Compatibility condition for pure nonlinear transverse waves (${\partial }_q{P}_{11}(1,q,0)=0$)

For a plane material-coordinate elastic wave with $u=(0,w(X_1,t),0)$, set $q=w_{,X_1}$. The longitudinal equation is $0={\partial }_{X_1}{P}_{11}(1,q,0)$. Arbitrary nonuniform pure transverse profiles thus require the displayed constitutive compatibility, which does not follow from isotropy alone. If it holds and ${\tau }^{\prime }(q)>0$, the shear equation closes as ${\rho }_0{w}_{tt}={\partial }_{X_1}\tau (q)$ with speed $c(q{)}^2={\tau }^{\prime }(q)/{\rho }_0$.

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 $\sqrt{(\mu +3h{q}^2)/{\rho }_0}$.