= Compatible nonlinear shear energy
{title2=$\tau(q)=\mu q+hq^3$}
An explicit objective <isotropic> example is $\mathcal W=\mu(S-1)/2+h(S-1)^2/4+(\lambda+\mu)(J-1)^2/2$, with $J=\det A$, $S=\operatorname{tr}(A^{\mathsf T}A)-2J$, and $\lambda,\mu>0$, $h\ge0$. On $A=[(a,q,0)^{\mathsf T},e_2,e_3]$, $S-1=(a-1)^2+q^2$ and $J=a$, so $P_{11}=0$ at $a=1$ while the shear <nominal stress> is the displayed cubic. The natural-state quadratic energy has <Lamé parameters> $\lambda,\mu$, and the shear speed is $\sqrt{(\mu+3hq^2)/\rho_0}$.
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