Abstract
The reader may have wondered why almost only RS corresponding to projections such as \(:\underline{n}\,\underline{n}\,\), 
, 
have appeared up to now. As a matter of fact, we have also encountered projections along more than one direction: the calculation of the angular dependence of the shear modulus G (8.22) of a 2D fabric involved the use of two vectors (8.18). The calculation of polycrystalline averages also involved terms like 
and 
(8.59). All these expressions come from the definition of engineering moduli (Sect. 4.7) or similar magnitudes that involve more than one direction in their projection.
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Notes
- 1.
What kind of engineering modulus, or which combination of them would this modulus correspond to?
- 2.
Same question!
- 3.
Can you calculate the latitudes of these parallels?
- 4.
Because the angle \(\psi \) does not appear in the third row of Euler’s \(\underset{{{{\backsim }}}}{L}\). A similar discussion as above can be made for the use of the columns of \(\underset{{{{\backsim }}}}{L}\) as unit projection vectors. In this case, it is \(\varphi \) the one that does not appear in the third column.
- 5.
Almost; see (9.10) below.
- 6.
There is a good reason to choose this name, but it is of no relevance for the purposes of this book.
- 7.
Can you write the \(\underset{{{{\backsim }}}}{L}\) for the transformation that brings the general \(\underline{\underline{A}}\) to the form (9.9)?
- 8.
It can also be brought to any of the two other variants in which \(\lambda \) and \(-\lambda \) occupy the elements 13 or 23. In the first case, it is the axis ②that points along \(\underline{u}\), in the second, it is ①.
- 9.
The sign of the vector product depends on the sign of the volume 3-form, in this case \(\underline{\underline{\underline{\epsilon }}}\), while the sign of the two-form onto which the vector product is mapped by duality does not.
- 10.
Three and not two are needed, because \(\underline{u}\) is not of unit modulus, so three components are necessary, or two to specify its arbitrary orientation, one to specify its magnitude.
- 11.
Or only five if absolute size is ignored.
- 12.
Why is this not surprising?
- 13.
Why?
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Laso, M., Jimeno, N. (2020). Transversality and Skew Symmetric Properties. In: Representation Surfaces for Physical Properties of Materials. Engineering Materials. Springer, Cham. https://doi.org/10.1007/978-3-030-40870-1_9
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