Shape-shifting and instabilities of plates and shells
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Slender structures like plates and shells -- for which at least one dimension is much smaller than the others -- are lightweight, flexible, and offer considerable strength with little material. As such, these structures are abundant in nature (e.g. flower petals, eggshells, and blood vessels) and design (e.g. bridge decks, fuel tanks, and soda cans). However, with slenderness comes suceptibility to large and often sudden deformations, which can be wildly nonlinear, as bending is energetically preferable to stretching. Though once considered categorically undesirable, these instabilities are often coveted nowadays in the engineering community. They provide mechanical explanations for observations in nature like the wrinkled structure of the brain or the snapping mechanism of the Venus fly trap, and when precisely controlled, enable the design of functional devices like artificial muscles or self-propelling microswimmers. As a prerequisite, these achievements require a thorough understanding of how thin structures "shape-shift" in response to stimuli and confinement. Advancing this fundamental knowledge is the goal of this thesis. In the first two chapters, we consider the shape-selection of shells and plates that are confined by their environment. The shells are made by residual swelling of silicone elastomers, a process that mimics differential growth, and causes initially flat structures to irreversibly morph into curved shapes. Flattening the central region forces further reconfiguration, and the confined shells display multi-lobed buckling patterns. These experiments, finite element (FE) simulations, and a scaling argument reveal that a single geometric confinement parameter predicts the general features of this shape-selection. Next, in experiments and molecular dynamics (MD) simulations, we constrain intrinsically flat sheets in the same manner, so that their center remains flat when we quasi-statically force them through a ring. In the absence of planar confinement, these sheets form a well-studied conical shape (the developable cone or d-cone). Our annular d-cone buckles circumferentially into patterns that are qualitatively similar to the confined shells, despite the distinct curvatures and loading methods. This is explained by the dominant role of confinement geometry in directing deformation, which we uncover via a scaling argument based on the elastic energy. There are also marked differences between the way plates and shells change shape, which we highlight when we investigate the rich dynamics of reconfiguration. In the final two chapters, we demonstrate how mechanics, geometry, and materials can inform the design of structures that use instabilities to function. We observe in experiments that dynamic loading causes a spherical elastomer shell to buckle at ostensibly subcritical pressures, following a substantial time delay. To explain this, we show that viscoelastic creep deformation lowers the critical load in the same predictable, quantifiable way that a growing defect would in an elastic shell. This work offers a pathway to introduce tunable, time-controlled actuation to existing mechanical actuators, e.g. pneumatic grippers. The final chapter aims at reducing the energy input required for bistable actuators, wherein snap-through instability is typically induced by a stimulus applied to the entire shell. To do so, we combine theory with 1D finite element simulations of spherical caps with a non-homogeneous distribution of stimuli--responsive material. We demonstrate that restricting the active area to the shell boundary allows for a large reduction in its size, while preserving snap-through behavior. These results are stimulus-agnostic, which we demonstrate with two sets of experiments, using residual swelling of bilayer silicone elastomers as well as a magneto-active elastomer. Our findings elucidate the underlying mechanics, offering an intuitive route to optimal design for efficient snap-through.
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