= Solution
Retain the <material frame indifference> assumption used in part (c), in addition to <material isotropy>. Put $p=u_{,1}$, $q=u_{,2}$ and $s=p^2+q^2$. For this <antiplane shear>,
$$
F=\begin{pmatrix}1&0&0\\0&1&0\\p&q&1\end{pmatrix},\qquad
B=\begin{pmatrix}1&0&p\\0&1&q\\p&q&1+s\end{pmatrix},\qquad J=1.
$$
The invariant identities give $I_B=II_B=3+s$ and $III_B=1$. Hence the restriction of the <strain energy density> is
$$
\boxed{W(Dy)=w(|Du|^2),\qquad w(s)=h(3+s,3+s,1).}
$$
<Antiplane energy does not determine the full stress>: the restriction gives one <scalar> energy along antiplane gradients. It does not determine the full <stress> derivative with respect to every entry of $F$.
Indeed, using the <isotropic hyperelastic stress invariant representation> and differentiating the energy, with $h_j$ denoting the corresponding <partial derivatives>, gives
$$
DW=2h_1F+2h_2(I_BI-B)F+2III_Bh_3F^{-T}.
$$
Using $S=J^{-1}TF^T$, the general objective <isotropic> <Cauchy stress tensor> is
$$
\boxed{S=\frac2J\left[h_1B+h_2(I_BB-B^2)+III_Bh_3I\right].}
$$
All <derivatives> are evaluated at the current invariant triple. The extra terms generally remain nonzero on antiplane gradients, so \b[the printed assertion that the full <stress> must be $\beta B$ does not follow].
A stress-free, objective <isotropic> counterexample is the compressible <neo-Hookean solid> $W(F)=\frac\mu2(|F|^2-3)-\mu\log J$, with $\mu>0$. Its derivative is $T=\mu F-\mu F^{-T}$, so on antiplane gradients
$$
S=\mu(B-I).
$$
Take $u(x_1,x_2)=x_1$. Then $B_{22}=1$ but $S_{22}=0$, whereas $B_{13}=1$ and $S_{13}=\mu$. The diagonal entry would force $\beta=0$, contradicting the shear entry. Thus this is not a matter of choosing the right <scalar> coefficient.
For the additional one-invariant constitutive assumption $W(F)=\widehat w(I_B)$, the stronger formula does hold, with
$$
\boxed{\beta(I_B,III_B)=\frac{2\widehat w'(I_B)}{\sqrt{III_B}}.}
$$
This is a valid special case, rather than a consequence of <material isotropy> alone. For the general restricted energy, the shear entries are still fixed: $S_{31}=2w'(s)u_{,1}$ and $S_{32}=2w'(s)u_{,2}$, since $w'=h_1+h_2$ along $(3+s,3+s,1)$. The full <normal stress> needs the constitutive information that the printed claim omits. If <material isotropy> alone is interpreted without <material frame indifference>, even the first reduction to $|Du|^2$ needs that additional <material frame indifference> assumption.
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