Dynamic behavior of elastic strips near shape transitions

Transient Amplification of Broken Symmetry in Elastic Snap-Through

A snap-through bifurcation occurs when a bistable structure loses one of its stable states and moves rapidly to the remaining state. For example, a buckled arch with symmetrically clamped ends can snap between an inverted and a natural state as the ends are released. A standard linear stability analysis suggests that the arch becomes unstable to asymmetric perturbations. Surprisingly, our experiments show that this is not always the case: symmetric transitions are also observed. Using experiments, numerics, and a toy model, we show that the symmetry of the transition depends on the rate at which the ends are released, with sufficiently fast loading leading to symmetric snap-through. Our toy model reveals that this behavior is caused by a region of the system's state space in which any initial asymmetry is amplified. The system may not enter this region when loaded fast (hence remaining symmetric), but will traverse it for some interval of time when loaded slowly, causing a transient amplification of asymmetry. Our toy model suggests that this behavior is not unique to snapping arches, but rather can be observed in dynamical systems where both a saddle-node and a pitchfork bifurcation occur in close proximity.

A dimensionally-reduced nonlinear elasticity model for liquid crystal elastomer strips with transverse curvature

Liquid crystalline elastomers (LCEs) are active materials that are of interest due to their programmable response to various external stimuli such as light and heat. When exposed to these stimuli, the anisotropy in the response of the material is governed by the nematic director, which is a continuum parameter that is defined as the average local orientation of the mesogens in the liquid crystal phase. This nematic director can be programmed to be heterogeneous in space, creating a vast design space that is useful for applications ranging from artificial ligaments to deployable structures to self-assembling mechanisms. Even when specialized to long and thin strips of LCEs - the focus of this work - the vast design space has required the use of numerical simulations to aid in experimental discovery. To mitigate the computational expense of full 3-d numerical simulations, several dimensionally-reduced rod and ribbon models have been developed for LCE strips, but these have not accounted for the possibility of initial transverse curvature, like carpenter's tape spring. Motivated by recent experiments showing that transversely-curved LCE strips display a rich variety of configurations, this work derives a dimensionally-reduced 1-d model for pre-curved LCE strips. The 1-d model is validated against full 3-d finite element calculations, and it is also shown to capture experimental observations, including tape-spring-like localizations, in activated LCE 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.