 2R ∂ x ∂t ∂x

(5.53)

1 ∂ 2 Mθθ ∂ 2w 2 ∂ 2 Mxθ Nθθ ∂ 2w ∂ 2 Mx x + 2 = ρh 2 + N¯ 2 + − 2 2 ∂x R ∂ x∂θ R ∂θ R ∂t ∂x

(5.54)

In the following, for comparison purposes, we present two sets of the nonlocal governing equations on the basis of the Sanders shell theory. The equations of Sanders are known as the best classical and first-order shear deformation theory [11].

5.2.2 Nonlocal Classical Shell Model Among the different nonlocal shell models, the most extensively used is the classical shell theory due to its simplicity. Nonlocal classical shell theory is based on the following set of assumptions which are known as the Kirchhoff hypotheses [12]. (a) The straight lines normal to the undeformed mid-surface remain straight in the deformed states and suffer no changes of length. This assumption results in zero 0 . transverse normal strain εzz (b) The transverse normal stress is negligible so that the plane stress assumption can be invoked.

98

5 Nonlocal Modelling of Nanoscopic Structures

(c) The straight lines normal to the undeformed mid-surface rotate so that they remain perpendicular to the mid-surface after the deformation. This implies that the transverse shear strains γx0z , and γx0z are zero. It should be noted that the transverse stresses tx z and t yz are not equal to zero because the transverse shear stress resultants are necessary for the equilibrium of the shell. Using these assumptions, the displacement expressions for the nonlocal stress resultants can be expressed as L H C Nx x

    A12 ∂v D11 ∂ 2 w ∂ 2w ∂u D12 ∂v (5.55) + +w − − 2 = A11 + 3 ∂x R ∂θ R ∂x2 R ∂θ ∂θ L H C Nθθ 

L H C N xθ = A66 L H C Mx x

  A22 ∂v ∂u + +w = A12 ∂x R ∂θ

∂v 1 ∂u + ∂x R ∂θ

 +

  D66 3 ∂v ∂ 2w 1 ∂u − 2 − R2 2 ∂ x ∂ x∂θ 2R ∂θ

(5.56)

(5.57)

    D12 ∂v ∂ 2w ∂ 2w D11 ∂u D12 ∂v (5.58) + 2 + w − D11 2 + 2 − 2 = R ∂x R ∂θ ∂x R ∂θ ∂θ ∂ 2w ∂ 2w D22 ∂v − + ( ) ∂x2 R 2 ∂θ ∂θ 2     1 ∂u D66 3 ∂v ∂ 2w 1 ∂u D66 ∂v + + −2 − = R ∂x R ∂θ R 2 ∂x ∂ x∂θ 2R ∂θ L H C Mθθ = −D12

L H C Mxθ

(5.59)

(5.60)

After substitution of Eqs. (5.55)–(5.60) into Eqs. (5.52)–(5.54), the governing equations for nonlocal classical shell model in terms of the axial, the circumferential and the radial displacements of the mid-surface, (u, v, w), are obtained as 

      A12 ∂ A66 D12 ∂2 A66 3D66 ∂ 2 D66 A11 2 + u+ + + 3 − v − ∂x R2 4R 4 ∂θ 2 R R R 4R 3 ∂ x∂θ   D11 ∂ 3 D66 ∂ 3 A12 ∂ D12 ∂ 3 − − w + − R ∂x R ∂x3 R 3 ∂ x∂ 2 θ R 3 ∂ x∂ 2 θ   ∂ 2u ∂ 2u (5.61) = L H C ρh 2 + N¯ 2 ∂t ∂x     A66 ∂ A12 D66 ∂2 21D66 ∂ 2 A22 ∂ 2 + + u + A + + 66 R R 4R 3 ∂ x∂θ ∂x2 4R 2 ∂ x 2 R 2 ∂θ 2  2  3 3 3  A22 ∂ D12 ∂ D22 ∂ D22 ∂ 5D66 ∂ v+ − 2 − 4 w + 4 − 2 2 2 3 R ∂θ R ∂θ R ∂ x ∂θ R ∂θ R 2 ∂ x 2 ∂θ   ∂ 2v ∂ 2v = L H C ρh 2 + N¯ 2 (5.62) ∂t ∂x

5.2 Nonlocal Cylindrical Shell Models

99



   5D66 ∂ 3 D22 ∂ 3 D11 ∂ 3 2D12 ∂ 3 D66 ∂ 3 A12 ∂ A22 ∂ + + u + v + − − R ∂x3 R ∂x R 3 ∂ x∂θ 2 R 2 ∂ x 2 ∂θ R 2 ∂ x 2 ∂θ R 4 ∂θ 3 R 2 ∂θ   2D12 ∂ 4 4D66 ∂ 4 D22 ∂ 4 A22 ∂4 D12 ∂ 2 + − D11 4 − − − 4 − 2 w R2 ∂ x 2 ∂x R 2 ∂ x 2 ∂θ 2 R 2 ∂ x 2 ∂θ 2 R ∂θ 4 R   2 2 ∂ w ∂ w = L H C ρh 2 + N¯ (5.63) ∂t ∂x2

The appropriate boundary conditions applicable to Sanders’ shell theory at x = 0 and x = l are expressed as [13] N x x = 0 or u = 0 3 Mxθ = 0 or v = 0 N xθ + 2R ∂ Mx x 2 ∂ Mxθ + = 0 or w = 0 ∂x R ∂θ ∂w Mx x = 0 or =0 ∂x

(5.64)

It should be noted that the nonlocal classical shell model cannot be used to study the deformations of a thick cylindrical shell. However, this theory has been extensively employed due to its simplicity.

5.2.3 Nonlocal First-Order Shear Deformation Theory It is well-known that the classical shell theory is inadequate to model thick cylindrical shells when the thickness-to-radius ratio is greater than 0.05. Therefore, the effect of transverse shear deformation is significant for the thick cylindrical shells. Since some one-dimensional nanoscopic structures have high values of thickness-todiameter ratio, the nonlocal first-order shear deformation theory must be used to take into account the transverse shear deformation. In the first-order shear deformation shell theory, the third part of the Kirchhoff hypotheses is removed. As a result, the straight lines normal to the undeformed mid-surface do not remain normal and so the transverse shear strains γθ0z , and γx0z are considered in the theory. According to the first-order shear deformation theory, the displacement expressions for the nonlocal stress resultants can be expressed as L H C Nx x

  A12 ∂v D11 ∂βx D12 ∂βθ ∂u + +w + + 2 = A11 ∂x R ∂θ R ∂x R ∂θ L H C Nθθ = A12

  A22 ∂v ∂u + +w ∂x R ∂θ

(5.65)

(5.66)

100

5 Nonlocal Modelling of Nanoscopic Structures

 L H C N xθ = A66

L H C Mx x

∂v 1 ∂u + ∂x R ∂θ



   D66 1 ∂βx ∂βθ 1 ∂v 1 ∂u + + + − R R ∂θ ∂x 2R ∂ x R ∂θ (5.67)

  ∂βx D11 ∂u D12 ∂v D12 ∂βθ = + 2 + w + D11 + 2 R ∂x R ∂θ ∂x R ∂θ L H C Mθθ = D12

L H C Mxθ

D22 ∂βθ ∂βx + 2 ∂x R ∂θ

(5.68)

(5.69)

     1 ∂βx 1 ∂u ∂βθ 1 ∂v 1 ∂u D66 ∂v + + D66 + + − = R ∂x R ∂θ R ∂θ ∂x 2R ∂ x R ∂θ (5.70)   v 1 ∂w − L H C Q θ = S44 βθ + (5.71) R ∂θ R   ∂w L H C Q x = S55 βx + ∂x

(5.72)

By substituting Eqs. (5.65)–(5.72) into Eqs. (5.36)–(5.40), the equations of motion for the first-order shear deformation theory can be expressed in terms of displacements and rotations as     2  A12 ∂ A66 D66 ∂2 A66 ∂ 2 3D66 ∂ 2 A11 2 + 2 u + + − v − ∂x R ∂θ 2 4R 4 ∂θ 2 R R 4R 3 ∂ x∂θ     D11 ∂ 2 A12 ∂w D66 ∂ 2 D12 D66 ∂ 2 βθ βx + + + + + R ∂x R ∂x2 2R 3 ∂θ 2 R2 2R 2 ∂ x∂θ   2 2 ∂ u ∂ u (5.73) = L H C ρh 2 + N¯ 2 ∂t ∂x      A66 A12 ∂ 2 u 2D66 ∂ 2 A22 ∂ 2 S44 v + + A66 + + − R R ∂ x∂θ R2 ∂ x 2 R 2 ∂θ 2 R2     2D66 ∂ 2 A22 S44 ∂w 2D66 ∂ 2 βx S44 + + βθ + + + R2 R 2 ∂θ R 2 ∂ x∂θ R ∂x2 R   ∂ 2v ∂ 2v ¯ (5.74) = L H C ρh 2 + N 2 ∂t ∂x

5.2 Nonlocal Cylindrical Shell Models

101

    S44 ∂2 A12 ∂u A22 ∂v S44 ∂ 2 A22 − + S55 2 + 2 2 − 2 w − + 2 R ∂x R2 R ∂θ ∂x R ∂θ R  2 2  ∂ w ∂βx ∂ w S44 ∂βθ (5.75) + S55 + = L H C ρh 2 + N¯ 2 ∂x R ∂θ ∂t ∂x     D11 ∂ 2 D12 D66 ∂ 2 3D66 ∂ 2 v u + + + R ∂x2 2R 3 ∂θ 2 R2 2R 2 ∂ x∂θ     ∂w ∂2 D12 D66 ∂ 2 + − S + − S + D 55 11 55 βx R2 ∂x ∂x2 R 2 ∂θ 2  3 2    ρh ∂ βx D12 D66 ∂ 2 βθ (5.76) = LH C + + R2 R ∂ x∂θ 12 ∂t 2     3D66 ∂ 2 D66 D66 ∂ 2 u D12 ∂ 2 βx S44 S44 ∂w + + + + v − 2R 2 ∂ x∂θ 2R ∂ x 2 R R ∂θ R R ∂ x∂θ    3 2  ρh ∂ βθ ∂2 D22 ∂ 2 + D66 2 + 3 − S44 βθ = L H C (5.77) ∂x R ∂θ 2 12 ∂t 2 The boundary conditions for these equations at x = 0 and x=l are as follows [14]. N x x = 0 or u = 0 3 Mxθ 2R 1 ∂ Mxθ Qx + R ∂θ Mx x Mxθ

N xθ +

= 0 or v = 0 = 0 or w = 0 = 0 or βx = 0 = 0 or βθ = 0

(5.78)

5.3 Nonlocal Plate Models Two-dimensional nanoscale systems or structures such as thin films, nanosheets and nanoplates are usually modelled by the nonlocal plate models. Specially, the graphene sheets can be modelled as continuous and homogeneous nanoplates (Fig. 5.3), wherein the empty spaces between the atoms are ignored. The nonlocal plate models are approximation of the three-dimensional nonlocal elasticity to two dimensions. In this section, the equations of motion of nanoplates are derived using the two most common plate models: the Kirchhoff plate model and the ReissnerMindlin plate model. In this way, the local Kirchhoff and Reissner-Mindlin plate models are reformulated using the nonlocal differential constitutive relations. The Kirchhoff model is applicable to a thin plate when the ratio of its thickness to the smaller lateral dimension is less than 0.05. Furthermore, the effect of transverse shear deformation is neglected in the Kirchhoff model whereas the Reissner-Mindlin

102

5 Nonlocal Modelling of Nanoscopic Structures

Fig. 5.3 Molecular structure of a graphene sheet and its equivalent plate model

model considers the effects of both rotary inertia and transverse shear deformation. Therefore, the Reissner-Mindlin model is applicable to moderately thick plates. To drive the equations, we introduce the Cartesian coordinate system (x, y, z) so that x and y lie in the mid-plane of the nanoplate and z is measured from the mid-plane. Now, let us consider a transversely isotropic nanoplate with thickness h, length L x in the x-direction and width L y in the y-direction. The displacements of an arbitrary point on the nanoplate are   ∂w(x, y, t) ∂w(x, y, t) + f (z) + θx (x, y, t) ∂x ∂x   ∂w(x, y, t) ∂w(x, y, t) + f (z) + θ y (x, y, t) U y (x, y, z, t) = v(x, y, t) − z ∂y ∂y (5.79) Uz (x, y, t) = w(x, y, t)

Ux (x, y, z, t) = u(x, y, t) − z

where u, v and w are the displacements of the point (x, y, 0) along the x, y and z coordinates respectively, while θx and θ y respectively denote the rotations in x z and yz planes of a line originally normal to the mid-plane before deformation. Furthermore, f (z) is the shape function which determine the distribution of the transverse shear strains and stresses along the thickness. Different shape functions presented in the literature are discussed by Mantari et al. [15]. According to Eq. (3.9), the strains are    2  2 ∂θ y ∂ 2w ∂ 2w ∂u ∂ w ∂θx ∂v ∂ w − z 2 + f (z) , ε − z + = + f (z) + yy ∂x ∂x ∂y ∂y ∂x ∂x2 ∂ y2 ∂ y2     ∂ f (z) ∂w ∂ f (z) ∂w + θx , γ yz = + θy γx z = ∂z ∂x ∂z ∂y   ∂θ y ∂v ∂ 2w ∂w ∂θx ∂u + − 2z + f (z) 2 + + , εzz = 0 (5.80) γx y = ∂y ∂x ∂ x∂ y ∂ x∂ y ∂y ∂x εx x =

In addition, the nonlocal stress and moment resultants are defined as

5.3 Nonlocal Plate Models

103 h

2 {N x x , N yy , N x y } =

{tx x , t yy , tx y }dz

(5.81)

{tx x , t yy , tx y }zdz

(5.82)

− h2 h

2 {Mx x , M yy , Mx y } = − h2 h

2 {Q x , Q y } =

{tx z , t yz }dz

(5.83)

− h2

The governing equations for the nonlocal stress and moment resultants are presented as follows. ∂ Nx y ∂ 2u ∂ Nx x + = ρh 2 ∂x ∂y ∂t

(5.84)

∂ N yy ∂ Nx y ∂ 2v + = ρh 2 ∂x ∂y ∂t

(5.85)

  ∂ Qy ∂ Qx ∂ 2w ∂ 2w ∂ 2w ∂ 2w ¯ ¯ ¯ + − Nx x 2 + 2 Nx y + N yy 2 + qz = ρh 2 ∂x ∂y ∂ x∂ y ∂x ∂y ∂t

(5.86)

∂ Mx y ∂ Mx x ρh 3 ∂ 2 θx + − Qx = ∂x ∂y 12 ∂t 2

(5.87)

∂ M yy ∂ Mx y ρh 3 ∂ 2 θ y + − Qy = ∂x ∂y 12 ∂t 2

(5.88)

where ρ is the mass density of the plate, qz denotes the distributed transverse load and N¯ x x , N¯ yy and N¯ x y are the applied in-plane compressive and shear forces measured per unit length. Using Eqs. (4.38), (5.81), (5.82) and (5.83), the nonlocal constitutive equations for the orthotropic nanoplate are given by ⎛

⎞

⎛

⎞

⎛

∂u ∂x ∂v ∂y

⎞

Nx x A11 A12 0 ⎜ ⎟ ⎟ L H ⎝ N yy ⎠ = ⎝ A12 A22 0 ⎠ ⎜ ⎝ ⎠ Nx y 0 0 A66 ∂v ∂u + ∂y ∂x ⎛ 2 ∂ w x ⎛ ⎞ + ∂θ ∂x2 ∂x B11 B12 0 ⎜ ∂θ ⎜ ∂2w + ∂ yy + ⎝ B12 B22 0 ⎠ ⎜ ∂ y2 ⎝ 0 0 B66 2 x 2 ∂∂x∂wy + ∂θ + ∂y

⎞

∂θ y ∂x

⎟ ⎟ ⎟ ⎠

(5.89)

104

5 Nonlocal Modelling of Nanoscopic Structures

⎛

⎞

⎛

⎞

⎛

Mx x D11 D12 0 ⎜ ⎜ L H ⎝ M yy ⎠ = − ⎝ D12 D22 0 ⎠ ⎜ Mx y 0 0 D66 ⎝ ⎛

⎞

⎛

H11 H12 0 ⎜ ⎜ + ⎝ H12 H22 0 ⎠ ⎜ 0 0 H66 ⎝  LH



Qy Qx

 =

S44 0 0 S55

∂2w ∂x2 ∂2w ∂ y2

2 ∂∂x∂wy 2

⎞ ⎟ ⎟ ⎟ ⎠

∂2w ∂x2

+

∂2w ∂ y2

+

2 ∂∂x∂wy + 2

∂θx ∂y

  ∂w + θ  y ∂y ∂w ∂x

⎞

∂θx ∂x ∂θ y ∂y

+

∂θ y ∂x

⎟ ⎟ ⎟ ⎠

(5.90)

(5.91)

+ θx

where h

2 {Ai j , Bi j , Di j , Hi j } =

{1, f (z), z 2 , z f (z)}Ci j dz

(i, j = 1, 2, 6) (5.92)

− h2 h

2 Sii = K s − h2

∂ f (z) Cii dz ∂z

(i = 4, 5)

(5.93)

in which K s denotes the shear correction factor and the plane stress-reduced stiffness coefficients Ci j are expressed as νy E x Ey Ex , C12 = , C22 = , 1 − νx ν y 1 − νx ν y 1 − νx ν y = G yz , C55 = G x z , C66 = G x y

C11 = C44

(5.94)

Here, E x and E y represent the Young moduli in the x- and y-directions respectively, νx and ν y are the associated Poisson ratios, G x y is the in-plane shear modulus and G x z and G zy are the transverse shear moduli. In addition, L H is the Helmholtz operator in the Cartesian coordinate, i.e.,  LH = 1

− l x2y λ2

∂2 ∂2 + ∂x2 ∂ y2

 (5.95)

where l x y is an external characteristic length. Equations (5.84)–(5.88) can be solved in conjunction with the following boundary conditions:

5.3 Nonlocal Plate Models

105

(N x x − N¯ x x )n x + (N x y − N¯ x y )n y = 0 or u is specified (N x y − N¯ x y )n x + (N yy − N¯ yy )n y = 0 or v is specified     ∂w ∂w ∂w ∂w + N¯ x y n x − N¯ x y + N¯ yy n y = 0 or w is specified Q x n x + Q y n y − N¯ x x ∂x ∂y ∂x ∂y n 2x Mx x + n 2y M yy + 2n x n y Mx y = 0 (M yy −

Mx x )n x n y + (n 2x

− n 2y )Mx y

or θn is specified = 0 or θs is specified

(5.96)

where θx = n x θn − n y θs , θ y = n y θn + n x θs

(5.97)

and n x and n y are the direction cosines of the unit normal to the boundary of the mid-plane. Equations (5.79)–(5.97) are the general equations of the nonlocal plate model. In the following, for the two different plate theories, the equations of motion in terms of displacements are derived on the basis of the foregoing nonlocal relations.

5.3.1 The Kirchhoff Plate Theory In this section, we derive the nonlocal thin plate model, which is based on the Kirchhoff following hypothesis: (a) Straight lines, initially perpendicular to the mid-plane before deformation, remain straight after the deformation. (b) The mid-plane does not undergo in-plane deformation and so the mid-plane remains unstrained. (c) The effect of transverse shear deformation is neglected. This implies that the straight lines initially normal to the undeformed mid-plane rotate so that they remain normal to the deformed mid-plane. The Kirchhoff hypothesis results in f (z)=0 and so the transverse shear strains γx z and γ yz are identically zero. Furthermore, in this plate model, it is assumed that the effect of rotary inertia is negligible. By neglecting the rotary inertia term (ρh 3 /12), and eliminating the transverse shear stress resultants, the five governing differential equations (5.84)–(5.88) can be reduced to the following equations: ∂ Nx y ∂ 2u ∂ Nx x + = ρh 2 ∂x ∂y ∂t

(5.98)

∂ N yy ∂ Nx y ∂ 2v + = ρh 2 ∂x ∂y ∂t

(5.99)

∂ 2 M yy ∂ 2 Mx y ∂ 2 Mx x + + 2 + qz = ∂x2 ∂ x∂ y ∂ y2   ∂ 2w ∂ 2w ∂ 2w ∂ 2w ¯ ¯ ¯ + N yy 2 + ρh 2 Nx x 2 + 2 Nx y ∂x ∂ x∂ y ∂y ∂t

(5.100)

106

5 Nonlocal Modelling of Nanoscopic Structures

The equations of motion (5.98)–(5.100) can be expressed in terms of displacements by substituting for the force and moment resultants from Eqs. (5.89) and (5.90). For thin nanoplates, the equations of motion are  2    ∂ u ∂ 2u ∂ 2u ∂v2 ∂ 2v + A66 = L H ρh 2 + A11 2 + A12 ∂x ∂ x∂ y ∂ y2 ∂ x∂ y ∂t  2    ∂ u ∂ 2v ∂ 2v ∂ 2u ∂ 2v A66 + 2 + A12 + A22 2 = L H ρh 2 ∂ x∂ y ∂x ∂ x∂ y ∂y ∂t

(5.101)

(5.102)

∂ 4w ∂ 4w ∂ 4w + 2(D + 2D ) + D 12 66 22 ∂x4 ∂ x 2∂ y2 ∂ y4   2 2 2 ∂ w ∂ w ∂ w ∂ 2w ¯ ¯ ¯ + N yy 2 − qz = 0 (5.103) +L H ρh 2 + N x x 2 + 2 N x y ∂t ∂x ∂ x∂ y ∂y D11

It is noted that Eq. (5.103) is not coupled to Eqs. (5.101) and (5.102), and it can be solved separately.

5.3.2 The Reissner-Mindlin Plate Theory In the derivation of Eqs. (5.101)–(5.103), it is assumed that the effects of both rotary inertia and shear deformation are very small so they are neglected. To improve the Kirchhoff plate theory, several researchers have attempted to account for the effects of the rotary inertia and the shear deformation. Reissner [16] and Mindlin [17] made the most important advance in this way. In this section, we discuss the essential features of the nonlocal Reissner-Mindlin plate theory. In the Reissner-Mindlin plate theory, the third part of the Kirchhoff hypotheses is removed, implying that the straight lines do not remain normal to the mid-plane after deformation. Therefore, we have f (z) = z, and the displacement components are assumed to have the form Ux (x, y, z, t) = u(x, y, t) + zθx (x, y, t) U y (x, y, z, t) = v(x, y, t) + zθ y (x, y, t) Uz (x, y, t) = w(x, y, t)

(5.104)

Using the displacement components (5.104), the strain-displacement relationships (5.80), the nonlocal constitutive equations (5.89)–(5.91) and the governing equations (5.84)–(5.88), the nonlocal governing differential equations of motion for the nanoplates based on the Reissner-Mindlin plate theory are derived as A11

 2    ∂ u ∂ 2u ∂ 2u ∂ 2v ∂ 2v ρh + A = L + A + 12 66 H ∂x2 ∂ x∂ y ∂ y2 ∂ x∂ y ∂t 2

(5.105)

5.3 Nonlocal Plate Models

   ∂ 2u ∂ 2v ∂ 2v ∂u 2 ∂ 2v + 2 + A12 + A22 2 = L H ρh 2 A66 ∂ x∂ y ∂x ∂ x∂ y ∂y ∂t     ∂θ y ∂θx ∂ 2w ∂ 2w S55 + S + + 44 ∂x ∂x2 ∂y ∂ y2   ∂ 2w ∂ 2w ∂ 2w ∂ 2w = L H ρh 2 + N¯ x x 2 + 2 N¯ x y + N¯ yy 2 − qz ∂t ∂x ∂ x∂ y ∂y   ∂ 2θy ∂ 2θy ∂ 2 θx ∂ 2 θx D11 2 + D12 + D66 + ∂x ∂ x∂ y ∂ y2 ∂ x∂ y   3 2   ρh ∂ θx ∂w = LH −S55 θx + ∂x 12 ∂t 2  2  ∂ 2θy ∂ 2θy ∂ θx ∂ 2 θx D66 + D + + D 12 22 ∂ x∂ y ∂x2 ∂ x∂ y ∂ y2   3 2   ρh ∂ θ y ∂w = LH −S44 θ y + ∂y 12 ∂t 2

107



(5.106)

(5.107)

(5.108)

(5.109)

It should be noted that the local Reissner-Mindlin plate theory is recovered when the parameter λ is identically set to zero. Furthermore, we note that the higher-order shear deformation plate models have not been found to be necessary for modelling the nanoplate [18].

5.3.3 Circular Nanoplate So far, significant effort has been focused on studying the mechanical characteristics of the ultrathin films having rectangular geometries, whereas ultrathin films with a circular shape have also various potential applications [19, 20]. Furthermore, it has been shown that the ultrathin films with circular shape have different distribution of the geometrical and mechanical properties compared to the rectangular configuration [21]. Therefore, it is necessary to develop an understanding of the behaviour of circular nanoplates. Generally, the deformations of circular nanoplates are classified into axisymmetric and asymmetric problems. For axisymmetric case, the material properties, the boundary conditions, and the loading are rotationally symmetric, and the deformation is, therefore, independent of the angular coordinate and the problem is reduced to a one-dimensional one. Unlike the rectangular nanoplates, the general formulations in detail are not presented in this case. Here we only present the nonlocal axisymmetric formulations on the basis of the Kirchhoff hypothesis so that the reader can generalize the approach. Let us consider a typical flat circular nanoplate clamped or simply-supported on a circular hole of radius a (Fig. 5.4). It is assumed that the origin is at the center of the hole. For all points of the mid-plane with a radial distance r , there are two principal radii of curvature R1 and R2 that are defined as

108

5 Nonlocal Modelling of Nanoscopic Structures

Fig. 5.4 Schematic illustration of the circular nanoplate

1 ∂ 2w 1 1 ∂w =− 2, =− R1 ∂r R2 r ∂r

(5.110)

where w is defection in the downward direction. Based on the nonlocal differential constitutive relations in the polar coordinates, the nonlocal stress couple resultants are obtained as  2   2  ∂ w ν ∂w (Mrr − Mtt ) 1 ∂ Mrr 2 2 ∂ Mrr Mrr − a λ = −κ −2 (5.111) + + ∂r 2 r ∂r r2 ∂r 2 r ∂r  2   2  ∂ Mtt ∂ w 1 ∂w (Mrr − Mtt ) 1 ∂ Mtt Mtt − a 2 λ2 = −κ ν + 2 (5.112) + + ∂r 2 r ∂r r2 ∂r 2 r ∂r wherein Mrr and Mtt are the radial and circumferential resultant moments, respectively, κ is bending rigidity of the nanoplate and ν denotes the Poisson ratio. For axisymmetric deformation, the governing equations of motion for the circular nanoplate subjected to uniform radial load Nˆ r are given by   ∂ 2w ∂w ∂ ∂(r Q r ) ˆ = ρhr 2 + Nr r ∂r ∂t ∂r ∂r

(5.113)

∂ Mrr + Mrr − Mtt − r Q r = 0 ∂r

(5.114)

r

5.3 Nonlocal Plate Models

109

where Q r is the transverse shear resultant. By using Eqs. (5.111), (5.112) and (5.114), the nonlocal governing equation is obtained as  2  1 ∂ ∂(r Q r ) 1 ∂(r Q r ) 2 2 ∂ − −a λ + 2 ∂r ∂r 2 r ∂r r ∂r  2   2 ∂ w 1 ∂w ∂ 1 ∂ = −κr + + ∂r 2 r ∂r ∂r 2 r ∂r

(5.115)

By substituting Eq. (5.113) into Eq. (5.115), we have ρhr

     2  ∂ 2w ∂ 2w ˆ r ∂ r ∂w − a 2 λ2 ∂ − 1 ∂ + 1 ˆ r ∂ r ∂w ρhr + N + N ∂r ∂r r ∂r ∂r ∂r ∂t 2 ∂r 2 r2 ∂t 2  2   2 ∂ w ∂ 1 ∂ 1 ∂w = −κr + + (5.116) r ∂r r ∂r ∂r 2 ∂r 2

The boundary conditions involve specifying one member of each of the following two pairs   ∂w or w r Q r − Nˆ r ∂r ∂w (5.117) r Mrr or ∂r

5.4 Nonlocal Membrane Theory for Spherical Shell The mechanical characteristics of spherical shell-like nanoscopic structures, such as different fullerene molecules and spherical shell resonators, have been studied during the last decades motivated by both fundamental and technological applications. Furthermore, spherical shell-like structures are of very numerous occurrence in biological organisms such as biological cells, types of viruses and liposomes. These nanoscopic structures are usually represented by isotropic momentless spherical shells. We now consider the derivation of the nonlocal membrane theory for spherical shells which has been used in many numerical simulations of different organic as well as inorganic nanoscopic structures. The theory expounded in this section follows the work of Ghavanloo and Fazelzadeh [22]. For a spherical shell-like nanoscopic structure with median radius R and effective wall thickness h, we use the spherical coordinates φ and θ (Fig. 5.5), where φ is the meridional angular coordinate and θ is the circumferential angular coordinate. It is assumed that the effective wall thickness is sufficiently small and consequently the resultant bending and twisting moments are negligible. This is known as a membrane approximation. This physically implies that the mid-surface of the spherical shell undergoes an extension only.

110

5 Nonlocal Modelling of Nanoscopic Structures

Fig. 5.5 Coordinate system definitions for a spherical membrane

The equations of motion of the spherical shell with the membrane approximation, which is executing a motion that is independent of the circumferential coordinate θ , are ∂ Nφφ ∂ 2u + (Nφφ − Nθθ ) cot φ = Rρh 2 ∂φ ∂t Rqr − (Nφφ + Nθθ ) = Rρh

∂ 2w ∂t 2

(5.118)

(5.119)

in which ρ is the density of the shell, u and w denote the tangential and the normal displacements respectively, and qr is the normal pressure. Furthermore, Nθθ and Nφφ are the nonlocal stress resultants. Using Eqs. (4.100) and (4.101), they can be expressed as a function of strains, i.e.,  Nφφ − R λ

2 2

1 ∂ 2 Nφφ 2Nφφ cot φ ∂ Nφφ 2cot 2 φ − + − (Nφφ − Nθθ ) R 2 ∂φ 2 R 2 ∂φ R2 R2



Eh (εφφ + νεθθ ) (5.120) 1 − ν2   1 ∂ 2 Nθθ 2Nθθ cot φ ∂ Nθθ 2cot 2 φ − Nθθ − R 2 λ2 2 + + (N − N ) φφ θθ R ∂φ 2 R 2 ∂φ R2 R2 Eh (νεφφ + εθθ ) (5.121) = 1 − ν2

=

where E is the Young modulus of the spherical shell and ν denotes Poisson’s ratio. Using Eq. (3.11), the strain components in Eqs. (5.120) and (5.121) are

5.4 Nonlocal Membrane Theory for Spherical Shell

  1 ∂u +w R ∂φ 1 = (u cot φ + w) R

111

εφφ = εθθ

(5.122)

Substituting from Eqs. (5.120), (5.121) and (5.122) into Eqs. (5.118) and (5.119) we obtain the equations of motion for the spherical shell in terms of displacements as  2  E ∂ u ∂u ∂w 2 (1 + ν) + cot φ − (ν + cot φ)u + ρ(1 − ν 2 )R 2 ∂φ 2 ∂φ ∂φ   ∂ 4u ∂ 3u ∂ 2u ∂ 3w ∂ 2u 2 2 (5.123) + cot φ + 2 (1 − cot φ) + 2 2 = 2 −λ ∂t ∂φ 2 ∂t 2 ∂φ∂t 2 ∂t ∂t ∂φ     E ∂u qr ∂ 2w + 2w + u cot φ − − ρ R 2 (1 − ν) ∂φ ρh ∂t 2    2 4 ∂ 3w ∂ w 1 ∂qr 2 1 ∂ qr − = −λ − + cot φ ρh ∂φ 2 ∂φ 2 ∂t 2 ρh ∂φ ∂φ∂t 2   2 ∂ w qr (5.124) − 2 −2 ρh ∂t From Eqs. (5.123) and (5.124), it is easily seen that the local membrane theory for momentless spherical shell is recovered if the parameter λ is set to zero.

5.5 Nonlocal Models for Nanocrystals In the last decades, nanocrystals have occupied the centre stage in pure and applied research due to their novel physical properties. Nanocrystals vary in size, morphology, composition of the material, structure and function. Among various available morphologies for nanocrystals, the spherical and elongated nanocrystals are the most studied structures. In this section, we construct the general radial nonlocal formulation which can be used to simulate the dynamics of both spherical and elongated nanocrystals. Let us consider an elastically homogeneous and isotropic nanosphere and also a circular long nanocylinder both of radius R. A spherical coordinate system (r , φ, θ ) and a cylindrical coordinate system (r , θ , z) are utilized to define the stress and deformation fields for the nanosphere and the long nanocylinder, respectively. For the radial deformation, the nonzero component of displacement is w(r, t). Therefore, the strain-displacement relations in terms of the radial displacement are: ε11 =

∂w w w , ε22 = (k − 1) , ε33 = ∂r r r

(5.125)

112

5 Nonlocal Modelling of Nanoscopic Structures

where subscripts 1, 2 and 3 denote r , θ and z in the cylindrical coordinate system, and r , φ and θ in the spherical coordinate system. Furthermore, for the nanocylinder k = 1 whereas k = 2 for the nanosphere. Furthermore, based on the nonlocal differential constitutive relations, the nonlocal stress-strain relations in curvilinear coordinate systems are expressed by  σ11 − R λ

2 2

 ∂ 2 σ11 2k ∂w kc12 w k ∂σ11 − 2 (σ11 − σ22 ) = c11 + + ∂r 2 r ∂r r ∂r r  2  ∂ σ22 2k k ∂σ22 + 2 (σ11 − σ22 ) σ22 − R 2 λ2 + 2 ∂r r ∂r r ∂w c11 w w + + c12 (k − 1) = c12 ∂r r r

(5.126)

(5.127)

where c11 and c12 denote the elastic constants which are expressed in terms of Young’s modulus E and Poisson’s ratio ν as follows c11 =

E(1 − ν) Eν , c12 = (1 + ν)(1 − 2ν) (1 + ν)(1 − 2ν)

(5.128)

The equation of motion for propagating elastic waves is: ∂σ11 k(σ11 − σ22 ) ∂ 2w + + fr = ρ 2 ∂r r ∂t

(5.129)

wherein ρ is density of the nanocrystals and fr is the body force in radial direction. Substituting Eqs. (5.126) and (5.127) into Eq. (5.129), we arrive at: 

  2  ∂ 4w k k ∂ 3w k ∂ 2w k ∂ fr 2 2 ∂ fr −R λ − 2 fr + − + ρR λ ∂r 2 ∂t 2 r ∂r ∂t 2 r 2 ∂t 2 ∂r 2 r ∂r r  2  k ∂ w k ∂w ∂ 2w − 2 w − ρ 2 + fr = 0 +c11 + (5.130) ∂r 2 r ∂r r ∂t 2 2

It should be noted that the order of Eq. (5.130) with respect to the parameter r is 2 and it is the same as that for the local model.

References 1. S.S. Rao, Vibration of Continuous Systems (John Wiley & Sons Inc., New Jersey, 2007) 2. S.P. Timoshenko, On the correction for shear of the differential equation for transverse vibrations of prismatic bars. Philos. Mag. 41, 744–746 (1921) 3. S.G. Kelly, Advanced Vibration Analysis (CRC Press, London, 2006) 4. J.R. Hutchinson, Shear coefficients for Timoshenko beam theory. J. Appl. Mech. 68, 87–92 (2001) 5. L.H. Donnell, Beams, Plates, and Shells (McGraw-Hill, New York, 1976)

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6. J.L. Sanders, Nonlinear theories for thin shells. Q. Appl. Math. 21, 21–36 (1963) 7. W. Flügge, Stresses in Shells (Springer, Berlin, 1960) 8. V.V. Novozhilov, Foundations of the Nonlinear Theory of Elasticity (Graylock Press, New York, 1953) 9. C.W. Bert, V. Birman, Parametric instability of thick, orthotropic, circular cylindrical shells. Acta Mech. 71, 61–76 (1988) 10. B. Gu, Y.W. Mai, C.Q. Ru, Mechanics of microtubules modeled as orthotropic elastic shells with transverse shearing. Acta Mech. 207, 195–209 (2009) 11. C.L. Dym, Introduction to the Theory of Shells: Structures and Solid Body Mechanics (Pergamon Press, New York, 1974) 12. E. Carrera, S. Brischetto, P. Nali, Plates and Shells for Smart Structures: Classical and Advanced Theories for Modeling and Analysis (John Wiley & Sons Inc., West Sussex, 2011) 13. H. Chung, Free vibration analysis of circular cylindrical shells. J. Sound Vib. 74, 331–350 (1981) 14. A.W. Leissa, Vibrations of Shells (NASA SP-288, Washington, 1973) 15. J.L. Mantari, A.S. Oktem, C. Guedes Soares, A new trigonometric shear deformation theory for isotropic, laminated composite and sandwich plates, Int. J. Solids Struct. 49, 43–53 (2012) 16. E. Reissner, The effect of transverse shear deformation on the bending of elastic plates. J. Appl. Mech. 12, A69–A77 (1945) 17. R.D. Mindlin, Influence of rotary inertia and shear on flexural motions of isotropic elastic plates. J. Appl. Mech. 18, 31–38 (1951) 18. B. Arash, Q. Wang, A review on the application of nonlocal elastic models in modeling of carbon nanotubes and graphenes. Comput. Mater. Sci. 51, 303–313 (2012) 19. B.A. Gurney, E.E. Marinoro, S. Pisana, Tunable graphene magnetic field sensor, US Patent 2011/0037464 A1 (2011) 20. A.M. Eriksson, D. Midtvedt, A. Croy, A. Isacsson, Frequency tuning, nonlinearities and mode coupling in circular mechanical graphene resonators. Nanotechnology 24, 395702 (2013) 21. F. Scarpa, S. Adhikari, A.J. Gil, C. Remillat, The bending of single layer graphene sheets: the lattice versus continuum approach. Nanotechnology 21, 125702 (2010) 22. E. Ghavanloo, S.A. Fazelzadeh, Nonlocal shell model for predicting axisymmetric vibration of spherical shell-like nanostructures. Mech. Adv. Mater. Struct. 22, 597–603 (2015)

Chapter 6

Elastic Properties of Carbon-Based Nanoscopic Structures

Despite the computational efficiency and the reasonable accuracy associated with the use of nonlocal elasticity theory, it requires a prior knowledge of the mechanical properties (e.g. elastic constants) and physical dimensions (e.g. effective thickness) of the system in order for the theory to be applied properly. Furthermore, there is no consent among different researchers regarding the choice of the nonlocal parameter. There has been disagreement on the adopted values of the nonlocal parameter and those of the mechanical properties, and as a result a number of inconsistencies are observed in the literature. In this chapter, we first provide a review of previous studies on the estimation of Young’s modulus of carbon nanotubes and then discuss the underlying reasons for the observed inconsistencies in the literature, and then clarify some of the long standing controversial issues related to the question of the effective wall thickness of carbon nanotubes. In this connection, we offer a discussion about the dispersion in the computed mechanical properties of carbon nanotubes, which has been attributed to the uncertainty in the value of their thickness, normally referred to as Yakobson’s paradox. Next, we present an accurate molecular mechanics-based model for predicting anisotropic elastic constants and the nonlocal parameters of the SWCNTs having an arbitrary chirality. Furthermore, the computed Young modulus of different types of nanoscopic structures including graphene sheets and fullerene molecules will be presented.

6.1 Young’s Modulus and Effective Wall Thickness of SWCNTs To comprehend the mechanical characteristics of a solid, one often employs specific quantities defined in continuum mechanics such as elastic moduli. The definition of the elastic moduli for a solid implies a spatial uniformity of the material, at least in a statistically averaged sense. However, it is well-known that a carbon nanotube © Springer Nature Switzerland AG 2019 E. Ghavanloo et al., Computational Continuum Mechanics of Nanoscopic Structures, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-3-030-11650-7_6

115

116

6 Elastic Properties of Carbon-Based Nanoscopic Structures

has a discrete atomistic structure whose wall comprises only of a limited number of atoms and so does not form a continuous spatial distribution. Nevertheless, to apply the existing continuum models and numerical techniques developed for continuum bodies to the carbon nanotubes, basic mechanical quantities such as Young’s and shear moduli, Poisson’s ratio and wall thickness have to be unambiguously defined [1]. 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 Δl F =E A l

(6.1)

where Δl is a small change of length, and the relative extension, Δl/l, is the unidirectional strain along the direction of the applied force. Experimental techniques are now available that allow for the application and measurement of forces in the pico- to micro-Newton range, so as observation can made of change of length in the Angstroms-to-micrometer range [2]. Hence, to estimate the Young modulus of the carbon nanotubes, the cross-section area A must be chosen accurately. For the carbon nanotubes, the cross-section area can be defined as 2π Rh in which R is the average radius of the nanotube and h is its effective wall thickness. The value of the radius is reasonably calculated by theoretical predictions (Eq. (2.6)) or experimental measurements. However, there is no consensus regarding the exact values of the effective thickness since a nanotube does not have a continuous wall. In some studies, it is assumed that the thickness of the nanotube is equal to the inter-planar spacing of two graphite layers (= 3.4 Å) [3–5]. This assumption may be of merit for MWCNTs, but it is rather ambiguous for SWCNTs. In contrast, Yakobson et al. [6] have determined the effective thickness of carbon nanotubes to be equal to 0.066 nm. Unfortunately, considerable inconsistencies have developed in the literature, wherein the reported values for the effective thickness of the SWCNTs has varied from 0.0617 to 0.69 nm [7, 8]. This large variations generates values of Young’s modulus of the SWCNTs ranging from 1 to 5.5 TPa, although most researchers agree that the Young modulus is approximately 1 to 2 TPa [3, 9]. Furthermore, the inconsistency in the reported values for Poisson’s ratio is large, varying from 0.12 to 0.28 [10]. Table 6.1 lists a few representative values of the effective wall thickness and the corresponding Young’s modulus of the SWCNTs which have been reported in the literature [3–6, 10–14]. As is seen from this table, although the values of Young’s modulus and the wall thickness are scattered in a rather wide range, the magnitude of the parameter Eh, known as the in-plane stiffness or surface Young’s modulus, is distributed in the vicinity of the value Eh = 360 J m−2 . Since the in-plane stiffness can be determined without knowing E and h separately, the variations resulting from the ill-defined nanotube thickness can be eliminated. Specifically, without defining the effective thickness of the SWCNTs, the in-plane stiffness can be directly calculated via various techniques such as the energy and the force methods [8].

6.1 Young’s Modulus and Effective Wall Thickness of SWCNTs

117

Table 6.1 The reported effective wall thickness and Young’s modulus of SWCNTs Authors Wall thickness h (nm) Young’s modulus E Eh (J m−2 ) (TPa) Krishnan et al. [3] Belytschko et al. [4] Li and Chou [5] Yakobson et al. [6] Vodenitcharova and Zhang [10] Lu [11] Zhou et al. [12] Pantano et al. [13] Cai et al. [14]

0.34 0.34 0.34 0.066 0.0617

1.25 0.94 1.01 5.5 4.88

425 320 343 363 300

0.34 0.074 0.075 0.32

0.974 5.1 4.84 1.06

332 377 363 340

In 2003, Vodenitcharova and Zhang [10] proposed a necessary condition namely that the effective wall thickness of SWCNTs cannot be greater than or equal to the theoretical diameter of a carbon atom (0.142 nm). Their argument was that the forces in a real SWCNT are experienced by a limited number of atoms, while in a continuum mechanics-based model the same forces must be experienced by a continuous wall. Hence, the effective wall thickness must be smaller than the atomic diameter; otherwise, the nanotube equilibrium state cannot be maintained [15]. Based on the Vodenitcharova and Zhang criterion, any of the previously assumed wall thicknesses larger than 0.142 nm should be excluded from future investigations.

6.2 Anisotropic Elastic Constants of SWCNTs The above mentioned previous investigations were mainly concerned about two fundamental elastic properties of a mechanical system, namely the longitudinal Young’s modulus and the Poisson ratio. In these studies, the SWCNT was regarded as an isotropic elastic medium characterized by two independent elastic constants. In isotropic models the elastic anisotropy due to curved structure of the nanotube is neglected. It is, however, obvious that due to the geometrical arrangement of the carbon atoms in an SWCNT, its carbon bonds in the longitudinal and circumferential directions are not deformed in the same manner. It is interesting to note that since the armchair and zigzag SWCNTs are symmetric about the longitudinal direction, then they exhibit a transverse-isotropic behaviour in the plane orthogonal to the longitudinal axis [16, 17]. For chiral nanotubes, however, since their carbon bonds are not symmetric about the longitudinal axis, they display a fully anisotropic behaviour in their mechanical properties. Generally, we can state that the SWCNT chirality induces the presence of anisotropy in its elastic properties. The modified Born rule was employed by Wu et al. [18] to formulate a finitedeformation shell theory based directly on the interatomic potential. It was shown

118

6 Elastic Properties of Carbon-Based Nanoscopic Structures

that those SWCNTs whose diameter is less than 1 nm display anisotropic mechanical properties. On the basis of the theory developed by Wu et al. [18], it has been argued by Peng et al. [19] that conventional isotropic shell models can not represent the SWCNTs since their constitutive relation embodies the coupling of tension and curvature on one hand, and bending and strain on the other. The orthotropic in-plane stress-strain relation for zigzag and armchair SWCNTs was employed by Ru [20] to formulate an anisotropic elastic shell model. This relation can shed light on the dependence of mechanical behaviour of small- radius nanotubes on chirality. A two-dimensional continuum model has been proposed in Ref. [21] which is capable of describing the anisotropic properties arising from the nanotube’s chirality. This model provides an insight into the axial-strain-induced torsion of chiral SWCNTs having actual sizes. An anisotropic shell model has also been developed on the basis of molecular mechanics for predicting the mechanical behaviour of SWCNTs [22]. The model couples the axial, the circumferential, and the torsional strains in SWCNTs. Here, we are concerned with obtaining the elastic properties of SWCNTs, having an arbitrary chirality, on the basis of molecular mechanics. To derive these properties, we adopt the approach given in Ref. [22]. The accurate modelling of the interatomic interaction between the carbon atoms is essential for the derivation of the anisotropic elastic properties of SWCNTs. Molecular mechanics is the standard methodology to describe these interactions. In this method, the total molecular potential energy UT of a molecular system is expressed as a sum of distinct bonded and non-bonded interactions [23]: UT = Uρ + Uθ + Uω + Uτ + UvdW + Ues

(6.2)

The bonded interactions refer to bond stretching, Uρ , and angular distortions, including bond angle bending, Uθ , bond inversion, Uω , and angle torsion, Uτ , while the non-bonded interactions refer to the van der Waals, UvdW and electrostatic, Ues terms. It is to be noted that for an SWCNT subject to an axial loading at small strain, it can be assumed that only the two energy terms namely, the bond stretching and angle variation terms, play significant role in the total molecular potential energy, and that the contribution of other terms may be neglected. Furthermore, near the equilibrium position, it has been shown via chemical calculations that the harmonic energy functions present a reasonable approximation to the potential energy of atomic interactions [24]. Consequently, the total potential energy function UT pertinent to the SWCNT can be expressed as U T = Uρ + Uθ =

1 i

2

K C−C (dri )2 +

1 j

2

CC−C (dθ j )2

(6.3)

where dri is the extension of the bond i and dθ j is the change in the bond angle j (Fig. 6.1). Moreover, K C−C is an elastic stiffness that accounts for the forcestretch relationship of the C–C bond, and CC−C is referred to as the effective stiffness associated with the angular distortion of the bond. These stiffness constants can be

6.2 Anisotropic Elastic Constants of SWCNTs

119

Fig. 6.1 Interatomic interaction in molecular mechanics theory: (a) elongation of the bond and (b) variation of the bond angle

obtained from ab initio quantum mechanical-based computations, or from empirical molecular potentials, or from the experimental data. For a typical atom, it forms three chemical bonds r1 , r2 , r3 and three associated bond angles θ1 , θ2 , θ3 (Fig. 6.2). The application of external loads will change the bond lengths by amounts dr1 , dr2 , dr3 and the three bond angles by the amounts dθ1 , dθ2 , dθ3 . The connections relating the external loads and the change in bond lengths and bond angles can be established via force equilibrium and the geometry of the SWCNT. Force equilibrium of bond extension and the moment equilibrium of three bonds lead to p = K C−C dr

(6.4)

Fig. 6.2 Schematic representation of a chiral type SWCNT (a) global structure, (b) side view and (c) top view of the local structure

120

6 Elastic Properties of Carbon-Based Nanoscopic Structures

q=

2CC−C ∗ M d aC−C

(6.5)

where p = ( p1 , p2 , p3 )T and q = (q1 , q2 , q3 )T are respectively the internal forces along and perpendicular to the C–C bonds, dr = (dr1 , dr2 , dr3 )T and d = (dφ1 , dφ2 , dφ3 )T . Furthermore, the matrix M∗ is defined by Mi∗j = −Nik∗ N ∗jk , i, j, k = 1, 2, 3 (sum over k)

(6.6)

where  Ni∗j

=

cos φi sin φk cos ψ j −sin φ j cos φk sin θ j

0

i = j = k i= j

(6.7)

The structural parameters θi , φi and ψi are depicted in Fig. 6.2 and are defined as functions of chiral indecis (n, m):  φ1 = arccos

2n + m



4π 2π , φ2 = + φ1 , φ 3 = + φ1 √ 2 2 3 3 2 n + nm + m π ψ1 = √ cos φ1 , 2 n + nm + m 2   π π + φ1 , cos ψ2 = √ 3 n 2 + nm + m 2   π π ψ3 = √ − φ1 cos 3 n 2 + nm + m 2

θi = arccos(sin φ j sin φk cos ψi + cos φ j cos φk ), i = j = k

(6.8)

(6.9) (6.10)

The axial, circumferential and shear strains can be calculated on the basis of the geometrical deformation of the SWCNT as follows [25]: εx x =

d[(2n + m)r1 cos φ1 − (n − m)r2 cos φ2 − (2m + n)r3 cos φ3 ] (6.11) √ 3aC−C n 2 + mn + m 2 εθθ =

2εxθ = γxθ =

d[mr1 sin φ1 − (n + m)r2 sin φ2 + nr3 sin φ3 ] √ √ 3aC−C n 2 + mn + m 2

(6.12)

d[(2n + m)r1 sin φ1 − (n − m)r2 sin φ2 − (2m + n)r3 sin φ3 ] √ 3aC−C n 2 + mn + m 2 (6.13)

Equations (6.11)–(6.13) cab be rewritten in a matrix form as ε=

1 B∗ p + J∗ d K C−C aC−C

(6.14)

6.2 Anisotropic Elastic Constants of SWCNTs

121

where ε = (εx x , εθθ , 2εxθ )T with B∗ and J∗ given by ⎞ ⎛ (2n√+ m) cos φ1 −(n − m) cos φ2 −(2m + n) cos φ3 √ √ 1 ⎠ (6.15) ⎝ B∗ = 3m sin φ1 − 3(n + m) sin φ2 3n sin φ3 Δ(n, m) (2n + m) sin φ1 −(n − m) sin φ2 −(2m + n) sin φ3 ⎞ ⎛ −(2n + m) sin φ1 √(n − m) sin φ2 (2m + n) sin φ3 √ √ 1 ⎠ (6.16) ⎝ J∗ = 3m cos φ1 − 3(n + m) cos φ2 3n cos φ3 Δ(n, m) (2n + m) cos φ1 −(n − m) cos φ2 −(2m + n) cos φ3

√ where Δ(n, m) = 3 n 2 + mn + m 2 . Employing Eq. (6.14), p can be expressed as a function of ε and d, p = K C−C aC−C B∗ −1 (ε − J∗ d)

(6.17)

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

(6.18)

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

(6.19)

The compatibility equations for a stress-free SWCNT are expressed as d[mr1 cos φ1 − (n + m)r2 cos φ2 + nr3 cos φ3 ] = 0

(6.20)

Rewriting Eqs. (6.18)–(6.20) in the matrix form leads to the following relation U∗ p + V∗ q + K C−C aC−C W∗ d = 0

(6.21)

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 V∗ = ⎝ sin φ1 sin φ2 sin φ3 ⎠ 0 0 0 ⎛ ⎞ 0 0 0 ⎠ 0 0 0 W∗ = ⎝ −m sin φ1 (n + m) sin φ2 −n sin φ3

(6.22)

(6.23)

(6.24)

Substitution of Eqs. (6.5) and (6.17) into Eq. (6.21), leads to  d = U∗ B∗ −1 J∗ −

2CC−C V∗ M∗ − W ∗ 2 K C−C aC−C

−1

U∗ B∗ −1 ε ≡ F∗ ε

(6.25)

122

6 Elastic Properties of Carbon-Based Nanoscopic Structures

The internal forces can be obtained as a function of the strains by substituting Eq. (6.25) into Eq. (6.17) giving p = K C−C aC−C B∗ −1 (I − J∗ F∗ )ε ≡ K C−C aC−C G∗ ε

(6.26)

wherein I is the identity matrix. From Eqs. (6.4) and (6.26), the changes in bond lengths can be obtained as dr = aC−C G∗ ε

(6.27)

Furthermore, the changes in bond angles dθ are given by dθ = Q∗ d

(6.28)

Q i∗j = −N ∗ji

(6.29)

where

By substituting Eq. (6.25), we have dθ = Q∗ F∗ ε ≡ R∗ ε

(6.30)

The surface density of the strain energy, i.e., the strain energy per unit area, can be obtained by the strain energy per atom divided by the area of an atom, i.e. 1 2

Π0 =

3

i=1

1 K (dri )2 2 C−C

+

3

j=1

1 C (dθ j )2 2 C−C

√ 2 3 3aC−C 4

(6.31)

The factor 1/2 inserted before the bond stretching energy of the representative atom is because the energy of bond stretching is shared between two atoms connected by the bond. Substituting Eqs. (6.27) and (6.30) into Eq. (6.31), the relationship between Π0 and ε is obtained as Π0 =

K C−C T ∗ T ∗ 2CC−C ε T R∗ T R∗ ε) √ (ε G G ε + 2 K C−C aC−C 3 3

(6.32)

The closed-form expression for the anisotropic surface elastic constants of the SWCNT is given by Yi j =

∂ 2 Π0 2 CC−C ∗ ∗ = √ (K C−C G li∗ G l∗j + 2 2 R R ) ∂εi ∂ε j aC−C li l j 3 3

(6.33)

In some special cases, including the zigzag (m = 0) and armchair (m=n) nanotubes and graphene sheets (m, n → ∞ ), a simple expression is obtained from Eq. (6.33)

6.2 Anisotropic Elastic Constants of SWCNTs

⎛ ⎜ ⎜ Y=⎜ ⎝

123

K C−C (λ∗ +3μ∗ ) √ 2 3(λ∗ +μ∗ )

K C−C (λ∗ −μ∗ ) √ 2 3(λ∗ +μ∗ )

0

K C−C (λ∗ −μ∗ ) √ 2 3(λ∗ +μ∗ )

K C−C (λ∗ +3μ∗ ) √ 2 3(λ∗ +μ∗ )

0

0

0

⎞

∗

K μ √ C−C ¯ ξ¯ μ∗ ) 3(λ+

⎟ ⎟ ⎟ ⎠

(6.34)

where for the zigzag and the armchair nanotubes, and the graphene sheet, we have respectively ⎧ 5−3 cos(π/n) ⎪ λ∗ = 14−2 ⎪ cos(π/n) ⎪ ⎪ ⎪ ⎪ C C−C ⎪ μ∗ = ⎨ K a2 C−C C−C

1 ⎪ ⎪ λ¯ = 6cos2 (π/2n) ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ξ¯ = [1+2 cos(π/2n)]2 9cos2 (π/2n) ⎧ ∗ 7−cos(π/n) λ = 34+2 ⎪ cos(π/n) ⎪ ⎪ ⎪ ⎪ C ⎪ ∗ C−C ⎪μ = ⎨ K a2 C−C C−C

⎪ ⎪ λ¯ = ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ξ¯ = [2+cos(π/2n)]2 [1+2 cos(π/2n)]2 ⎧ ∗ λ = 16 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ μ∗ = CC−C2 K a

4−cos2 (π/2n) 2[1+2 cos(π/2n)]2

C−C C−C

⎪ ⎪ λ¯ = 16 ⎪ ⎪ ⎪ ⎩ ξ¯ = 1

(6.35)

For the computation of the numerical results, the required basic quantities that must be appropriately defined are the elastic constants associated with the bond stretching and angular distortion of the bond, whose values are listed as K C−C = 742 N/m and CC−C = 1.42 nN nm respectively [23]. Figure 6.3 displays the variations of surface elastic constants of the SWCNTs having different chiralities with diameter. It can be seen that while the surface Young’s modulus Y11 (= Y22 ) and the surface shear modulus (Y33 ) increase, the Y12 (= Y21 ) modulus decreases rather fast for small radii nanotubes when the diameter is increased. Furthermore, while the surface Young’s modulus for zigzag nanotubes is much more sensitive to the diameter compared to that for armchair nanotubes, the surface shear modulus for armchair nanotubes is much more sensitive to the diameter compared to that for zigzag nanotubes. On the other hand, by increasing the nanotube’s chiral angle, Y11 (= Y22 ) increases while Y33 and Y12 (= Y21 ) is reduced. It is also seen that Y13 (= Y31 = −Y23 = −Y32 ) is indeed very small (two orders smaller

124 Fig. 6.3 Chirality dependent anisotropic elastic properties of SWCNTs. (a) Y11 (= Y22 ), (b) Y12 (= Y21 ), (c) Y13 (= Y31 = −Y23 = −Y32 ), and (d) Y33

6 Elastic Properties of Carbon-Based Nanoscopic Structures

6.2 Anisotropic Elastic Constants of SWCNTs

125

than others). As can be expected, the chirality- and size-dependence may be ignored when the diameter is larger than 2.0 nm, and the elastic properties approach the limit values for graphene sheets.

6.3 Nonlocal Parameter for Carbon Nanotubes An important issue in the field of nonlocal elasticity theory is the estimation of the length scale coefficient or the nonlocal parameter. Compared to the strong interest shown in improving the nonlocal structural models for carbon nanotubes, the estimation of this parameter, e0 , has received scant attention. Obtaining an estimate of this parameter forms one of the most important steps in the application of nonlocal elasticity theory to nanoscopic structures. Due to the complexity in performing the experiments to obtain the value of this parameter, different theoretical approaches including Lattice Dynamics (LD) [26, 27], MD simulations [28–30] and molecular structural mechanics [31–33] have been used to obtain an estimate of this parameter in various nonlocal models. Initially, Eringen [26] proposed the value of e0 ≈ 0.39 employing the matching of the dispersion curves using the plane wave nonlocal theory, and the Born-Karman model of lattice dynamics applied at the Brillouin zone boundary. Furthermore, Eringen also proposed the value of e0 ≈ 0.31 in his investigation of Rayleigh surface wave employing the nonlocal continuum mechanics and lattice dynamics. Using MD simulations of the SWCNTs, Wang and Hu [34] recommended the value of √ e0 = 1/ 12 ≈ 0.288 for the wave propagation in the nonlocal Euler and Timoshenko beam models. Zhang et al. [35] evaluated the value of 0.82 for this parameter by comparing the theoretical buckling strain obtained via the nonlocal thin shell model and strains obtained via molecular mechanics simulations. On the basis of the wave propagation analysis, Wang [36] estimated that e0 L i < 2.0 nm for frequencies greater than 10 THz. It was found that the adopted value is dependent on the crystal structure in the lattice dynamics. By curve fitting the results from MD simulations and the results obtained from the nonlocal Donnell model for buckling of SWCNTs, the values of e0 for different chiral angles have been predicted [37]. The range of values obtained was from a minimum of 0.546 for a (15, 4) chiral shell to a maximum of 1.043 for a (11, 9) chiral shell. The radial shell buckling of the MWCNTs was analysed by adopting the average stress in the equilibrium equations, with a value e0 ≈ 0.456 [38]. Comparison between the nonlocal Timoshenko beam theory and the MD simulation results for free vibration [28] has shown that the values of e0 ranged between 0 and 19 depending on the aspect ratio, boundary conditions and mode shapes of the nanoscopic structures. Furthermore, it was physically explained that e0 allocates a part of the kinetic energy to the strain energy. The values of e0 for SWCNTs and DWCNTs were obtained [39] by comparison of the wave dispersion obtained from the nonlocal Flügge shell theory and MD simulation. The estimated value of e0 for transverse wave in (10,10)@(15,15) DWCNT was found to be 0.6. Furthermore, values of e0 = 0.2 and 0.23 were predicted for propagation of torsional wave in (10,10)

126

6 Elastic Properties of Carbon-Based Nanoscopic Structures

and (15,15) SWCNTs, respectively. Employing the MD simulation results, Zhang et al. [40] adopted the e0 value as 0.3 and 3 respectively in the nonlocal shell and beam models to predict the critical strains of axially compressed SWCNTs. In another study [41], the results from the MD simulations were matched with those from the nonlocal shell model, and the nonlocal parameter for vibrations of DWCNTs with different boundary conditions was extracted. It was concluded that the chirality does not have an appreciable effect on the calibrated values of e0 . It should be noted, however, that further theoretical studies [29, 31, 33] have contradicted this conclusion. Modelling via molecular structural mechanics has led to derivation of explicit expressions pertaining to e0 of the zigzag and armchair SWCNTs [31]. The values obtained were respectively 0.3221 and 0.3458 for the zigzag and armchair SWCNTs. In a further study, the following relations were obtained for e0 of the zigzag and armchair CNTs employing the bending characteristics of the CNTs [29], i.e., For zigzag SWCNTs: For armchair SWCNTs:

R + 0.2024 aC−C R e0 = 0.052 + 0.1528 aC−C

e0 = 0.0297

(6.36)

where the details for obtaining these relations are given in Ref. [29]. The Padé approximantion and several beam theories were employed to obtain simple analytical expressions for e0 related to geometrical properties and vibration modes √ [42]. It was √ found that the value of e0 for the vibration of CNTs ranges from 1/ 12 to 1/ 6 depending on the geometry and vibration modes. The first systematic attempt to develop a new formulation for predicting the value of e0 of the SWCNTs having arbitrary chirality was made by Ghavanloo and Fazelzadeh [33]. It was concluded that rather than adopting a fixed value for the e0 , its predicted values depended on the tube’s chirality, diameter, the aspect ratio and the wave mode shapes. In the next section, we present the derivation of e0 for SWCNTs having an arbitrary chirality as given in Ref. [33]. On the basis of what has been reviewed above, it can be stated that the magnitudes of the e0 are scattered owing to the effects of various factors such as the nanotube’s diameter, its chirality, its aspect ratio (length/diameter) and its boundary conditions. Furthermore, e0 is different for diverse physical problems and there is no rigorous methodology developed so far to estimate it, while the extension of the previous methods to different cases being quite difficult.

6.4 Evaluation of the Nonlocal Parameter for SWCNTs In this section, we consider how to estimate the Eringen’s nonlocal parameter for SWCNTs having an arbitrary chirality. To begin with, let us first state the nonlocal constitutive relations for an anisotropic SWCNT in polar coordinate, i.e.,

6.4 Evaluation of the Nonlocal Parameter for SWCNTs

127



tx x

 ∂ 2 tx x 1 ∗ 1 ∂ 2 tx x ∗ ∗ = (E 11 − (e0 L i ) + 2 εx x + E 12 εθθ + E 13 εxθ ) (6.37) ∂x2 R ∂θ 2 h  2  ∂ tθθ 1 ∂ 2 tθθ 2 + − 2 tθθ tθθ − (e0 L i )2 ∂x2 R 2 ∂θ 2 R 1 ∗ ∗ ∗ εx x + E 22 εθθ + E 23 εxθ ) (6.38) = (E 12 h  2  ∂ txθ 1 ∂ 2 txθ 1 + − t txθ − (e0 L i )2 xθ ∂x2 R 2 ∂θ 2 R2 1 ∗ ∗ ∗ = (E 13 εx x + E 23 εθθ + E 33 εxθ ) (6.39) h 2

where the constants E i∗j are given by [20] ⎛

⎞

⎛

H¯ 1 H¯ 2 0 ⎜ E∗ ⎟ ⎜ 0 H¯ 1 − H¯ 2 ⎜ 22 ⎟ ⎜ ⎜ ∗ ⎟ ⎜ ⎜ ⎜ E 12 ⎟ ⎜ H¯ 4 0 0 ⎜ ⎟ ⎜ E∗ ⎟ = ⎜ ¯ 0 H5 0 ⎜ 33 ⎟ ⎜ ⎜ ∗ ⎟ ⎜ ⎝ E 13 ⎠ ⎜ ⎝ 0 0 −0.5 H¯ 2 ∗ E 23 0 0 −0.5 H¯ 2 ∗ E 11

H¯ 3 H¯ 3

0

⎞

⎞ ⎛ ⎟ 1 0 ⎟ ⎟ ⎜ sin 2ϑ ⎟ ⎟ ⎜ − H¯ 3 0 ⎟ ⎟ ⎜ cos 2ϑ ⎟ ⎟ ⎟⎜ ⎟ ⎜ − H¯ 3 0 ⎟ ⎟ ⎝ cos 4ϑ ⎠ ⎟ ¯ 0 − H3 ⎠ sin 4ϑ 0 H¯ 3

(6.40)

∗ and all constants H¯ i (i = 1, 2, 5) are related to four in-plane elastic constants (C11 , ∗ ∗ ∗ C12 , C22 and C33 ) via

1 ∗ ∗ ∗ ∗ + 3C22 + 2C12 + 4C33 ) H¯ 1 = (3C11 8 1 ∗ ∗ − C22 ) H¯ 2 = (C11 2 1 ∗ ∗ ∗ ∗ + C22 − 2C12 − 4C33 ) H¯ 3 = (C11 8 1 ∗ ∗ ∗ ∗ + C22 + 6C12 − 4C33 ) H¯ 4 = (C11 8 1 ∗ ∗ ∗ ∗ + C22 − 2C12 + 4C33 ) H¯ 5 = (C11 8

(6.41) (6.42) (6.43) (6.44) (6.45)

Here, the four independent elastic constants are specified by ∗ ∗ = C22 = C11

Eh Ehν Eh ∗ ∗ , C12 = , C33 = 1 − ν2 1 − ν2 2(1 + ν)

(6.46)

It is to be noted that a chiral SWCNT has a chiral angle difference of less than π/12 relative to either a zigzag or an armchair SWCNT. Therefore, ϑ represents the

128

6 Elastic Properties of Carbon-Based Nanoscopic Structures

rotation angle between a chiral SWCNT and either a zigzag or an armchair SWCNT (−π /12 ≤ ϑ ≤ π/12 ), in either the clockwise or the counter clockwise direction. Consequently, according to this definition and Eq. (2.7), the rotation angle is given by ϑ = cos

−1





2n + m

−

√ 2 n 2 + mn + m 2

π 12

(6.47)

Equations (6.37)–(6.39) can be rewritten in the matrix form as 1 ∗ E ε h

L∗ t =

(6.48)

where t = (tx x , tθθ , txθ )T and  2 ⎧ 2 ∂ ⎪ 1 − (e L ) + 0 i ⎪ ∂x2 ⎪ ⎪ ⎪  2 ⎪ ⎪ ⎨ 1 − (e0 L i )2 ∂ 2 + ∂x ∗ Lij =  2 ⎪ ⎪ ⎪ 1 − (e0 L i )2 ∂∂x 2 + ⎪ ⎪ ⎪ ⎪ ⎩ 0

1 ∂2 R 2 ∂θ 2



1 ∂2 R 2 ∂θ 2

−

2 R2

1 ∂2 R 2 ∂θ 2

−

1 R2

 

i = j =1 i = j =2

(6.49)

i = j =3 i = j

From a different perspective, and based on the Eq. (6.33), we obtain the equivalent continuum constitutive relation for the SWCNT as t=

1 Yε h

(6.50)

Equations (6.48) and (6.50) are combined and the atomistic-continuum relation is obtained as L∗ Yε = E∗ ε

(6.51)

According to the linear Flügge shell theory [43], the strain-displacement relationships for a circular cylindrical shell are ⎛

⎞

⎛

εx x ⎜ ⎝ εθθ ⎠ = ⎜ ⎝ 2εxθ

∂ ∂x

0

−z ∂∂x 2

0

1 ∂ R ∂θ

z (∂ R 2 ∂θ 2

( R1 −

z )∂ R 2 ∂θ

(1 + Rz ) ∂∂x

2

1 R

−

2

⎛ ⎞ ⎟ u ⎝ ⎠ − 1) ⎟ ⎠ v w

∂ − 2z R ∂ x∂θ 2

⎞

(6.52)

6.4 Evaluation of the Nonlocal Parameter for SWCNTs

129

where u, v and w refer to the mid-surface displacements along the x, θ and z coordinates respectively. Substituting Eq. (5.52) into Eq. (6.51) and simplifying the resultant equation, the following relation is obtained ⎛

⎞⎛ ⎞ u ⎟⎜ ∗ ∗ ⎟ ⎝ ⎠ L ∗22 Y22 L ∗22 Y23 ⎠ ⎝ 0 Υ22 Υ23 ⎠ v w ∗ ∗ ∗ L ∗33 Y23 L ∗33 Y33 Υ31 Υ32 Υ33 ∗ ∗ ⎞ ⎛ ∗ ∗ ⎞ ⎛ ⎞ E 12 E 13 Υ11 0 Υ13 u ∗ ∗ ⎟ ⎜ ∗ ∗ ⎟ ⎝ ⎠ E 22 E 23 Υ23 ⎠ ⎝ 0 Υ22 ⎠ v w ∗ ∗ ∗ ∗ ∗ E 23 E 33 Υ31 Υ32 Υ33

L ∗11 Y11 L ∗11 Y12 L ∗11 Y13

⎜ ∗ ⎝ L 22 Y12 L ∗33 Y13 ⎛ ∗ E 11 ⎜ ∗ = ⎝ E 12 ∗ E 13

⎞⎛

∗ Υ11

0

∗ Υ13

(6.53)

wherein ∗ Υ11 ∗ Υ31

 2  ∂ z ∂ ∂2 1 ∂ 1 ∗ ∗ ∗ , Υ13 = −z 2 , Υ22 = , Υ23 = − 2 = −1 , ∂x ∂x R ∂θ R R ∂θ 2     ∂ z 1 z ∂ 2z ∂ 2 w ∗ ∗ − 2 , Υ32 , Υ33 (6.54) = = 1+ =− R R ∂θ R ∂x R ∂ x∂θ

For a harmonic wave field in the SWCNT, the displacement field can be written in a complex form as ⎛ ⎞ ⎛ ⎞ U¯ u ⎝ v ⎠ = ⎝ V¯ ⎠ exp(iλξ x + isθ + iωt) w W¯

(6.55)

where U¯ , V¯ and W¯ are the displacement amplitudes in the x, θ , z directions, respectively, λξ (λξ = ξ π/L C N T , ξ is the half-axial wave number and L C N 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. (6.53) and after making some simplifications, we get ⎛

∗ ¯ ∗ ¯ ∗ A1 Y12 − E 12 A1 Y13 − E 13 A¯ 1 Y11 − E 11

⎞

⎜ ⎟ ⎜ A¯ 2 Y12 − E ∗ A¯ 2 Y22 − E ∗ A¯ 2 Y23 − E ∗ ⎟ 12 22 23 ⎠ ⎝ ∗ ¯ ∗ ¯ ∗ A¯ 3 Y13 − E 13 A3 Y23 − E 23 A3 Y33 − E 33 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ iλξ 0 zλξ 2 U¯ 0 ⎜ ⎟   1 1 1 2 ⎜ ⎟ ⎝ ¯ ⎠ = ⎝ 0 ⎠ (6.56) 0 is + z + 1 s ×⎝ V 2 R R R ⎠ 0 W¯ 1 z z 2z is − is iλ + iλ λ s ξ ξ R R2 R R ξ

130

6 Elastic Properties of Carbon-Based Nanoscopic Structures

where  A¯ 1 = 1 + (e0 L i )2 λξ 2 +  A¯ 2 = 1 + (e0 L i )2 λξ 2 +  2 ¯ A3 = 1 + (e0 L i ) λξ 2 +

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

(6.57)

For the non-trivial solution, the determinant of this set of equations must be zero, i.e. ⎞ ⎛ ∗ ¯ ∗ ¯ ∗ A1 Y12 − E 12 A1 Y13 − E 13 A¯ 1 Y11 − E 11 ⎟ ⎜ ∗ ¯ ∗ ¯ ∗ ⎟ ¯ if λξ = 0 (6.58) det ⎜ ⎝ A2 Y12 − E 12 A2 Y22 − E 22 A2 Y23 − E 23 ⎠ = 0 ∗ ¯ ∗ ¯ ∗ ¯ A3 Y13 − E 13 A3 Y23 − E 23 A3 Y33 − E 33 The three associated eigenvalues for the system are e0x x L i , e0θθ L i and e0xθ L i , and these are functions of chirality (n, m) of the SWCNT and the longitudinal and circumferential wavenumbers. To evaluate the e0 of the SWCNTs having different geometrical characteristics, the following parameters are used: Eh = 360 TPa nm, ν = 0.16, K C−C = 742 N/m and CC−C = 1.42 nN nm. Figure 6.4a–c demonstrate the predicted values of e0 of the SWCNTs related to the first longitudinal mode of wave propagation (ξ = 1, s = 0). Similar results, shown in Fig. 6.5a–c, refer to the case of the first coupled longitudinal-torsional mode (ξ = 1, s = 1). To elucidate the effect of the aspect ratio (L C N T /2R), the predicted values of e0 for the first longitudinal mode and the first coupled longitudinal-torsional mode are plotted versus the aspect ratio in Fig. 6.6 for two armchair SWCNTs. It is seen from Fig. 6.6a that, for the case of the first coupled longitudinal torsional mode, the value of e0 in the longitudinal direction is about 0.43 and is not sensitive to the aspect ratio. For the case of longitudinal mode, however, the aspect ratio imposes a significant effect on the longitudinal nonlocal parameter. The longitudinal nonlocal parameter increases linearly with an increase in the aspect ratio. Furthermore, as shown in Fig. 6.6b, c, the other nonlocal parameters (e0θθ and e0xθ ) are not sensitive to the aspect ratio.

6.5 Anisotropic Mechanical Properties of Graphene Sheets Many investigators have considered the mechanical behaviour of the graphene sheet assuming isotropic properties for this material. Consequently they had to consider only two elastic parameters, e.g. Young’s modulus and Poisson’s ratio. Similar to SWCNTs, a wide range of values of the Young modulus and Poisson’s ratio,

6.5 Anisotropic Mechanical Properties of Graphene Sheets

131

Fig. 6.4 Variations of nonlocal parameter with diameter for different types of SWCNTs. (a)–(c) first longitudinal mode (ξ = 1, s = 0)

as well as the effective thickness, have been reported in the literature [7, 44–49] (see Table 6.2). The large scatter in the value of the Young modulus has been listed for the first time in Ref. [50]. Poot and van der Zant [51] have indicated that a graphene flake is stiffer along the principle direction than the other directions. Furthermore, for graphene sheets, anisotropic mechanical properties were observed along different load directions [52–56]. These anisotropic properties are related to the hexagonal

132

6 Elastic Properties of Carbon-Based Nanoscopic Structures

Fig. 6.5 Variations of nonlocal parameter with diameter for different types of SWCNTs. (a)–(c) first coupled longitudinaltorsional mode (ξ = 1, s = 1)

morphology of the unit cell of the graphene. Thus, ambiguities arise as to whether the graphene sheet is an isotropic structure, and by how much its mechanical properties vary along different directions? Employing MD simulations, the anisotropic elastic properties of single-layered graphene sheets with armchair and zigzag helicities were predicted by Shen et al. [57].

6.5 Anisotropic Mechanical Properties of Graphene Sheets

133

Fig. 6.6 Variations of nonlocal parameters with aspect ratio; (a) e0x x , (b) e0θ θ and (c) e0xθ

In these simulations, the large-scale atomic/molecular massively parallel simulator (LAMMPS) software [58] was employed together with the adaptive intermolecular reactive empirical bond order potential (AIREBO) [59]. Two types of single-layered graphene sheets having different values of aspect ratios were considered, namely; an armchair sheet with aspect ratios A R = 1.97, 1.44 and 1.01, and a zigzag sheet with aspect ratios A R = 1.95, 1.45 and 0.99. Four independent material parameters are required to completely describe the elastic behaviour of the graphene sheets. These parameters are denoted by E 11 , E 22 , G 12 and ν12 , respectively. To determine these properties, three MD simulation tests including uniaxial tension, shear load and transverse uniform pressure tests were performed in the temperature range from

134

6 Elastic Properties of Carbon-Based Nanoscopic Structures

Table 6.2 Elastic properties of the graphene sheets listed from the literature Authors Wall thickness h (nm) Young’s modulus E Poisson’s ration (TPa) Kudin et al. [44] Reddy et al. [45] Armchair sheets Zigzag sheets Wang et al. [46] Huang et al. [7] Hemmasizadeh et al. [47] Sakhaee-Pour [48] Armchair sheets Zigzag sheets Chiral sheets Shokrieh and Rafiee [49]

0.335

1.029

0.149

0.34 0.34 0.0665 0.0874−0.0618 0.0811−0.0574 0.34

1.095–1.125 1.106–1.201 5.07 2.69−3.81 2.99−4.23 1.01

0.445–0.498 0.442–0.465 0.158 0.142 0.397 0.19

0.34 0.34 0.34 0.33

1.042 1.040 0.992 1.04

1.285 1.441 1.129 –

Table 6.3 Material properties and geometrical parameters of single-layered graphene sheets at room temperature Armchair Armchair Armchair Zigzag Zigzag Zigzag sheet 1 sheet 2 sheet 3 sheet 1 sheet 2 sheet 3 Properties E 11 (TPa) E 22 (TPa) G 12 (TPa) ν12 Geometrical parameters L x (nm) L y (nm) h (nm)

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

300–700 K. Furthermore, direct computation of the bending deflections via MD simulation, uniquely yields the effective thickness of single-layered graphene sheets [57]. The results, including the values of Young’s moduli along the principle axis, and shear modulus and Poisson’s ratio, obtained at the room temperature are listed in Table 6.3.

6.6 Effective Young’s Modulus of Fullerene Molecules

135

6.6 Effective Young’s Modulus of Fullerene Molecules Compared to the studies performed on the elastic properties of the carbon nanotubes and the graphene sheets, the study of the Young modulus of the fullerene molecules has received very scant attention. In most researches, the elastic properties of the graphene sheet have been utilized to explore the mechanical characteristics of the fullerenes [60–63]. Ruoff and Ruoff [64, 65] made the first investigation to predict the bulk modulus of C60 . Compressive mechanical properties of C60 fullerene molecule at different temperatures were studied by Shen [66] via the MD simulation using the Tersoff-type interatomic potential energy function. The compressive mechanical properties of different fullerene molecules including C20 , C60 , C80 and C180 using a quantum MD simulation were also investigated [67]. Giannopoulos et al. [68] studied the radial elastic stiffness of different fullerene molecules using the spring-based structural mechanics method. It was found that the radial Young’s modulus decreased non-linearly with an increase in the radius of fullerene. The elastic modulus of C60 molecule was calculated by Jamal-Omidi et al. [69] employing the finite element method. Shell elements were used to model the fullerene structures and the elastic properties of these elements were adopted from graphene sheets. Combination of the MD simulations and continuum shell theory was used as a new approach to determine the Young modulus of nine different spherical fullerenes, from C60 to C2000 [70]. Here, to find the effective Young’s modulus of the fullerenes, the strain energy due to the application of a uniform tensile load was calculated from the MD simulation, and then a continuum spherical shell model was utilized to relate the strain energy and the tensile load. Energy equivalence between the MD simulation and the continuum spherical shell model was employed to compute the effective Young’s modulus of the fullerenes. In the MD simulations, the LAMMPS software and the AIREBO force field were employed to obtain the strain energy at room temperature. To evaluate the Young modulus, the parameters used were; ν = 0.3 and h = 0.182 nm. Since all fullerenes do not form perfect spheres, a fullerene may be modelled as a sphere with the equivalent radius R f . It was shown that a fullerene having more than 100 atoms, the difference between its area and its equivalent sphere is less than one percent [71]. Table 6.4 lists the equivalent radius for the fullerenes based on the relaxed positions of atoms. Figure 6.7 displays the calculated Young’s modulus as a function of the equivalent radii of the fullerenes. It is seen that Young’s modulus is a decreasing function of the radius and it converges gradually to the Young modulus of an ideally flat graphene sheet. However, the C60 fullerene displays an irregular result. It is worth noticing that the irregularity in the Young modulus is attributed to the assumption of the thin-walled shell theory. Therefore, it can be concluded that the thin shell theory is not applicable for the case of small fullerenes.

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6 Elastic Properties of Carbon-Based Nanoscopic Structures

Fig. 6.7 Predicted Young’s modulus of fullerenes with different radii Table 6.4 Calculated radii for some spherical fullerenes

Number of atoms (N ) 60 180 240 540 720 980 1280 1620 2000

Radius (Å) 3.549 6.042 6.960 10.361 11.940 13.921 15.896 17.873 19.853

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Part II

Chapter 7

Computational Modelling of the Vibrational Characteristics of Zero-Dimensional Nanoscopic Structures

Insight into the vibrational characteristics of zero-dimensional nanoscopic structures is of fundamental interest, since it can be used to predict their geometrical and material properties. Zero-dimensional nanoscopic structures are nano-sized particles with all their three dimensions restricted to a few tens of nanometers. Investigation of these nanoscopic structures has prompted a growing research endeavour in diverse fields including nanolubrication, nanomanufacturing, nanocoatings and nanocomposites [1]. In this chapter, we consider the nonlocal vibration analysis of zero-dimensional nanoscopic structures. An overview of the current literature discussing the vibration characteristics of zero-dimensional nanoscopic structures is presented first. We then discuss the application of the nonlocal models to the investigation of the vibration properties of the spherical fullerene molecules and nanoparticles.

7.1 Historical Review The applicability of the nonlocal elasticity theory to the description of the vibrational behaviour of zero-dimensional nanoscopic structures was not addressed until 2012. The first contribution to the modelling of these types of nanoscopic structures by using the nonlocal elasticity theory was made by Ghavanloo and Fazelzadeh [2, 3]. However, only a limited amount of work has been devoted to the mathematical modelling of the spherical nanoscopic structures via the application of the nonlocal elasticity. In this section, we review the studies performed to determine the natural frequencies of zero-dimensional nanoscopic structures on the basis of the nonlocal elasticity approach.

© Springer Nature Switzerland AG 2019 E. Ghavanloo et al., Computational Continuum Mechanics of Nanoscopic Structures, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-3-030-11650-7_7

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7 Computational Modelling of the Vibrational Characteristics of Zero-Dimensional …

7.1.1 Axisymmetric Vibration of Spherical Nanoshells The vibrational analysis of spherical nanoshells, such as different fullerene molecules, has been performed with the aid of experimental and computational methods [4–9] in order to ascertain the pattern of the vibrational modes and the variability of the natural frequencies. Application of the nonlocal elasticity theory to spherical nanoshells was initially addressed by Fazelzadeh and Ghavanloo [2], so as to formulate the axisymmetric vibration of a fluid-filled spherical membrane nanoshell. In that study, it was assumed that the nanoshell was completely filled with a compressible and inviscid fluid. Zaera et al. [10] utilized the nonlocal spherical shell model to analyse the free axisymmetric vibrations of empty spherical nanoshells. In another study, the axisymmetric vibrational behaviour of ten spherical fullerenes, namely; C60 , C180 , C240 , C540 , C960 , C1500 , C2160 , C2940 , C3840 , and C4860 was investigated in the framework of nonlocal elastic theory [11]. An analytical expression, based on the nonlocal shell theory was proposed to determine the natural frequencies of spherical fullerene molecules. Furthermore, it was shown that the results obtained from the nonlocal shell model were in in good agreement with those reported by experimental and atomistic-based studies. In the aforementioned studies [2, 10, 11], isotropic momentless spherical shells were taken into account and the bending and twisting moments were neglected. Vila et al. [12] extended the previous studies by including the bending effects. They obtained the natural frequencies and the corresponding modal shapes of closed thin spherical nanoshells.

7.1.2 Breathing-Mode of Spherical Nanoparticles Mechanical resonance experiments are regarded as very efficient nondestructive methods to characterize nanometer-scale objects. The Raman spectroscopy technique [13] and the time-resolved pump-probe experiments [14] have been the main tools to study the vibrations of the nanoparticles. Breathing-mode of the nanoparticles corresponds to synchronous collective vibration of all atoms in the radial direction. This mode is identified as the A1g mode [15]. To evaluate the breathing vibrational mode of nanoparticles with a radius larger than 2 nm, the classical continuum model, which was proposed by Lamb [16], has been usually used. The applicability of the Lamb’s model for the description of rather large nanoparticles has been confirmed in atomistic-based calculations [17, 18]. However, recent studies have shown that the prediction of the vibration characteristics when using the Lamb formulations may not produce the correct results when the nanoparticles are composed of a few number of atomic layers [19]. Motivated by this idea, Ghavanloo and co-workers [20–23] developed a set of nonlocal continuum models to study the breathing vibrational mode of isolated and embedded nanoparticles. It was shown that the suggested nonlocal model with an appropriate nonlocal parameter could predict the frequencies of breathing-modes in good agreement with the results obtained from MD simulations

7.1 Historical Review

145

and also the experimental results for several nanoparticles. They also showed that the inverse relationship between the breathing frequency and the nanoparticle radius was bound to fail when the nanoparticle radius was smaller than 1.5 nm. To sum up the results from these works, it was noticed that the findings point directly to the possibility of employing the nonlocal continuum modelling as an effective way in simulating the nanoparticles.

7.2 Axisymmetric Vibrational Modes of Spherical Fullerene Molecules The nonlocal membrane theory for spherical shell, described in Sect. 5.4, is employed to estimate the frequency of axisymmetric vibrational modes of spherical fullerenes. Based on Eqs. (5.123) and (5.124) and setting qr = 0, the governing equations for the vibration of the fullerenes are obtained as  2  E ∂ u ∂u ∂w 2 (1 + ν) + cot φ − (ν + cot φ)u + R 2 ρ(1 − ν 2 ) ∂φ 2 ∂φ ∂φ   ∂ 4u ∂ 3u ∂ 2u ∂ 3w ∂ 2u 2 2 (7.1) + cot φ + 2 (1 − cot φ) + 2 2 = 2 −λ ∂t ∂φ 2 ∂t 2 ∂φ∂t 2 ∂t ∂t ∂φ   E ∂u ∂ 2w + 2w + u cot φ + 2 2 ρ R (1 − ν) ∂φ ∂t  4 3 2  ∂ w ∂ w ∂ w (7.2) = λ2 + cot φ −2 2 ∂φ 2 ∂t 2 ∂φ∂t 2 ∂t It should be noted that Eqs. (7.1) and (7.2) are not identical with those proposed by Ghavanloo and Fazelzadeh [11] and Rafii-Tabar et al. [24]. The differences arise due to the fact that the components of the Laplacian of stress tensor, (∇ 2 t)φφ and (∇ 2 t)θθ , have been approximated by the Laplacian of stress tenser components, ∇s2 tφφ and ∇s2 tθθ . Notwithstanding this point, we will show that the approximate formulations can be used with reasonable accuracy in some special cases. In free vibration analysis, it is assumed that the two displacements u and w vary harmonically with respect to the time variable t as follows u(φ, t) = U (φ) exp( jωt)

(7.3)

w(φ, t) = W (φ) exp( jωt)

(7.4)

√ where j = −1 and ω is the natural frequency. Substitution of Eqs. (7.3) and (7.4) into Eqs. (7.1) and (7.2), leads to the following equations, i.e.,

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7 Computational Modelling of the Vibrational Characteristics of Zero-Dimensional …

dU d 2U dW (1 + ν) + cot φ − (ν + cot 2 φ)U + dφ 2 dφ dφ  2   dU dW 2 2 d U 2 ¯ 2 + U (1 − cot φ) + 2 =0 (7.5) + cot φ +Ω (1 − ν ) U −λ dφ 2 dφ dφ  2   dU d W dW + cot φ + 2W + U cot φ − Ω¯ 2 (1 − ν) W − λ2 − 2W =0 dφ dφ 2 dφ (7.6) where Ω¯ 2 =

R 2 ρω2 E

(7.7)

The solutions of Eqs. (7.5) and (7.6) may be assumed in the form U (φ) = U¯ Pn1 (φ)

(7.8)

W (φ, t) = W¯ Pn (φ)

(7.9)

where U¯ and W¯ are unknown coefficients and, Pn and Pn1 are the Legendre function and the associated Legendre function of first kind and first order, respectively. The associated Legendre polynomials are defined by Pn1 (φ) = −

d Pn (φ) dφ

(7.10)

It is known that the Legendre functions satisfy the following differential equation: d 2 Pn d Pn + n(n + 1)Pn = 0 + cot φ dφ 2 dφ

(7.11)

where n is a non-negative integer number. Similarly, the associated Legendre functions are the solutions of the associated Legendre differential equation: d P1 d 2 Pn1 + cot φ n + [n(n + 1) − (1 + cot 2 φ)]Pn1 = 0 2 dφ dφ

(7.12)

Substituting Eqs. (7.8) and (7.9) into Eqs. (7.5) and (7.6) and using Eqs. (7.10)(7.12), yields the following algebraic eigenvalue problem   0 (1 − ν)[1 + λ2 (n(n + 1) + 2)] 2 ¯ × Ω −2(1 − ν 2 )λ2 −(1 − ν 2 )[1 + λ2 (n(n + 1) − 2)]      U¯ n(n + 1) 2 U¯ = (7.13) ¯ −n(n + 1) + 1 − ν − (1 + ν) W W¯

7.2 Axisymmetric Vibrational Modes of Spherical Fullerene Molecules

147

For a non-trivial solution, the determinant of the coefficient matrix in Eq. (7.13) must vanish. Hence, one gets the frequency equation as Γ4 Ω¯ 4 + Γ2 Ω¯ 2 + Γ0 = 0

(7.14)

where Γ0 = n(n + 1) − 2, Γ2 = −{λ2 [(n 2 + n)(n 2 + n + 1) + 2ν(n 2 − 1) − 6] + n(n + 1) + (1 + 3ν)}, 2

Γ4 = (1 − ν 2 )((1 + λ2 n 2 + λ2 n) − 4λ4 )

(7.15)

Moreover, it follows from Eq. (7.13) that U¯ n =

(1 + ν) − 2Ω¯ 2 (1 − ν 2 )λ2 W¯ n −n(n + 1) + 1 − ν + Ω¯ 2 (1 − ν 2 )[1 + λ2 (n(n + 1) − 2)]

(7.16)

so that the mode shapes are given by U (φ) =

(1 + ν) − 2Ω¯ 2 (1 − ν 2 )λ2 W¯ n Pn1 (φ) −n(n + 1) + 1 − ν + Ω¯ 2 (1 − ν 2 )[1 + λ2 (n(n + 1) − 2)] (7.17) W (φ, t) = W¯ n Pn (φ)

(7.18)

It is clear from Eq. (7.14) that, for each n, there are two natural frequencies. Thereup. fore, we have an upper branch (Ωn ) and a lower branch (Ωnlow. ) of the frequency spectrum. When n = 0, the frequency of the lower branch is imaginary and so it represents a spurious frequency. However, the upper branch in this case corresponds to a pure radial vibration and is referred to as the breathing-mode or pulsating mode. Its corresponding natural frequency is up.

Ω0

= +

(1 + 3ν) − λ2 (2ν + 6) 2(1 − ν 2 )(1 − 4λ4 )  (λ2 [2ν + 6] − (1 + 3ν))2 + 8(1 − ν 2 )(1 − 4λ4 ) 2(1 − ν 2 )(1 − 4λ4 )

(7.19)

Although the frequency of the breathing-mode is higher than all of the frequencies of the lower branch, this frequency would normally be considered as the fundamental frequency. In addition, it is interesting to note that Ω1 has a null frequency and so this mode corresponds to a rigid-body motion. To justify the applicability of the nonlocal model, we compare the computed frequencies of the C60 molecules obtained from Eq. (7.14) with some of the existing numerical and experimental results. Numerical calculations require an appropriate choice of the basic quantities. The in-plane stiffness Eh = 363 Jm2 [25], Poisson’s ratio ν = 0.145 [26] and the average radius R = 0.355 nm [27] are assumed. In addition, the mass surface density

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7 Computational Modelling of the Vibrational Characteristics of Zero-Dimensional …

Table 7.1 Comparison between the frequencies of C60 (cm−1 ) obtained from the nonlocal model and the existing theoretical and experimental results Method Vibration mode up. up. up. up. Ω0 Ω1 Ω2 Ω3 Ω2low. Nonlocal model (λ = 0.15) Raman spectroscopy [28] Local continuum model [5] Force-constant model [29] Molecular mechanics [6] Molecular dynamics [30]

488 496 496 492 490 483

585 576 607 589 587 565

801 774 833 788 850 —

1034 1183 1109 1208 1187 —

237 273 273 269 269 213

which is the mass of atoms per unit area of a molecule, can be expressed as ρh =

m c nc 4π R 2

(7.20)

where m c (= 1.9943 × 10−26 kg) is the mass of the carbon atom and n c denotes the number of carbon atoms. For C60 , we have ρh = 0.7756 × 10−7 g/cm2 . Table 7.1 lists the computed values of the frequencies of different modes of C60 molecule and those reported via the Raman spectroscopy [28], together with the existing theoretical results [5, 6, 29], and molecular simulations results [30]. As can be inferred from this table, the predicted numerical results obtained from the nonlocal model are in reasonable agreement with the results reported by the experimental and atomisticbased studies. Figure 7.1 displays the dimensionless frequencies of the lower and the upper branches for different values of the nonlocal parameter and ν = 0.3. Note that the mode numbers n are discrete and so only those points corresponding to the integer values of n are physically meaningful. It is seen that the frequencies of both upper and lower branches decrease with an increase in the nonlocal coefficient λ. The decreasing rate of frequencies are not the same for all modes, and so the curves are not parallel to each other. Furthermore, it is observed that the frequencies of up. the breathing-mode (Ω0 ) and the translational mode (Ω1low. ) are not very sensitive to the parameter λ. The influence of the nonlocal parameter starts to play a major role for n larger than 3. Actually there are notable differences between the local and nonlocal theories for large values of n, even for small values of λ. This means that the application of the local model to the analysis of the fullerene molecules would lead to an overprediction of the frequencies. As was stated above, there are two distinct natural frequencies and corresponding mode shapes for each value of n. The mode shapes of axisymmetric vibration are shown in Fig. 7.2 for n from 0 through 5 and ν = 0.3. It is observed that the mode shapes of the upper and the lower branches for the same n are similar, even though their vibration frequencies differ greatly. Furthermore, although we have not shown here, the mode shape is not affected by the nonlocal parameter. Let us finally consider the comparison between the approximate formulations developed for the frequencies of upper and lower branches in Refs. [11, 24]

7.2 Axisymmetric Vibrational Modes of Spherical Fullerene Molecules

149

Fig. 7.1 Frequency spectra for different values of nonlocal parameter; (a) upper branch and (b) lower branch

and those proposed here. In those studies, the frequencies of the upper and lower braches were given by 1 + 3ν + n(n + 1) 2(1 − ν 2 )[1 + λ2 n(n + 1)] [1 + 3ν + n(n + 1)]2 − 4(1 − ν 2 )[n(n + 1) − 2] + 2(1 − ν 2 )[1 + λ2 n(n + 1)]

Ωnup. =

1 + 3ν + n(n + 1) 2(1 − ν 2 )[1 + λ2 n(n + 1)] [1 + 3ν + n(n + 1)]2 − 4(1 − ν 2 )[n(n + 1) − 2] − 2(1 − ν 2 )[1 + λ2 n(n + 1)]

(7.21)

Ωnlow. =

(7.22)

To compare the proposed model (Eq. (7.14)) with the approximate formulations (Eqs. (7.21) and (7.22)), Table 7.2 lists the frequencies obtained from the exact and

150

7 Computational Modelling of the Vibrational Characteristics of Zero-Dimensional …

Fig. 7.2 Modal shapes for: (a) n = 0, (b) n = 1, (c) n = 2, (d) n = 3, (e) n = 4, and (f) n = 5

7.2 Axisymmetric Vibrational Modes of Spherical Fullerene Molecules

151

Table 7.2 Comparison between frequencies obtained from the exact and approximate models for ν = 0.3 λ = 0.00

Vibration mode up.

Ω0

up.

Ω1

up.

Ω2

up.

Ω3

Ω2low.

Ω3low.

λ = 0.05

λ = 0.10

λ = 0.15

λ = 0.20

Exact model

1.6903

1.6861

1.6737

1.6535

1.6265

Approximate model

1.6903

1.6903

1.6903

1.6903

1.6903

Relative error (%)

0.00

0.25

0.99

2.23

3.92

Exact model

2.0702

2.0599

2.0300

1.9829

1.9221

Approximate model

2.0702

2.0650

2.0498

2.0251

1.9920

Relative error (%)

0.00

0.25

0.98

2.13

3.64

Exact model

2.8533

2.8270

2.7525

2.6410

2.5061

Approximate model

2.8533

2.8322

2.7714

2.6783

2.5624

Relative error (%)

0.00

0.18

0.69

1.41

2.25

Exact model

3.8102

3.7493

3.5832

3.3500

3.0897

Approximate model

3.8102

3.7543

3.6003

3.3810

3.1320

Relative error (%)

0.00

0.13

0.48

0.93

1.37

Exact model

0.7348

0.7307

0.7187

0.7000

0.6761

Approximate model

0.7348

0.7293

0.7137

0.6897

0.6599

Relative error (%)

0.00

0.19

0.70

1.47

2.40

Exact model

0.8700

0.8584

0.8262

0.7796

0.7260

Approximate model

0.8700

0.8573

0.8221

0.7720

0.7152

Relative error (%)

0.00

0.13

0.50

0.97

1.49

approximation approaches. Computation of the frequencies of different modes shows that the natural frequencies obtained from Eqs. (7.21) and (7.22) have relative errors of less than 4% compared with those obtained from Eq. (7.14). Furthermore, it is seen from the table that the relative errors decrease with increasing n. Consequently, Eqs. (7.21) and (7.22), which were obtained by approximating the components of the Laplacian of the stress tensor, can be used with a reasonable accuracy.

7.3 Vibration of Spherical Nanoparticles 7.3.1 Radial Vibration of Isolated Nanoparticles Based on Eq. (5.130) and setting k = 2 and fr = 0, the governing equation for the free vibration of an isolated nanoparticle is obtained as

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7 Computational Modelling of the Vibrational Characteristics of Zero-Dimensional …

 ρR λ

2 2

∂ 4w 2 ∂ 3w 2 ∂ 2w + − ∂r 2 ∂t 2 r ∂r ∂t 2 r 2 ∂t 2





∂ 2 w 2 ∂w 2w + c11 − 2 + ∂r 2 r ∂r r 2 ∂ w −ρ 2 =0 ∂t



(7.23)

For harmonic motion with frequency ω, the solution is sough in the form w(r, t) = W (r ) exp( jωt)

(7.24)

Substituting Eq. (7.24) into Eq. (7.23), the following equation is derived:   2 dW 2 d2W 2 + Bs − 2 W = 0 + dr 2 r dr r

(7.25)

where

Bs = ω

ρ c11 − ρ R 2 ω2 λ2

(7.26)

Equation (7.25) is the spherical Bessel differential equation and its general solution is [31] W (r ) = C1

J3/2 (Bs r ) Y3/2 (Bs r ) + C2 √ √ r r

(7.27)

where C1 and C2 are unknown constants and, J3/2 and Y3/2 are the first and the second kind Bessel functions of order 1.5 respectively. Since the center of nanoparticle could not have an infinite displacement, we must set C2 = 0 to remove the infinite value √ of Y3/2 (Bs r )/ r when r = 0. The resulting equation is W (r ) = C1

J3/2 (Bs r ) √ r

(7.28)

For an isolated nanoparticle, it is assumed that the stress-free boundary condition occurs at the surface of the nanoparticle. Therefore, we have  dW  2c12 W (R) =0 c11 +  dr r =R R

(7.29)

Substituting Eq. (7.28) into Eq. (7.29) yields the characteristic equation for the natural frequencies as   c12 R Bs J5/2 (R Bs ) − J3/2 (R Bs ) 1 + 2 =0 c11

(7.30)

7.3 Vibration of Spherical Nanoparticles Table 7.3 Material properties for Au and Ag Material ρ/g cm3 Au Ag

19.283 10.50

153

Elastic constants (1011 N/m2 ) c11 c12 1.9244 1.2399

1.6298 0.9367

Table 7.4 Comparison between the breathing frequency of Ag and Au nanoparticles predicted by the nonlocal model and those reported in Refs. [33, 34] Material Radius (nm) ω (cm−1 ) Ref. Nonlocal model Previous studies (λ = 0.15) Ag

1.5

33.0

34.0

Au

2 2.4

24.7 19.3

27.6 18.0

3.4

13.6

13.0

Mankad et al. [33] Mankad et al. [34]

By solving Eq. (7.30), all radial natural frequencies of the nanoparticle can be obtained. Note that the lowest frequency is the fundamental frequency which corresponds to the breathing-mode. To compute the numerical results, we need to give an appropriate definition of the elastic constants of the nanoparticles. The values of material properties of Ag and Au nanoparticles are listed in Table 7.3, taken from [32]. The validity of the nonlocal approach is tested via several experimental data. The calculated results are listed in Table 7.4, along with the experimental data reported in the literature [33, 34]. 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 studies. Figure 7.3 displays the results of computation of the fundamental radial frequency of Au and Ag nanoparticles with respect to the inverse radius for four different values of the nonlocal parameter (e0 L i = Rλ). It can be seen that the local frequencies (Lamb’s model) are inversely proportional to the size of the nanoparticles. However, the inverse relationship between the frequency and the nanoparticle radius is not available for ultra-small nanoparticles due to the small-scale effect. Furthermore, the nonlocal predictions are always smaller than their local counterparts. Physically this may be due to the nonlocal interactions between the atom situated at the reference point and all other atoms that induce surface compression. This compression reduces the equivalent structural rigidity of the nanoparticle and downshifts (redshifts) its frequency. A similar explanation has been pointed to by Eringen [35]. In addition, it can be seen from this figure that the difference between the local and nonlocal models is enhanced when the size is decreased.

154

7 Computational Modelling of the Vibrational Characteristics of Zero-Dimensional …

Fig. 7.3 Variations of the breathing-mode frequencies for (a) Au and (b) Ag nanoparticles as a function of 1/R

7.3.2 Vibration Under Radial Body Force In this subsection, the radial vibration of a nanoparticle subject to two types of body force are considered: a centrifugal force, frc , and a Lorentz’s force due to magnetic field, frL . The centrifugal force is defined as [36]: frc = ρΩ 2 w

(7.31)

where Ω is the speed of rotation. In addition, the Lorentz force is calculated from [22] frL

=

αrL

∂ ∂r



∂w 2w + ∂r r

 (7.32)

where αrL is constant which is a function of magnetic permeability and the initial magnetic field. Substituting Eq. (7.31) into Eq. (5.130) and setting k = 2, we have

7.3 Vibration of Spherical Nanoparticles

155



  2  ∂ 4w 2 ∂ 3w 2 ∂ 2w 2 ∂w 2w 2 2 2 ∂ w − ρΩ R λ − 2 ρR λ + − + ∂r 2 ∂t 2 r ∂r ∂t 2 r 2 ∂t 2 ∂r 2 r ∂r r  2  2 ∂ w 2 ∂w 2w ∂ w +c11 + (7.33) − 2 − ρ 2 + ρΩ 2 w = 0 ∂r 2 r ∂r r ∂t 2 2

Substituting Eq. (7.24) into Eq. (7.33), the following equation is derived:   d2W 2 dW 2 2 + ςs − 2 W = 0 + dr 2 r dr r

(7.34)

where ςs2 =

ρ(ω2 + Ω 2 ) c11 − ρ R 2 λ2 (ω2 + Ω 2 )

(7.35)

Here, ςs is the corresponding eigenvalue and it can be determined by   c12 =0 Rςs J5/2 (Rςs ) − J3/2 (Rςs ) 1 + 2 c11

(7.36)

Therefore, the analytical frequency relation for the radial breathing-mode frequency of the nanoparticle, when the nonlocal effect and the centrifugal force are taken into account, is expressed as ω2 =

c11 ςs2 − ρΩ 2 (1 + R 2 λ2 ςs2 ) ρ(1 + R 2 λ2 ςs2 )

(7.37)

In a similar manner to the case of the centrifugal force, the governing equation for vibration of the nanoparticle subject to a circumferential magnetic field is obtained as  4  ∂ w ∂ 2w 2 ∂ 3w 2 ∂ 2w 2 2 − ρ ρR λ + − ∂r 2 ∂t 2 r ∂r ∂t 2 r 2 ∂t 2 ∂t 2  2  2  ∂ ∂ w 2 ∂w 2w 2 2 ∂ 2 2 L − 2 − 2 −R λ αr + + ∂r 2 r ∂r r ∂r 2 r ∂r r  2   2  ∂ w 2 ∂w 2w ∂ w 2 ∂w 2w L − 2 + αr − 2 = 0 (7.38) +c11 + + ∂r 2 r ∂r r ∂r 2 r ∂r r Therefore, in this case, the analytical expression for the breathing-mode frequency is obtained as ω2 =

ςs2 (c11 + αrL ) + R 2 λ2 ςs4 αrL ρ(1 + R 2 λ2 ςs2 )

(7.39)

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7 Computational Modelling of the Vibrational Characteristics of Zero-Dimensional …

7.3.3 Frequency Analysis of Nanoparticle Subject to Surface Periodic Excitation In this case, all equations are similar to those reported in Sect. 7.3.1 except for the boundary condition (7.29). We assume that a harmonic pressure is applied at the surface of the nanoparticle, thus σ11 (R) = σrr (R) = −P exp( jωt)

(7.40)

Since we have no knowledge of σ22 at the surface, we try to eliminate it from the set of the constitutive equations at the surface i.e.,   4 ∂w  2c12 w(R) (7.41) (σ11 (R) − σ22 (R)) = c11 + σ11 (R) + R λ  2 R ∂r r =R R    4 ∂w  (c11 + c12 )w(R) σ22 (R) − R 2 λ2 2 (σ11 (R) − σ22 (R)) = c12 + R ∂r r =R R (7.42) 

2 2

Subtracting Eq. (7.42) from Eq. (7.41) and rearranging, the following relation is obtained.    ∂w  w(R) (c11 − c12 ) − (7.43) (1 + 6λ2 )(σ11 (R) − σ22 (R)) = ∂r r =R R Using Eq. (7.41) and (7.43), and applying the boundary condition (7.40), we obtain   4(c11 − c12 )λ2 w(R) 2c12 + − P exp( jωt) = R 1 + 6λ2     ∂w  4(c11 − c12 )λ2 c11 − (7.44) + ∂r r =R 1 + 6λ2 Using Eq. (7.28) and after rearranging of Eq. (7.44), we obtain F RF =

c11 C1 1 = R 3/2 P C f Bs R J5/2 (Bs R) − (2 cc1211 + 1)J3/2 (Bs R)

(7.45)

where Cf = 1−

4(1 − 1+

c12 )λ2 c11 6λ2

(7.46)

7.3 Vibration of Spherical Nanoparticles

157

Fig. 7.4 Variations of the FRF with the excitation frequency for Au nanoparticle with radius 1 nm

Note that Eq. (7.45) is defined as the frequency response function (FRF) of the system. Although the FRF is defined as the non-dimensional ratio of the response and force, the frequency response function is independent of both of them. It is wellknown that the roots of the denominator of the FRF are the natural frequencies of the system. To simplify the numerical analysis, the magnitude of the FRF is converted into its decibel scale defined as:    c11 C1  (7.47) Γd B = 20log10  3/2  R P To illustrate the small-scale effects, the variations of the FRF as a function of the excitation frequency for the Au nanoparticles are shown in Fig. 7.4. As expected, the FRF approaches infinity when the excitation frequency is close to the natural frequencies. The physical meaning of this is that the resonance phenomenon occurs. An interesting implication of the nonlocal elasticity approach is that the resonance frequencies get closer to each other at higher frequencies. It is also seen from Fig. 7.4 that the peaks of the curves are equally spaced throughout the frequency domain for e0 L i = Rλ = 0. Furthermore, it is observed that the FRF has a local minimum between two consecutive resonance frequencies. The frequency at which this phenomenon occurs is called minima.

References 1. D. Guo, G. Xie, J. Luo, Mechanical properties of nanoparticles: basics and applications. J. Phys. D: Appl. Phys. 47, 013001 (2014) 2. S.A. Fazelzadeh, E. Ghavanloo, Coupled axisymmetric vibration of nonlocal fluid-filled closed spherical membrane shell. Acta Mech. 223, 2011–2020 (2012)

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3. E. Ghavanloo, S.A. Fazelzadeh, Nonlocal elasticity theory for radial vibration of nanoscale spherical shells. Eur. J. Mech. A Solids 41, 37–42 (2013) 4. R.E. Stanton, M.D. Newton, Normal vibrational modes of Buckminsterfullerene. J. Phys. Chem. 92, 2141–2145 (1988) 5. L.T. Chadderton, Axisymmetric vibrational modes of fullerene C60 . J. Phys. Chem. Solids 54, 1027–1033 (1993) 6. S. Adhikari, R. Chowdhury, Vibration spectra of fullerene family. Phys. Lett. A 375, 2166–2170 (2011) 7. J.H. Lee, B.S. Lee, F.T.K. Au, J. Zhang, Y. Zeng, Vibrational and dynamic analysis of C60 and C30 fullerenes using FEM. Comput. Mater. Sci. 56, 131–140 (2012) 8. H. Nejat Pishkenari, P. Ghaf Ghanbari, Vibrational analysis of the fullerene family using Tersoff potential. Curr. Appl. Phys. 17, 72–77 (2017) 9. E. Ghavanloo, S.A. Fazelzadeh, H. Rafii-Tabar, A computational modeling of Raman radial breathing-like mode frequencies of fullerene encapsulated inside single-walled carbon nanotubes. J. Mol. Model. 23, 48 (2017) 10. R. Zaera, J. Fernández-Sáez, J.A. Loya, Axisymmetric free vibration of closed thin spherical nano-shell. Compos. Struct. 104, 154–161 (2013) 11. E. Ghavanloo, S.A. Fazelzadeh, Nonlocal shell model for predicting axisymmetric vibration of spherical shell-like nanostructures. Mech. Adv. Mater. Struct. 22, 597–603 (2015) 12. J. Vila, R. Zaera, J. Fernández-Sáez, Axisymmetric free vibration of closed thin spherical nanoshells with bending effects. J. Vib. Control 22, 3789–3806 (2016) 13. H. Guo, L. He, B. Xing, Applications of surface-enhanced Raman spectroscopy in the analysis of nanoparticles in the environment. Environ. Sci. Nano 4, 2093–2107 (2017) 14. A. Crut, P. Maioli, N. Del Fatti, F. Vallee, Time-domain investigation of the acoustic vibrations of metal nanoparticles: size and encapsulation effects. Ultrasonics 56, 98–108 (2015) 15. 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, 312–329 (2015) 16. H. Lamb, On the vibrations of a spherical shell. Proc. Lond. Math. Soc. s1–14, 50–56 (1882) 17. D.B. Murray, L. Saviot, Acoustic vibrations of embedded spherical nanoparticles. Phys. E 26, 417–421 (2005) 18. V. Mankad, S.K. Gupta, P.K. Jha, N.N. Ovsyuk, G.A. Kachurin, Low-frequency Raman scattering from Si/Ge nanocrystals in different matrixes caused by acoustic phonon quantization. J. Appl. Phys. 112, 054318 (2012) 19. J. Wang, Y. Gao, M.Y. Ng, Y.C. Chang, Radial vibration of ultra-small nanoparticles with surface effects. J. Phys. Chem. Solids 85, 287–292 (2015) 20. E. Ghavanloo, S.A. Fazelzadeh, Radial vibration of free anisotropic nanoparticles based on nonlocal continuum mechanics. Nanotechnology 24, 075702 (2013) 21. S.A. Fazelzadeh, E. Ghavanloo, Radial vibration characteristics of spherical nanoparticles immersed in fluid medium. Modern Phys. Lett. B 27, 1350186 (2013) 22. E. Ghavanloo, S.A. Fazelzadeh, T. Murmu, S. Adhikari, Radial breathing-mode frequency of elastically confined spherical nanoparticles subjected to circumferential magnetic field. Phys. E 66, 228–233 (2015) 23. E. Ghavanloo, A. Abbasszadehrad, Frequency domain analysis of nano-objects subject to periodic external excitation. Iran. J. Sci. Technol. Trans. Mech. Eng. (2018). https://doi.org/10. 1007/s40997-018-0178-5 24. H. Rafii-Tabar, E. Ghavanloo, S.A. Fazelzadeh, Nonlocal continuum-based modeling of mechanical characteristics of nanoscopic structures. Phys. Rep. 638, 1–97 (2016) 25. M. Todt, F.G. Rammerstorfer, F.D. Fischer, P.H. Mayrhofer, D. Holec, M.A. Hartmann, Continuum modeling of van der Waals interactions between carbon onion layers. Carbon 49, 1620– 1627 (2011) 26. D. Kahn, K.W. Kim, M.A. Stroscio, Quantized vibrational modes of nanospheres and nanotubes in the elastic continuum model. J. Appl. Phys. 89, 5107–5111 (2001) 27. E. Ghavanloo, S.A. Fazelzadeh, Oscillations of spherical fullerenes interacting with graphene sheet. Phys. B 504, 47–51 (2017)

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Chapter 8

Modelling the Mechanical Characteristics of One-Dimensional Nanoscopic Structures

This chapter is concerned with the study of the mechanical characteristics of one-dimensional nanoscopic structures (ODNSs) that are described on the basis of the nonlocal models presented in Chap. 5. ODNSs are materials with only one of their dimensions outside the nanometer range. In the last decades, different ODNSs including nanowires, nanorods, nanotubes, nanobelts and nanoribbons of a variety of materials have been successfully synthesized and employed. These structures can be (a) metallic, ceramic, or polymeric, (b) amorphous or crystalline, (c) isolated or embedded within a continuum [1]. They are exotic basic material units and always possess unique mechanical characteristics including extraordinary small dimensions, and flexibility and mechanical strength. Furthermore, ODNSs represent the smallest-dimension structures that can efficiently transport charge carriers, and so they are expected to play significant roles in the functioning of integrated nanoscale devices [2]. In addition, these nanoscopic structures are ideal for exploring a large number of unique phenomena at the nanoscale [3]. As a result, ODNSs have stimulated a great deal of interest due to their enormous technological importance as well as prompting numerous advances in both fundamental and applied sciences. They have found extensive potential technological applications in nanoelectronic, nanooptoelectronic, nanoelectro-mechanical systems, nanocomposite materials, biological and chemical sensing and as probe tips in novel probe microscopy [4, 5]. Furthermore, ODNSs will play a very significant role as building blocks for the next-generation of nanoscale devices. From an engineering viewpoint, understanding the mechanical characteristics of ODNSs is an essential prerequisite for the reliable design of such devices. Therefore, a variety of theoretical, computational and experimental methods have been developed to investigate the mechanical properties and behaviour of ODNSs. In this chapter, the recently developed experimental techniques to study the mechanical behaviour of ODNSs will be first reviewed. We then discuss the theoretical modelling of the mechanical behaviour of nanorods and nanowires, including their vibrational behaviour, and their static deformations, employing the © Springer Nature Switzerland AG 2019 E. Ghavanloo et al., Computational Continuum Mechanics of Nanoscopic Structures, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-3-030-11650-7_8

161

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nonlocal models. Since the nanotubes, as a special family of ODNSs, deserve special attention, therefore, the mechanical modelling of nanotubes will be discussed separately in Chap. 9.

8.1 Measurement of Mechanical Characteristics of One-Dimensional Nanoscopic Structures Property measurements of ODNSs are extremely challenging and difficult tasks to perform because of the small sizes involved. Over the past two decades, several experimental techniques for the measurement of mechanical properties, specifically the elastic properties, of ODNSs have been developed, including the nanoscale tensile test, the beam bending technique, the electromechanical resonance technique and the nanoindentation technique [6]. In this section, we briefly review the most important measurement techniques.

8.1.1 Nanoscale Tensile Test A tensile test is a standard and reliable technique for simultaneously observing and measuring the mechanical characteristics of macroscale specimens. It has been shown that a tensile test can also be implemented at the nanoscale [7, 8]. However, application of this measurement technique at the nanoscale involves immense challenges due to the extreme difficulty in manipulation, loading, and force and deformation sensing in experiments. The first attempt to perform the tensile testing of ODNSs was carried out by Yu et al. [9]. In that study, two ends of a multi-walled carbon nanotube (MWCNT) were placed between two opposing AFM tips. One of the AFM tips was rigid, while the other was very soft and was used as a load sensor. The stiff AFM tip was moved and consequently the MWCNT was pulled along its axis. The elongation was video recorded by SEM images at each loading step. The schematic of the nanoscale tensile test is shown in Fig. 8.1. That study attracted extensive attention and subsequently a series of publications have appeared by several research groups. However, these tensile tests encounter significant challenges since the specimens must be clamped at both ends of the probes.

8.1.2 Beam Bending Techniques Beam bending techniques are typically grouped into two categories; the cantilever beam bending and the three-point bending. In the first approach, one end of the ODNS is fixed rigidly to a substrate, while a known force is applied at the other end of the

8.1 Measurement of Mechanical Characteristics …

163

Fig. 8.1 Schematic illustration of a nanoscale tensile test

Fig. 8.2 Schematic illustration of a cantilever beam bending test

Fig. 8.3 Schematic illustration of a three-point bending test

specimen (Fig. 8.2). The cantilever beam bending was first adopted for some ODNSs in 1997 by Wong et al. [10]. In the alternative approach, the three-point bending, the ODNS is clamped at both ends, and loaded transversely with a concentrated load applied in the middle of the ODNS. In practice, ODNSs were deformed transversely by the AFM probe (Fig. 8.3). As the AFM probe is moved downward, the bending load applied to the nanomaterial could be measured [11]. Such a set up was used for the first time by Salvetat and co-workers [12] to quantify the bending modulus of carbon nanotubes (CNTs). It should be noted that the bending tests at the nanoscale are easier than the nanoscale tensile tests because the sample preparation is relatively simple for the former tests. Furthermore, the bending deformation is usually large enough and so

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8 Modelling the Mechanical Characteristics …

it may be measured by SEM images. However, these experimental techniques have the following limitations [13]: (1) Since the bending stiffness of ODNSs is much smaller than their tensile stiffness, high force resolution is required for nanoscale bending tests; (2) The bending test is very sensitive to uncertainties in cross-section of ODNSs; (3) Compared to the nanoscale tensile test, it is not straightforward particularly during the large deformation.

8.1.3 Mechanical Resonance Technique Mechanical resonance testing is a nondestructive technique for measuring the mechanical properties of ODNSs. The schematic representation of this technique is depicted in Fig. 8.4. In this technique, the cantilever ODNS can be excited mechanically or electrically. The resonance phenomenon occurs when the excitation frequency matches the natural frequency of the ODNS. After the measurement of the resonance frequencies, the dimensions of ODNS are measured from scan images. Using the measured resonance frequency and dimensions, the bending modulus of the ODNS is calculated. The first calculation of the elastic bending-modulus of ODNSs by using the mechanical resonance testing was reported in 1999 by Poncharal et al. [14]. Subsequently this method was utilized for various ODNSs such as single-walled carbon nanotubes (SWCNTs) [15], MWCNTs [16] and nanowires [17, 18]. One advantage of using this technique is that the testing method is particularly suitable for in-situ testing. Furthermore, only the resonance excitation instrument is only required for such experiments, and no further devices are required. There are, of course, some disadvantages in the application of the mechanical resonance testing. For example, this technique can only measure the bending stiffness of ODNSs that are permanently attached to a substrate or a probe.

Fig. 8.4 Schematic illustration of the mechanical resonance technique

8.1 Measurement of Mechanical Characteristics …

165

8.1.4 Nanoindentation Technique Nanoindentation is the leading technique for a quantitative characterization of the mechanical properties of nanoscopic structures [19]. This method was introduced in 1992 [20] to determine the hardness and elastic modulus of materials. In this technique, a prescribed load is applied to an indenter in contact with the specimen. During nanoindentation testing, the applied load and penetration depth are simultaneously recorded and consequently the load-displacement curve is obtained. The hardness and elastic modulus of ODNS can be determined from the load-displacement curve. It should be noted that the nanoindentation technique is similar to the three-point bending test described above. The main difference between the two techniques is that, in the three-point bending test, ODNSs are suspended while ODNSs stay on top of a flat substrate in the nanoindentation technique. We refer the reader to the reviews on the nanoindentation [19, 21].

8.1.5 Challenges in the Mechanical Characterisation of ODNSs In recent years, experimental investigations at the nanoscale have been rapidly developed to obtain the mechanical characteristics of ODNSs. However, mechanical testing remains a challenge because two characteristic dimensions of the specimen are within the nanometer range [22]. The main challenges in measuring the mechanical characteristics of ODNSs include: (1) specimen fabrication, manipulation and positioning of specimens with nanometer resolution and high throughput; (2) application and measurement of forces in the nano-Newton (nN) range and (3) measurement of displacement with nanometer (nm) accuracy. Furthermore, these mechanical tests are extremely difficult and expensive to perform. As a result, the current experimental data are either insufficient or inconsistent. Therefore, most studies on the mechanical characteristics of ODNSs rely mainly on theoretical analyses and numerical simulations.

8.2 Vibration of Nanorods One of the most familiar type of ODNSs is a nanorod or a nanowire. The difference between a nanorod and a nanowire is related to their aspect ratios (length-to-diameter ratio). Nanowires have high aspect ratios, e.g., larger than 5, while the aspect ratios of nanorods are in the range of 3–5 [23, 24]. In the case of the nanorods, two specific vibrational modes have been distinguished. These are the extensional or axial mode (Fig. 8.5a) and the breathing-mode (Fig. 8.5b) [25]. The extensional mode is characterised by the periodic extension and contraction of the nanorods along their

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8 Modelling the Mechanical Characteristics …

Fig. 8.5 Two vibrational modes of a cylindrical nanorod: a extensional mode and b breathing-mode

central-lines. The breathing-mode of the nanorods corresponds to a synchronous and symmetric in-phase displacements of all atoms in the radial direction [26]. In this section, we provide the theoretical framework for modelling the extensional and breathing-modes of the nanorods on the basis of the nonlocal elasticity theory.

8.2.1 Extensional Vibration 8.2.1.1

Nonlocal Differential Approach

The nonlocal differential model is widely used to investigate the size-dependent behaviour of a nanorod due to its simplicity. Based on Eq. (5.21) and setting p = 0, the governing equation for the extensional vibration of the nanorod is obtained as EA

  2 4 ∂ u ∂ 2u 2 2 ∂ u = ρ A − l λ ∂x2 ∂t 2 ∂t 2 ∂ x 2

(8.1)

For the free vibrational analysis, it is assumed that the displacement u varies harmonically with respect to time and, therefore, the expression for the axial displacement is: u(x, t) = X (x) exp( jωt)

(8.2)

√ where j = −1, ω denotes the circular frequency and X (x) represents the spatial function. Substituting Eq. (8.2) into Eq. (8.1) leads to d2 X + β2 X = 0 dx2

(8.3)

8.2 Vibration of Nanorods

167

where β2 =

ρω2 (E − ρl 2 λ2 ω2 )

(8.4)

The general solution of Eq. (8.4) can be expressed as X (x) = C1 sin(βx) + C2 cos(βx)

(8.5)

in which C1 and C2 are unknown constants which can be evaluated from the boundary conditions. Let us first consider a nanorod of length l which is clamped at both ends. The appropriate boundary conditions for this case are u(0, t) = 0 → X (0) = 0 u(l, t) = 0 → X (l) = 0

(8.6)

Applying the boundary conditions to the solution (8.5), a set of homogeneous algebraic equations with respect to the constants C1 and C2 is obtained. 

0 1 sin(βl) cos(βl)



C1 C2

 =

  0 0

(8.7)

In order C12 + C22 to be different from zero, the following determinant must vanish:    0 1    sin(βl) cos(βl)  = 0

(8.8)

Hence, in this case, we have sin(βl) = 0

(8.9)

The nonlocal extensional frequencies of the nanorod can be calculated on the basis of Eq. (8.4) by computing the roots of Eq. (8.9). The resultant equation is  nπ ωn = l

E ρ(1 + λ2 n 2 π 2 )

n = 1, 2, 3, ...

(8.10)

Other boundary conditions, to which the nanorod could be subject, are clampedfree conditions. In this case, we have u(0, t) = 0 → X (0) = 0     2 2 2 dX  =0 N (l, t) = 0 → E − ρλ l ω d x x=l

(8.11)

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8 Modelling the Mechanical Characteristics …

Table 8.1 Comparison between the fundamental periods of extensional vibration of the nonlocal model and those computed from MD simulations for gold nanorods T (ps) Nanorod Length (nm) Local model Nonlocal mode MD simulation λ=0 λ = 0.13 [27] Sample 1 Sample 2 Sample 3 Sample 4 Sample 5 Sample 6

17.26 21.33 25.39 29.43 33.49 37.52

21.55 26.64 31.71 36.75 41.82 46.86

23.28 28.77 34.25 39.70 45.18 50.61

23.45 28.77 34.26 39.73 45.12 50.61

Using the boundary conditions (8.11), the nonlocal extensional frequencies of the cantilever nanorod are obtained as follow  4E (2n − 1)π n = 1, 2, 3, ... (8.12) ωn = 2 2l ρ(4 + λ (2n − 1)2 π 2 ) The validity of the nonlocal model for the extensional vibration of nanorods can be checked against the results reported in the literature. Based on Eq. (8.10), the fundamental period of extensional vibration of clamped-clamped nanorods, n = 1, can be written as  ρ(1 + λ2 π 2 ) Text = 2l (8.13) E In Table 8.1, the predicted fundamental periods of extensional vibration of several gold nanorods are compared with the results obtained from MD simulations [27]. The following values are used in the numerical calculations: Young’s modulus E = 49.5 GPa and density ρ = 19.3 g/cm3 [27]. The results show that the nonlocal model could predict the MD simulations results better than the local model.

8.2.1.2

Nonlocal Integral Approach

Here, the extensional vibration of nanorods is studied on the basis of the nonlocal integral model. By neglecting the distributed axial force, p, from Eq. (5.19), we can derive the governing equation which describes the extensional vibration of a nanorod as ⎡ l ⎤  ∂ ⎣ ∂u(ξ, t) ⎦ ∂ 2 u(x, t) EA α(x − ξ ) (8.14) dξ = ρ A ∂x ∂ξ ∂t 2 0

8.2 Vibration of Nanorods

169

As explained in Chap. 4, for the one-dimensional problem, the most commonly used kernel function is   |x − ξ | 1 exp − (8.15) α(x − ξ ) = 2lλ lλ Substituting Eq. (8.15) into Eq. (8.14), we have ⎡ EA

∂ ⎣ ∂x

⎤   |x − ξ | ∂u(ξ, t) ∂ 2 u(x, t) 1 exp − dξ ⎦ = ρ A 2lλ lλ ∂ξ ∂t 2

l 0

(8.16)

The integro-differential equation (8.16) can be rewritten as EA lλ

  x − ξ ∂u(ξ, t) 1 exp dξ 2lλ lλ ∂ξ

l x

−

EA lλ

x 0

  ξ − x ∂u(ξ, t) ∂ 2 u(x, t) 1 exp dξ = ρ A 2lλ lλ ∂ξ ∂t 2

(8.17)

Differentiating Eq. (8.17) with respect to x yields EA E A ∂u(x, t) + 2 2 − 2 2 l λ ∂x l λ +

EA l 2 λ2

l

x 0



x −ξ 1 exp 2lλ lλ

  1 ξ − x ∂u(ξ, t) exp dξ 2lλ lλ ∂ξ 

x

∂u(ξ, t) ∂ 3 u(x, t) dξ = ρ A ∂ξ ∂ x∂t 2

(8.18)

To eliminate the integral terms from Eq. (8.18), we can again apply differentiation with respect to x. The resultant equation is   2 4 ∂ 2 u(x, t) ∂ u(x, t) 2 2 ∂ u(x, t) EA = ρA −l λ ∂x2 ∂t 2 ∂ x 2 ∂t 2

(8.19)

with the following boundary conditions ⎡ −

E A ∂u(0, t) EA + 2 2⎣ l 2 λ2 ∂ x l λ

l 0



−ξ 1 exp 2lλ lλ



⎤ ∂u(ξ, t) ⎦ ∂ 3 u(0, t) dξ = ρ A ∂ξ ∂ x∂t 2 (8.20)

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8 Modelling the Mechanical Characteristics …

⎤ ⎡ l    EA ⎣ ξ − l ∂u(ξ, t) ⎦ ∂ 3 u(l, t) 1 E A ∂u(l, t) + 2 2 exp dξ = ρ A − 2 2 l λ ∂x l λ 2lλ lλ ∂ξ ∂ x∂t 2 0

(8.21) The integral terms in Eqs. (8.20) and (8.21) can be eliminated by using Eq. (8.17). Therefore, the boundary conditions are rewritten as EA

∂ 2 u(0, t) ∂ 3 u(0, t) ∂u(0, t) − lλρ A + l 2 λ2 ρ A =0 2 ∂x ∂t ∂ x∂t 2

(8.22)

∂ 2 u(l, t) ∂u(l, t) ∂ 3 u(l, t) 2 2 + lλρ A + l λ ρ A =0 ∂x ∂t 2 ∂ x∂t 2

(8.23)

EA

Equations (8.22) and (8.23) are known as constitutive boundary conditions [28]. These boundary conditions will be discussed in Chap. 12. By substituting Eq. (8.2) into Eqs. (8.22)–(8.23) and simplifying the resulting equations, we obtain  d X  (E − l 2 λ2 ρω2 ) + lλρω2 X (0) = 0 d x x=0  d X  (E − l 2 λ2 ρω2 ) − lλρω2 X (l) = 0 d x x=l

(8.24) (8.25)

Furthermore, we have two standard boundary conditions given by X (0) = X (l) = 0

(8.26)

Using Eq. (8.5) and the boundary conditions (8.24)–(8.26), the only solution is X (x) = 0. This means that there are no nontrivial solutions and the fully nonlocal integral model is an ill-posed problem. A similar conclusion has been stated in Refs. [28–30]. To overcome this limitation, some effective techniques have been developed. One of these techniques will be discussed in the next subsection. 8.2.1.3

Two-Phase Nonlocal Integral Approach

To eliminate the ill-posed nature of the fully nonlocal integral model, a mixture of local-nonlocal constitutive relation has been frequently utilized in the literature [28–32] on the basis of the original proposal by Eringen [33]. The local-nonlocal constitutive mixture can be defined by a convex combination of local and nonlocal phases. Therefore, a modified kernel function can be proposed for the mixture of the local-nonlocal relation as   |x − ξ | (1 − η1 ) exp − (8.27) α(x − ξ ) = η1 δ(|x − ξ |) + 2lλ lλ

8.2 Vibration of Nanorods

171

where δ(|x − ξ |) is the Dirac delta function and η1 is a nonnegative material constant. The phase parameter η1 can vary from 0 to 1. It should be noted that the fully nonlocal model is recovered by setting η1 = 0, and the local model corresponds to η2 = 0. Substituting Eq. (8.27) into Eq. (8.14), we have ⎡ l ⎤    |x − ξ | ∂u(ξ, t) ∂ u(x, t) (1 − η1 ) ∂ ⎣ η1 + exp − dξ ⎦ ∂x2 2lλ ∂ x lλ ∂ξ 2

0

ρ ∂ 2 u(x, t) = E ∂t 2

(8.28)

Initially, let us remove the modulus in the integrand, i.e., ⎡ x ⎤      l (1 − η1 ) ∂ ⎣ ξ − x ∂u(ξ, t) x − ξ ∂u(ξ, t) ⎦ exp dξ + exp dξ 2lλ ∂ x lλ ∂ξ lλ ∂ξ x

0

∂ 2 u(x, t) ρ ∂ 2 u(x, t) − η = 1 E ∂t 2 ∂x2

(8.29)

Then differentiating Eq. (8.29) with respect to x twice yields 2(1 − η1 ) ∂ 2 u(x, t) (1 − η1 ) − + 2l 2 λ2 ∂x2 2l 3 λ3 +

(1 − η1 ) 2l 3 λ3



l exp

x −ξ lλ

x





x exp 0

ξ −x lλ



∂u(ξ, t) dξ ∂ξ

∂u(ξ, t) ρ ∂ 4 u(x, t) ∂ 4 u(x, t) dξ = − η 1 ∂ξ E ∂ x 2 ∂t 2 ∂x4 (8.30)

Eliminating the integral terms from Eqs. (8.29) and (8.30), the partial differential equation for the extensional vibration of the nanorod can be written as η1

∂ 4 u(x, t) 1 ∂ 2 u(x, t) ρ ∂ 4 u(x, t) ρ ∂ 2 u(x, t) − − + = 0 (8.31) ∂x4 l 2 λ2 ∂ x 2 E ∂ x 2 ∂t 2 El 2 λ2 ∂t 2

with the boundary conditions ∂ 3 u(0, t) ∂ 2 u(0, t) ∂u(0, t) + η lλ + (1 − η1 ) 1 3 2 ∂x ∂x ∂x ρ 2 2 ∂ 3 u(0, t) ρlλ ∂ 2 u(0, t) − =0 + l λ E ∂ x∂t 2 E ∂t 2 −η1l 2 λ2

(8.32)

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8 Modelling the Mechanical Characteristics …

Table 8.2 Standard boundary conditions of the nanorods End conditions Boundary conditions Clamped-clamped Clamped-free

u(0, t) = u(l, t) = 0

 2 u(0, t) = 0 and N (l, t) = η1 ∂u(l,t) + η1 ∂ ∂u(l,t) − ∂x x2

ρ ∂ 2 u(l,t) E ∂t 2

∂ 3 u(l, t) ∂ 2 u(l, t) ∂u(l, t) − η lλ + (1 − η1 ) 1 3 2 ∂x ∂x ∂x ρ 2 2 ∂ 3 u(l, t) ρlλ ∂ 2 u(l, t) + =0 + l λ E ∂ x∂t 2 E ∂t 2

 lλ = 0

−η1l 2 λ2

(8.33)

It is interesting to note that we have two other specified standard end conditions given in Table 8.2. Substituting Eq. (8.2) into Eqs. (8.31)–(8.33) leads to   ρω2 1 d4 X d 2 X ρω2 − − η1 4 + X =0 dx dx2 E l 2 λ2 El 2 λ2    d 3 X  d 2 X  d X  −η1l 2 λ2 + η lλ + (1 − η ) 1 1 d x 3 x=0 d x 2 x=0 d x x=0  ρω2 2 2 d X  ρlω2 λ − + l λ X (0) = 0 E dx  E

(8.34)

(8.35)

x=0

   d 3 X  d 2 X  d X  − η lλ + (1 − η ) 1 1 d x 3 x=l d x 2 x=l d x x=l  ρω2 2 2 d X  ρlω2 λ l λ X (l) = 0 − −  E d x x=l E −η1l 2 λ2

(8.36)

Introducing the dimensionless variables x X X¯ = , x¯ = , ω¯ = ωl l l



ρ E

(8.37)

Eqs. (8.34)–(8.36) can then be written as   ω¯ 2 d 4 X¯ d 2 X¯ 1 2 ω ¯ − 2 X¯ = 0 + − 4 2 2 d x¯ d x¯ λ λ    3 ¯  d 2 X¯  d X¯  2 d X + η1 λ 2  + (1 − η1 ) −η1 λ d x¯ 3 x=0 d x¯ x=0 d x¯ x=0 ¯ ¯ ¯  ¯  dX  −ω¯ 2 λ2 + ω¯ 2 λ X¯ (0) = 0 d x¯  η1

x=0 ¯

(8.38)

(8.39)

8.2 Vibration of Nanorods

173

   d 3 X¯  d 2 X¯  d X¯  −η1 λ − η1 λ 2  + (1 − η1 ) d x¯ 3 x=1 d x¯ x=1 d x¯ x=1 ¯ ¯ ¯  ¯  d X  −ω¯ 2 λ2 − ω¯ 2 λ X¯ (1) = 0 d x¯  2

(8.40)

x=1 ¯

The general solution of Eq. (8.38) is given by X¯ (x) ¯ = C1 sin(r1 x) ¯ + C2 cos(r1 x) ¯ + C3 exp(r2 x) ¯ + C4 exp(r2 x) ¯

(8.41)

where  r1 =

2(2η1 − 1)λ2 ω¯ 2 + λ4 ω¯ 4 + 1 + λ2 ω¯ 2 − 1 2η1 λ2

(8.42)

2(2η1 − 1)λ2 ω¯ 2 + λ4 ω¯ 4 + 1 − λ2 ω¯ 2 + 1 2η1 λ2

(8.43)

 r2 =

Furthermore, C1 , C2 , C3 and C4 are unknown constants to be determined from the boundary conditions. Although exact solutions can be found for both clampedclamped and clamped-free boundary conditions, they will not, however, be sought here. Alternatively, we present the approximate solutions reported by Zhu and Li [30]. The approximate solutions pertinent to the clamped-clamped boundary conditions can be expressed as [30] ω¯ = nπ(1 − α1 λ − α2 λ2 − α3 λ3 )

n = 1, 2, 3, ...

(8.44)

where √ α1 = 2(1 − η1 )   1 √ √ √ α2 = (1 − η1 ) n 2 π 2 (1 + η1 ) − 8(1 − η1 ) 2   1 √ √ √ α3 = (1 − η1 ) 24(1 + η1 − 2 η1 ) + n 2 π 2 (4η1 − 5 η1 − 11) 3 (8.45) In a similar way, the nonlocal extensional frequencies of the clamped-free boundary conditions, on the basis of two-phase nonlocal elasticity, can be expressed as ω¯ =

  (2n − 1)π 1 − β1 λ − β2 λ2 − β3 λ3 2

n = 1, 2, 3, ...

(8.46)

174

8 Modelling the Mechanical Characteristics …

Table 8.3 Computed fundamental periods of extensional vibration for clamped-clamped gold nanorods T (ps) Nanorod Length (nm) Eq. (8.44) η1 = 0.01, λ = 0.04 MD simulation [27] Sample 1 Sample 2 Sample 3 Sample 4 Sample 5 Sample 6

17.26 21.33 25.39 29.43 33.49 37.52

23.37 28.88 34.37 39.84 45.34 50.79

23.45 28.77 34.26 39.73 45.12 50.61

where √ β1 = (1 − η1 ) π2 √ 2 (2n − 1)2 (1 − η1 ) − (1 − η1 ) β2 = 8  1 √ √ 2 2 β3 = β1 1 + η1 − 2 η1 + (2n − 1) π (4η1 − 11 η1 − 11) (8.47) 24 Table 8.3 lists a set of selected values of the fundamental periods of extensional vibration (2π/ω) for clamped-clamped gold nanorods, calculated by Eq. (8.44) and an MD simulation reported in Ref. [27]. The results show that the two-phase nonlocal integral approach with the properly selected phase parameter and nonlocal scale coefficient can provide a remarkably accurate prediction of the extensional vibration of nanorods. The above formalism has been applied [30] to the computation of the natural frequencies of both clamped-clamped and clamped-free end conditions. Table 8.4 lists a set of selected values of the dimensionless natural frequencies computed from Eq. (8.44) for clamped-clamped nanorods. Similar results obtained from Eq. (8.46) are listed in Table 8.5 for clamped-free nanorods. The results show that the natural frequencies decrease as the nonlocal scale coefficient increases and the phase parameter decreases. It should be noted that the local-nonlocal mixture method is an effective approach to eliminate the ill-posed nature of the problem for η1 > 0. However, it will tend to an ill-posed problem anyway for η1 → 0, where the full nonlocal model is recovered.

8.2.2 Breathing Vibration The nonlocal model for nanocrystals, described in Sect. 5.5, is employed to derive the breathing vibration of nanorods. Based on Eq. (5.130) and setting k = 1, the governing equation for the breathing vibration of the nanorod is obtained as

8.2 Vibration of Nanorods

175

Table 8.4 Computed fundamental periods of extensional vibration for clamped-clamped gold nanorods λ η1 = 0.001 η1 = 0.010 η1 = 0.100 η1 = 1.000 n=1 n=2 n=1 n=2 n=1 n=2 n=1 n=2 0.0005 0.0010 0.0050 0.0100 0.0500

3.1385 3.1355 3.1111 3.0805 2.8392

6.2771 6.2709 6.2199 6.1523 5.5298

3.1388 3.1359 3.1132 3.0846 2.8570

6.2775 6.2718 6.2242 6.1607 5.5636

3.1394 3.1373 3.1199 3.0979 2.9163

6.2789 6.2745 6.2378 6.1879 5.6880

Table 8.5 Dimensionless natural frequencies of clamped-free nanorods λ η1 = 0.001 η1 = 0.010 η1 = 0.100 n=1 n=2 n=1 n=2 n=1 n=2 0.0005 0.0010 0.0050 0.0100 0.0500

1.5700 1.5693 1.5632 1.5555 1.4943

 ρR λ

2 2

−ρ

4.7101 4.7078 4.6884 4.6622 4.3880

1.5701 1.5694 1.5637 1.5566 1.4992

4.7103 4.7081 4.6900 4.6654 4.4036

∂ 4w 1 ∂ 3w 1 ∂ 2w + − ∂r 2 ∂t 2 r ∂r ∂t 2 r 2 ∂t 2

1.5703 1.5697 1.5654 1.5600 1.5153



4.7108 4.7091 4.6952 4.6758 4.4599

 + c11

3.1416 3.1416 3.1416 3.1416 3.1416

6.2832 6.2832 6.2832 6.2832 6.2832

η1 = 1.000 n=1 n=2 1.5708 1.5708 1.5708 1.5708 1.5708

4.7124 4.7124 4.7124 4.7124 4.7124

∂ 2 w 1 ∂w 1 − 2w + ∂r 2 r ∂r r



∂ 2w =0 ∂t 2

(8.48)

Using the method of separation of variables, the displacement w can be written as: w(r, t) = W (r ) exp( jωt)

(8.49)

Substitution of the solution (8.49) into governing equation (8.48) leads to d2W 1 dW +W + dr 2 r dr

 Bc2

1 − 2 r

 =0

(8.50)

where ρω2 R 2 c11

(8.51)

    dW 1 r + W Bc2 r − =0 dr r

(8.52)

Bc =

√

Ω

R 1 − Ω 2 λ2

, Ω2 =

Multiplying Eq. (8.50) by r gives d dr

176

8 Modelling the Mechanical Characteristics …

Employing a proper change of variables, Eq. (8.52) reduces to the Bessel equation with general solution: W (r ) = C1 J1 (Bc r ) + C2 Y1 (Bc r )

(8.53)

where J1 and Y1 are first and second kind Bessel functions of order one respectively, and C1 and C2 are constants to be determined using the boundary conditions. Since the displacement at the origin is not infinite, the coefficient of the Bessel function of the second kind must be zero, so we have W (r ) = C1 J1 (Bc r )

(8.54)

For the stress-free boundary condition, we obtain the following equation:  W (R) dW  =0 + a12 dr r =R R

(8.55)

where a12 =

c12 c11

(8.56)

Substituting Eq. (8.54) into the boundary condition (8.55), the frequency equation is obtained as:     Ω Ω Ω − J1 √ J2 √ √ (1 + a12 ) = 0 (8.57) 1 − Ω 2 λ2 1 − Ω 2 λ2 1 − Ω 2 λ2 By solving Eq. (8.56), we can obtain the natural frequencies of the nanorods. It should be noted that the lowest frequency is known as the breathing-mode frequency. To demonstrate the relevance of the nonlocal model to predict the breathing-mode frequency of nanorods with different diameters, the calculated breathing frequencies of Si nanorods are displayed in Fig. 8.6 and compared with the results given by Trejo et al. [34] based on ab initio calculations. For numerical calculations in this figure, we adopted c11 = 1.6578 × 1011 N/m2 , c12 = 0.6394 × 1011 N/m2 , ρ = 2.331 g/cm3 [35]. It can be seen that the nonlocal results are in reasonable agreement with the results reported in Ref. [34]. Figure 8.7 shows the variations of the dimensionless frequency, Ω, with the nonlocal coefficient, λ, for different values of the transverse to longitudinal elastic moduli ratio, a12 . The results show that the frequencies increase with an increase in the transverse-to-longitudinal elastic moduli ratio.

8.3 Static Deformation of Nanorods Under Axial Loadings

177

Fig. 8.6 Comparison between the breathing mode frequencies of silicon nanorods obtained from the nonlocal model and ab initio calculations

Fig. 8.7 Variations of the dimensionless breathing frequency of the nanorod with the nonlocal coefficient

8.3 Static Deformation of Nanorods Under Axial Loadings 8.3.1 Nonlocal Differential Approach In this subsection, the static deformation of nanorods under axial loadings will be analysed on the basis of the nonlocal differential approach. For this purpose, it is assumed that the variables in Eq. (5.21) are time independent and so the governing equation becomes

178

8 Modelling the Mechanical Characteristics …

Table 8.6 Analytical solutions for static deformation of nanorods Loading condition Clamped-clamped Clamped-free p(x) = p0 p(x) =

p0 x l

EA

u(x) =

p0 x 2E A (l

u(x) =

p0 x 2 6E Al (l

− x) − x 2)

u(x) =

p0 x E A (l

u(x) =

p0 x 2 2 E Al (l λ

− x2 )

d 2 p(x) d 2 u(x) + p(x) − l 2 λ2 =0 2 dx dx2

+

l2 2

−

x2 6 )

(8.58)

The solution of the nonlocal governing equation (5.85) can be obtained as [36] l 2 λ2 1 u(x) = C1 + C2 x + p(x) − EA EA

x (x − s) p(s)ds

(8.59)

0

The constants C1 and C2 in solution (8.59) are determined from the boundary conditions. In addition, using Eqs. (5.8) and (5.16), we obtain N (x) = E A

dp(x) du(x) − l 2 λ2 dx dx

(8.60)

In Table 8.6, the analytical solutions for static deformation of nanorods under two loadings with various boundary conditions are listed. From the table it is seen that the static deformation of nanorods under a constant axial load does not exhibit the small-scale effects for both of the end conditions. In addition, the solution of the nonlocal nanorod with clamped-clamped ends, under a linear axial load, is found to be identical to the local solution. This behaviour has been defined as a nonlocal paradox. This paradox, and some strategies to overcome it, will be addressed in Chap. 12.

8.3.2 Two-Phase Nonlocal Integral Approach Since no nontrivial solution exists for the fully nonlocal integral model [37], we analyse the static deformation of nanorods on the basis of the two-phase nonlocal integral approach. Using Eqs. (5.16) and (8.27), the equilibrium equation is obtained as ⎛ ⎡ ⎤⎞   l |x − ξ | du(ξ ) ⎥⎟ d ⎜ du(x) (1 − η1 )E A ⎢ + dξ ⎦⎠ + p(x) = 0 ⎝η1 E A ⎣ exp − dx dx 2lλ lλ dξ 0

(8.61)

8.3 Static Deformation of Nanorods Under Axial Loadings

179

For a given external axial load p(x), after integrating both sides of the governing equation (8.61), we obtain the following integral equation ⎡ du(x) (1 − η1 ) ⎣ + dx 2η1lλ

l 0

⎤   x |x − ξ | du(ξ ) p(s) exp − dξ ⎦ = − ds + C1 lλ dξ η1 E A 0

(8.62) Notice that Eq. (8.62) is of the general form l exp(m |x − ξ |)y(ξ )dξ = f (x)

y(x) + κ

(8.63)

0

where du(x) (1 − η1 ) 1 y(x) = , κ= , m = − , f (x) = − dx 2η1 lλ lλ

x 0

p(s) ds + C1 η1 E A

(8.64)

From a mathematical point of view, Eq. (8.63) is known as the Fredholm integral equation of second kind and it can be reduced to the following second-order linear nonhomogeneous ordinary differential equation d 2 f (x) d 2 y(x) + m(2κ − m)y(x) = − m 2 f (x) dx2 dx2

(8.65)

with the boundary conditions for Eq. (8.65) having the form  dy  + my(0) = d x x=0  dy  − my(l) = d x x=l

 d f  + m f (0) d x x=0  d f  − m f (l) d x x=l

(8.66) (8.67)

The general solution of Eq. (8.65) is given by y(x) = C2 cosh(κ1 x) + C3 sinh(κ1 x) + f (x) x 2κm − sinh[κ1 (x − s)] f (s)ds κ1 0

with

(8.68)

180

8 Modelling the Mechanical Characteristics …

κ1 =

 m(m − 2κ)

(8.69)

By imposing the boundary conditions (8.66) and (8.67), the constants C2 and C3 are expressed as 2κκ1 m C2 =

l

cosh[κ1 (l − s)] f (s)ds − 2κm 2

0

l

sinh[κ1 (l − s)] f (s)ds

0

κ12 sinh(κ1l) − 2mκ1 cosh(κ1l) + m 2 sinh(κ1l) (8.70) C3 = −

m C2 κ1

(8.71)

Using Eqs. (8.64) and (8.68), the axial displacement of the nanorods is obtained as x u(x) = y(s)ds + C4 (8.72) 0

where C4 is the integration constant. The two unknown constant C1 and C4 can be determined by using the standard boundary conditions as Clamped-clamped u(0) = u(l) = 0

  du  d 2 u  Clamped-free u(0) = 0 and N (l) = η1 E A + λl( p(l) + η1 E A 2  d x x=l dx 

(8.73) )=0 x=l

(8.74) In the following subsection, as an example, we present the applicability of the two-phase nonlocal integral approach to model the nanoscale tensile test of the nanorods.

8.3.3 Nanorod in Tension Let us consider a nanorod subjected to boundary force F = σ¯ A applied at its end sections so that a uniform tensile stress σ¯ is induced in the nanorod (Fig. 8.8). Substituting ε(x) = du/d x in Eq. (8.62) and setting p(x) = 0, we obtain ⎤ ⎡ l    |x − ξ | (1 − η1 ) ⎣ ε(ξ )dξ ⎦ = C1 ε(x) + exp − (8.75) 2η1lλ lλ 0

Furthermore, the two-phase nonlocal integral constitutive equation is

8.3 Static Deformation of Nanorods Under Axial Loadings

181

Fig. 8.8 Schematic illustration of a nanorod in tension

⎡ η1 Eε(x) +

(1 − η1 )E ⎣ 2lλ

l

⎤   |x − ξ | ε(ξ )dξ ⎦ = σ (x) exp − lλ

(8.76)

0

Substituting the uniform tensile stress σ¯ into Eq. (8.76) and comparing with Eq. (8.75), the unknown constant C1 is obtained as C1 =

F AEη1

(8.77)

Following the procedure described in Sect. 8.3.2, the solution of Eq. (8.75) is of the type    x x + C3 sinh ε(x) = C2 cosh √ √ lλ η1 lλ η1    x F F(1 − η1 ) 1 − cosh + − √ AEη1 AEη1 lλ η1 

(8.78)

where      √ − sinh λ√1η1 η1 1 − cosh λ√1η1 F(1 − η1 )       C2 = AEη1 √ sinh λ√1η1 + 2 η1 cosh λ√1η1 + η1 sinh λ√1η1 (8.79) C3 =

√ η1 C 2

(8.80)

Using Eq. (8.72), the exact solution for the axial displacement u(x) can be expressed as

182

8 Modelling the Mechanical Characteristics …

Fig. 8.9 Effect of the phase parameter; a strain and b displacement

   x x √ + C lλ η cosh √ √ 3 1 lλ η1 lλ η1   F(1 − η1 ) √ Fx x + + C4 + lλ η1 sinh √ AE AEη1 lλ η1

√ u(x) = C2 lλ η1 sinh



(8.81)

By enforcing a zero displacement at x = 0, the integration constant C4 is obtained as √ C4 = −C3lλ η1 = −C2 lλη1 (8.82) The influences of the phase parameter and the nonlocal coefficient on the strain and deflection of the nanorod under a uniform tensile stress are plotted in Figs. 8.9 and 8.10, respectively. In Fig. 8.9, the nonlocal coefficient is fixed at 0.25 and the results are shown for four values of the phase parameter. In Fig. 8.10, the phase parameter is fixed at 0.02 and λ = 0.25, 0.30, 0.35 and 0.40. It should be noted that an increasing trend in the strain values in the nanorod end portions is observed. This behaviour is known as the boundary effect [38]. Furthermore, it is seen that the strain value at the boundary decreases with an increase in the phase parameter and it disappears

8.3 Static Deformation of Nanorods Under Axial Loadings

183

Fig. 8.10 Effect of the nonlocal coefficient; a strain and b displacement

when η1 = 1 which corresponds to the local solution. We can also conclude that the strain value at the boundary does not depend on the nonlocal coefficient. Finally, the results show that the displacement of the nanorod increases as the nonlocal coefficient increases and the phase parameter decreases.

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6. G. Stan, R.F. Cook, Mechanical properties of one-dimensional nanostructure, in Scanning Probe Microscopy in Nanoscience and Nanotechnology, ed. by B. Bhushan (Springer, Heidelberg, 2010), pp. 571–611 7. M.A. Haque, M.T.A. Saif, A review of MEMS-based microscale and nanoscale tensile and bending testing. Exp. Mech. 43, 248–255 (2006) 8. X. Li, X. Ling, L. Sun, L. Liu, D. Zeng, Q. Zheng, Measurement of mechanical properties of one-dimensional nanostructures with combined multi-probe platform. Compos. Part B 43, 70–75 (2012) 9. M.F. Yu, O. Lourie, M.J. Dyer, K. Moloni, T.F. Kelly, R.S. Ruoff, Strength and breaking mechanism of multiwalled carbon nanotubes under tensile load. Science 287, 637–640 (2000) 10. E.W. Wong, P.E. Sheehan, C.M. Lieber, Nanobeam mechanics: elasticity, strength, and toughness of nanorods and nanotubes. Science 277, 1971–1975 (1997) 11. S.K. Jeon, H.S. Jang, O.H. Kwon, S.H. Nahm, Mechanical test method and properties of a carbon nanomaterial with a high aspect ratio. Nano Converg. 3, 29 (2016) 12. J.P. Salvetat, A.J. Kulik, J.M. Bonard, G.A.D. Briggs, T. Stockli, K. Metenier, S. Bonnamy, F. Beguin, N.A. Burnham, L. Forro, Elastic modulus of ordered and disordered multiwalled carbon nanotubes. Adv. Mater. 11, 161–165 (1999) 13. M.A. Haque, M.T.A. Saif, A review of MEMS-based microscale and tensile and bending testing. Exp. Mech. 43, 248–255 (2003) 14. P. Poncharal, Z.L. Wang, D. Ugarte, W.A. de Heer, Electrostatic deflections and electromechanical resonances of carbon nanotubes. Science 283, 1513–1516 (1999) 15. L. Gao, Z.L. Wang, Z. Bai, W.A. de Heer, L. Dai, M. Gao, Nanomechanics of individual carbon nanotubes from pyrolytically grown arrays. Phys. Rev. Lett. 85, 622–625 (2000) 16. X.L. Wei, Y. Liu, Q. Chen, M.S. Wang, L.M. Peng, The very low shear modulus of multiwalled carbon nanotubes determined simultaneously with the axial Youngs modulus via in situ experiments. Adv. Funct. Mater. 18, 1555–1562 (2008) 17. D.A. Dikin, X. Chen, W. Ding, G. Wagner, R.S. Ruoff, Resonance vibration of amorphous SiO2 nanowires driven by mechanical or electrical field excitation. J. Appl. Phys. 93, 226–230 (2003) 18. C.Q. Chen, Y. Shi, Y.S. Zhang, J. Zhu, Y.J. Yan, Size dependence of Youngs modulus in ZnO nanowires. Phys. Rev. Lett. 96, 075505 (2006) 19. H. Nili, K. Kalantar-zadeh, M. Bhaskaran, S. Sriram, In situ nanoindentation: probing nanoscale multifunctionality. Prog. Mater. Sci. 58, 1–29 (2013) 20. W.C. Oliver, G.M. Pharr, Improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments. J. Mater. Res. 7, 1564–1580 (1992) 21. X. Li, B. Bhushan, K. Takashima, C.W. Baek, Y.K. Kim, Mechanical characterization of micro/nanoscale structures for MEMS/NEMS applications using nanoindentation techniques. Ultramicroscopy 97, 481–494 (2003) 22. Y. Zhu, C. Ke, H.D. Espinosa, Experimental techniques for the mechanical characterization of one-dimensional nanostructures. Exp. Mech. 47, 7–24 (2007) 23. L. Tsakalakos, Nanostructures for photovoltaics. Mat. Sci. Eng. R 62, 175–189 (2008) 24. K. Kiani, Nonlocal-integro-differential modeling of vibration of elastically supported nanorods. Physica E 83, 151–163 (2016) 25. K. Yu, P. Zijlstra, J.E. Sader, Q.H. Xu, M. Orrit, Damping of acoustic vibrations of immobilized single gold nanorods in different environments. Nano Lett. 13, 2710–2716 (2013) 26. 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, 312–329 (2015) 27. Y. Gan, Z. Sun, Z. Chen, Extensional vibration and size-dependent mechanical properties of single-crystal gold nanorods. J. Appl. Phys. 118, 164304 (2015) 28. G. Romano, R. Barretta, M. Diaco, F.M. de Sciarra, Constitutive boundary conditions and paradoxes in nonlocal elastic nanobeams. Int. J. Mech. Sci. 121, 151–156 (2017) 29. J. Fernández-Sáez, R. Zaera, Vibrations of Bernoulli-Euler beams using the two-phase nonlocal elasticity theory. Int. J. Eng. Sci. 119, 232–248 (2017)

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Chapter 9

Modelling the Mechanical Characteristics of Carbon Nanotubes: A Nonlocal Differential Approach

Computational modelling of the mechanical characteristics of carbon nanotubes forms one of the most intensively studied topics in the nanomechanics field. The underlying motivation for this interest is evident. Carbon nanotubes, due to their unique and exotic mechanical properties, play a pivotal role in nanoscience and in all fields of nanotechnology, and in materials science and technology. A thorough understanding of their mechanical characteristics, including their static deformation properties, their buckling and vibrational behaviour, is therefore crucial for the design and construction of nanodevices. Owing to the extensive research output pertinent to the use of the nonlocal approach in modelling the mechanics of carbon nanotubes, we have divided the material in this chapter into two subsections; with the first subsection providing the mechanical characteristics of large aspect ratio SWCNTs, which are modelled using the nonlocal beam theories, while in the second subsection, we employ the nonlocal shell models to theoretically study the mechanical behaviour of carbon nanotubes. In the present chapter, we use the nonlocal differential approach owing to its wide and successful use in solving various mechanical problems related to carbon nanotubes. Some challenges facing the applications of the nonlocal differential approach to carbon nanotubes will be discussed in Chap. 12.

9.1 Classification of Carbon Nanotubes Classification of carbon nanotubes into different categories is a necessary prerequisite for developing appropriate nonlocal models. The standard classification is based upon the length scales of the carbon nanotubes. Generally, the various length scales used are the carbon–carbon bond length, aC−C , the radius R, the length L C N T and the effective thickness, h C N T . The ratio of the length to the radius, L C N T /R, i.e., the aspect ratio, is used to segregate the nanotubes into a beam or a shell. The largeaspect-ratio nanotubes, with L C N T /R well above 10, can be modelled by a nanoscale beam. Furthermore, the ratio of the nanotube thickness to its radius, h C N T /R can © Springer Nature Switzerland AG 2019 E. Ghavanloo et al., Computational Continuum Mechanics of Nanoscopic Structures, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-3-030-11650-7_9

187

188

9 Modelling the Mechanical Characteristics of Carbon Nanotubes …

be used to determine if a nanotube is to be considered as a thin or a thick shell. A criterion for distinguishing thin shells is also given as 1 hC N T < R 20

(9.1)

This type of classification reveals which continuum-based model is more appropriate for modelling the global behaviour of the carbon nanotubes. A detailed discussion about this classification has been given [1, 2].

9.2 Large Aspect Ratio SWCNTs: Nanobeams The nonlocal beam theory was first employed by Sudak [3] and Peddieson et al. [4] to study the mechanical behaviour of large aspect ratio carbon nanotubes. They revealed that the nonlocal beam theory could potentially play a useful role in nanotechnology applications. Based on their conclusion, several studies have been performed in this field to describe the mechanical behaviour of the carbon nanotubes [5–10]. Since the existing literature on this topic is vast and rapidly expanding, it is not feasible to review all the relevant published materials. The interested reader is referred to the reviews by Arash and Wang [11], Rafii-Tabar et al. [12], Eltaher et al. [13], Askari et al. [14] and Behera and Chakraverty [15]. Furthermore, the investigation of the mechanical characteristics of carbon nanotubes via the nonlocal beam theories is also presented in the books by Gopalakrishnan and Narendar [16] and Karlicic et al. [17]. In this section, we will describe the static and dynamic behaviour of the SWCNTs in the framework of the nonlocal beam theories, presented earlier in Sect. 5.1. In addition, we provide the expression for large deformation of nanobeams subject to end forces according to the nonlocal beam theory.

9.2.1 Bending Deformations of SWCNTs Subject to Transverse Loading 9.2.1.1

Nonlocal Euler–Bernoulli Beam Theory

Let us consider the linear bending deformation of SWCNTs under the influence of a static transverse load on the basis of the nonlocal Euler–Bernoulli beam theory. For static bending problem, P and ∂ 2 w/∂t 2 can be set to zero in Eq. (5.22) and so the partial differential equation becomes EI

2 d 4w 2 2d q − q + L λ =0 C N T dx4 dx2

(9.2)

9.2 Large Aspect Ratio SWCNTs: Nanobeams

189

Furthermore, using Eqs. (5.9) and (5.17), the bending moment is given by M = −E I

d 2w − L C2 N T λ2 q dx2

(9.3)

In the following, we consider three types of boundary conditions, including simply-supported, clamped-clamped and clamped-free ends. According to Eq. (5.18), the boundary conditions for the simply-supported SWCNT may be written as w(0) = w(L C N T ) = 0, M(0) = M(L C N T ) = 0

(9.4)

For the clamped-clamped SWCNT, the boundary conditions are   dw  dw  = =0 w(0) = w(L C N T ) = 0, d x x=0 d x x=L C N T

(9.5)

Finally, for the clamped-free nanobeam, the boundary conditions are   dw  d M  w(0) = = 0, M(L C N T ) = =0 d x x=0 d x x=L C N T

(9.6)

The general solution of Eq. (9.2) is given by x L C2 N T λ2 w(x) = D1 + D2 x + D3 x + D4 x − (x − s)q(s)ds EI 0 ⎛ s ⎞ x  1 (x − s) ⎝ (s − y)q(y)dy ⎠ ds + EI 2

0

3

(9.7)

0

where D1 , D2 , D3 and D4 are unknown constants which can be evaluated from the boundary conditions. Substituting Eq. (9.7) into Eq. (9.3), the bending moment is obtained as x M(x) = −2E I D3 − 6E I D4 x −

(x − s)q(s)ds

(9.8)

0

For reasons of simplicity, only the bending deformations of SWCNTs with simplysupported boundary conditions are considered here. To solve this problem, two general approaches are described. In the first approach, the solution is directly obtained by applying the boundary conditions. Substituting Eqs. (9.7) and (9.8) into Eq. (9.4), a set of homogeneous algebraic equations with respect to the unknown constants is obtained.

190

9 Modelling the Mechanical Characteristics of Carbon Nanotubes … ⎛

⎞⎛ ⎞ 1 0 0 0 D1 3 ⎜ 1 LC N T L2 ⎟ ⎜ ⎟ LC N T CNT ⎜ ⎟ ⎜ D2 ⎟ ⎝ 0 0 −2E I ⎠ ⎝ D3 ⎠ 0 D4 0 0 −2E I −6E I L C N T ⎛ ⎜ ⎜ ⎜ ⎜ =⎜ ⎜ ⎜ ⎝

2 2 LC NT λ EI

L C N T

(L C N T − s)q(s)ds −

0

1 EI

0 L C N T

(L C N T − s)

0

0

0 L C N T



s

(L C N T − s)q(s)ds



⎞

⎟ (s − y)q(y)dy ds ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

(9.9)

0

By solving Eq. (9.9), the constants are given by D1 = 0 D3 = 0

D2 =

L C N T λ2 LC NT + 6E I EI

 LC N T (L C N T − s)q(s)ds 0

1 − E I LC NT

LC N T

(L C N T − s) ⎝

0

D4 = −

1 6E I L C N T

⎛

s

⎞ (s − y)q(y)dy ⎠ ds

0 LC N T

(L C N T − s)q(s)ds

(9.10)

0

The deflection of the nonlocal beam can be determined by substituting these coefficients into Eq. (9.7). In the second approach, the solution is obtained by using the Fourier infinite series. The displacement w which satisfies the simply-supported boundary conditions can be written as w(x) =

∞  n=1

Wn sin

nπ x LC NT

(9.11)

where Wn is the displacement amplitude. The distributed load q(x) can also be written as ∞  nπ x q(x) = Q n sin (9.12) L CNT n=1 where Q n is associated with various types of loads and can be obtained by applying the inverse Fourier transformation:

9.2 Large Aspect Ratio SWCNTs: Nanobeams

Qn =

191

LC N T

2

q(x) sin

LC NT

nπ x dx LC NT

(9.13)

0

Substitution of the expansions for w and q from Eqs. (9.11) and (9.12) into Eq. (9.2), leads to Wn =

Q n L C4 N T (1 + n 2 π 2 λ2 ) E I n4π 4

(9.14)

Therefore, the static deflection is expressed as: w(x) =

∞  Q n L C4 N T (1 + n 2 π 2 λ2 ) nπ x sin 4π 4 E I n L CNT n=1

(9.15)

In the following, the bending deflection of a simply-supported nanobeam subject to a uniform transverse load, q0 , is considered in order to compare between the two approaches. Using the first approach, an analytical expression is derived as w1 (x) =

q0 L C4 N T EI



1 λ2 + 24 2



x LC NT

−

 x3 λ2 x 2 x4 − + 2 L C2 N T 12L C3 N T 24L C4 N T (9.16)

Furthermore, using Eqs. (9.13) and (9.15), we obtain w2 (x) =

∞ q0 L C4 N T  4(1 + (2n − 1)2 π 2 λ2 ) (2n − 1)π x sin E I n=1 LC NT (2n − 1)5 π 5

(9.17)

To compare the two approaches, Fig. 9.1 shows the static deformation of the SWCNT obtained from the first and the second approaches. The results of the second approach is only calculated for one-term (n = 1). It is observed that the results from both approaches agree quite well. It should be noted that some bending solutions of the nonlocal elastic beams have been found to be identical to the local solutions, i.e., the small-scale effect is not present in the nonlocal solutions. These paradoxical results appear when the following problems are solved: (a) Bending of the clamped-clamped nanobeams under uniform or linear distributions of transverse loads. (b) Bending of the clamped-free nanobeams under end-point load or couple. We note that the nonlocal paradox and some strategies to overcome it will be discussed in Chap. 12.

192

9 Modelling the Mechanical Characteristics of Carbon Nanotubes …

Fig. 9.1 Comparison between deflections obtained from the two approaches

9.2.1.2

Nonlocal Timoshenko Beam Theory

For nanotubes with an aspect ratio (length-to-diameter ratio) between 10 and 50, with extensive nanotechnology applications, the contribution of the rotation of the crosssection is significant and must therefore be taken into account. It has been shown that [18–21] the mechanical behaviour of the SWCNTs obtained from the Timoshenko beam theory reflects a truer representation of the mechanical characteristics. Based on Eqs. (5.34) and (5.35) and setting P = 0 and ∂ 2 w/∂t 2 , the governing equations for bending in the nonlocal Timoshenko beam theory are obtained as

 ∂ 2 w ∂ψ ∂ 2q + q − L C2 N T λ2 2 = 0 + 2 ∂x ∂x ∂x

 2 ∂w ∂ ψ +ψ =0 E I 2 − G AK s ∂x ∂x

G AK s

(9.18) (9.19)

In addition, the explicit expressions for the nonlocal bending moment and the nonlocal shear force can be obtained by using Eqs. (5.9), (5.10), (5.27) and (5.28) as dψ − L C2 N T λ2 q(x) M = EI dx

 dw dq + ψ − L C2 N T λ2 Q = G AK s dx dx

(9.20) (9.21)

9.2 Large Aspect Ratio SWCNTs: Nanobeams

193

Various types of boundary conditions can be set up for these equations. For a nanobeam clamped at both ends, we have w(0) = w(L C N T ) = 0, ψ(0) = ψ(L C N T ) = 0

(9.22)

whereas for a clamped-free nanobeam, the boundary conditions are w(0) = ψ(0) = 0, M(L C N T ) = Q(L C N T ) = 0

(9.23)

and the mathematical description of a simply-supported nanobeam is represented by Eq. (9.4). Now, let us consider the solution of Eqs. (9.18) and (9.19) for the simplysupported nanobeam. In this connection, we use the second approach, described in Sect. 9.2.1.1. The displacement w, the rotation and the load q can be written as: w(x) =

∞ 

Wn sin

nπ x LC NT

(9.24)

Ψn cos

nπ x LC NT

(9.25)

Q n sin

nπ x LC NT

(9.26)

n=1

ψ(x) =

∞  n=1

q(x) =

∞  n=1

By substituting Eqs. (9.24)–(9.26) into Eqs. (9.18) and (9.19), we obtain  nπ Wn + Ψn + Q n (1 + λ2 n 2 π 2 ) = 0 LC NT

 n2π 2 nπ − E I Ψn 2 − G AK s Wn + Ψn = 0 LC NT LC NT

nπ − G AK s LC NT

(9.27)

(9.28)

Therefore, we obtain w(x) =

∞  1+ n=1

E I n2π 2 G AK s L C2 N T

ψ(x) = −



Q n L C4 N T (1 + λ2 n 2 π 2 ) nπ x sin E I n4π 4 LC NT

∞  Q n L C3 N T (1 + λ2 n 2 π 2 ) nπ x cos 3 3 EIn π LC NT n=1

(9.29)

(9.30)

It should be noted that both the inclusion of transverse shear and the nonlocal constitutive equations increase the deflection. However, for the simply-supported nanobeam considered here, the inclusion of the transverse shear has no effect on the rotation ψ.

194

9 Modelling the Mechanical Characteristics of Carbon Nanotubes …

9.2.2 Buckling of SWCNTs Under a compressive axial loading, a carbon nanotube suddenly undergoes a large deformation and it is said to have buckled. Usually carbon nanotubes with large aspect ratios undergo beam buckling [22]. In the case of beam buckling, the nanotube keeps its circular cross-section and buckles sideways as a whole, in a similar fashion to a beam (Fig. 9.2 taken from Ref. [23]). According to Eq. (5.22), the governing equation for an SWCNT subjected to a compressive loading is given as d 4w EI 4 + P dx

4 d 2w 2 2d w − L λ C N T dx2 dx4

 =0

(9.31)

It should be noted that the boundary conditions are the same as those in the local beam theory. The general solution of Eq. (9.31) can be easily derived as: w(x) = D1 + D2 x + D3 cos(κ x) + D4 sin(κ x)

(9.32)

where κ2 =

P E I − P L C2 N T λ2

(9.33)

Furthermore, D1 , D2 , D3 and D4 are unknown constants and can be evaluated by the two boundary conditions at each end of the nanobeam, which may be written in the following general forms    2  3  dw  0 d w 0 d w + + s¯3 + s¯4 d x x=0 d x 2 x=0 d x 3 x=0    2  3   0 0 dw  0 d w 0 d w sˆ1 w(0) + sˆ2 + sˆ3 + sˆ4 d x x=0 d x 2 x=0 d x 3 x=0    2  3  dw  1 d w 1 d w s¯11 w(L C N T ) + s¯21 + s ¯ + s ¯ 3 4 d x x=L C N T d x 2 x=L C N T d x 3 x=L C N T    2  3   1 1 dw  1 d w 1 d w sˆ1 w(L C N T ) + sˆ2 + sˆ3 + sˆ4 dx  dx2  dx3  s¯10 w(0)

s¯20

x=L C N T

x=L C N T

=0 =0 =0 = 0 (9.34)

x=L C N T

where s¯i0 , sˆi0 , s¯i1 and sˆi1 (i = 1, 2, 3, 4) are the parameters whose values for different usual boundary conditions are listed in Table 9.1. By substituting Eq. (9.32) into the boundary conditions given by Eq. (9.34), a set of homogeneous algebraic equations with respect to the constants D1 , D2 , D3 and D4 is obtained

9.2 Large Aspect Ratio SWCNTs: Nanobeams

195

Fig. 9.2 Beam-like buckling of a zigzag nanotube as reported by Merli et al. [23]. Reprinted with permission from the publisher Table 9.1 Nonzero values of parameters s¯i0 , sˆi0 , s¯i1 and sˆi1 used in Eq. (9.34) Boundary conditions

s¯i0

sˆi0

s¯i1

Simply-supported (S-S) Clamped-free (C-F) Clamped-clamped (C-C)

s¯10 s¯10 s¯10

sˆ30 sˆ20 sˆ20

s¯11 s¯31 s¯11

=1 =1 =1

=1 =1 =1

sˆi1 =1 =1 =1

⎞ ⎛ ⎞ D1 0 ⎜ D2 ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎜ ⎟ = S4×4 ⎝ D3 ⎠ ⎝ 0 ⎠ D4 0

sˆ31 = 1 sˆ21 = κ 2 , sˆ41 = 1 sˆ21 = 1

⎛

(9.35)

where S11 = s¯10 , S12 = s¯20 , S13 = s¯10 − s¯30 κ 2 , S14 = κ(¯s20 − s¯40 κ 2 ) S21 = sˆ10 , S22 = sˆ20 , S23 = sˆ10 − sˆ30 κ 2 , S24 = κ(ˆs20 − sˆ40 κ 2 ) S31 = s¯11 , S32 = L C N T s¯11 + s¯21 S33 = s¯11 cos(κ L C N T ) − s¯21 κ sin(κ L C N T ) − s¯31 κ 2 cos(κ L C N T ) + s¯41 κ 3 sin(κ L C N T ) S34 = s¯11 sin(κ L C N T ) + s¯21 κ cos(κ L C N T ) − s¯31 κ 2 sin(κ L C N T ) − s¯41 κ 3 cos(κ L C N T ) S41 = sˆ11 , S42 = L C N T sˆ11 + sˆ21 S43 = sˆ11 cos(κ L C N T ) − sˆ21 κ sin(κ L C N T ) − sˆ31 κ 2 cos(κ L C N T ) + sˆ41 κ 3 sin(κ L C N T ) S44 = sˆ11 sin(κ L C N T ) + sˆ21 κ cos(κ L C N T ) − sˆ31 κ 2 sin(κ L C N T ) − sˆ41 κ 3 cos(κ L C N T ) (9.36) For nontrivial solutions of the constants D1 − D4 , the determinant of the above matrix is set equal to zero: |S| = 0

(9.37)

196

9 Modelling the Mechanical Characteristics of Carbon Nanotubes …

Using Eq. (9.37), we can determine the characteristic equation, mode shape and the eigenvalue for the boundary conditions. The characteristic equations and the expression of mode shapes for different boundary conditions are presented in Ref. [24]. The eigenvalue is derived as follows κ=

¯ βπ LC NT

(9.38)

where β¯ has different values for different boundary conditions, i.e., β¯ = n for the simply-supported nanobeams, β¯ = 2n for the clamped-clamped nanobeams and β¯ = (2n − 1)/2 for the clamped-free nanobeams. It should be noted that n is a nonnegative integer. From Eq. (9.33), the buckling load for SWCNTs on the basis of the nonlocal Euler–Bernoulli beam model can be obtained as PEB =

β¯ 2 π 2 E I L C2 N T (1 + λ2 β¯ 2 π 2 )

(9.39)

Furthermore, the critical buckling load occurs at n = 1. As in the nonlocal Euler– Bernoulli beam model, the governing equations for buckling based on the nonlocal Timoshenko beam theory are

G AK s



2 4  d 2 w dψ d w 2 2d w =0 − P + − L λ CNT dx2 dx dx2 dx4

 ∂w ∂ 2ψ E I 2 − G AK s +ψ =0 ∂x ∂x

(9.40) (9.41)

In this case, we only consider the transverse motion of the beam with simplysupported boundary conditions. Using the second approach of solution and Eqs. (9.24) and (9.25), the following system of algebraic equations are obtained: 

) −G AK s Ln2 π + P( Ln2 π + L C2 N T λ2 Ln4 π ) −G AK s Lnπ CNT 2

2

2

CNT

 0 = 0

2

4

CNT

4

CNT

G AK s Lnπ CNT

E I Ln2 π + G AK s 2

2



Wn Ψn



CNT

(9.42)

By assuming the non-trivial solution of Eq. (9.42), the buckling load of the nonlocal Timoshenko nanobeam is calculated from PT =

L C2 N T



EI G AK s

n2π 2 E I  n2 π 2 + 1 (1 + λ2 n 2 π 2 ) 2 L

(9.43)

CNT

It can be clearly seen that the buckling load becomes smaller by a factor (E I n 2 π 2 /G AK s L C2 N T + 1) in the nonlocal Timoshenko beam theory compared to that in the nonlocal Euler–Bernoulli beam model.

9.2 Large Aspect Ratio SWCNTs: Nanobeams

197

9.2.3 Postbuckling Behaviour of Highly Deformable Nanobeams Large aspect ratio carbon nanotubes are very flexible, and they can undergo large deflection under a compressive axial loading and still remain elastic, i.e., behaving just like an elastica. Furthermore, in practical applications such as in nanotweezers, as atomic force microscope tips and as nanoprobes, the large deformation of the nanotubes in the postbuckling stage has been observed experimentally. A thorough understanding of the postbuckling behaviour of nanotubes is important for improving the design of atomic force microscope tips, nanoprobes, nanoneedles and nanoactuators. In this subsection, the nonlinear governing equations and the procedure to obtain the closed-form solution for the postbuckling behaviour of a cantilever nanotube subject to a compressive load at its free end is presented on the basis of nonlocal model. Consider a cantilever SWCNT of length L CNT and flexural rigidity E I subject to an axial force P at its tip as shown in Fig. 9.3. The Cartesian coordinate system X − Y is chosen and located at the fixed end. The tangential angle between the SWCNT axis and the X direction is denoted by ψ. Using trigonometrical relations applied to a differential element d S (Fig. 9.4), the geometrical relations are dX = cos(ψ) dS dY = sin(ψ) dS

(9.44) (9.45)

Furthermore, the differential equations of a nanotube can be obtained on the basis of the equilibrium of the deformed element d S, i.e., dT = −V dψ

Fig. 9.3 A cantilever SWCNT subject to a compressive load P

(9.46)

198

9 Modelling the Mechanical Characteristics of Carbon Nanotubes …

Fig. 9.4 Free body diagram of a differential element

dV =T dψ dM =V dS

(9.47) (9.48)

where T , V and M are respectively the normal force, the shear force, and the bending moment. The boundary conditions of the problem are X (S = 0) = 0, Y (S = 0) = 0, ψ(S = 0) = 0, ψ(S = L C N T ) = ψ B M(S = L C N T ) = 0, T (S = L C N T ) = −P cos(ψ B ) V (S = L C N T ) = −P sin(ψ B ) (9.49) In addition, the relationship between the bending moment and the curvature in the framework of the nonlocal elasticity is defined as M − L C2 N T λ2

d2 M dψ = EI d S2 dS

(9.50)

To simplify the analysis, the following dimensionless quantities are defined: x=

X

, y=

Y

, s=

S

, p¯ =

LC NT LC NT LC NT 2 V LC NT T L C2 N T M LC NT , v¯ = , t¯ = m¯ = EI EI EI

P L C2 N T EI (9.51)

Substitution of Eq. (9.51) into Eqs. (9.44)–(9.50) gives dx = cos(ψ) ds

(9.52)

9.2 Large Aspect Ratio SWCNTs: Nanobeams

199

dy = sin(ψ) ds d t¯ = −¯v dψ d v¯ = t¯ dψ d m¯ = v¯ ds d 2 m¯ dψ m¯ − λ2 2 = ds ds

(9.53) (9.54) (9.55) (9.56) (9.57)

Furthermore, the load and the boundary and conditions are x(s = 0) = 0, y(s = 0) = 0, ψ(s = 0) = 0, ψ(s = 1) = ψb m(s ¯ = 1) = 0, t¯(s = 1) = − p¯ cos(ψ B ), v¯ (s = 1) = − p¯ sin(ψ B ) (9.58) Using Eqs. (9.54) and (9.55) and (9.58), the normal and the shear forces are obtained as functions of ψ: t¯ = − p¯ cos(ψ)

(9.59)

v¯ = − p¯ sin(ψ)

(9.60)

Equations (9.56), (9.57) and (9.60) are combined and the curvature relation is obtained as: m¯ = [1 − λ2 p¯ cos(ψ)]

dψ ds

(9.61)

Multiplying each term of Eq. (9.61) by d m/ds ¯ and using Eqs. (9.56) and (9.60), we obtain m¯

d m¯ dψ = −[1 − λ2 p¯ cos(ψ)] p¯ sin (ψ) ds ds

(9.62)

It is convenient to define a new variable Φ such that: Φ = 1 − λ2 p¯ cos(ψ)

(9.63)

Using Eq. (9.63), Eq. (9.62) is rewritten as d ds



Φ2 2 m¯ + 2 = 0 λ

(9.64)

200

9 Modelling the Mechanical Characteristics of Carbon Nanotubes …

By integrating Eq. (9.64) with respect to s, we obtain m¯ 2 = c1 −

Φ2 λ2

(9.65)

where c1 is a constant and can be determined from the proper boundary condition. Substitution of Eq. (9.63) into Eq. (9.65), yields m¯ 2 = c1 −

1 2 (1 − λ2 p¯ cos(ψ)) λ2

(9.66)

Taking the square root of Eq. (9.66) and after rearrangement we have  m¯ =

c1 −

1 (1 − λ2 p¯ cos(ψ))2 λ2

(9.67)

By enforcing zero bending moment at ψ = ψ B , the constant c1 is calculated as c1 =

1 2 (1 − λ2 p¯ cos(ψ B )) λ2

(9.68)

To find the postbuckling configuration of the nanobeams, it is necessary to obtain the x and y as functions of ψ. Using Eqs. (9.52), (9.53), (9.61), (9.67) and (9.68), it can be shown that ψ 

s(ψ) = 0

ψ x(ψ) = 0

ψ y(ψ) = 0

(1 − λ2 p¯ cos(χ ))dχ c1 −

1 (1 λ2

(9.69)

− λ2 p¯ cos(χ ))2

(1 − λ2 p¯ cos(χ )) cos(χ )dχ  c1 − λ12 (1 − λ2 p¯ cos(χ ))2

(9.70)

(1 − λ2 p¯ cos(χ )) sin(χ )dχ  c1 − λ12 (1 − λ2 p¯ cos(χ ))2

(9.71)

Equations (9.70) and (9.71) are the closed-form expressions for the postbuckling configuration of the nanobeams as a function of a single unknown parameter ψ B . Using the end condition ψ(s = 1) = ψ B , the unknown parameter can be found as ⎧

ψB dχ ⎪ =0 ⎨ 1 − 0 √2 p(cos(χ)−cos(ψ ¯ B )) ψ

B 2 λ(1−λ p¯ cos(χ))dχ ⎪ ⎩1 − √ 2 0

(1−λ2 p¯ cos(ψ

B ))

−(1−λ2 p¯ cos(φ))2

for λ = 0 = 0 for λ > 0

(9.72)

9.2 Large Aspect Ratio SWCNTs: Nanobeams

201

Fig. 9.5 Postbuckling shapes of a nanobeam with some prescribed values of p¯

It is to be noted that Eqs. (9.70)–(9.72) contain the elliptic integrals, and so to obtain the postbuckling shapes of an SWCNT these equations must be solved numerically by different approaches such as the false position method, the finite difference method or the Newton method etc. Figure 9.5 shows the postbuckling shapes of the nanobeam subject to different values of a compressive load. In this figure, the dimensionless nonlocal scale coefficient, λ, is fixed at 0.2. As expected, the deflection of the nanobeam increases with an increase in the force magnitude. Furthermore, it is observed that the tip-angle increases and converges to the angle of the applied load, when the force magnitude increases. It is noted that the tip-angle of the nanobeam does not exceed the value of the angle of the applied load. The postbuckling load-deflection curves of the nanobeams for different values of the nonlocal scale coefficient are displayed in Fig. 9.6. It can be seen that, as the postbuckling load increases, the maximum transverse displacement, yend , goes up sharply. After each curve reaches its peak point at a postbuckling applied load, the maximum deflection decreases when the load is increased. Further results and discussion can be found in Ref. [25].

9.2.4 Linear Flexural Vibrations of Carbon Nanotubes The study of vibration of carbon nanotubes is of great interest in the nanotechnology applications owing to the relevance of this topic in the investigation of the electronic and optical properties of carbon nanotubes. Furthermore, the vibrational characteristics of carbon nanotubes 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

202

9 Modelling the Mechanical Characteristics of Carbon Nanotubes …

Fig. 9.6 Postbuckling load-deflection curves for different values of the nonlocal scale coefficient

properties. Hence, finding the natural frequencies is a fundamental step in the prediction of the dynamical response of a system. In this subsection, we will consider the applications of the nonlocal Euler–Bernoulli and Timoshenko beam models in the analysis of linear flexural vibrations of carbon nanotubes. For the free vibration problem, it is assumed that the transverse distributed force and the applied axial compressive load are equal to zero. As a result, the governing equation for the linear free vibration of carbon nanotubes on the basis of the Euler–Bernoulli beam model is derived as 

2 4 ∂ w ∂ 4w 2 2 ∂ w =0 (9.73) − LC NT λ EI 4 + ρA ∂x ∂t 2 ∂ x 2 ∂t 2 In addition, the nonlocal bending moment is given by M(x, t) = L C2 N T λ2 ρ A

∂ 2w ∂ 2 w(x, t) − EI 2 ∂t ∂x2

(9.74)

Let the solution of Eq. (9.73), which satisfies the simply-supported boundary conditions, be w(x, t) =

∞ 

Wn sin

n=1

nπ x exp( jωt) LC NT

(9.75)

Therefore, the natural frequencies are given by n2π 2 ωn = 2 LC NT



EI  ρ A 1 + λ2 n 2 π 2 

(9.76)

9.2 Large Aspect Ratio SWCNTs: Nanobeams

203

In a manner similar to the Euler–Bernoulli beam, the equations of motion from the nonlocal Timoshenko beam theory are given by 

2  4 ∂ w ∂ 2 w ∂ψ 2 2 ∂ w = ρA + − LC NT λ G AK s ∂x2 ∂x ∂t 2 ∂ x 2 ∂t 2

 

2 4 ∂ 2ψ ∂w ∂ ψ 2 2 ∂ ψ E I 2 − G AK s − L λ + ψ = ρI CNT ∂x ∂x ∂t 2 ∂ x 2 ∂t 2

(9.77)

(9.78)

For simply-supported boundary conditions, the displacement and rotation can be represented by ∞ 

w(x, t) =

Wn sin

nπ x exp( jωt) LC NT

(9.79)

Ψn cos

nπ x exp( jωt) LC NT

(9.80)

n=1

ψ(x, t) =

∞  n=1

Substituting Eqs. (9.79) and (9.80) into Eqs. (9.77) and (9.78), we have ⎛ ⎝

n2 π 2 nπ LC NT L C2 N T 2 Er g n 2 π 2 nπ L C N T G K s L C2 N T

⎞ +1

⎠

Wn Ψn

 =

 

ρω2 1 0 Wn (1 + λ2 n 2 π 2 ) 0 r g2 Ψn G Ks (9.81)

where r g2 = I /A. By solving the eigenvalue problem given by Eq. (9.81), the natural frequencies of simply-supported SWCNTs can be obtained on the basis of the nonlocal Timoshenko beam model. For any value of n, there are two natural frequencies. The smaller value corresponds to the bending deformation mode and the larger value corresponds to the shear deformation mode.

9.3 Low Aspect Ratio SWCNTs: Cylindrical Nanoshells Both global and local deformations manifest themselves in low aspect ratio SWCNTs. Consequently, the nanotube should be modelled as a cylindrical elastic shell rather than an elastic beam [26]. The mechanical characteristics of the SWCNTs, particularly during bending [27], buckling under compressive loads [28, 29] and vibrations [30, 31] have been studied using various elastic shell models. In this section, the buckling and the vibrations of the SWCNTs are presented on the basis of the nonlocal shell theories discussed in Sect. 5.2.

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9 Modelling the Mechanical Characteristics of Carbon Nanotubes …

9.3.1 Shell-Like Buckling of SWCNTs The local Euler–Bernoulli beam model can be employed to investigate the buckling behaviour of the SWCNTs having a large aspect ratio (L C N T /R > 20), while the nonlocal version of this model does not provide a better prescription for predicting the buckling of these types of SWCNTs [32]. On the other hand, for SWCNTs with moderate aspect ratios (16 < L C N T /R < 20), it becomes necessary to apply the nonlocal beam theories to better predict their buckling behaviour. For short SWCNTs having large diameters (L C N T /R < 16), it has also been shown that the nonlocal shell model, with an appropriate small-scale parameter, could well predict the critical buckling strains in good agreement with the results obtained by MD simulations. These points are summarized in Fig. 9.7 taken from Ref. [29]. In the case of shell buckling, the SWCNT undergoes a wavy deformation to lower the applied compressive energy, while at the same time the SWCNT axis maintains a straight geometry just as in a thin-shell buckling case (Fig. 9.8 taken from [33]).

Fig. 9.7 The variations of the critical buckling strains and modes with aspect ratio for (5, 5) SWCNTs as reported by Silvestre et al. [29]. Reprinted with permission from the publisher Fig. 9.8 Change of shape of a (10, 10) SWCNT due to axial buckling as reported by Zhang et al. [33]. Reprinted with permission from the publisher

9.3 Low Aspect Ratio SWCNTs: Cylindrical Nanoshells

205

Table 9.2 Computed critical buckling strains for (5, 5) SWCNTs L C N T /2R MD simulation Sanders shell theory First-order shear deformation theory 2.4 3.14 4.24 6.08 7.93

0.08146 0.07528 0.06992 0.06507 0.05499

0.08729 0.08288 0.07858 0.07509 0.05949

0.08415 0.07916 0.07553 0.07225 0.05604

As discussed previously in Sect. 5.2, different shell theories have been widely used in research publications to investigate the buckling characteristics of SWCNTs. In the great majority of previously published works the local and nonlocal Donnell shell models were used due to their inherent simplicity [29, 34]. However, Silvestre et al. [29] have shown that the Sanders shell theory could correctly model the buckling behaviour of SWCNTs, while the local and nonlocal Donnell shell models fail to do so. In addition, in some studies [35, 36] it has been shown that the local firstorder shear deformation theory is the robust shell model for predicting the buckling behaviour of SWCNTs. In particular, Wang et al. [36] compared the critical buckling strains of SWCNTs obtained from the local Sanders shell theory with the local firstorder shear deformation theory and MD simulations. Some results of their analysis are reproduced in Table 9.2. According to these results, they have pointed out that the results calculated from the first-order shear deformation theory are quite acceptable and, therefore, there is no need to resort to the nonlocal continuum theories to study the buckling of SWCNTs. However, as can be inferred from this table, the buckling strains are overestimated within the local models. Consequently, in the following, the nonlocal Sanders thin shell theory is employed for the buckling analysis of SWCNTs. Let us first derive the elastic buckling equations pertinent to an SWCNT by starting from Eqs. (5.61)–(5.63), with ∂ 2 u/∂t 2 = ∂ 2 v/∂t 2 = ∂ 2 w/∂t 2 = 0, 

   

 ∂ ∂2 A66 3D66 ∂ 2 D66 A12 A66 D12 v − − A11 2 + u + + + R R ∂x R2 4R 4 ∂θ 2 R3 4R 3 ∂ x∂θ    A12 ∂ D11 ∂ 3 D12 ∂ 3 ∂ 2u D66 ∂ 3 ¯ − + − (9.82) N − w = L HC R ∂x R ∂x3 R 3 ∂ x∂ 2 θ R 3 ∂ x∂ 2 θ ∂x2 

    A66 ∂ ∂2 21D66 ∂ 2 A22 ∂ 2 D22 ∂ 2 A12 D66 v u + A66 2 + + 2 + 4 + + 3 2 2 2 2 R R 4R ∂ x∂θ ∂x 4R ∂ x R ∂θ R ∂θ 

  A22 ∂ ∂2v 5D66 ∂ 3 D12 ∂ 3 D22 ∂ 3 + − (9.83) − 2 − 4 w = L H C N¯ 2 R 2 ∂θ R ∂ x 2 ∂θ R ∂θ 3 R 2 ∂ x 2 ∂θ ∂x

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9 Modelling the Mechanical Characteristics of Carbon Nanotubes …



 D11 ∂ 3 D66 ∂ 3 A12 ∂ u + 3 − R ∂x3 R ∂ x∂θ 2 R ∂x   2D12 ∂ 3 A22 ∂ 5D66 ∂ 3 D22 ∂ 3 v + − 2 + + 4 R 2 ∂ x 2 ∂θ R 2 ∂ x 2 ∂θ R ∂θ 3 R ∂θ   D12 ∂ 2 ∂4 2D12 ∂ 4 4D66 ∂ 4 D22 ∂ 4 A22 + − D11 4 − − − 4 − 2 w R2 ∂ x 2 ∂x R 2 ∂ x 2 ∂θ 2 R 2 ∂ x 2 ∂θ 2 R ∂θ 4 R

2  w ∂ = L H C N¯ 2 (9.84) ∂x

where A11 = A22 = Y11 , A12 = Y12 , A66 = Y33 Y11 h 2 Y12 h 2 Y33 h 2 , D12 = , D66 = D11 = D22 = 12 12 12

(9.85)

in which Yi j are the surface elastic constants of an SWCNT. Generally, it is difficult to find the exact analytical solution for the buckling of SWCNTs with general boundary conditions due to the complexity of the situation, and coupling of the governing differential equations. Since the analytical solution could be found for the case of simply-supported boundary, here we only analyse this boundary condition. For this boundary condition, we have w = v = 0 at both ends of shells. The displacement fields that satisfy these essential boundary conditions at x = 0 and L C N T can be written as: u(x, θ ) = U¯ cos(kq x) cos(kθ θ )

(9.86)

v(x, θ ) = V¯ sin(kq x) sin(kθ θ )

(9.87)

w(x, θ ) = W¯ sin(kq x) cos(kθ θ )

(9.88)

where U¯ , V¯ and W¯ are the displacement amplitudes in the x, θ , z directions, respectively, kq (kq = qπ /L C N T , q is the half-axial wave-number) is the wave-number along the axial direction, kθ is the wave-number in the circumferential direction. This displacement field offers a good way to solve the couple problems. Substitution of Eqs. (9.86)–(9.88) into Eqs. (9.82)–(9.84), yields a set of algebraic equations for U¯ , V¯ and W¯ as ⎛

⎞ ⎛ ⎞ U¯ 0 ¯ q , kθ , N¯ )3×3 ⎝ V¯ ⎠ = ⎝ 0 ⎠ B(k 0 W¯ wherein B¯ is a nonsymmetric matrix whose elements are given by

(9.89)

9.3 Low Aspect Ratio SWCNTs: Cylindrical Nanoshells

207





 A66 3D66 kθ2 2 2 2 2 ¯ kθ − 1 + μ kq + 2 − B11 = A11 kq + N¯ kq2 R2 4R 4 R 

A66 D12 A12 D66 kq kθ + + 3 − B¯ 12 = − R R R 4R 3

 D12 A12 D11 3 D66 kq − kq − + kq kθ2 B¯ 13 = − R R R3 R3 

A12 A66 D66 kq kθ B¯ 21 = − + + R R 4R 3  



 21D66 A22 D22 kθ2 2 2 2 2 ¯ kq + + 4 kθ − 1 + μ kq + 2 B22 = A66 + N¯ kq2 4R 2 R2 R R 

A22 D22 3 D12 5D66 ¯ kq2 kθ + B23 = 2 kθ + 4 kθ + R R R2 R2 A12 D11 3 D66 kq − k − kq kθ2 B¯ 31 = − R R q R3  A22 D22 3 2D12 5D66 ¯ kq2 kθ + B32 = 2 kθ + 4 kθ + R R R2 R2

 2D12 A22 D12 2 4D66 D22 4 ¯ + B33 = 2 + 2 kq + D11 kq + kq2 kθ2 + 4 kθ4 R R R2 R2 R

 2  k − 1 + μ2 kq2 + θ2 (9.90) N¯ kq2 R where μ = L C N T λ. For the non-trivial solution, the determinant of this set of equations must be zero, i.e., det B¯ = 0

(9.91)

Solving the resulting eigenvalue problem, the critical buckling loads of the SWCNTS can be obtained. For elastic buckling analysis, we require the minimum load. Table 9.3 lists a set of selected values of the buckling loads for (8, 8) SWCNTs, calculated from MD simulation [37] and from the nonlocal Sanders thin shell theory. A reasonable agreement is obtained, which confirms the accuracy as well as the validity of the present formulation. It is to be noted that the material properties of the (8, 8) SWCNTs are obtained from the formulation given in Sect. 6.2. Furthermore, in our numerical calculation, according to Wang et al. [38], a suitable choice for the value of the effective thickness of SWCNT h = 0.0665 nm was used.

9.3.2 Vibrational Properties of SWCNTs The carbon atoms in an SWCNT undergo constant vibrations about their mean positions. The quantum of these vibratory motion is known as phonon and it propagates

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9 Modelling the Mechanical Characteristics of Carbon Nanotubes …

Table 9.3 Comparison between the results obtained from the nonlocal shell model and from the MD simulation for critical axial buckling loads of (8, 8) SWCNTs L C N T /2R Ansari et al. [37] MD simulation (nN) Nonlocal Sanders thin shell theory with λ = 0.01 (nN) 6.47 8.3 10.1 13.7 17.3 20.9 24.5 28.1 31.6

31.4797 20.1078 13.9871 7.6032 4.3763 2.9126 2.0916 1.7015 1.3669

30.8379 19.6266 13.5119 7.3784 4.5526 3.0364 2.1369 1.5643 1.1892

through the SWCNT. Generally, the phonons are characterized by the frequency wave vector dispersion. The phonon dispersion in SWCNTs is complicated and a mathematically involved process [39]. Unfortunately, the full phonon dispersion of SWCNTs cannot be experimentally determined, and can be obtained via a force-constant model [40]. The calculated phonon dispersion of a (10, 10) SWCNT obtained from the force-constant model is shown in Fig. 9.9. In the last decade, it has been shown that the phonon dispersion of SWCNTs could be calculated by continuum-based models [30, 31, 41, 42]. Here the calculation of the phonon dispersion is performed in the framework of the nonlocal Sanders thin shell theory. For the vibration problem, N¯ can be set to zero in Eqs. (5.61)–(5.63) leading to the partial differential equations

     A66 A12 ∂ A66 D12 ∂2 3D66 ∂ 2 D66 + + − − u + v A11 2 + ∂x R2 4R 4 ∂θ 2 R R R3 4R 3 ∂ x∂θ   D11 ∂ 3 D66 ∂ 3 A12 ∂ D12 ∂ 3 − − w + − R ∂x R ∂x3 R 3 ∂ x∂ 2 θ R 3 ∂ x∂ 2 θ  2

4  ∂ u ∂ u 1 ∂ 4u 2 = ρh (9.92) − μ + ∂t 2 ∂ x 2 ∂t 2 R 2 ∂θ 2 ∂t 2





    ∂2 A66 ∂ 21D66 ∂ 2 A22 ∂ 2 D22 ∂ 2 A12 D66 v u + A + + + + + 66 R R 4R 3 ∂ x∂θ ∂x2 4R 2 ∂ x 2 R 2 ∂θ 2 R 4 ∂θ 2   5D66 ∂ 3 A22 ∂ D12 ∂ 3 D22 ∂ 3 + w − − − R 2 ∂θ R 2 ∂ x 2 ∂θ R 4 ∂θ 3 R 2 ∂ x 2 ∂θ  2

4  1 ∂ 4v ∂ v ∂ v = ρh − μ2 + 2 2 2 (9.93) 2 2 2 ∂t ∂ x ∂t R ∂θ ∂t

9.3 Low Aspect Ratio SWCNTs: Cylindrical Nanoshells

209

Fig. 9.9 Calculated phonon dispersion curves of a (10, 10) SWCNT obtained from the force-constant model as reported by Popov [40]. Reprinted with permission from the publisher



 D11 ∂ 3 D66 ∂ 3 A12 ∂ u + 3 − R ∂x3 R ∂ x∂θ 2 R ∂x   5D66 ∂ 3 D22 ∂ 3 2D12 ∂ 3 A22 ∂ + + 4 v + − 2 R 2 ∂ x 2 ∂θ R 2 ∂ x 2 ∂θ R ∂θ 3 R ∂θ   ∂4 D12 ∂ 2 2D12 ∂ 4 4D66 ∂ 4 D22 ∂ 4 A22 w + − D − − − − 11 R2 ∂ x 2 ∂x4 R 2 ∂ x 2 ∂θ 2 R 2 ∂ x 2 ∂θ 2 R 4 ∂θ 4 R2  2

4  ∂ w ∂ w 1 ∂ 4w 2 = ρh (9.94) − μ + ∂t 2 ∂ x 2 ∂t 2 R 2 ∂θ 2 ∂t 2

For the simply-supported boundary conditions, the displacements can be written as u(x, θ, t) = U¯ cos(kq x) cos(kθ θ ) exp( jωt)

(9.95)

v(x, θ, t) = V¯ sin(kq x) sin(kθ θ ) exp( jωt)

(9.96)

w(x, θ, t) = W¯ sin(kq x) cos(kθ θ ) exp( jωt)

(9.97)

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9 Modelling the Mechanical Characteristics of Carbon Nanotubes …

By substituting Eqs. (9.95)–(9.97) into Eqs. (9.92)–(9.94), we obtain ⎛

⎞⎛ ⎞ ⎛ ⎞⎛ ⎞ U¯ U¯ K 11 K 12 K 13 M11 0 0 ⎝ K 21 K 22 K 23 ⎠ ⎝ V¯ ⎠ = ω2 ⎝ 0 M11 0 ⎠ ⎝ V¯ ⎠ 0 0 M11 K 31 K 32 K 33 W¯ W¯

(9.98)

where  A66 3D66 kθ2 K 11 = + − R2 4R 4 

A66 D12 A12 D66 + + 3 − kq kθ K 12 = − R R R 4R 3 

A12 D11 3 D12 D66 K 13 = − + 3 kq kθ2 kq − k − R R q R3 R 

A66 D66 A12 kq kθ + + K 21 = − R R 4R 3  

21D66 A22 D22 2 kq + K 22 = A66 + + 4 kθ2 4R 2 R2 R 

A22 D22 3 D12 5D66 kq2 kθ K 23 = 2 kθ + 4 kθ + + R R R2 R2 A12 D11 3 D66 K 31 = − kq kθ2 kq − k − R R q R3  A22 D22 3 2D12 5D66 kq2 kθ K 32 = 2 kθ + 4 kθ + + R R R2 R2 

A22 D12 2 2D12 4D66 D22 4 kq2 kθ2 + 4 kθ4 K 33 = 2 + 2 kq + D11 kq + + 2 2 R R R R R

 2  k θ (9.99) M11 = ρh 1 + μ2 kq2 + 2 R

A11 kq2

Solving Eq. (9.98) yields three eigenfrequencies for each (kq , kθ ), giving three different vibration modes. Figure 9.10, shows the effect of the small-scale parameter, μ, on the axisymmetric (kθ = 0) phonon dispersion of a (15, 15) armchair SWCNT. For numerical simulation purpose, we assume that ρh = 0.7718 × 10−7 g/cm2 . The longitudinal, radial and torsional modes are denoted by L, R and T respectively. The results from Fig. 9.10 indicate 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 small-scale parameter on the frequencies of the SWCNT is dependent on the axial wavelength. This effect increases with decreasing axial wavelength (increasing kq ). In addition, the phonon-dispersion relations of the (15, 15) SWCNT are shown in Fig. 9.11 for kθ = 1 and 2.

9.3 Low Aspect Ratio SWCNTs: Cylindrical Nanoshells

211

Fig. 9.10 Comparisons of frequency curves of a (15, 15) SWCNT with different values of the small-scale parameter for the axisymmetric case

Since SWCNTs have a great potential in the construction of nanodevices, their cutoff frequencies are of particular significance. These cut-off frequencies or frequencies of axially uniform vibrational modes can be derived by setting q = 0. The cut-off frequencies versus circumferential wave-number kθ are listed in Table 9.4 for an (8, 8) armchair SWCNT for different values of the scale coefficient μ = 0, 0.1 and 0.2 nm. It is observed that only one cut-off frequency is independent of the scale coefficient i.e., is free of the scale effect, and this exists at kθ = 0 (axisymmetric motion). Furthermore, for vibrations with kθ = 1, two cut-off frequencies exist. Three cut-off frequencies exist only when kθ is set at kθ ≥ 2.

9.3.3 Radial Breathing Mode The structure of carbon nanotubes can be studied via the Raman spectroscopy as an important nondestructive tool. A typical Raman spectrum of an SWCNT is depicted in Fig. 9.12. Generally, in the Raman spectroscopy of SWCNTs, there exist four important vibrational modes [43]. The first mode is the radial breathing mode (RBM) that corresponds to a symmetric in-phase synchronous displacements of all the carbon atoms in the radial direction resembling the breathing of the nanotube. The frequencies corresponding to the RBM are usually in the range 100–500 cm−1 . The second mode is the D band mode around 1300 cm−1 , occurring from the dispersive double

212

9 Modelling the Mechanical Characteristics of Carbon Nanotubes …

Fig. 9.11 Dispersion curves of a (15, 15) SWCNT for a kθ = 1 and b kθ = 2

resonance processes associated with defects [44]. The third mode is the tangential mode (the G band) arising from vibrations of two adjacent carbon atoms located along the nanotube or in the circumferential direction. This mode is usually in the range 1500–1600 cm−1 [45]. The fourth mode is the G’-band which is the overtone of the D-band and appears in the range 2500–2900 cm−1 [46].

9.3 Low Aspect Ratio SWCNTs: Cylindrical Nanoshells

213

Table 9.4 Cut-off frequencies (cm−1 = Hertz/speed of light in cm/s) for an (8, 8) armchair SWCNT kθ μ = 0.0 nm μ = 0.1 nm μ = 0.2 nm 0 1 2 3 4 5 6 7 8 9 10

0.00 0.00 20.27 57.31 109.87 177.66 260.61 358.68 471.85 600.12 743.47

0.00 136.96 273.91 410.87 547.82 684.78 821.73 958.69 1095.64 1232.60 1369.55

213.58 302.05 477.87 676.02 881.55 1090.30 1300.73 1512.14 1724.18 1936.65 2149.44

0.00 0.00 19.02 50.15 88.43 130.64 174.78 219.72 264.83 309.81 354.53

0.00 134.69 257.00 359.55 440.93 503.54 551.12 587.27 614.93 636.33 653.08

213.58 297.05 448.37 591.59 709.54 801.73 872.37 926.30 967.70 999.79 1024.97

0.00 0.00 16.31 38.43 61.66 84.72 107.36 129.60 151.52 173.18 194.64

0.00 128.50 220.46 275.56 307.47 326.54 338.51 346.40 351.82 355.69 358.54

213.58 283.41 384.63 453.39 494.77 519.92 535.83 546.37 553.65 558.86 562.71

Fig. 9.12 Raman spectrum of a typical SWCNT

The RBM is known as a fingerprint of SWCNTs, and is the routinely accessible optical mode for identifying the diameter and chirality of nanotubes using the Raman spectroscopy [47, 48]. The discrete atomic structure of the carbon nanotubes leads to a chirality dependence of the RBM frequency. By determining the frequency of the RBM by the resonance Raman spectroscopy and comparing with theoretical calculations, it is possible to assign (n, m) indices to individual SWCNTs, i.e., the most frequently used optically based experimental technique to properly assign the chirality indices of SWCNTs is the resonance Raman spectroscopy [49]. The prediction of the RBM frequency of SWCNTs has formed a subject of deep interest and a challenging research topic over the past decades. Here we propose an explicit expression for determining the RBM frequency of the SWCNTs with arbitrary chirality. Since the RBM vibration is axisymmetric (i.e.

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9 Modelling the Mechanical Characteristics of Carbon Nanotubes …

u = v = 0 and ∂/∂ x = ∂/∂θ = 0), the dynamic equations (5.52)–(5.54) for the RBM vibration are reduced to −

Nθθ ∂ 2w = ρh 2 R ∂t

(9.100)

where h

2 Nθθ =

tθθ dz

(9.101)

− h2

In addition, the nonlocal constitutive relations are

 2 (t − t ) = trr + (e0 L i )2 rr θθ R2 

2 2 tθθ − (e0 L i ) (trr − tθθ ) = r2

Y12 εθθ h

(9.102)

Y22 εθθ h

(9.103)

wherein εθθ =

w R

(9.104)

Solving Eqs. (9.102) and (9.103) for tθθ , we obtain tθθ = 

Y22 h

1−

+

2(e0 L i )2 Y12 h(R 2 +2(e0 L i )2 )

4(e0 L i )4 R 2 (R 2 +2(e0 L i )2 )

+

2(e0 L i )2 R2



w R

(9.105)

Therefore, the symmetric in-phase vibration of all carbon atoms of the SWCNT in the radial direction is governed by 0 L i ) Y12 Y22 + (R2(e 2 +2(e L )2 ) ∂ 2w w 0 i  ρh 2 +  =0 4(e0 L i )4 2(e0 L i )2 R 2 ∂t 1 − R 2 (R 2 +2(e L )2 ) + R 2 2

0

(9.106)

i

The solution of this differential equation provides an analytical relation, linking the RBM frequency to the geometrical parameters of the SWCNT as 

2(e0 L i )2 Y12 (n,m)

Y22 (n, m) + (R 2 (n,m)+2(e L )2 ) 1 1 0 i  ω R B M (n, m) = ! 4(e0 L i )4 2(e0 L i )2 R(n, m) ρh 1− + R 2 (n,m)(R 2 (n,m)+2(e0 L i )2 )

R 2 (n,m)2

(9.107)

9.3 Low Aspect Ratio SWCNTs: Cylindrical Nanoshells

215

Table 9.5 Comparison of the RBM frequency calculated by the nonlocal formula and the previous results (n, m) Local model Nonlocal model Zhang et al. [50] e0 L i /R = 0 e0 L i /R = 0.1 (Calc.) (Expt.) (16, 7) (17, 5) (13, 10) (14, 8) (16, 4) (17, 2) (12, 9) (13, 7) (14, 5) (15, 3) (16, 1) (11, 8) (12, 6) (13, 4) (14, 2)

145.0 148.0 148.3 153.6 161.4 163.6 162.3 168.4 173.5 177.0 178.9 179.2 186.4 192.1 195.7

147.0 149.9 150.0 155.3 163.3 165.5 164.2 170.4 175.5 179.1 181.0 181.3 188.5 194.3 198.0

151 154 154 158 165 167 166 171 176 179 181 181 187 192 195

150 154–156 155 158 164 165–166 165–171 171–173 175 179 179–182 183 189 193–196 195

To validate the proposed formulation, the obtained results are compared with the experimental and numerical results reported by Zhang et al. [50]. A comparison between the values obtained from the nonlocal model and the previous values for the RBM frequency of different SWCNTs is given in Table 9.5. From these results in Table 9.5, it is clear that the results obtained from the nonlocal model are in good agreement with those found either computationally or experimentally.

References 1. V.M. Harik, Mechanics of carbon nanotubes: applicability of the continuum-beam models. Comput. Mater. Sci. 24, 328–342 (2002) 2. H. Rafii-Tabar, Computational Physics of Carbon Nanotubes (Cambridge University Press, Cambridge, 2008) 3. L.J. Sudak, Column buckling of multiwalled carbon nanotubes using nonlocal continuum mechanics. J. Appl. Phys. 94, 7281–7287 (2003) 4. J. Peddieson, G.R. Buchanan, R.P. McNitt, Application of nonlocal continuum models to nanotechnology. Int. J. Eng. Sci. 41, 305–312 (2003) 5. 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, 130–135 (2006) 6. Q. Wang, K.M. Liew, Application of nonlocal continuum mechanics to static analysis of microand nano-structures. Phys. Lett. A 363, 236–242 (2007) 7. P. Lu, H.P. Lee, C. Lu, P.Q. Zhang, Application of nonlocal beam models for carbon nanotubes. Int. J. Solids Struct. 44, 5289–5300 (2007)

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8. J.N. Reddy, Nonlocal theories for bending, buckling and vibration of beams. Int. J. Eng. Sci. 45, 288–307 (2007) 9. M. Aydogdu, A general nonlocal beam theory: its application to nanobeam bending, buckling and vibration. Phys. E 41, 1651–1655 (2009) 10. R. Barretta, F.M. de Sciarra, A nonlocal model for carbon nanotubes under axial loads. Adv. Mater. Sci. Eng. 2013, 360935 (2013) 11. B. Arash, Q. Wang, A review on the application of nonlocal elastic models in modeling of carbon nanotubes and graphenes. Comput. Mater. Sci. 51, 303–313 (2012) 12. H. Rafii-Tabar, E. Ghavanloo, S.A. Fazelzadeh, Nonlocal continuum-based modeling of mechanical characteristics of nanoscopic structures. Phys. Rep. 638, 1–97 (2016) 13. M.A. Eltaher, M.E. Khater, S.A. Emam, A review on nonlocal elastic models for bending, buckling, vibrations, and wave propagation of nanoscale beams. Appl. Math. Model. 40, 4109– 4128 (2016) 14. H. Askari, D. Younesian, E. Esmailzadeh, L. Cveticanin, Nonlocal effect in carbon nanotube resonators: a comprehensive review. Adv. Mech. Eng. 9, 1–24 (2017) 15. L. Behera, S. Chakraverty, Recent researches on nonlocal elasticity theory in the vibration of carbon nanotubes using beam models: a review. Arch. Comput. Methods Eng. 24, 481–494 (2017) 16. S. Gopalakrishnan, S. Narendar, Wave Propagation in Nanostructures: Nonlocal Continuum Mechanics Formulations (Springer, Switzerland, 2013) 17. D. Karlicic, T. Murmu, S. Adhikari, M. McCarthy, Nonlocal Structural Mechanics (WileyISTE, London, 2016) 18. C.M. Wang, Y.Y. Zhang, X.Q. He, Vibration of nonlocal Timoshenko beams. Nanotechnology 18, 105401 (2007) 19. N. Khosravian, H. Rafii-Tabar, Computational modelling of a non-viscous fluid flow in a multiwalled carbon nanotube modelled as a Timoshenko beam. Nanotechnology 19, 275703 (2008) 20. 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, 225404–225413 (2008) 21. E. Ghavanloo, S.A. Fazelzadeh, Flow-thermoelastic vibration and instability analysis of viscoelastic carbon nanotubes embedded in viscous fluid. Phys. E 44, 17–24 (2011) 22. 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. Comput. Mater. Sci. 79, 619–626 (2013) 23. R. Merli, C. Lázaro, S. Monleón, A. Domingo, A molecular structural mechanics model applied to the static behavior of single-walled carbon nanotubes: new general formulation. Comput. Struct. 127, 68–87 (2013) 24. C.M. Wang, C.Y. Wang, J.N. Reddy, Exact Solutions for Buckling of Structural Members (CRC Press, Florida, 2005) 25. M.A. Maneshi, E. Ghavanloo, S.A. Fazelzadeh, Closed-form expression for geometrically nonlinear large deformation of nano-beams subjected to end force. Eur. Phys. J. Plus 133, 256 (2018) 26. C.Q. Ru, Elastic models for carbon nanotubes, in Encyclopaedia of Nanoscience and Nanotechnology, vol. 2, ed. by H.S. Nalwa (American Scientific, Stevenson Ranch, 2004), pp. 731–744 27. 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 Struct. 11, 2122–2140 (2014) 28. 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. Dyn. 7, 629–645 (2007) 29. N. Silvestre, C.M. Wang, Y.Y. Zhang, Y. Xiang, Sanders shell model for buckling of singlewalled carbon nanotubes with small aspect ratio. Compos. Struct. 93, 1683–1691 (2011)

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30. C.Y. Wang, L.C. Zhang, An elastic shell model for characterizing single-walled carbon nanotubes. Nanotechnology 19, 195704 (2008) 31. E. Ghavanloo, S.A. Fazelzadeh, Vibration characteristics of single-walled carbon nanotubes based on an anisotropic elastic shell model including chirality effect. Appl. Math. Model. 36, 4988–5000 (2012) 32. Y.Y. Zhang, C.M. Wang, W.H. Duan, Y. Xiang, Z. Zong, Assessment of continuum mechanics models in predicting buckling strains of single-walled carbon nanotubes. Nanotechnology 20, 395707 (2009) 33. 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, 074304 (2006) 34. C.M. Wang, Y.Y. Zhang, Y. Xiang, J.N. Reddy, Recent studies on buckling of carbon nanotubes. Appl. Mech. Rev. 63, 030804 (2010) 35. D.D.T.K. Kulathunga, K.K. Ang, J.N. Reddy, Accurate modeling of buckling of single- and double-walled carbon nanotubes based on shell theories. J. Phys. Condens. Matter 21, 435301 (2009) 36. C.M. Wang, Z.Y. Tay, A.R. Chowdhuary, W.H. Duan, Y.Y. Zhang, N. Silvestre, Examination of cylindrical shell theories for buckling of carbon nanotubes. Int. J. Struct. Stab. Dyn. 11, 1035–1058 (2011) 37. R. Ansari, S. Sahmani, H. Rouhi, Rayleigh–Ritz axial buckling analysis of single-walled carbon nanotubes with different boundary conditions. Phys. Lett. A 375, 1255–1263 (2011) 38. L.F. Wang, Q.S. Zheng, J.Z. Liu, Q. Jiang, Size dependence of the thin-shell model for carbon nanotubes. Phys. Rev. Lett. 95, 105501 (2005) 39. H.S.P. Wong, D. Akinwande, Carbon Nanotube and Graphene Device Physics (Cambridge University Press, Cambridge, 2011) 40. V.N. Popov, Carbon nanotubes: properties and application. Mat. Sci. Eng. R 43, 61–102 (2004) 41. M. Mitra, S. Gopalakrishnan, Vibrational characteristics of single-walled carbon-nanotube: time and frequency domain analysis. J. Appl. Phys. 101, 114320 (2007) 42. 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, 1191– 1203 (2011) 43. 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, 312–329 (2015) 44. C. Thomsen, S. Reich, Double resonant Raman scattering in graphite. Phys. Rev. Lett. 85, 5214–5217 (2000) 45. L. Li, T. Chang, Explicit solution for G-band mode frequency of single-walled carbon nanotubes. Acta Mech. Solida Sin. 22, 571–583 (2009) 46. M.S. Dresselhaus, P.C. Eklund, Phonons in carbon nanotubes. Adv. Phys. 49, 705–814 (2000) 47. H. Kuzmany, W. Plank, M. Hulman, C. Kramberger, A. Gruneis, T. Pichler, H. Peterlik, H. Kataura, Y. Achiba, Determination of SWCNT diameters from the Raman response of the radial breathing mode. Eur. Phys. J. B 22, 307–320 (2001) 48. J. Maultzsch, H. Telg, S. Reich, C. Thomsen, Radial breathing mode of single-walled carbon nanotubes: optical transition energies and chiral-index assignment. Phys. Rev. B 72, 205438 (2005) 49. P.T. Araujo, P.B.C. Pesce, M.S. Dresselhaus, K. Sato, R. Saito, A. Jorio, Resonance Raman spectroscopy of the radial breathing modes in carbon nanotubes. Phys. E 42, 1251–1261 (2010) 50. D. Zhang, J. Yang, M. Li, Y. Li, (n, m) assignments of metallic single-walled carbon nanotubes by Raman spectroscopy: the importance of electronic Raman scattering. ACS Nano 10, 10789– 10797 (2016)

Chapter 10

Application of Nonlocal Elasticity Theory to Modelling of Two-Dimensional Structures

The mechanical properties of a graphene layer are fixed by the strength and stability of its carbon bonds. In a graphene sheet, every carbon atom is available for bonding with three other nearest neighbour atoms, forming strong planar σ -bonds with them. In graphite, however, this layer is rather very weakly bonded to other layers via vertical π -bonds [1]. The exotic mechanical properties of graphene play a significant role in various applications, and they also affect the other properties of graphene; for instance, its mechanical deformation promotes changes in its electronic structure [2]. Thus, the mechanical properties of graphene must be carefully studied to enhance its application potential. Even though the discovery of the graphene sheet has a very short history, it is very significant that its mechanical properties have been studied in an unusually significant number of papers employing the nonlocal continuum theory. In this chapter, the mechanical behaviour of the graphene sheets will be discussed via the nonlocal plate models. Simple analytical relations are derived to determine their bending behaviour, the natural frequencies and the buckling loads.

10.1 Bending Deformation of Graphene Sheets Under a Transverse Load Let us consider a rectangular flat nanoplate with thickness h, length L x in the xdirection and width L y in the y-direction. For the bending deformation of a flat graphene sheet, the in-plane compressive and shear forces and ∂ 2 w/∂t 2 must be set to zero in Eq. (5.103). The resultant partial differential equation then becomes

© Springer Nature Switzerland AG 2019 E. Ghavanloo et al., Computational Continuum Mechanics of Nanoscopic Structures, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-3-030-11650-7_10

219

220

10 Application of Nonlocal Elasticity Theory to Modelling …

∂ 4w ∂ 4w ∂ 4w D11 4 + 2(D12 + 2D66 ) 2 2 + D22 4 ∂x ∂x ∂ y ∂y   2 2  ∂ qz ∂ qz = qz − (e02 L i2 ) + ∂x2 ∂ y2

(10.1)

It is assumed that the load distributed over the surface of the graphene sheet is described by a double sine series: ∞  ∞ 



mπ x qz (x, y) = qmn sin Lx m=1 n=1







nπ y sin Ly

(10.2)

where L y L x qmn =

qz (x, y) sin 0

mπ x nπ y sin d xd y Lx Ly

(10.3)

0

Since we want to find the closed form solution, the simply-supported boundary conditions are only considered. The boundary conditions for a simply-supported graphene sheet are w(0, y) = w(L x , y) = w(x, 0) = w(x, L y ) = 0 Mx x (0, y) = Mx x (L x , y) = M yy (x, 0) = M yy (x, L y ) = 0

(10.4)

Using the Navier method, the solution of Eq. (10.1) takes on the form w(x, y) =

∞  ∞  m=1 n=1



mπ x Amn sin Lx





nπ y sin Ly

 (10.5)

where m and n are the half wave numbers in the x and y directions, respectively, and Amn is the amplitude of displacement and can be determined from the condition of equilibrium. By substituting Eqs. (10.2) and (10.5) into Eq. (10.1), the coefficients of Amn are obtained as Amn =

qmn [L 4x L 4y + e02 L i2 (L 2x L 4y m 2 π 2 + L 4x L 2y n 2 π 2 )] D11 L 4x L 4y m 4 π 4 + 2(D12 + 2D66 )L 2x L 2y m 2 n 2 π 4 + D22 L 4x n 4 π 4

(10.6)

10.2 Buckling of Single-Layered Graphene Sheets The in-plane deformation of a graphene sheet subject to a compressive load is very small prior to the onset of buckling instability. Different types of buckling are observed in graphene sheets, due to their relatively small bending rigidity, see

10.2 Buckling of Single-Layered Graphene Sheets

221

Fig. 10.1 Different types of the buckling; a rippling mode and b wrinkling mode. Reprinted from Ref. [3] with permission from publisher

Fig. 10.1 taken from Ref. [3]. Here the applicability of the nonlocal plate theories for prediction of the buckling characteristics of the graphene sheets having various shapes is presented.

10.2.1 Rectangular Shape For buckling analysis under in-plane normal load using the Kirchhoff plate theory, the time derivative, the distributed transverse load and N¯ x y in Eq. (5.103) are set to zero. Therefore, the resultant equation is ∂ 4w ∂ 4w ∂ 4w D11 4 + 2(D12 + 2D66 ) 2 2 + D22 4 ∂x ∂x ∂ y ∂y    2 2   2 ∂ w ∂ 2w ∂ ∂ 2 2 ¯ ¯ + 1 − (e0 L i ) + 2 N x x 2 + N yy 2 = 0 ∂x2 ∂y ∂x ∂y

(10.7)

Let N¯ x x = N¯ , N¯ yy = Λ N¯ , where N¯ is the buckling load and Λ is the load ratio. Substituting Eq. (10.5) into Eq. (10.7), we have 2 66 22 4 L x m 2n2 + D +2D n L2 D D 11 11 x y  (10.8) (e02 L i2 ) 2 L 2x L4 4 2 2 + L 2 π m + (1 + Λ)m n L 2 + Λn 4 L 4x

L2

m 4 L 2y + 2

π D11 N¯ = L 2y m 2 + Λn 2 L 2x 2

L 2y



D12 D11

x

y

y

or, in an equivalent form, π 2 D11 N¯ = K L 2y

(10.9)

222

10 Application of Nonlocal Elasticity Theory to Modelling …

where K is the buckling load parameter defined by 2 66 22 4 L x m 2n2 + D +2D n L2 D11 D11 x y  2 (e02 L i2 ) 2 L L4 x + L 2 π m 4 + (1 + Λ)m 2 n 2 L 2 + Λn 4 L 4x

L2

K =

m 4 L 2y + 2 L2

m 2 + Λn 2 L 2x

y



D12 D11

x

y

(10.10)

y

To simplify the analysis, the following dimensionless quantities are defined: β=

Di j Lx e0 L i , λ= , di j = Ly Lx D11

(10.11)

Substitution of Eq. (10.11) into Eq. (10.10) gives K =

m 4 β12 + 2(d12 + 2d66 )m 2 n 2 + d22 n 4 β 2 m 2 + Λn 2 β 2 + λ2 π 2 (m 4 + (1 + Λ)m 2 n 2 β 2 + Λn 4 β 2 )

(10.12)

Expression (10.12) gives all values of K corresponding to different values of m and n. From among these values one must select the smallest, which will be the critical value. It can be shown that the smallest value of K is obtained for n = 1. Consequently, the equation for K takes the form K =

m 4 β12 + 2(d12 + 2d66 )m 2 + d22 β 2 m 2 + Λβ 2 + λ2 π 2 (m 4 + (1 + Λ)m 2 β 2 + Λβ 2 )

(10.13)

It is clear that for a given β the critical buckling load parameter is obtained by selecting m so that it minimizes Eq. (10.13). For the case of a uniaxial loading (Λ = 0), the variations of the buckling load parameter as a function of the aspect ratio β for λ = 0.00, 0.05 and 0.10 are shown in Fig. 10.2. Referring to this figure, the magnitude of K for any value of the aspect ratio and the nonlocal parameter can be readily found. Furthermore, Fig. 10.3 illustrates the effects of the aspect ratio and the load ratio on the buckling load parameter. Note that the material properties are obtained based on Eqs. (6.34) and (6.35). For other boundary conditions, the buckling of the graphene sheet can be obtained by different numerical approaches such as the finite element method, the Galerkin method, the differential quadrature method and etc.

10.2.2 Circular Shape On the basis of the nonlocal axisymmetric plate theory described in Sect. 5.3.3, the governing equation of the buckling of a circular graphene sheet is expressed as

10.2 Buckling of Single-Layered Graphene Sheets

223

Fig. 10.2 Variations of the buckling load parameter as a function of the aspect ratio and the nonlocal parameter

Fig. 10.3 Variations of the buckling load parameter as a function of the aspect ratio and the load ratio for λ = 0.05

2  2    d d w dw dw 1 1 d 2 2 r 2 + r −a λ + 2 − dr dr 2 r dr r dr dr  2   2 d w 1 dw 1 d d (10.14) + + = −κr dr 2 r dr dr 2 r dr Nˆ r



d dr

It is convenient to define two new variables g(r ) =

d 2 w 1 dw + dr 2 r dr

(10.15)

224

10 Application of Nonlocal Elasticity Theory to Modelling …

Nˆ r

Ξ2 =

(10.16)

(κ−a 2 λ2 Nˆ r )

Substituting Eqs. (10.15) and (10.16) into Eq. (10.14), we obtain 1 ∂g ∂2g + + Ξ 2g = 0 2 ∂r r ∂r

(10.17)

The general solution of Eq. (10.17) is [4] g(r ) = C1 J0 (Ξr ) + C2 Y0 (Ξr )

(10.18)

where C1 and C2 are unknown constants, J0 and Y0 are the Bessel functions of the first and second kind of zero orders, respectively. Substituting Eq. (10.18) into Eq. (10.15) and integrating the resultant relation with respect to r , we have C1 C2 C3 dw = J1 (Ξr ) + Y1 (Ξr ) + dr Ξ Ξ r

w(r ) = −

C1 C2 J0 (Ξr ) − 2 Y0 (Ξr ) + C3 ln(r ) + C4 Ξ2 Ξ

(10.19)

(10.20)

where C3 and C4 are constants of integration. Since w(r ) must be finite for all values of r , thus the coefficients of the two terms ln(r ) and Y0 (Ξr ), having singularities at the origin (r = 0), must be zero. As a result, the solution can be rewritten as w(r ) = −

C1 J0 (Ξr ) + C4 Ξ2

(10.21)

To determine the buckling load of the circular graphene sheet, the boundary conditions must be known. Here the fully clamped boundary conditions are considered. Mathematical implementation of the boundary conditions at r = a is expressed as follows. dw =0 (10.22) w(a) = 0, dr r =a Using the general solution (10.21), we obtain

(Ξ a) − J0Ξ 1 2 J1 (Ξ a) Ξ

0



C1 C4

 =

  0 0

(10.23)

10.2 Buckling of Single-Layered Graphene Sheets

225

For a nontrivial solution of Eq. (10.23), we have J1 (Ξ a) = 0

(10.24)

which is the stability criterion. This equation must be solved to obtain the parameter Ξ . The first five values are [5] Ξ1 a = j1 = 3.8317 Ξ2 a = j2 = 7.0156 Ξ3 a = j3 = 10.1735 Ξ4 a = j4 = 13.3237 Ξ5 a = j5 = 16.4706

(10.25)

Using the definition of the parameter Ξ , the lowest buckling load is obtained as Nˆ r =

j12 κ (a 2 + a 2 j12 λ2 )

(10.26)

10.3 Deformation of Geometrically Imperfect Circular Graphene Sheets Experimental evidence together with computer-based molecular simulations have suggested that at nonzero temperature the graphene sheet does not display a flat geometry [6–8]. Few results are, however, available that relate to the mechanical behaviour of the nonflat circular graphene sheets. By applying the nonequilibrium MD simulation method, Neek-Amal and Peeters [9] have studied the stability behaviour of a circular graphene sheet subject to a radial load. It was shown that, by application of a radial stress to the graphene sheet, following some small strain, the sheet buckled and was bent into a geometry, called a graphene nano-bowl that was stable thermodynamically. In this section, we employ the nonlocal elasticity theory to investigate the deformation properties of a geometrically imperfect circular graphene sheet subject to a uniform radial load. We assume that the load is symmetrically distributed and hence it would be sufficient to consider the axisymmetric deformations only. The circular graphene sheet that we consider is assumed to have a small amount of initial curvature. We obtain the closed-form solution for deformations of a suspended graphene sheet clamped on a circular hole by solving the governing equation and boundary conditions. Consider Fig. 10.4 whereby a typical nonflat graphene sheet is clamped on a circular hole of radius a. To obtain a mathematical formulation for this problem, we denote the radial position, the downward deflection and the initial imperfection by

226

10 Application of Nonlocal Elasticity Theory to Modelling …

Fig. 10.4 Schematic illustration of a geometrically imperfect circular graphene sheet subject to a radial compressive load

r , w and w0 , respectively. Furthermore, we assume that the graphene sheet is subject to a uniform radial load Nr . Using Eq. (5.110), the principal radii of curvature are ψ d 2w dψ 1 1 dw 1 , = =− 2 = =− R1 dr dr R2 r dr r

(10.27)

where ψ is the slop. From the differential constitutive relations, the relations between the change in the curvatures and the bending moments can be obtained as

d 2 Mrr (Mrr − Mtt ) 1 d Mrr Mrr − a λ −2 + dr 2 r dr r2   d(ψ − ψ0 ) ν(ψ − ψ0 ) + =κ dr r



2 2



d 2 Mtt (Mrr − Mtt ) 1 d Mtt Mtt − a λ +2 + dr 2 r dr r2   d(ψ − ψ0 ) (ψ − ψ0 ) + =κ ν dr r

(10.28)



2 2

(10.29)

10.3 Deformation of Geometrically Imperfect Circular Graphene Sheets

227

where ψ0 represents the slop of the initial configuration of the graphene sheet. For axisymmetric deformation of circular plates, the equilibrium equation is given by Mrr − Mtt d Mrr + = Qr dr r

(10.30)

Furthermore, using Eq. (5.113), the shearing force at any point is given by Q r = −Nr ψ

(10.31)

By using Eqs. (10.28)–(10.31), the nonlocal governing equation is obtained as     κΞ 2 d 2 ψ0 ψ0 1 dψ 1 1 dψ0 d 2ψ 2 (10.32) +ψ Ξ − 2 = − 2 + + dr 2 r dr r Nr dr 2 r dr r where Ξ was defined similar to Eq. (10.16). Here, we have assumed that the imperfection can be described by the relation 

r2 w0 (r ) = W 1 − 2 a

2 (10.33)

with W being the imperfection amplitude at the center. Using ψ0 = dw0 /dr together with substitution of Eq. (10.33) into Eq. (10.32), leads to the governing equation   κΞ 2 32W r 1 dψ 1 d 2ψ 2 = − + ψ Ξ + − dr 2 r dr r2 Nr a 4

(10.34)

This equation represents the inhomogeneous version of Bessel’s equation and its general solution being the sum of the homogeneous solution, ψh and the particular solution ψ p . The homogeneous solution of Eq. (10.34) is given by ψh (r ) = C1 J1 (Ξr ) + C2 Y1 (Ξr )

(10.35)

where C1 and C2 are unknown constants, and J1 and Y1 are the order one Bessel functions of first and second kinds respectively. Since the displacement at the origin is not infinite, the coefficient of the Bessel function of the second kind must be zero and so we have ψh (r ) = C1 J1 (Ξr )

(10.36)

Furthermore, the particular solution is ψ p (r ) = −

32κ W r Nr a 4

(10.37)

228

10 Application of Nonlocal Elasticity Theory to Modelling …

Therefore, the rigorous solution is obtained as ψ(r ) = C1 J1 (Ξr ) −

32κ W r Nr a 4

(10.38)

It follows, by integration, that w(r ) =

C1 16κ W r 2 J0 (Ξr ) + + C3 Ξ Nr a 4

(10.39)

where J0 is the zero order Bessel function of first kind and C3 is the constant of integration. The boundary conditions are w(a) = 0

(10.40)

ψ(a) = 0

(10.41)

Using Eqs. (10.38) and (10.41), we obtain C1 =

32κ W a Nr a 3 J1 (Ξ a)

(10.42)

Furthermore, the constant C3 is obtained as follows: C3 = −

32κ W a 16κ W J0 (Ξ a) − Nr a 3 Ξ J1 (Ξ a) Nr a 2

(10.43)

Substitution of Eqs. (10.42) and (10.43) into Eq. (10.39), leads to the closedform solutions of axisymmetric deformation of a geometrically imperfect circular graphene sheet w(r ) =

16κ W (a 2 − r 2 ) 32κ W a [J (Ξr ) − J (Ξ a)] − 0 0 Nr a 3 Ξ J1 (Ξ a) Nr a 4

(10.44)

The deformation profiles along the radial direction of the graphene sheet corresponding to five different values of uniform radial compressive force Nr are computed on the basis of Eq. (10.44) and the values κ = 1.46 eV, ν = 0.149 and λ = 0 nm, a = 10 nm and W = 0.25 nm, and are shown in Fig. 10.5. It is to be noted that the deformation profile shows the initial configuration of the graphene sheet for a zero compressive force. Furthermore, as expected, the results show that the deflection increases with an increase in the radial force. This implies that a larger radial force

10.3 Deformation of Geometrically Imperfect Circular Graphene Sheets

229

Fig. 10.5 Deformation profiles of a geometrically imperfect circular graphene sheet subject to different radial loads

Fig. 10.6 Variations of load Nr with the net central deflection for different values of imperfection amplitude

results in a deeper nano-bowl. Figure 10.6 shows the variations of the load Nr with the net central deflection (δ = w(0) − W ) for various values of W that characterize the imperfection and with λ = 0.0625 and a = 8 nm. It is observed that as the initial imperfection tends to zero, W → 0, the load-deflection curve asymptotically approaches that of a flat graphene sheet. Further results and discussion can be found in Ref. [10].

230

10 Application of Nonlocal Elasticity Theory to Modelling …

10.4 Transverse Vibration of Graphene Sheets Experimental studies have shown that graphene sheets can be mechanically actuated and that the resonant frequencies can be extracted using optical interferometry [11]. Vibrational behaviour of the graphene sheets has been the subject of extensive investigations due to its importance in different technological applications in such fields as electronics, optics, sensor devices, and nano-electromechanical devices [12, 13]. In this section, we consider the vibrational behaviour of the graphene sheets by employing the nonlocal plate models.

10.4.1 Transverse Free Vibration of Rectangular Graphene Sheet A single-layered graphene sheet can be approximately modelled by a thin elastic nanoplate [14]. The equation of motion for the transverse free vibration of a thin elastic nanoplate can be expressed as ∂ 4w ∂ 4w ∂ 4w + 2(D + 2D ) + D 12 66 22 ∂x4 ∂ x 2∂ y2 ∂ y4   2  4 ∂ w ∂ w ∂ 4w =0 +ρh − (e02 L i2 ) + 2 2 2 2 2 ∂t ∂ x ∂t ∂ y ∂t D11

(10.45)

For a rectangular graphene plate with edge lengths L x and L y , there are eight boundary conditions for every case. Two cases of boundary conditions are discussed below: (a) fully simply-supported, and (b) two opposite edges simply-supported and the other two edges having any arbitrary conditions. For the fully simply-supported rectangular graphene sheet, the well-known exact solution for the free vibration analysis is obtained by the Navier method. In this method, the solution is presented by w(x, y, t) =

∞  ∞  m=1 n=1

 Amn sin

mπ x Lx



 sin

nπ y Ly

 exp( jωt)

(10.46)

Substitution of Eq. (10.46) into Eq. (10.45) and using the dimensionless parameters defined in Eq. (10.11), gives 2 ωmn =

π 4 D11 m 4 + 2(d12 + 2d66 )β 2 m 2 n 2 + d22 β 4 n 4 ρh L 4x 1 + λ2 (m 2 π 2 + β 2 n 2 π 2 )

(10.47)

The fundamental natural frequency can be obtained by letting m = 1 and n = 1. Note that the amplitude Amn cannot be determined from the linear eigenvalue

10.4 Transverse Vibration of Graphene Sheets

231

problem. For a square graphene sheet, Eq. (10.47) becomes 2 = ωmn

π 4 D11 m 4 + 2(d12 + 2d66 )m 2 n 2 + d22 n 4 ρh L 4x 1 + λ(m 2 π 2 + n 2 π 2 )

(10.48)

and the fundamental natural frequency is 2 = ω11

π 4 D11 1 + 2(d12 + 2d66 ) + d22 ρh L 4x 1 + 2λπ 2

(10.49)

The first four vibration mode shapes of the square nanoplates corresponding to the natural frequencies ωmn , given by Eq. (10.48), are shown in Fig. 10.7. It is seen that only one peak is observed at the center of the square nanoplate. Furthermore, it is clear that the number of peaks increases as the order of the vibration mode becomes higher. For the second type of boundary condition, the Levy type solution method is usually used. Assuming that the simply-supported edges are at x = 0 and x = L x , the solution is presented by w(x, y, t) =

∞  m=1



mπ x f (y) sin Lx

 exp( jωt)

(10.50)

Substituting Eq. (10.50) into Eq. (10.45) and using a specific value of m yields a4

d4 f d2 f + a + a0 f = 0 2 d y4 d y2

(10.51)

where m2π 2 − ρhω2 e02 L i2 L 2x   m4π 4 m2π 2 a0 = D11 4 − ρhω2 1 + e02 L i2 2 Lx Lx a4 = D22 , a2 = 2(D12 + 2D66 )

(10.52)

The solutions of Eq. (10.51) can be expressed as f (y) = C1 cosh(η1 y) + C2 sinh(η1 y) + C3 cos(η2 y) + C4 sin(η2 y) (10.53) where

η1 =

    −a + a 2 − 4a a 4 0  2 2 2a4

(10.54)

232

10 Application of Nonlocal Elasticity Theory to Modelling …

Fig. 10.7 Mode shapes of a square simply-supported nanoplate

η2 =

    a + a 2 − 4a a 4 0  2 2 2a4

(10.55)

and the constants C1 , C2 , C3 and C4 must be determined from the boundary conditions at y = 0 and y = L y . The boundary conditions yield the characteristic equation from which ω is determined. For example, if the two edges are simply-supported and the other two are clamped, we have f (0) = f (L y ) = 0,

d f d f = =0 dy y=0 dy y=L y

(10.56)

10.4 Transverse Vibration of Graphene Sheets

233

By substituting Eq. (10.53) into the boundary conditions, the eigenvalue problem may be expressed as ⎛

⎞⎛ ⎞ 1 0 1 0 C1 ⎜ cosh(η1 L y ) sinh(η1 L y ) ⎟ ⎜ C2 ⎟ cos(η L ) sin(η L ) 2 y 2 y ⎜ ⎟⎜ ⎟ ⎝ ⎠ ⎝ C3 ⎠ 0 η2 0 η1 η1 sinh(η1 L y ) η1 cosh(η1 L y ) −η2 sin(η2 L y ) η2 cos(η2 L y ) C4 ⎛ ⎞ 0 ⎜0⎟ ⎟ =⎜ (10.57) ⎝0⎠ 0 In order to avoid a trivial solution (C1 = C2 = C3 = C4 = 0), the determinant of the coefficient matrix in Eq. (10.57) must vanish. This condition gives the characteristic equation.

10.4.2 Viscoelastic Graphene Sheets Embedded in a Viscoelastic Medium In materials such as nanocomposites, graphene sheets can be embedded as inclusions in polymer and metallic media displaying a viscoelastic property [15]. Generally, in a viscoelastic medium, a portion of the deformation energy is recoverable while the other portion is cannot be recovered. Consequently attention should be directed to the analysis of their vibration in viscoelastic media. Viscoelastic properties of a graphene oxide nanoplate have been reported experimentally by Su et al. [16]. Clear hysteresis loops are observed in the tensile tests on this nanoplate, indicating that it has a viscoelastic property. In this subsection, we investigate the vibration characteristics of viscoelastic graphene sheets embedded in a viscoelastic medium by using the nonlocal orthotropic Kirchhoff’s plate theory to model the viscoelastic graphene sheets. The viscoelastic medium is modelled as the Kelvin–Voigt foundation. Our formulations follow the work of Pouresmaeeli et al. [17]. The main difficulty of this type of problem is related to the correct mathematical description of the foundation. There exist various models of viscoelastic foundations. The simplest is the viscoelastic Winkler model, wherein the foundation is described as a system of parallel springs and dashpots. On the basis of this model, the distributed transverse load is expressed as qz = −K w w − Cw

∂w ∂t

(10.58)

where K w and Cw are the modulus and the damping factor of the linear viscoelastic Winkler foundation, respectively. In addition, on the basis of the Kelvin–Voigt model,

234

10 Application of Nonlocal Elasticity Theory to Modelling …

in elastic materials with viscoelastic structural damping coefficient g, the flexural rigidities D11 , D12 , D22 and D66 are replaced with the operators D11 (1 + g∂/∂t), D12 (1 + g∂/∂t), D22 (1 + g∂/∂t) and D66 (1 + g∂/∂t), respectively [18]. Application of these operators to Eq. (5.103) and using Eq. (10.58), one can write  4   ∂ w ∂ 4w ∂ 5w ∂ 5w D11 +g 4 +g 2 2 + 2(D12 + 2D66 ) ∂x4 ∂ x ∂t ∂ x 2∂ y2 ∂ x ∂ y ∂t  4     4 ∂ w ∂ 2w w ∂ 5w ∂ ∂ 4w 2 2 + ρh +g 4 − (e0 L i ) + 2 2 +D22 ∂ y4 ∂ y ∂t ∂t 2 ∂ x 2 ∂t 2 ∂ y ∂t   2 2  w w ∂ ∂ +K w w − (e02 L i2 ) + ∂x2 ∂ y2    3 ∂w ∂ 3w ∂ w − (e02 L i2 ) + =0 (10.59) +Cw ∂t ∂ x 2 ∂t ∂ y 2 ∂t 

For the simply-supported boundary conditions, the solution of Eq. (10.59) has the form w(x, y, t) =

∞  ∞ 

 Amn sin

m=1 n=1

mπ x Lx



 sin

nπ y Ly

 exp(st)

(10.60)

where s is the complex eigenfrequency. Substituting Eq. (10.60) into Eq. (10.59), the eigenfrequency is obtained as s1,2 = −α¯ 1 ±



α¯ 12 − α¯ 2

(10.61)

where 

D11 mLπ4 + 2(D12 + 2D66 ) mLπ2 nLπ2 + D22 nLπ4 x x y y    α¯ 1 = n2 π 2 2 2 m2π 2 2ρh 1 + e0 L i L 2 + L 2 x y    n2 π 2 2 2 m2π 2 Cw 1 + e0 L i L 2 + L 2 y  2x 2   + 2 2 2ρh 1 + e02 L i2 mLπ2 + nLπ2 4

4

2

x

2

2

2

4

4

g

(10.62)

y

D11 mLπ4 + 2(D12 + 2D66 ) mLπ2 nLπ2 + D22 nLπ4 x x y y    α¯ 2 = n2 π 2 2 2 m2π 2 ρh 1 + e0 L i L 2 + L 2 x y    n2 π 2 2 2 m2π 2 K w 1 + e0 L i L 2 + L 2 y  x   + n2 π 2 2 2 m2π 2 ρh 1 + e0 L i L 2 + L 2 4



4

2

x

y

2

2

2

4

4

(10.63)

10.4 Transverse Vibration of Graphene Sheets

For convenience, the following dimensionless parameters are defined:  ρh K w L 4x Cw L 2x , K∗ = , C∗ = √ s ∗ = s L 2x D11 D11 ρh D11  Di j e0 L i g D11 Lx , μ= G= 2 , β= , di j = L x ρh Lx Ly D11

235

(10.64)

Substituting Eq. (10.64) into Eq. (10.61), the dimensionless eigenfrequency becomes    2 (10.65) s ∗ = −ωmn ξmn ± j 1 − ξmn It is to be noted that the imaginary part of this eigenfrequency corresponds to the frequency and the real part corresponds to the damping effect. Here, ωmn and ξmn are the dimensionless undamped frequency and the damping ratio, respectively. These can be expressed as m 4 π 4 + 2(d12 + 2d66 )β 2 m 2 n 2 π 4 + d22 β 4 n 4 π 4 1 + μ2 (m 2 π 2 + β 2 n 2 π 2 ) K ∗ {1 + μ2 (m 2 π 2 + β 2 n 2 π 2 )} + 1 + μ2 (m 2 π 2 + β 2 n 2 π 2 ) G(m 4 π 4 + 2(d12 + 2d66 )β 2 m 2 n 2 π 4 + d22 β 4 n 4 π 4 ) = 2ωmn {1 + μ2 (m 2 π 2 + β 2 n 2 π 2 )} C ∗ {1 + μ2 (m 2 π 2 + β 2 n 2 π 2 )} + 2ωmn {1 + μ2 (m 2 π 2 + β 2 n 2 π 2 )}

2 = ωmn

ξmn

(10.66)

(10.67)

For ξmn > 1 the vibration is over-damped, while for ξmn = 1 it is critically damped and for 0 < ξmn < 1 it is under-damped. Figure 10.8 displays the variations of imaginary and real parts of the fundamental dimensionless eigenfrequency (n = m = 1), and the damping ratio as a function of the nonlocal parameter corresponding to different values of the damping coefficient of the surrounding medium, and K ∗ = 100, G = 0.01 and β = 1. It is observed that the frequency is significantly influenced by the damping coefficient C ∗ and the nonlocal parameter μ. It is also seen that the imaginary part of the eigenfrequency becomes zero when μ = 0.17 for C ∗ = 35, indicating that the system is critically damped for this value. The fundamental frequency and damping of a viscoelastic nanoplate are shown as a function of the nonlocal parameter in Fig. 10.9a, b respectively, for different values of the viscoelastic structural damping coefficient G = 0.05, 0.06, 0.065 and 0.075. For numerical calculations in this case, we take K ∗ = 100, C ∗ = 20, β = 1 and n = m = 1. Furthermore, the regions of ξ11 > 1, ξ11 = 1, and ξmn < 1 for different values of the viscoelastic structural damping are plotted in Fig. 10.9c. It is obvious that the frequency is significantly affected by the structural damping coefficient G

236 Fig. 10.8 Effect of the damping coefficient on the vibration of the nanoplate; a Imaginary part of the eigenfrequency, b Real part of the eigenfrequency and c Damping ratio

10 Application of Nonlocal Elasticity Theory to Modelling …

10.4 Transverse Vibration of Graphene Sheets Fig. 10.9 Variations of the eigenfrequency and the damping ratio with the nonlocal parameter for different values of the structural damping; a imaginary part, b real part and c damping ratio

237

238

10 Application of Nonlocal Elasticity Theory to Modelling …

and the nonlocal parameter μ. As expected, the frequency of the nanoplate decreases with the increment of the nonlocal parameter and the structural damping coefficient. Moreover, it can be found that the system is critically damped for the cases of μ = 0.097 for G = 0.065 and μ = 0.25 for G = 0.075.

10.5 Application of the Nonlocal Anisotropic Elasticity to Modelling Graphene Sheets Employing the linear differential operator for the nonlocal anisotropic elasticity presented in Sect. 4.5, the equation of motion of the graphene sheets is derived as ∂ 4w ∂ 4w ∂ 4w D11 4 + 2(D12 + 2D66 ) 2 2 + D22 4 ∂x ∂x ∂ y ∂y   ∂ 2w ∂2 ∂2 ∂ 2w ρh 2 + N¯ x x 2 + 1 − L 11 2 − L 22 2 ∂x ∂y ∂t ∂x  2 2 ∂ w ∂ w + N¯ yy 2 − qz = 0 +2 N¯ x y ∂ x∂ y ∂y

(10.68)

Let N¯ x x = N¯ , N¯ yy = Λ N¯ and N¯ x y = 0, and substituting Eqs. (10.2) and (10.46) into Eq. (10.68), we have

 m4π 4 n4π 4 m 2π 2 n2π 2 Amn D11 4 + 2(D12 + 2D66 ) 2 + D22 4 Lx L x L 2y Ly   m2π 2 n2π 2 m2π 2 ω2 ρh Amn + N¯ Amn − 1 + L 11 2 + L 22 2 Lx Ly L 2x  2 2 π n +Λ N¯ 2 Amn + qmn = 0 Ly

(10.69)

For the bending problem, by setting N¯ and ω to go to zero in Eq. (10.69), the static deflection is obtained    2 2 2 2 nπ y x ∞  ∞ 1 + L 11 mLπ2 + L 22 nLπ2 qmn sin mπ sin  Lx Ly x y w(x, y) = (10.70) m 2 π 2 n2 π 2 m4π 4 n4 π 4 D11 L 4 + 2(D12 + 2D66 ) L 2 L 2 + D22 L 4 m=1 n=1 x

x

y

y

For the buckling problem, qmn and ω are set to zero. Thus, we obtain  N¯ =

D11 mLπ4 + 2(D12 + 2D66 ) mLπ2 nLπ2 + D22 nLπ4  x  x 2 2y y m π n2 π 2 m2π 2 n2 π 2 1 + L 11 L 2 + L 22 L 2 + Λ L2 L2 4

4

2

x

y

2

2

x

2

4

y

4

(10.71)

10.5 Application of the Nonlocal Anisotropic Elasticity to Modelling Graphene Sheets

239

Finally, for the free vibration problem, it is assumed that qmn and N¯ are set equal to zero. As a result, the natural frequencies are given by

ωmn

  4 4 2 2 2 2 4 4  D11 mLπ4 + 2(D12 + 2D66 ) mLπ2 nLπ2 + D22 nLπ4  x x y y   = m2π 2 n2 π 2 ρh 1 + L 11 L 2 + L 22 L 2 x

(10.72)

y

References 1. H.S.P. Wong, D. Akinwande, Carbon Nanotube and Graphene Device Physics (Cambridge University Press, Cambridge, 2011) 2. K.M. Liew, Y. Zhang, L.W. Zhang, Nonlocal elasticity theory for graphene modeling and simulation: prospects and challenges. J. Model. Mech. Mater. 1, 20160159 (2017) 3. H. Rafii-Tabar, E. Ghavanloo, S.A. Fazelzadeh, Nonlocal continuum-based modeling of mechanical characteristics of nanoscopic structures. Phys. Rep. 638, 1–97 (2016) 4. C.R. Wylie, Advanced Engineering Mathematics (McGraw-Hill, London, 1960) 5. G.N. Watson, Theory of Bessel Functions (Cambridge University Press, Cambridge, 1944) 6. J.C. Meyer, A.K. Geim, M.I. Katsnelson, K.S. Novoselov, T.J. Booth, S. Roth, The structure of suspended graphene sheets. Nature 446, 60–63 (2007) 7. A. Fasolino, J.H. Los, M.I. Katsnelson, Intrinsic ripples in graphene. Nat. Mater. 6, 858–861 (2007) 8. W. Bao, F. Miao, Z. Chen, H. Zhang, W. Jang, C. Dames, C.N. Lau, Controlled ripple texturing of suspended graphene and ultrathin graphite membranes. Nat. Nanotechnol. 4, 565–566 (2009) 9. F.M. Neek-Amal, F.M. Peeters, Buckled circular monolayer graphene: a graphene nano-bowl. J. Phys. Condens. Matter 23, 045002 (2011) 10. E. Ghavanloo, Axisymmetric deformation of geometrically imperfect circular graphene sheets. Acta Mech. 228, 3297–3305 (2017) 11. D. Garcia-Sanchez, A.M. van der Zande, A. San Paulo, B. Lassagne, P.L. McEuen, A. Bachtold, Imaging mechanical vibrations in suspended graphene sheets. Nano Lett. 8, 1399–1403 (2008) 12. S.A. Fazelzadeh, E. Ghavanloo, Nanoscale mass sensing based on vibration of single-layered graphene sheet in thermal environments. Acta Mech. Sin. 30, 84–91 (2014) 13. A. Nag, A. Mitra, S.C. Mukhopadhyay, Graphene and its sensor-based applications: a review. Sens. Actuator A-Phys. 270, 177–194 (2018) 14. R. Chowdhury, S. Adhikari, F. Scarpa, M.I. Friswell, Transverse vibration of single-layer graphene sheets. J. Phys. D: Appl. Phys. 44, 205401 (2011) 15. I. Srivastava, Z.Z. Yu, N.A. Koratkar, Viscoelastic properties of graphene-polymer composites. Adv. Sci. Eng. Med. 4, 10–14 (2012) 16. 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, 2804–2809 (2012) 17. S. Pouresmaeeli, E. Ghavanloo, S.A. Fazelzadeh, Vibration analysis of viscoelastic orthotropic nanoplates resting on viscoelastic medium. Compos. Struct. 96, 405–410 (2013) 18. A.D. Drozdov, Viscoelastic Structures: Mechanics of Growth and Aging (Academic Press, San Diego, 1998)

Chapter 11

Nonlocal Elasticity Models for Mechanics of Complex Nanoscopic Structures

In the previous chapters, we have focused on the applications of the nonlocal elasticity models to the study of the mechanical characteristics of simple nanoscopic structure. In this chapter, we will survey the nonlocal continuum-based studies concerned with the mechanical behaviour of more complex nanoscopic structures. There are three main sections in this chapter. In Sect. 11.1, the application of the nonlocal models to nanoscale devices such as resonant sensors is described. Section 11.2 summarizes the application of the nonlocal models to some biological nanoscopic structures. Finally, Sect. 11.3 is concerned with the application of the models to the investigation of the properties of other types of nanoscopic structures including nanopeapods, nanotube heterojunctions, carbon honeycombs, Y-junction carbon nanotube and piezoelectric nanoscopic structures.

11.1 Application of the Nonlocal Models to Nanoscale Devices 11.1.1 Resonant Sensors Nanosensors are chemically or mechanically activated sensors that can be utilized to detect the presence of nanoparticles and biomolecules or to monitor the development of physical parameters such as temperature and pressure at the nanoscale. Among numerous existing physical and chemical sensing techniques for detection of properties of nanoscopic structures, resonance-based sensors form a class of novel devices in the field of nanometrology [1]. Their detection criteria are established on the basis of the measurement of the resonant frequency shift of the sensor which is sensitive to the position, size and shape of the attached object.

© Springer Nature Switzerland AG 2019 E. Ghavanloo et al., Computational Continuum Mechanics of Nanoscopic Structures, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-3-030-11650-7_11

241

242

11.1.1.1

11 Nonlocal Elasticity Models for Mechanics of Complex Nanoscopic Structures

Carbon Nanotube-Based Resonant Sensors

Experimental evidences show that biological structures, such as DNA, bacteria, proteins, enzymes, etc., can be immobilized on the surface of carbon nanotubes [2–4]. This has motivated the development of carbon nanotube-based nanomechanical sensor [5]. The idea of using carbon nanotubes (CNTs) as highly sensitive mass sensors was first proposed by Poncharal et al. [6]. The first attempt to use the nonlocal elasticity theory in modelling carbon nanotube-based resonant sensors was made by Lee et al. [7]. They studied the frequency shift and sensitivity of carbon nanotube-based cantilever sensor with an attached mass employing the nonlocal Euler-Bernoulli beam theory. In another theoretical study, the axial vibration of SWCNT-based mass sensors was investigated by Aydogdu and Filiz [8] on the basis of the nonlocal rod model. The potential application of a DWCNT as a mass sensor was first explored by Shen et al. [9] using the nonlocal Timoshenko beam theory. In that study, the mass sensor was modelled as a cantilever beam carrying a mass at the free end of the inner tube. They assumed that the inner and outer tubes had different lengths and the interaction between the two tubes was described by the van der Waals force. Their results revealed that the DWCNT-based mass sensor had a noticeable advantage with respect to SWCNT-based mass sensors. To ensure the realistic applications of a cantilever SWCNT as a mass sensor, a mathematical framework according to the nonlocal Euler-Bernoulli beam theory was developed and compared with molecular mechanics [10]. Two configurations of the added mass, namely, a point mass and a distributed mass were considered as physically realistic models (see Fig. 11.1, taken from [10]). Reasonable agreements between the molecular mechanics simulations and the nonlocal beam model were observed. As a result, it was concluded that the nonlocal beam model was applicable to SWCNT-based mass sensors. A generalized study of the SWCNT serving as a virus or bacterium sensor, including the complex contributions of the nonlocality and the surface effects, was presented

Fig. 11.1 Two configurations of mass attached to the SWCNT. Reprinted with permission from the publisher

11.1 Application of the Nonlocal Models to Nanoscale Devices

243

Fig. 11.2 Schematic illustration of a clamped-clamped carbon nanotube with an attached mass under a tensile load as reported by Natsuki et al. [13]. Reprinted with permission from the publisher

by Elishakoff et al. [11]. They demonstrated that these effects greatly influence the sensitivity of the virus sensor. In order to increase the sensitivity of the sensors, Natsuki and co-workers [12, 13] proposed the idea of using the double-end-fixed carbon nanotubes under axial tensile loads. Figure 11.2, taken from [13], shows a schematic diagram of the clamped-clamped carbon nanotube carrying an attached concentrated mass and subject to an axial tensile force. According to the results obtained, the feasibility of the proposed idea was proved, and its potential applicability was demonstrated. In another studies, Eltaher et al. [14] and Li et al. [15] showed that the sensitivity of SWCNT-based sensors was of the order of zeptogram (1 zg = 10−21 g). For a more comprehensive discussion about the application of the nonlocal models to the SWCNT-based mass sensors, we refer the reader to the rather extensive reviews on the subject [5, 16].

11.1.1.2

Graphene-Based Resonant Sensors

The first vibration analysis of graphene-based mass sensors was made by Shen et al. [17] using the nonlocal Kirchhoff plate theory. They discussed the effects of mass and position of the attached nanoobject on the frequency shift of the graphene sheet. The numerical results obtained revealed that the sensitivity of graphene-based sensors was of the order of zeptogram, of the same order as obtained in Refs. [18, 19]. The nonlocal plate theory was utilized to model the graphene sheet as a nanomechanical sensor, and it was compared with a molecular mechanics-based modelling employing the universal force field model [20]. Reasonable agreement between the results from the nonlocal continuum based-model and those from the molecular mechanics simulations were observed (see Fig. 11.3, taken from [20]). Consequently, it was concluded that the nonlocal theory can be used to model graphene-based mass sensors. That work was, however, limited to attached masses with a line shape distribution. Lee et al. [21] have analysed the mass detection of a graphene-based nanoresonator within the framework of the nonlocal theory. They included the contribution of the small-scale effect, size and aspect ratio of the graphene sheet on the sensitivity of the sensor. The plausibility of double-layered graphene sheets acting as resonant mass sensors was explored by Natsuki et al. [22] and Shen et al. [23]. In most of previous publications, the effect of temperature change on the frequency shift of the graphene-based mass sensors has not been taken into account,

244

11 Nonlocal Elasticity Models for Mechanics of Complex Nanoscopic Structures

Fig. 11.3 Identification of mass from the variations in the frequency-shift of a cantilevered SLGS resonator for two possible arrangements of attached objects as reported by Murmu and Adhikari [20]. Reprinted with permission from the publisher

while we know that thermal gradient has a significant effect on the performance and sensitivity of such sensors. Motivated by this fact, Fazelzadeh and Ghavanloo [24] investigated the vibration characteristics of graphene-based mass sensors with multiple attached nanoparticles in thermal environments. In that study, the sensor was modeled via a simply-supported rectangular orthotropic nanoplate having multiple attached nanoparticles placed at different locations (Fig. 11.4). Since the physical dimensions of the nanoparticles were assumed to be very small compared to the in-plane dimensions of the single layer graphene sheet (SLGS), they were modelled as point particles. From that investigation, it can be surmised that the sensitivity of the mass sensor increases with an increase in the temperature difference while the nonlocal effect reduces the frequency and increases the relative frequency shift (see Fig. 11.5, taken from [24]).

11.1 Application of the Nonlocal Models to Nanoscale Devices

245

Fig. 11.4 SLGS with multiple attached nanoparticles; a molecular representation and b an equivalent continuum model [24]. Reprinted with permission from the publisher

Fig. 11.5 Combined effects of the nonlocal parameter (μ = e0 a), the temperature difference (θ) and the mass of the nanoparticle on the variations of the relative frequency shift as reported in Ref. [24]. Reprinted with permission from the publisher

246

11 Nonlocal Elasticity Models for Mechanics of Complex Nanoscopic Structures

SLGSs as resonant sensors for detection of ultra-fine nanoparticles was investigated by Jalali and coworkers [25, 26] by combining MD simulations with the nonlocal elasticity approach. To simultaneously consider the geometric nonlinearity, the spatial nonlocality and the atomic interactions between the SLGS and the nanoparticles, they used a nonlinear nonlocal plate model carrying an attached mass-spring system. In the MD simulations, the interactions between carbon-carbon, metal-metal and metal-carbon atoms were described by the AIREBO potential, the embedded atom method and the Lennard-Jones potential respectively. By matching the frequency shifts obtained from the nonlocal model and the MD simulation, having the same vibration amplitude, a proper value of the small-scale parameter was computed. Furthermore, they also clarified how the inclusion of nonlinearity, nonlocality and the distribution of attached nanoparticles affected the frequency shifts of the sensor. Finally, we should mention that the application of the nonlocal plate models to the graphene-based resonant sensors is thoroughly discussed in Chap. 10 of the book by Karlicic et al. [27].

11.1.2 Nanoturbine In the course of the past few years, attention has been focused on the development of fabricated rotating nanodevices, such as nanogears and nanoturbines. Since the design of nanodevices that can produce controlled unidirectional rotation forms an important issue in basic and applied nanotechnology, Li et al. [28] designed a nanoturbine composed of an SWCNT and graphene sheets that could be driven by fluid flow (Fig. 11.6). They studied the rotational motion of the proposed nanoturbine quantitatively by using MD simulations. Their results show that the rotation rate of the nanoturbine is much smaller than a macroscopic counterpart with the same driving

Fig. 11.6 Schematic illustration of the nanoturbine introduced by Li et al. [28]. Reprinted with permission from the publisher

11.1 Application of the Nonlocal Models to Nanoscale Devices

247

flow. This conceptual design has opened a new perspective for further development of rotary nanodevices. In recent years, Ghadiri and co-workers [29–31] performed a set of nonlocal plate modelling of the vibration analysis of a nanoturbine blade. In these studies, the rotating effects on the free bending vibration of the nanoturbine blade were examined. It should be noted that the research on the nanoturbines is still in the primary stages and it is hoped that it will expand in the near future.

11.2 Mechanical Characteristics of Biological Nanoscopic Structures Biological structures exhibit all levels of hierarchical structures from macroscopic to nanoscopic length scales. Despite the knowledge about the building blocks and the architecture of biological nanoscopic structures, the prediction of the mechanics of these structures even at the most elementary level is challenging [32]. Thus, accurate mechanical modelling of these nanoscopic structures is essential for our understanding of their mechanical behaviour. In recent years, some researchers have used the nonlocal elasticity approaches to model the mechanical characteristics of biological nanoscopic structures. In this section, we review some of the results in the literature regarding the mechanics of biological structures.

11.2.1 Mechanical Behaviour of Protein Microtubules The biomechanical characteristics of eukaryotic cells are mainly dependent on their cytoskeleton structure. The cytoskeleton is a highly dynamic structure which exerts forces and generates movements without any major chemical changes. It consists of three major protein threadlike biopolymers, namely, actin filaments, intermediate filaments, and microtubules [33, 34]. All cytoskeletal components have a particular physical behaviour and special structures well adapted to their own functions [35]. Among these biopolymers, the microtubules are the most rigid of the cytoskeletal filaments and have the most complex structure. The microtubules are hollow cylindrical tubes with outer and inner diameters of about 25 and 17 nm, respectively, and tens of microns long. These nanoscopic structures are built from αβ-tubulin heterodimers, the so-called protofilaments [35]. The αβ dimers in cellular microtubules are most frequently arranged as 13 protofilaments (see Fig. 11.7, taken from [36]). The microtubules are essential for various fundamental physiological processes such as separating the chromosomes during cell division, facilitating long-distance intracellular transport of organelles and proteins, providing mechanical strength to maintain the shape of the cell, supporting the motor proteins to generate the force required for cell movement, and acting as important targets for anticancer drugs [37–40]. In all these processes, their mechanical characteristics are very vital to accomplishment of the functions. Thus, the mechanics of the microtubules have become a subject of primary interest in numerous studies.

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11 Nonlocal Elasticity Models for Mechanics of Complex Nanoscopic Structures

Fig. 11.7 Structure of a 13-3 microtubule as reported in Ref. [36]. Reprinted with permission from the publisher

When the microtubule is long, or has a large aspect ratio, it can be modelled realistically by using various beam theories such as the Euler-Bernoulli and the Timoshenko beam theories [34, 41–43]. As the first attempt to capture the small-scale effect, the nonlocal Timoshenko beam model for the persistence lengths of microtubules with the consideration of the small-scale effect was carried out by Gao and Lei [44]. Furthermore, the buckling behaviour of these structures was investigated. The variations of the persistence length of microtubules as a function of their contour lengths are shown in Fig. 11.8, taken from [44]. It is found that the inclusion of nonlocality has significant effects on the persistence length of short microtubules. Furthermore, it is observed that the persistence length turns into a constant value for a very long microtubule. Civalek and co-workers [45–48] have reported a series of studies to investigate the bending, buckling and vibration characteristics of the microtubules on the basis of the nonlocal beam models. Moreover, Heireche et al. [49] analysed the vibration characteristics of microtubules by employing the nonlocal Timoshenko beam model and using the wave propagation approach. In another study, the relevance of the local and nonlocal beam theories to model the deformation mechanisms of the cantilever microtubules with large inter-protofilament sliding was examined [50]. In addition, microtubules exhibit many shell-like mechanical phenomena [51]. Shen [52, 53] analysed the buckling and postbuckling behaviour of the microtubules by employing an orthotropic nonlocal shell model. In addition, the torsional buckling

11.2 Mechanical Characteristics of Biological Nanoscopic Structures

249

Fig. 11.8 Persistence length of microtubules as a function of their contour lengths as reported by Gao and Lei [44]. Reprinted with permission from the publisher

and postbuckling of the microtubules in thermal environments was analysed [54]. Gao and An [55] investigated the buckling of microtubules on the basis of the nonlocal anisotropic elastic shell model and the Stoke flow theory. Figure 11.9, taken from [55], shows the comparison of the buckling load of microtubules obtained from the Timoshenko beam, and the isotropic and anisotropic shell models. It is found that the buckling load based on the anisotropic shell is smaller than those obtained from the Timoshenko beam model and the isotropic shell model. Recently, the nonlocal orthotropic shell model was developed to study the wave propagation in microtubules and to obtain the complete analytical formulae of wave velocities [56].

Fig. 11.9 Comparison of the buckling load obtained from different models as reported by Gao and An [55]. Reprinted with permission from the publisher

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11.2.2 Analysis of Lipid Tubules Self-assembled lipid tubules are hollow cylindrical supramolecular structures made up by rolling up lipid bilayers [57]. The diameters of self-assembled lipid tubules span the range between 10 nm and 2.0 µm depending on the nature of the lipid molecules and the condition under which the self-assembly occurs [58]. Due to their exotic physical and chemical properties, these structures are widely used as controlled release systems in drug delivery [59], a template for the synthesis of onedimensional inorganic materials [60] and a substrate for the helical crystallization of proteins [61] etc. Since the discovery of lipid tubules from bilayer membranes [62], a substantial amount of research has been performed related to their morphology and molecular structure. However, few results are available in the literature on the mechanical properties of the lipid tubules, while an understanding of their mechanical characteristics is necessary for developing their applications in the nanotechnology. In 2011, Shen [63] studied several mechanical behaviour of lipid tubules including the postbuckling, nonlinear bending and nonlinear vibration on the basis of the nonlocal Euler-Bernoulli beam theory. This was the first study in this field reported in the literature. In another study, the linear free vibration in the pre/postbuckled regions and nonlinear dynamic stability of lipid tubules with in-plane movable ends were investigated by Zhong et al. [64] using the nonlocal Euler-Bernoulli beam theory. The application of the nonlocal elasticity approaches to modelling of the lipid tubules is at its early stages, and has not attracted much attention yet.

11.2.3 Low Frequency Vibrational Modes of Spherical Viruses Viruses are one of the smallest biological structures known in Nature and come in a variety of shapes and sizes including spherical, spherocylindrical, rod-like and conical shapes [65]. Viruses are mainly classified as RNA or DNA types depending on the type of nucleic acids composing them. These nanometric structures have found potential applications in diverse fields of the rapidly developing nanotechnologies [66]. There is a need to investigate the vibration characteristics of these viruses due to their importance in design of the virus based nanotemplates. The vibrational modes of viruses were studied by Babincová et al. [67] using ultrasound technique with GHz frequency for destroying the HIV virus particles. Theoretical estimate of the vibrational frequencies of spherical virus particles was obtained by Ford [68] using two different approaches. The dispersion relations pertinent to the lowest vibrational frequencies of tobacco mosaic virus (TMV), and the M13 bacteriophage immersed in air and water, have been reported by Balandin and Fonoberov [69]. Only one study could be found in the literature concerning the mechanical modelling of the spherical viruses on the basis of the nonlocal elasticity theory. The axisymmetric vibrational behaviour of the plant virus cowpea chlorotic mottle

11.2 Mechanical Characteristics of Biological Nanoscopic Structures

251

Fig. 11.10 Structure of the cowpea chlorotic mottle virus

virus (CCMV), which is one spherical member of the viral capsid, was studied by Ghavanloo and Fazelzadeh [70]. The CCMV capsid is a nearly spherical shell with a diameter of 28 nm (Fig. 11.10).

11.2.4 Persistence Length of Collagen Molecules Collagen or tropocollagen molecule is one of the most protein-based structural material in the human body. This molecule is responsible for the mechanical stability, elasticity and strength of many biological tissues such as ligaments, tendons, and bones. The molecule has the ability to adapt its properties in response to external stimuli, such as mechanical loads [71]. Collagen molecule is formed from three helical polypeptide chains which are turned around each other forming a three-stranded rod-like molecule (Fig. 11.11) with a diameter of about 1.4 nm and a length of about 300 nm [72]. The flexural rigidity of the collagen molecules is an essential parameter in understanding the mechanics of collagen fibers and tissues. The flexural rigidity of a molecular chain is usually determined by its persistence length. This length describes the chains resistance to thermal forces [73]. In the course of the past two decades, this length has been estimated via different techniques such as electron microscopy analysis, optical tweezers, hydrodynamic method, etc. Unfortunately, considerable inconsistencies exist in the literature, wherein the reported obtained values for the length of have varied from 14 to 180 nm [74]. Therefore, the proposition of a method for predicting the persistence length of this molecule and also explaining the experimental results is necessary. In this connection, Ghavanloo [75] developed a nonlocal viscoelastic beam model in conjunction with the mean field Langevin equation to predict the persistence

Fig. 11.11 Molecular structure of a collagen molecule

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length of the collagen molecules in a solvent. An explicit formula for the persistence length of the collagen molecule with the consideration of the small-scale effect, the viscoelastic properties of the molecule, the loading frequency, and the viscosity of the solvent was presented. The numerical results clearly showed the persistence length of the collagen molecule depends on its contour length. It was found that the persistence length of the molecule is smaller than its contour length, implying that physically the molecule is relatively flexible.

11.3 Other Types of Nanoscopic Structures 11.3.1 Nanotube-Based Heterojunctions Topological defects appearing in the hexagonal lattice network of the carbon nanotubes can promote a change of their chirality. A heterojunction can be constructed by joining two carbon nanotubes with different chiralities. This is done by introducing a heptagon and pentagon pair [76, 77] (Fig. 11.12). For instance, a metallic type SWCNT can be attached to a semiconducting type SWCNT, leading to the appearance of a set of new functions, different from those of the constituent SWCNTs. Therefore, fabrication of nanotube-based heterojunctions may offer new possibilities in such fields as nanoelectronics. Filiz and Aydogdu [78] have used the nonlocal continuum-based model to predict the longitudinal vibration of nanotube-based heterojunctions. General equations for the longitudinal motion of n-segmented carbon nanotubes were obtained and the natural frequencies of carbon nanotubes with two segments were computed. Two different boundary conditions; namely, the fixed end-fixed end and fixed-free boundary conditions, were considered. It was found that the longitudinal vibration frequencies of carbon nanotube-based heterojunctions were highly over estimated within the classical model since the effect of the small length scale was not taken into account.

Fig. 11.12 Molecular structure of carbon nanotube-based heterojunctions as reported by Filiz and Aydogdu [78]. Reprinted with permission from the publisher

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253

Fig. 11.13 Molecular structure of a nanopeapod

11.3.2 Nanopeapods Another form of carbon-based nanostructure is the so-called nanopeapod. This is constructed by encapsulating C60 fullerenes within SWCNTs (Fig. 11.13). These hybrid molecules were first observed by Smith et al. [79, 80] using high resolution transmission electron microscopy while purifying and annealing SWCNTs. The encapsulated C60 fullerenes may change the physical, electrical and mechanical properties of the SWCNT. In recent years, both theoretical and experimental studies have been performed to obtain the characteristics of SWCNTs encapsulated by fullerenes [81–84]. The nonlocal Donnell’s shell model has been employed to predict the structural instability of the nanopeapods subject to two different loading conditions; namely radially applied external pressure [85] and torsional moments [86]. A pristine (10,10) SWCNT and a C60 @(10,10) nanopeapod were considered. The computed results for the (10,10) SWCNT were compared with the results obtained from MD simulation. The critical loads for the C60 @(10,10) nanopeapod were predicted. The simulation results for the critical torsional moment of the C60 @(10,10) nanopeapod and the (10,10) SWCNT are shown in Fig. 11.14, taken from [86]. The numerical results demonstrate that due to the presence of the C60 fullerenes in the nanotube the stability resistance increased by more than 100%, indicating good agreement with some MD simulation results.

11.3.3 Y-Junction Carbon Nanotubes Since the year 2000, the synthesis of Y-junction carbon nanotubes (YJCNTs) has been reported in a series of publications. These nanoscopic structures can be potentially utilized as fluidic components in complex fluid networks and nanoscale lab-on-a-chip applications [87]. Figure 11.15, taken from [88], shows the HRTEM image and the molecular structure model of a YJCNT. Another application of the nonlocal elasticity theory is to model the YJCNTs. The nonlinear vibration characteristics of YJCNTs conveying viscous fluid and subject to a longitudinal magnetic field was investigated by Ghorbanpour-Arani and Zarei [89] using the nonlocal Euler-Bernoulli beam model and the von Karman

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Fig. 11.14 Variations of the computed critical torsional moment of a C60 @(10,10) nanopeapod and a (10,10) SWCNT with L = 12.62 nm and the scale coefficient η (= e0 a). Reproduced from Ref. [86] with permission from the publisher

Fig. 11.15 HRTEM image and molecular structure model of a YJCNT reported in Ref. [88] with permission from the publisher

nonlinear strain approximation. In another studies, they also considered the vibration of YJCNTs embedded in a visco-Pasternak foundation and conveying nanomagnetic viscous fluid under a 2D magnetic field [90, 91].

11.3.4 Piezoelectric Nanoscopic Structures The nonlocal elasticity theory has been applied to study piezoelectric materials. Wang and Wang [92] considered the bending and the electromechanical coupling behaviour of piezoelectric nanowires using combined nonlocal piezoelectric theory and the surface elasticity theory. Employing the nonlocal piezoelectric cylindrical

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255

shell theory, Ghorbanpour-Arani et al. [93] studied the electro-thermo-mechanical buckling of double-walled boron nitride nanotubes embedded in a bundle of CNTs. The buckling and postbuckling behaviour of piezoelectric nanobeams were investigated by Liu et al. [94], using the piezoelectric nonlocal Timoshenko beam theory. Further studies were carried out by Ke and co-workers [95, 96] on the basis of the nonlocal piezoelectric Timoshenko beam theory to investigate the thermoelectricmechanical vibration of the piezoelectric nanobeams. They found that the natural frequency of the piezoelectric nanobeam was quite sensitive to the thermoelectricmechanical loadings. In a further study [97], the analytical solution for the natural frequencies of the piezoelectric nanoplate was developed by the same group. For further discussion, we refer the reader to the rather extensive review on the subject [98].

11.3.5 Carbon Honeycombs In recent years, new stable type of carbon-based honeycombs with a threedimensional superstructure were experimentally synthesized [99]. Some theoretical investigations have shown that the carbon-based honeycombs possess outstanding mechanical properties [100, 101]. These honeycombs consist of a periodic array of honeycomb cells with six walls, each of which is a single-layer graphene sheet (Fig. 11.16). In a combined MD simulations and the nonlocal-based modelling [102], the buckling behaviours of the carbon-based honeycombs were investigated. The MD simulations were performed with the LAMMPS software and the AIREBO force field was used to model the interactions among the carbon atoms. It was reported that the results obtained from the nonlocal model fitted the MD results very well when the parameter e0 was set at 0.67. Furthermore, the MD simulation results revealed

Fig. 11.16 Molecular structure of a periodic carbon-based honeycomb reported in Ref. [101] with permission from the publisher

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that the small-scale effects are able to significantly affect the buckling behaviour of the carbon-based honeycombs.

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Chapter 12

Recent Developments and Future Challenges in the Application of Nonlocal Elasticity Theory

So far, we have provided very comprehensive general formulations of the standard nonlocal continuum-based models and have considered their detailed applications in the investigation of various mechanical characteristics and functioning of a variety of nanoscopic structures. Recent developments in the nonlocal continuum mechanics have led to the emergence of some new ideas in the specialized literature. These ideas are different from the conventional ideas discussed so far in this book. To complement and complete our discussions about the nonlocal elasticity approaches, we include in this chapter some of the important concepts that have been proposed in these latest studies.

12.1 Nonlocal Paradox As described in the previous chapters, generally, two distinct nonlocal elasticity approaches, namely the nonlocal integral-based and the nonlocal differential-based approaches, have been reported in the literature. The integral-based approach can be considered as a strong nonlocal model whereas the differential-based approach can be classified as weak nonlocal model [1]. Recently, some inconsistencies and paradoxical results were observed in the nonlocal elasticity approaches. These finding have motivated the researchers in this field to deal with these paradoxical results. The nonlocal paradoxes can appear when the results from the nonlocal differential-based models under different loading and boundary conditions are compared. The nonlocal solutions of some static problems of the nanorods and the nanobeams have found to be identical to the local ones, and so the nonlocal effects are absent in these solutions. Some cases were discussed in Chaps. 8 and 9. This type of the nonlocal paradox was first reported by Peddieson et al. [2]. In 2015, Li et al. [3] resolved the problems that were performed by Peddieson et al. [2] and calculated the same results. Besides, they also pointed out that the nonlocal equivalent stiffness of a nanobeam may be enhanced or weakened, depending on the specific applied © Springer Nature Switzerland AG 2019 E. Ghavanloo et al., Computational Continuum Mechanics of Nanoscopic Structures, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-3-030-11650-7_12

261

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loads. Other types of paradoxes can be recognized when the vibrational behaviour of nanobeams is studied using the nonlocal differential-based model. For all boundary conditions, except the cantilever conditions, the nonlocal beam models predict a softening effect (i.e. lower frequencies) and the frequencies decrease when the nonlocal scale coefficient increases. However, the fundamental frequencies of the cantilever beams predicted by the nonlocal models are higher than those predicted by the local models [4, 5]. This paradoxical result may be a consequence of nonselfadjointness of the problem. The self-adjointness of Eringen’s nonlocal elasticity will be explained in Sect. 12.2. An additional paradox can be observed when comparing the results of the nonlocal integral-based and the nonlocal differential-based approaches. Benvenuti and Simone [6] observed that the nonlocal behaviour of a bar in tension using the differentialbased model is not consistent with the results obtained from the integral-based model. In another study, Fernández-Sáez et al. [7] developed a general method to solve the problem of the static bending of the Euler-Bernoulli beam on the basis of the Eringen integral model. Their results also revealed the inconsistencies between the nonlocal integral-based and the nonlocal differential-based models for various boundary and load conditions. Hence, they concluded that the nonlocal differential-based model is not equivalent to the nonlocal integral-based model because certain boundary conditions are not properly fulfilled. Recently, Romano et al. [8] have proposed the concept of constitutive boundary conditions and showed that the fulfilment of these boundary conditions is a necessary requirement for the nonlocal beam models. In order to overcome these paradoxes, several efforts have been made. As a first attempt, Challamel and Wang [1] proposed a new hybrid nonlocal model such that the local and nonlocal curvatures were simultaneously introduced into the new constitutive relation. In another attempt, Challamel and co-workers [9–11] proposed discrete microstructural models to overcome the paradoxes. Their model for the beam problem is discussed in Sect. 12.5. In addition, Khodabakhshi and Reddy [12] developed a two-phase integro-differential nonlocal model and suggested a general finite element formulation to solve the problem. More importantly, the paradoxical cantilever beam problem was solved with this model. However, they could not fully solve the cantilever paradox as their model appeared to give a different static response for consecutive values of the nonlocal parameter. Moreover, they introduced the simply supported beam paradox. In another study, Fernández-Sáez et al. [7] formulated the static bending of the Euler-Bernoulli beam using the Eringen integral constitutive equation to solve the paradox presented by Peddieson et al. [2]. Koutsoumaris et al. [13] developed the integral model with the modified kernel, which normalized and defined in a finite domain, for the static bending of beams. They showed that the integral-based model with modified kernel does not give rise to paradoxes as the model of the nonlocal differential-based approach does. Recently, as an alternative approach, Romano and Barretta [14] developed a stress-driven integral constitutive law and claimed that it provides a natural way to get well-posed nonlocal elastic problems. This model will be discussed in Sect. 12.6. By reviewing the previous studies, we can now conclude that the reason for the existence of these paradoxes resides in the inconsistency of forming the natural and

12.1 Nonlocal Paradox

263

essential nonlocal boundary conditions. Thus, as it was demonstrated in all these studies on the paradoxes, the nonlocal differential-based models can give correct solutions for the nonlocal field problems under only one condition related to a correct formulation of the nonlocal boundary conditions.

12.2 Self-adjointness of Eringen’s Nonlocal Elasticity One issue that has been a subject of debate about the applicability of the nonlocal differential constitutive relation, Eq. (4.38), is whether or not the nonlocal models are self-adjoint, and the governing equations can be derived via a single-energy functional. The possibility of the nonself-adjointness of the Eringen model and the impossibility of constructing the underlying quadratic energy functional for such nonlocal beam models has been raised by Reddy in a series of important papers [15, 16]. Later, Adali [17–19], showed that the governing equations of the Eringen type nonlocal elasticity that is applied to the beam and plate models can be derived from a single-energy functional. It has been shown [20] that it is possible to formulate an energy function for the bending, vibration, and buckling problems via the application of the nonlocal Euler-Bernoulli beam model, as well as using the higher order beam models. Challamel and co-workers [21] found that the Eringen nonlocal elasticity reveal self-adjointness in some problems having specific boundary conditions. The following conclusions may be drawn from their study: (a) The application of the nonlocal beam model to the buckling problem leads to self-adjointness under all boundary conditions. (b) Regardless of circumstances, the nonlocal Euler-Bernoulli beam model is obtainable from a single-energy function for the case of simply-supported boundary conditions. (c) The vibration analysis of the cantilever nanobeams via the nonlocal beam model is a nonself-adjoint problem. The implication of this is that it is not possible to build an associated functional energy, and the problem is clearly nonconservative. Furthermore, a surprising and paradoxical result that emerges in this case is that the small length scale effect tends to stiffen the local system, in contrast to the softening tendency observed for all the other boundary conditions.

12.3 Nonlocal Constitutive Boundary Conditions Although there have been significant successes with the use of nonlocal differentialbased beam models, and a great deal of potentiality is inherent in them, it should be clearly understood that these models do suffer from some limitations. It has been known that in the modelling of boundary conditions for such structures we require very precise definitions and accurate treatment. It should be mentioned that

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due to difficulties in deriving correct nonlocal boundary condition expressions, previous studies have opted for local boundary conditions instead of the nonlocal ones. Moreover, the fulfilment of natural boundary conditions has posed a challenge when solving the nonlocal beam problems [22]. As explained in Sect. 12.1, Romano and co-workers [8] introduced the constitutive boundary conditions for the EulerBernoulli beam model and provided the proof that their fulfilment by the bending field provides the necessary and sufficient conditions to guarantee the existence and uniqueness of the solution of the integral equation that defines the corresponding elastic curvature. Furthermore, they concluded that the paradoxical results originate from the lack of compatibility between the constitutive boundary conditions and the equilibrium conditions that are imposed on the bending field. Here, we present the constitutive boundary conditions for two types of kernels. In the original form of the nonlocal constitutive relation due to Eringen [23], the nonlocal elastic law for the Euler-Bernoulli beam model is formulated by relating the bending moment Mλ and the curvature κ using the integral convolution b Mλ =

α(x − y)E I (y)κ(y)dy

(12.1)

a

where a and b denote end-points of the nanobeam, and E I is the bending rigidity. Using the kernel (4.43), we have x Mλ = a

b + x

1 exp 2λL e



 y−x E I (y)κ(y)dy λL e

  1 x−y E I (y)κ(y)dy exp 2λL e λL e

(12.2)

Differentiating Eq. (12.2) with respect to x yields d Mλ 1 = dx λL e 1 − λL e

b x

x a

  1 x−y E I (y)κ(y)dy exp 2λL e λL e 1 exp 2λL e



 y−x E I (y)κ(y)dy λL e

(12.3)

Eliminating the terms involving integrals from Eq. (12.3), and differentiation with respect to x leads to Mλ (x) − λ2 L 2e

d 2 Mλ = E I (x)κ(x) dx2

(12.4)

12.3 Nonlocal Constitutive Boundary Conditions

265

where the homogeneous constitutive boundary conditions are  d Mλ  =0 Mλ (a) − L e λ d x x=a  d Mλ  Mλ (b) + L e λ =0 dx 

(12.5)

x=b

It is to be noted that Eq. (12.4) is utilized in the static and dynamic modelling of nanobeams, without reference to the constitutive boundary conditions. Similarly, if use is made of the bi-Helmholtz kernel (4.51), the solution of the constitutive integral equation (12.1) then provides the unique solution of the constitutive differential equation: Mλ (x) − L 2e (λ21 + λ22 )

4 d 2 Mλ 4 2 2 d Mλ + L λ λ = E I (x)κ(x) e 1 2 dx2 dx4

(12.6)

where the homogeneous constitutive boundary conditions are   d 2 Mλ  d Mλ  − L e (λ1 + λ2 ) + Mλ (a) = 0 d x 2 x=a d x x=a   d 2 Mλ  d Mλ  L 2e λ1 λ2 + L (λ + λ ) + Mλ (a) = 0 e 1 2 d x 2 x=b d x x=b    d 3 Mλ  d 2 Mλ  d Mλ  L 2e λ1 λ2 − L (λ + λ ) + =0 e 1 2 d x 3 x=a d x 2 x=a d x x=a    d 3 Mλ  d 2 Mλ  d Mλ  2 L e λ1 λ2 + L e (λ1 + λ2 ) + =0 d x 3 x=b d x 2 x=b d x x=b L 2e λ1 λ2

(12.7)

12.4 Nonlocal Fully Intrinsic Equations The application of the constitutive boundary conditions to nonlocal beam models is very complicated since these conditions are based on the nonlocal force and moment. Motivated by these idea, Tashakorian and co-workers [5] developed a nonlocal fully intrinsic equations for modelling free vibration of beams by considering the constitutive boundary conditions. The fully intrinsic beam equations include a set of first-order partial differential equations of motion. The displacements and rotations are not included in these formulations [24]. The dynamics of beams accommodating the small-scale effect can be determined by using the nonlocal fully intrinsic equations without ever computing the actual displacements and rotations along the beam. The constitutive boundary conditions can now be handled easily in the framework of the intrinsic formulations. In this section, we review the nonlocal fully intrinsic Euler-Bernoulli beam equations.

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Fig. 12.1 Differential beam element

In order to develop the fully intrinsic beam equations, let us consider an element of a beam in the b1 -b2 plane (Fig. 12.1). The force equation of motion in the b2 direction gives ∂ 2u2 ∂ F2 + f2 = ρ A 2 ∂ x1 ∂t

(12.8)

where u 2 denotes the transverse displacement in the b2 direction, F2 is the shear force, f 2 is the external transverse force per unit length, ρ is the mass density and A is the cross sectional area of the beam. Furthermore, the moment equilibrium about the b3 axis leads to ∂ M3 + F2 = 0 ∂ x1

(12.9)

The linear velocity can be written as V2 =

∂u 2 ∂t

(12.10)

On the basis of the Euler-Bernoulli assumption, one can also write the slope of the beam (θ3 ) as θ3 =

∂u 2 ∂ x1

(12.11)

Therefore, the angular velocity can be written as follows Ω3 =

∂ 2u2 ∂θ3 = ∂t ∂t∂ x1

(12.12)

12.4 Nonlocal Fully Intrinsic Equations

267

In addition, the relation between the displacement and the curvature is κ3 =

∂ 2u2 ∂ x12

(12.13)

Finally, the nonlocal fully intrinsic equations for the linear vibration of the nanobeam according to the Euler-Bernoulli beam model are ∂ F2 ∂ V2 − f2 = ρA ∂ x1 ∂t

(12.14)

∂ M3 = −F2 ∂ x1

(12.15)

∂ V2 = Ω3 ∂ x1

(12.16)

∂Ω3 ∂κ3 = ∂ x1 ∂t

(12.17)

Furthermore, using Eqs. (12.4), (12.14) and (12.15), the nonlocal intrinsic constitutive relation is obtained as   λ2 L 2e ∂ V2 M3 (12.18) + ρA − f2 κ3 = EI EI ∂t Equations (12.14)–(12.18) can be solved by using the proper boundary conditions. For a clamped-clamped nanobeam, the boundary conditions may be written as V2 (0) = 0, Ω3 (0) = 0, V2 (L e ) = 0, Ω3 (L e ) = 0

(12.19)

For a clamped-simply supported nanobeam, the boundary conditions are V2 (0) = 0, Ω3 (0) = 0, V2 (L e ) = 0, M3 (L e ) − λL e F2 (L e ) = 0

(12.20)

For a simply-supported nanobeam, the boundary conditions are V2 (0) = 0, M3 (0) + λL e F2 (0) = 0, V2 (L e ) = 0, M3 (L e ) − λL e F2 (L e ) = 0

(12.21)

Finally, for a clamped-free nanobeam, the boundary conditions are V2 (0) = 0, Ω3 (0) = 0, M3 (L e ) = 0, F2 (L e ) = 0

(12.22)

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Several numerical methods, such as the finite difference, the finite element and the differential quadrature methods can be easily used to solve these equations and boundary conditions. Although the nonlocal fully intrinsic beam equations provide a suitable framework for considering the constitutive boundary conditions, however, the paradoxical results for the vibration of cantilever nanobeams are still present. For a detailed discussion about this topic, we refer the interested reader to Ref. [5].

12.5 Microstructure-Based Nonlocal Model In 1920, Hencky [25] proposed a discrete beam model comprising rigid beam segments connected by frictionless hinges and elastic rotational springs. Challamel and co-workers [9–11] developed a microstructure-based nonlocal model on the basis of the Hencky model. In this section, the microstructure-based nonlocal beam model is briefly introduced. In this connection, let us consider a lattice beam of length L comprising of n rigid beams with length a (L = na) as shown in Fig. 12.2 (taken from Ref. [26]). It is assumed that the rigid beams are connected together by elastic rotational springs. The rotational stiffness is C = E I /a for the intermediate springs and C1 = 2E I /a for the spring at a clamped end [26]. In addition, the total mass M = ρ AL is uniformly lumped at each node of the microstructured model. There fore the nodal mass is m = M/n at intermediate nodes and m/2 at the ends. The beam is subject to an axial force denoted by P = σ0 A, and a uniform distributed transverse force q.  The Lagrangian, L, for the microstructured beam model with different boundary conditions is defined as For simply-supported ends:   n−1  n−1  w j+1 − 2w j + w j−1 2 1  dw j 2 1  L= m − C 2 j=1 dt 2 j=1 a 

+

  n−1 n−1 w j+1 − w j 2  1 Pa + qaw j 2 j=0 a j=1

(12.23)

For clamped-clamped ends: 

L= +

  n−1  n−1  w j+1 − 2w j + w j−1 2 1  dw j 2 1  m − C 2 j=1 dt 2 j=1 a

 2     n−1 n−1 w j+1 − w j 2  w1 wn−1 2 1 1 1 Pa + qaw j − C1 − C1 2 j=0 a 2 a 2 a j=1 (12.24)

12.5 Microstructure-Based Nonlocal Model

269

Fig. 12.2 Microstructured beam model with various boundary conditions: (a) simply-supported ends, (b) clamped ends, (c) clamped-simply supported ends and (d) clamped-free ends. Reprinted from Ref. [26] with permission from publisher

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For clamped-simply support ends:   n−1  n−1  w j+1 − 2w j + w j−1 2 1  dw j 2 1  L= m − C 2 j=1 dt 2 j=1 a 

  2  n−1 n−1 w j+1 − w j 2  w1 1 1 + Pa + qaw j − C1 2 j=0 a 2 a j=1

(12.25)

It should be noted that the first node of the beam is numbered 0, and the last node of the beam is numbered n. For all mentioned boundary conditions, we have w0 = wn = 0. To derive the equations of motion, the Lagrange equation d dt





∂L ∂ w˙ i





∂L − =0 ∂wi

(12.26)

is used where the superdot denotes differentiation with respect to time t. Using Eq. (12.26), the equations of motion for the discrete system are obtained as

m



m



m

d 2 w1 EI + 3 (w3 − 4w2 + 5w1 ) dt 2 a P CA + (w2 − 2w1 ) + 2 w1 = qa a a

for i = 1

d 2 wi EI + 3 (wi+2 − 4wi+1 + 6wi − 4wi−1 + wi−2 ) 2 dt a P + (wi+1 − 2wi + wi−1 ) = qa for i = 2, 3, ..., n − 2 a

d 2 wn−1 EI + 3 (wn−3 − 4wn−2 + 5wn−1 ) dt 2 a CB P + (wn−2 − 2wn−1 ) + 2 wn−1 = qa a a

for i = n − 1

(12.27)

(12.28)

(12.29)

where C A and C B are the parameters whose values for different boundary conditions are listed in Table 12.1.

Table 12.1 Values of parameters C A and C B used in Eqs. (12.27) and (12.29) Boundary conditions CA CB Simply-supported (S-S) Clamped-simply supported (C-S) Clamped-clamped (C-C)

0 C1 C1

0 0 C1

12.5 Microstructure-Based Nonlocal Model

271

For the clamped-free ends, the Lagrangian can be written as   n−1  n−1  w j+1 − 2w j + w j−1 2 1  dw j 2 1  L= m − C 2 j=1 dt 2 j=1 a 

  n−1 n−1 w j+1 − w j 2  1 1 + Pa + qaw j + qawn 2 j=0 a 2 j=1  2 2  w1 1 1 dwn − C1 + m 2 a 4 dt

(12.30)

At the nodes i = 1, n − 1 and n, we have the following equations

m



m

d 2 w1 EI + 3 (w3 − 4w2 + 5w1 ) dt 2 a P C1 + (w2 − 2w1 ) + 2 w1 = qa a a

for i = 1

d 2 wn−1 EI + 3 (wn−3 − 4wn−2 + 5wn−1 − 2wn ) dt 2 a P + (wn−2 − 2wn−1 + wn ) = qa for i = 2, 3, ..., n − 1 a

1 d 2 wn EI m 2 + 3 (wn − 2wn−1 + wn−2 ) 2 dt a P qa for i = n + (wn−1 − wn ) = a 2

(12.31)

(12.32)

(12.33)

It should be noted that, for i = 2, 3, , n − 2, the equations of motion are exactly the same as Eq. (12.28). The discrete equations can be transformed into an equivalent continuum equation. The following relations between the discrete displacement, wi , and the equivalent continuous displacement w(x) holds for a sufficiently smooth deflection function wi = w(xi ) = w(x) wi+1 = w(xi + a) = w(x + a) =

  ∞  ak d k w dw = exp a k! d x k dx k=0

(12.34) (12.35)

In addition, the pseudo-differential operators can be introduced as [9] wi+1 − 2wi + wi−1

      dw dw 2 a dw − 2w + exp − a = 4 sin h = exp a dx dx 2 dx (12.36)

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wi+2 − 4wi+1 + 6wi − 4wi−1 + wi−2         dw dw dw dw − 4 exp a + 6w − 4 exp − a + exp 2a = exp 2a dx dx dx dx   a dw (12.37) = 16sinh4 2 dx Therefore, substituting Eqs. (12.36) and (12.37) into Eq. (12.28), the generalized bending problem is governed by the following pseudo-differential equation ρA

    P ∂ 2w EI 4 a ∂w 2 a ∂w + 4 =q + 16 sinh sinh ∂t 2 a4 2 ∂x a2 2 ∂x

(12.38)

The pseudo-differential operator can be efficiently approximated via the Padé approximantion as   ∂2w 4 ∂x2 2 a ∂w = sin h + O(a 2 ) a2 ∂ 2 w a2 2 ∂x 1 − 12 2 ∂x

  ∂4w 16 ∂x4 4 a ∂w 2 = sin h 2 + O(a ) 4 a 2 ∂x 2 2 a ∂ w 1 − 12 ∂x2

(12.39)

(12.40)

Substituting Eqs. (12.39) and (12.40) into Eq. (12.38), multiplying the resultant equation by (1 − (a 2 /12 )(∂ 2 w/∂ x 2 ))2 and neglecting the fourth-order term in a, Eq. (12.38) can then be efficiently approximated by the linear differential equation given by       a2 ∂ 2 ∂ 2w a2 ∂ 2 ∂ 2w a2 ∂ 2q ∂ 4w + ρA 1 − = q− EI 4 + P 1 − ∂x 12 ∂ x 2 ∂ x 2 6 ∂ x 2 ∂t 2 6 ∂x2 (12.41) which can be considered as the microstructure-based nonlocal model. Eq. (12.41) is equivalent with the nonlocal differential-based approach for specific values of a. The application of Eq. (12.41) to the bending, the buckling and the vibration analyses has been completely discussed by Challamel et al. [9]. They have shown that some paradoxial results mentioned in Sect. 12.1 were resolved by using the microstructurebased nonlocal model. The interested reader can find additional details on this topic in Refs. [9, 11, 27]. Finally, it should be noted that other types of microstructured nonlocal models for lattice membrane and plates have been developed by Challamel and co-workers [28, 29].

12.6 Stress-Driven Nonlocal Integral Model

273

12.6 Stress-Driven Nonlocal Integral Model In 2017, a new classification of nonlocal elasticity was proposed by Romano and Barretta [14]. According to this classification, two different definitions of fully nonlocal integral elasticity exist. The first is a strain-driven nonlocal integral model which was adopted by Eringen [30]. In this nonlocal model, the nonlocal stress is defined by an integral convolution of elastic strain with an averaging kernel dependent on a nonlocal parameter. This model has been successfully used in some nonlocal problems defined on unbounded domains that the involved fields that vanish at infinity. Furthermore, the strain-driven nonlocal integral was replaced with an equivalent differential equation under boundary conditions that vanish at infinity. This equivalent differential formulation was later utilized to study the mechanical characteristics of nanoscopic structures over bounded continuous domains. A comprehensive review on the application of the nonlocal differential-based models for modelling size-effects in nanoscopic structures can be found in Ref. [31]. However, the integral convolutions cannot be replaced with differential relations for bounded domains, unless suitable constitutive boundary conditions are added [8]. In addition, since the constitutive boundary conditions are incompatible with equilibrium conditions, the associated elastic problems are ill-posed. As a result, serious difficulties and unexpected results emerge. The second way to define a fully nonlocal integral model was suggested by Romano and Barretta [14]. This model is known as a stress-driven nonlocal integral model. In this new model, the roles of stress and elastic strain fields are exchanged. Therefore, the stress-driven nonlocal constitutive equation is written as  εinlj (x1 , x2 , x3 , t) =

      α(x, x , λ)E i−1 jkl σkl (x 1 , x 2 , x 3 , t)d x 1 d x 2 d x 3

(12.42)

V

where E −1 is the elastic compliance tensor. It is stated [14] that the stress-driven nonlocal model is well-posed for structures defined on bounded domains and suffer from no inconsistencies. This approach was applied for several static and dynamic problems of the nanobeams [32–36]. In contrast to the strain-driven nonlocal models, the results calculated from the stress-driven model exhibit a hardening behaviour when nonlocality is increasing. However, to our knowledge, their conclusions have not been confirmed by other researchers. Judgments about these findings, must await reliable experiments and/or atomistic-based calculations.

12.7 Concluding Remarks and Future Prospects In this book, we have covered comprehensively all the areas of the application of the nonlocal elasticity models to nanoscopic structures as far as possible. Notwithstanding the undoubted successes and inherent potential power of the nonlocal elasticity

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approaches, it should also be remarked that they also suffer from some limitations. For example, they lack the ability to address the thermal characteristics of the nanoscopic structures, while the MD simulations have definite advantage for application in this area. Furthermore, the well-posed nonlocal model, which can resolve all the mentioned nonlocal paradoxes, has not been proposed yet. By the end of the 2016, it seemed that the research on the nonlocal elasticity theory has reached a saturation point, while there exist several open questions which have remained unanswered. One important lingering question is the correct choice of the nonlocal parameter, since the data obtained by different researchers on the magnitude of this parameter are not yet unitary and unambiguous. From all these data we can surmise that the value of the nonlocal parameter depends in a crucial way on the state of motion, the boundary condition and the mode shapes. Future research should help elucidate this question. Another unclarified issue in current literatures is how the nonlocal effect influences the equivalent stiffness of the nanostructures. Some researchers assert that the nonlocal effect reduces the equivalent stiffness while some other references predict an enhancement in the equivalent stiffness. Therefore, research on this area must continue. Finally, it should be noted that one of our future works includes the modelling of the new proposed nanoscopic structures such as graphene helicoid, graphyne, twin graphene, nanospring and etc. by using the nonlocal elasticity approaches.

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