= Solution
{c}
Multiplying the <deformation gradient> by its <matrix transpose> gives the <left Cauchy-Green deformation tensor> and its <trace>:
$$
\mathbf B=\begin{pmatrix}
1&0&0\\
0&1+\tau^2r^2&\tau r\\
0&\tau r&1
\end{pmatrix},\qquad I_1=3+\tau^2r^2.
$$
The derivative of the <strain energy density> is
$$
\hat w'(I_1)=\mu_1+\mu_2 I_1^n.
$$
For brevity put $g(r)=\mu_1+\mu_2(3+\tau^2r^2)^n$. Part (a) gives the <Cauchy stress tensor>, in the current <cylindrical coordinate system> <orthonormal basis>, as
$$
\boxed{\mathbf T=\begin{pmatrix}
2g-p&0&0\\
0&2g(1+\tau^2r^2)-p&2g\tau r\\
0&2g\tau r&2g-p
\end{pmatrix}.}
$$
In particular, the end-face <shear stress> is $T_{\theta z}=2g\tau r$. The undetermined <pressure> $p$ affects the normal <stresses> but does not affect this <shear stress>. No normal <traction> conditions have been specified to determine $p$.
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