Fracture toughness of semi-regular lattices

Previous studies have shown that the kagome lattice has a remarkably high fracture toughness. This architecture is one of eight semi-regular tessellations, and this work aims to quantify the toughness of three other unexplored semi-regular lattices: the snub-trihexagonal, snub-square and elongated-triangular lattices. Their mode I fracture toughness was obtained with finite element simulations, using the boundary layer technique. These simulations showed that the fracture toughness 𝐾 𝐼𝑐 of a snub-trihexagonal lattice scales linearly with relative density ̄𝜌 . In contrast, the fracture toughness of snub-square and elongated-triangular lattices scale as ̄𝜌 1 . 5 , an exponent different from other prismatic lattices reported in the literature. These numerical results were then compared with fracture toughness tests performed on Compact Tension specimens made from a ductile polymer and produced by additive manufacturing. The numerical and experimental results were in excellent agreement, indicating that our samples had a sufficiently large number of unit cells to accurately measure the fracture toughness. This result may be useful to guide the design of future experiments.


Introduction
Lattice materials are not only light, stiff and strong, but they also have a high fracture toughness (O'Masta et al., 2017;Gu et al., 2019;Liu et al., 2020).Optimising the architecture of lattice materials to maximise their elastic modulus and strength has been the subject of many investigations, and a few highly efficient designs have now been identified (Berger et al., 2017;Tancogne-Dejean et al., 2018;Hsieh et al., 2019).These designs have properties that are close to the theoretical bounds on elastic modulus and strength, leaving marginal room for further improvements.In contrast, the fracture toughness is unbounded and many architectures have remained unexplored.
The effect of architecture on the fracture toughness has been documented for a few prismatic (2D) lattices.Analytical studies (Ashby, 1983;Gibson and Ashby, 1997;Chen et al., 1998;Lipperman et al., 2007;Quintana-Alonso and Fleck, 2009;Berkache et al., 2022), Finite Element (FE) simulations (Fleck and Qiu, 2007;Quintana-Alonso and Fleck, 2007;Romijn and Fleck, 2007), and experiments (Huang and Gibson, 1991;Quintana-Alonso et al., 2010;Seiler et al., 2019) have shown that the fracture toughness of elastic-brittle lattices can be expressed as: where ρ is the relative density of the lattice,  is the length of the cell walls,   is the tensile strength of the parent material, and the constants  and  are topology-dependent and listed in Table 1 for five prismatic lattices (which are shown in Fig. 1).This scaling law was later extended to ductile lattices by Tankasala et al. (2015).They assumed that the parent material follows the Ramberg-Osgood relationship, where the strain  and stress  are related by: where  0 and  0 are the yield strain and stress, respectively, and  is the strain-hardening exponent.Their simulations showed that the effect of ductility can be captured by adding a term in Eq. ( 1), which becomes: where   is the failure strain of the parent material.Their results showed that the constant  ′ can be sensitive to the degree of strain hardening ; therefore, it is not necessarily equal to  in Eq. ( 1).In contrast, the exponent  was identical for both elastic-brittle and ductile lattices.Even though fracture toughness tests have been conducted on a few ductile lattices (Alsalla et al., 2016;O'Masta et al., 2017;Gu et al., 2018Gu et al., , 2019;;Daynes et al., 2021;Li et al., 2021), there are, to the best of our knowledge, no direct comparison between Eq. ( 3) and experiments.
The results in Table 1 show that the exponent  may take three different values.In general, bending-dominated architectures, such as https://doi.org/10.Williams (1979).

Table 1
Constants  and  in Eq. (1) for different prismatic lattices.Source: Data collected from Fleck and Qiu (2007) and Romijn and Fleck (2007)  the hexagonal lattice, have  = 2 (Gibson and Ashby, 1997).Otherwise, stretching-dominated topologies, like the triangular lattice or the 3D octet truss, have an exponent  = 1 (Fleck and Qiu, 2007;O'Masta et al., 2017).The kagome lattice, however, has an unusual behaviour: despite being stretching-dominated, it has a lower value of , making it significantly tougher than other architectures at low relative densities.The kagome lattice is also geometrically different from other architectures listed in Table 1.The hexagonal, square, and triangular lattices are classified as regular tessellations, meaning that they are made from a single regular polygon, see Fig. 1.In contrast, the kagome lattice is assembled from two regular polygons and is therefore classified as a semi-regular tessellation (Williams, 1979).There are seven other semiregular tessellations, see Fig. 1, and this work aims to quantify their fracture toughness.
Our study will focus on snub-trihexagonal, snub-square and elongated-triangular lattices (see Fig. 1) as they are the stiffest and strongest semi-regular tessellations (Omidi and St-Pierre, 2022).The other four semi-regular lattices are bending-dominated; therefore, their fracture toughness is expected to be low and comparable to the hexagonal lattice with  = 2.We will show that the snub-trihexagonal has a fracture toughness similar to the triangular lattice with  = 1, whereas the snub-square and elongated-triangular lattices exhibit a unique behaviour with  = 1.5, an exponent different from other prismatic lattices listed in Table 1.Our study includes both FE simulations and experiments: the predictions will be used to calibrate Eq. (3), which will then be compared to fracture toughness tests.
This article is structured as follows.The numerical modelling approach and the testing procedure are described in Section 2.Then, the numerical and experimental results are presented in Section 3, followed by a discussion in Section 4.

Numerical modelling approach
The fracture toughness of each lattice was predicted using Finite Element (FE) simulations.All simulations were done with the implicit solver of the commercial software Abaqus and assuming finite strain.We used the boundary layer method, which was introduced by Schmidt and Fleck (2001) and then used in many other studies (Fleck and Qiu, 2007;Romijn and Fleck, 2007;Tankasala et al., 2015;Gu et al., 2018), to ensure that our results can be directly compared to those presented in Table 1.
For each tessellation, we used a square domain with a side length of 300, where  is the length of a cell wall.The domain contained an initial crack in the negative  1 direction, as shown in Fig. 2a.A detailed view of the position of the initial crack is given in Fig. 2bd for each architecture.Additional simulations (not included here) showed that moving the crack tip to a different cell had a negligible effect on the fracture toughness.Also, numerical experimentation on a snub-trihexagonal lattice indicated that the fracture toughness is fairly insensitive to the crack orientation, see Appendix A.
All bars were meshed using Timoshenko beam elements (B21 code in Abaqus); 50 elements per bar were used around the crack tip ( ≤ 30, see Fig. 2a), whereas 10 elements per bar were used elsewhere.A mesh convergence analysis revealed that further refinements had a negligible effect on the predicted fracture toughness.Each node on the outer boundary of the domain had an applied displacement based on the   field, see Fig. 2a.The snub-trihexagonal lattice is isotropic (Omidi and St-Pierre, 2022) and therefore, the displacement field ( 1 and  2 ) had the form: where  and  are the polar coordinates of each node (see Fig. 2a); the functions   (, ) are specified in Williams (1952); and the shear modulus  and Poisson's ratio  of the snub-trihexagonal lattice are given in Omidi and St-Pierre (2022).Otherwise, the displacement field for the orthotropic snub-square and elongated-triangular lattices was given by (Liu et al., 1998): where the functions   are detailed in Liu et al. (1998) and the three non-dimensional parameters are: for orthotropic lattices (Quintana-Alonso et al., 2010).The elastic moduli  1 ,  2 ; the shear modulus  12 ; and the Poisson's ratios  12 ,  21 are given in Omidi and St-Pierre (2022) for both snub-square and elongated-triangular lattices.Numerical experimentation showed that prescribing the material rotation associated with the asymptotic   field to the boundary nodes or leaving them free to rotate has a negligible effect on the predicted fracture toughness.This was also observed by Romijn and Fleck (2007) and Tankasala et al. (2015); therefore, only translations were prescribed in our analysis and rotational degrees-of-freedom were left free.In all cases, the cell wall material was assumed to follow the Ramberg-Osgood relationship detailed in Eq. ( 2).The degree of strain hardening  and the failure strain   were varied in the simulations while keeping the yield strain  0 = 0.02 and the yield strength  0 = 45 MPa fixed.These values of  0 and  0 are representative of the polymer used later in the experiments, see Section 2.2.Finally, the fracture toughness   corresponds to the value of   when the maximum strain in any element reaches the failure strain   .

Specimen design, manufacturing and testing
Fracture toughness tests were performed to corroborate the numerical simulations.All tests were done on Compact Test (CT) specimens, and their dimensions are given in Fig. 3 for each topology.The width  , and crack length  were slightly different for each lattice, but  (2018).All samples had a depth  = 15 mm in the prismatic direction.
For each architecture, three values of relative density were produced, ρ = 0.2, 0.25, and 0.3.This was done by keeping the bar length fixed to  = 6 mm, and changing the cell wall thickness  as indicated in Table 2 (the relationship between ρ and ∕ is given in Omidi and St-Pierre (2022) for each topology).Note that additional FE simulations were conducted to ensure that the CT samples had a sufficiently large number of unit cells to provide an accurate measurement of the fracture toughness.This analysis is detailed in Appendix B. All samples were manufactured by additive manufacturing; more specifically, by stereolithography using a Form 3L machine from Formlabs.First, the geometry was created in Abaqus and a stl file was exported to the Form 3L machine.Second, the specimen was printed with a layer thickness of 50 μm and using the Formlabs Clear resin.All samples were printed with their prismatic axis perpendicular to the printing bed.After printing, 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.
All CT samples were tested using a MTS electromechanical testing machine with a capacity of 30 kN and with a constant displacement rate of 2 mm/min.Both the force and load-line displacement were recorded by the testing machine.For the isotropic snub-trihexagonal lattice, the fracture toughness was calculated according to (ASTM E1820, 2018): where   is the maximum force; the dimensions ,  and  are given in Fig. 3; and the function  (∕ ) is given in ASTM E1820 (2018).Bao et al. (1992) showed that Eq. ( 7) can be extended to orthotropic materials by including a dimensionless function  ().Therefore, the fracture toughness of snub-square and elongated triangular lattices was evaluated by: where the value of the function  () is given in Appendix C and varies with the degree of orthotropy , which is defined in Eq. ( 6).Both Eqs. ( 8) are based linear elastic fracture mechanics, but this is supported by the measured responses, which are presented in Section 3.2.
Tensile tests were conducted to measure the response of the Clear resin used to manufacture all CT samples.Following the procedure detailed above, dog-bone specimens were fabricated with dimensions comparable to those of the cell walls in the CT samples.The tensile specimens had a gauge length of 10 mm, a width of 10 mm, and a thickness of 0.50 mm.Ten tests were conducted at a strain-rate of 5⋅10 −4 s −1 and the average material properties were: a Young's modulus   = 2.0±0.10GPa, a yield strength  0 = 45±3 MPa, and a failure strain   = 0.10 ± 0.02.A measured stress-strain curve is given in Fig. 4 for a sample with properties close these average values.

Numerical results
The fracture toughness of each lattice, predicted with FE simulations, is plotted as a function of relative density in Figs. 5 and 6. Results are shown for different values of failure strain   in Fig. 5, while keeping the degree of strain hardening fixed at  = 13.In contrast, Fig. 6 shows the effect of the strain hardening exponent , for a fixed value   = 0.1.In both figures, the relative density ρ ≥ 0.1 to ensure that buckling does not occur before fracture (Shaikeea et al., 2022).In all cases, the first cell wall to fracture is the vertical bar in front of the crack tip, as shown on the deformed meshes in Fig. 7.
Clearly, increasing the failure strain   the fracture toughness of all three lattices, see Fig. 5.For example, the fracture toughness of a snub-trihexagonal lattice increases by 127% when an elastic-brittle parent material is replaced by a ductile solid with   = 0.1 and  = 13.This increase is sensitive to architecture; being 87% for the snubsquare and 50% for the elongated-triangular lattice.On the other hand, increasing  decreases the fracture toughness, see Fig. 6.This reduction, however, saturates around  = 33 as the response of the parent material becomes elastic perfectly-plastic, see Eq. ( 2).
The results in Figs. 5 and 6 were used to find the parameters ,  ′ and  for the scaling laws introduced earlier in Eqs. ( 1) and (3).The results, summarised in Table 3, show that  ′ varies significantly with  for both the snub-square and elongated-triangular lattices.This is, however, not the case for the stretching-dominated snub-trihexagonal lattice.These observations are in-line with the results of Tankasala et al. (2015); their simulations showed that  ′ is sensitive to  for the diamond and hexagonal lattices, whereas  ′ is roughly constant for the stretching-dominated triangular lattice.
Next, we turn our attention to the exponent  of the scaling law, see Eqs. (1) and (3).The fracture toughness of a snub-trihexagonal lattice scales linearly with relative density, which gives  = 1, see Figs. 5a and 6a.In contrast,  = 1.5 for snub-square and elongated-triangular lattices.Results in Figs. 5 and 6 show that this scaling is insensitive to the failure strain   and the degree of strain hardening .

Experimental results
Force versus displacement curves are plotted in Fig. 8 for the three semi-regular lattices with a relative density ρ = 0.25.In all cases, the response is linear up to the peak force   , which corresponds to the first fracture event.Photographs showing the deformation of the samples before and after fracture are given in Fig. 9. Multiple bars fail in the first fracture event, starting with the vertical cell wall ahead of    the crack tip, as predicted numerically (see Fig. 7).The tests on samples with ρ = 0.2 and 0.3 are not shown here, but they had a similar crack propagation path and also had a linear response, which justifies the use of Eqs. ( 7) and ( 8) to calculate the fracture toughness.
The normalised fracture toughness is plotted in Fig. 10 as a function of relative density for the three architectures considered in this study.Two samples were tested for each geometry and both data points are included in Fig. 10.In general, the scatter is small; the average difference between tests is 4%, and the largest difference is 11%.This can be attributed to the variability of the failure strain of the polymer, which is   = 0.10 ± 0.02 as mentioned in Section 2.2.
The measurements are also compared to FE simulations in Fig. 10.These numerical results are reproduced from Fig. 5 and correspond to the case where  0 = 45 MPa,  0 = 0.02,   = 0.1, and  = 13.These material properties were obtained by fitting the measured tensile response of the polymer, and we can see in Fig. 4 that this Ramberg-Osgood description follows closely the measured stress-strain curve.The scaling law in Eq. ( 3), which is fitted on these numerical results, is also included in Fig. 10 for completeness.There is a very good  remarkable considering that the specimens tested had about 10 times fewer unit cells than the FE simulations (compare the dimensions in Figs.2a and 3).These results suggest that the fracture toughness can be accurately measured with CT samples where the width  and length  of the cell walls are such that  ∕ ≈ 30 or higher.Interestingly, the numerical simulations done by Gu et al. (2019) suggested a similar size requirement for the 3D octet truss.

Discussion
In this section, we compare the fracture toughness of the three semi-regular lattices to that of regular tessellations.This is done by contrasting parameters  and  in Table 3 with the data given in Table 1.First, we can see that the snub-trihexagonal lattice has very similar performances to the triangular tessellation; they both have  = 1, but the value of  is 8% lower for the snub-trihexagonal lattice.Second, the results show that the snub-square and elongatedtriangular lattices have a unique behaviour since they are the only prismatic lattices with  = 1.5.The mechanisms leading to  = 1.5 can be described with an analytical model, which is detailed below.This analysis is based on the work done by Tankasala et al. (2015) for isotropic topologies and it is extended here for orthotropic lattices.
Consider a snub-square or elongated-triangular lattice with a semiinfinite crack loaded in mode I as shown in Fig. 2. In the vicinity of the crack tip, the  -integral is related to the macroscopic stress  22 and strain  22 by:  ∝  22  22  1 . (9) Both the snub-square and elongated-triangular lattices are stretchingdominated when loaded in the  2 direction (Omidi and St-Pierre, 2022).Therefore, the axial tensile stress   and strain   in a bar close to the crack tip are related the macroscopic stress and strain as: Substituting these expressions in Eq. ( 9) yields: Next, we can estimate the toughness   by setting   =   at a distance  1 = .Neglecting the linear term in Eq. ( 2), the corresponding stress M. Omidi and L. St-Pierre   ≈  0 (  ∕ 0 ) 1∕ and the above expression becomes: The relationship between the  -integral and the stress intensity factor   for orthotropic materials is detailed in Suo et al. (1991).The expression depends on the degree of anisotropy, but we show in Appendix D that for the snub-square and elongated-triangular lattices it can be approximated as: where  2 is the elastic modulus in  2 and  12 is the in-plane shear modulus of the lattice.For both snub-square and elongated-triangular lattices,  2 ∝ ρ  and  12 ∝ ρ3   (Omidi and St-Pierre, 2022).Using these expressions and substituting Eq. ( 12) in ( 13) yields: which is in the form of Eq. ( 3) and includes the correct exponent  = 1.5.To summarise, the above analysis shows that the constant  = 1.5 is due to two contributing factors.First, the zone of tensile deformation close to the crack tip, which is visible in Fig. 7 and reflected in Eq. ( 10).Second, the orthotropic behaviour of both snub-square and elongated-triangular lattices, which leads to Eq. ( 13).Finally, our numerical and experimental results show that the snubsquare is tougher than the elongated-square lattice, see Fig. 10, and this can also be explained with the above analytical expressions.Omidi and St-Pierre (2022) showed that the snub-square lattice has a much higher shear modulus  12 than the elongated-square, even though both lattices have a similar elastic modulus  2 .We can see from Eq. ( 13) how a higher shear modulus  12 will lead to a higher fracture toughness   .

Conclusion
The fracture toughness of three ductile semi-regular lattices was investigated using FE simulations and experiments.We found that the snub-trihexagonal lattice has a fracture toughness that scales linearly with relative density ρ and is similar to that of a triangular lattice.In contrast, the fracture toughness of snub-square and elongatedtriangular lattices scales as ρ1.5 , where the exponent of 1.5 is unique amongst prismatic lattices loaded in mode I.We showed analytically that this is a consequence of the tensile deformation at the crack tip and the orthotropic behaviour of these two tessellations.For the three architectures considered, the fracture toughness predicted by FE simulations was in excellent agreement with experiments performed on CT samples produced by additive manufacturing.This demonstrates that it is possible to accurately measure the fracture toughness of ductile lattice materials even though experiments are done with significantly fewer unit cells that what is typically used in FE simulations.Our results will be beneficial for the design of specimens in future experimental studies, and the development of guidelines to measure the fracture toughness of lattice materials.
This study was limited to the onset of fracture and did not cover the resistance to crack growth (R-curve).Recent work by Tankasala and Fleck (2020) and Hsieh et al. (2020) has shown that some architectures, such as the triangular lattice, have a rising R-curve and future work is needed to determine if the three semi-regular lattices considered in this study also exhibit a strong resistance to crack propagation.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.crack tip.The mode I stress intensity factor was extracted using the contour integral method implemented in Abaqus.
The value of  () is given in Table C.2 for both orthotropic lattices.The value of  is close to unity for the snub-square lattice, but significantly higher for the elongated-triangular topology.This can be attributed to the fact that the shear modulus of the snub-square lattice is significantly higher than that of the elongated-triangular lattice, and this leads to important differences in the values of , see Table C.2 and Eq. ( 6).

Appendix D. Relationship between 𝑱 and 𝑲 𝑰 for orthotropic lattices
For a linear elastic orthotropic solid, the  -integral is related to the mode I stress intensity factor by (Suo, 1990;Suo et al., 1991): where the parameters  and  are defined in Eq. ( 6) and depend upon the elastic moduli  1 ,  which is the relationship used in Eq. ( 13).

Fig. 1 .
Fig. 1.Examples of prismatic lattices: there are three regular and eight semi-regular tessellations.Regular lattices are made from a single regular polygon, whereas semi-regular tessellations are assembled from multiple regular polygons.The nomenclature is based on Williams (1979).

Fig. 2 .
Fig. 2. (a) Domain used in the finite element predictions.The dashed red line indicate the position of the initial crack for (b) snub-trihexagonal, (c) snub-square, and (d) elongated-triangular lattices.

Fig. 5 .
Fig. 5. Normalised fracture toughness as a function of relative density for (a) snub-trihexagonal, (b) snub-square, and (c) elongated-triangular lattices.Results are shown for a strain hardening exponent  = 13 and different values of failure strain   .

Fig. 6 .
Fig. 6.Normalised fracture toughness as a function of relative density for (a) snub-trihexagonal, (b) snub-square, and (c) elongated-triangular lattices.Results are shown for a failure strain   = 0.1 and different values of strain hardening exponent .

Fig. 8 .
Fig. 8. Force versus load-line displacement recorded during fracture toughness tests.Responses are shown for a relative density ρ = 0.25.

Fig. A. 1 .
Fig. A.1.Three selected configurations to quantify the effect of crack orientation on the fracture toughness of a snub-trihexagonal lattice.

Fig. A. 2 .
Fig. A.2. Effect of crack orientation on the fracture toughness of an elastic-brittle snub-trihexagonal lattice.

Fig. B. 1 .
Fig. B.1.Normalised fracture toughness as a function of the normalised number of cells in a CT sample.The dimensions used in experiments are shown with ⊗. Results are shown for a relative density ρ = 0.3.

Table 2
Values of relative density ρ, slenderness ratio ∕, and bar thickness  used in experiments.All samples had a bar length  = 6 mm.
selected to ensure that ∕ ≈ 0.25, as recommended in ASTM E1820

Table 3
Parameters ,  ′ , and  for the scaling laws in Eqs.(1) and (3).Results for  ′ are given for different values of strain hardening exponent , whereas  corresponds to an elastic-brittle material.

Table C .1
(Omidi and St-Pierre, 2022) snub-square and elongated-triangular lattices.The moduli are expressed as a function of the relative density ρ and of the elastic modulus   of the parent material(Omidi and St-Pierre, 2022).Value of the function  () used in Eq. (8) to evaluate the fracture toughness of snub-square and elongated-triangular lattices.
2 ; the shear modulus  12 ; and the Poisson's ratios  12 ,  21 of the lattice.With the elastic properties listed in Table C.1, we find that: where the above simplification is based on the fact that  12 ≪  2 for low values of ρ, see Table C.1.Finally, substituting Eq. (D.2) in (D.1) returns: