Existence of Solutions of Plane Traction Problems for Ideal Composites

The theory of plane deformations of ideal fiber-reinforced composites involves hyperbolic equations, but boundary data are specified as in elliptic problems. When the surface tractions are given at every boundary point of a plane region, and thus given at two points on each characteristic, it is not obvious that the problem is well-set. We show that under the usual global equilibrium conditions on prescribed tractions, a solution does exist. This is done by reducing the problem to an integral equation whose kernel depends on the shape of the region, locating the spectrum of eigenvalues, and then invoking standard results of the Hilbert-Schmidt theor