Abstract
This chapter covers mathematical background needed for the theoretical analyses in later chapters. A brief introduction to vector and tensor analysis is provided, with tensor manipulations done using dyadic notation. Basic tensor operations, special tensors, coordinate transformation, and tensor calculus are discussed.
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Notes
- 1.
- 2.
A stress component depends on its direction of action and the orientation of the area on which it acts.
- 3.
In this book, we consider only the effects of geometry on tensor components while ignoring dynamic effects caused by relative motion between different reference frames.
- 4.
Tensor equations valid for all coordinate systems also can be written in indicial notation, although some physical intuition may be lost. In fact, for easier mathematical manipulations, many authors prefer to use indices, although some refer incorrectly to scalar components (e.g., a i) as vectors. In this book, we use both direct and indicial notation.
- 5.
Throughout this book, the summation convention applies only to the indices i, j, k, l, m, n with the default range being 1,  2,  3 (in 3D space). Thus, summation over x is not implied for a xx.
- 6.
By default, we set x = x 1, y = x 2 and z = x 3 in Cartesian coordinates.
- 7.
In general, base vectors need not be unit vectors, nor must they be orthogonal to each other, but these cases are not considered in this book.
- 8.
Some authors use the notation a ⊗b for the tensor product.
- 9.
Drew (1961) is an excellent resource for dyadic and polyadic analysis.
- 10.
In this chapter, lowercase bold letters denote first-order tensors (vectors), and uppercase bold letters denote second-order tensors (dyadics).
- 11.
To review eigenvalue problems, please see any standard reference on matrix algebra.
- 12.
In Cartesian coordinates, the components of a vector are the orthogonal projections of the vector onto the coordinate axes (Fig. 2.1). A physical interpretation for the components of a tensor cannot be visualized as easily.
References
Drew TB (1961) Handbook of vector and polyadic analysis. Reinhold Publishing Corporation, New York
Einstein A (1916) The foundation of the general theory of relativity. Annalen Phys 14:769–822
Flugge W (1972) Tensor analysis and continuum mechanics. Springer, New York
Holzapfel GA (2000) Nonlinear solid mechanics: a continuum approach for engineering. Wiley, New York
Malvern LE (1969) Introduction to the mechanics of a continuous medium. Prentice-Hall, Englewood Cliffs
Simmonds JG (1994) A brief on tensor analysis. Springer, New York
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Taber, L.A. (2020). Vector and Tensor Analysis. In: Continuum Modeling in Mechanobiology. Springer, Cham. https://doi.org/10.1007/978-3-030-43209-6_2
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DOI: https://doi.org/10.1007/978-3-030-43209-6_2
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