Deformations without bending: explicit examples. (English) Zbl 1414.74020

Mladenov, Ivaïlo M. (ed.) et al., Proceedings of the 20th international conference on geometry, integrability and quantization, Sts. Constantine and Elena (near Varna), Bulgaria, June 2–7, 2018. Sofia: Avangard Prima; Sofia: Bulgarian Academy of Sciences, Institute of Biophysics and Biomedical Engineering. Geom. Integrability Quantization 20, 246-254 (2019).
Summary: Here we consider an interesting class of free of bending deformations of thin axial symmetric shells subjected to uniform normal pressure. The meridional \(k_{\mu}\) and the parallel \(k_{\pi}\) principal curvatures of the middle surfaces of such shells obey the non-linear relationship \(k_{\mu}=2ak_{\pi}^2+3k_{\pi}\), \(a=\text{const}\). These non-bending shells depend on two arbitrary parameters, which are the principal radii \(r_{\mu}\) and \(r_{\pi}\) of some fixed parallel of the shell. Besides, these surfaces have no closed form description in elementary functions. Our principle aim here is to present such a parameterization of the whole class of non-bending closed surfaces by making use of the canonical forms of the elliptic integrals. The obtained explicit formulas are then applied for the derivation of the basic geometrical characteristics of these surfaces.
For the entire collection see [Zbl 1408.53003].


74K25 Shells
74A10 Stress
53A04 Curves in Euclidean and related spaces
53A05 Surfaces in Euclidean and related spaces
33E05 Elliptic functions and integrals
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