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Yet a fundamental tool in these fields is precisely the notion of a holomorphic function, which can represent many classical flows in the plane: uniform flow, of course, but also sources and sinks (corresponding to flow in the plane with one point removed), a point vortex, and flow around a wedge. Formulas due to the German Heinrich Blasius (1883–1970) can be used to calculate the overall mechanical loads (resultant force and moment) exerted on the surface of an obstacle immersed in an incompressible ideal fluid.
Fracture mechanics
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When a solid material is subjected to loads, it deforms and eventually fails. Some materials break without significant apparent deformation, such as glass and ceramics, whereas others bend considerably before fracturing, including copper and steels. Conformal maps are immensely useful for studying this phenomenon.
Fracture mechanics developed considerably during the second half of the 20th century, particularly after a series of accidents caused by construction defects that went undetected in a range of structures, from ships such as the Liberty ships of the Second World War to aircraft such as the Comet. In two dimensions, elasticity problems can be solved using complex analysis to find Airy stress functions for a branched or unbranched crack. Solutions can also be constructed beyond the planar case, for circular cracks, for example, and even for loads applied to the faces of more intricately shaped cracks.
