The completeness of Maxwell's stress function representation

Consideration is given to the completeness of Maxwell's solution of the equations of equilibrium of the linear theory of elasticity in which the second order symmetric tensor field representing the stress in the reference configuration of an elastic body is set equal to the curl of the curl of a matrix of three arbitrary smooth functions. The Maxwell solution is obtained from Beltrami's solution, in which the self-equilibrated stress field is equal to the curl of a curl of a symmetric tensor field, and it is shown that the previously demonstrated completeness of Beltrami's solution in the case of smooth (four times continuously differentiable) equilibrated stress fields implies the completeness of the Maxwell solution. The Maxwell representation is also found to be valid for some only twice differentiable stress fields if an extended definition of the curl curl operator is applied.