The mechanics of fluids and the mechanics of solids represent the two major areas of physics and applied mathematics that meet in continuum mechanics, a field that forms the foundation of civil and mechanical engineering. This unified approach to the teaching of fluid and solid mechanics focuses on the general mechanical principles that apply to all materials. Students who have familiarized themselves with the basic principles can go on to specialize in any of the different branches of continuum mechanics. This text opens with introductory chapters on matrix algebra, vectors and Cartesian tensors, and an analysis of deformation and stress. Succeeding chapters examine the mathematical statements of the laws of conservation of mass, momentum, and energy as well as the formulation of the mechanical constitutive equations for various classes of fluids and solids. In addition to many worked examples, this volume features a graded selection of problems (with answers, where appropriate). Geared toward undergraduate students of applied mathematics, it will also prove valuable to physicists and engineers.
Author(s): A. J. M. Spencer
Publisher: Dover
Year: 2004
Language: English
Commentary: Unabridged republication of the 1980 edition by Longman group UK limited
Pages: 191
Preface
1: Introduction 1
1.1 Continuum mechanics 1
2: Introductory matrix algebn 4
2.1 Matrices 4
2.2 The summation convention 6
2.3 Eigenvalues and eigenvectors 8
2.4 The Cayley-Hamilton theorem 11
2.5 The polar decomposition theorem 12
3: Vectors and carteslan tensors 14
3.1 Vectors 14
3.2 Coordinate transformations 16
3.3 The dyadic product 19
3.4 Cartesian tensors 20
3.5 Isotropic tensors 22
3.6 Multiplication of tensors 23
3.7 Tensor and matrix notation 25
3.8 Invariants of a second-order tensor 27
3.9 Deviatoric tensors 31
3 .10 Vector and tensor calculus 31
4: Particle kinematics 33
4.1 Bodies and their configurations 33
4.2 Displacement and velocity 36
4.3 Time rates of change 37
4.4 Acceleration 39
4.5 Steady motion. Particle paths and streamlines 41
4.6 Problems 42
5: Stress 44
5 .1 Surface traction 44
5 .2 Components of stress 45
5 .3 The traction on any surface 46
5 .4 Transformation of stress components 49
5 .5 Equations of equilibrium 51
5 .6 Principal stress components, principal axes of stress
and stress invariants 52
5.7 The stress deviator tensor 56
5.8 Shear stress 57
5.9 Some simple states of stress 57
5 .10 Problems 60
6: Modons and deformadom 63
6.1 Rigid-body motions 63
6.2 Extension of a material line element 66
6.3 The deformation gradient tensor 68
6.4 Finite deformation and strain tensors 70
6.5 Some simple finite deformations 74
6.6 Infinitesimal strain 78
6. 7 Infinitesimal rotation 82
6.8 The rate-of-deformation tensor 83
6.9 The velocity gradient and spin tensors 85
6.10 Some simple flows 87
6.11 Problems 88
7: Con1enadon laws 91
7 .1 Conservation laws of physics 91
7 .2 Conservation of mass 91
7 .3 The material time derivative of a volume integral 96
7.4 Conservation of linear momentum 97
7.5 Conservation of angular momentum 98
7.6 Conservation of energy 100
7.7 The principle of virtual work 102
7.8 Problems 103
8: Linear constitutive equations 104
8.1 Constitutive equations and ideal materials 104
8.2 Material symmetry 106
8.3 Linear elasticity 110
8.4 Newtonian viscous fluids 116
8.5 Linear viscoelasticity 118
8.6 Problems 120
9: Further analysis of finite deformation 123
9.1 Deformation of a surface element 123
9.2 Decomposition of a deformation 125
9.3 Principal stretches and principal axes of deformation 127
9.4 Strain invariants 130
9.5 Alternative stress measures 132
9.6 Problems 134
10: Non-linear constitutive equations 136
10.1 Non-linear theories 136
10.2 The theory of finite elastic deformations 136
10.3 A non-linear viscous fluid 142
10.4 Non-linear viscoelasticity 144
10.5 Plasticity 145
10.6 Problems 149
11: Cylindrical and spherical polar coordinates 153
11.1 Curvilinear coordinates 153
11.2 Cylindrical polar coordinates 153
11.3 Spherical polar coordinates 161
11.4 Problems 167
Appendix. Representation theorem for an isotropic tensor function 170
Answers 173
Further reading 179
Index 180
