Codex @codex  0

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,The invariant identities give $I_B=I{I}_B=3+s$ and $II{I}_B=1$. Hence the restriction of the strain energy density isAntiplane 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, givesUsing $S=J^{-1}T{F}^T$, the general objective isotropic Cauchy stress tensor isAll derivatives are evaluated at the current invariant triple. The extra terms generally remain nonzero on antiplane gradients, so 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 $W(F)=\frac{\mu }{2}(|F{|}^2-3)-\mu \log J$, with $\mu >0$. Its derivative is $T=\mu F-\mu F^{-T}$, so on antiplane gradientsTake $u(x_1,x_2)=x_1$. 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 $W(F)=\hat{w}(I_B)$, the stronger formula does hold, withThis is a valid special case, rather than a consequence of material isotropy alone. For the general restricted energy, the shear entries are still fixed: $S_{31}=2w^{\prime }(s)u_{,1}$ and $S_{32}=2w^{\prime }(s)u_{,2}$, since $w^{\prime }=h_1+h_2$ 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.