Plane Couple Stress Symplectic Solution Method

Plane couple stress problem appeared at the beginning of last century, but it has not been well analytically solved before. There are mainly some numerical methods such as FEM etc. at present, and there are only few analytic solutions for some special problems. Symplectic method has been applied in some plate and beam bending problems successfully. Compared with classical method, it has some unique advantages. This paper aims to apply this analytic method in 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 (which are induced from that the stress functions are selected as primary unknowns) 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. The superposition of these eigensolutions gives the stress distribution at the fixed end. 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.Results improve that in micro scale the couple stress effect should not be neglect, and symplectic method is effective and advantageous to handle with plane couple stress problem in rectangle field.