Cartesian Tensors in Engineering Science

Cartesian 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