Non-classical effects in straight-fibre and tow-steered composite beams and plates

Multilayered composites are widespread in load-bearing structures of the aeronautical and wind energy industries. Increasingly, advanced composites are spreading into the mass-market automotive sector, where the lightweight advantages of composites improve structural efficiencies and thereby enable a new generation of electric cars. Composite laminates are mostly employed in thin-walled semi-monocoque structures as the manufacturing processes, such as pre-preg curing and resin infusion, are amenable to this type of construction. However, their imminent diversification to new applications will benefit from extending the range of possible laminate configurations in terms of layer material properties, stacking sequences and laminate thicknesses, as well as the nature of service loading. Such a diversification can add significant complexity when, for example, the layer material properties differ by multiple orders of magnitude or when the composite comprises of relatively thick cross-sections. In case of the former, the structural response is non-intuitive and cannot be modelled adequately using classical lamination theory. The latter adds non-classical effects due to transverse shearing and transverse normal stresses, which are particularly pernicious due to the lack of reinforcing material in the stacking direction and can lead to the delamination of layers. Reliable design of these multilayered structures requires tools for accurate stress analysis that account for these non-classical higher-order effects. Despite offering high fidelity, threedimensional (3D) finite element models are prohibitive for iterative design studies due to their high computational expense. Consequently, a large number of approximate higher-order twodimensional (2D) theories have been formulated over the last decades, with the aim of predicting accurate 3D stress fields while maintaining superior computational efficiency. The majority of these formulations have focused on purely displacement-based approaches that typically require post-processing steps to recover accurate transverse stresses. The work presented here uses the Hellinger-Reissner mixed-variational principle to derive a higher-order 2D equivalent single-layer formulation that predicts variationally consistent 3D stress fields in laminated beams and plates with 3D heterogeneity, i.e. laminates comprised of layers with material properties that differ by multiple orders of magnitude and that also vary continuously in-plane. The formulation is shown to be accurate to within a few percent of 3D elasticity and 3D finite element solutions. A novelty of the present approach is that the computational expense is reduced by basing all stress fields on the same set of unknowns. Furthermore, by enforcing Cauchy’s equilibrium equations in the variational statement via Lagrange multipliers, and then solving the ensuing governing equations in the strong form using spectral methods, boundary layers in the 3D stress fields are captured robustly. The present formulation is then used to ascertain the relative effects of transverse shear, transverse normal and zig-zag deformations. By studying non-traditional materials and stacking sequences with pronounced transverse anisotropy, the results presented herein provide physical insight into the governing factors that drive non-classical effects, with the aim of aiding the intuition of structural engineers in preliminary design stages. Finally, to showcase a possible application, the model is applied in an optimisation study that tailors the through-thickness stress fields in a beam in order to reduce the likelihood of delaminations. In the author’s opinion, the general formulation presented herein is well-suited for accurate and computationally efficient stress analysis in industrial applications.