Thin Structures With Imposed Metric

Abstract

We consider thin structures with a non necessarily realizable imposed metric, that only depends on the surface variable. We give a unified presentation of the three main limit models. We establish the generalized membrane model and we show, by means of an algebraic proof, that the internal membrane energy vanishes on short maps of the metric restricted to the plane. We recall that a generalized bending model can occur only when this reduced metric admits sufficiently regular isometric immersions. When the entries R12·· of the Riemannian curvature tensor are null, this bending energy can vanish; then the next model is necessarily a generalized von Kármán model whose minimum is zero if and only if the three-dimensional metric is flat.Résumé. Nous considérons des structures finesà métrique imposée, dépendant des variables planaires, non nécessairement réalisable. Nous donnons une présentation synthétique des trois principaux modèles limites. Nousétablissons un modèle de membrane généralisé et montrons que sonénergie interne s'annule pour les déformations non expansives de la métrique restreinte au plan. Nous rappelons qu'un modèle de flexion généralisé ne peut apparaître que si cette métrique réduite admet des immersions isométriques suffisamment régulières, et que, dès lors que les composantes R12·· du tenseur de courbure sont nulles, l'énergie de flexion peut s'annuler. Le modèle suivant est alors nécessairement un modèle de von Kármán généralisé dont le minimum est nul si et seulement si le tenseur de courbure est nul, c'est-à-dire si la métrique de départ est plate.

Covering a surface with pre-stressed ribbons : from theory to nano-structures fabrication

“…Introduced by an heuristic method in Danescu et al (2013) and later reconsidered in the framework of small-strains and large-rotations in Danescu and Ionescu (2021), the geodesic curvature represents the key concept for the design of 3D structures from planar pre-stresses films. From a different point of view, the equilibirum shape of a pre-stressed material was investigated by using dimension reduction in Le Dret and Raoult (1995); Friesecke et al (2002aFriesecke et al ( ,b, 2006; de Benito Delgado and Schmidt (2020); Wang et al (2019); de Benito Delgado, Miguel and Schmidt, Bernd (2021) leading to a hierarchy of non-linear elastic models Lewicka and Raoult (2018).…”

Shell design from planar pre-stressed structures

“…Recent work in [6,7] on various approximations of the 3D elasticity with incompatible pre-strain/stress provided a hierarchy of nonlinear elastic models that fulfill the requirement (i) but which cannot solve the design problem stated in (ii). From a different perspective, the pioneering work in [8][9][10] built a theory of incompatible surface growth that includes geometric nonlinearities and finite stretch, both core ingredients for a realistic description of the relaxed shapes.…”