= Incompressible inextensible-fibre plane strain
{title2=$F=R(\theta)\begin{pmatrix}1&\gamma\\0&1\end{pmatrix}$}
Let $e=(\cos\theta,\sin\theta)$ be the current fibre direction and $n=(-\sin\theta,\cos\theta)$. <Inextensibility> fixes the first column of the <deformation gradient> to $e$, and <incompressibility> fixes the second to $\gamma e+n$. Commuting mixed derivatives gives $\gamma_{,1}=\theta_{,1}$ and $\theta_{,2}=\gamma\theta_{,1}$. Thus $\gamma=\theta+f(X_2)$, and curves tangent to $(-\gamma,1)$ map to straight transverse curves with constant $\theta$.
Back to article page
