Antiplane shear has displacement normal to a plane and independent of that normal coordinate. In an isotropic linear elastic body, and , together with their symmetric partners, are the only nonzero stress tensor entries. Static balance without body force reduces to the Laplace equation for .
For subsonic motion , an antiplane shear displacement satisfies . The displayed rescaling makes it a harmonic function of , so locally for a holomorphic function . Set . The Cauchy-Riemann equations then give . Keeping this potential normalization explicit avoids losing a factor of in the dynamic stress field. At sonic or supersonic speeds the elliptic harmonic reduction does not apply.
A harmonic conjugate satisfying and makes a holomorphic function. Its derivative packages the two antiplane shear stresses into the displayed complex field. On multiply connected domains a global potential additionally requires the appropriate periods to vanish; prescribed displacement jumps can instead be represented by branches of the potential.
For a crack on the real axis under applied shear traction , write for the additional complex stress potential for antiplane shear derivative. Reflection of the induced displacement gives . Traction-free crack faces therefore require ; intact portions have . This is a scalar Riemann-Hilbert problem. Endpoint regularity, single-valued displacement and the absence of additional remote loads fix its homogeneous terms.
For the moving semi-infinite crack, take the branch cut of on the negative real axis and ahead of the tip. With steadily moving antiplane shear, face tractions imply . The antisymmetric displacement response gives . Hence has jump . The Sokhotski–Plemelj theorem gives the displayed Cauchy integral, assuming convergence and no extra remote homogeneous loading. For it is purely imaginary, so . At fixed co-moving loading this ahead-of-tip stress is independent of speed; the displacement still has the factor .
For real , this analytic homogeneous field has real boundary values on both faces of the negative-axis crack, so it adds no face shear traction. Ahead of the tip it adds stress . Its stress tends to zero at infinity, but its primitive grows like , producing unbounded remote displacement and infinite total elastic energy. Thus merely requiring vanishing far stress does not uniquely fix the face-loaded crack solution. Excluding an added remote stress-intensity field, for example through the localized-load bounded-displacement far condition, removes this homogeneous ambiguity. Ordinary inverse-square-root tip stress remains allowed and has finite energy in any bounded neighbourhood of the tip.
For two semi-infinite cracks separated by a ligament, choose on the ligament. Its upper crack values are on the right and on the left. Multiplication by converts the Hilbert problem for an antiplane crack to an additive jump; the Sokhotski–Plemelj theorem gives the displayed solution, assuming convergence. A homogeneous term transfers an extra resultant through the ligament and has logarithmic displacement at infinity. The usual correction with no additional remote resultant excludes it. The far-field normalization is essential to uniqueness.
Choose at infinity, cut on the finite crack. Its upper crack value is , so has jump . The Sokhotski–Plemelj theorem gives the displayed Cauchy integral solution. It decays as for integrable weighted traction. A possible homogeneous term would give a logarithmic displacement period; it is removed for a dislocation-free finite crack with no extra remote loading.
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Antiplane shear refers to a specific type of shear deformation in a material where the displacement occurs perpendicular to the plane of interest. In this context, "antiplane" indicates that the shear strain is considered in a direction that is perpendicular to the principal plane of stress or definition of the problem. In three-dimensional elasticity, problems can often be simplified by focusing on one specific type of deformation.