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On a nonlinear elastic shell system in liquid crystal theory: generalized Willmore surfaces and Dupin cyclides. (English) Zbl 1186.74079

Summary: An elastic membrane model of smectic A liquid crystal deformation is derived ab initio via a variational approach. The well-determined nature of the resulting nonlinear model equations reveals that the deformed states of the liquid crystal lamellae can only adopt privileged geometries. These are shown to generalize classical and novel ‘integrable’ geometries associated with Willmore, linear Weingarten and ‘membrane’ O surfaces. The main result establishes that, remarkably, the membrane model admits layered parallel Dupin cyclide structures of the kind originally observed by Friedel and Grandjean in their pioneering experiments of 1910 and subsequently elaborated upon by Friedel in 1922 and later by Bragg.

MSC:

74K25 Shells
53A05 Surfaces in Euclidean and related spaces
76A15 Liquid crystals
82D30 Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses)
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