= Hencky stress relations
{c}
{title2=$p-2k\phi=\mathrm{constant}\ \text{on }\alpha,\quad p+2k\phi=\mathrm{constant}\ \text{on }\beta$}
In <plane strain> <perfect plasticity> with $\sigma_{11}=-p+k\sin2\phi$, $\sigma_{22}=-p-k\sin2\phi$ and $\sigma_{12}=-k\cos2\phi$, equilibrium gives
$$
\nabla p=2k\begin{pmatrix}\cos2\phi&\sin2\phi\\\sin2\phi&-\cos2\phi\end{pmatrix}\nabla\phi.
$$
The <slip line> tangent directions $(\cos\phi,\sin\phi)$ and $(-\sin\phi,\cos\phi)$ are eigenvectors with eigenvalues $+1,-1$. Taking directional derivatives proves the two <Hencky stress relations>. These sign choices must be changed consistently if the shear-stress convention changes.
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