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Classical Continuum Mechanics (Applied and Computational Mechanics) PDF

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Classical Continuum Mechanics APPLIED AND COMPUTATIONAL MECHANICS A Series of Textbooks and Reference Books Founding Editor J.N. Reddy Continuum Mechanics for Engineers, Forth Edition G. Thomas Mase, Ronald E. Smelser & Jenn Stroud Rossmann Dynamics in Engineering Practice, Eleventh Edition Dara W. Childs, Andrew P. Conkey Advanced Mechanics of Continua Karan S. Surana Physical Components of Tensors Wolf Altman, Antonio Marmo De Oliveira Continuum Mechanics for Engineers, Third Edition G. Thomas Mase, Ronald E. Smelser & Jenn Stroud Rossmann Classical Continuum Mechanics, Second Edition Karan S. Surana For more information about this series, please visit: https://www.crcpress.com/ Applied-and-Computational-Mechanics/book-series/CRCAPPCOMMEC Classical Continuum Mechanics Karan S. Surana Department of Mechanical Engineering The University of Kansas Lawrence, Kansas Second edition published 2022 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 © 2022 Karan S. Surana First edition published by CRC Press 2014 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 publisher 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 it in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, access www.copyright.com or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. For works that are not available on CCC, please contact mpkbookspermissions@tandf. co.uk Trademark notice: Product or corporate names may be trademarks or registered trademarks and are used only for identification and explanation without intent to infringe. ISBN: 978-0-367-61296-2 (hbk) ISBN: 978-0-367-61521-5 (pbk) ISBN: 978-1-003-10533-6 (ebk) DOI: 10.1201/9781003105336 Typeset in CMR10 by KnowledgeWorks Global Ltd. To My beloved family Abha, Deepak, Rishi, Yogini, and Riya Contents PREFACE xvii ABOUT THE AUTHOR xix LIST OF ABBREVIATIONS xxi 1 INTRODUCTION 1 2 CONCEPTS AND MATHEMATICAL PRELIMINARIES 9 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2.1 Einstein notations . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2.2 Index notations and Kronecker delta . . . . . . . . . . . . . 11 2.2.3 Vector and matrix notations . . . . . . . . . . . . . . . . . . 12 2.2.4 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3 Permutation tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.4 (cid:15)(cid:15)(cid:15)-δδδ identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.5 Operations using vector, matrix and Einstein notations . . . . . . . 16 2.5.1 Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.5.2 Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.5.3 Factoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.5.4 Contraction (or trace). . . . . . . . . . . . . . . . . . . . . . 17 2.6 Reference frame and reference frame transformation . . . . . . . . . 19 2.7 Coordinate frames and transformations . . . . . . . . . . . . . . . . 21 2.7.1 Cartesian frame and orthogonal coordinate transformations . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.7.2 Curvilinear coordinates (or frame) . . . . . . . . . . . . . . . 25 2.8 Curvilinear frames, covariant and contravariant bases . . . . . . . . 26 2.8.1 Covariant basis . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.8.2 Contravariant basis . . . . . . . . . . . . . . . . . . . . . . . 29 2.8.3 Alternatewaytovisualizecovariantandcontravariantbases ggg˜ ,ggg˜i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 i 2.9 Scalars, vectors and tensors . . . . . . . . . . . . . . . . . . . . . . . 31 2.10 Coordinate transformations, definitions and operations . . . . . . . 32 2.10.1 Coordinate transformation T . . . . . . . . . . . . . . . . . . 33 2.10.2 Induced transformations . . . . . . . . . . . . . . . . . . . . 34 vii viii CONTENTS 2.10.3 Isomorphismbetweencoordinatetransformationsandinduced transformations . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.10.4 Transformations by covariance and contravariance . . . . . . 36 2.10.5 The tensor concept: Covariant and contravariant tensors . . 40 2.11 Tensorsinthree-dimensionalx-frame, tensoroperations, orthogonal coordinate transformations and invariance . . . . . . . . . . . . . . 47 2.11.1 Tensors in Cartesian x-frame . . . . . . . . . . . . . . . . . . 47 2.11.2 Tensor operations . . . . . . . . . . . . . . . . . . . . . . . . 48 2.11.3 Transformations of tensors defined in orthogonal frames due to orthogonal coordinate transformation . . . . . . . . . . . 51 2.11.4 Invariants of tensors. . . . . . . . . . . . . . . . . . . . . . . 54 2.11.5 Hamilton-Cayley theorem . . . . . . . . . . . . . . . . . . . 59 2.11.6 Differential calculus of tensors . . . . . . . . . . . . . . . . . 60 2.12 Some useful relations . . . . . . . . . . . . . . . . . . . . . . . . . . 65 2.13 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3 KINEMATICS OF MOTION, DEFORMATION AND THEIR MEASURES 71 3.1 Description of motion . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.2 Material particle displacements . . . . . . . . . . . . . . . . . . . . . 72 3.3 Lagrangian, Eulerian descriptions and descriptions in fluid mechanics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.3.1 Lagrangian or referential description of motion . . . . . . . . 73 3.3.2 Eulerian or spatial description of motion . . . . . . . . . . . 74 3.3.3 Descriptions in fluid mechanics. . . . . . . . . . . . . . . . . 75 3.3.4 Notations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.4 Material derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.4.1 Material derivative in Lagrangian or referential description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.4.2 Material derivative in Eulerian or spatial description . . . . 77 3.5 Acceleration of a material particle . . . . . . . . . . . . . . . . . . . 79 3.5.1 Lagrangian or referential description . . . . . . . . . . . . . 79 3.5.2 Eulerian or spatial description . . . . . . . . . . . . . . . . . 79 3.6 Deformation Gradient Tensor . . . . . . . . . . . . . . . . . . . . . . 80 3.7 Continuousdeformationofmatter, restrictionsonthedescriptionof motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 3.8 Change of description, co- and contra-variant measures . . . . . . . 84 3.9 Notations for covariant and contravariant measures . . . . . . . . . 85 3.10 Deformation, measures of length and change in length . . . . . . . . 85 3.10.1 Covariant measures of length and change in length . . . . . 85 3.10.2 Contravariant measures of length and change in length . . . 87 3.11 Covariant and contravariant measures of finite strain in Lagrangian and Eulerian descriptions . . . . . . . . . . . . . . . . . . . . . . . . 88 3.11.1 Covariant measures of finite strains . . . . . . . . . . . . . . 90 3.11.2 Contravariant measures of finite strains . . . . . . . . . . . . 94 3.12 Changes in strain measures due to rigid rotation of frames . . . . . 98 3.12.1 Change in covariant Lagrangian descriptions of strain when x are changed to x(cid:48) due to rigid rotation . . . . . . . . . . . 100 i i CONTENTS ix 3.12.2 Changes in contravariant Eulerian measures of strains when x are changed to x(cid:48) due to rigid rotation . . . . . . . . . . . 102 i i 3.12.3 Change in covariant Lagrangian measures of strain when x¯ i are changed to x¯(cid:48) by rigid rotation [Q¯] . . . . . . . . . . . . 103 i 3.12.4 ChangesinEulerianmeasuresofstrainwhenx¯ arechanged i to x¯(cid:48) by rigid rotation [Q] . . . . . . . . . . . . . . . . . . . 105 i 3.13 Invariants of strain tensors . . . . . . . . . . . . . . . . . . . . . . . 107 3.14 Expanded form of strain tensors . . . . . . . . . . . . . . . . . . . . 108 3.14.1 Green’s strain measure: [ε ] . . . . . . . . . . . . . . . . . . 108 [0] 3.14.2 Almansi strain measure : [ε¯[0]] . . . . . . . . . . . . . . . . . 109 3.15 Physical meaning of strains . . . . . . . . . . . . . . . . . . . . . . . 109 3.15.1 Extensions and stretches parallel to ox ,ox ,ox axes in 1 2 3 the x-frame: covariant measure of strain in Lagrangian description usingεεε , Green’s strain in Lagrangian [0] description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 3.15.2 Extensions and stretches parallel to x¯-frame axes using [ε¯[0]] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 3.15.3 Angles between the fibers or material lines . . . . . . . . . . 116 3.16 Small deformation, small strain deformation physics . . . . . . . . . 119 3.16.1 Green’s strain: Lagrangian description . . . . . . . . . . . . 119 3.16.2 Almansi strain tensor: Eulerian description . . . . . . . . . . 121 3.17 Additive and multiplicative decompositions of deformation gradient tensor [J] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 3.17.1 Additive decomposition of [J] . . . . . . . . . . . . . . . . . 123 3.17.2 Multiplicative decomposition of [J]: Polar decomposition into stretch and rotation tensor . . . . . . . . . . . . . . . . 124 3.17.3 Strain measures in terms of [S ], [S ] and [R] . . . . . . . . . 131 r l 3.18 Invariantsof[C ],[B[0]],[S ]and[S ]intermsofprincipalstretches [0] r l of [S ] and [S ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 r l 3.18.1 Principal stretches of [S ] and [S ] . . . . . . . . . . . . . . . 132 r l 3.18.2 Principal invariants of [C ] in terms of λr . . . . . . . . . . 133 [0] i 3.18.3 Principal Invariants of [S ] . . . . . . . . . . . . . . . . . . . 134 r 3.18.4 Principal invariants of [B[0]] in terms of λl . . . . . . . . . . 135 i 3.18.5 Principal Invariants of [S ] . . . . . . . . . . . . . . . . . . . 136 l 3.19 Deformation of areas and volumes . . . . . . . . . . . . . . . . . . . 136 3.19.1 Areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 3.19.2 Volumes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 3.19.3 Integral form of dA¯ over ∂V¯ . . . . . . . . . . . . . . . . . 139 { } 3.20 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 4 DEFINITIONS AND MEASURES OF STRESSES 147 4.1 Concept of stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 4.2 Cut Principle of Cauchy. . . . . . . . . . . . . . . . . . . . . . . . . 147 4.3 Definition of stress on area dA¯ . . . . . . . . . . . . . . . . . . . . 148 n 4.4 Cauchy stress tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 4.4.1 Force balance . . . . . . . . . . . . . . . . . . . . . . . . . . 151 4.4.2 Moment of Forces . . . . . . . . . . . . . . . . . . . . . . . . 153 4.4.3 Cauchy principle. . . . . . . . . . . . . . . . . . . . . . . . . 154

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