= 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 $\dot\lambda\geq0$. 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 $-f_ba_{,1}+f_aa_{,2}=f_bb_{,2}+f_aa_{,2}=0$. Nonzero regular yield gradient is needed for a characteristic direction.
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