Compatible nonlinear shear energy ($\tau (q)=\mu q+h{q}^3$)

An explicit objective isotropic example is $\mathcal{W}=\mu (S-1)/2+h(S-1{)}^2/4+(\lambda +\mu )(J-1{)}^2/2$, with $J=\det A$, $S=\mathrm{tr}(A^{\mathsf{T}}A)-2J$, and $\lambda ,\mu >0$, $h\ge 0$. 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 $\sqrt{(\mu +3h{q}^2)/{\rho }_0}$.