  In the theory of elasticity, the constraint of compatibility conditions on
displacement field is not equivalent to the property of displacement field. The
difference may broaden the possibility of solutions to elasticity problems,
which means that the solution methods involved in solving the compatibility
equations like stress-based solution method and stress function method may be
invalid. The study argues the validity of the solution methods involved in
solving the compatibility equations. It is concluded that the absence of
constraints on local rigid body rotation causes the non-uniqueness of
displacement fields for a rectangular beam purely bent and a cantilever loaded
at the end.
