Symplectic Solution for a Plane Couple Stress Problem
Based on the Pro-Hellinger-Reissner variational principle of plane couple stress problem, the dual PDEs are proposed corresponding to the force method extension. The duality solution methodology is thus extended to plane couple stress problem, and then the method of separation of variables and eigenfunction expansion in the symplectic space is used to find the analytical solutions. A long strip domain plate with both lateral edges free, fixed at one end and under simple tension at the far end, is solved analytically. The solution is composed of the inhomogeneous boundary condition solutions and the superposition of eigensolutions of homogeneous lateral boundary conditions. The method of separation of variables is used for the dual PDEs, from which the eigen-root transcendental equation is solved and the corresponding eigenvector functions are obtained. The boundary conditions at the fixed end are derived via the variational method. Numerical results show that due to the effect of couple stress, the stress distribution is no longer infinity as given by the classical theory of elasticity at the corner of fixed end.
Tong Yi-Zhang and Jang-Kyo Kim
W. X. Zhong et al., "Symplectic Solution for a Plane Couple Stress Problem", Advanced Materials Research, Vol. 9, pp. 143-152, 2005