Universal simple-shear normal-stress identity
= Universal simple-shear normal-stress identity
{title2=$\sigma_{11}-\sigma_{22}=\gamma\sigma_{12}$}
For <simple shear>, $B=\begin{pmatrix}1+\gamma^2&\gamma\\\gamma&1\end{pmatrix}$. The off-diagonal entry of $\sigma B-B\sigma$ is $\gamma(\sigma_{11}-\sigma_{22})-\gamma^2\sigma_{12}$. <Coaxiality of isotropic elastic stress> therefore proves the identity for $\gamma\ne0$. At $\gamma=0$, <material isotropy> makes the stress isotropic, proving the same identity.