= Solution
{c}
The <material frame of a rod> obeys the <director kinematics of a rod>
$$
\mathbf d_i'=\mathbf u\times\mathbf d_i.
$$
In the stress-free <reference configuration>, the <constitutive equation> gives $u_1=0$, $u_2=\hat u$, and $u_3=0$, since all three stiffnesses are positive. Thus $\mathbf u=\hat u\mathbf d_2$. Its <cross product> with $\mathbf d_2$ vanishes, so
$$
\boxed{\mathbf d_2'=\hat u\mathbf d_2\times\mathbf d_2=\mathbf0.}
$$
Consequently this director is a fixed vector in space, even though the other two directors rotate along the <elastic rod>.
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