= Length-preserving directions and principal directions of simple shear
{title2=$F=I+\gamma e_1\otimes e_2,\quad N_2(2N_1+\gamma N_2)=0$}
For $\gamma>0$, direct multiplication gives $|FN|^2-|N|^2=\gamma N_2(2N_1+\gamma N_2)$. Thus length-preserving line directions form the union of two planes in direction space: $N_2=0$ and $2N_1+\gamma N_2=0$. Their intersection is the third coordinate axis. This concerns segment length rather than pointwise fixed lines; a segment can move and turn while keeping its length.
The two nontrivial <principal stretches> are $v_\pm=(\sqrt{4+\gamma^2}\pm\gamma)/2$, with $v_+v_-=1$. The maximizing reference direction is proportional to $(1,v_+,0)$. Consequently it tends to $(1,1,0)$ as $\gamma$ tends to zero from above. At zero itself all stretches equal one, so every direction maximizes length; the limit selects a direction from a degenerate endpoint. This follows from the largest eigenvector of the <right Cauchy-Green deformation tensor>.
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