= Uniaxial sheet stretching stiffness
{title2=$Y_{\rm 2D}=Eh$}
The \b[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> $h$,
$$
\boxed{Y_{\rm 2D}=Eh,\qquad N=Y_{\rm 2D}\varepsilon.}
$$
Here $N=F/w$ for tensile <force> $F$ and sheet width $w$. Since <Hooke's law> gives $\sigma=E\varepsilon$ and $F=\sigma wh$, division by $w$ gives the result. $Y_{\rm 2D}$ has units $\mathrm{N/m}$, while $E$ has units $\mathrm{N/m^2}$.<Development of Constitutive Equations for Continuum, Beams and Plates by Tomasz Wierzbicki>[§4.1, Equation4.1 and Figure4.1: uniaxial stress and lateral contraction]{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 $Y_{\rm 2D}=100000\,\mathrm{N/m}$. At strain $\varepsilon=0.001$, $N=100\,\mathrm{N/m}$, $F=1\,\mathrm N$ and extension $\delta=\varepsilon L=50\,\mu\mathrm m$. The specimen's axial stiffness is $F/\delta=Y_{\rm 2D}w/L=20000\,\mathrm{N/m}$, which depends on its width and gauge length.
A tensile measurement can infer $Y_{\rm 2D}$ 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>.<Development of Constitutive Equations for Continuum, Beams and Plates by Tomasz Wierzbicki>[Equation4.16 and §4.2: plane-stress constitutive relations]{ref} The loading condition is part of the meaning of the reported stiffness.
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