= Solution
For antiplane <displacement> $u=(0,0,w)$ the only nonzero <strain> components are $e_{xz}=w_x/2$ and $e_{yz}=w_y/2$. Commutation of <mixed partial derivatives> gives the necessary local compatibility condition
$$
\boxed{\partial_ye_{xz}-\partial_xe_{yz}=0.}
$$
The supplied <stress> is symmetric, independent of $z$ and trace-free. Its first two <equilibrium> equations are automatic; its third gives
$$
\partial_x\left(-\frac{\mu y}{x^2+y^2}\right)+
\partial_y\left(\frac{\mu x}{x^2+y^2}\right)=0
$$
away from the origin. <Antiplane shear> has zero trace, so isotropic <Hooke's law> reduces to $\boxed{e=\tau/(2\mu)}$. Hence
$$
\boxed{w_x=-\frac y{x^2+y^2},\qquad w_y=\frac x{x^2+y^2}.}
$$
Locally this integrates to $w=\arg(x+iy)+C$, represented on a suitable branch by $\arctan(y/x)+C$. The <strain> condition is satisfied at every point of the <punctured plane>, but the <displacement> cannot be made globally single-valued there:
$$
\boxed{\oint_{r=R}(w_x\,dx+w_y\,dy)=2\pi.}
$$
A <gradient> of a globally single-valued smooth <displacement> would instead integrate to zero around any <closed curve>. Local compatibility is therefore not sufficient on this <multiply connected region>.
A physical construction is a Volterra <screw dislocation>: cut the cylinder along a radial half-plane ending on its axis, shift one cut face relative to the other parallel to the axis, and rejoin. The axial mismatch is the <Burgers vector>. In the given normalization its magnitude is $2\pi$; with a dimensional mismatch $b$ one uses $w=b\arg(x+iy)/(2\pi)$. An intact pristine solid cannot reach this state by a smooth single-valued elastic deformation alone. The axis is a <dislocation core> where the continuum elastic field is singular and must be regularized by microscopic structure or <plastic deformation>. On the punctured domain the differential condition holds locally; globally its nonzero <circulation> detects the defect. In a distributional description, that incompatibility is concentrated on the axis. This is <local strain compatibility versus screw-dislocation circulation>.
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