= Solution
Use orthonormal cylindrical bases in the <reference configuration> and <current configuration>. The <deformation gradient> maps a radial increment $dR$ to $r'dR$, an azimuthal arc $R\,d\Theta$ to $r\,d\Theta$, and an axial increment $dZ$ to itself. Therefore
$$
\boxed{F=r' e_r\otimes E_R+\frac rR e_\theta\otimes E_\Theta+e_z\otimes E_Z.}
$$
Differentiating the radius relation gives $r'=R/r$. Its three <principal stretches> are
$$
\boxed{\lambda_r=\frac Rr,\qquad\lambda_\theta=\frac rR,\qquad\lambda_z=1.}
$$
Their product is one, as required by <elastic incompressibility>. The symbols called “strains” in the source are these stretch ratios; engineering <strain> would instead subtract one. The axial stretch is one because this is <plane strain>.
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