On the accuracy of localised 3D stress fields in tow-steered laminated composite structures

Abstract

Variable Angle Tow (VAT) composites offer increased freedom for tailoring material properties compared to traditional straight-fibre composites. This increased freedom leads to greater design flexibility for enhanced structural performance but comes at the cost of more complex, spatially-varying displacement, strain and stress fields. To maximise the utility of VAT composites, a computationally efficient, yet accurate, numerical framework is needed. To this end, we employ a modelling approach that builds upon the recently developed, hierarchical Serendipity Lagrange finite elements. Three-dimensional (3D) stress distribution is obtained using the present modelling technique and verified against 3D finite element solutions, as well as other formulations available in the literature. A key advantage of the present approach is the ability to predict accurate 3D stress fields efficiently, i.e. with reduced computational effort, including around local features such as geometric, kinematic or constitutive boundaries. Moreover, the present work concerns the peculiarities of commonly used mathematical expressions for describing spatially varying fibre orientations across VAT laminates. The presence of an absolute value in the function used to describe fibre orientation can lead to discontinuities in fibre angle slope and curvature. In turn, these discontinuities lead to mathematical singularities in the constitutive relations along the laminate. If this singularity is not appropriately modelled as a boundary of the continuum, but rather as an interior point of the continuum, stresses may be predicted inaccurately. Compared to other models in the literature, our method is capable of unveiling detailed 3D stresses readily and accurately also in the vicinity of this singularity.

Efficient 3D Stress Capture of Variable-Stiffness and Sandwich Beam Structures

“…In the authors' previous work [35,40], the displacement-based Unified Formulation model, based on the hierarchical UF-RZT 50 B4 1 SL5 10,721 10 8 10 6 70 B4 1 SL6 19,412 10 8 * Time and space complexities associated with a pre-conditioner conjugate gradient algorithm (iterative solver) [40,44]. * * b denotes the bandwidth of a matrix.…”

“…The layer-wise model is obtained by removing stiffness terms, K τϕi j , K φsi j and K φϕi j , that account for the zig-zag kinematics. The explicit form for the matrix, K τsi j , can be found in Appendix A of [35] and the expression for other matrices, as required by ESL-ZZ models, are given as…”

“…Moreover, discrepancies in the transverse normal stresses for some laminates were observed when compared with 3D FE solutions. The authors' recent work on VS laminates [35] solves this disparity in the transverse stress results by adopting a LW model of the Serendipity Lagrange expansion (SLE) based Unified Formulation (UF). The LW approach of the displacement-based formulation was shown to yield accurate localized 3D stress fields.…”

“…For VS laminate D, since all layers have fiber orientation angle (T 0 ) value at the mid-span with either 0 • or 90 • , the transverse shear stress distribution is continuous and differentiable, and therefore, the transverse normal stress correlates well for all models as shown in Figure 5f. The reader is referred to the authors' previous work [35] for more insight on these intricacies in modeling variable-stiffness laminates.…”

Effect of Fiber Orientation Path on the Buckling, Free Vibration, and Static Analyses of Variable Angle Tow Panels

“…The advantages of the layerwise method in the analysis of VAT and sandwich beam structure were also demonstrated by Patni et al [36]. Specifically, the authors considered the three-dimensional (3D) stress distribution derived using hierarchical serendipity Lagrange finite elements and reported on the enhanced structural performance of VAT structures compared to that of traditional straight-fiber composites [37]. Several studies have reported on the application of the CUF in various cases involving plates, shells, beams, thermoelastics, piezo-electric problems, and aeroelastic applications [38,39,40,41,42].…”