Mechanics of Nanocomposites Mechanics of Nanocomposites Homogenization and Analysis Authored by Farzad Ebrahimi, Ali Dabbagh Published [2020] by CRC Press 6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742 and by CRC Press 2 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN © 2020 Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, LLC Reasonable efforts have been made to publish reliable data and information, but the author and pub- lisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged, please write and let us know so we may rectify in any future reprint. 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ISBN: [978-0-367-25433-9] (hbk) ISBN: [978-0-429-31679-1] (ebk) Typeset in Times by Cenveo® Publisher Services Dedication “To my honorable parents” Farzad Ebrahimi “To my dearests: my honorable parents and my beloved sisters” Ali Dabbagh Table of Contents Preface.......................................................................................................................xi Acknowledgment ...................................................................................................xiii About the Authors ....................................................................................................xv Part 1 Preliminary requirements Chapter 1 Introduction ..........................................................................................3 1.1 An Introduction to Composites and Nanocomposites ...............3 1.2 Applications of Nanocomposites in Industry ..........................17 Chapter 2 Mathematical Tools ............................................................................ 21 2.1 An Introduction to Elasticity ...................................................21 2.2 Kinematic Relations of Continuous Systems ..........................24 2.2.1 Classical Shell Theory................................................24 2.2.2 First-Order Shear Deformation Shell Theory ............34 2.2.3 First-Order Shear Deformation Beam Theory ...........40 2.2.4 Third-Order Shear Deformation Beam Theory ......... 47 2.2.5 Refined Higher-Order Beam Theory ......................... 52 2.2.6 Third-Order Shear Deformation Plate Theory ...........60 2.2.7 Refined Higher-Order Plate Theory ...........................69 2.3 Homogenization of Nanocomposites .......................................80 2.3.1 Material Properties of Single-Layered CNTR Nanocomposites ......................................................... 81 2.3.2 Material Properties of Single-Layered GPLR Nanocomposites .........................................................83 2.3.3 Material Properties of Single-Layered GOR Nanocomposites .........................................................85 2.3.4 Material Properties of Multi-Layered Nanocomposites .........................................................87 2.3.5 Material Properties of Porous Single-Layered Nanocomposites .........................................................89 2.3.6 Material Properties of Viscoelastic Nanocomposites .................................................... 92 Part 2 Static analyses Chapter 3 Buckling Analysis of Embedded GOR Nanocomposite Shells .........97 3.1 Problem Definition ..................................................................97 3.2 Numerical Results and Discussion ..........................................97 3.3 Concluding Remarks ............................................................. 104 vii viii Table of Contents Chapter 4 Thermally Influenced Stability Behaviors of GOR Nanocomposite Cylinders ................................................................105 4.1 Problem Definition ................................................................105 4.2 Numerical Results and Discussion ........................................105 4.3 Concluding Remarks ............................................................. 111 Chapter 5 Thermal Buckling Behaviors of Sandwich Beams Covered by FG-CNTR Nanocomposite Facesheets .......................................113 5.1 Problem Definition ................................................................ 113 5.2 Numerical Results and Discussion ........................................ 114 5.3 Concluding Remarks .............................................................119 Chapter 6 Thermo-Magnetically Influenced Static Stability of FG-GOR Nanocomposite Beams ................................................121 6.1 Problem Definition ................................................................ 121 6.2 Numerical Results and Discussion ........................................121 6.3 Concluding Remarks .............................................................126 Chapter 7 Effects of Axial Excitation and Non-Uniform Magnetic Field on the Buckling Behavior of FG-GOR Nanocomposite Beams..........127 7.1 Problem Definition ................................................................127 7.2 Numerical Results and Discussion ........................................127 7.3 Concluding Remarks .............................................................132 Chapter 8 Thermo-Mechanical Buckling Analysis of Multilayered FG-GPLR Nanocomposite Plates ....................................................133 8.1 Problem Definition ................................................................133 8.2 Numerical Results and Discussion ........................................ 133 8.3 Concluding Remarks ............................................................. 136 Chapter 9 On Buckling Behaviors of FG-GOR Nanocomposite Plates Rested on Elastic Substrate ..............................................................139 9.1 Problem Definition ................................................................ 139 9.2 Numerical Results and Discussion ........................................139 9.3 Concluding Remarks ............................................................. 145 Chapter 10 Thermal Buckling Analysis of Embedded GOR Nanocomposite Plates ......................................................................147 10.1 Problem Definition ................................................................147 10.2 Numerical Results and Discussion ........................................ 147 10.3 Concluding Remarks ............................................................. 152 Table of Contents ix Part 3 Dynamic analyses Chapter 11 Thermo-Mechanical Vibration Analysis of Sandwich Beams with CNTR Nanocomposite Facesheets...............................157 11.1 Problem Definition ................................................................ 157 11.2 Numerical Results and Discussion ........................................ 158 11.3 Concluding Remarks ............................................................. 165 Chapter 12 Thermally Affected Vibrational Responses of Multilayered FG-GPLR Nanocomposite Beams..............................167 12.1 Problem Definition ................................................................167 12.2 Numerical Results and Discussion ........................................ 167 12.3 Concluding Remarks ............................................................. 171 Chapter 13 Thermo-Magnetically Influenced Vibration Analysis of GPLR Nanocomposite Beams .....................................................173 13.1 Problem Definition ................................................................ 173 13.2 Numerical Results and Discussion ........................................173 13.3 Concluding Remarks ............................................................. 180 Chapter 14 Damped Vibration Analysis of Multilayered FG-GPLR Nanocomposite Beams Embedded on Viscoelastic Substrate ...........................................................................................181 14.1 Problem Definition ................................................................ 181 14.2 Numerical Results and Discussion ........................................181 14.3 Concluding Remarks ............................................................. 185 Chapter 15 Influence of External Non-Uniform Magnetic Field on the Natural Frequency of FG-GOR Nanocomposite Beams ...............................................................................................187 15.1 Problem Definition ................................................................187 15.2 Numerical Results and Discussion ........................................ 187 15.3 Concluding Remarks ............................................................. 192 Chapter 16 Thermo-Mechanical Vibration Analysis of Multilayered FG-GPLR Nanocomposite Plates ....................................................193 16.1 Problem Definition ................................................................193 16.2 Numerical Results and Discussion ........................................ 193 16.3 Concluding Remarks ............................................................. 198