Three-dimensional asymptotic finite element method for anisotropic inhomogeneous and laminated plates

An asymptotic finite element model for anisotropic inhomogeneous and laminated plates is developed within the framework of three-dimensional elasticity. The formulation begins with a Hellinger-Reissner functional in which the displacements and transverse stresses are taken to be the functions subject to variation. By means of asymptotic expansion the H-R functional for the problem is decomposed into functionals of various orders from which the asymptotic finite element equations are derived. In the multilevel computations the transverse stresses and displacements may be interpolated independently, and the midplane displacements are the only unknown nodal degree-of-freedoms (DOF) in the system equations, thus the total DOF at each level is less than that of a homogeneous Kirchhoff plate. The stiffness matrix remains unchanged ; the one generated at the leading-order level is always used at subsequent levels. The formulation is three-dimensional yet requires only two-dimensional finite element discretization with no need of interpolation in the thickness direction. The through-thickness effect can be accounted for in a consistent and hierarchical manner. Numerical comparisons with the benchmark solutions show that the method is effective in modeling of multilayered composite plates.