Equivalence between Cauchy-Riemann & planar elastostatic equation
Wednesday 18 May, 2022at 15:15 - 16:00
Abstract: In many engineering fields, such as e.g. the study of machine elements and the design of bridges, it is vital to have a clear sense of the distribution of internal forces in the structure at hand. If the domain of interest is planar, such as the cross-section of a beam or a cog, one commonly resorts to the field of planar elasticity theory. In particular, if the problem is time-independent we use the elastostatic equation whose solution yields the deformation and thereby the stress distribution in the domain.
The equation is most commonly solved using numerical methods. However, due to this method breaking down in concave geometries, it is vital to have some exact solutions to use for e.g. benchmarking. Kolosov and Mushkelisvhili first popularized these at the turn of the last century, where they used complex potential methods to construct solutions. We shall follow this work spiritually and prove that the elasticity equation is equivalent to the inhomogeneous Cauchy-Riemann equation. With this equivalence in place, we can use well-known solutions for the complex problem to construct physical solutions.