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On examples to a conjecture of de Saint Venant. (English) Zbl 0794.73027

Torsion of cylindrical bars in the framework of classical elasticity leads to elliptic boundary value problems. The displacement is given by a harmonic function whereas the stresses are related to the gradient of this function. There is a unique maximum for the harmonic function, whose location is called the center of twist (torsion). The points where the stress is maximized, i.e., the directional derivative along the exterior normal achieves its maximum modulus, are called dangerous points. Saint- Venant suggested that the dangerous points are situated on the boundary. The author [J. Elasticity 22, No. 1, 57-61 (1989; Zbl 0693.73005)], and M. Ramaswamy [Nonlinear Anal., Theory Methods Appl. 15, No. 9, 891- 894 (1990; Zbl 0737.73024)] have given counterexamples to show that the conjecture is false. An example is given here, which raises doubt about the validity of a theorem of A. A. Kosmodem’yanskii [Sov. Math., Dokl. 39, No. 1, 112-114 (1989); translation from Dokl. Akad. Nauk SSSR 304, No. 3, 546-548 (1989; Zbl 0692.35036)], which implies that the stress is maximal on a boundary point, where the distance from the boundary to the center of twist has a local minimum.

MSC:

74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74B05 Classical linear elasticity
35Q72 Other PDE from mechanics (MSC2000)
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References:

[1] Saint Venant, B. De, Mémoires sur la torsion des prismes, Mémoires présentés par divers savants à l’académie des sciences de l’institut impérial de France, 14, 233-560 (1856), 2 Sér.
[2] Sweers, G., A counterexample with a convex domain to a conjecture of de Saint Venant, J. Elasticity, 22, 57-61 (1989) · Zbl 0693.73005
[3] Ramaswamy, M., A counterexample to the conjecture of Saint Venant by reflection methods, Diff. Integral Eqns, 3, 653-662 (1990) · Zbl 0718.73021
[4] Kosmodem’yanskiǐ, Jr. A.A., The behaviour of solutions of the equation Δ \(u\) = -1 in convex domains, Soviet Math. Dokl., 39, 112-114 (1989)
[5] Makar-Limanov, L. G., Solution of Dirichlet’s problem for the equation Δ \(u\) = -1 in a convex region, Math. Notes Acad. Sci. USSR, 9, 195-206 (1971) · Zbl 0222.31004
[6] Ramaswamy, M., On a counterexample to a conjecture of Saint Venant, Nonlinear Analysis, 15, 891-894 (1990) · Zbl 0737.73024
[7] Kawohl, B., On the location of maxima of the gradient for solutions to quasilinear elliptic problems and a problem raised by Saint Venant, J. Elasticity, 17, 195-206 (1987) · Zbl 0624.73011
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