Antiplane perfect plasticity

In antiplane perfect plasticity, velocity $w=v_3$ and shear stresses $a={\sigma }_{13},b={\sigma }_{23}$ depend only on $x_1,x_2$. Force balance is $a_{,1}+b_{,2}=0$, the fixed yield surface is $f(a,b)=k$, and the associated flow rule is $\nabla w=2\dot{\lambda }(f_a,f_b)$ with $\dot{\lambda }\ge 0$. Differentiating the yield equation in each coordinate and combining it with equilibrium shows that $a,b,w$ are constant along characteristic curves tangent to $(-f_b,f_a)$. For example $-f_b{a}_{,1}+f_a{a}_{,2}=f_b{b}_{,2}+f_a{a}_{,2}=0$. Nonzero regular yield gradient is needed for a characteristic direction.

A centred fan consists of straight characteristic curves emanating from one point. Stress and antiplane velocity are constant on each ray and depend on the polar angle $\varphi$. The radial tangent condition is $(\cos \varphi ,\sin \varphi )\parallel (-f_b,f_a)$, so $f_a\cos \varphi +f_b\sin \varphi =0$. A boundary ray carrying a traction-free crack-face stress $b=0$ has slope $\tan {\varphi }_b=-f_a/f_b$, interpreted as vertical when $f_b=0$. For the elliptic yield condition $a^2/A^2+b^2/B^2=1$, a forward fan has $a=-A^2\sin \varphi /H$, $b=B^2\cos \varphi /H$, $H=(A^2{\sin }^2\varphi +B^2{\cos }^2\varphi {)}^{1/2}$. The boundary rays are vertical; the adjacent rear regions have constant stresses $(\mp A,0)$. Velocity is a function of angle whose magnitude requires further loading data.