Geometric elasticity · Snap-through

Arch Snap-Through

Why does a buckled strip jump to its mirror shape, and what sets how fast it goes? A thin strip clamped at both ends and pushed together buckles into an arch with two stable shapes, arched up and arched down. Push its middle down and the force first rises, then falls. Once holding the strip would take a pull, no balanced shape is left nearby, so it jumps to the inverted arch. A deep arch twists into an S shape before it jumps; a shallow one is pressed flat first.

Push the middle of a buckled strip clamped at both ends. The force rises, peaks and falls; when the blade would have to pull, the strip snaps to the inverted arch and rings. The force curve is set against the continuous Föppl–von Kármán solution, and the snap and the ringing against linear stability.

Largest blade force on the strip $16.5\,B/L^{2}$

Growth rate of the snap

\lambda

, measured $21.1/t^{*}$

Ringing angular frequency

\omega

after the snap, measured $43/t^{*}$

Confinement: end shortening over stretchability $d/S=300$

Model assumptions

The strip follows the dynamic beam equation with midline stretching of Pandey, Moulton, Vella and Holmes (EPL, 2014), in their units: lengths in the strip length, forces in the bending stiffness over the length squared, and time in the bending time of the strip. It is 48 finite-difference segments with clamped, flat ends and a fixed end shortening. The blade is a frictionless point contact that can only push. The push is quasi-static: each frame is the equilibrium with the midpoint at the blade, and when the blade would have to pull, the strip is released into the dynamic equation. The uniform viscous damping is not in the paper; it stands in for air and material losses. Gravity, twist, the width of the strip and plasticity are left out. Deflections on the canvas are stretched vertically by the factor shown there.

Orange is the push from above and blue the push from below, both computed on the strip; cream dashes are the continuous solution and the predicted growth.

More numbers and the saved record

Largest force of the continuous solution

16.6\,B/L^{2}

Largest force of the push from below

16.5\,B/L^{2}

Growth rate from linear stability, with damping

21.2/t^{*}

Antisymmetric vibration of the arch, with damping

44.1/t^{*}

Symmetric vibration of the arch, with damping

59.5/t^{*}

Rise of the unloaded arch

0.0891\,L

Midpoint deflection when the strip leaves the blade

-2.19\times10^{-4}\,L

Relative energy drift of a free vibration without damping

5.05\times10^{-6}

Time step of the dynamic equation

1.29\times10^{-4}\,t^{*}

The JSON record holds the parameters, both simulated pushes with their release, snap and ringing, and, kept apart, the continuous force curve, the linear-stability rates and the published reference values.

Start with the essentials

Focus question: Why does a buckled strip jump to its mirror shape, and what sets how fast it goes?
One-sentence intuition: Under displacement control the strip follows a branch of balanced shapes until that branch stops being stable. There the nearest stable shape is the inverted arch, and the strip falls into it at a rate set by the unstable mode of the last shape it held.

Core mathematical model

Dynamic beam with midline stretching

\frac{\partial^{2}w}{\partial t^{2}}+\gamma\,\frac{\partial w}{\partial t}+\frac{\partial^{4}w}{\partial x^{4}}+\tau^{2}\,\frac{\partial^{2}w}{\partial x^{2}}=f\,\delta(x),\qquad -\tfrac{1}{2}<x<\tfrac{1}{2}

Pandey et al. (2014), Eq. 1, with the damping $\gamma$ added here. Lengths are in $L$ , forces in $B/L^{2}$ and time in $t^{*}=L^{2}(\rho_{\mathrm{s}}hb/B)^{1/2}$ . The clamps hold $w=0$ and $\partial w/\partial x=0$ at both ends.

Compression from the end shortening

S\,\tau^{2}=d-\frac{1}{2}\int_{-1/2}^{1/2}\Bigl(\frac{\partial w}{\partial x}\Bigr)^{2}dx,\qquad S=\frac{h^{2}}{12L^{2}}

Hooke's law for the midline (Pandey et al., Eq. 2) integrated along the strip. The stretchability $S$ compares bending with stretching; the ratio $d/S$ alone decides the shape of the response.

Three regimes of the confinement

\Bigl(\frac{d}{S}\Bigr)_{\mathrm{Euler}}=4\pi^{2}\approx 39.48,\qquad \Bigl(\frac{d}{S}\Bigr)_{\mathrm{twist}}=\tau_{\max}^{2}\approx 80.76

Below the Euler value the strip stays straight. Between the two values the arch is pressed flat before it snaps; above the second it twists into an S shape first. $\tau_{\max}\approx 8.99$ solves $\tan(\tau/2)=\tau/2$ and is the load of the first antisymmetric clamped mode.

Force on the twisted branch

f=-207.75\,w_{0},\qquad f_{\mathrm{c}}=-129.53\,\bigl(d-80.76\,S\bigr)^{1/2}

Pandey et al., Eq. 3: once the strip twists, the force falls in proportion to the midpoint deflection and reaches zero when the midpoint does. Solving the same equations again here gives 207.745 and 129.530.

How fast the snap grows

\lambda=-\frac{\gamma}{2}+\sqrt{\frac{\gamma^{2}}{4}+\sigma^{2}},\qquad \sigma\approx 24.11\ \text{from the twisted state}

A small departure $a\,e^{\lambda t}$ from the last shape the blade held obeys $\ddot{a}+\gamma\dot{a}=\sigma^{2}a$ . Pandey et al. find $\sigma\approx 24.113$ for the twisted state; this strip of 48 segments gives 24.04.

Common difficulties

The largest force is not where it snaps

Typical misconceptionThe strip snaps when the blade pushes hardest.

Better mental modelWith the blade position controlled, the strip passes the force peak and keeps a balanced shape. It snaps only where the force falls to zero, because beyond that point the blade would have to pull. Under a fixed dead load the jump would instead start at the force peak.

The speed is not just the wave speed

Typical misconceptionA snap is as fast as sound crossing the strip.

Better mental modelThe time scale is the bending time of the strip, and the rate within it comes from the unstable mode of the last balanced shape. Close to the buckling threshold that mode is weak, so the jump is slow.

What the model leaves out

Typical misconceptionA real plastic strip behaves exactly like this one.

Better mental modelThe strip follows the dynamic beam equation with midline stretching of Pandey, Moulton, Vella and Holmes (EPL, 2014), in their units: lengths in the strip length, forces in the bending stiffness over the length squared, and time in the bending time of the strip. It is 48 finite-difference segments with clamped, flat ends and a fixed end shortening. The blade is a frictionless point contact that can only push. The push is quasi-static: each frame is the equilibrium with the midpoint at the blade, and when the blade would have to pull, the strip is released into the dynamic equation. The uniform viscous damping is not in the paper; it stands in for air and material losses. Gravity, twist, the width of the strip and plasticity are left out. Deflections on the canvas are stretched vertically by the factor shown there.

Run the experiment

01
Watch a deep arch twist
Start from Deep arch twists, then snaps and watch the first push.
What to observe: The force rises to about 16.5, then the arch leans into an S shape and the force falls in a straight line to zero as the midpoint reaches the clamps' level. The strip then snaps, with a growth rate of 21.1 against 21.2 from linear stability.
02
Make the arch shallow
Choose Shallow arch flattens, then snaps, where the confinement is 60.
What to observe: The arch stays symmetric and is pressed flat. The peak force is only 1.95 and the snap starts from the flat strip, growing at 13.9 against 14.0 predicted.
03
Get close to buckling
Choose Barely buckled: a slow snap, with the confinement at 45, just above the Euler value of about 39.5.
What to observe: The jump grows at only 7.18, a third of the deep arch. Near the threshold the flat state is barely unstable, so the strip leaves it slowly.
04
Let it ring
Choose Little damping: it rings.
What to observe: After the jump the inverted arch rings for many cycles. The measured angular frequency, 46.1, sits close to the 44.2 of the first antisymmetric vibration mode of the arch.
05
Push it yourself
Drag the top blade down slowly, then drag the lower blade up quickly.
What to observe: Your push draws its own yellow force curve over the computed one. Whatever the speed, the strip leaves the blade and jumps once the force reaches zero.

Arch Snap-Through

How does a buckled strip snap through?

Core mathematical model

Dynamic beam with midline stretching

Compression from the end shortening

Three regimes of the confinement

Force on the twisted branch

How fast the snap grows

Common difficulties

The largest force is not where it snaps

The speed is not just the wave speed

What the model leaves out

Run the experiment

Watch a deep arch twist

Make the arch shallow

Get close to buckling

Let it ring

Push it yourself

Related experiments and articles

Geometric elasticity: how thin things bend, buckle and snap