Geometric elasticity · Growth and shape

Non-Euclidean Plate

Why does a disc that grows into a curved geometry stay flat when thick but turn into a dome when thin? Growth can ask a sheet for a curved geometry that no flat sheet can have, such as rings that grow too short for their radius. A thick disc stays flat and carries the mismatch as stretching. Below a threshold thickness bending is cheaper, so the disc buckles into a dome with almost no stretching left. The thinner the disc, the closer that dome comes to a spherical cap.

Growth can give a sheet a target geometry that no flat sheet has, here that of a sphere. Grow the disc and watch: a thick one stays flat and stretches, a thin one buckles into a dome that is nearly free of stretching. The buckling thickness, the energies and the dome height are set against the stretch-free cap and the published threshold.

Thickness below which the flat disc buckles $h_{\mathrm{B}}=0.299$

Share of the energy in stretching, grown disc $7.24\,\%$

Dome height over the stretch-free cap $0.999$

Growth

K

at which this disc leaves the plane $K_{\mathrm{B}}=0.0186$

Model assumptions

The disc is the axisymmetric non-Euclidean plate of Efrati, Sharon and Kupferman (J. Mech. Phys. Solids, 2009): an annulus of geodesic radii 0.1 to 1.1 whose target metric has constant Gaussian curvature K, with the reduced energy of their Eq. 4.2 and the prefactor set to one, as in their Fig. 4.1. It is 80 radial elements, and each frame of the growth is an equilibrium found by Newton's method. Only shapes with rotational symmetry are allowed, so wrinkles, dimples and saddles of negative curvature are out of reach, and growth is imposed rather than computed. The probe is a stiff spring on the rise of the inner rim; after it lets go, the disc relaxes by overdamped gradient flow, a path to the nearest equilibrium rather than the motion of a real sheet with inertia.

Orange is stretching and blue bending, green their sum, all computed on the disc; cream dashes are the flat disc and the stretch-free cap.

More numbers and the saved record

Stretching content of the flat disc

0.0143

Bending content of the stretch-free cap

0.541

Crossover thickness where the two are equal

0.163

Buckling thickness over the square root of K

0.299

Elastic energy of the grown disc

2.69\times10^{-5}

Height of the grown dome

0.541

Height of the stretch-free cap

0.541

Radial elements

80

The JSON record holds the parameters, every equilibrium of the growth and of the thickness sweep, the buckling thresholds, and, kept apart, the flat-disc and stretch-free limits and the published threshold.

Physics tutorial

How does growth make a flat disc curve?

BackgroundLeaves, flowers and gel sheets that swell unevenly are given a geometry by their growth rather than a shape. When that geometry is curved, no flat sheet can have it, and the sheet must either stretch or bend away from the plane.

Why it mattersEvery element of the disc is told the same lengths whatever its thickness. Yet a thick disc stays flat and a thin one becomes a dome: thickness decides only which of the two costs, stretching or bending, is cheaper to pay.

Start with the essentials

Focus question: Why does a disc that grows into a curved geometry stay flat when thick but turn into a dome when thin?
One-sentence intuition: Stretching costs energy in proportion to the thickness and bending in proportion to its cube. A thick disc pays for the curved geometry by stretching in the plane; once the disc is thinner than a threshold, bending into a near-isometric dome is cheaper, and the stretching left over falls much faster than the bending as the disc thins.

Core mathematical model

The target geometry

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

Lengths along a radius are kept, but a ring at geodesic radius $r$ should have circumference $2\pi\Phi(r)$ , shorter than $2\pi r$ . That is the metric of a sphere of radius $1/\sqrt{K}$ (Efrati et al., Sec. 4).

Stretching plus bending

E=\int_{r_{0}}^{r_{1}}\bigl(h\,w_{S}+h^{3}\,w_{B}\bigr)\,\Phi\,dr,\qquad w_{S}=\epsilon_{1}^{2}+\epsilon_{2}^{2}+2\nu\,\epsilon_{1}\epsilon_{2}

Efrati et al., Eq. 4.2, for a surface of revolution with radius $\phi(r)$ and slope $\zeta$ . The strains $\epsilon_{1}=\phi'^{2}(1+\zeta^{2})-1$ and $\epsilon_{2}=\phi^{2}/\Phi^{2}-1$ measure the mismatch with the target lengths; $w_{B}$ is the bending density. The disc here is the annulus $0.1\le r\le 1.1$ .

The stretch-free cap

\phi=\Phi,\qquad H_{0}=\frac{\cos\sqrt{K}r_{0}-\cos\sqrt{K}r_{1}}{\sqrt{K}},\qquad E_{B0}=\tfrac{2}{3}(1+\nu)\bigl(\cos\sqrt{K}r_{0}-\cos\sqrt{K}r_{1}\bigr)

A zone of the sphere has exactly the target lengths, so its stretching is zero. Its height is $H_{0}$ , and its bending content $E_{B0}$ follows because the bending density is $\tfrac{2}{3}(1+\nu)K$ everywhere on it. For $K=1$ and $\nu=0.5$ , $H_{0}=0.541$ and $E_{B0}=0.541$ .

Two thicknesses

h_{C}=\sqrt{E_{S}^{\infty}/E_{B0}}\approx 0.163,\qquad h_{B}\approx 0.299

Setting the flat disc's stretching $h\,E_{S}^{\infty}$ equal to the cap's bending $h^{3}E_{B0}$ gives the crossover $h_{C}$ . Linear stability of the flat disc gives the buckling thickness $h_{B}$ ; Efrati et al. report about 0.3 and note that $h_{C}$ is smaller.

Common difficulties

Thin does not mean weak here

Typical misconceptionA thin disc buckles because it is too weak to hold its shape.

Better mental modelEvery thickness is asked for the same curved lengths. Thickness only sets the price of bending against stretching, which grows as the thickness squared, so below a threshold the disc chooses to bend.

No equal sharing

Typical misconceptionAt equilibrium stretching and bending share the energy about equally.

Better mental modelIn the energy chart, as the thickness falls from 0.037 to 0.01, the stretching per thickness falls 27-fold, from $4.1\times10^{-5}$ to $1.5\times10^{-6}$ , while the bending part falls only 12-fold, from $5.5\times10^{-4}$ to $4.7\times10^{-5}$ . Efrati et al. call this lack of equipartition surprising; it is why very thin sheets behave as if they cannot stretch.

What the model leaves out

Typical misconceptionReal leaves and gel discs look exactly like this.

Better mental modelThe disc is the axisymmetric non-Euclidean plate of Efrati, Sharon and Kupferman (J. Mech. Phys. Solids, 2009): an annulus of geodesic radii 0.1 to 1.1 whose target metric has constant Gaussian curvature K, with the reduced energy of their Eq. 4.2 and the prefactor set to one, as in their Fig. 4.1. It is 80 radial elements, and each frame of the growth is an equilibrium found by Newton's method. Only shapes with rotational symmetry are allowed, so wrinkles, dimples and saddles of negative curvature are out of reach, and growth is imposed rather than computed. The probe is a stiff spring on the rise of the inner rim; after it lets go, the disc relaxes by overdamped gradient flow, a path to the nearest equilibrium rather than the motion of a real sheet with inertia.

Run the experiment

01
Grow a thin disc
Start from Thin disc grows into a cap and watch one growth cycle.
What to observe: The disc leaves the plane almost at once, at a target curvature of 0.0186, and ends as a dome 0.999 as high as the stretch-free cap. Only 7.2 % of its energy is stretching, and the color stays dark.
02
Make it thick
Choose Thick disc stays flat, where the thickness is 0.45.
What to observe: The disc grows all the way to K = 1 without leaving the plane and glows with stretching. Its thickness is above the buckling thickness of 0.299, so the flat state stays stable.
03
Sit just past the threshold
Choose Just past the buckling threshold, with thickness 0.26.
What to observe: The disc stays flat until K = 0.760, then rises gently. At K = 1 the dome is only 0.577 of the cap and stretching still holds 59.7 % of the energy.
04
Turn it inside out
Choose Cap pops inside out.
What to observe: The probe pushes the inner rim down. The energy climbs to about six times that of the cap, then drops by half at a dome height of about −0.21 as the outer part flips. Let go, and the disc relaxes into the mirror cap with the same energy, $8.48\times10^{-5}$ .
05
Push the rim yourself
Drag the yellow ring at the inner rim down through the disc, then let go.
What to observe: Let go early and the dome springs back; push past the snap and it settles inside out. Your path is drawn in blue on the energy chart.

Non-Euclidean Plate

How does growth make a flat disc curve?

Core mathematical model

The target geometry

Stretching plus bending

The stretch-free cap

Two thicknesses

Common difficulties

Thin does not mean weak here

No equal sharing

What the model leaves out

Run the experiment

Grow a thin disc

Make it thick

Sit just past the threshold

Turn it inside out

Push the rim yourself

Related experiments and articles

Geometric elasticity: how thin things bend, buckle and snap