Generalized Cesaro formulas and third order compatibility equations

We consider the classical problem of elasticity theory concerning the conditions of compatibility deformations, which ensure the determination of a continuous field of displacements of an elastic body by the deformation field. We construct generalized Cesaro representations that allow one to define the displacement field through integro-differential operators on the components of the strain tensor deviator with an accuracy up to quadratic polynomials. It has been established that the quadratures both for the pseudovector of local rotations and for the volume change deformation are completely determined by the deformation deviator field. We present the conditions for the existence of the listed quadratures, which are written in the form of five third differential order coincidence equations with respect to the five components of the strain tensor-deviator.