Demonstrating Invalidity of Solution Methods Involved in Solving Compatibility Equations

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.