Analytical Solution for the Bending Problem of Micropolar Plates Based on the Symplectic Approach

An analytical solution for the bending problem of micropolar plates is derived based on the symplectic approach. By applying Legendre's transformation, we obtain the Hamiltonian canonical equation for the bending problem of a micropolar plate. Utilizing the method of separation of variables, the homogeneous Hamiltonian canonical equation can be transformed into an eigenvalue problem of the Hamiltonian operator matrix. We derive the eigensolutions of the eigenvalue problem for the simply supported, free, and clamped boundary conditions at the two opposite sides. Based on the adjoint symplectic orthogonal relation of the eigensolutions, the solution of the bending problem of the micropolar plate is expressed as a series expansion of eigensolutions. Numerical results confirm the validity of the present approach for the bending problem of micropolar plates under various boundary conditions and demonstrate the capability of the proposed approach to capture the size-dependent behavior of micropolar plates.