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 $0\le \gamma \le \pi /2$. On the upper free face, tensile tangential stress selects $p=-k$, $\varphi =\pi /4-\gamma$. The adjacent centred fan has radial $\beta$ lines and circular $\alpha$ lines: $\varphi =\vartheta -\pi /2$, $p=-k+2k[\vartheta -(3\pi /4-\gamma )]$, for $\pi /4\le \vartheta \le 3\pi /4-\gamma$. It joins a constant stress region containing the forward axis, where $\varphi =-\pi /4$ 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.