Keywords: multilayered plates, finite elements, higher-order formulations, accuracy.

This paper presents for the first time a numerical assessment of higher-order multilayered plate finite elements formulations derived within the "Unified Formulation" (UF) proposed by Carrera [1]. The UF enables the implementation of a large number of formulations for the through-thickness behaviour within the axiomatic two-dimensional modeling of plates and shells. In this work, attention is focussed on formulations deriving from the classical displacement approach arising from the Principle of Virtual Displacements (PVD). The laminate description is performed by employing either layer-dependent variables (LayerWise descriptions, LW) or variables defined at multilayer level and thus referred to the reference surface of the plate (Equivalent Single Layer descriptions, ESL). Linear up to fourth-order expansions of the unknowns in the thickness direction can be chosen for both descriptions. In ESL formulations, an additional Degree of Freedom (DOF) may be introduced to take account for the change of slope in the displacement distributions at the layers' interfaces (Zig-Zag-shape). In compact notation, the behaviour of the unknown in each layer of the plate can be written as

where the usual summation convention is employed, denotes the order of the expansion in the thickness direction and is the local thickness coordinate defined at layer level. The functions can be defined at layer level (LW description) or on the reference surface of the shell (ESL description). The most common models for multilayered plates, namely the Classical Laminate Theory (CLT) as well the First-Order Shear Deformation Theory (FSDT) are recovered by the means of appropriate trace operators. The distribution of the displacements in the plane of the plate is discretized according to standard Finite Element (FE) procedures. Linear four-noded and quadratic nine-noded lagrangian elements have been considered. In total, more than 25 finite elements for multilayered plates have been implemented and compared.
In order to allow a systematic evaluation of the numerical properties of the considered FE, the analysis is restricted to one case study for which Pagano obtained the exact three-dimensional elasticity solution [2]. A simply supported rectangular plate with symmetric, cross-ply lay-up made of three layers and subjected to a bisinusoidal transverse pressure load has been taken as basis for the numerical assessment. Different thickness ratios as well as different numerical quadrature techniques for the transverse stress components have been considered. A further computer code implements the UF within a closed form solution technique satisfying the domain equations as well as the boundary conditions in an exact manner. The closed-form solution is free from any numerical difficulties associated to discretization issues: the only accuracy limits are therefore due to the employed assumptions for the through-thickness behaviour.

As a result, comparisons between the FE solutions, analytical closed-form and exact elasticity solution can be made. The assessment has shown that the numerical difficulties due to the FE approximation are limited to the well-known shear locking phenomenon, which could be circumvented by the use of reduced/selective quadrature techniques. The convergence analyses have shown that the proposed FE converge to the analytical solutions obtained with the related thickness assumptions. A quadratic order of convergence with respect to the total number of employed FE could be established for the nine-noded element, whereas the bilinear element converges with a first-order rate. In general, it turned out that the thickness assumptions employed for the multilayered plate does not influence the behaviour on the reference surface: the FE discretization and the thickness assumptions are decoupled. Since the errors introduced by these two modelling techniques are independent, it can happen that these errors cancel out each other. In this case, a coarse FE mesh associated with a lower-order thickness expansion can yield better results than a coarse mesh associated to a refined through-thickness description. The results of this work encourage to exploit the unique features of the UF in combination with the FEM for a systematic study of multilayered plates.

1

E. Carrera, "Theories and finite elements for multilayered plates and shells: A unified compact formulation with numerical assessment and benchmarking", Arch. Comp. Meth. Eng., 10, 215-296, 2003. doi:10.1007/BF02736224

2

N. J. Pagano, "Exact Solutions for Rectangular Bidirectional Composites and Sandwich Plates", J. of Compos. Mat., 4, pp. 20-34, 1970. doi:10.1016/0010-4361(70)90076-5

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