= Elastic sheet
An \b[elastic sheet] is a body whose <thickness> $h$ is small relative to its lateral dimensions and which responds elastically to <deformation>. Its geometry separates in-plane stretching from out-of-plane bending. A thin <elastic plate> is a useful model when the assumptions of <linear elasticity> and small through-thickness <strain> apply.
The different thickness dependences explain much of a sheet's behaviour. For uniform uniaxial loading with free lateral contraction, <Hooke's law> gives $\sigma=E\varepsilon$. Multiplying by thickness gives the tensile <force> per unit width $N=Eh\varepsilon$: the <uniaxial sheet stretching stiffness> is $Eh$. For a homogeneous, isotropic thin <elastic plate>, its <bending stiffness> is
$$
B=\frac{Eh^3}{12(1-\nu^2)},
$$
where $E$ is <Young's modulus> and $\nu$ is <Poisson's ratio>.<Capillary Origami: Spontaneous Wrapping of a Droplet with an Elastic Sheet by Charlotte Py and others>[p.156103-2: isometric bending energy density and plate stiffness preceding Equation1]{ref}
For an original scaling example, halve $h$ while keeping material and lateral dimensions fixed. The uniaxial tensile stiffness halves, but the <bending stiffness> falls to one eighth. Thin sheets can therefore become much easier to bend without becoming equally easy to stretch. In simple bending to radius $R$, a layer a distance $z$ from the middle surface has strain approximately $z/R$; choosing $h=0.1\,\mathrm{mm}$ and $R=10\,\mathrm{mm}$ gives outer-layer strain about $0.005$, or $0.5\%$.
The 2007 <capillary origami> study made this competition observable by placing a <water> <droplet> on cut <polydimethylsiloxane> sheets and following their folding during <evaporation>. Whether a sheet enclosed the drop depended on size, thickness and outline.<Capillary Origami: Spontaneous Wrapping of a Droplet with an Elastic Sheet by Charlotte Py and others>[pp.156103-1–156103-2: preparation and Figure1]{ref} The <sheet elastocapillary energy ratio> compares the resulting capillary and bending <energy> scales.
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