Continuum Mechanics

Undergraduate text opens with introductory chapters on matrix algebra, vectors and Cartesian tensors, and an analysis of deformation and stress; succeeding chapters examine laws of conservation of mass, momentum, and energy as well as the formulation of mechanical constitutive equations. 1992 edition.

Preface

1: Introduction

1

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1.1 Continuum mechanics

1

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2: Introductory matrix algebra

4

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2.1 Matrices

4

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2.2 The summation convention

6

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2.3 Eigenvalues and eigenvectors

8

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2.4 The Cayley-Hamilton theorem

11

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2.5 The polar decomposition theorem

12

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3: Vectors and cartesian tensors

14

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3.1 Vectors

14

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3.2 Coordinate transformations

16

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3.3 The dyadic product

19

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3.4 Cartesian tensors

20

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3.5 Isotropic tensors

22

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3.6 Multiplication of tensors

23

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3.7 Tensor and matrix notation

25

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3.8 Invariants of a second-order tensor

27

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3.9 Deviatoric tensors

31

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3.10 Vector and tensor calculus

31

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4: Particle kinematics

33

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4.1 Bodies and their configurations

33

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4.2 Displacement and velocity

36

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4.3 Time rates of change

37

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4.4 Acceleration

39

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4.5 Steady motion. Particle paths and streamlines

41

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4.6 Problems

42

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5: Stress

44

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5.1 Surface traction

44

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5.2 Components of stress

45

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5.3 The traction on any surface

46

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5.4 Transformation of stress components

49

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5.5 Equations of equilibrium

51

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5.6 Principal stress components, principal axes of stress and stress invariants

52

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5.7 The stress deviator tensor

56

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5.8 Shear stress

57

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5.9 Some simple states of stress

57

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5.10 Problems

60

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6: Motions and deformations

63

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6.1 Rigid-body motions

63

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6.2 Extension of a material line element

66

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6.3 The deformation gradient tensor

68

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6.4 Finite deformation and strain tensors

70

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6.5 Some simple finite deformations

74

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6.6 Infinitesimal strain

78

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6.7 Infinitesimal rotation

82

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6.8 The rate-of-deformation tensor

83

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6.9 The velocity gradient and spin tensors

85

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6.10 Some simple flows

87

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6.11 Problems

88

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7: Conservation laws

91

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7.1 Conservation laws of physics

91

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7.2 Conservation of mass

91

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7.3 The material time derivative of a volume integral

96

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7.4 Conservation of linear momentum

97

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7.5 Conservation of angular momentum

98

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7.6 Conservation of energy

100

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7.7 The principle of virtual work

102

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7.8 Problems

103

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8: Linear constitutive equations

104

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8.1 Constitutive equations and ideal materials

104

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8.2 Material symmetry

106

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8.3 Linear elasticity

110

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8.4 Newtonian viscous fluids

116

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8.5 Linear viscoelasticity

118

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8.6 Problems

120

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9: Further analysis of finite deformation

123

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9.1 Deformation of a surface element

123

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9.2 Decomposition of a deformation

125

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9.3 Principal stretches and principal axes of deformation

127

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9.4 Strain invariants

130

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9.5 Alternative stress measures

132

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9.6 Problems

134

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10: Non-linear constitutive equations

136

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10.1 Non-linear theories

136

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10.2 The theory of finite elastic deformations

136

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10.3 A non-linear viscous fluid

142

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10.4 Non-linear viscoelasticity

144

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10.5 Plasticity

145

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10.6 Problems

149

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11: Cylindrical and spherical polar coordinates

153

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11.1 Curvilinear coordinates

153

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11.2 Cylindrical polar coordinates

153

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11.3 Spherical polar coordinates

161

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11.4 Problems

167

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Appendix. Representation theorem for an isotropic tensor function

170

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Answers

173

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Summary

Table of Contents

Supplemental Materials

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Continuum Mechanics

Summary

Table of Contents

Supplemental Materials

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