Abstract
So far in our discussion of stress we have made as much use as possible of stress vectors, because vector quantities are relatively easy to understand and to visualize. We have pointed out, however, that there is a fundamental distinction between describing stress on a single plane using a stress vector and describing stress on the infinite collection of planes at a point using a stress tensor. We now describe the tensor further, in particular the significance of the tensor components. Fortunately this aspect of stress can still be discussed in terms of stress vectors. As we will see, the tensor components of stress are nothing more than the components of three specially chosen stress vectors.
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Notes and References
Reference axes are typically designated x, y, and z by older books in the mechanics literature—e.g., Nadai (1950)—and by Jaeger (1969), Ramsay (1967), Johnson (1970), and Hubbert (1972). In this book the x1, x2, x3 system and tensor notation is also introduced at an early stage because this will make it easier for students to consult books in the recent physical and engineering literature, which make extensive or exclusive use of tensor notation (e.g., Nye, 1957; Fung, 1969; Malvern, 1969; Mase, 1970).
An introduction to the concept of stress, including a discussion of the difference between forces and stresses, is given by Johnson (1970, pp. 175–184).
The difference between the sign conventions for shearing stresses used in Mohr diagrams and in tensor treatments is discussed by Brace (1968, pp. 59–61).
The procedure for finding the magnitude and orientations of the principal stresses from any set of tensor components is discussed by Durelli et al. (1958, pp. 17–24), Ramsay (1967, pp. 33–35), and by Jaeger (1969, pp. 12–13).
The necessary equalities between pairs of shear components, indicated by Equations 10.5, are derived more rigorously by Bombolakis (1968, pp. 36–37).
Tensor notation is explained by Hawkins (1963, pp. 22–23), Nye (1964, pp. 7, 8), Malvern (1969, pp. 14, 15), and Mase (1970, pp. 8–10).
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© 1976 Springer-Verlag New York Inc.
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Means, W.D. (1976). Tensor Components of Stress. In: Stress and Strain. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-9371-9_10
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DOI: https://doi.org/10.1007/978-1-4613-9371-9_10
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-07556-3
Online ISBN: 978-1-4613-9371-9
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