Representations of ℂ, biharmonic vector fields, and the equilibrium equation of planar elasticity

Abstract

The equilibrium equation for an elastic body subjected to surface forces asserts the linear dependence of the Laplacian and the gradient of the divergence of the vector field which gives the displacement at each point. James Clerk Maxwell (1831–1879) was the first to point out that the component functions of such a field are biharmonic, i.e., their Laplacians are harmonic functions. Using only algebraic tools familiar to advanced undergraduates we show that the usual complex variable representation of two-variable biharmonic functions falls naturally out of a power series construction based on matrix representations of ℂ. Under the assumption of linear stress and strain components, this construction is then used to describe the solutions to the planar equilibrium equation in terms of the geometry of the Moebius plane.