Page 7 - Introduction to Continuum Mechanics

vi Contents

2B8 Trace of a Tensor 22
2B9 Identity Tensor and Tensor Inverse 23
2B10 Orthogonal Tensor 24
2B11 Transformation Matrix Between Two Rectangular
Cartesian Coordinate Systems 26
2B12 Transformation Laws for Cartesian Components of Vectors 28
2B13 Transformation Law for Cartesian Components of a Tensor 30
2B14 Defining Tensors by Transformation Laws 32
2B15 Symmetric and Antisymmetric Tensors 35
2B16 The Dual Vector of an Antisymmetric Tensor 36
2B17 Eigenvalues and Eigenvectors of a Tensor 38
2B18 Principal Values and Principal Directions of Real Symmetric Tensors 43
2B19 Matrix of a Tensor with Respect to Principal Directions 44
2B20 Principal Scalar Invariants of a Tensor 45

Part C Tensor Calculus 47
2C1 Tensor-valued functions of a Scalar 47
2C2 Scalar Field, Gradient of a Scalar Function 49
2C3 Vector Field, Gradient of a Vector Field 53
2C4 Divergence of a Vector Field and Divergence of a Tensor Field 54
2C5 Curl of a Vector Field 55
Part D Curvilinear Coordinates 57

2D1 Polar Coordinates 57
2D2 Cylindrical Coordinates 61
2D3 Spherical Coordinates 63
Problems 68

Chapter 3 Kinematics of a Continuum 79
3.1 Description of Motions of a Continuum 79
3.2 Material Description and Spatial Description 83
3.3 Material Derivative 85
3.4 Acceleration of a Particle in a Continuum 87
3.5 Displacement Field 92
3.6 Kinematic Equations For Rigid Body Motion 93
3.7 Infinitesimal Deformations 94
3.8 Geometrical Meaning of the Components of the Infinitesimal Strain Tensor 99