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Sandbox Physics
Geometric elasticity · A course with experiments

L1 · Chapter 1

Why a thin sheet bends instead of stretching

Stretching strains every layer of a strip by the same amount, so its cost grows in proportion to the thickness. Bending strains each layer in proportion to its distance from the middle, so its cost grows as the cube of the thickness. The ratio of the two, the stretchability S=h2/12L2S=h^{2}/12L^{2}, is about one ten-millionth for a sheet of paper. A thin structure therefore takes any shape that lets it avoid stretching, and that one preference drives everything else in this course.

Sandbox Physics ·

A strip and its two ways to deform

Take a strip of length LL, width bb and thickness hh, made of a material with Young’s modulus EE. Put xx along the strip and zz across its thickness, with z=0z=0 on the middle surface. A strain is a change of length divided by the length; it has no unit.

Pull the strip so that every layer lengthens by the same strain ε\varepsilon. Hooke’s law gives a stress EεE\varepsilon on the cross-section hbhb, so the strip acts as a spring of stiffness Ehb/LEhb/L. Its stored energy is the spring energy below, and it grows in proportion to hh.

Us=12 Ehb L ε2U_{\mathrm{s}}=\tfrac{1}{2}\,Ehb\,L\,\varepsilon^{2}

Why bending costs the cube of the thickness

Now bend the strip into an arc of radius ρ\rho, or curvature κ=1/ρ\kappa=1/\rho, without changing the length of its middle surface. A layer at height zz lies on an arc of radius ρ+z\rho+z, so its strain is z/ρ=zκz/\rho=z\kappa: the outer layers stretch, the inner ones shorten, and the middle keeps its length.

The energy per unit volume of a layer is 12E(zκ)2\tfrac12E(z\kappa)^{2}. Adding the layers from −h/2-h/2 to h/2h/2 brings in ∫z2 dz=h3/12\int z^{2}\,dz=h^{3}/12, which defines the bending stiffness BB. A plate held from contracting sideways has B=Eh3/12(1−ν2)B=Eh^{3}/12(1-\nu^{2}) with Poisson ratio ν\nu, as Pandey et al. use for a shell; it still grows as the cube of hh.

Ub=12 B L κ2,B=Eb∫−h/2h/2z2 dz=Ebh312U_{\mathrm{b}}=\tfrac{1}{2}\,B\,L\,\kappa^{2},\qquad B=Eb\int_{-h/2}^{h/2}z^{2}\,dz=\frac{Ebh^{3}}{12}

One number compares them: the stretchability

Ask which stretching strain costs as much as bending the whole strip into an arc of radius LL. Setting 12EhbLε2\tfrac12EhbL\varepsilon^{2} equal to 12B/L\tfrac12B/L gives ε2=h2/12L2\varepsilon^{2}=h^{2}/12L^{2}. Pandey et al. (2014) call this ratio the stretchability SS. For paper 0.1 mm thick and 100 mm long it is 8.3×10−88.3\times10^{-8}: a strain of only 0.03 % already costs as much as bending the whole sheet into an arc of radius 100 mm.

Measured in units of B/LB/L, the energy of a strip whose ends are pushed together by a fraction dd of its length has a bending part and a stretching part weighted by 1/S1/S. The bracket is the strain left in the middle surface after the extra length taken up by the slope w′w' of the deflection w(x)w(x); the next chapter derives it. Efrati, Sharon and Kupferman (2009) write the same split for plates: stretching times hh plus bending times h3h^{3}.

UB/L=12∫−1/21/2w′′2 dx+12S(d−12∫−1/21/2w′2 dx)2,S=h212L2\frac{U}{B/L}=\frac12\int_{-1/2}^{1/2}w''^{2}\,dx+\frac{1}{2S}\Bigl(d-\frac12\int_{-1/2}^{1/2}w'^{2}\,dx\Bigr)^{2},\qquad S=\frac{h^{2}}{12L^{2}}

Worked example

Pushing the ends of the Lab strip together

The deep-arch preset of the arch Lab has h/L=0.03h/L=0.03, so S=0.032/12=7.5×10−5S=0.03^{2}/12=7.5\times10^{-5}, and its ends are pushed together by d=0.0225d=0.0225, 2.25 % of the length. Compare two ways to take up that shortening. In each, decide which term of the energy survives.

Staying straight keeps w=0w=0, so only the stretching term remains: d2/2S=0.02252/(1.5×10−4)=3.375d^{2}/2S=0.0225^{2}/(1.5\times10^{-4})=3.375 in units of B/LB/L. Bending without any stretch needs a shape whose slope uses up the whole shortening. The clamped shape w=12w0(1+cos⁡2πx)w=\tfrac12w_{0}(1+\cos2\pi x) has 12∫w′2dx=π2w02/4\tfrac12\int w'^{2}dx=\pi^{2}w_{0}^{2}/4 and 12∫w′′2dx=π4w02\tfrac12\int w''^{2}dx=\pi^{4}w_{0}^{2}; setting the first equal to dd gives a bending energy of 4π2d=0.8884\pi^{2}d=0.888.

Bending is 3.80 times cheaper. The two energies are equal when d=8π2Sd=8\pi^{2}S, a shortening of 5.9×10−35.9\times10^{-3} for this strip, and of only 6.6×10−66.6\times10^{-6}, about 0.66 µm over 100 mm, for the sheet of paper. That is why a sheet buckles at the slightest push from its ends.

Ustraight=d22S=3.375,Ubent=4π2d=0.888(units of B/L)U_{\text{straight}}=\frac{d^{2}}{2S}=3.375,\qquad U_{\text{bent}}=4\pi^{2}d=0.888\qquad\Bigl(\text{units of }B/L\Bigr)

The same choice in a sheet on water and in a growing disc

A sheet floating on water and pushed from its edges cannot shorten either, so it wrinkles. Its bending competes with the weight of the liquid it lifts, which sets the wrinkle wavelength 2π(B/ρg)1/42\pi(B/\rho g)^{1/4}; Diamant and Witten (2011) solved exactly how those wrinkles gather into a single fold.

A disc that grows more at its centre than at its rim faces the same choice in two dimensions. A thick disc pays by stretching and stays flat; a thin one bends into a dome. Chapters 5 and 6 work this out with the disc Lab, where the thickness slider moves the disc from one choice to the other.

An experiment to think through

Predict, change, explain

  1. In the arch Lab the deep-arch preset has h/L=0.03h/L=0.03 and a confinement d/S=300d/S=300. Predict d/Sd/S if you double the thickness at the same end shortening, and whether the strip still has to bend much to take up the shortening.
  2. Choose “Deep arch twists, then snaps”, watch one push, then set “Thickness over length” to 0.06 and watch again.
  3. Explain the new confinement readout and the shallower arch using S=h2/12L2S=h^{2}/12L^{2} and the two energies of the worked example.
Press a clamped arch until it snaps Grow a flat disc into a dome

The arch Lab solves the clamped beam equation with midline stretching on 48 segments, pushed at the middle by a blade that can only press. 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. A plastic ruler is 300 mm long and 1 mm thick. What is its stretchability?

Read an explained answer

S=(1/300)2/12=9.3×10−7S=(1/300)^{2}/12=9.3\times10^{-7}. A strain of only S≈0.1 %\sqrt{S}\approx0.1\,\% costs as much as bending the ruler into an arc of radius 300 mm.

2. Halve the thickness of a strip. By what factor do its stretching stiffness, its bending stiffness and its stretchability change?

Read an explained answer

The stretching stiffness Ehb/LEhb/L halves, the bending stiffness Ebh3/12Ebh^{3}/12 falls eightfold, and SS falls fourfold. A thinner strip is softer in both ways, but much softer in bending.

3. For the shallow-arch preset, h/L=0.06h/L=0.06 and d=0.018d=0.018. Which is cheaper: staying straight or bending without stretch?

Read an explained answer

S=3×10−4S=3\times10^{-4}, so staying straight costs d2/2S=0.54d^{2}/2S=0.54 and bending without stretch costs 4π2d=0.7114\pi^{2}d=0.711. Straight wins this comparison, yet the Lab strip still buckles: a mixture of some compression and some bending costs only 0.477. The next chapter finds that mixture.

Sources to read alongside

  1. 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.

  2. 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.

  3. Diamant & Witten, Compression induced folding of a sheet: an integrable system, Physical Review Letters 107, 164302 (2011), arXiv:1107.5505v1

    Read in full: the wrinkle wavelength of a sheet on liquid, the exact fold shapes (Eq. 12), the exactly quadratic fall of the pressure as the sheet is pushed in, and the point where the symmetric fold touches itself, at a pressure of about 0.040 in their units.

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