An Extension of the Complex Variable Method in Plane Elasticity to Domains with Corners: A Notch Problem

A severe limitation on the complex variable method in plane elasticity is the requirement that the region in which the stresses and strains are sought admits a rational conformal mapping into the half-plane or the unit circle. Of course, this limitation has not prevented the solution of many fundamental and interesting problems. However, the extension of this method to more general regions remains a source of debate. We here present an attempt in this direction. If the conformal mapping of the region is an algebraic function, it is still possible to find mathematical conditions defining the complex-valued stress function, but its explicit representation now requires the solution of a singular, linear, complex-valued, integro-differential equation.