Geometrically controlled snapping transitions in shells with curved creases

Abstract

Curvature and mechanics are intimately connected for thin materials, and this coupling between geometry and physical properties is readily seen in folded structures from intestinal villi and pollen grains to wrinkled membranes and programmable metamaterials. While the well-known rules and mechanisms behind folding a flat surface have been used to create deployable structures and shape transformable materials, folding of curved shells is still not fundamentally understood. Shells naturally deform by simultaneously bending and stretching, and while this coupling gives them great stability for engineering applications, it makes folding a surface of arbitrary curvature a nontrivial task. Here we discuss the geometry of folding a creased shell, and demonstrate theoretically the conditions under which it may fold smoothly. When these conditions are violated we show, using experiments and simulations, that shells undergo rapid snapping motion to fold from one stable configuration to another. Although material asymmetry is a proven mechanism for creating this bifurcation of stability, for the case of a creased shell, the inherent geometry itself serves as a barrier to folding. We discuss here how two fundamental geometric concepts, creases and curvature, combine to allow rapid transitions from one stable state to another. Independent of material system and length scale, the design rule that we introduce here explains how to generate snapping transitions in arbitrary surfaces, thus facilitating the creation of programmable multistable materials with fast actuation capabilities.C urved shells are generally used to enhance structural stability (1-3), because the coupling between bending and stretching makes them energetically costly to deform. The consequences of this coupling are seen in both naturally occurring scenarios, such as intestinal villi and pollen grains (4, 5), and find use in man-made structures such as programmable metamaterials (6-9). When these shells have multistable configurations, the transition between them is opposed by geometrically enhanced rigidity resulting from the dominant stretching energy. Often, even for relatively small range of deformation, stretching leads to the high forces and rapid acceleration associated with a "snap-through" transition in many natural and man-made phenomena (10-17). For example, Venus flytraps (Dionaea muscipula) use this mechanism to generate a snapping motion to close their leaves (11), hummingbirds (Aves: Trochilidae) twist and rotate their curved beaks to catch insect prey (14), and engineered microlenses use a combination of bending and stretching energy to rapidly switch from convex to concave shapes to tune their optical properties (12). Despite the ability to engineer bistability and snapping transitions in a variety of systems by using prestress or material anisotropy (18-24), a general geometric design rule for creating a snap between stable states of arbitrary surfaces does not exist. This stands in stark contrast to the well-known rules and consequences fo...

Nonlinear mechanics of rigidifying curves

“…And show that this deformation is energetically favored in the experimental conditions of [12]. This last point is discussed further in the conclusion in * e-mail: csantang@physics.umass.edu Sec.…”

“…If so, that would seem to be at odds with experiments described in Ref. [12], in which it was shown that a shell can be folded continuously across an asymptotic curve without any stretching at all, while folding across a non-asymptotic curve would require traversing a stretching energy barrier.…”

“…The two solutions have opposite signs of normal curvature, which is required for existence of folding isometries [12,20].…”

Folded isometric deformations and banana-shaped seedpod

“…In parallel, the description of folds on plates has significantly been improved in the last 20 years: an energy for straight folds has been rigorously derived and validated [7,8] and some examples of curved folds have been treated [9]; current focus is rather on more complicated Miura-Ori structures [10][11][12][13][14][15]. Folds on shells have comparatively received much less attention, two topics having been well-studied: the rigidity induced by folds on shells [16] and the ridge bounding a dimple on a sphere for which a rigorous energy has been derived [17,18]. Once again classical differential geometry could be of great use for the understanding of folded shells.…”

O elemento finito T6-3i na análise de placas e dinâmica de cascas.

“…Os resultados calculados pelo PEFSYS apresentam valores oscilantes a medida que se aproxima da borda, quanto mais grosseira a malha, maior essa oscilação. Exemplos numéricos da deformação de um domo fino são apresentados, incluindo domos com vincos (BENDE et al, 2015). O elemento finito triangular de casca usado, o T6-3i, oferece grande flexibilidade para a geração das malhas curvas não estruturadas, bem como bom resultados.…”