On the Positive Definiteness of the Poincaré–Steklov Operator for Elastic Half-Plane

Abstract

The Poincaré–Steklov operator that maps normal stresses to normal displacements on a part of a half-plane boundary is studied. A boundary value problem is formulated to introduce the associated Poincaré–Steklov operator. An integral representation based on the solution to the Flamant problem for an elastic half-plane subjected to a concentrated normal force is given for the operator under consideration. It is found that the properties of the Poincaré–Steklov operator depend on the choice of kinematic conditions specifying the rigid-body displacements of the half-plane. Positive definiteness conditions of the Poincaré–Steklov operator are obtained. It is shown that a suitable scaling of the computational domain can be used to provide the positive definiteness of this operator.