On the positive definiteness of the Poincaré–Steklov operator for elastic half-plane

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 displacements of the half-plane as a rigid body. 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.