A Cauchy like problem in plane elasticity

Abstract

Let $\mathrm{\Omega }$ be a bounded domain in the plane, representing an elastic body. Let ${\mathrm{\Gamma }}_0$ be a portion of the boundary $\mathrm{\Gamma }$ of $\mathrm{\Omega }$, ${\mathrm{\Gamma }}_0$ being assumed to be paralled to the $x$-axis. It is proposed to determine the stress field in $\mathrm{\Omega }$ from the displacements and surface stresses given on ${\mathrm{\Gamma }}_0$. Under the assumption of plane stress, it is shown that $u_x+u_y$ is a harmonic function. An Airy stress function is introduced, from which the stress field is computed.