Snap-buckling in asymmetrically constrained elastic strips

Elastic Snap-Through Instabilities Are Governed by Geometric Symmetries

Many elastic structures exhibit rapid shape transitions between two possible equilibrium states: umbrellas become inverted in strong wind and hopper popper toys jump when turned inside out. This snap through is a general motif for the storage and rapid release of elastic energy, and it is exploited by many biological and engineered systems from the Venus flytrap to mechanical metamaterials. Shape transitions are known to be related to the type of bifurcation the system undergoes, however, to date, there is no general understanding of the mechanisms that select these bifurcations. Here we analyze numerically and analytically two systems proposed in recent literature in which an elastic strip, initially in a buckled state, is driven through shape transitions by either rotating or translating its boundaries. We show that the two systems are mathematically equivalent, and identify three cases that illustrate the entire range of transitions described by previous authors. Importantly, using reduction order methods, we establish the nature of the underlying bifurcations and explain how these bifurcations can be predicted from geometric symmetries and symmetry-breaking mechanisms, thus providing universal design rules for elastic shape transitions.

Dynamic behavior of elastic strips near shape transitions

Elastic strips provide a general motif for studying shape transitions. When actuated through rotation of its boundaries, a buckled strip exhibits, depending on the direction of rotation, three types of shape transitions: buckling, algebraic snap-through, or exponential snap-through. The transition dynamics is linked to the character of the bifurcation, which, in turn, is disclosed by the normal form of the system, but deriving normal forms is challenging. Recent work has used asymptotic methods to obtain this form for algebraic snap-through, but, to date, there is no methodology for extending this analysis to other transitions. Here we introduce a method to analyze the dynamic characteristics of an elastic strip near a transition and extend, in a straightforward manner, the previously proposed asymptotic analysis to exponential snap-through and buckling transitions. Importantly, we show that these normal forms dictate all the dynamic characteristics of the elastic strip near a shape transition. Our analysis provides reliable tools to diagnose and anticipate elastic shape transitions.

Snap buckling of bistable beams under combined mechanical and magnetic loading

We investigate the mechanics of bistable, hard-magnetic, elastic beams, combining experiments, finite-element modelling (FEM) and a reduced-order theory. The beam is made of a hard magneto-rheological elastomer, comprising two segments with antiparallel magnetization along the centreline, and is set into a bistable curved configuration by imposing an end-to-end shortening. Reversible snapping is possible between these two stable states. First, we experimentally characterize the critical field strength for the onset of snapping, at different levels of end-to-end shortening. Second, we perform three-dimensional FEM simulations using the Riks method to analyse high-order deformation modes during snapping. Third, we develop a reduced-order centreline-based beam theory to rationalize the observed magneto-elastic response. The theory and simulations are validated against experiments, with an excellent quantitative agreement. Finally, we consider the case of combined magnetic loading and poking force, examining how the applied field affects the bistability and quantifying the maximum load-bearing capacity. Our work provides a set of predictive tools for the rational design of one-dimensional, bistable, magneto-elastic structural elements. This article is part of the theme issue 'Probing and dynamics of shock sensitive shells'.

Autonomous snapping and jumping polymer gels

Snap-through buckling is commonly used in nature for power-amplified movements. While natural examples such as Utricularia and Dionaea muscipula can autonomously reset their snapping structures, bio-inspired analogues require external mediation for sequential snap events. Here we report the design principles for self-repeating, snap-based polymer jumping devices. Transient shape changes during the drying of a polymer gel are exploited to generate mechanical constraint and an internal driving force for snap-through buckling. Snap-induced shape changes alter environmental interactions to realize multiple, self-repeating snap events. The underlying mechanisms are understood through controlled experiments and numerical modelling. Using these lessons, we create snap-induced jumping devices with power density outputs (specific power ≈ 312 W kg^-1) that are similar to high-performing jumping organisms and engineered robots. These results provide the demonstration of an autonomous, self-repeating, high-speed movement, marking an important advance in the development of environmental energy harvesting, high-power motion that is important for microscale robots and actuated devices.

Creep trajectory transition of a nonstationary viscoelastic model onto a single rate parameter

A system of three-variable differential equations, which has a nonstationary trajectory transition through the control of a single rate parameter, is formulated. For the nondimensional system, the critical trajectory creeps before a transition in a long-lasting plateau region in which the velocity vector of the system hardly changes and then diverges positively or negatively in finite time. The mathematical model well represents the compressive viscoelasticity of a spring-damper structure simulated by the multibody dynamics analysis. In the simulation, the post-transition behaviors realize a tangent stiffness of the self-contacted structure that is polarized after transition. The mathematical model is reduced not only to concisely express the abnormal compression problem, but also to elucidate the intrinsic mechanism of creep-to-transition trajectories in a general system.

Twist-Induced Snapping in a Bent Elastic Rod and Ribbon

Snapping of a slender structure is utilized in a wide range of natural and manmade systems, mostly to achieve rapid movement without relying on musclelike elements. Although several mechanisms for elastic energy storage and rapid release have been studied in detail, a general understanding of the approach to design such a kinetic system is a key challenge in mechanics. Here we study a twist-driven buckling and fast flip dynamics of a geometrically constrained ribbon by combining experiments, numerical simulations, and an analytical theory. We identify two distinct types of shape transitions: A narrow ribbon snaps, and a wide ribbon forms a pair of localized helices. We construct a phase diagram and explain the origin of the boundary, which is determined largely by the geometry. We quantify the effects of gravity and clarify the timescale dictating the rapid flipping. Our study reveals the unique role of geometric twist-bend coupling in the fast dynamics of a thin constrained structure, which has implications for a wide range of biophysical and applied physical problems.

Snap-Buckling Motivated Controllable Jumping of Thermo-Responsive Hydrogel Bilayers

Responsive hydrogel actuators have promising applications in diverse fields. Most hydrogel actuators are limited by slow actuation or shape transformations. This work reports on snap-buckling motivated jumping of thermoresponsive hydrogel bilayers. The bilayers are composed of poly(NIPAM- co-DMAPMA)/clay hydrogel with different lower critical solution temperatures in each layer, and thus undergo slow reversible curling/uncurling at temperature changes. The gels are adhesive to numerous materials including aluminum. The adhesion between the gels and an aluminum ratchet is utilized to constrain the thermoresponsive deformation of the bilayers to store elastic energy. When the accumulated elastic energy overwhelms the gel-aluminum adhesion, snap-buckling takes place to abruptly release the accumulated energy, which motivates the bilayer to jump. The jumping direction, start time, height, and distance are controlled by the geometry of the bilayers or the ratchet. This work paves a novel way for the rapid actuation of responsive hydrogels in a controlled manner and may stimulate the development of novel hydrogel devices.