Abstract
To get much farther with the analysis of strain, we now have to introduce four kinds of equations by which a deformation can be specified. We then explain what is meant by displacement and deformation gradients, and this leads directly, in the next four chapters, to simple formulae that allow us to calculate the tensor components of strain.
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Notes and References
In this book quantities like ∂u/∂x, the gradients of the displacement field, are called displacement gradients, and quantities like ∂X/∂x, the gradients of the coordinate transformation field, are called deformation gradients. This follows the usage of Truesdell and Toupin (1960, pp. 245, 247), Eringen (1967, p. 11), Malvern (1969, p. 156), Mase (1970, p. 80) and Johnson (1970, p. 197). However, these terms are sometimes used in different ways. For example, Fung (1969, p. 112) calls quantities like ∂u/∂x deformation gradients.
At this point in our analysis of strain, there are two alternative directions for further study. One approach is to take the deformation gradients themselves and to show how these quantities can be used to work out such things as the principal strains or the strain of lines in any given initial orientation. This is the approach explained by Jaeger (1969, pp. 23–34), Howard (1968), and Ramsay (1967, pp. 55–65). The other approach, the one embarked upon here, is to use the deformation or displacement gradients to calculate the strain and deformation tensors, from which such quantities as the principal strains may be subsequently obtained. This approach, while it is one step more complicated, has the advantage that it gives beginning students further exposure to tensor methods and should help them ultimately to make use of books like Nye (1964), Fung (1969), and Malvern (1969).
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© 1976 Springer-Verlag New York Inc.
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Means, W.D. (1976). Displacement and Deformation Gradients. In: Stress and Strain. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-9371-9_18
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DOI: https://doi.org/10.1007/978-1-4613-9371-9_18
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-07556-3
Online ISBN: 978-1-4613-9371-9
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