Abstract
Here, we return to the plane approximations briefly introduced at the end of Chap. 5, with the aim of exploring in more detail the nature of those solution strategies. Just as the torsion problem can be solved by using a suitably chosen scalar potential, the plane-stress and plane-strain approximations can also be reduced to finding a scalar potential. However, unlike the torsion case, in which the potential was essentially a harmonic function, this time the potential turns out to be bi-harmonic. Several representative examples are used to illustrate the general theory. A number of general facts about the bi-harmonic equation are relegated to Appendix E, which the reader is encouraged to study concomitantly with the present chapter.
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- 1.
Note that, by setting \(\varvec{e}_1\rightarrow \varvec{b}\) and \(\varvec{e}_2\rightarrow \varvec{c}\), we can think of this reduced identity tensor as \(\varvec{I}_2\equiv \varvec{I}-\varvec{a}\otimes \varvec{a}\), where \(\varvec{I}\) is the full identity tensor associated with the 3D vector space \(\mathscr {V}\).
- 2.
\(\varvec{n}_{\perp }\) is obtained by a \(90^{\circ }\) clockwise rotation of the tangent vector \(\varvec{\tau }\).
- 3.
The solution given here is a particular case of a more general result obtained in 1892 by Alfred-Aimé Flamant, a French engineer and pupil of B. de Saint-Venant.
- 4.
See (E.55) for the general form of the Airy function in polar coordinates.
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Coman, C.D. (2020). Two-Dimensional Approximations. In: Continuum Mechanics and Linear Elasticity. Solid Mechanics and Its Applications, vol 238. Springer, Dordrecht. https://doi.org/10.1007/978-94-024-1771-5_8
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