Triply periodic minimal surfaces based honeycomb structures with tuneable mechanical responses

ABSTRACT We proposed a novel class of honeycomb structures inspired by triply periodic minimal surfaces (TPMS) architecture with tuneable mechanical responses. The design procedure based on the level-set approximation was first introduced for two TPMS-based honeycombs, namely G-Honeycomb and P-Honeycomb. A numerical model was developed and validated by experimental results to evaluate the mechanical properties of G-Honeycomb and P-Honeycomb lattices. The results showed that P-Honeycomb exhibited higher elastic modulus and plateau stress than G-Honeycomb at various relative densities. Meanwhile, both TPMS-based honeycombs showed higher in-plane elastic modulus than conventional square honeycombs, although they had the same nodal connectivity. The idea of designing lattices with tuneable mechanical responses was achieved by introducing a density gradient or creating a G-P hybrid structure with the topology of G-Honeycomb and P-Honeycomb in different regions. The graded structures showed gradual stiffening and progressive collapse under compression, while the G-P hybrid structure can exhibit distinct properties at different compressive strains.


Introduction
Lattice structures have attracted increasing attention since they exhibit promising mechanical properties, such as lightweight, high specific stiffness, specific strength and energy absorption ability (Ashby and Gibson 1997;Ashby et al. 2001;Schaedler and Carter 2016;Peng et al. 2020), leading to the application of these structures in protective armour (Imbalzano et al. 2016;Imbalzano et al. 2018), thermal management (Wong et al. 2009), bone tissue engineering (Zhang, Fang, and Zhou 2017), energy absorption (Tran and Peng 2021;Peng and Tran 2020), heat dissipation (Catchpole-Smith et al. 2019;Aremu et al. 2017), structural components (Peng et al. 2021;Wickramasinghe et al. 2022).The mechanical properties of lattice structures are determined predominantly by the unit cell topology and material properties (Ashby 2006).Over the last decade, many researchers have explored different architectures to optimise the mechanical properties of lattice structures (Rashed et al. 2016;Wang et al. 2018;Wang et al. 2021;Daynes et al. 2017;Cao et al. 2020;Lan and Tran 2021).
The honeycomb structure is a popular form of the lattice structure.Many studies have investigated the mechanical responses and crashworthiness of honeycomb structures experimentally (Lin et al. 2021;Wu et al. 2017), analytically (Yu et al. 2016;Wang and McDowell 2004) and numerically (Zhang and Zhang 2013;Sun et al. 2018), while the mechanical responses of many new cell topologies have been explored (Zhang et al. 2021;Abdullahi and Gao 2020;Ma, Li, and Xie 2020).Over recent years, many researchers have focused on improving the mechanical properties of traditional square and hexagonal honeycombs by distributing the materials efficiently.Novel structures inspired by biological structures have been widely considered in the design of honeycomb structures.For example, Zhang et al. (Zhang et al. 2021) introduced a new kind of jointbased hierarchical honeycomb with self-similar cells.They investigated the crushing deformation of the hierarchical honeycomb structures and found that they showed better stability than traditional honeycombs.Abdullahi and Gao (Abdullahi and Gao 2020) designed honeycomb structures with randomised cell sizes based on Voronoi tessellations.Their results showed that the Voronoi-based honeycomb structures presented progressive and stable deformation under out-of-plane compression and produced better crashworthiness performances than regular multi-cell honeycombs.Ma et al. (Ma, Li, and Xie 2020) investigated the crashworthiness of thin-walled bio-inspired multi-cellular corrugated structures.These structures presented controllable and promising responses to the traditional ones.
Triply periodic minimal surfaces (TPMS) inspired structures have attracted increasing attention from researchers.It can be designed and optimised as a novel class of lattice structures for lightweight and high mechanical properties (Han and Che 2018;Al-Ketan and Abu Al-Rub 2019;Abueidda et al. 2020;Wang et al. 2020).Mathematically, TPMS is defined as a surface with zero mean curvature, three-dimensional (3D) periodicity, and smooth topology by minimising the surface area with a given boundary.Practically, TPMS can be approximated by level-set approximation equations (Maskery et al. 2017).Lattice structures with a similar topology as TPMS can be created based on two strategies.In the design of the network phase of TPMS structures, one sub-domain of the space partitioned by the surface is filled with a solid while the matrix phase is obtained by thickening the surface (Al-Ketan and Abu Al-Rub 2019).The mechanical properties of lattices based on TPMS made of stainless steel (Yan et al. 2014;Li, Xiao, and Song 2021;Al-Ketan, Rowshan, and Abu Al-Rub 2018), aluminium alloy (Yan et al. 2014;Yan et al. 2015), polymer (Maskery et al. 2018;Cao et al. 2021;Zhang et al. 2018) and concrete (Nguyen- Van et al. 2020;Nguyen-Van et al. 2022) have been evaluated.These structures have been found to present exceptional mechanical properties compared to traditional strut-based lattices.Meanwhile, owing to the nature of mathematically designed unit cells, the topology of structures can be tailored locally (Plocher and Panesar 2020;Choy et al. 2017;Maskery et al. 2018;Al-Ketan, Lee, and Abu Al-Rub 2021).For example, (Maskery et al. 2018) presented that lattices based on TPMS can be easily designed with a graded relative density.They have found that the graded structures showed a tailorable collapse process and advantage of energy absorption under compression compared to equivalent uniform lattices.Al-Ketan et al. (Al-Ketan, Lee, and Abu Al-Rub 2021) proposed a design procedure for stochastic cellular structures based on TPMS.These structures presented superior mechanical properties compared to the periodic ones at high relative densities.Most recently, Maskery and Ashcroft (Maskery and Ashcroft 2020) introduced the idea of designing honeycomb structures based on TPMS.However, these TPMS approximation equationbased honeycomb structures are yet to be investigated.
With the aim of uncovering novel honeycomb structures with promising and tuneable mechanical responses, this work explored the potential of designing TPMS-based honeycomb structures.Firstly, the design procedure of two types of TPMS-based honeycombs, namely G-Honeycomb and P-Honeycomb, the fabrication and testing approach, and numerical simulations are explained.Then, the numerical simulations are validated by the experimental results, followed by a comprehensive investigation of the mechanical properties of G-Honeycomb and P-Honeycomb based on numerical simulations.The mechanical properties of these structures are compared with that of square honeycombs obtained analytically.After that, the tuneable mechanical responses of TPMS-based honeycombs are demonstrated by functionally graded and hybrid designs.Finally, the major findings of this study are summarised.

Design of honeycomb structures based on TPMS architecture
Previous research has presented that TPMS structures can be designed based on several strategies, including parametric, implicit and boundary functions.In this work, honeycomb structures are designed based on the level-set approximation of TPMS structures.Firstly, level-set approximation equations can define TPMS in three-dimensional space following: where c is the parameter controlling the isovalue offset from zero level-set.Then, TPMS lattice structures can be designed based on the expression: where the intervals [−c, c] determine the volume fraction of the solid phase in the structure.The TPMS gyroid and primitive surfaces are defined using the nodal function: f gyroid (x, y, z) = cos (w x x) sin (w y y) + cos (w y y) sin (w z z ) f primitve (x, y, z) = cos (w x x) + cos (w y y) + cos (w z z), (4) where x, y and z are spatial coordinates in the threedimensional Cartesian coordinate.w defines the periodicities of TPMS function, where n i controls how many unit cells are along the x, y, z directions.L i defines the dimensions of the lattice in the corresponding directions.
New two-dimensional structures can be introduced by substituting z = 0 to eliminate the z periodicity in Eq. (3) and Eq. ( 4) and substituting them into Eq.( 2): The 2D space is separated into two regions by Eq. ( 6) and Eq. ( 7).One of these regions is continuous areas, while the other is comprised of periodic isolated areas.The design process of the TPMS gyroid-based honeycomb is demonstrated in Figure 1.The grey area in Figure 1c presents the regions which are not connected, while the white area is the continuous region.A novel family of honeycomb-like structures can be produced by considering the continuous region as solid and the separated regions as void.The relative density (RD), r * , of the structure is determined based on the volume fraction of the solid regions, V solid , with respect to the total volume, V, of the structure: A MATLAB script was developed to compute the isosurface based on triangular mesh according to the implicit trigonometric functions of the structure.Stereolithography (STL) files for 3D printing and mesh preparation of numerical simulation are then generated.The unit cell size of the gyroid is designed to be 20 mm to facilitate the accurate control of relative density and the 3D printing process.The correlation between the level-set value, c, and the relative density, r * , of the structure is examined and presented in Figure 2a.A linear relation between the level-set value and the relative density is found for both structures, while the gyroid-based honeycomb shows a slightly higher relative density compared to that of the primitive-based one with the same levelset value.The geometry of gyroid (G-honeycomb) and primitive (P-honeycomb) based honeycomb structures at various relative densities are illustrated in Figure 2b and Figure 2c, respectively.

Fabrication and mechanical testing
The  evaluate the base material property.The uniaxial compression test is performed on the lattice specimens to obtain the stress-strain curve to validate the later numerical simulations.An Instron 5900R universal testing machine with a 30 kN load cell is utilised, while a digital camera is used to capture the deformation of lattice samples during the test.The samples are compressed at a loading rate of 1 mm/ min up to a strain of 0.7.Three samples of each design are tested to ensure the repeatability of the results.

Numerical simulation
Finite element (FE) analysis is utilised to evaluate the mechanical properties of G-Honeycomb and P-Honeycomb lattices at various relative density (RD) and their tuneable mechanical responses.Commercial finite element code ABAQUS/Explicit 2020 is used to develop the numerical model to simulate the uniaxial compression tests.The dimensions of the FE models for the different gyroid structures are identical to the experimental samples used in the compression tests.The G-Honeycomb and P-Honeycomb lattices are modelled with plane strain assumption as these structures are based on 2D surfaces to reduce the computational cost.Each unit cell in these lattices is discrete by 2,337 4-node bilinear plane strain quadrilateral, reduced integration, and hourglass control CPE4R elements after a mesh sensitivity analysis to ensure the numerical results are not depending on the mesh density.
The constitutive model of the base material used in the structure is assumed to be elastic and perfectlyplastic, based on the tensile tests of the 3D printed tensile samples with the same orientation as the G-Honeycomb and P-Honeycomb lattice samples.Table 1 summarises the mechanical properties of the printed base material.Two platens are modelled and simplified to rigid.The bottom test platen is fixed, and the compressive loading is applied to the lattice by end-shortening displacement of the top platen.
The contact between parts is described by hard formulation for normal behaviour and penalty method with a friction coefficient of 0.3 for tangential behaviour.
The reaction forces on the platen and the displacements of the platen are extracted for further evaluation.

Performance evaluation method
The elastic modulus of the structure is calculated from the slope of the initial linear region of the compression stress-strain curve.The energy absorption efficiency, h, is calculated using: The densification strain, 1 d , is then obtained when the maximum energy absorption efficiency is reached: The plateau stress, s p , is calculated as the mean stress over the strain range from 1 = 0.1 up to 1 d to eliminate the influences of the elastic stage and the final densification stage: According to the analytical model for cellular structures proposed by Gibson and Ashby (Gibson 2003), the mechanical properties can be predicted dependent on the relative density (ρ * ) according to the scaling law: where ∅ is the property of interest, ∅ s is the property of the bulk material, n is the density exponent, and C accounts for geometric factors.The Gibson-Ashby equation is utilised to fit the results and provide predictions.

Validation of numerical results
The G-Honeycomb and P-Honeycomb lattices with a relative density of 0.25 are fabricated by techniques mentioned previously.These samples are tested under quasi-static uniaxial compression tests.The experimental stress-strain curves are plotted alongside the results obtained by numerical simulations in Figure 3. G-Honeycomb and P-Honeycomb lattices show typical stress-strain curves of cellular structures under compressive loadings.A three-stage response for both lattices is observed: an initial linear elastic stage where the elastic modulus is determined, followed by a plastic plateau stage and the final densification stage.Compared to G-Honeycomb, P-Honeycomb exhibits higher stiffness and plateau stress, and fluctuation of stress measured in the plateau stage is found for P-Honeycomb.It  Figures 4 and 5 present the deformations of the G-Honeycomb lattice and P-Honeycomb lattice, respectively, captured both experimentally and numerically.A uniform deformation mode is found for G-Honeycomb, while cell rotation and wall compaction stiffening mechanism is observed.The stress distribution on the structure from FE simulation reveals that before reaching a strain of 0.15, the cell walls of G-Honeycomb undergo elastic and plastic bending.By a strain of 0.2, some cell walls start to come into contact with others.Overall, the FE simulations predict the deformation of the G-Honeycomb lattice under  compression accurately.By comparison, the deformation of P-Honeycomb is dominated by localised buckling of the cell walls.Localised buckling occurs on the cell walls when loaded to a strain of 0.2, which is also confirmed by the FE simulations.As shown in Figure 5b, shear bands form on the wall of cells located at the +45 • regions.
Upon further loading on the structure, the upper portion of P-Honeycomb starts to crush with buckling of the cell walls (see Figure 5a), which could also explain the fluctuation in the plateau stress for P-Honeycomb, as each drop of the stress in the plateau region (see Figure 3a) corresponds to the buckling of cell walls.At a strain of 0.15, the deformation of the P-Honeycomb tends to be unsymmetric, but the FE simulation predicts symmetrical deformation.It could be attributed to a number of factors.For example, the FDM fabrication of the samples may introduce defects on the lattice, which could affect the location of localised buckling, leading to an unsymmetric deformation.Meanwhile, there could be slight misalignment in the compression test, which can lead to an unsymmetric deformation of the sample.To verify this assumption, an artificial misalignment of the compression platens is introduced in the numerical model.Figure 6 presents the comparison between deformations of the P-Honeycomb obtained by experiments and numerical simulations.It can be seen that the slight misalignment of the compression platens in numerical simulation can induce the P-Honeycomb to collapse unsymmetrically.
Overall, the FE model can accurately predict and validate the mechanical response of G-Honeycomb and P-Honeycomb lattices.The following section of this study will evaluate the mechanical properties and tuneable responses of G-Honeycomb and P-Honeycomb lattices based on FE simulations.

Mechanical response of TPMS-based honeycomb
The in-plane mechanical properties of G-Honeycomb and P-Honeycomb lattices are analysed in this section.It is noticeable that the G-Honeycomb and P-Honeycomb present similar connectivity compared to the square honeycomb, where each node is connected by four cell walls perpendicularly.This similarity is shown in Figure 7a-c.Thus, an analytical analysis is established based on the square honeycomb to compare the mechanical properties of G-Honeycomb and P-Honeycomb with that of the square honeycomb.Considering a representative cell wall of square honeycomb with a length of l, a thickness of t and a width of b, as illustrated in Figure 7d.The force applied on each cell wall at the node is Therefore, the bending moment applied at the node is Based on Euler-Bernoulli beam theory, the deflection of the cell wall could be estimated as.
where E s is elastic modulus of the base material.
The compressive strain of the cell wall can be expressed as Then, the elastic modulus of the square honeycomb can be calculated by The normalised elastic modulus of the square honeycomb is given by With the increase of compression load, plastic hinges will form at the nodes of the cell walls.Considering the rotation of the plastic hinge to be f, the plateau plastic collapse stress from the upper bond theorem can be determined from where the fully plastic moment M p , is given by s ys bt 2 (Wang and McDowell 2004).The plateau plastic collapse stress from the lower bond theorem occurs when the bending moment acting on the node reaches the fully plastic moment M p .Both upper and lower bond plastic plateau stress  leads to normalised plateau stress of The effective elastic modulus and plateau stress of G-Honeycomb and P-Honeycomb lattices with relative densities from 0.1-0.4 are evaluated based on the FE model validated in the previous section.The Gibson-Ashby scaling law is employed to fit the results from FE simulations, and the results are summarised alongside the analytical solutions for the square honeycomb, as presented in Figure 8.Both G-Honeycomb and P-Honeycomb present higher normalised elastic modulus than the square honeycomb, although they have similar nodal connectivity.As shown in Figure 8a, the normalised elastic modulus of P-Honeycomb shows a greater dependence on the relative density than that of the other two structures, which increases significantly with the increase of the relative density.G-Honeycomb presents lower in-plane stiffness than the P-Honeycomb one, which could be attributed to the twisting of the individual unit cells.
Regarding density-plateau stress scaling relationships, G-Honeycomb and P-Honeycomb show different behaviours than square honeycombs.Both G-Honeycomb and P-Honeycomb exhibit a density exponent close to 3, while the analytical prediction of square honeycomb is 2. Similar to the elastic modulus, P-Honeycomb presents higher plateau stress than G-Honeycomb (see Figure 8b).Overall, it can be concluded that G-Honeycomb and P-Honeycomb can produce higher in-plane stiffness and plateau stress compared to square honeycomb at the same relative density as materials are efficiently relocated around the nodes and cell walls by the TPMS functions and the smooth nature of TPMS structures.

Parametric study: design with graded relative density
The relative density of G-Honeycomb and P-Honeycomb depends on level-set values in level-set approximation equations, and there are density-property scaling relationships for these structures.Tuneable mechanical responses can be achieved by changing the density distribution across the structure.To demonstrate this feature, the constant level-set value, c, is replaced with a spatially dependent function, c(y), so that the relative density satisfies, r * = 0.4 − 0.003 * y. (20) The structure has a relative density of 0.4 at the bottom and 0.1 at the top, respectively, resulting in an average relative density of 0.25.The mechanical responses of G-Honeycomb and P-Honeycomb with graded relative density are studied based on FE simulations.The stress-strain curves of graded structures are plotted against that of uniform structures in Figure 9a and d.A shift from the three-stage response to a gradual stiffening response is found for both G-Honeycomb and P-Honeycomb, as no plateau region is observed for both graded structures.Meanwhile, the graded P-Honeycomb shows fluctuations in the stress as the uniform one (see Figure 9d), which could also be associated with the buckling of individual cell walls.Figure 9b and c compare uniform and graded G-Honeycomb deformation and stress distribution.By introducing the density gradient, the deformation of the G-Honeycomb lattice changes from a uniform deformation mode to a layer-by-layer crushing mode.With increasing the compressive strain, the highest stress is developed in the regions with low relative density, which is marked as the crushed region in Figure 9c.The cell walls in these regions start to buckle or yield, leading to a layer-by-layer crushing deformation mode (see Figure 9c).The structure shows stiffening behaviour at high compressive strain with compactions of more cell walls.Figure 9e and f show the deformation and stress distribution of uniform and graded P-Honeycomb, respectively.A layer-by-layer crushing deformation mode is found for the graded structure.Higher stress forms on cell walls with low relative density, and they start to buckle with the increase of compressive strain, which is marked as the crushed region in Figure 9f.By comparison, higher stress is formed on the wall of cells located at the ±45°regions for the uniform structure.It can be concluded that the strain-strain response and deformation mode of G-Honeycomb and P-Honeycomb lattices can be tuned by introducing a desirable spatially dependent function for its relative density.

Parametric study: design with hybrid geometry
The previous section presents that G-Honeycomb and P-Honeycomb show distinct mechanical properties.These level-set approximation equation-based structures enable the design of hybrid structures in which one cell topology transitions into another.By combining G-Honeycomb and P-Honeycomb, the hybrid lattice can present the properties of these structures at the different compressive strains.To demonstrate this interesting feature, here we introduced a simple G-P hybrid structure defined by. Figure 10a compares stress-strain curves of G-Honeycomb, P-Honeycomb and G-P hybrid structures.The response of the hybrid structure under compressive loading can be divided into three regions.Initially, the hybrid structure shows a similar elastic modulus to that of P-Honeycomb.As the stiffer nature of P-Honeycomb, the compressive load is mainly transferred through the P-Honeycomb region initially.Then, it is followed by a plateau region with similar stress as G-Honeycomb.It could be attributed to the deformation mainly occurring in the G-Honeycomb region, as illustrated in Figure 10b-i, due to G-Honeycomb presenting lower in-plane stiffness compared to P-Honeycomb.Upon further compression, the compaction of cell walls happens, causing the stiffening of G-Honeycomb.
The deformation of the hybrid structure shifts to the P-Honeycomb region, as illustrated in Figure 10b-ii.The hybrid structure presents a second elastic response with a similar modulus as G-Honeycomb, followed by a plateau region with similar plateau stress as P-Honeycomb.When the compaction of cell walls in the P-Honeycomb region occurs, the hybrid structure finally reaches the densification stage, as presented in Figure 10b-iii.It is also noticeable that there is no stress concentration in the transition region of the hybrid structure, indicating that the sigmoid function introduced can achieve a smooth transition from one topology to another.Figure 10c-e compares the elastic modulus, plateau stress and energy absorption of different structures.The hybrid structure shows a similar initial elastic modulus and energy absorption as P-Honeycomb, while the plateau stress is an average of G-Honeycomb and P-Honeycomb.
Here, the results demonstrate that introducing a desirable boundary function G(x, y) to the G-P hybrid structure can achieve the mechanical properties of both G-Honeycomb and P-Honeycomb at different compressive strains.This ability expands the design freedom and designated responses that TPMS-based honeycomb structures can achieve.

Conclusions
Novel honeycomb-like structures were designed based on the level-set approximation of TPMS.Two different topologies, namely G-Honeycomb and P-Honeycomb, were investigated in this work.A numerical model was developed and validated by experimental results to predict the mechanical properties of these structures.Associated density-elastic modulus, density-plateau stress scaling relationships and structural deformation were obtained.The tuneable mechanical responses of these structures were also demonstrated by designing with graded relative density and hybrid topologies.The significant findings of this research are summarised as follows: . The strategies to design 2D honeycomb-like structures inspired by the level-set approximation method of generating TPMS structures were introduced.Linear correlations were found between the level-set values and relative density of both G-Honeycomb and P-Honeycomb.At the same time, P-Honeycomb presented a slightly higher relative density than G-Honeycomb with the same level-set value. .Both G-Honeycomb and P-Honeycomb showed a linear elastic, plateau, and densification three-stage behaviour under compressive loadings.G-Honeycomb exhibited a uniform deformation mode with twisting of unit cells.In contrast, the deformation of P-Honeycomb was dominated by localised buckling and yielding of the cell walls, leading to the fluctuation of plateau stress measured.P-Honeycomb exhibited higher elastic modulus and plateau stress than G-Honeycomb at various relative densities.Although G-Honeycomb and P-Honeycomb had the same nodal connectivity as square honeycomb, their elastic modulus overperformed the analytical prediction of square honeycomb. .By replacing the constant level-set value with a spatially dependent function, a gradient can be introduced to the relative density of the structure.The linear elastic, plateau, and densification threestage behaviour under compressive loadings of G-Honeycomb and P-Honeycomb can be turned into a gradual stiffening response, and they presented tailorable and progressive failure modes.These features could be promising for controlling the structural stiffness under dynamic loading.
. By combining the level-set approximation equations to define G-Honeycomb and P-Honeycomb structures with a sigmoid function, a G-P hybrid structure can be created with a smooth transition from one topology to another.The G-P hybrid structure presented distinct properties at different compressive strains.Besides, no stress concentration was observed in the transition region, indicating that the method to create a G-P hybrid structure would not compromise the performance of the structure.
Overall, the investigations into novel honeycomb structures based on TPMS can provide insights into designing lightweight structures for various applications.These structures can present high in-plane stiffness compared to traditional square honeycomb with the same weight.They may also be found to exhibit tuneable mechanical responses to loadings.For example, they can provide a gradual stiffening and layer-by-layer crushing response by introducing density gradient, while distinct responses at different compressive strains by hybrid design.These interesting features continue to be a key benefit compared with traditional honeycomb structures.

Notes on contributors
Chenxi Peng is a PhD candidate at the School of Engineering, RMIT University.His research interests focus on the multifunctional properties of lattice structures fabricated by additive manufacturing.
Pier Marzocca is a professor in Aerospace Engineering at RMIT University and director of Sir Lawrence Wackett Defence & Aerospace Centre.His research efforts focus mainly on aerothermo-elasticity, fluid-structure interactions, linear/nonlinear structural dynamics, aeroelastic stability and control.
Phuong (Jonathan) Tran is a director of the digital construction laboratory at RMIT University in Melbourne.His research interests include biomimicry design and additive manufacturing.He has published over 120 peer-reviewed journal articles in related fields.
Figure 1.(a) Two dimensional Unit cell defined by the function f gyroid , defined in equation 3(b) An unit cell defined by the function U G−honeycomb from equation (6); (c) An illustration of the 2D space separated by U G−honeycomb .

Figure 2 .
Figure 2. (a) The correlation between level-set value, c, and the relative density, r * , of two honeycomb types.The topology of the honeycomb structures at various relative densities are illustrated in (b) gyroid-based and (c) primitive-based honeycomb structures.
could be attributed to the local buckling of the cell walls, which are revealed by deformations of the lattices captured during experiments and FE simulations.Meanwhile, G-Honeycomb reaches the densification stage at a strain of around 0.4, while densification happens for P-Honeycomb at around 0.6.When comparing the results obtained by FE simulations with experimental ones, an overall good agreement is achieved between them for both G-Honeycomb and P-Honeycomb.The FE simulation captures the three-stage response well, although a slight discrepancy is found in the densification stage.

Figure 3 .
Figure 3.Comparison between stress-strain curves from experiments and FE simulations: (a) G-honeycomb and (b) P-honeycomb.

Figure 4 .
Figure 4. (a) Deformation of G-Honeycomb lattice under compression; (b) Deformation and finite element stress contours of the G-Honeycomb lattice under quasi-static compression.

Figure 5 .
Figure 5. (a) Deformation of the P-Honeycomb lattice under compression; (b) Deformation and finite element stress contours of P-Honeycomb lattice under compression.

Figure 6 .
Figure 6.(a) Deformation of the P-Honeycomb lattice under compression; (b) Deformation and finite element stress contours of P-Honeycomb lattice subjected to misaligned compression.

Figure 8 .
Figure 8. Comparisons between G-Honeycomb, P-Honeycomb, and square honeycomb for density (a) normalised modulus and (b) plateau stress scaling relationships.

U
hybrid = aU G−Honeycomb + (1 − a)U P−Honeycomb , (21) where a is the function to control the distribution of two different cell types in the hybrid structure.A sigmoid function is introduced as a = 1 1 + e −kG(x,y) , (22) where G(x, y) is a function defining the transition boundary, k is the constant to control the width of the transition region.With the sigmoid function defining the transition boundary, the topology can smoothly transit from G-Honeycomb to P-Honeycomb.With G(y) = y − 50 and k = 1, a hybrid lattice consisting of P-Honeycomb in the top half and G-Honeycomb in the bottom half is created.The relative density of the hybrid structure, P-Honeycomb and G-Honeycomb region is 0.25.The mechanical response of the hybrid structure is evaluated based on the FE model.

Figure 9 .
Figure 9. (a) compressive stress-strain curves of uniform G-Honeycomb, FE stress contour of (b) uniform and (c) graded G-Honeycomb; (d) compressive stress-strain curves of uniform P-Honeycomb, FE stress contour of (e) uniform and (f) graded P-Honeycomb.

Figure 10 .
Figure 10.(a) comparison between compressive stress-strain curves of G-Honeycomb, P-Honeycomb and G-P hybrid structure, (b) FE stress contour of hybrid structure at different stages; comparison between (c) elastic modulus, (d) plateau stress and (d) energy absorption of G-Honeycomb, P-Honeycomb and G-P hybrid structure.

Table 1 .
Mechanical properties of the 3D-printed base material.