An Invariant Model for any Composite Plate Theory and FEM Applications: The Generalized Unified Formulation

Abstract

The Generalized Unifled Formulation (GUF) presented here is a modern approach for the Finite Element analysis of sandwich plates and in general multilayered structures. GUF is a comprehensive formulation which includes practically all possible axiomatic theories. Any type of theory with any combination of orders of expansion for the difierent displacement variables can be obtained from the expansion of six 1 £ 1 arrays (the kernels or fundamental nuclei of the Generalized Unifled Formulation). Each of the displacement variables is independently treated and difierent orders of expansions for the difierent unknowns can be chosen. Since inflnite combinations can be freely chosen for the displacements, GUF allows the user to write, with a single invariant formulation implemented in a single FEM code, 1 3 Higher-order Shear Deformation Theories (HSDT), 1 3 Zig-Zag Theories (ZZT) and 1 3 Layer-Wise Theories (LWT). The six independent fundamental nuclei are formally invariant with respect to the order used for the expansion or with respect to the type of theory. GUF is a powerful versatile tool: the user can freely decide the type of theory (e.g., LWT or ZZT) and the orders used in the expansions. Thus, a code based on GUF can be fast (less terms can be used in the expansions of the unknwons) when the case under investigation is less demanding or when a large number of runs is required (optimization problems or probabilistic studies) and the software can be very accurate (layerwise models with high orders used for the unknowns) when the problem is very challenging and the accuracy is more important than the CPU time. GUF can be implemented in existing commercial codes and \intelligent software can be developed. The codes can, in fact, use any theory or order of expansion to attempt the convergence in case of complex three-dimensional problems with localized efiects without the actual need of solid elements. This is another advantage because the number of Degrees of Freedoms could be reduced without loosing in computational e‐ciency and with the advantage of having the two dimensional mesh typical of plate and shell models. The Generalized Unifled Formulation can be also adopted for mixed variational statements. In such cases the number of independent kernels would be difierent but their size would still be 1 £ 1 arrays. Multifleld problems such as thermoelastic applications and multilayered plates embedding piezo-layers can also be analyzed as well. This paper assesses bending of sandwich structures. Analytical and elasticity solution are compared. The efiect of the Zig-Zag form of the displacement is discussed.