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 a