Compatibility of Deformations and the Thrice Differentiability of the Displacement Field

The question of the necessary class of smoothness of solutions to quasi-static problems in the mechanics of a deformable solid in terms of displacements is discussed. It is shown that in order for the equations of compatibility of deformations to become identities when substituting displacements into them, the existence of some third mixed derivatives of displacements is required. For a linearly elastic compressible isotropic elastic medium, a counterexample is given in which the displacement field, being a doubly differentiable solution to the boundary value problem for the system of Lamé equations in the entire domain, is not a solution to the problem in displacements at all points of this domain.