Symplectic solutions for orthotropic micropolar plane stress problem

Abstract

The symplectic approach was utilized to derive solutions to the orthotropic micropolar plane stress problem. The Hamiltonian canonical equation was first obtained by applying Legendre’s transformation and the Hamiltonian mixed energy variational principle. Then, by using the method of separation of variables, the eigenproblem of the corresponding homogeneous Hamiltonian canonical equation was derived. Subsequently, the corresponding eigensolutions for three kinds of homogeneous boundary conditions were derived. According to the adjoint symplectic orthogonality of the eigensolutions and expansion theorems, the solutions to this plane stress problem were expressed as a series expansion of these eigensolutions. The numerical results for the orthotropic micropolar plane stress problem under various boundary conditions were presented and validated using the finite element method, which confirmed the convergence and accuracy of the proposed approach. We also investigated the relationship between the size-dependent behaviour and material parameters using the proposed approach. Furthermore, this approach was applied to analyze lattice structures under an equivalent micropolar continuum approximation.