An efficient method for design of lattice core sandwich structures with superior buckling strength under compression

ABSTRACT Buckling failure is a major concern in lightweight lattice core sandwich structures (LCSSs). This study proposes an efficient method for designing LCSSs with superior buckling strength under uniaxial compression. In this approach, the buckling-resistant non-uniform shapes of single struts with different slenderness and inclination are first identified, and then they are directly used to replace uniform struts in LCSSs. This method can save considerable design time compared to conventional methods, which require conducting optimizations on entire LCSSs. Numerical results demonstrate that four designed representative non-uniform LCSSs can gain over 15% improvement in buckling strength compared to their counterpart uniform LCSSs. This improvement is even comparable to the solutions from optimizations on entire LCSSs. Two LCSSs with non-uniform struts are 3D printed, and the test results validate at least 10% improvement in compressive strength. Based on the proposed method, various LCSSs with superior buckling strength could be designed for different loadings.


Introduction
Lattice core sandwich structures (LCSSs) are comprised of periodic lattice cores embedded in two face plates.LCSSs have promising properties, including lightweight and high stiffness/strength-to-weight ratios (Wadley, Fleck, and Evans 2003;Xiao et al. 2021), high energy absorption performance (Nemat-Nasser et al. 2007;Jin et al. 2019), thermal protection (Cheng et al. 2016) and acoustic insulation (Liu et al. 2019), as well as unusual functional properties (Salari-Sharif, Schaedler, and Valdevit 2018;Du Plessis et al. 2022;Li et al. 2021).Hence, they show great potential in various engineering applications, such as lightweight aerospace and automobile components (Xiao et al. 2021;Rashed et al. 2016).Since the lattice core plays a critical role in the overall properties of LCSSs, design of its geometries has attracted significant attention.One of the major design problems is to create LCSSs with high buckling resistance, especially for LCSSs composed of slender struts (or thin plates), which are prone to buckling failure.
It has been proved that truss lattice cores have higher buckling strength than their plate-based and shell-based counterparts in low volume fractions (Andersen, Wang, and Sigmund 2021).Various strategies have been developed to improve the buckling strength of truss lattices.For instance, size optimization was carried out to optimize the section sizes of uniform struts in octet truss (Gorguluarslan et al. 2016), double-diagonal square lattice (Fernandes et al. 2021) and macroscopic truss structures (Madah and Amir 2019).However, the buckling strength of uniform struts is usually suboptimal (Cox 1992).To further improve the buckling strength, non-uniform struts with varying section sizes were used instead.This strategy was first developed for the design of a single column.Olhoff and Rasmussen (1977) proposed a variational method to design the optimal clamped-clamped column with non-uniform sections, which led to a 32.62% higher critical buckling load than the uniform case.Later, Szyszkowski and Watson (1988) and Manickarajah, Xie, and Steven (2000) developed numerical optimization methods for frame structures in which the non-uniform struts were represented in a piece-wise manner.Wang et al. (2019Wang et al. ( , 2020) ) developed optimization methods on streamline stiffener paths to maximize the buckling load of non-uniform curved gridstiffened composite structures.Two-scale topology optimization techniques were developed to design graded lattice structures with minimal mechanical and thermal compliance (Montemurro, Refai, and Catapano 2022;Bertolino and Montemurro 2022).The authors (Zhang et al. 2018;Zhang 2019) introduced a Fourier series (FS) representation model into the optimization framework, which enables a smooth representation of non-uniform struts.
The fabrication of LCSSs with non-uniform struts is a challenging issue.Because of the nonuniformity and periodicity of the struts in LCSSs, traditional manufacturing technologies such as deformation forming and snap-fit technologies may not be feasible (Feng, Wu, and Yu 2016;Wadley, Fleck, and Evans 2003).Rapidly developed additive manufacturing (AM) technologies have realized various LCSSs through the integrated fabrication of lattice cores and face plates (Li et al. 2021;Ye, Bi, and Hu 2020;Gautam, Idapalapati, and Feih 2018;Wu et al. 2018).Moreover, AM has enabled the fabrication of LCSSs with highly precise and complex strut and nodal shapes to give exceptional mechanical properties.For instance, by tuning the strut profiles, the fabricated truss lattices exhibit improved Young's modulus, buckling strength and energy absorption, and reduced stiffness anisotropy (Zhang et al. 2018;Tancogne-Dejean, Spierings, and Mohr 2016;Tancogne-Dejean and Mohr 2018;Qi et al. 2019;Cao et al. 2018;Beharic, Egui, and Yang 2018).
Most existing design methods require repeated executions of optimization procedures for the entire LCSSs with different topologies and geometric sizes.The optimization usually needs hundreds of iterations and costs tremendous computing time to solve state equations.However, according to the results on different lattice cores, the obtained non-uniform struts have similar profiles and buckling modes (Zhang 2019).Exploring this similarity is crucial for rapidly designing LCSSs with different topologies without the cumbersome execution of optimization procedures.
This study aims to develop an efficient strategy for designing LCSSs with superior buckling strength under compression without the execution of optimization procedures on the entire lattice cores.The key idea of the proposed design strategy is to break the optimization of the entire lattice into suboptimization problems of single struts.First, the shapes of the non-uniform single struts with the highest buckling strength are investigated under different inclination angles, slenderness ratios and buckling modes.The buckling-resistant non-uniform shapes are represented using FS.The obtained non-uniform struts with different slenderness ratios and inclination angles exhibit highly similar shapes.More importantly, it is verified that the struts created by scaling the FS coefficients for the vertical strut have almost the same buckling strength improvement as those obtained by solving the optimization.On this basis, one design strategy to create LCSSs is to directly replace the uniform struts with the non-uniform struts.For cases in which multiple buckling modes are triggered in LCSSs, different sets of FS coefficients are used according to the buckling modes.To validate the proposed method, four LCSSs with non-uniform struts are designed: the Kagome, body-centred cubic (BCC), vertical struts reinforced body-centred cubic (BCCZ) and six-fold (Wang and Sigmund 2020) truss lattice cores.The numerical results show that the LCSSs with non-uniform struts gain 16.63-20.89%improvements in critical buckling load compared to their uniform counterparts.It is worth noting that such improvements are very close to the optimization solutions on the entire LCSSs, but the proposed method needs little design effort compared with the optimization method once the buckling-resistant non-uniform struts have been determined.The improvement is further verified by uniaxial compression tests on the three-dimensionally (3D)-printed BCC and six-fold truss LCSS specimens.
The remainder of this article is organized as follows.Section 2 illustrates the FS-based geometric representation method to model smooth non-uniform struts in LCSSs.In Section 3, the shapes of non-uniform single struts that maximize the first two orders of buckling eigenvalues are identified, which are used as basic elements for the design of non-uniform lattice cores in Section 4. Uniform and non-uniform six-fold truss LCSS specimens are 3D printed and tested to validate the proposed design strategy in Section 5.The advantages and limitations of the proposed method, as well as potential future research, are discussed in Section 6.

Representation of LCSSs
Four representative truss lattice cores in single-layer LCSSs are considered in this work; namely, the Kagome, BCC, BCCZ and six-fold truss lattices.Figure 1 presents their geometries, where the core height (i.e. the distance between two face plates) is denoted by h, and the inclination angle θ of each strut is defined as the angle between the vertical direction and the strut axis.All of the struts in Kagome and BCC have the same θ, while BCCZ and six-fold truss lattice include inclined and vertical struts.In both uniform and non-uniform struts, the slenderness ratio is denoted by S = l/2r eff , where r eff = V/π l is the effective radius, with l and V for the strut length and volume, respectively.
To represent the struts with smoothly varying sections, the FS-based implicit representation model developed in Zhang et al. (2018) is adopted.Here, a strut with a non-uniform section is described  by a continuous FS function r(ξ ), which defines the radius r at local coordinate ξ along the axis.Eliminating sine terms due to the section symmetry and truncating the first N terms, r(ξ ) is stated by where α n are the FS coefficients.This model can describe plenty of shapes using the first several terms (Zhang et al. 2018).As a particular case, the uniform strut is obtained by keeping α 0 but setting other parameters to be zero.The strut volume can be analytically formulated as In the global coordinate system, a strut along any direction can be generated via coordinate transformation on Equation (1).As multiple struts could intersect at single joints, the blending operation for the implicit function (Storm et al. 2013) is used to ensure smooth intersection.More specifically, given a lattice core composed of M struts, the representation function of the blended structure (x, y, z) is expressed as where a determines the smoothness of intersections and exp( ) is the exponent function.In this work, a is fixed at 10, which guarantees the smoothness of intersections from multiple struts.The lattice cores with non-uniform struts and smooth joints are generated by iso-surface (x, y, z) = 0, as shown in Figure 2.

Explicit identification of buckling-resistant non-uniform struts
In the first step, the buckling-resistant non-uniform shapes of single struts with various inclination angles and slenderness ratios are identified.For this purpose, numerical optimization is conducted on single struts; this is formulated by Maximize where the design variables are the FS coefficients {α 1 , . . ., α N }, ν i denotes the ith eigenmode, and K 0 and K denote the initial stiffness matrix and differential stress stiffness matrix, respectively.The objective function is to maximize either the first-order or second-order eigenvalue λ j .The optimization workflow is conducted using Python-Abaqus scripting, as shown in Figure 3.In the optimization, the first six FS terms in Equation ( 1) are used to represent the strut shapes, i.e.N = 6, since previous work shows that increasing one additional term generates slightly increased eigenvalues but leads to significantly increased computing costs (Zhang et al. 2018).The trust-region method (Conn, Gould, and Toint 2000), a metaheuristic algorithm for minimizing nonlinear multivariate scalar functions with or without constraints, is adopted for its convenient integration with the Python SciPy package (Virtanen et al. 2020).The finite difference method with a step size of 0.01 is used to obtain the gradient of the objective function with respect to the design variables.The optimization loop is terminated when the maximum change of design variables in two subsequent loops is smaller than 0.01.After the buckling-resistant non-uniform struts have been obtained, they are used to replace uniform struts in LCSSs according to the buckling modes.In this off-line process, no optimization procedure is required on the entire LCSSs.
As no volume constraint is imposed, problem ( 4) is formulated to obtain the ratios of α i /α 0 (i = 1 . . .6) for the non-uniform struts (i.e. the strut shape) for a specific eigenmode, rather than an optimized solution for a specific strut volume.After optimization, the obtained parameters α 0 ∼ α 6 can be proportionally varied for any desired strut volumes, but their ratios are fixed.In addition, in the FS-based representation, the strut volume is mainly determined by α 0 , while α 1 ∼ α 6 have little effect.By inserting the obtained coefficients in Tables S1-S4 (see Section S1 in the supplementary material) into Equation ( 2), it is found that the contribution of α 1 ∼ α 6 to the volume is less than 3% of that contributed by α 0 .This implies that when fixing α 0 in optimization, the volume of the optimized strut can vary within a small range.In this way, the buckling characteristics of the obtained struts can be comparable to those of struts obtained by conducting optimizations with a volume constraint.For a clamped-clamped column with a height of 30 mm, volume of 53 mm 3 and N = 6 FS terms, the results obtained using the proposed method gain 27.66% buckling strength improvement, while the results from the optimization method with a volume constraint (Zhang et al. 2018) show a 26.83% improvement.
In finite element (FE) analysis, geometric models in the form of triangular surface mesh are generated by iso-surface extraction on the implicit functions.After a surface remeshing, the volumetric mesh (free mesh) with linear tetrahedral elements (C3D4) is generated using the Abaqus built-in function 'convert tri to tet'.The average element sizes for struts and BCCZ lattice cores are 0.2 mm and 0.3 mm, resulting in 40-50 k elements [30 k degrees of freedom (DOFs)] and 700 k elements (400 k DOFs), respectively.The element size is selected according to the mesh refinement study to balance the simulation accuracy and computational costs (see Section S4 in the supplementary material).The clamped-clamped boundary condition (BC) is imposed on struts and lattice cores by constraining all DOFs on the nodes at the bottom face and the x-and y-translational DOFs on the nodes at the top face.The subspace eigensolver in Abaqus is used to solve the eigenvalue buckling problems.In addition to FE methods, some novel and accurate numerical methods are proposed to calculate the preand post-buckling behaviour.For instance, the differential quadrature method is effective for providing accurate buckling loads and buckling modes of thin rectangular functionally graded carbon nanotube-reinforced composite plates (Jiao et al. 2019).Moreover, a robust Bézier-based multi-step method, developed in Kabir and Aghdam (2019), is used to investigate the nonlinear vibration and post-buckling of composite beams.
The optimization problem is first solved to obtain the buckling-resistant non-uniform shapes of a vertical strut with θ = 0 • and h = 30 mm, where the loading aligns with the strut axis.During optimization, the average diameter of strut α 0 is fixed at 1.5 and α 1 ∼ α 6 are taken as the design variables, with upper and lower bounds of 0.2 and −0.2, respectively.The initial guess is {α i } = 0. Figure 4 presents the obtained non-uniform strut shapes, and the corresponding FS coefficients are given in Table 1.For the first-and second-order buckling modes, the non-uniform struts have 27.66% and 24.46% improvement in buckling strength, respectively.Such improvements can also be verified by the more uniform strain energy distributions of the non-uniform sections compared with the uniform ones.This improvement is slightly lower than the theoretical value of 32% for the first-order eigenvalue (Olhoff and Rasmussen 1977).One possible reason for this deficiency may be the limited number of FS terms, and the optimal shapes could be obtained by including higher-order FS terms.
Next, the non-uniform struts with various slenderness ratios S, varying from 10 to 30, are generated by optimization.From Figure 5(a), it can be seen that the non-uniform struts with different slenderness ratios from optimization exhibit highly similar shapes, and their FS coefficients are proportionally related (see Tables S1 and S2 in Section S1 in the supplementary material).In addition, Table 1.Fourier series (FS) coefficients of the buckling-resistant nonuniform struts with respect to the first-and the second-order eigenvalues.

Eigenvalue
Fourier Series coefficients First-order 1.500, 0.001, 0.177, 0.001, −0.072, 0.001, 0.037 Second-order 1.500, 0.004, −0.021, −0.168, 0.026, −0.029, −0.058 when varying the inclination angle θ from 0°to 60°, the non-uniform struts obtained by optimization have almost the same enlarged and narrowed sections along the axis, as seen in Figure 6.These results imply that the buckling-resistant non-uniform strut shapes with different slenderness and inclination angles are closely related.Therefore, the non-uniform strut shapes could be efficiently built by scaling only one set of FS coefficients with the target volume.Here, the FS coefficients in Table 1 are used to create non-uniform struts with any S and θ by the scaling method, and their eigenvalue improvements are compared with those obtained by optimization.In the case of S ≥ 15, the scaled struts have almost the same strength improvement as the struts obtained by optimization, where the relative difference is less than 1% (see Figure 5(b) and (c)).When the struts become thicker, the scaled struts show marginally less improvement than the non-uniform struts from the optimization.In particular, the scaled struts have 2% and 4% decreased improvement for the first and second-order buckling modes, respectively.Similar conclusions are found for the scaled struts with any θ (Figure 6(b) and (c)).Note that such a design method avoids the cumbersome optimization process on the entire LCSS, and therefore can save considerable computational costs.In the next section, the scaled struts will be used to design LCSSs for pursuing superior buckling strength.

Design of LCSSs with non-uniform struts
This section develops an efficient design method to design LCSSs with superior buckling strength.The underlying concept is to replace the uniform struts in LCSSs with the non-uniform struts obtained in Section 3. FE analysis on the LCSSs with uniform struts is first required to identify the  buckling mode of each strut, and then the flowchart in Figure 7 is followed to design LCSSs with high buckling strength.
The Kagome and BCC lattice cores include identical struts with the same inclination angle and slenderness ratio, which implies that all the struts simultaneously buckle in the same mode under compression loads.Notice that all of the struts intersect in the centre joint, and thus the second-order eigenmode is triggered (Figure 4(b)).Accordingly, the non-uniform struts are generated by scaling  the FS coefficients about the second-order eigenvalue in Table 1.The designed non-uniform Kagome and BCC cores are presented in Figure 8(a) and (b).To maintain the target volume, the FS coefficients for the Kagome and BCC cores are scaled by a factor of 0.975 and 0.979, respectively.From Table 2, it is found that the scaled FS coefficients are close to those obtained by running optimization on the entire lattice cores, which require over 200 and 312 iterations for the Kagome and BCC lattices, respectively.In contrast, the BCCZ and six-fold truss lattice core comprises two sets of struts which have different inclination angles and slenderness ratios.More importantly, they have different topologies and therefore the struts are triggered in different buckling modes.To pursue the highest buckling strength, the strut radii are adjusted such that all of the struts buckle simultaneously.To achieve this, an analytical method is developed to determine the radius ratio of two sets of struts (see details in Section S2 in the supplementary material).The analysis reveals that this radius ratio is relevant to the buckling mode, which is 1.5, in accordance with the first-and second-order eigenmodes (Equation S6 in Section S2).After tuning the radii of the uniform struts, they are replaced by the non-uniform struts according to the eigenmodes.The non-uniform BCCZ and six-fold truss lattice cores are shown in Figure 8(c) and (d), and the FS coefficients are given in Table 3.In the optimization of these two LCSSs, the number of design parameters is doubled, so the initial design is set to the FS coefficients obtained from the proposed method to reduce the computational cost.As a result, the optimization processes cost 135 iterations for both cases.Again, it is emphasized that the replacement of the non-uniform struts is achieved according to the excited buckling mode rather than the inclination angle.For example, in the six-fold truss lattice core, the central vertical strut has a second-order buckling mode, while the others have a first-order buckling mode.Although they have the same inclination angle, the non-uniform struts are created using different sets of coefficients.
Optimization on single struts is the most time-consuming step of the proposed method, which requires a sufficient number of elements to obtain optimized non-uniform struts.However, compared to the conventional methods, which conduct optimization procedures on entire LCSSs, the proposed method could save considerable time.This is mainly because single struts have far fewer DOFs than the entire LCSSs if the same accuracy is required with an identical mesh resolution.Moreover, this off-line method can be used for the non-uniform struts of various LCSSs with different topological strut connections, and this step does not need to consume any design time.

Numerical verification
The buckling strength of the designed single-layer LCSSs is verified through eigenvalue buckling analysis.The LCSSs are composed of an isotropic and linear elastic base material, with a Young's modulus of 1.41 GPa and Poisson's ratio of 0.35.The obtained critical buckling loads (denoted by P cr ) are compared to those of LCSSs with uniform struts and the non-uniform solutions obtained by conducting optimization on the entire LCSSs (Zhang et al. 2018).
The buckling modes and critical buckling loads are shown in Figure 9, and comparisons of the improvements are summarized in Table 4.It can be seen that all of the non-uniform lattice cores exhibit identical buckling modes to their uniform counterparts, and they possess improved buckling strength.Both non-uniform Kagome and BCC LCSSs designed using the proposed method have 20.57% improved P cr compared with the uniform LCSSs, and this improvement is almost the same  as that obtained using the optimization method on the entire LCSSs.The non-uniform BCCZ and six-fold truss LCSSs from the proposed method show improvements of 20.89% and 16.63%, respectively, compared to the uniform ones.These improvements are only 2.40% and 8.34% lower than the optimized results.The simulation results of higher-order eigenmodes show that the lowest eigenvalues of the two sets of struts differ by only 4.79% and 11.77% for the designed BCCZ and six-fold truss lattice cores, indicating that simultaneous buckling is roughly guaranteed.Furthermore, a nonlinear post-buckling analysis is conducted on the single-layer LCSS with 3 × 3 arrayed BCCZ lattices.The built-in Riks method in Abaqus is used to account for geometric nonlinearity as well as the elastic-plastic behaviour of the solid constituent, the properties of which are obtained from the tensile test, as shown in Section S3 in the supplementary material.The first-order eigenmode is introduced as the initial imperfection and imposed on the model (Panettieri et al. 2021).The load-displacement curves related to one reference point on the top surface are plotted in Figure 10(a), and the von Mises stress contours at the compressive displacement of 0.8 mm are shown in Figure 10(b).The results show that both arrayed LCSSs buckle in the same way as the lattice unit cell (Figure 9) because of the uniform compression and clamped BCs.Moreover, the peak load of the proposed non-uniform design (1352 N) outperforms its uniform counterpart (1161 N) by 16.38%.The results verify that not only the single lattice units but also the arrayed LCSSs can gain improved buckling strength, based on the designed non-uniform struts in Table 1.

Fabrication and testing methods
The BCC and six-fold truss LCSSs are 3D printed and experimentally tested for verification of the improved buckling strength.The unit cells of LCSSs with uniform and non-uniform lattice cores designed by the proposed method were fabricated using a selective laser sintering (SLS) system Farsoon 402P from nylon powders FS3300PA without fibre reinforcement, with a layer thickness of 0.1 mm.The SLS-fabricated nylon materials have nearly isotropic elasticity, high strength and high toughness (Dizon et al. 2018;Xu et al. 2019), as desired for validation of the elastic buckling behaviour.Each specimen included a lattice core with the design parameters shown in Table 5 and two face plates with a thickness of 3 mm.All the specimens were 3D printed in the same orientation to minimize the influence of material anisotropy.The surface roughness of the printed struts was evaluated by an RH-2000 High-Resolution 3D Optical Microscope.The representative uniform and non-uniform specimens are shown in Figure 11.It is worth mentioning that the printed LCSSs differ from the numerical results in Section 4 because the strut radius was increased to ensure manufacturability with the SLS system; thus, the BCC LCSS was scaled to 40 mm in height to reduce the influence of printing defects on the mechanical properties.
At least three specimens were fabricated for each design.The specimens were tested under compression loads using an Instron E10000 universal testing system with a 5 kN load cell for BCC and a 10 kN load cell for six-fold truss LCSSs.A quasi-static out-of-plane compressive strain of 0.001/s was applied through displacement control.The compressive strain was calculated as the compressive displacement divided by the initial core height.The deformation and failure mechanism of specimens during the entire compression test were captured by a camera.The strength was determined from the peak load of the load-displacement curves, and the stiffness was measured by the maximum slope prior to the load peak.
Mechanical properties of the FS3300PA material were evaluated according to standard tensile tests based on ASTM D638-14 (ASTM International 2014).Tensile specimens were printed along 0°, 45°  and 90°to the build direction with the same process parameters and five tests were conducted for each direction.The Young's moduli along the 0°, 45°and 90°directions are 1.31 ± 0.05, 1.37 ± 0.09 and 1.57 ± 0.03 GPa, with a relative difference of less than 20%.More details on the tensile tests can be found in Section S3 (see supplementary material).
Nonlinear FE analysis was also conducted to evaluate the failure mechanism and to understand the discrepancies between the numerical and experimental results.Following the nonlinear FE analysis in Section 4, the Riks method was used with both the geometric nonlinearity and elastic-plastic material behaviour of FS3300PA.Surface roughness in terms of the porous surface layer was estimated as 80 μm from the surface morphology images of the optical microscope (see Section 5.3), which was then modelled by uniformly reducing the radius of all struts in the analyses.The first-order eigenmode was taken as the initial imperfection and imposed on the model (Panettieri et al. 2021).Finally, the load-displacement curves were extracted from the reference point of the top surface and plotted against the experimental data for comparison.

Compression test results
The testing load-displacement curves and deformations are shown in Figure 12.The peak loads from test data and nonlinear FE analyses, as well as the critical buckling load from linear buckling analysis, are listed in Table 6.For the BCC LCSSs (Figure 12(a)), the experimental peak loads reach 56.4 ± 0.8 N and 67.0 ± 2.9 N for uniform and non-uniform designs, respectively, resulting in an improvement in strength of 18.77%.From the results, it is found that: (1) the linear buckling FE analysis overestimates the experimental buckling strength owing to printing defects; and (2) the nonlinear FE analysis incorporating material plasticity and the estimated surface roughness results in good agreement with the experimental results in terms of peak loads and stress-strain curves.It is also found that the deformations of the uniform LCSSs agree well with linear buckling analysis results, while those of the  In linear buckling FE analysis, the strength is represented by the critical buckling load; while in nonlinear FE analysis and the experimental results, it is calculated as the peak load from load-displacement curves.
non-uniform cases are slightly different from the FE results in Figure 9, which could be attributed to the plastic deformation in the thin regions.
For the six-fold truss LCSS, the recorded deformations subjected to applied displacements of 0.75-1 mm clearly present two types of buckling modes for both uniform and non-uniform designs.The outer vertical struts buckle in the first-order mode, while the centre vertical and inclined struts buckle in the second-order mode.Such a buckling mode agrees well with the numerical results in Figure 9(d).The experimental peak loads of the uniform and non-uniform specimens are 416.9 ± 32.1 N and 458.6 ± 22.9 N, respectively.Consequently, the non-uniform design gains an improvement in buckling strength of about 10.00%.Again, for the six-fold truss LCSS, the linear FE analysis overestimates the buckling strength, while the nonlinear FE model with printing defects provides a good agreement with the experimental results.
In summary, both experimental and nonlinear FE analysis results demonstrate the improved buckling strength of the non-uniform designed BCC and six-fold truss LCSSs over their uniform counterparts.Owing to printing defects such as surface roughness, the improvement in the peak loads is smaller than the linear buckling predictions.The experimental variations are shaded in the load-displacement curves (Figure 12).The standard deviation of the measured compressive peak load is less than 5% for BCC and 8% for six-fold truss LCSSs, indicating good repeatability of the tests.Conversely, the non-uniform LCSSs exhibit reduced stiffness, by 14.58% and 10.88%, respectively.The combined high strength and low stiffness of non-uniform LCSSs leads to a higher strain at the buckling failure than for the uniform cases.

Printing defect characterization and error analysis
As shown in Table 6, the peak loads in the experiments are lower than the critical buckling loads in the linear buckling analysis.The peak load for the six-fold truss LCSS can reach only about 45% of the critical buckling load.However, by taking both the material plasticity and surface roughness of 80 μm into account in the nonlinear FE analysis, this discrepancy becomes much lower, at less than 8%.In addition, the strength improvement matches well with the experiments.This indicates that printing defects such as roughness and material yielding play a critical role in the buckling strength of 3D printed LCSSs.
To characterize the surface roughness in terms of a porous surface layer of SLS fabricated struts, the failure locations and fractured cross-sections of the tested specimens are examined by an optical microscope, as shown in Figure 13.Although the LCSSs with both uniform and non-uniform struts fail by buckling, the struts have different broken modes in the post-buckling region.For the two sixfold truss LCSSs, the fracture locations are shown in Figure 13(a) and (b).The results indicate that the uniform struts are broken by bending at the middle and clamped ends of the struts, while the non-uniform struts are broken in shear modes in thin regions.For the BCC LCSSs with both uniform and non-uniform struts, no fracture is observed before a compressive displacement of 3 mm, as shown in Figure 12(a), indicating that they fail by plastic deformation.Figure 13(c) and (d) depicts two representative regions in the fractured cross-sections, which show that the dense inner regions are surrounded by a layer of partially melted powder.Surface roughness in terms of the porous surface layer is widely observed in laser powder bed fusion fabricated structures (Fu et al. 2022).In this work, the measured thicknesses of the porous surface layer range from 40 μm to 200 μm.Since the measured thicknesses are not negligible compared with the minimum radius of the non-uniform struts (570 μm), the reductions in the compressive strength of printed LCSSs are prominent.Nevertheless, the effectiveness of the proposed method in improving buckling strength is validated by the experimental results.

Discussion and conclusion
The aim of this work is to explore the design strategy that breaks the optimization of the entire LCSS into suboptimization problems of single struts.The buckling-resistant non-uniform single struts are utilized to assemble LCSSs instead of conducting optimization procedures on the entire LCSSs.Since the computational cost of a single strut is significantly lower than that of the entire LCSS, the design method can greatly improve the design efficiency.Both numerical and experimental results validated that the proposed method can improve the compressive strength of various types of LCSS by more than 10%.
In this study, the non-uniform strut with a clamped-clamped BC subjected to uniaxial compression is obtained, which can be used for single-layer LCSSs under uniform compression.However, LCSSs that are subjected to complex loading cases under various BCs are usually required in engineering applications.For instance, multiple loads may be imposed on LCSSs, where the loadings on each lattice core unit can be different, and the buckling characteristics of each lattice unit may be affected by their neighbouring units.In this case, the clamped-clamped BCs on single struts with uniaxial compression may not be enough to generate optimal shapes.To extend the proposed method for complex loading and BCs, two further studies should be accomplished.At the strut level, the influence of various BCs on the buckling modes, loads and optimized shapes of single struts needs to be investigated.Elastic BCs, which are mimicked by twisting springs or periodic BCs, can be imposed to account for the stiffness contribution from deformed struts in neighbouring unit cells or from different loadings.On this basis, a database can be set up to include FS coefficients to represent buckling-resistant nonuniform struts for various loadings and BCs.At the lattice structural level, the uniform struts under local compressive loads can be identified by carrying out a static analysis, and then they are replaced by the obtained non-uniform struts in the database according to their local BCs and buckling modes.The improvement in buckling strength reaches the maximum when all compressive struts buckle simultaneously; otherwise, the improvement could decrease.
As one of the most widely used support-free 3D printing techniques, SLS greatly expands the design freedom on LCSSs.However, printing defects exist owing to the powder-bed AM process and fine geometric features.Since the buckling behaviour of slender structures is sensitive to geometric imperfections, the partially sintered powders (shown in Figure 13(b) and (c)) play a detrimental role in experimental validation.Similar observations are also reported in selective laser melting printed metallic lattices (Refai et al. 2020).In addition, geometric defects, including strut thickness deviations, strut waviness, and non-flat and non-parallel face plates, can also lead to reduced stiffness and strength in experiments.In Figure 12, the partially sintered powders are modelled by thickness reduction in nonlinear simulations, which results in a good agreement in compressive strength.The discrepancy in experimental stiffness compared with the simulation results is attributed to the geometric defects.To minimize the discrepancy between the simulation and experimental results, quantitative investigation of the printing defects and X-ray micro-computed tomography-based modelling of imperfect structures can achieve high simulation accuracy (Liu et al. 2017;Lei et al. 2019).The high-fidelity modelling of imperfect lattice structures is a promising research direction to comprehensively investigate the influence of printing defects and their distributions on the buckling behaviour of lightweight LCSSs.
In conclusion, this work proposes an efficient design methodology to create LCSSs with superior buckling strength under compression without running optimization procedures on entire LCSSs.This work first identifies the buckling-resistant non-uniform shapes of single struts against the firstand second-order eigenvalues and explicitly represents them using FS coefficients.Then, scaled nonuniform struts are used to replace the uniform ones in LCSSs, where the replacement fully considers different buckling modes of struts and the scaling ensures identical strut volumes.This method can save considerable design time compared to the conventional methods, which conduct optimizations on entire LCSSs.The buckling characteristics of four types of lattice cores have been studied.Both numerical and experimental results demonstrate that the BCC and six-fold truss LCSSs designed by the proposed method can gain at least 18% and 10% improvement in the compressive buckling strength.More importantly, the improvements are almost the same as the results achieved by conducting optimizations on the entire LCSSs.In future research, the generalization of the proposed method to complex lattice structures and load cases in real engineering applications, as well as the sensitivity of the buckling behaviour of LCSSs to printing defects, will be investigated.

Disclosure statement
No potential conflict of interest was reported by the authors.

Figure 4 .
Figure 4. Illustration of the obtained non-uniform single struts: (a) uniform strut and its first-and second-order buckling modes; and non-uniform shapes and buckling modes with maximized (b) first-order eigenvalue and (c) second-order eigenvalue.

Figure 5 .
Figure 5.Effect of slenderness on the improvement in eigenvalues: (a) profiles of struts with different slenderness ratios obtained by optimization, and comparisons of (b) first-order and (c) second-order eigenvalue improvements of the struts obtained by the scaling method and optimization.

Figure 6 .
Figure 6.Effect of inclination angle on improvement of eigenvalues: optimization results for (a) first-order and (b) second-order eigenvalues under different angles, and (c) comparisons of the eigenvalue improvement of the non-uniform struts obtained by the scaling method and by optimization.

Figure 7 .
Figure 7. Schematic illustration of the proposed design method for creating lattice core sandwich structures (LCSSs) with superior buckling strength, taking the six-fold truss lattice core as an example.

Figure 9 .
Figure 9. Buckling modes and critical buckling loads of (a) Kagome, (b) body-centred cubic (BCC), (c) vertical-struts reinforced body-centred cubic (BCCZ), and (d) six-fold truss lattice cores with uniform struts (first row), non-uniform struts from the proposed method (second row) and from optimization (third row).Contour plots show the normalized strain energy density.

Figure 10 .
Figure 10.Nonlinear finite element (FE) analysis results of 3 × 3 vertical-struts reinforced body-centred cubic (BCCZ) lattice arrays: (a) load-displacement curves of uniform and non-uniform BCCZ lattice arrays; (b) deformations of uniform and non-uniform BCCZ lattice arrays at the displacement of 0.8 mm.Contour plots show the von Mises stress σ v .

Figure 12 .
Figure 12.Nonlinear finite element (FE) analysis and experimental load-displacement curves with shaded experimental variations and compressive deformations of (a) body-centred cubic (BCC) and (b) six-fold truss lattice core sandwich structures (LCSSs) with uniform struts (left) and proposed non-uniform struts (right).

Figure 13 .
Figure 13.Broken modes and fractured cross-sections of lattice cores: (a, b) fracture locations of uniform and non-uniform sixfold truss lattice core sandwich structures (LCSSs); (c, d) two images of the fractured cross-sections of a six-fold truss LCSS lattice core, in which the dense regions [highlighted by the dark (red) dashed lines] are surrounded by a layer of partially melted powders [highlighted by the pale (yellow) dashed lines].

Table 3 .
Design parameters of uniform and non-uniform vertical-struts reinforced body-centred cubic (BCCZ) and six-fold truss lattice core sandwich structures.

Table 5 .
Design parameters of body-centred cubic (BCC) and six-fold truss lattice core sandwich structures for selective laser sintering fabrication and testing.

Table 6 .
Comparison of the compressive strength of body-centred cubic (BCC) and six-fold truss lattice core sandwich structures from finite element (FE) analysis and experimental results.