Keywords: layerwise theory, mixed formulation, least-squares formulation, finite element model, multilayered composite plate.

A layerwise finite element model is developed in a mixed least-squares formulation for static analysis of multilayered composite plates. An axiomatic type approach is used, with a layerwise variable description of displacements, transverse stresses and in-plane strains, in a truly (partial) mixed formulation. As demonstrated by early three-dimensional elasticity solutions [1], multilayered composite structures may exhibit complicating effects introduced by anisotropic behaviour, such as high transverse deformability, zig-zag effects and interlaminar continuity. In this respect, layerwise models are much better suited to accurately capture such effects. In fact, mixed formulations are conveniently useful to completely and a priori fulfil C⁰_z requirements, i.e., both displacements and transverse stresses must be C⁰ continuous functions in the thickness z-direction, due to compatibility and equilibrium reasons. Furthermore, the motivation for adopting a mixed least-squares formulation as well is that it leads to a variational unconstrained minimization problem, where the finite element approximating spaces can be chosen independently, as opposed to mixed weak form models.

The assessment of the present model results is primarily based on a comprehensive comparison with the three-dimensional elasticity solutions. Further assessment is also facilitated by comparison with other finite element results, above all, the layerwise mixed weak form model by Carrera [2]. The numerical examples focus on the static analysis of the simply supported square multilayered composite plate (0/90/90/0), under a bi-sinusoidal transverse load, with a range of side-to-thickness ratios a/h = 2,4,10,20,50,100,500. Altogether, it is shown that the present model is able to obtain quite accurate results in very good agreement with the three-dimensional solutions. Specifically, for moderately thick to very thin plates a/h = 10,...,500, the present model achieves highly accurate results for all variables, in fact better than Carrera model, even though the z-expansion order is of third-order in the present model and fourth-order in the Carrera model. For very thick plates a/h = 2,4, the present model also reaches rather accurate results, but less so for the in-plane normal stresses. This is precisely where the Carrera model provides better results, which suggests that the present model might benefit from a further refinement, layerwise, whether by adding more (numerical) layers or by increasing the z-expansion order. In effect, such results are expected for the near future. Finally, it is important to underline that unlike the Carrera model, the present layerwise mixed least-squares model is shown to be insensitive to shear-locking altogether.

1

N.J. Pagano, "Exact solutions for rectangular bidirectional composites and sandwich plates", Journal of Composite Materials, 4, 20-34, 1970. doi:10.1016/0010-4361(70)90076-5

2

E. Carrera, L. Demasi, "Classical and advanced multilayered plate elements based upon PVD and RMVT. Part 2: Numerical implementations", International Journal for Numerical Methods in Engineering, 55, 253-291, 2002. doi:10.1002/nme.493

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