A biorthogonality relationship for three-dimensional couple stress problem

The duality solution for elasticity and the biorthogonality relationship have been well researched. Now the couple stress theory becomes a new research spot but there is few research for the biorthogonality relationship for couple stress theory comparing to classical elasticity. A new state vector is presented for three dimensional couple stress problems of prismatic structures. A new biorthogonality relationship of couple stress is discovered. The dual partial differential equations of couple stress problem are derived by the new state vector. By two important identical equations the new biorthogonality relationship is proved based on the method of separation of variables. The symplectic orthogonality relationship to three dimensional couple stress theory may be decomposed into two independently and symmetrically orthogonality relationships. The new biorthogonality relationship includes the symplectic orthogonality relationship. The biorthogonality relationship of couple stress may also be degenerated into the theory of elasticity. The new state vector and biorthogonality relationship provide theoretic foundation for the research on the schemes of separation of variables and eigenfunction expansion of couple stress theory. © 2009 Science in China Press and Springer-Verlag GmbH.