 Elastic sheet

An 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 $\sigma =E\epsilon$. Multiplying by thickness gives the tensile force per unit width $N=Eh\epsilon$: the uniaxial sheet stretching stiffness is $Eh$. For a homogeneous, isotropic thin elastic plate, its bending stiffness iswhere $E$ is Young's modulus and $\nu$ is Poisson's ratio.^[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.^[ref] The sheet elastocapillary energy ratio compares the resulting capillary and bending energy scales.