Refined modeling of composite plates with in‐plane heterogeneity

Abstract

This article is concerned with the mechanical behavior of heterogeneous composite plates with elastic moduli and layered geometries varying through the thickness direction and periodically along the in-plane directions. By using the concept of decomposition of the rotation tensor, we first formulate the three-dimensional elastic problem in an intrinsic form for application to the geometrically nonlinear problem. The variational asymptotic method is then exercised, leading to simultaneous homogenization and dimensional reduction to construct an effective and simple model suitable for plates with in-plane periodicity. In particular, having obtained the refinement terms up to the second order, we develop a refined plate model, namely a generalized Reissner-Mindlin model that is capable of capturing the transverse shear deformations. In order to deal with realistic and complex plate geometries and constituent materials efficiently and conveniently, the proposed model is implemented into a single unified formulation suitable for incorporation into a commercial analysis package. Numerical results computed herein for a few examples are compared to similar results available in the literature to demonstrate the application and performance of the present model.

Nonlinear homogenization of heterogeneous periodic plates of Reissner-Mindlin type

“…Another way for solving this type of problem is firstly to propose a two-scale expression of the potential energy (Lewiński, 1991;Lee et al, 2014). But this problem is more complex that the initial scale problem, and so to the author's knowledge there exist two possibilities for solving it in a tractable way.…”

“…The first one is the formal asymptotic expansion method used by Lewiński (1991). The second one is the variational asymptotic method proposed by Sutyrin (1997), Berdichevskii (1979) and Lee et al (2014). Anyway it will seem to be difficult to extend this approach in nonlinear setting.…”

“…The method of homogenization proposed by Lee et al (2014) is deduced from the introduction of the double scale asympytotic expansion method into a new double scale variational formulation. The developments given by this method are valid for the macro-scale Reissner--Mindlin behavior and restricted to small deformations and large rotations and displacements.…”

On the homogenization of a nonlinear shell

“…In this vein, Larsson and Landervik [47] propose a new model for homogenization of a heterogeneous shell based on a second-order expansion in thickness of the current position of the shell and finally a simplified assumption on the initial geometry of the shell. The method of homogenization valid for periodic heterogeneous plates, proposed by Lee et al [48], gives interesting results but it seems to be restricted to small deformations and large rotations and displacements. As a consequence, this approach seems to be intractable in the fully nonlinear setting.…”

Some specific aspects of linear homogenization shell theory

“…In this vein, Larsson and Landervik [23] propose a new model for homogenization of heterogeneous shells based on a second-order expansion in thickness of the current position of the shell and finally a simplified assumption on the initial geometry of the shell. The method of homogenization valid for periodic heterogeneous plates proposed by Lee et al [24] gives an interesting result but it seems to be restricted to small deformations and large rotations and displacements. This is due to fact that the simplification linked to the application of variational-asymptotic method [25,26] runs only under the small deformation assumption.…”