Stiffness characterization of curved structures made of lattice materials: Additive manufacturing,measurements and numerical estimations

Abstract

Most experimental studies that characterize the mechanical properties of lattice structures have used samples with straight macro-shapes, such as cylinders or prisms. However, real-world applications often demand more complex geometries. As a first step towards addressing this, lattice structures with curved macro-shapes were designed with various cellular topologies and arrangements and subjected to transverse loads to evaluate their effective stiffness. This evaluation allows the characterization of the effect of the topology and arrangement on the stiffness of the latticed curved beams; obtaining a thorough understanding of the role of various design parameters to tune the effective stiffness of lattice structures. Load-displacement curves were obtained using laboratory tests, homogenization techniques, and finite element analysis. Experimental tests were done on two groups of additively manufactured samples subjected transversally at the midpoint. We analyzed the effective transverse stiffness of linearly arranged cellular topologies (square, hexagonal, and re-entrant) used to form curved beams. It was found that the hexagonal lattice had a 9.80 N/mm stiffness that was approximately 67% stiffer than the softest, which was the square lattice oriented at 45°. Then, each topology was modified for radial arrangement for further study on how conformal arrangements influence the effective transverse stiffness. The square lattice in a conformal (radial) arrangement had a 20.68 N/mm stiffness, that was approximately 130% stiffer than its linear arrangement counterpart.

Introduction

Lattice materials could be employed in various applications where specific material arrangements are needed within a structure or component. ¹ Lattices can be formed from the repetition of the smallest unit, known as unit cell. Depending on the type of elements that constitute them, there could be various topologies. By selecting the appropriate topology and adjusting the geometric parameters, designers can adjust the mechanical properties of the structure, tuning it for specific applications. For instance, lattice tubes have been designed for energy absorption, ² internal lattice cylinders for non-pneumatic wheels,^3,4 lattices for electromagnetic isolation in vehicle components, ⁵ and lightweight lattice structures for drone components. ⁶

The effective mechanical properties of lattice materials are dependent not only on the base material they are made of, but also on the topology, and shape of their constituent elements. Pan et al. ⁷ demonstrated that lattices with graded thickness in their unit cells exhibit higher energy absorption than lattices with uniform thickness. Gibson ⁸ analyzed porous structures in various types of wood, finding that the topology plays a crucial role in the material's mechanical properties. Gibson and Ashby ⁹ explored how the orientation of unit cells affects lattice stiffness, noting that it can either increase or decrease depending on the topology. Shen et al. ¹⁰ studied the effect of wall thickness on lattice elasticity, concluding that thicker walls result in a higher elastic modulus. Tsuyoshi et al. ¹¹ compared lattice samples made from square, hexagonal, and re-entrant hexagonal topologies, observing different Poisson's ratios—positive, near- zero, and negative, respectively, under tension. Hence, both stiffness and Poisson's ratio can be controlled by selecting the appropriate unit cell topology.

The mechanical properties of cellular and lattice structures are typically characterized using samples with a macro-shape of either a prism or a cylinder. A macro-shape refers to the external geometry containing the unit cells. For example, Ho et al. ¹² examined lattice structures with cubic macro-shapes for air vehicle applications, while Alketan et al. ¹³ applied surface unit cells (TPMS) over cubic shapes for vertical displacement loading. Zheng et al. ¹⁴ tested ultralight stiff lattices with cubic macro-shapes, and other studies, such as those by Ren et al. ¹⁵ and Nikolaos et al., ¹⁶ used beam macro-shapes to investigate buckling under transverse loading. Additionally, Li and Shen ¹⁷ and Dongseok et al. ¹⁸ focused on stress concentration in beams subjected to vertical loading, varying unit cell density to enhance stiffness. These studies show that common macro-shapes for testing lattice behavior are cubes and straight beams. However, recent research has explored alternative macro-shapes to uncover new lattice properties and applications. For example, Dezianian and Azadi ¹⁹ used cylindrical macro-shapes for lattice topology optimization in non-pneumatic tires, adapting the lattice to a curved arrangement. Jaehyung et al. ²⁰ employed cylindrical shapes for flexible cellular spokes in tires, and Yoder et al. ²¹ investigated hexagonal and re-entrant hexagonal lattices adapted to long macro-shapes. Nguyen et al. ²² researched conformal lattice design, where the direction of the unit cells is adjusted to follow an irregular macro-shape.

The mentioned works represent the state of the art for the analysis of lattice applied over conventional and non-conventional macro-shapes. However, the analysis of these works is limited due to the lack of lattice arrangement variation. The arrangement of a lattice can be modified by the way the unit cells interconnect to conform the lattice. This modification can be done by changing the direction angle of the unit cells pattern ²³ or by changing the direction of each unit cell independently to adapt a lattice to an irregular body. ²⁴ Arrangement variation can change the mechanical behavior of the lattice without altering the dimensions of the unit cells. ²⁵ This presents an open issue for the enhancement of mechanical characteristics of a lattice applied over non-conventional macro-shapes without altering the topology, volume fraction, or the dimensions of the unit cell.

The objective of this work is to investigate the mechanical behavior of lattices applied to curved beams, focusing on different unit cell topologies and arrangements. Specifically, square, hexagonal, and re-entrant lattices are analyzed, as each offers a distinct Poisson's ratio: zero, positive, and negative, respectively. The study examines the impact of unit cell arrangement on the curved beam, including a case where the arrangement follows the curvature of the beam's edges. First, simulated vertical displacement tests were conducted on curved beams, assessing vertical displacement, stress concentration, and stiffness. Then, the curved beams were additively manufactured and tested experimentally under transverse loading at the midpoint. Finally, the deformation shapes and stress concentrations of the different geometries and arrangements were analyzed to identify the stiffest structure.

Materials and methods

This section details the design parametrization for the curved beams and the unit cell topologies for analysis. The aspects related to finite element analysis and additive manufacturing are also given here.

Design of lattice curved beam

The curved beam used as reference through the remaining of this work is shown in Figure 1 and was computationally modelled using the SOLIDWORKS^® (2021 Dassault Systems, Vélizy, France). The radius and thickness were chosen to fit within the printing capabilities of an Ultimaker 2+ additive manufacturing (AM) machine.

Figure 1.

Curved beam dimensions.

The curved beams were then filled with the cellular topologies shown in Figure 2. The dimensions of these three unit cells to be applied to the macro-shape are also shown in Figure 2. These topologies exhibit different properties under tension, e.g., in their Poisson's ratios. The square topology has a Poisson's ratio close to zero, while the hexagonal and re-entrant hexagonal unit cells have positive and negative Poisson's ratios, respectively. ¹¹ The selected dimensions for each topology ensure a volume fraction of 60 ± 1% for the beams across all models and samples. To characterize the influence of arrangement orientation on effective stiffness, different orientation angles (β), measured relative to the x-axis, were used (see the right hand side on Figure 2). Each topology is considered in two orientation angles: 0° and 45° for the square topology, 0° and 30° for the hexagonal topology, and 0° and 90° for the re-entrant hexagonal topology.

Figure 2.

Unit cell topologies, parameters and dimensions.

To evaluate how the cellular pattern arrangement affects the effective beam stiffness, two configurations of the square topology were proposed: linear and radial. A linear arrangement involves a distribution of the unit cells along the x-axis and y-axis regardless of the macro-shape. A radial arrangement consists of angular distribution of unit cells that follow the macro-shape (curved in this case) of the beam. The radial arrangement could also be known as a conformal arrangement ²⁶ as it adapts to the shape of the intended macro-shape.

Consider the square topology as an example, its integration into the curved lattice beam in linear arrangement is illustrated on the top part of Figure 3. The process involves intersecting the curved beam macro-shape with the lattice created using each topology and orientation angle. The result is a curved beam filled with the designed lattice topologies. The inset on the bottom part of Figure 3, illustrates the other lattice arrangement variations studied.

Figure 3.

Process to generate a curved beam with a linear arrangement of unit cells.

An effect of applying bidimensional lattices to a curved macro-shape is the appearance of incomplete unit cells at the edges as exemplified in Figure 4. These incomplete unit cells are defined as open cells due to the open walls that they generate in the latticed curved beam. On the other hand, in the middle zone of the latticed curved beam, the unit cells interconnect with each other generating a continuous lattice. Each topology applied on a linear arrangement on the macro-shape generates open cells at the edges that affect the continuity of the lattice.

Figure 4.

Open cells and lattice continuity definition.

The process for applying a radial (conformal) lattice arrangement to the curved beam macro-shape could be exemplified as shown in Figure 5. The first step involves creating a square lattice in linear arrangement filling a straight beam with the same width as the curved beam macro-shape. The length of the straight beam corresponds to the radius of the curved beam macro-shape, and the unit cells used in the straight beam have the same dimensions as those in the linear arrangement. Then, the straight beam is bent using the bending function in SolidWorks^® to form a 180° arc with the desired radius. Similarly to the technique of Zhao et, al. ²⁷ that used crystal patterns to change the nodes of interconnected unit cells, our radial arrangement allows us to change the interconnection of the unit cells without altering the unit cell dimensions. Changing the mechanical behavior of the lattice without variation on the unit cell size, topology and lattice volume fraction.

Figure 5.

Process to generate a curved beam with a radial (conformal) arrangement of unit cells.

A visual comparison between the linear and radial arrangements using the square lattice is presented in Figure 6. Note in the linear arrangement, Figure 6(a), the open unit cells of the lattice at the edges of the curve. Note also the 2.86° angle formed in the radial arrangement, Figure 6(b), due to the bending to conform the lattice. The black dotted line indicates how the corresponding cell wall results in the same curvature as the beam defined in Figure 5, while the walls aligned with the beam radius remain straight (blue dotted line). The square unit cells in the radial arrangement maintain the same length along the x-axis and cell wall thickness as those in the linear arrangement.

Figure 6.

Square lattice unit cells in a) linear arrangement and b) radial arrangement.

Characterization of effective properties through numerical homogenization

Similarly to approaches reported in,^28,29 the stiffness of the curved structures was also analyzed by means of homogenization. This type of analyses enables a comparison of the effective properties towards the possibility of reducing computational resources by modeling solid structures rather than those with micro-structures. It also exposes the effect or the role of the edge effects of using lattice structures in structures with curved edges. The effective mechanical properties for homogenization were characterized using ANSYS^® Material designer (v2022, National Instruments, USA) using the unit cells defined in Figure 2 for the square, hexagonal and re-entrant lattices. The software tool uses periodic boundary conditions to obtain the effective properties of the unit cells along multiple directions to obtain the Young's modulus (E), shear modulus (G) and Poisson's ratio (V) in the x-, y- and z-axes. The solid curved beams were modelled and subjected to the loading and boundary conditions as their lattice counterparts (this will be described in Section 2.4) but fed with the effective properties. These were meshed with linear hexahedrons of size 0.7 mm.

Additive manufacturing of curved lattice samples

Curved beam samples were additively manufactured via Fused Filament Fabrication with an Ultimaker^® 2 + . The samples manufactured are shown in Figure 7 corresponding to the different topologies and arrangements used in experimentation. The material used for printing was Ultimaker^® polylactic acid (PLA) in black, with a 100% infill and a 0.1 mm layer profile.

Figure 7.

a) Additively manufactured beams different topologies and arrangements and b) sample located in the texture analyzer machine.

Mechanical characterization, equipment and conditions

Curved lattice beams were subjected to transverse loading in a Perten Instruments^® TVT6700 texture analyzer. Figure 7 depicts the boundary conditions of the loading scenario used. The input displacement (red color) is applied at the superior edge A of the curved beams as seen in Figure 7. Pin support (blue color) are applied at the inferior edges B of the curved beam that are closest to the arc center. Pin support are fixed along the y- and z-axes but free along the x-axis. The inferior edges closest to the arc center were selected for the application of the pin support to recreate the deformation of the beam on the texture analyzer machine, as other edges would separate from the floor while the closest ones stay fixed in y- and z-axes. Towards the characterization of the effective stiffness, the tests are divided in three groups. The first group characterizes the role of topology in linear arrangements. The second group characterizes the role of cellular pattern orientation, two angles for each topology were studied. For these two groups, the vertical input displacement is set to 10 mm. The third group characterizes the influence of the arrangements, linear or radial, of the square unit cell with a 0° orientation angle. The vertical displacement for these is 6.6 mm, a lower value than in the first analysis to avoid exceeding the force limits of the texture analyzer, as the lattice curved samples in this group exhibit higher stiffness.

The finite element models (FEM) of the curved beams were built in ANSYS^® Workbench (v2022, National Instruments, USA) to subject them to the linear elastic loading scenarios previously described. Boundary conditions were set shown in Figure 8, and the beams were meshed using Tetrahedral elements. The material properties of PLA and the mesh size used are provided in Table 1. The automatic convergence option was applied to determine the final element size. From the FE models, vertical displacements, forces, and stress distributions were obtained. Effective stiffness S is calculated with the ratio between the force F and the displacement d as $S=F/d$ .

Figure 8.

Loading case boundary conditions for a compression test with transversal displacement applied at A. y- and z-axes fixed at B but x-axis is kept free in a) radial arrangement beams and b) linear arrangement beams.

Table 1.

Computational simulation parameters.

Parameter	Value
Mesh size	0.5 mm
PLA Young's modulus	800 MPa
PLA Poisson's ratio	0.3

Experimental tests were run on triplicates of each design. The mean and standard deviation of the data were calculated for each topology and orientation angle. Vertical displacement and force data were measured and exported. These experimental results were then compared to those obtained from finite element simulations.

Results and discussion

The set of results that allow us to relate the design parameters of the topologies and the arrangements to the effective stiffness are presented and discussed next.

Effect of cell topology in linear arrangement

The average load-displacement curves for the three topologies oriented at an angle of 0° are presented in Figure 9. Each curve exhibited a linear region before yielding. The length of this linear region varies depending on the cellular material topology arrangement used. The stiffness was calculated from this linear region, defined from 0 to a vertical displacement of 1.2 mm for all topologies. The load obtained from the FEMs is also included in Figure 9 with a dotted red line for comparison.

Figure 9.

Experimental average and finite element load-displacement curves for curved beams with square, hexagon and re-entrant unit cells in linear arrangement and β = 0◦.

The effective stiffness values calculated are provided in Figure 10. Curved beams with hexagonal cellular topology exhibited the highest stiffness, 3.23% greater than those with square lattice and 33.59% higher than those with re-entrant topology. The minimum error between experimental results and FE estimations was seen in the square topology with β = 45°, with a percentage error of 1.68%. The maximum error was seen in the hexagon topology with β = 30° with a percentage error of 9.49%. Data variability was minimal, with the standard deviation remaining below ±0.5 N/mm. To explain why hexagonal topology is stiffer than the other lattices, a visual comparison of the deformation shapes of the latticed curved beams is shown in Figure 11.

Figure 10.

Effective stiffness comparison.

Figure 11.

Stress maps and deformation comparison between simulated and experimental beams with linear arrangement and β = 0◦.

The deformed shapes of the lattice arrangements on the additively manufactured beams closely resemble those observed in the FE simulations. While the square lattice is typically stiffer than bending-dominated structures, such as the hexagonal or re-entrant lattices, when subjected to in-plane compressive or tensile loads, this was not the case in the present study. Structures subjected to loads that induce bending experience stretching and compression (depending on the direction of bending). Meaning that the resulting stiffness behavior of the lattices cannot be explained only by analyzing if their unit cells have a bending-dominated or a stretch-dominated deformation.

A better explanation for the stiffness behavior of the beams can be obtained from the analysis of the open unit cells that resulted at the edges of the curved beam macro-shapes as seen in Figure 11. These open cells consist of incomplete unit cells, which were cut off at the edges in order to follow the intended macro-shape. These cells were more evident in the beams made of the square and re-entrant topologies. Open cells break the lattice continuity, that is defined as the sequence of complete unit cells that compose the lattice. The lack of lattice continuity makes gaps in the stress distribution and compromises the structure's capacity to withstand loads. To compare and analyze the effect of the lattice continuity on each topology, the percentage of open cells for each topology is shown in Table 2. It is seen that the hexagonal lattices have a lower percentage of open cells, which grants them a higher lattice continuity and a higher stiffness. As well, due to the low lattice continuity along the arc-length of the curved macro-shape, the stiffness of the square and re-entrant lattices was lower.

Table 2.

Open cells percentage for lattice curved beams for each topology.

Topology	Total cells	Open cells	Open cells percentage
Square 0°	231	63	27.27%
Square 45°	235	85	36.17%
Hexagon 0°	272	69	25.36%
Hexagon 30°	279	62	22.22%
Re-entrant 0°	285	89	31.22%
Re-entrant 90°	281	91	32.38%

The differences in stiffness values between the simulations and experiments could be attributed to imperfections in the additive manufacturing process. In additively manufactured beams, the design of a solid body is transformed into a sequence of layers, reducing homogeneity. Additionally, imperfections can be observed in Figure 11, where features that were straight in the computational models appear slightly curved in the additively manufactured samples.

Effect of pattern orientation angle in linear arrangement

In this subsection, the impact of the orientation angle on the stiffness of latticed curved beams with a linear arrangement is analyzed. Figure 12 presents the load-displacement curves for each topology, comparing the two orientation angles studied for each. Note that the topology with higher sensibility to the orientation angle is the square lattice (see Figure 12(a)), while the difference between the other topologies are minimal or negligible as shown in Figure 12(b) and 11(c), respectively.

Figure 12.

Average experimental load-displacement curves for all curved beams with unit cells in linear arrangement. a) square unit cells, b) hexagonal unit cells, c) re- entrant unit cells.

The effective stiffness values obtained from this orientation angle comparison are also shown in Figure 10. The square lattice exhibited the largest difference in effective stiffness, with an approximately 40% change when the orientation angle was modified. In contrast, the hexagonal and re-entrant honeycomb topologies showed only minor variations in stiffness, with differences of approximately 1% and 0.6%, respectively.

To further investigate the stiffness difference resulting from changes in the orientation angle, Figure 13 includes the CAD models and corresponding stress maps of the curved beams, contrasted with the additively manufactured samples in key regions. As noted in the previous section, square and re-entrant topologies display open cells at the edges of the curved beam, whereas the hexagonal unit cells (especially in the 30° arrangement) maintain better lattice continuity, avoiding open cells.

Figure 13.

Stress maps and deformation comparison between simulated and experimental beams with linear arrangement and β > 0◦.

The largest stiffness difference due to orientation angle changes was observed in the square topology, and this can be attributed to two factors. First, when oriented at 45°, the middle region near the loading point lacked lattice continuity, as seen in Figure 13. In contrast, when oriented at 0°, this region exhibited more lattice continuity, which could result in a higher load withstanding structure. Second, the square lattice exhibits in-plane directional dependence. Previous studies ³⁰ have shown that a square lattice is stretch-dominated when loaded parallel to cell wall direction but bending-dominated at other orientations. Therefore, any change in angle significantly affects the effective stiffness. These differences could be attributed to the fact that the only topology that changes its deformation mechanism when changing its orientation is that of the square lattice. Both the reentrant and the hexagonal are bending-dominated structures, regardless of their orientation with respect to the loading direction. It is then expected that variations in stiffness when using them will be minimal. On the other hand, the square lattice, when changing from the 0° orientation (stretch-dominated) to 45° (bending-dominated) the effective Younǵs modulus can drop by 30% (for the relative density tested here). These results will be further discussed in the next section.

Minimal stiffness differences were observed in the hexagonal topology samples, likely due to the near in-plane isotropy of the hexagonal honeycomb structure, resulting from its rotational symmetry. ³¹ A similar explanation can be applied to the re-entrant and hexagonal structures, as they both maintain similar lattice continuity, as shown in Figure 11 and 12. The impact of lattice continuity along the arc length is further discussed in the next section.

Minimal stiffness differences were observed in the curved beams with hexagonal unit cells. This could be attributed to the hexagonal lattice being nearly in-plane isotropic as per its degrees of rotational symmetry. Another important feature that could lead to similar stiffness results in beams with re-entrant and hexagonal unit cells, is the similar lattice continuity where the load is applied (see Figures 10 and 12). Then, similarity in lattice continuity results in similar load bearing capacity and, thus, similar stiffness.

Transverse response of curved beams with homogenized effective properties

For a deeper analysis on the effect of open cells on the stiffness of the curved beam, a comparison analysis between latticed models and solid beams homogenized with equivalent material properties was made. The effective mechanical properties of the unit cells, considering them to be orthotropic materials, are listed in Table 3. Note that, hexagonal and re-entrant topologies have a symmetric variation on some properties (E_x, E_y, V_yz, V_xz, G_yz, G_xz) due to their rotational symmetry and the volume fraction used.

Table 3.

Equivalent orthotropic properties for each topology and orientation.

Unit cell	Ex (MPa)	Ey(MPa)	Ez(MPa)	Vxy	Vyz	Vxz	Gxy(MPa)	Gyz(MPa)	Gxz(MPa)
Square 0°	412.06	411.91	659.48	0.03	0.18	0.18	398.40	217.31	217.33
Square 45°	295.37	295.58	659.48	0.73	0.06	0.06	388.58	348.15	348.19
Hexagon 0°	374.68	401.70	572.45	0.68	0.04	0.05	415.45	205.24	220.68
Hexagon 30°	401.70	374.68	572.45	0.68	0.05	0.04	415.45	220.68	205.24
Re-entrant 0°	320.78	318.57	240.58	−0.39	0.07	0.12	345.51	348.57	351.49
Re-entrant 90°	318.57	320.78	240.58	−0.39	0.12	0.07	345.51	351.49	348.57

Finite element models were used to characterize the stiffness of curved beams made of an equivalent material. As an example of the resulting stress maps, a comparison between the curved beam with square lattice at β = 45° and the solid beam made from the equivalent material with properties listed in Table 3 is presented in Figure 14(a). The deform shapes of the beams are similar, and the stresses concentrate on analogous areas. The corresponding effective stiffness calculated for these cases is compared to those obtained from the lattice beams in Figure 15. Overall, homogeneous beams with equivalent properties resulted stiffer than those modeled with the lattice structures. These differences could be attributed to the effect of the open cells. Note in Figure 15 that the curved beam with square lattice at β = 45° has a considerable number of open cells at the edges as reported in Table 2. The resulting lattice continuity in the square topology at β = 45° makes 28.1% softer than their solid homogeneous counterpart.

Figure 14.

a) Stress maps of lattice curved beam of square topology with β = 45° and its equivalent material counterpart and b) deform shapes scaled to 2:1 of open cells on top of the beam.

Figure 15.

Effective stiffness comparison between latticed curved beams and homogeneous equivalent materials using finite element simulation.

As the response of the curved structures under the transverse load is bending, the top edge of the beam is in compression while the bottom edge is in tension. Due to the lack of interconnectivity of the unit cells in the structure (see Figure 14(b)), these edges will deform more by opening in these regions, reducing their stiffness. To exhibit this, we have included deform shapes scaled by a factor of 2 in Figure 14(b). This behavior is analogous to what one would expect when a structure with struts oriented at 45°, and no structural boarders, is stretched. All the elements close to the edges will open, as these offer low stiffness due to the open segments.

This suggests that an application of the lattices over curved beams while avoiding open cells could lead to structures with ∼ 30% higher stiffness; here 28.1% for the Square at 45° and 21.7% for the Square at 0° as presented in Figure 15. To reduce the effects of open cells at the edges, increasing the number of unit cells within the structure. If one needs to keep the same volume fraction, adjustments on the design parameters dimensions will be required, making this approach challenging both computationally and experimentally. So, another alternative could be having lattice structures that adapt to the outer shape. Lattices that can conform to the outer shape of the structure they are filling could result in superior properties; this will be further discussed in the next section.

Effect of lattice arrangement on lattice beams

The role of the type of arrangement in the mechanical properties of these curve structures was characterized using FE models of the three topologies in radial arrangement. These models are shown in Figure 16 along with the stress distribution obtained from the loading and boundary conditions explained in Section 2. In all three topologies, the radial arrangement completely avoids having incomplete unit cells at the edges of the latticed beam: this results in a structure with a better lattice continuity when compared with their linear arrangement counterparts. Due to the loading conditions, and the bending response of the structure, the upper and lower segments of the beams carry most of the load. This is reflected in higher stress concentrations as seen in Figure 16.

Figure 16.

Finite element simulations of square, hexagonal and re-entrant latticed curved beams on radial arrangement.

The effective stiffness characterized from the models shown in Figure 16 is plotted in Figure 17. Here it is compared with that corresponding to the linear arrangement samples. Regardless of the topology, the radial arrangement of the unit cells increases the stiffness of the model approximately a 100% more than the linear arrangement, mainly due to two aspects. First, the number of open cells at the edges, and second the alignment of matter (or cell walls) that follow the outer shape of the samples. The latter enhances the loading bearing capacity of the structures. This exhibits how this conformability of the cell walls to the outer shape allows having superior properties without the increasing the volume fraction drastically.

Figure 17.

Finite element effective stiffness of radial and β = 0° linear arrangement latticed curved beams.

For validation of the finite element results, experimental test was made for square latticed curved beams of both arrangements. Figure 18a and 18(b) present the average force-displacement curves for the curved beams constructed from the square topology, with linear and radial arrangements, respectively. The dotted red line highlights the F.E. force curve from the linear regions from which the stiffness is calculated (in the range 0.0 ≤ d ≤ 6.6 mm).

Figure 18.

Load-displacement curves for beams with square unit cells. a) Average and standard deviation of experimental curves in linear arrangement (β = 0◦), b) average and standard deviation of experimental curves in radial arrangement.

The effective stiffness of the radial arrangement for the square lattice is 20.12 ± 0.13, while the stiffness of the linear arrangement is 8.74 ± 0.24. The radial arrangement is the only one among the designs tested here that ensures lattice continuity along the entire arc length. As a result, there is a significant increase in stiffness—131.41% higher compared to the linear arrangement. This substantial improvement in stiffness can primarily be attributed to the way the lattice is now arranged. As shown in Figure 19, the radial arrangement achieves lattice continuity even at the edges. These edge regions are not added separately but are integral to the radial conformal arrangement, and they bear most of the load, as reflected in the stress maps in Figure 19. Consequently, no open cells are present in the radial arrangement. In contrast, non-conformal arrangements, such as the linear ones, do not provide this advantage. In these cases, additional boundary edges were not incorporated, which would have taken on most of the load-bearing responsibility, potentially distorting the true contribution of the cellular topology to the effective stiffness.

Figure 19.

Stress maps and deformation comparison for the linear and radial arrangements.

Conclusions

The stiffness of architected curved beams was investigated both computationally and experimentally using additively manufactured samples. The study characterized variations in cell topology, orientation angle, and arrangement when these structures were subjected to transverse loading. Among all the variations, the radial arrangement exhibited the highest stiffness, which was attributed to the lattice continuity that enhanced its load-bearing capacity. The orientation angle had a lesser effect on the stiffness of the hexagonal and re-entrant honeycomb structures, due to their rotational symmetry and in-plane isotropic properties. Within the linear regime, the stiffness values were contrasted to simulation results from finite element (FE) models, showing good agreement, since all errors were below 10%. The observed discrepancies were attributed to manufacturing limitations and imperfections. This study focused on three types of 2D structures, which provided valuable insights. However, to fully understand the mechanical properties of cellular structures, especially when they are used to fill complex macro-shapes, further research is required. Future studies could explore additional topologies, macro-shapes, and loading conditions. Since engineering and other applications often involve more complex macro-shapes, this study is relevant for expanding the range of applications for cellular and architected materials.