Curvature-Induced Instabilities of Shells

Abstract

Induced by proteins within the cell membrane or by differential growth, heating, or swelling, spontaneous curvatures can drastically affect the morphology of thin bodies and induce mechanical instabilities. Yet, the interaction of spontaneous curvature and geometric frustration in curved shells remains poorly understood. Via a combination of precision experiments on elastomeric spherical shells, simulations, and theory, we show how a spontaneous curvature induces a rotational symmetry-breaking buckling as well as a snapping instability reminiscent of the Venus fly trap closure mechanism. The instabilities, and their dependence on geometry, are rationalized by reducing the spontaneous curvature to an effective mechanical load. This formulation reveals a combined pressurelike term in the bulk and a torquelike term in the boundary, allowing scaling predictions for the instabilities that are in excellent agreement with experiments and simulations. Moreover, the effective pressure analogy suggests a curvature-induced subcritical buckling in closed shells. We determine the critical buckling curvature via a linear stability analysis that accounts for the combination of residual membrane and bending stresses. The prominent role of geometry in our findings suggests the applicability of the results over a wide range of scales.PACS numbers: 02.40. Yy, 87.17.Pq, 87.10.Pq Owing to their slender geometry, thin elastic shells display intriguing mechanical instabilities. Perhaps the most iconic example is the buckling of a spherical shell under pressure -a catastrophic situation that often leads to structural failure [1,2]. Instabilities and shape changes are also fundamental during the development and morphogenesis of thin tissue [3,4]. To control and evolve shape, Nature heavily relies on internal stimuli such as growth, swelling, or active stresses [5,6]. If the stimulus varies through the thickness of the shell, it generally induces a change of the spontaneous (or natural) curvature of the tissue [7]. Examples are the ventral furrow formation in Drosophila [8] or the fast closure mechanism invoked by the Venus fly trap to catch prey [9]. Harnessing similar concepts for technological applications, internal stimuli were also suggested as a means to design adaptive metamaterials [10] and soft robotics actuators [11]. To describe the mechanics of slender structures with arbitrary stimuli, classical shell mechanics was extended recently to model bodies that do not possess a stressfree configuration [12][13][14], leading to the non-Euclidean shell theory [15]. Despite recent progress [16,17], the role of curvature-altering stimuli, and their interplay with geometric frustration and instabilities in thin, initially curved shells, remains poorly understood.In this Letter, we combine precision experiments with non-Euclidean shell theory to reveal how curvature stimuli induce mechanical instabilities in spherical shells. Our experiments demonstrate symmetry-breaking as well as snap-through shape transitions depending on the amoun...

Gait-optimized locomotion of wave-driven soft sheets

“…2D). This field exerts active mechanical stress on the sheet by direct modification of the reference tensors described above, consistent with prior work on the active deformation of thin shells: 33,34,38 % aexp(2ac)% a, % b -Àbc% a,…”

Static bistability of spherical caps

“…This parameter may now be interpreted as the lateral size of the shell, L, compared to the width of the typical boundary layer that is induced by the competition between bending and stretching, (hR) 1/2 . If the shell is large compared to this boundary layer, we expect it should remain bistable, but if the boundary layer becomes too large then it will instead be monostable, similar to the threshold for bistability in spherical caps subjected to an evolving natural curvature [24].…”

“…More recently, structures that are able to switch between two different configurations have attracted interest for applications in which morphing between states is desirable, for example in mechanical metamaterials and origami structures [1,7,14] or in actuators [3]. Various mechanisms have been proposed by which the system may be forced between two stable states including loading via magnetic forces [18,29], fluid flow [2,12,31], changes in lateral confinement [13,11] or, more generally, variations in the natural curvature of a structure [23,24].…”

Shallow shell theory of the buckling energy barrier: From the Pogorelov state to softening and imperfection sensitivity close to the buckling pressure

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