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More About This Title Fundamentals of Continuum Mechanics
English
A concise introductory course text on continuum mechanics
Fundamentals of Continuum Mechanics focuses on the fundamentals of the subject and provides the background for formulation of numerical methods for large deformations and a wide range of material behaviours. It aims to provide the foundations for further study, not just of these subjects, but also the formulations for much more complex material behaviour and their implementation computationally.
This book is divided into 5 parts, covering mathematical preliminaries, stress, motion and deformation, balance of mass, momentum and energy, and ideal constitutive relations and is a suitable textbook for introductory graduate courses for students in mechanical and civil engineering, as well as those studying material science, geology and geophysics and biomechanics.
- A concise introductory course text on continuum mechanics
- Covers the fundamentals of continuum mechanics
- Uses modern tensor notation
- Contains problems and accompanied by a companion website hosting solutions
- Suitable as a textbook for introductory graduate courses for students in mechanical and civil engineering
English
English
Nomenclature xv
Introduction 1
PART ONE MATHEMATICAL PRELIMINARIES 3
1 Vectors 5
1.1 Examples 9
1.1.1 9
1.1.2 9
Exercises 9
Reference 11
2 Tensors 13
2.1 Inverse 15
2.2 Orthogonal Tensor 16
2.3 Principal Values 16
2.4 Nth-Order Tensors 18
2.5 Examples 18
2.5.1 18
2.5.2 18
Exercises 19
3 Cartesian Coordinates 21
3.1 Base Vectors 21
3.2 Summation Convention 23
3.3 Tensor Components 24
3.4 Dyads 25
3.5 Tensor and Scalar Products 27
3.6 Examples 29
3.6.1 29
3.6.2 29
3.6.3 29
Exercises 30
Reference 30
4 Vector (Cross) Product 31
4.1 Properties of the Cross Product 32
4.2 Triple Scalar Product 33
4.3 Triple Vector Product 33
4.4 Applications of the Cross Product 34
4.4.1 Velocity due to Rigid Body Rotation 34
4.4.2 Moment of a Force P about O 35
4.5 Non-orthonormal Basis 36
4.6 Example 37
Exercises 37
5 Determinants 41
5.1 Cofactor 42
5.2 Inverse 43
5.3 Example 44
Exercises 44
6 Change of Orthonormal Basis 47
6.1 Change of Vector Components 48
6.2 Definition of a Vector 50
6.3 Change of Tensor Components 50
6.4 Isotropic Tensors 51
6.5 Example 52
Exercises 53
Reference 56
7 Principal Values and Principal Directions 57
7.1 Example 59
Exercises 60
8 Gradient 63
8.1 Example: Cylindrical Coordinates 66
Exercises 67
PART TWO STRESS 69
9 Traction and Stress Tensor 71
9.1 Types of Forces 71
9.2 Traction on Different Surfaces 73
9.3 Traction on an Arbitrary Plane (Cauchy Tetrahedron) 75
9.4 Symmetry of the Stress Tensor 76
Exercise 77
Reference 77
10 Principal Values of Stress 79
10.1 Deviatoric Stress 80
10.2 Example 81
Exercises 82
11 Stationary Values of Shear Traction 83
11.1 Example: Mohr–Coulomb Failure Condition 86
Exercises 88
12 Mohr’s Circle 89
Exercises 93
Reference 93
PART THREE MOTION AND DEFORMATION 95
13 Current and Reference Configurations 97
13.1 Example 102
Exercises 103
14 Rate of Deformation 105
14.1 Velocity Gradients 105
14.2 Meaning of D 106
14.3 Meaning of W 108
Exercises 109
15 Geometric Measures of Deformation 111
15.1 Deformation Gradient 111
15.2 Change in Length of Lines 112
15.3 Change in Angles 113
15.4 Change in Area 114
15.5 Change in Volume 115
15.6 Polar Decomposition 116
15.7 Example 118
Exercises 118
References 120
16 Strain Tensors 121
16.1 Material Strain Tensors 121
16.2 Spatial Strain Measures 123
16.3 Relations Between D and Rates of EG and U 124
16.3.1 Relation Between Ė and D 124
16.3.2 Relation Between D and U̇ 125
Exercises 126
References 128
17 Linearized Displacement Gradients 129
17.1 Linearized Geometric Measures 130
17.1.1 Stretch in Direction N 130
17.1.2 Angle Change 131
17.1.3 Volume Change 131
17.2 Linearized Polar Decomposition 132
17.3 Small-Strain Compatibility 133
Exercises 135
Reference 135
PART FOUR BALANCE OF MASS, MOMENTUM, AND ENERGY 137
18 Transformation of Integrals 139
Exercises 142
References 143
19 Conservation of Mass 145
19.1 Reynolds’ Transport Theorem 148
19.2 Derivative of an Integral over a Time-Dependent Region 149
19.3 Example: Mass Conservation for a Mixture 150
Exercises 151
20 Conservation of Momentum 153
20.1 Momentum Balance in the Current State 153
20.1.1 Linear Momentum 153
20.1.2 Angular Momentum 154
20.2 Momentum Balance in the Reference State 155
20.2.1 Linear Momentum 156
20.2.2 Angular Momentum 157
20.3 Momentum Balance for a Mixture 158
Exercises 159
21 Conservation of Energy 161
21.1 Work-Conjugate Stresses 163
Exercises 165
PART FIVE IDEAL CONSTITUTIVE RELATIONS 167
22 Fluids 169
22.1 Ideal Frictionless Fluid 169
22.2 Linearly Viscous Fluid 171
22.2.1 Non-steady Flow 173
Exercises 175
Reference 176
23 Elasticity 177
23.1 Nonlinear Elasticity 177
23.1.1 Cauchy Elasticity 177
23.1.2 Green Elasticity 178
23.1.3 Elasticity of Pre-stressed Bodies 179
23.2 Linearized Elasticity 182
23.2.1 Material Symmetry 183
23.2.2 Linear Isotropic Elastic Constitutive Relation 185
23.2.3 Restrictions on Elastic Constants 186
23.3 More Linearized Elasticity 187
23.3.1 Uniqueness of the Static Problem 188
23.3.2 Pressurized Hollow Sphere 189
Exercises 191
Reference 194
Index 195
English
“Motivated students will benefit from this systematic, disciplined and concise treatment of the fundamentals of continuum mechanics. Many practitioners will also appreciate the logical organization, and the lucid descriptions of such matters as the distinctions between the various common stress and strain measures.” (Pure and Applied Geophysics, 1 November 2015)
