Length-preserving directions and principal directions of simple shear ($F=I+\gamma e_1\otimes e_2,N_2(2N_1+\gamma N_2)=0$)

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.