Delayed bifurcation in elastic snap-through instabilities

TL;DR

This work analyzes delayed snap-through in an elastic arch driven by a time-varying end-shortening, revealing that the snap-through event lags behind the static fold due to dynamic loading. By combining a geometrically nonlinear elastic model with a quasi-linear approximation and a rigorous asymptotic reduction, the authors derive a universal amplitude equation for the near-fold dynamics that encompasses inertia and damping through a single parameter $\Lambda$. They show that the bottleneck near the saddle-node leads to a finite-time blow-up of the reduced amplitude, providing explicit scaling laws for the time and parameter lag in both underdamped and overdamped regimes, and validate these results against full elastica simulations. The findings offer general insights and scaling laws for delayed bifurcation in elastic instabilities, with potential applications to design and control of bistable structures under dynamic loading.

Abstract

We study elastic snap-through induced by a control parameter that evolves dynamically. In particular, we study an elastic arch subject to an end-shortening that evolves linearly with time, i.e. at a constant rate. For large end-shortening the arch is bistable but, below a critical end-shortening, the arch becomes monostable. We study when and how the arch transitions between states and show that the end-shortening at which the fast 'snap' happens depends on the rate at which the end-shortening is reduced. This lag in snap-through is a consequence of delayed bifurcation and occurs even in the perfectly elastic case when viscous (and viscoelastic) effects are negligible. We present the results of numerical simulations to determine the magnitude of this lag as the loading rate and the importance of external viscous damping vary. We also present an asymptotic analysis of the geometrically-nonlinear problem that reduces the salient dynamics to that of an ordinary differential equation; the form of this reduced equation is generic for snap-through instabilities in which the relevant control parameter is ramped linearly in time. Moreover, this asymptotic reduction allows us to derive analytical results for the observed lag in snap-through that are in good agreement with the numerical results of our simulations. Finally, we discuss scaling laws for the lag that should be expected in other examples of delayed bifurcation in elastic instabilities.

Figures (9)

• Figure 1: Schematic showing the bifurcation structure that generically underlies elastic snap-through: for different values of a control parameter, $\mu$ (horizontal axis), an elastic structure is either bistable or monostable. (Here the vertical axis is a scalar 'amplitude' that describes the equilibrium states, e.g. the maximum displacement.) One of the stable states (solid curve) disappears at a saddle-node bifurcation, labelled $\mu=\mu_\mathrm{fold}$ (drawn as a filled circle), where it merges with an unstable state (dotted curve). In each panel the dynamics of snap-through depend on the path taken in parameter space, illustrated by the blue arrows, which begin at the filled square. (a) gomez2017critical considered the dynamics with a fixed value of the bifurcation parameter $\mu=\mu_\mathrm{fold}+\Delta\mu_0$, with $\Delta\mu_0 < 0$, by starting from a shape close to the equilibrium shape at the bifurcation point. (b) In this paper, we consider an alternative scenario in which the system begins in a stable equilibrium state well before the bifurcation point. The parameter $\mu$ then evolves in time at a finite rate. In particular, we seek to determine the delay in the control parameter at snap-through, labelled $|\Delta \mu_{\mathrm{eff}}|$, observed with linear parameter variation, i.e. $\mu=\mu_\mathrm{fold}+\dot{\mu} \:t$.
• Figure 2: A schematic diagram of the problem considered in this paper. An elastic strip of length $L$ is confined by two clamps that are separated by a horizontal distance $L-\Delta L$ (the strip's width is directed into the page) forming an arch. One clamp imposes $\theta = \alpha$, while the other imposes $\theta=0$; here $\theta$ is the inclination of the strip to the horizontal. For a given value of $\alpha$, the arch may be bistable or monostable depending on the value of $\Delta L$: in the bistable scenario it can adopt either the natural state (red curve) or the inverted state (blue curve). If monostable, it can only adopt the natural state. In this paper we consider the dynamics of the snap-through from the inverted to natural states resulting from variation in $\Delta L$ at a constant rate $\dot{\Delta L}<0$ (as indicated by the arrow).
• Figure 3: The steady bifurcation behaviour of the arch (determined from solutions of the moment balance \ref{['eqn:ElasticaND']} subject to \ref{['eqn:ThetaBC']} and \ref{['eqn:IntConstraints']}). Presented are raw numerical results for (a) the vertical displacement of the midpoint, $Y(1/2)$, and (b) the horizontal force in the arch, $N_X$, for different angles of inclination, $\alpha$ (as described in the legend). When rescaled as suggested by the quasi-linear theory, these raw data collapse onto a universal bifurcation structure as $\alpha\to0$. The predictions of the quasi-linear model (described by eqns \ref{['eqn:LinMidPoint']} and \ref{['eqn:LinConstraintDetail']}) are also shown as black curves (the dashed branch is unstable). (Note that in (c) the unstable branch lies above the fold point, but in (d) the unstable branch lies below the fold point and has the largest compressive force $-N_X=\tau^2$.)
• Figure 4: The evolution of the midpoint displacement, $Y(1/2,T)$, and the horizontal force resultant, $N_X$, as functions of the distance beyond the quasi-statics snap-through threshold, $\mu_\mathrm{fold}(\alpha)-\mu(T)=|\dot{\mu}|T$. We vary the loading rates $\dot{d}=\dot{\Delta L} \: t_\ast/L=\alpha^2\dot{\mu}$ (shown by the legend) while fixing $\alpha=\pi/6$. In each plot, the results of dynamic simulations are shown for different loading rates $\dot{d}$ (given by the legend), together with the quasi-static bifurcation diagram (solid and dashed red curves). (a)--(b) Numerical results with large damping, $\nu=100$, show that there is a delay in the transition between the stable solution branches as $\mu$ varies; the size of this delay depends on the loading rate. (c)--(d) Numerical results with a significantly smaller damping, $\nu=0.01$, show a qualitatively similar delay in the transition. (In this case snap-through is followed by significant underdamped oscillations; for clarity only the start of these are plotted.) The prevalence of this delay with both small and large damping shows that the phenomenology is a feature of the snap-through transition, rather than the presence of damping alone.
• Figure 5: The size of the delay in snap-through for different loading rates, $\dot{d}$, and damping coefficient, $\nu$ (indicated by symbol thickness, with the thinnest having $\nu=10^{-3}$ and each increment representing an increase by a factor of $10$ up to $\nu=10^3$ for the thickest symbols). Numerical results of the fully-nonlinear problem with different inclination angles, $\alpha$ (indicated by the symbol shape), are shown. Here the time delay for snap-through, $T_\mathrm{snap}$, is defined as the time after $\mu=\mu_\mathrm{fold}$ at which the midpoint of the arch reaches its maximum (see the label in fig. \ref{['fig:RawTrajectories']}c). In (a) raw results are shown, while (b) shows that the duration of snap-through is not controlled by the damping $\nu$ alone. (Note that in (a) the $x$-axis shows $|\dot{d}|$, while in (b) $|\dot{\mu}|=\dot{d}/\alpha^2$ is used.)