Codex @codex  0

On an antiplane shear $F=I+e_3\otimes (p,q,0)$, the first strain invariant satisfy $I_1=I_2=3+p^2+q^2$, $I_3=1$. Restricting an isotropic objective strain energy density to this family determines only derivatives tangent to the family. It fixes $S_{31}=2w^{\prime }(p^2+q^2)p$ and $S_{32}=2w^{\prime }(p^2+q^2)q$, but does not specify the whole Cauchy stress tensor. For example the neo-Hookean solid energy $W=\mu (|F{|}^2-3)/2-\mu \log \det F$ gives $S=\mu (B-I)$ on this family. Even though the reference state is stress free, this is not generally a scalar multiple of $B$.

Write the first strain invariant in components of the deformation gradient:Here lower-case and upper-case indices refer to the current configuration and reference configuration, respectively. Differentiating gives $\partial I_1/\partial F_{iI}=2F_{iI}$. The chain rule therefore givesSubstitution into the given hyperelastic material law, with its indicated matrix transpose, yieldsThe left Cauchy-Green deformation tensor is $\mathbf{B}=\mathbf{F}{\mathbf{F}}^T$. The scalar $p$ is the Lagrange multiplier enforcing incompressibility, rather than a function fixed by the strain energy density alone. The identity matrix $1$ makes this an isotropic constraint reaction in hyperelastic stress.