Cartesian Tensors in Engineering Science
The Commonwealth and International Library: Structures and Solid Body Mechanics Division
- 1st Edition - January 1, 1966
- Imprint: Pergamon
- Author: L. G. Jaeger
- Editor: B. G. Neal
- Language: English
- Paperback ISBN:9 7 8 - 0 - 0 8 - 0 1 1 2 2 1 - 3
- eBook ISBN:9 7 8 - 1 - 4 8 3 1 - 3 8 7 2 - 5
Cartesian Tensors in Engineering Science provides a comprehensive discussion of Cartesian tensors. The engineer, when working in three dimensions, often comes across quantities… Read more
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Request a sales quoteCartesian Tensors in Engineering Science provides a comprehensive discussion of Cartesian tensors. The engineer, when working in three dimensions, often comes across quantities which have nine components. Variation of the components in a given plane may be shown graphically by a familiar construction called Mohr's circle. For such quantities it is always possible to find three mutually perpendicular axes, called principal axes, with respect to which the six ""paired up"" components are all zero. Such quantities are called symmetric tensors of the second order. The student may at this stage be struck by the fact that the physical quantities with which he normally deals have either one component, three components or nine components, being respectively scalars, vectors, and what have just been called second order tensors. The family of quantities having 1, 3, 9, 27, … components does exist. It is the tensor family in three dimensions. The book discusses the ""tests"" a given quantity must pass in order to qualify as a member of the family. The products of tensors, elasticity, and second moment of area and moment of inertia are also covered. Although written primarily for engineers, it is hoped that students of various branches of physical science may find this book useful.
List of Principal Symbols
Chapter 1. Cartesian Axes. Scalars and Vectors
Suffix Notation
Right-handed and Left-handed Axes
Direction Cosines
Rotation of Axes
Transformation of Vector Components
Dummy Suffices
Operation of a Matrix on a Column Vector
Summary
Examples for Solution
Chapter 2. Properties of Direction Cosine Arrays. Second and Higher Order Tensors
The Array of Direction Cosines
Normalization Conditions and Orthogonality Conditions Expressed in Matrix Notation and also in Suffix Notation
The Dummy Suffix Rule
The Kronecker Delta
The Determinant of the Array of Direction Cosines
Second Order Tensors—transformation of Components
Products of Vector Components
The Stress Tensor
Higher Order Tensors
Summary
Examples for Solution
Chapter 3. Symmetric Second Order Tensors
Mohr's Circle for the Symmetric Second Order Tensor in Two Dimensions
Other Examples of Mohr's Circle
The Flexibility Tensor
Principal Axes
Two or more Principal Components Equal
The Isotropic Tensor
The Kronecker Delta as a Second Order Tensor
Summary
Examples for Solution
Chapter 4. The Products of Tensors
Product of Two Tensors
Product of Any Number of Tensors
Contraction
The Invariants of a Second Order Tensor
The Second Order Tensor as a Vector Operator
The Levi-Civita Density
Scalar Product and Vector Product of Two Vectors
The Vector Operator V
Eigenvectors of a Second Order Tensor
Summary
Examples for Solution
Chapter 5. Elasticity
The Stress Tensor
The Strain Tensor and the Appropriate Definition of Shear Strain
Linear Elastic Behavior
Homogeneity
Isotropy
Relationships between Stress Components and Strain Components
Product of Stress and Strain Components
Strain Energy
Energy of Dilatation and Energy of Distortion
Summary
Examples for Solution
Chapter 6. Second Moment of Area and Moment of Inertia. Dynamics
Bending of Beams - The Second Moment of Area Tensor
Motion of a Rigid Body—the Moment of Inertia Tensor
Momentum and Moment of Momentum
Newton's Second Law
Inertial Axes and Embedded Axes
Euler's Equations of Motion
Gyroscopes
Summary
Examples for Solution
Appendix
Linear Transformation
Square Matrices and Column Matrices
Matrix Multiplication - Row into Column Rule
The Unit Matrix
Reciprocal of a Matrix
The Reversal Rule
Transposition
Orthogonal Matrix
- Edition: 1
- Published: January 1, 1966
- No. of pages (eBook): 124
- Imprint: Pergamon
- Language: English
- Paperback ISBN: 9780080112213
- eBook ISBN: 9781483138725