= Solution
For <antiplane shear>, the infinitesimal displacement is $(0,0,w(x_1,x_2))$. The isotropic <linear elasticity> constitutive law gives $\sigma_{13}=\mu w_{,1}$ and $\sigma_{23}=\mu w_{,2}$, together with the symmetric entries $\sigma_{31},\sigma_{32}$. All other <stress tensor> components vanish. Static force balance without <body force> becomes $\mu(w_{,11}+w_{,22})=0$, so $w$ satisfies the <Laplace equation>.
Locally choose a <harmonic conjugate> $\phi$ by $\phi_{,1}=w_{,2}$ and $\phi_{,2}=-w_{,1}$. Compatibility follows from the <Laplace equation>. These are the <Cauchy-Riemann equations> for $F=\phi+iw$, and hence
$$
F'(z)=\phi_{,1}+iw_{,1}=w_{,2}+iw_{,1},\qquad \boxed{\sigma_{23}+i\sigma_{13}=\mu F'(z).}
$$
Thus the <complex stress potential for antiplane shear> supplies both stresses and, by integration, the displacement. A global potential must have the displacement periods appropriate to the cracked body; these periods also help fix homogeneous solutions.
For the additional crack-induced field, set $G=\mu F'$ and $p(t)=\sigma_{23}^A(t,0)$. Only $p$ loads the prospective faces, since $\sigma_{13}^A$ is tangential to their normal. Denote upper and lower boundary values by $G^+,G^-$. The imposed cancellation of face <tractions> is $\operatorname{Re}G^\pm=-p$. The traction data and geometry are invariant under the displacement reflection $w(x_1,x_2)\mapsto-w(x_1,-x_2)$. For the induced correction with no independently added loading, uniqueness up to a rigid translation allows the odd displacement choice. Then $w_{,2}$ is even, $w_{,1}$ is odd, and $G^-=\overline{G^+}$ on the real axis. Therefore the <Hilbert problem for an antiplane crack> is
$$
\boxed{G^++G^-=-2p\text{ on cracks},\qquad G^+-G^-=0\text{ on intact portions}.}
$$
The second condition expresses continuation through uncracked material. There $G$ is real and the additional $\sigma_{13}$ vanishes. This scalar <Riemann-Hilbert problem> is supplemented by at most inverse-square-root stress singularities at the crack tips and the usual correction with no additional remote resultant or imposed dislocation. Without a far-field or period normalization, the face conditions alone permit homogeneous terms. The formulas below select this normalized induced field and explicitly identify the excluded terms.
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