Classical Continuum Mechanics (Applied and Computational Mechanics)

532 Pages·2021·6.647 MB·English

by  Karan S. Surana

Classical Continuum
Mechanics
APPLIED AND COMPUTATIONAL MECHANICS
A Series of Textbooks and Reference Books
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Karan S. Surana
Physical Components of Tensors
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Karan S. Surana
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Classical Continuum
Mechanics
Karan S. Surana
Department of Mechanical Engineering
The University of Kansas
Lawrence, Kansas
Second edition published 2022
by CRC Press
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© 2022 Karan S. Surana
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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
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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