Uniaxial sheet stretching stiffness ($Y_{2D}=Eh$)

The uniaxial sheet stretching stiffness is the coefficient relating axial force per unit width to axial engineering strain in a homogeneous thin elastic sheet under uniaxial stress with free lateral contraction. With Young's modulus $E$ and thickness $h$,Here $N=F/w$ for tensile force $F$ and sheet width $w$. Since Hooke's law gives $\sigma =E\epsilon$ and $F=\sigma wh$, division by $w$ gives the result. $Y_{2D}$ has units $N/m$, while $E$ has units $N/{\mathrm{m}}^2$.^[ref]

For an original example, choose $E=2\mathrm{GPa}$, $h=50\mu \mathrm{m}$, $w=10\mathrm{mm}$ and gauge length $L=50\mathrm{mm}$. Then $Y_{2D}=100000N/m$. At strain $\epsilon =0.001$, $N=100N/m$, $F=1\mathrm{N}$ and extension $\delta =\epsilon L=50\mu \mathrm{m}$. The specimen's axial stiffness is $F/\delta =Y_{2D}w/L=20000N/m$, which depends on its width and gauge length.

A tensile measurement can infer $Y_{2D}$ from the slope of $F/w$ against $\delta /L$. It needs known dimensions, reasonably uniform loading and freedom for lateral contraction; unloading tests elastic recovery. If lateral strain is instead constrained to zero while through-thickness stress remains zero, the plane stress equations give coefficient $Eh/(1-{\nu }^2)$, with $\nu$ Poisson's ratio.^[ref] The loading condition is part of the meaning of the reported stiffness.