An asymptotic analysis of anisotropic heterogeneous plates with consideration of end effects

Abstract

A finite element-based asymptotic analysis tool is developed for general anisotropic plates. The formulation begins with three-dimensional equilibrium equations in which the thickness coordinate is scaled by the characteristic length of the plate. This allows us to split the equations into two parts, such as the one-dimensional microscopic equations and the two-dimensional macroscopic equations, via the virtual work concept. The one-dimensional microscopic analysis yields the through-the-thickness warping function at each level, which does not require two-dimensional macroscopic analysis. The two-dimensional macroscopic equations provide the governing equations of the plate at each level in a recursive form. These can be solved in an orderly manner, in which proper macroscopic boundary conditions should be incorporated. The displacement prescribed boundary condition is obtained by introducing the orthogonality condition of asymptotic displacements to the plate fundamental solutions. In this way, the end effects of the plate are kinematically corrected without applying the sophisticated decay analysis method. The developed asymptotic analysis method is applied to semiinfinite plates with simply supported and clamped-free boundary conditions. The results obtained are compared to those of three-dimensional FEM, three-dimensional elasticity, and Reissner-Mindlin plate theory. The usefulness of the present method is also discussed.

Analytical Asymptotic Solutions for Rectangular Laminated Composite Plates

“…Because of the asymptotic convergence (Kim, 2009;Kim et al, 2008), the zeroth order displacements and each order displacements are assumed by…”

“…The order of magnitude of body forces and applied forces is presupposed by following the definitions given by Buannic and Cartraud (2001) and Kim (2009).…”

“…(8), a fundamental solution which has an analytical form, since (Kim et al, 2008), can be calculated. The detailed procedure is given by Kim (2009). In the present formal asymptotic method, the problems are divided into two parts: macroscopic and microscopic problems.…”

“…Yu et al (2002) and Yu (2005) used the variational asymptotic method to derive a Reissner-Mindlin-like plate theory using through-the-thickness finite element analysis. Meanwhile, the decay method or the averaged boundary conditions were used to determine the interior solution for displacement prescribed boundaries (Buannic and Cartraud, 2001;Duva and Simmonds, 1992;Kim, 2009;Kim et al, 2008). The generalized averaged boundary conditions were successively employed to generic anisotropic beams, which demand much less efforts than the decay analysis method.…”

“…Furthermore, this analysis prompted researchers to consider the three-dimensional composite coupling characteristics accurately. Recently, Kim (2009) developed the formal asymptotic method-based plate analysis, called the FAMPA, while accounting for the averaged boundary conditions. The solutions calculated were accurate in both simply-supported and clamped boundary conditions with reasonable computational efforts.…”