Generalized Cesàro Formulas and Third-Order Compatibility Equations

We consider the classical problem of elasticity theory concerning the conditions of strain compatibility, which ensure the determination of a continuous field of displacements of an elastic body by the strain field. We construct generalized Cesàro representations that allow defining the displacement field through integrodifferential 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 bulk strain are completely determined by the strain deviator field. We present the conditions for the existence of the listed quadratures, which are written in the form of five third differential order compatibility equations for the five components of the strain deviator tensor.