Geometric elasticity · A course with experiments

L2 · Chapter 5

Growth that no flat sheet can hold

Growth that varies across a sheet tells each small piece how long to be, not where to go. Suppose it makes the rings of a disc shorter than $2\pi r$ while keeping the radii. Those are the lengths on a sphere, and Gauss showed that the curvature of a surface can be read from lengths alone. A plane has none, so a flat disc cannot have these lengths. It must stretch in its plane, or bend out of it into a piece of sphere, which has the lengths exactly. The Lab’s disc, asked for $K=1$ , is offered a cap of height 0.541.

Sandbox Physics · 2026-10-30

Growth sets lengths, not a shape

Efrati, Sharon and Kupferman (2009) start from a disc of gel that shrinks more at its rim than at its centre. Each piece shrinks the same way through the thickness, so the gel has no reason to bend up or down. What it does acquire is a new set of preferred distances between neighbouring points: a target metric.

If some surface in space has those distances, the sheet can take that shape with no strain in its midplane. If none does, every shape the sheet can take is strained somewhere. Efrati et al. call such a sheet a non-Euclidean plate, and they write its elastic energy as a stretching term, measured against the target metric, plus a bending term.

Reading curvature from circles

Take a disc where distances along a radius are unchanged but the ring at radius $r$ should have circumference $2\pi\Phi(r)$ . The target metric is then $d\bar{s}^{2}=dr^{2}+\Phi^{2}d\theta^{2}$ , Eq. 4.1 of Efrati et al., and its Gaussian curvature is $K=-\Phi''/\Phi$ . On a flat sheet $\Phi=r$ and $K=0$ .

The Lab chooses $\Phi=\sin(\sqrt{K}\,r)/\sqrt{K}$ , which gives the same $K$ everywhere. For small $r$ its rings fall short of $2\pi r$ by a fraction $Kr^{2}/6$ . A ruler and a tape measure on the sheet are enough to find $K$ .

d\bar{s}^{2}=dr^{2}+\Phi(r)^{2}\,d\theta^{2},\qquad K=-\frac{\Phi''}{\Phi},\qquad \Phi=\frac{\sin(\sqrt{K}\,r)}{\sqrt{K}}\approx r\Bigl(1-\frac{Kr^{2}}{6}\Bigr)

Why no flat sheet can comply

Gauss proved that bending a surface without stretching it never changes its Gaussian curvature, because that curvature is fixed by lengths measured within the surface. A flat sheet has $K=0$ , and no amount of bending without stretching can give it $K=1$ . Lying flat, it must stretch.

The sphere of radius $1/\sqrt{K}$ does have the target lengths: the ring at distance $r$ from the pole, measured along the surface, has radius $\sin(\sqrt{K}\,r)/\sqrt{K}$ . So the disc can avoid stretching by becoming a zone of that sphere, a cap. That cap is still bent, so it still stores energy: Efrati et al. stress that a non-Euclidean plate has no state free of stress. Its mirror image, the cap turned inside out, has the same lengths and the same bending, so it is an equally good choice.

A curved sheet resists being flattened

Once the sheet has curved, the same argument makes it stiff. Push a spherical shell of radius $R$ inward by $w$ and every ring gets shorter by a fraction $w/R$ . That is stretching, and it costs energy in proportion to $Eh\,(w/R)^{2}$ , far more than bending a flat plate by the same amount.

Pandey et al. (2014) use exactly this extra term in their Eq. 9 for vibrating spherical caps, the toy poppers that leap off a table. In the Lab, turning the cap inside out first raises its energy to about six times that of the cap, because every shape in between misses the target lengths.

\rho_{s}h\,\frac{\partial^{2}w}{\partial t^{2}}+B\,\nabla^{4}w+\frac{Eh}{R^{2}}\,w=0

Worked example

The Lab’s disc at full growth

The Lab’s disc is the ring $0.1\le r\le1.1$ , asked for $K=1$ . Its outer rim should be $2\pi\sin1.1=5.600$ long; a flat rim at radius 1.1 is $2\pi\times1.1=6.912$ . The target is 19.0 % short, close to the small-curvature estimate $Kr^{2}/6=20.2\,\%$ . Its target area, $2\pi(\cos0.1-\cos1.1)=3.402$ , is 9.8 % less than the flat ring’s 3.770.

The stretch-free option is the zone of the unit sphere between polar distances 0.1 and 1.1. Its height is $H_{0}=\cos0.1-\cos1.1=0.541$ . In the Lab, the thin preset ends as a dome 0.999 as high as this cap. For Efrati et al.’s gel, $K=0.11\ \mathrm{cm^{-2}}$ is the curvature of a sphere of radius $1/\sqrt{0.11}=3.02\ \mathrm{cm}$ .

\frac{2\pi\sin1.1}{2\pi\times1.1}=\frac{5.600}{6.912}=0.810,\qquad H_{0}=\cos0.1-\cos1.1=0.541

An experiment to think through

Predict, change, explain

Before playing “Thin disc grows into a cap”: will the disc rise to the full cap height $H_{0}=0.541$ , or stay well below it? And will the inside-out cap have more energy, less, or the same?
In the non-Euclidean plate Lab play “Thin disc grows into a cap”, then “Cap pops inside out”. Compare the final dome heights and the energy readouts before and after the flip.
Use Gauss’s result and the sphere’s ring radius $\sin(\sqrt{K}\,r)/\sqrt{K}$ to explain why the thin disc ends almost exactly on the cap, and why the mirror cap costs the same energy.

Grow a flat disc into a dome

The disc Lab finds equilibrium shapes of the axisymmetric energy of Efrati, Sharon and Kupferman on 80 radial elements as growth is imposed.

Try explaining it yourself

1. On Efrati et al.’s gel, $K=0.11\ \mathrm{cm^{-2}}$ . How long is the ring 2 cm from the centre, measured along the sheet?

Read an explained answer

$2\pi\sin(0.3317\times2)/0.3317=11.66\ \mathrm{cm}$ , against $4\pi=12.57\ \mathrm{cm}$ on a flat sheet: 7.2 % short. The estimate $Kr^{2}/6=7.3\,\%$ is close because $\sqrt{K}\,r=0.66$ is still small.

2. The preset “Gentle growth, shallow cap” asks for $K=0.3$ . What is the stretch-free cap height, and how short is the rim?

Read an explained answer

$H_{0}=(\cos0.0548-\cos0.6025)/0.5477=0.319$ . The rim target is $2\pi\sin(0.6025)/0.5477=6.501$ against 6.912, 5.9 % short, so this disc needs a much smaller rise.

3. A sheet of paper rolls into a cylinder without stretching, but cannot wrap a ball. Explain both with Gaussian curvature.

Read an explained answer

A cylinder curves in one direction only, so its Gaussian curvature, the product of the two principal curvatures, is zero, the same as the flat sheet. A ball has $K=1/R^{2}>0$ , and Gauss’s theorem forbids reaching it without changing lengths, so the paper must crumple or tear.

Sources to read alongside

Pandey, Moulton, Vella & Holmes, Dynamics of snapping beams and jumping poppers, EPL 105, 24001 (2014), arXiv:1310.3703v2
Read in full, with the supplementary information: the dynamic beam equation and the integrated Hooke law (Eqs. 1 and 2), the three ranges of d/S and the force on the twisted branch (Eq. 3), the linear stability problem (Eqs. 4 and 5), the growth rate 24.113 and the near-threshold formulas (Eqs. 6 and 7), the ringing frequencies 42.38 and 44.36, the popper equation (Eq. 9) and Figs. 1 to 5.
Efrati, Sharon & Kupferman, Elastic theory of unconstrained non-Euclidean plates, Journal of the Mechanics and Physics of Solids 57, 762 (2009), arXiv:0810.2411v1
Read in full: the definition of a non-Euclidean plate and why it cannot be both flat and strain-free (Sec. 3), the energy as stretching times h plus bending times h cubed, the axisymmetric reference metric and energy (Eqs. 4.1 and 4.2), the buckling thickness of about 0.3, the smaller crossover thickness and the missing equipartition (Sec. 4.2, Figs. 4.1 and 4.2), and the gel discs shaped by uneven shrinking to a Gaussian curvature of 0.11 per square centimetre.

Growth sets lengths, not a shape

Reading curvature from circles

Why no flat sheet can comply

A curved sheet resists being flattened

The Lab’s disc at full growth

Predict, change, explain

Try explaining it yourself

1. On Efrati et al.’s gel, K=0.11 cm−2K=0.11\ \mathrm{cm^{-2}}K=0.11 cm−2. How long is the ring 2 cm from the centre, measured along the sheet?

2. The preset “Gentle growth, shallow cap” asks for K=0.3K=0.3K=0.3. What is the stretch-free cap height, and how short is the rim?

3. A sheet of paper rolls into a cylinder without stretching, but cannot wrap a ball. Explain both with Gaussian curvature.

Sources to read alongside

1. On Efrati et al.’s gel, $K=0.11\ \mathrm{cm^{-2}}$ . How long is the ring 2 cm from the centre, measured along the sheet?

2. The preset “Gentle growth, shallow cap” asks for $K=0.3$ . What is the stretch-free cap height, and how short is the rim?