Design of pre-stressed plate-strips to cover non-developable shells

• Published in: European journal of mechanics, A, Solids Vol. 95; p. 104609
• Main Authors: Danescu, Alexandre, Ionescu, Ioan R.
• Format: Journal Article
• Language: English
• Published: Berlin Elsevier Masson SAS 01-09-2022; Elsevier BV; Elsevier
• Subjects: Curvature; Differential Geometry; Engineering Sciences; Large rotations and small strains; Mathematics; Mechanics; Mechanics of materials; Non-developable strip-surfaces; Nonlinear elasticity; Pre-stress; Shell design; Solid mechanics; Structural mechanics; Toruses; Nonlinear elasticity; Non-developable strip-surfaces; Shell design; Large rotations and small strains; Pre-stress
• ISSN: 0997-7538; 1873-7285
• DOI: 10.1016/j.euromechsol.2022.104609

In this paper we address the following design problem: what is the shape of a plate and the associated pre-stress that relaxes toward a given three-dimensional shell ? As isometric transformations conserve the gaussian curvature, three-dimensional non-developable shells cannot be obtained from the relaxation of pre-strained plates by using isometric transformations only. Overcoming this geometric restriction, including small-strains and large rotations, solves the problem for small areas only. This paper dispenses with the small-area restriction to cover three-dimensional shells fully by using shell-strips. Since shell-strips have an additional geometric parameter, we show that under suitable assumptions that relate the width of the strip to the curvature of the shell, we are able to design arbitrary shell surfaces by covering them with shell-strips. As an illustration, we provide optimized covers of the sphere in a variety of different surface-strips relaxed from plate-strips with homogeneous and isotropic pre-stress. Moreover, we propose the design of the torus, of the helicoid and of the non-developable Möbius band, which requires inhomogeneous and anisotropic pre-stress.
•Obtain 3D shells using pre-stress relaxation starting from planar pre-stressed ribbons.•Design of both planar shape and pre-strain needed to cover non-developable shells.•Solution to design problems for to the sphere, the torus and the Mobius band