A novel parameter to tailor the properties of prismatic lattice materials

Lattice materials are extremely efficient in combining high stiffness and strength at low densities. Their architecture is a periodic assembly of bars, which, in most cases, all have the same length and cross-section. This is, however, suboptimal since the level of stress is not the same in all bars. To take these variations into account, we propose to design prismatic lattices with two different bar thicknesses. The ratio of these two thicknesses introduces a new parameter in the design of lattices. Analytical expressions are developed to capture the effect of this new parameter on the elastic modulus, failure mode and compressive strength of hexagonal and triangular lattices. This analytical work is then validated by finite element simulations and experiments performed on polymer lattices fabricated by additive manufacturing. This new parameter offers two advantages in the design of prismatic lattices. First, the thickness ratio can be used to vary the properties of a lattice without changing its relative density. Second, it allows to stiffen and strengthen the lattice along a specific loading direction and therefore, controls the degree of anisotropy. This work opens new possibilities to tailor the mechanical properties of prismatic lattices, and facilitates the creation of new materials by design.


Introduction
Lattice materials can be designed to have precise, and often unique, mechanical properties [1,2].Their properties are a function of three parameters: (i) the parent material from which the lattice is made of, (ii) its topology and (iii) its relative density ρ (which represents the volume fraction of material) [3].For example, the elastic modulus of a lattice can be expressed as [4]: where   is the Young's modulus of the parent material, whereas  and  are two constants dependent upon the topology of the lattice.
Optimising the architecture to maximise the elastic modulus has been the subject several investigations, and a number of highly efficient isotropic topologies have been identified recently [5][6][7].Fabricating these optimised topologies has become significantly easier with the recent and rapid development of additive manufacturing [8,9].On one hand, this technology facilitated the production of existing concepts, such as truss lattices [10][11][12][13][14]; shell lattices [15,16]; and hierarchical topologies [17][18][19][20][21][22].On the other hand, additive manufacturing and rapid prototyping technologies opened new opportunities to customise the design of lattices and enhance their properties.One example is the development of ultralight micro-lattices that are strengthen by material size effects [23][24][25][26][27].Other examples include lattices with tapered beams [28][29][30], or an origami architecture [31] to increase their energy absorption; topologies combining two dissimilar materials to achieve high strength and high compressive strains [32]; and architectures with wavy struts to enhance their tensile fracture strain [33,34].
Additive manufacturing also offers the possibility to optimise the properties of lattices by changing the distribution of material.Previous studies on this topic can be categorised in two groups depending if the optimisation is done at (i) the structural length-scale or (ii) the unit cell level.In the former group, the properties of a lattice structure are optimised for specific loading conditions by spatially varying the relative density [35][36][37][38][39].These are commonly referred to as graded architectures.In the latter group, computational topology optimisation is used to design the unit cell of the lattice, based on a set of constraints and objective functions [40][41][42][43].While these optimisation approaches can lead to highly efficient topologies, they are computationally expensive and often restricted to elastic properties only.These limitations highlight the need to investigate analytically the effect of mass distribution in lattices, and this is the subject of this study.
In this work, we propose to tailor the properties of prismatic lattices by introducing two different bar thicknesses,  1 and  2 , as shown in Fig. 1.For most loading cases, these two bars are carrying different loads and therefore, optimising their thickness accordingly has the potential to increase performances.This geometrical change introduces a new design parameter, the thickness ratio t =  1 ∕ 2 .We will show https://doi.org/10.1016/j.ijmecsci.2022.107079Received 20 August 2021; Received in revised form 18 December 2021; Accepted 10 January 2022 below that t has a pronounced effect on the properties: it transforms the constant  in Eq. ( 1) into a function of t, and this offers an opportunity to modify the properties without changing the relative density ρ of the lattice.The objective of the paper is to quantify the effect of t on the elastic modulus, compressive strength and failure modes of prismatic lattices.Our investigation focuses on hexagonal and triangular lattices since they have strikingly different behaviours: the former is bending-dominated whereas the latter is stretching-dominated.
The in-plane properties of triangular and hexagonal lattices with t = 1 have been investigated both analytically and experimentally [44][45][46].Hexagonal lattices with t = 2 have also been the subject of many investigations [47][48][49][50][51][52][53][54].These are expanded honeycombs, fabricated by gluing and pulling apart strips of materials, and therefore, their different bar thicknesses are simply a consequence of the manufacturing route.Nonetheless, the bar thickness has been considered as a design parameter in a number of studies.For example, hexagonal lattices with cell walls that vary in thickness along their length (thicker at the nodes than in the middle) can be stiffer and stronger than their uniform counterparts [55][56][57][58].Others, have design honeycombs with a gradient in bar thickness to increase the transverse shear modulus [59] or, in most cases, the energy absorption capacity [60][61][62][63][64].Note that in these graded honeycombs, the bar thickness is varied at different locations inside a structure (thereby modifying structural properties), which is different from the approach proposed here where the distribution of material is changed within the unit cell (tuning its material properties).Another approach to improve the properties of honeycombs has been to completely re-design their geometry, see the review of Qi et al. [65].This, however, produces stepwise changes in properties, whereas the thickness ratio t proposed in this study enables to vary properties in a continuous manner.
This paper is structured as follows.Analytical expressions are developed in Section 2 to capture the effect of t on the mechanical properties of hexagonal and triangular lattices.These analytical results are presented in Section 3 and then validated using finite element simulations in Section 4. The study is complemented by experiments performed on samples manufactured by additive manufacturing, and these are presented in Section 5.In these experiments, t is varied by changing both  1 and  2 while keeping the relative density fixed.Finally, the main advantages of t are discussed in Section 6.

Analytical modelling
In this section, analytical expressions are derived for the elastic modulus and strength of prismatic lattices with different bar thicknesses, see Fig. 1.Triangular and hexagonal topologies are considered in turn, and equations are derived for uniaxial compression in  1 and  2 directions.Both lattices are assumed to be made from a linear elastic, perfectly plastic material, characterised by a Young's modulus   and a yield strength   .The cell walls of the stretching-dominated triangular lattice are considered to behave as pin-jointed trusses, whereas those of the bending-dominated hexagonal topology are modelled as Euler-Bernoulli beams.These assumptions generally acceptable provided that lattices have a relative density ρ ≤ 0.3 [4] and that the cell walls of the hexagonal lattice are sufficiently slender ( 1 ∕ and  2 ∕ are less than 0.2) to neglect axial and shear deformations [44].

Triangular lattice
Consider the triangular lattice shown in Fig. 1a, where all bars have a length , but not the same thickness.The horizontal bars have a thickness  1 , whereas the diagonal struts have a thickness  2 .Therefore, the relative density of the triangular lattice is given by: . (2)

Loading in 𝑥 1 direction
First, consider the triangular lattice loaded in the  1 direction by a uniform compressive stress  1 .The triangular lattice is stretchingdominated, and the axial forces in bars 1 and 2 are given by [45]: respectively, where  is the out-of-plane dimension, and the negative sign indicates a compressive force.We emphasise here that the axial forces  1 and  2 are independent of the bar thickness.Since only bar 1 is carrying load, the nominal compressive strain of the lattice is given by: where   is the Young's modulus of the parent material.The elastic modulus of the lattice in the  1 direction is  1 =  1 ∕ 1 , which gives: where the thickness ratio t =  1 ∕ 2 .Next, we turn our attention to the compressive strength of the triangular lattice.Two collapse mechanisms are possible: (i) elastic buckling and (ii) yielding.Elastic buckling will occur when the compressive load in a bar reaches the Euler buckling load [66]: where  is the second moment of area and  is the end constraint factor ( = 1 for pinned joints, whereas  = 2 when both ends are fixed).When compressed in the  1 direction, only bar 1 is loaded in compression (see Eq. ( 3)), therefore setting  1 =   gives us the elastic buckling strength: where the subscript  and the superscript [ 1 ] are used to indicate that bar 1 is buckling elastically.It is important to mention that the end constraint factor  varies with t.It is possible to show analytically that  decreases from 2 to 1 as t increases from 0 to 5.This lengthy analysis relies on beam-column theory [67,68] and is provided in Appendix A.1.The collapse mode is anticipated to change from elastic buckling to yielding as the relative density is increased.The yield strength of the lattice is obtained by setting  1 =  1   , which gives: Above, the subscript  and the superscript [ 1 ] are used to indicate that yielding occurs in bar 1.Finally, the lattice will collapse in the mode associated with the lowest stress, and therefore, its compressive strength is given by: 1, , Consider the triangular lattice loaded in the  2 direction by uniform compressive stress  2 .For this loading scenario, the axial forces in bars 1 and 2 are [45]: respectively.The nominal compressive strain  2 can be obtained by equating the internal strain energy to the work done by the external stress  2 .For the unit cell in Fig. 1a, the internal strain energy is: whereas the work done by the external stress is  = 2 √ 3 2  2  2 .Setting  =  and solving for  2 returns: The elastic modulus in the  2 direction is simply  2 =  2 ∕ 2 , and this yields: The compressive strength of the triangular lattice in the  2 direction is controlled by either elastic buckling or yielding.Elastic buckling will occur when  2 =   , which yields: 2  32 Again, the end constraint factor  varies with t and its value is provided in Appendix A.1.Otherwise, yielding can occur in bar 1 or 2 since both struts are carrying load, see Eq. (10).Yielding in bar 1 will occur when  1 =  1   , whereas bar 2 will yield when  2 =  2   .This gives us: 2, = 3 ρ  and ( 15) Finally, the compressive strength  2 of the triangular lattice is governed by the collapse mode which occurs at the lowest stress and this can be expressed as: 2, ,

Hexagonal lattice
The hexagonal lattice is shown in Fig. 1b and its relative density is given by:

Loading in 𝑥 1 direction
Consider the hexagonal lattice loaded in the  1 direction by a uniform compressive stress  1 .If the axial deformation of bar 1 is neglected, the hexagonal lattice deforms by bending bar 2, see Fig. 1b.Based on Gibson and Ashby [44], the nominal compressive strain is: and the elastic modulus of the hexagonal lattice is  1 =  1 ∕ 1 , which yields: When the hexagonal lattice is compressed along the  1 direction, three collapse modes are possible: (i) elastic buckling of bar 1, (ii) yielding in bar 1, and (iii) plastic collapse in bar 2. The axial force in bar 1 is: and elastic buckling will occur when  1 is equal to the Euler buckling load, which gives: Again, the end constraint  varies with t and its value is derived in Appendix A.2. Next, the stress causing yielding in bar 1 is obtained by setting  1 =    1 and this returns: Plastic collapse in bar 2 will occur when the bending moment reaches the fully plastic moment   =    2 ∕4.Adapting the expression of Gibson and Ashby [44] gives us: 1, = 9 2 Finally, the collapse mode associated with the lowest stress will dictate the compressive strength of the lattice: 1, , 1, ,

Loading in 𝑥 2 direction
When the hexagonal lattice is loaded by a uniform compressive stress  2 in the  2 direction, it deforms by bending bar 2. The nominal compressive strain is obtained by adapting the result of Gibson and Ashby [44], which gives: The elastic modulus in the  2 direction becomes: Comparing this result to Eq. ( 20) reveals that the hexagonal lattice has the same elastic modulus in both  1 and  2 directions.For compression in the  2 direction, there is a single failure mode: plastic collapse in bar 2. Therefore, by adapting the expression of Gibson and Ashby [44], we find that the compressive strength  2 of the hexagonal lattice is: Finally, note that all analytical expressions presented in this section are in agreement with previous work done by Gibson and Ashby [44] and Wang and McDowell [45]: the properties of uniform triangular and hexagonal lattices can be recovered by setting t = 1, whereas those of expanded honeycombs (with double thickness vertical walls) can be retrieved with t = 2.

Analytical results
Analytical results are presented in this section to quantify the effect of t on the elastic modulus, collapse mode and compressive strength of both triangular and hexagonal lattices.

Elastic modulus
The effect of t on the elastic modulus of the triangular lattice is shown in Fig. 2a.The elastic modulus  1 is plotted in red, whereas  2 is in black.The triangular lattice is stretching-dominated and therefore, both moduli are normalised by ρ  .Consequently, the results are independent of the relative density ρ and of the choice of parent material (  ), see Eq. ( 5), (13).
The results in Fig. 2a show that  1 increases monotonically with increasing t.On the other hand, the elastic modulus  2 reaches a maximum value of 0.36 ρ  at t = 0.5, which corresponds to  2 = 2 1 .This peak was anticipated since the axial force  2 = −2 1 , see Eq. (10).The main advantage offered by t is the possibility to vary the elastic modulus of the lattice while keeping its relative density ρ fixed.For the triangular lattice,  1 can be varied over a wide range of values, increasing more than five-fold when t is varied from 0.25 to 3. Of course, increasing  1 by increasing t also leads to a reduction in  2 .Note, however, that  1 increases more rapidly than  2 decreases for t > 0.5.For example, compare the elastic moduli of a triangular lattice with t = 3 to those of a conventional lattice with t = 1.The lattice with t = 3 has a modulus  1 80% higher than that with t = 1, whereas the reduction in  2 is only 35%.Therefore, the trade-off between  1 and  2 can be advantageous.
The elastic modulus of the hexagonal lattice is plotted as a function of t in Fig. 2b.The hexagonal lattice is bending-dominated and therefore its elastic modulus is normalised by ρ3   to ensure that the results are insensitive to both ρ and   , see Eq. ( 20), (27).In contrast with the triangular lattice, the hexagonal topology has the same elastic modulus in both directions, see Eq. ( 20) and (27).In addition, the elastic modulus increases monotonically as t decreases.These two observations can be explained by the fact that the hexagonal lattice deforms primarily by bending bar 2 when loaded along either  1 or  2 .Again, varying t allows to change the elastic modulus over a wide range of values:  1 and  2 change by an order of magnitude when t varies from 0.25 to 3. Note that the effect of t on the elastic modulus is more important for the hexagonal lattice than for the triangular topology.This is because the expression of  1 for the hexagonal lattice contains terms in t 3 , whereas for triangular lattice  1 contains only terms in t.

Collapse modes
Failure maps are presented in Fig. 3 to illustrate how the collapse mode changes as a function of ρ and t.These maps are sensitive to the choice of parent material, and the results in Fig. 3 are for   = 200 GPa and   = 250 MPa, which is representative of structural steel.
Two collapse modes are possible when the triangular lattice is compressed in  1 : elastic buckling or yielding of bar 1, see Fig. 3a.As expected, elastic buckling is predominant at low values of relative density (less than ρ ≈ 0.07) and when t is inferior to approximately 0.25.On the other hand, three collapse modes can occur when the triangular lattice is compressed in  2 : elastic buckling in bar 2, and yielding of bar 1 or 2, see Fig. 3b.Again, elastic buckling takes place at low values of relative density.Yielding takes place in bar 1 when t < 0.5, and in bar 2 otherwise.
Failure maps for the hexagonal lattice are given in Fig. 3c and d for compression in  1 and  2 , respectively.In both directions, most geometries fail by plastic collapse of bar 2. For compression in  1 , the hexagonal lattice can also fail by elastic buckling or yielding of bar 1, but these modes are active only for very low values of ρ or t.
Finally, the influence of the parent material on the collapse mode can be evaluated by comparing Fig. 3 to 4, where the failure maps are plotted for   = 2.4 GPa and   = 41 MPa.These material properties are those of a polymer, which will be used in the experiments reported below (Section 5).For both lattices, changing the parent material from steel to a polymer significantly expands the predominance of elastic buckling.This is particularly striking for the triangular lattice, where nearly all geometries fail by elastic buckling when the parent material is a polymer.

Compressive strength
The influence of t on the compressive strength is shown in Fig. 5 for the triangular lattice and in Fig. 6 for the hexagonal topology.In both figures, the results are given for steel lattices (  = 200 GPa and   = 250 MPa) with ρ = 0.001 (in part a) and ρ = 0.2 (in part b).These two values of relative density were selected to cover all failure modes for steel lattices, see Fig. 3.
Triangular lattices with ρ = 0.001 fail by elastic buckling in both  1 and  2 directions, see Fig. 3. Accordingly, the compressive strength is normalised by ρ3   in Fig. 5a; therefore, the results are valid for any triangular lattice that fails by elastic buckling, see Eq. ( 7), (14).The compressive strength  1 increases with increasing t, whereas  2 decreases with increasing t.The failure mode of triangular lattices switches to yielding when its relative density is increased to ρ = 0.2, see Fig. 3. Consequently, its compressive strength has been normalised by ρ  in Fig. 5b to ensure that the results are representative of any triangular lattice that fails by yielding, see Eq. ( 8), (15), (16).The compressive strength  1 increases with increasing t, but, in contrast,  2 exhibit a peak strength  2, = 0.6 ρ  when t = 0.5.This peak is associated with a change in failure mode: yielding occurs in bar 1 for t < 0.5 and in bar 2 otherwise, see Fig. 3.
The compressive strength of an hexagonal lattice with ρ = 0.001 is shown in Fig. 6a, where results are plotted on a double -axis.On the left axis,  1 is normalised by ρ3   because the lattice fails by elastic buckling, see Fig. 3c.On the right axis,  2 is normalised by ρ2   since the hexagonal lattice collapses by plastic bending, see Fig. 3d.These normalisations are based on Eq. ( 22) and ( 28) to ensure that the results are insensitive to ρ and the material properties (as long as the failure modes remain unchanged).Clearly,  2 increases monotonically with decreasing t.In contrast, the strength along  1 displays a peak  1, = 0.147 ρ3   at t = 1.11.This peak is the result of two competing effects.On one hand, increasing t increases the elastic buckling resistance according to Eq. ( 22).On the other hand, the end constraint  decreases with increasing t:  decreases from 1 to 0.1 when t varies from 0 to 5, see Appendix A.2.
Finally, the compressive strength of the hexagonal lattice with ρ = 0.2 is plotted in Fig. 6b, There are three different collapse modes when ρ = 0.2 (see Fig. 3c) and therefore it is impossible to have a unique normalisation in this case.Consequently, both  1 and  2 are normalised by   in Fig. 6b.The hexagonal lattice has the same strength in both directions when t ≥ 0.27 because plastic collapse of bar 2 is the operative failure mode, see Fig. 3c,d.For this failure mode, the compressive strength ( 1 =  2 ) increases with decreasing t.Then, when t < 0.27,  1 decreases with decreasing t, and this is due to a change in failure mode, see Fig. 3c.

Description of the finite element model
Finite Element (FE) simulations were performed to verify the analytical results presented in Section 3.All simulations were done with the commercial software Abaqus (version 2017) and using the standard solver (Static, Implicit step in Abaqus).Periodic unit cells, shown in Fig. 1, were used for each topology.In all cases, the bar length was kept fixed to  = 10 mm, but the bar thicknesses  1 and  2 were modified to vary ρ and t.The cell walls were discretised using Timoshenko beam elements (B21 in Abaqus notation): we used 30 elements per bar for the triangular lattice and 70 elements per bar for the hexagonal topology.A mesh convergence study revealed that further mesh refinements had a negligible effect on the results (a difference less than 0.3%).A geometric imperfection was included in all simulations; it had the shape of the first eigenmode and its amplitude was set to 5% of the bar thickness expected to buckle.The parent material was modelled as an elastic perfectly plastic solid.The elastic regime was linear and isotropic, characterised by a Young's modulus   = 200 GPa and a Poisson's ratio   = 0.26, up to a yield strength   = 250 MPa.
Periodic boundary conditions were imposed with the following constraint equations [69,70]:  where   and  are the difference in displacement and rotation, respectively, between corresponding points of the unit cell;   is the macroscopic nominal strain tensor; and   is the displacement vector connecting two corresponding points of the unit cell.The compressive response of the lattice along  1 was obtained by prescribing the value of  11 , and the work conjugate  11 was calculated by Abaqus assuming that  22 = 0. Likewise, the compressive response along  2 was obtained by imposing  22 , and computing  22 provided that  11 = 0.

Comparison between simulations and analytical results
Finite Element predictions for  1 and  2 are compared to analytical results in Fig. 2 for both triangular and hexagonal lattices.The FE data was obtained for ρ = 0.001, but the normalisation used in Fig. 2 is such that the results are insensitive to the choice of relative density even up to ρ = 0.2.Clearly, there is an excellent agreement between FE simulations and analytical equations for both  1 and  2 , and over the entire range of t considered here.This is true for both triangular and hexagonal topologies.
Next, FE predictions of the compressive strength are compared to analytical results in Figs. 5 and 6 for triangular and hexagonal lattices, respectively.In both figures, there is an excellent agreement between FE and analytical results.In all cases, the failure mode predicted analytically corresponded to the one observed in FE simulations.Based on the results shown in Figs. 2, 5 and 6, we conclude that the analytical expressions derived in Section 2 are validated by FE simulations.

Experiments
Experiments were also conducted to assess the accuracy of the analytical predictions derived in Section 2. The procedure used to manufacture the samples is described below.Then, the measured responses and observed failure modes are reported.Finally, the experimental results are compared to our analytical predictions.

Specimen manufacture
All samples were fabricated by additive manufacturing; more specifically, using a Formlabs Form 2 machine, which uses stereolithography to cure a photo-polymerising resin.Both triangular and hexagonal lattices were manufactured and their dimensions are given in Fig. 7. Lattice materials exhibit size effects, but the number of cells in each sample is sufficiently large to ensure that the elastic modulus is within 12% of the infinite bulk limit [46,71].For each topology, three specimens were prepared with t = 0.5, 1, and 2, while keeping the relative density fixed.This was done by varying the bar thicknesses  1 and  2 as indicated in Table 1, and keeping the bar length fixed at  = 10 mm.In all cases, the specimens had a depth of 15 mm in the prismatic direction.Tests were conducted in both  1 and  2 directions, and the geometries are indicated on the failure maps presented earlier in Fig. 4.
The specimens were fabricated as follows.First, the geometry of the sample was created in Abaqus and a stl file was exported to the Form 2 machine.Second, the specimen was printed with a layer thickness of 0.025 mm and using the Formlabs Clear resin.All samples were printed with their prismatic axis perpendicular to the printing bed.Finally, when the print was completed, the lattice was washed in an isopropyl alcohol (IPA) solution and post-cured under UV light at a temperature of 60 • C for 30 min, as recommended in the Formlabs documentation.
Tensile tests were conducted to characterise the behaviour of the Clear resin used to manufacture the lattices.Following the procedure detailed above, dog-bone specimens were fabricated with a gauge length of 33 mm and a width of 6 mm, in accordance with the  14).Ten tests were performed: five with a thickness of 1.5 mm and five with a thickness of 3 mm.The tensile response was, however, insensitive to the thickness of the dog-bones.From these ten tensile tests, conducted at 10 −3 s −1 , the average material properties and their standard deviation were: a Young's modulus   = 2.43 ± 0.18 GPa, a 0.2% yield strength   = 40.9± 2.4 MPa, and an ultimate tensile strength of 62.0 ± 2.4 MPa.These values are close to those reported by Formlabs (which are a Young's modulus of 2.80 GPa and an ultimate tensile strength of 65 MPa).Finally, a measured stress-strain curve is given in Fig. 8; this test had properties very close to the averages given above.

Compressive responses and collapse modes
The printed specimens had no visible imperfections, such as broken or wavy bars.All samples were tested in compression using a MTS electromechanical testing machine with a capacity of 30 kN.The lattices were crushed at a constant rate of 0.05 mm/s, corresponding to a nominal strain rate of approximately 7 ⋅ 10 −4 s −1 .Both the compressive force and displacement were recorded by the testing machine.
The compressive responses of triangular lattices with ρ = 0.17 are shown in Fig. 9a and b for compression along  1 and  2 , respectively.In each plot, responses for t = 0.5, 1 and 2 are compared.All samples had a linear elastic regime up to a peak stress, followed by a gradually softening response.To show the failure modes, photographs of all tests are given in Fig. 10.All specimens compressed in the  1 direction failed by elastic buckling of bar 1.In contrast, elastic buckling occurred in bar 2 for all triangular lattices loaded in  2 .These failure modes are in agreement with our analytical predictions, see the failure maps in Fig. 4a, b.The measured responses for the hexagonal lattice compressed in  1 are shown in Fig. 9c.The sample with t = 0.5 has a linear elastic response up to the peak stress.This specimen failed by elastic buckling of bar 1, see the photograph given in Fig. 11a.This is the same buckling mode observed previously for uniform honeycombs [44].In contrast, the response of specimens with t = 1 and 2 was characterised by an initial linear elastic regime, followed by a non-linear regime, before reaching the peak stress.These geometries failed by plastic collapse of bar 2, see Fig. 11b, c, where localisation begins at the centre of the specimen.This deformation mode is practically identical to the one observed previously for expanded metallic honeycombs [49,50].Similarly, all samples failed by plastic collapse of bar 2 when the hexagonal lattice was compressed in  2 , see the responses in Fig. 9d and photographs in Fig. 11d,e,f.The failure modes observed for the hexagonal lattice are in agreement with the maps presented earlier in Fig. 4c, d.

Comparison between experiments and analytical/FE predictions
The measured elastic moduli are compared to our analytical predictions in Fig. 2.There are multiple data points since the majority of our tests were repeated three times to assess their variability.The experimental scatter is low; in fact, the relative deviation is inferior to 10% for 92% of our tests.For both lattices, there is a good agreement between the measured and analytical values of  1 and  2 .The largest discrepancy is observed for the elastic modulus  1 of triangular lattices with ρ = 0.29 and t = 2, where the measurements are on average 17% higher than the analytical prediction.This is mainly due to a size effect: when the number of cells  across the width of triangular lattice is small, its modulus  1 is ( + 1)∕ times higher than its asymptotic value [46].In our experiments,  = 8 and therefore, the modulus  1 is 12.5% higher due to this size effect.Considering this, the difference between experiments and analytical prediction is less than 5%.
Next, the measured compressive strength is compared to our analytical results in Fig. 12.Here, the compressive strength is normalised by the yield strength of the parent material   = 41 MPa.Again, the scatter is small on the measured compressive strength; 86% of our tests have a relative deviation of less than 10%.The largest discrepancy between experimental and analytical results is observed for the triangular lattice compressed in  1 with ρ = 0.29 and t = 2, see Fig. 12b.This sample is expected to fail by yielding (see the map in Fig. 4a) and a factor that may explain the discrepancy is the fact that the polymer used in the experiments has a strain hardening response (see Fig. 8) which is neglected in the analytical model.Furthermore, the discrepancies are not attributed to the fact that our analytical model neglects the stress concentrations at the nodes.To reach this conclusion, we compared the results of 2D plane stress simulations (not reported here) with those obtained with beam elements and found a negligible difference between the two approaches.Nonetheless, there is, for both topologies, a reasonable agreement between the measured compressive strength and the analytical predictions shown in Fig. 12.
Finally, the observed deformation modes are compared to FE simulations in Figs. 10 and 11 for the triangular and hexagonal lattices, respectively.Recall that FE simulations were done using beam elements, but their profile was rendered in Figs. 10 and 11 for visualisation purposes only.For both topologies, there is a good agreement between the observed and simulated deformation modes.This is true for all values of t and for both loading directions.

Discussion
In this paper, we introduced a new parameter t to customise the in-plane properties of triangular and hexagonal lattices.The thickness ratio t does not affect the behaviour of a lattice (whether is it stretchingor bending-dominated) but it does increase the number of possible failure modes (there is an additional failure mode for both triangular and hexagonal lattices).The main advantage of t is that it enables to vary the mechanical properties of a lattice without changing its density.This is illustrated in Fig. 13, where the normalised elastic modulus  1 ∕  and the normalised strength  1 ∕  of the triangular lattice are plotted as a function of relative density and for selected values of t.Both  1 and  1 increase monotonically with increasing t and therefore, the maximum increase in performances will depend on the highest value of t that can be manufactured.Using a fairly conservative estimate that  1 = 5 2 already leads to significant increases in properties: a triangular lattice with t = 5 is more than two times stiffer and up to 3.8 times stronger than a conventional design with t = 1, see Fig. 13.
The properties of the hexagonal lattice can also be increased by varying t.For this topology, however, it is  2 and  2 that increase monotonically with decreasing t, see Fig. 14.Again, varying the bar thickness by a factor of 5 ( 2 = 5 1 ⇒ t = 0.2) can lead to significant changes in properties: a hexagonal lattice with t = 0.2 is more than 2.5 times stiffer and 1.8 times stronger than its counterpart with t = 1.
The results in Figs. 13 and 14 show how the properties along a specific direction can be increased by varying t, and this is ideal for applications where the load is concentrated in a single direction.Increasing the properties along one direction, however, tends to reduce the properties in the perpendicular direction.This can be used in an advantageous way: t can be used to control the degree of anisotropy of a lattice.Elastic anisotropy is usually quantified using the Zener index [72], but it requires components of the elasticity tensor that were outside the scope of this work.To overcome this, we propose to quantify anisotropy here using the ratio  1 ∕ 2 , which has an intuitive physical meaning.For the triangular lattice, the anisotropy ratio  1 ∕ 2 Fig. 10.Photographs showing the failure modes of a triangular lattice with ρ = 0.17 and selected values of t.All samples compressed in  1 (left column) fail by elastic buckling of bar 1, whereas those compressed in  2 (right column) fail by elastic buckling of bar 2. The instant when each photograph was taken is indicated in Fig. 9a, b.For scale, all bars have a length of 10 mm.The deformation modes obtained from FE simulations using beam elements are also included for comparison.
of the can be obtained by dividing Eq. ( 5) by (13), which yields: Therefore, the anisotropy ratio  1 ∕ 2 varies linearly with t.This relationship is plotted in Fig. 15 and it is reasonably well corroborated by our experiments.Recall that conventional triangular lattices with t = 1 are transversely isotropic,  1 =  2 [44,45,70].Introducing t, however, expands the design space and allows us to create lattices with a wide range of  1 ∕ 2 , see Fig. 15.Unfortunately, this advantage is limited to the triangular lattice since  1 =  2 for the hexagonal lattice, see Eq. ( 20) and ( 27).Finally, it is important to note that even though our analysis focused on triangular and hexagonal lattices, the concept proposed in this paper is applicable to other topologies.For example, we show in Supplementary material how the same approach can be applied to a kagome lattice.The concept could also be extended to three-dimensional lattices such as the octet truss.For 3D topologies, all bars in a given plane would have a diameter  1 whereas others would have a diameter  2 , and therefore, the properties could be adjusted by controlling the ratio  1 ∕ 2 .

Conclusions
We examined the performances of prismatic lattices with cell walls having two different thicknesses.The ratio of these two thicknesses introduced a new design parameter t.The influence of t on the mechanical properties of triangular and hexagonal lattices was first investigated analytically.Then, these analytical expressions were (i) validated using finite element simulations, and (ii) corroborated by experiments conducted on polymer samples produced by additive manufacturing.
The results showed that this new parameter t can be used to vary the properties of a lattice while keeping its relative density fixed.The thickness ratio also controlled the degree of anisotropy: varying t typically increased the properties in one direction but reduced them in the perpendicular direction.Our analysis indicated that this tradeoff can be advantageous since the increase in one direction is often more important than the reduction in the other direction.For example, the elastic modulus of the triangular lattice increased by 80% in one direction and decreased by only 35% in the perpendicular direction, when t was increased from 1 to 3. In conclusion, the thickness ratio t introduced in this paper offers a new way to tailor the properties of prismatic lattices.Our analysis focused on triangular and hexagonal

Appendix A. End constraint factor
The elastic buckling strength of hexagonal lattices has been studied extensively for in-plane uniaxial [44,68,[73][74][75] and biaxial [47,48,[76][77][78][79][80] compression.The approach employed here is based on a small strain analysis of a perfect periodic unit cell, as in [44,68], and provides an upper bound for the buckling load [76].Our analysis is similar to that of Fan et al. [68] who derived analytically the end constraint factor  for different prismatic lattices.Their work, however, was limited to lattices with t = 1, and here we extend their study to capture the influence of t on the end constraint factor .
The method used to calculate  relies on the stiffness matrix of a single bar.Consider a bar of length  loaded by an axial compressive force  , as shown in

Fig. 1 .
Fig. 1.Geometry of (a) triangular and (b) hexagonal lattices, where the horizontal bars, numbered 1, have a thickness  1 and the diagonal bars, numbered 2, have a thickness  2 .The periodic unit cells used in finite element simulations are delimited by the red dashed lines.

Fig. 2 .
Fig. 2. Normalised elastic modulus as a function of t =  1 ∕ 2 for (a) triangular and (b) hexagonal lattices.The modulus  1 is in red, and  2 is in black.

Fig. 3 .
Fig. 3. Failure maps for prismatic lattices made from steel (  = 200 GPa and   = 250 MPa).Results are given for the triangular lattice compressed in (a)  1 and (b)  2 ; and the hexagonal lattice compressed in (c)  1 and (d)  2 .

Fig. 4 .
Fig. 4. Failure maps for prismatic lattices made from a polymer (  = 2.4 GPa and   = 41 MPa).Results are given for the triangular lattice compressed in (a)  1 and (b)  2 ; and the hexagonal lattice compressed in (c)  1 and (d)  2 .The geometries tested are marked with different symbols depending on the failure mode.In all cases, the observed failure modes were in agreement with our analytical predictions.

Fig. 7 .
Fig. 7. Dimensions of the samples tested: triangular lattice compressed in (a)  1 and (b)  2 ; and hexagonal lattice compressed in (c)  1 and (d)  2 .All samples had a bar length  = 10 mm and a depth of 15 mm in the prismatic direction.All dimensions are in mm.

Fig. 8 .
Fig. 8. Tensile response of the polymer used to manufacture all samples.
Fig. A.1a. Next, assume that small rotations   and   are imposed at ends  and , respectively, and a small transverse displacement  is prescribed, see Fig. A.1b.The corresponding bending moments   and   , and shear force  are given by [67]:where   is the elastic modulus and  is the second moment of area of the bar.In addition, s and  * are:s = (1 + ) and  * = 2 s −   2    , (A.2)

Table 1
Values of relative density ρ and thickness ratio t covered in the experiments.