Topology optimization of hierarchical honeycomb structures to reduce stress concentration, deformation, and deflection

Abstract

The fast growth of additive manufacturing (AM) has aided the design and manufacturing process of complex structures with customized mechanical properties for use in various engineering sectors. Design for additive manufacturing (DfAM) is a top priority research topic, particularly for lattice-structured shapes such as honeycombs (HCs). Adding hierarchical structures can enhance the mechanical properties of HCs but complicate the design, needing advanced AM technologies for production. In this study, optimal Ti6Al4V(ELI) hierarchical honeycomb (HC) lattice designs were numerically modelled, numerically crushed, and additively manufactured and were then investigated with regard to their mechanical properties for each method. At the onset of the study, a stress-strain curve was developed to better represent cellular and lattice design deformation mechanisms and analytical models were also developed to characterize natural cellular structures and lattice design behaviour. The study reviewed open-access literature on analytical models for polygon-based lattice structures, grading them based on mechanical properties such as strength, stiffness, and deformation. Four polygonal hollow cells were graded in decreasing transverse stiffness order: circular, hexagonal, triangular, and square. These cells were configured into Ti6Al4V(ELI) lattice designs and numerically modelled in compression scenarios for planar in-plane uniaxial loading in the ABAQUS package. The hexagonal polygon was subjected to out-of-plane and in-plane uniaxial compression loads to compare the bending and buckling behaviour of finite element (FE) models to analytical models. The triangular polygon had the greatest load-bearing capacity and stiffness of all polygons modelled. The hexagonal model generated deformations due to compression, similar to those reported in literature. The critical buckling loads for analytical HC models were found to be below the yield stress for (1-, 1.125-, and 1.25-mm wall thicknesses) and above the yield stress for all FE HC models. The effective stiffness of HC models increased with increasing (t/L) ratio. Ti6Al4V(ELI) hierarchical HC models with hexagonal, circular, and triangular polygons at their vertices were also analyzed in this work through ABAQUS software. The models were compared to regular HC models for out-of-plane loads in uniaxial compression tests. The hierarchical HC model with hexagonal substructures ranked higher in terms of load-bearing capacity and stiffness for loads applied in the in-plane x-direction. For loading in the in-plane y-direction, the model with circular substructures ranked highest. The highest load-bearing capacity and stiffness were observed for in-plane loading in the x-direction. Deflections were smaller for loading in the two in-plane directions than in the out-of-plane direction. In-plane loading at the vertices led to high stress concentrations at the loading points. Loading the models directly at the apex regions led to the highest stress and deflection magnitudes. Hierarchical HCs with lower sharp geometric changes minimized stress concentrations. Quasi-static numerically crushed hierarchical HCs were investigated regarding the impact of their topologies on deformation response modes and energy absorption characteristics in ABAQUS. The findings showed that bending deformation dominated under x-directional load while bending and buckling loads influenced deformation response modes under y-directional load. The material at the vertices of these structures failed first for both loading directions. The buckling of the vertically positioned cell walls caused instability under the y-directional load, causing the structure to rotate along the z-axis instead of being crushed along the y-axis. Some deformed cells showed the Poisson's ratio phenomenon, and the load-displacement curve for loading the regular HC in the x-axis showed a low oscillatory response. The study also presented an alternative method for generating effective iterations at the vertices and edges in planar HCs of Ti6Al4V(ELI) through the Altair-Optistruct modelling package. Mesh convergence tests were conducted on planar unit hexagonal cells, followed by topology and shape optimization and numerical analysis. The findings were compared with the numerical analysis before topology optimization. Acceptable predictions were obtained for the planar numerical models, with material reductions of 30% and 8% for the planar unit hexagon cell and HC model, respectively. The maximum stresses in these numerical models after shape optimization were reduced by 58% and 4%, respectively. Previous works have investigated HC structures, but the hierarchical design presented here offers significant improvements in load-bearing capacity and energy-absorption, as demonstrated through quasi-static crushing tests. Vertex-based hierarchical HCs were created and tested for surface quality and crushing behaviour in quasi-static scenarios. At the onset, zero-, first-, and second-order hierarchical Ti6Al4V(ELI) HCs were designed and produced using direct metal laser sintering (DMLS) technology. These Ti6Al4V(ELI) specimens were then quasi-statically crushed with an MTS CriterionTM, Model 43 universal testing machine to assess the mechanical behaviour regarding the HC topologies with various levels of hierarchy. The experimental results were compared with finite element predictions from ABAQUS, showing strong correlation in the mechanisms of deformation, stress distribution, and modes of failure, thereby validating the numerical models. This validation confirmed the accuracy of the topology and shape optimisation numerical models used thus, ensuring that the optimised designs are both mechanically robust and manufacturable. Surface roughness in hierarchical HCs increased with the hierarchy level of the structure. Hierarchical HCs exhibited primary deformation through beam bending, with pronounced bucklin -order HCs showed an average peak failure load of approximately 19.75 kN and 21.25 kN for x- and y-direction, respectively. While first-order HCs reached 23.25 kN and 25.25 kN for x- and y-direction, respectively. Second-order HCs reached 25.50 kN and 27.50 kN for x- and y-direction, respectively. These values were derived from load-displacement curves obtained in both the x- and y-directions. The improvement in failure loads with increasing hierarchy highlighted the mechanical benefits of hierarchical structuring, particularly the role of vertex-based geometry in distributing stress and delaying collapse-mechanisms. As the level of hierarchy increased, the variation in failure loads between the x and y loading directions decreased. This reduction made it difficult to rank the first- and second-order hierarchical HCs based on their response to loads applied along these two different loading directions. However, the overall trend observed in this work, confirmed that higher-order hierarchical designs contributed positively to structural integrity. Notable differences in load peaks were observed in zero-order specimens, particularly between x1 and x2, with the latter showing lower peaks, likely due to failure along an inclined line, possibly caused by flaws or poor sintering at cell wall separation points. This study introduces novel mechanisms of deformation in hierarchical HC designs and provides critical insights into their mechanical behaviour under various loading conditions. These findings reveal how hierarchical structuring not only improves peak failure loads but also introduces controlled, progressive collapse mechanisms that enhance energy dissipation and structural resilience under multi-directional loading. Therefore, the optimised hierarchical HCs are anticipated to enhance the performance of structural components across the biomedical, automotive, and aerospace industries by significantly improving their load-bearing capacity, and strength-to-weight ratio. g under y-direction loading. Post-peak load drops, observed in an oscillatory manner, were attributed to vertex fractures following initial bending or buckling deformation. The collapse under compressive loads followed a progression from bending and/or buckling to Poisson's ratio-driven lateral expansion, concluding with sliding along inclined and horizontal lines for x- and y-direction loading, respectively. Crushing occurred sequentially, beginning with the lowest hierarchy order and progressing upwards. Zero-order HCs demonstrated the lowest failure loads, with first-order HCs showing an increase and second-order HCs exhibiting the highest failure loads. The mechanical performance of the hierarchical honeycomb (HC) structures was assessed through quasi-static compression testing along multiple orientations. Zero-, first-, and second-order hierarchical HCs exhibited progressively increasing peak failure loads, indicating enhanced load-bearing capacity with the introduction of multiscale geometry. Specifically, zero