= Centred slip-line fan at a traction-free V-notch
For a symmetric reentrant notch opening towards negative $x_1$, let its semi-angle be $0\leq\gamma\leq\pi/2$. On the upper free face, tensile tangential stress selects $p=-k$, $\phi=\pi/4-\gamma$. The adjacent centred fan has radial $\beta$ lines and circular $\alpha$ lines: $\phi=\vartheta-\pi/2$, $p=-k+2k[\vartheta-(3\pi/4-\gamma)]$, for $\pi/4\leq\vartheta\leq3\pi/4-\gamma$. It joins a constant stress region containing the forward axis, where $\phi=-\pi/4$ and $p=-k(1+\pi-2\gamma)$. Therefore $\sigma_{22}=k(2+\pi-2\gamma)$ in that fully plastic notch-tip region. This is a local stress field, not a claim that an arbitrary specimen is plastically yielded everywhere.
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