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
Preface xiii
Nomenclature xv
Introduction 1 (2)
PART ONE MATHEMATICAL PRELIMINARIES 3 (66)
1 Vectors 5 (8)
1.1 Examples 9 (4)
1.1.1 9 (1)
1.1.2 9 (1)
Exercises 9 (2)
Reference 11 (2)
2 Tensors 13 (8)
2.1 Inverse 15 (1)
2.2 Orthogonal Tensor 16 (1)
2.3 Principal Values 16 (2)
2.4 Nth-Order Tensors 18 (1)
2.5 Examples 18 (3)
2.5.1 18 (1)
2.5.2 18 (1)
Exercises 19 (2)
3 Cartesian Coordinates 21 (10)
3.1 Base Vectors 21 (2)
3.2 Summation Convention 23 (1)
3.3 Tensor Components 24 (1)
3.4 Dyads 25 (2)
3.5 Tensor and Scalar Products 27 (2)
3.6 Examples 29 (2)
3.6.1 29 (1)
3.6.2 29 (1)
3.6.3 29 (1)
Exercises 30 (1)
Reference 30 (1)
4 Vector (Cross) Product 31 (10)
4.1 Properties of the Cross Product 32 (1)
4.2 Triple Scalar Product 33 (1)
4.3 Triple Vector Product 33 (1)
4.4 Applications of the Cross Product 34 (2)
4.4.1 Velocity due to Rigid Body 34 (1)
Rotation
4.4.2 Moment of a Force P about O 35 (1)
4.5 Non-orthonormal Basis 36 (1)
4.6 Example 37 (4)
Exercises 37 (4)
5 Determinants 41 (6)
5.1 Cofactor 42 (1)
5.2 Inverse 43 (1)
5.3 Example 44 (3)
Exercises 44 (3)
6 Change of Orthonormal Basis 47 (10)
6.1 Change of Vector Components 48 (2)
6.2 Definition of a Vector 50 (1)
6.3 Change of Tensor Components 50 (1)
6.4 Isotropic Tensors 51 (1)
6.5 Example 52 (5)
Exercises 53 (3)
Reference 56 (1)
7 Principal Values and Principal Directions 57 (6)
7.1 Example 59 (4)
Exercises 60 (3)
8 Gradient 63 (6)
8.1 Example: Cylindrical Coordinates 66 (3)
Exercises 67 (2)
PART TWO STRESS 69 (26)
9 Traction and Stress Tensor 71 (8)
9.1 Types of Forces 71 (2)
9.2 Traction on Different Surfaces 73 (2)
9.3 Traction on an Arbitrary Plane 75 (1)
(Cauchy Tetrahedron)
9.4 Symmetry of the Stress Tensor 76 (3)
Exercise 77 (1)
Reference 77 (2)
10 Principal Values of Stress 79 (4)
10.1 Deviatoric Stress 80 (1)
10.2 Example 81 (2)
Exercises 82 (1)
11 Stationary Values of Shear Traction 83 (6)
11.1 Example: Mohr-Coulomb Failure 86 (3)
Condition
Exercises 88 (1)
12 Mohr's Circle 89 (6)
Exercises 93 (1)
Reference 93 (2)
PART THREE MOTION AND DEFORMATION 95 (42)
13 Current and Reference Configurations 97 (8)
13.1 Example 102 (3)
Exercises 103 (2)
14 Rate of Deformation 105 (6)
14.1 Velocity Gradients 105 (1)
14.2 Meaning of D 106 (2)
14.3 Meaning of W 108 (3)
Exercises 109 (2)
15 Geometric Measures of Deformation 111 (10)
15.1 Deformation Gradient 111 (1)
15.2 Change in Length of Lines 112 (1)
15.3 Change in Angles 113 (1)
15.4 Change in Area 114 (1)
15.5 Change in Volume 115 (1)
15.6 Polar Decomposition 116 (2)
15.7 Example 118 (3)
Exercises 118 (2)
References 120 (1)
16 Strain Tensors 121 (8)
16.1 Material Strain Tensors 121 (2)
16.2 Spatial Strain Measures 123 (1)
16.3 Relations Between D and Rates of EG 124 (5)
and U
16.3.1 Relation Between E and D 124 (1)
16.3.2 Relation Between D and U 125 (1)
Exercises 126 (2)
References 128 (1)
17 Linearized Displacement Gradients 129 (8)
17.1 Linearized Geometric Measures 130 (2)
17.1.1 Stretch in Direction N 130 (1)
77.7.2 Angle Change 131 (1)
17.1.3 Volume Change 131 (1)
17.2 Linearized Polar Decomposition 132 (1)
17.3 Small-Strain Compatibility 133 (4)
Exercises 135 (1)
Reference 135 (2)
PART FOUR BALANCE OF MASS, MOMENTUM, AND 137 (30)
ENERGY
18 Transformation of Integrals 139 (6)
Exercises 142 (1)
References 143 (2)
19 Conservation of Mass 145 (8)
19.1 Reynolds' Transport Theorem 148 (1)
19.2 Derivative of an Integral over a 149 (1)
Time-Dependent Region
19.3 Example: Mass Conservation for a 150 (3)
Mixture
Exercises 151 (2)
20 Conservation of Momentum 153 (8)
20.1 Momentum Balance in the Current State 153 (2)
20.1.1 Linear Momentum 153 (1)
20.1.2 Angular Momentum 154 (1)
20.2 Momentum Balance in the Reference 155 (3)
State
20.2.1 Linear Momentum 156 (1)
20.2.2 Angular Momentum 157 (1)
20.3 Momentum Balance for a Mixture 158 (3)
Exercises 159 (2)
21 Conservation of Energy 161 (6)
21.1 Work-Conjugate Stresses 163 (4)
Exercises 165 (2)
PART FIVE IDEAL CONSTITUTIVE RELATIONS 167 (28)
22 Fluids 169 (8)
22.1 Ideal Frictionless Fluid 169 (2)
22.2 Linearly Viscous Fluid 171 (6)
22.2.1 Non-steady Flow 173 (2)
Exercises 175 (1)
Reference 176 (1)
23 Elasticity 177 (18)
23.1 Nonlinear Elasticity 177 (5)
23.1.1 Cauchy Elasticity 111 (67)
23.1.2 Green Elasticity 178 (1)
23.1.3 Elasticity of Pre-stressed Bodies 179 (3)
23.2 Linearized Elasticity 182 (5)
23.2.1 Material Symmetry 183 (2)
23.2.2 Linear Isotropic Elastic 185 (1)
Constitutive Relation
23.2.3 Restrictions on Elastic Constants 186 (1)
23.3 More Linearized Elasticity 187 (8)
23.3.1 Uniqueness of the Static Problem 188 (1)
23.3.2 Pressurized Hollow Sphere 189 (2)
Exercises 191 (3)
Reference 194 (1)
Index 195
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