Stress characteristics for an anisotropic yield curve ($d(\sigma -l)=0\text{on }\alpha ,d(\sigma +l)=0\text{on }\beta$)

Parameterize a fixed yield surface by stress-space arc length $l$ with ${\xi }_{,l}=-\cos 2\varphi$ and ${\tau }_{,l}=-\sin 2\varphi$. For mean normal stress $\sigma$, force balance gives $\nabla \sigma =M\nabla l$, where $M=\left(\begin{array}{cc}\cos 2\varphi & \sin 2\varphi \\ \sin 2\varphi & -\cos 2\varphi \end{array}\right)$. Its eigenvectors along angles $\varphi$ and $\varphi +\pi /2$ have eigenvalues $+1$ and $-1$, proving the displayed slip line invariants.