On the Efficient and Accurate Non-linear Computational Modeling of Multilayered Bending Plates. State of the Art and a Novel Proposal: The \(2\text {D}+\) Multiscale Approach

After conducting a comprehensive historical review of presently established methods for computational modeling of multilayered bending plates, the present work introduces a novel 2D multiscale strategy, termed the 2D+ approach. The proposed approach is based on the computational homogenization formalism and is envisaged to serve as an appealing alternative to current methodologies for modeling multilayered plates in bending-dominated situations. Such structural elements involve modern and relevant materials, such as laminated composites characterized by the heterogeneous distribution of low-aspect-ratio layers showing substantial non-linear mechanical behavior across their thickness.

Within this proposed approach, the 2D plate mid-plane constitutes the macroscopic scale, while a 1D filament-like Representative Volume Element (RVE), orthogonal to the plate mid-plane and spanning the plate thickness, represents the mesoscopic scale. Such RVE, in turn, is capturing the non-linear mechanical behavior throughout the plate thickness at each integration point of the 2D plate-midplane finite element mesh. The chosen kinematics and discretization at the considered scales are particularly selected to (1) effectively capture relevant aspects of non-linear mechanical behavior in multilayered plates under bending-dominated scenarios, (2) achieve affordable computational times (computational efficiency), and (3) provide accurate stress distributions compared to the corresponding high-fidelity 3D simulations (computational accuracy).

The proposed strategy aligns with the standard, first-order, hierarchical multiscale setting, involving the linearization of the macro-scale displacement field along the thickness. It employs an additional fluctuating displacement field in the RVE to capture higher-order behavior, which is computed through a local 1D finite element solution of a Boundary Value Problem (BVP) at the RVE. A notable feature of the presented 2D+ approach is the application of the Hill–Mandel principle, grounded in the well-established physical assumption imposing mechanical energy equivalence in the macro and meso scales. This links the 2D macroscopic plate and the set of 1D mesoscopic filaments, in a weakly-coupled manner, and yields remarkable computational savings in comparison with standard 3D modeling. Additionally, solving the resulting RVE problem in terms of the fluctuating displacement field allows the enforcement of an additional condition: fulfillment of linear momentum balance (equilibrium equations). This results in a physically meaningful 2D-like computational setting, in the considered structural object (multilayered plates in bending-dominated situations), which provides accurate stress distributions, typical of full 3D models, at the computational cost of 2D models.