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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