= Solution
Put $a=\sigma_{13}$, $b=\sigma_{23}$, $w=v_3$, and write $f_a=\partial f/\partial a$, $f_b=\partial f/\partial b$. In <antiplane perfect plasticity>, <force balance> and the fixed <yield surface> give
$$
a_{,1}+b_{,2}=0,\qquad f(a,b)=k.
$$
Differentiate the second relation in $x_2$. Along the proposed <characteristic curve>,
$$
\frac{da}{ds}
=\alpha(-f_ba_{,1}+f_aa_{,2})
=\alpha(f_bb_{,2}+f_aa_{,2})=0.
$$
Similarly, differentiating the yield relation in $x_1$ gives
$$
\frac{db}{ds}
=\alpha(-f_bb_{,1}+f_ab_{,2})
=-\alpha(f_bb_{,1}+f_aa_{,1})=0.
$$
The <associated flow rule> and $\dot\epsilon_{i3}=w_{,i}/2$ give $w_{,1}=2\dot\lambda f_a$, $w_{,2}=2\dot\lambda f_b$. Consequently
$$
\frac{dw}{ds}
=2\alpha\dot\lambda(-f_bf_a+f_af_b)=0.
$$
Thus \b[both shear <stresses> and the antiplane <velocity> are constant on each characteristic]. Assume a regular yield gradient $(f_a,f_b)\ne0$. Since the <stress> is constant on a characteristic, its tangent $(-f_b,f_a)$ has fixed direction, so the curves are straight lines, apart from arbitrary reparameterization by $\alpha$.
In a <centred antiplane plastic fan> those lines emanate from the crack tip. Hence
$$
a=a(\phi),\qquad b=b(\phi),\qquad w=w(\phi),
$$
and a radial line at angle $\phi$ from the $x_1$ axis must have
$$
(\cos\phi,\sin\phi)\parallel(-f_b,f_a),\qquad
f_a\cos\phi+f_b\sin\phi=0.
$$
The crack faces have normal in the $x_2$ direction, so traction freedom requires $b=0$ there. The fan joins the adjacent constant crack-face <stress> region along the characteristic carrying that <stress>. Its boundary line therefore satisfies
$$
\boxed{\tan\phi_b=-\frac{f_a}{f_b}\quad\text{at }b=0,}
$$
with the direction interpreted directly when the denominator vanishes. The original PDF reverses this ratio. Its displayed ratio is $dx_1/dx_2$, whereas the tangent of an angle from the $x_1$ axis is $dx_2/dx_1$. The distinction cannot be removed while retaining both its stated characteristic direction and its angle convention.
For the elliptic <yield surface>, with $A,B>0$,
$$
f_a=2a/A^2,\qquad f_b=2b/B^2.
$$
On the forward fan $-\pi/2<\phi<\pi/2$, choose the loading branch $b>0$. The radial tangent condition gives $a/A^2=-(b/B^2)\tan\phi$. Substituting in the yield equation and choosing the positive square root gives exactly
$$
\boxed{a=\frac{-A^2\tan\phi}{(B^2+A^2\tan^2\phi)^{1/2}},\qquad
b=\frac{B^2}{(B^2+A^2\tan^2\phi)^{1/2}}.}
$$
For the endpoints, use the equivalent nonsingular expressions
$$
a=-\frac{A^2\sin\phi}{H(\phi)},\qquad
b=\frac{B^2\cos\phi}{H(\phi)},\qquad
H(\phi)=(A^2\sin^2\phi+B^2\cos^2\phi)^{1/2}.
$$
They give $(a,b)=(-A,0)$ on the upper boundary ray and $(A,0)$ on the lower one. The correct boundaries are \b[$\phi=\pm\pi/2$], the vertical line through the tip. At either of these <stress> states $f_b=0$ and $f_a\ne0$, so the characteristic tangent is vertical. The printed boundary ratio would instead give a horizontal line, providing a concrete counterexample to that intermediate assertion.
Extend the upper and lower rear sectors by the constant <stresses> $(-A,0)$ and $(A,0)$ respectively; these satisfy the traction-free crack faces and match the fan <stresses> continuously. The reversed loading has all <stresses> negated. To check equilibrium within the forward fan, differentiation gives
$$
a_{,\phi}=-\frac{A^2B^2\cos\phi}{H^3},\qquad
b_{,\phi}=-\frac{A^2B^2\sin\phi}{H^3},
$$
and hence $a_{,1}+b_{,2}=(-\sin\phi\,a_{,\phi}+\cos\phi\,b_{,\phi})/r=0$. The yield equation is satisfied identically as well.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2005/iii/paper-78-plastic-fan.png]
{title=Radial characteristics and vertical boundaries of the elliptic antiplane fan}
Finally $\nabla w=w'(\phi)e_\phi/r$, while $(f_a,f_b)=2e_\phi/H$. The <associated flow rule> reduces to
$$
\boxed{\dot\lambda=\frac{H(\phi)w'(\phi)}{4r},\qquad w'(\phi)\geq0}
$$
for this <stress> branch. Any such angular <velocity> field is constant along the rays and has constant values in the adjacent rear sectors. Its magnitude and angular variation require <velocity> or loading data beyond the specified traction-free faces; the local <stress> fan alone does not determine them.
Back to article page
