Some Properties of the Boundary Value Problem of Linear Elasticity in Terms of Stresses

The relation between the classical formulation of linear elastic problems in displacements and the stress formulation proposed by Pobedria is studied. It is shown that if the Navier and Pobedria differential operators are elliptic then corresponding boundary value problems are equivalent. The values of parameters for which Pobedria's boundary value problem has the Fredholm property are found. The homogeneous Pobedria's system is considered as a spectral problem with Poisson's ratio as a spectral parameter. The points of the essential spectrum are found and classified. The example of solving Pobedria's system for the Lamé problem for a spherical shell is presented.