The intent of this expository lecture is to draw attention to an interesting two-dimensional mathematical model arising in solid mechanics involving a single second-order linear or quasilinear partial differential equation. This model has the virtue of relative mathematical simplicity without loss of essential physical relevance. Anti-plane shear deformations are one of the simplest classes of deformations that solids can undergo. In anti-plane shear (or longitudinal shear, generalized shear) of a cylindrical body, the displacement is parallel to the generators of the cylinder and is independent of the axial coordinate. This is the Mode III fracture mode for crack problems. In recent years, considerable attention has been paid to the analysis of anti-shear deformations within the context of various constitutive theories (linear and nonlinear) of solid mechanics. Such studies were largely motivated by the promise of relative analytic simplicity compared with plane problems since the governing equations are a single second-order linear or quasilinear partial differential equation rather than higher order or coupled systems of partial differential equations. Thus the anti-plane shear problem plays a useful role as a pilot problem, within which various aspects of solutions in solid mechanics may be examined in a particularly simple setting. In this lecture, recent developments on these issues are described for both linear and nonlinear solid mechanics.
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