The properties of lattice materials are strongly influenced by their nodal connectivity; yet, previous studies have focused mainly on topologies with a single vertex configuration. This work investigates the potential of demi-regular lattices, with two vertex configurations, to outperform existing topologies, such as triangular and kagome lattices. We used finite element simulations to predict the fracture toughness of three elastic-brittle demi-regular lattices under mode I, mode II, and mixed-mode loading conditions. The fracture toughness of two demi-regular lattices scales linearly with relative density ¯ ρ, and outperform a triangular lattice by 15% under mode I and 30% under mode II. Their mixed-mode fracture envelopes are also up to 43% larger than that of a triangular lattice. The third demi-regular lattice has a fracture toughness that scales with√¯ρand matches the remarkable toughness of a kagome lattice. These findings open new avenues for the development of tougher lightweight materials. Keywords: Cellular materials, Fracture, Toughness, Finite element analysis, Demi-regular tessellations Lattice materials are well known for their ability to be stiff and strong at low densities [1–6], but their architecture can also be tailored to achieve a high fracture toughness [7– 10]. While the stiffness and strength of lattices are limited by bounds, and architectures approaching those limits have been identified [11–13], their fracture toughness is not theo- retically bounded. Relatively few topologies have been investigated so far in the quest to maximise fracture toughness; therefore, the aim of this study is to discover tougher lattice materials by exploring novel architectures. The influence of topology on fracture toughness has been documented predominantly for ∗Corresponding author: luc.st-pierre@aalto.fi Preprint submitted to Elsevier May 28, 2023 Table 1: Constants DI,DII, and din (1) for planar transversely isotropic lattice materials. These topologies are shown in Fig. 1 and 2. Topology DI DII d Reference Hexagonal 0.800 0.370 2 [19] Triangular 0.500 0.380 1 [19] Snub-trihexagonal 0.460 — 1 [31] Kagome 0.212 0.133 0.5 [19] A 0.570 0.510 1 This study B 0.580 0.490 1 This study C 0.210 0.150 0.5 This study planar lattices. Analytical [14–18], numerical [19–25] and experimental [26–29] studies have shown that the fracture toughness of an elastic-brittle lattice can be expressed as: KIc σts√ ℓ=DI¯ρdandKIIc σts√ ℓ=DII¯ρd, (1) where KIcandKIIcare the fracture toughness under mode I and II, respectively; ¯ ρis the relative density of the lattice; ℓis the length of the cell walls; σtsis the tensile strength of the parent material; and the constants DI,DII, and dare given in Table 1 for planar transversely isotropic lattices, which are shown in Fig. 1. Orthotropic topologies, such as the square or snub-square lattices, have also been investigated but they do not outperform the toughest isotropic lattices [20, 21, 27, 30, 31]. The exponent dhas a strong effect on the fracture toughness: a lower dleads to a higher fracture toughness at low relative densities ¯ ρ, see (1). The exponent dis insensitive to the choice of parent material [32], and strongly influenced by the lattice’s nodal connectivity (the number of bars meeting at each joint). With three bars per joint, the hexagonal lattice (Fig. 1a) deforms by bending, and this leads to a particularly low fracture toughness with d= 2 [14, 15, 33, 34]. In contrast, the triangular lattice (Fig. 1b), with six bars per joint, and the snub-trihexagonal architecture (Fig. 1c), with five bars per joint, are stretching-dominated meaning that their members carry predominantly axial stresses. This mode of deformation leads to a fracture toughness that scales linearly with relative density, with an exponent 2 𝓍2 𝓍1(a) (b) (c) (d)Figure 1: Four planar topologies listed in Table 1: (a) hexagonal, (b) triangular, (c) snub-trihexagonal, and (d) kagome lattices. In all cases, the thick black line denotes the position of the initial crack considered in previous studies [19, 31]. d= 1 [19, 31, 35]. Finally, the kagome lattice (Fig. 1d), with four bars per joint, has an unusual behaviour. It is stretching-dominated, as stiff and strong as a triangular lattice [2], but its fracture toughness scales with the square-root of relative density ( d= 0.5), making it remarkably tough at low values of ¯ ρ. This exceptional toughness is due to a crack tip blunting phenomenon caused by localised bending deformation [19, 32]. The kagome lattice is the only known topology with d= 0.5, and it is unclear if other architectures could match or exceed its toughness. All architectures investigated so far share a similar characteristic: they have a unique vertex configuration. This means that each vertex inside a given lattice is surrounded by the same sequence of polygons, see Fig. 1. This is an important limitation considering that nodal connectivity has such a strong effect on fracture toughness. To broaden the search for tougher architectures, we turn our attention to demi-regular lattices, which have two different vertex configurations. Three examples are shown in Fig. 2: demi-regular lattices A and B have joints with five and others with six bars, whereas the vertices of tessellation C have either four or five struts. These three demi-regular lattices were recently shown to be 3 (a) (b) (c) (d) 𝓍2 𝓍1𝑟 𝜃 (𝑡,ℓ) 300ℓ300ℓ(𝑢𝑖,𝜔)Figure 2: Demi-regular lattices (a) A, (b) B, and (c) C. The thick black line indicates the position of the initial crack, and fracture sites under mixed-mode loading are shown with |,||, and|||symbols. (d) Domain and coordinate system used in our finite element predictions. stretching-dominated and transversely isotropic, with an elastic modulus that is comparable but slightly lower than that of kagome and triangular lattices [36]. The aim of this study is to characterise the fracture behaviour of these three demi-regular lattices and evaluate if they are tougher than other stretching-dominated architectures, such as kagome and triangular lattices. The fracture toughness of each demi-regular lattice was predicted using Finite Element (FE) simulations; more specifically, with the static implicit solver of the commercial software Abaqus. Our modelling approach relied on the boundary layer method, as used in previous studies [19, 21, 31, 34, 35], to ensure that our results can be compared directly those in Table 1. For each demi-regular lattice, a large square domain was created with a side length of 300ℓ, where ℓis the length of a cell wall (see Fig. 2d). This domain included a semi-infinite crack along the negative x1axis and a detailed view showing the position of the crack tip for 4 each topology is given in Fig. 2. Numerical experimentation showed that moving the crack tip to another cell or changing the crack orientation have a negligible effect on the fracture toughness. We show, in supplementary material, that the crack orientation considered in Fig. 2 is the one associated with the lowest mode I fracture toughness. The cell walls were meshed using Timoshenko beam elements (B21 in Abaqus notation). A fine mesh size of ℓ/30 was used within a 60 ℓ×60ℓarea centered at the crack tip, whereas a coarser mesh size of ℓ/10 was employed elsewhere. Further mesh refinements had an imperceptible effect on the results. The relative density ¯ ρwas varied from 0.01 to 0.2 by changing the strut thickness twhile keeping the strut length ℓfixed. The relationship between ¯ρandt/ℓis given in supplementary material for each demi-regular lattice. Each node on the domain’s perimeter was subjected to a displacement corresponding to the asymptotic crack tip solution from linear elastic fracture mechanics. The displacement field included two translations, u1andu2, and an in-plane rotation ω, see Fig. 2d. Expressions foru1,u2, and ωare lengthy and provided in supplementary material. They are functions of the nodal coordinates ( r, θ); the elastic properties of the lattice given in [36]; and the applied stress intensify factors KIandKIIfor modes I and II, respectively. Our approach is identical to that used by Fleck and Qiu [19] to predict the fracture toughness of other planar transversely isotropic lattices. The cell walls were modeled as an elastic-brittle material, characterised by a Young’s modulus Esand Poisson ratio νs, up to a tensile fracture strength σts. In our simulations, the fracture toughness KIc(orKIIc) corresponded to the value of KI(orKII) when the maximum tensile stress in any element of the lattice reached the strength of the parent material σts. The fracture toughness of each demi-regular lattice is plotted as a function of relative density in Fig. 3a for mode I and Fig. 3b for mode II. In each plot, the fracture toughness is normalised by σts√ ℓbased on the scaling law given in (1). Deformed meshes for each topology with ¯ ρ= 0.05 are given in Fig. 4 for both modes I and II. The fracture toughness of demi-regular lattices A and B scales linearly with relative density, see Fig. 3, corresponding to an exponent d= 1 in (1). In contrast, the fracture toughness of tessellation C scales with√¯ρ, giving d= 0.5. With a lower value of d, lattice C 5 A B C1 1 A B C(a) 0.003 𝜌ത0.3 𝐾𝐼𝑐 𝜎𝑡𝑠ℓ 0.2 0.01(b) 0.003 𝜌ത0.3 𝐾𝐼𝐼𝑐 𝜎𝑡𝑠ℓ 0.2 0.011 0.5 1 0.5 1 1Figure 3: Fracture toughness under (a) mode I and (b) mode II, as a function of relative density ¯ ρ. Results are given for demi-regular lattices A, B, and C. is significantly tougher than topologies A and B at low relative densities. This holds true for both KIcandKIIcas the exponent dis the same for both modes I and II. The lower value ofdfor topology C is due to crack tip blunting, which does not occur in tessellations A and B, see Fig. 4. The results in Fig. 3 were used to evaluate the constants DIandDIIin (1) and their values are given in Table 1. For each demi-regular lattice, DI> D IImeaning that fracture toughness is higher for mode I than mode II. The differences are, however, sensitive to topology; the ratios KIc/KIIcare 1.12, 1.18 and 1.40 for tessellations A, B, and C, respectively. The fracture envelope under mixed-mode loading is plotted in Fig. 5a for demi-regular lattices A and B, and in Fig. 5b for tessellation C. Based on the results in Fig. 3, the stress intensity factors KIandKIIare normalised here by σts¯ρ√ ℓfor lattices A and B, and by σts√¯ρℓ for tessellation C. This normalisation ensures that each fracture envelope is independent of relative density ¯ ρ. Each envelope is formed by two or three straight lines, where each segment corresponds to a different fracture location. The segments are labeled |,||, and|||in Fig. 5, and the corresponding fracture locations are shown in Fig. 2. The fracture envelopes of lattices A and B are close to a quarter circle, whereas that of tessellation C has two straight sides, see Fig. 5. For topologies A and B, fracture takes place in the vertical bar in front of the crack tip when KI≫KIIand moves off the crack plane 6 (a) (b) Mode I Mode II (c)Figure 4: Deformed meshes for demi-regular lattices (a) A, (b) B, and (c) C. Results are shown for mode I (left) and mode II (right) loading. All lattices have a relative density ¯ ρ= 0.05. 7 00.020.040.060.080.10.120.140.160.18 0 0.05 0.1 0.15 0.2 0.250° Kagome 00.10.20.30.40.50.6 0 0.1 0.2 0.3 0.4 0.5 0.6Demi A x1 Demi B x1 Triangular 𝐾𝐼𝐼𝑐 𝜎𝑡𝑠𝜌̅ℓ 𝐾𝐼𝑐 𝜎𝑡𝑠𝜌̅ℓ(a) (b) 𝐾𝐼𝑐 𝜎𝑡𝑠ℓ𝜌̅KagomeC 𝐾𝐼𝐼𝑐 𝜎𝑡𝑠ℓ𝜌̅A B TriangularFigure 5: Mixed-mode fracture envelopes: (a) demi-regular lattices A and B are compared to a triangular tessellation and (b) demi-regular lattice C is compared to a kagome tessellation. The segments labelled |,||, and|||correspond to different fracture sites, which are shown in Fig. 2. when KIIincreases, see Fig. 2. In contrast, the fracture sites in tessellation C are always off the crack plane and located a few cells away from the crack tip. The fracture behaviour of demi-regular lattices A and B is very similar to that of a triangular lattice. These three topologies have a fracture toughness that scales linearly with relative density ( d= 1), see Fig. 3 and [19]. The bars in a triangular lattice carry high axial stresses and minimal bending stresses [19], and we noticed the same for demi-regular lattices A and B. This is characteristic of lattices with a high nodal connectivity: tessellations A and B have either five or six bars per vertex, and similarly, a triangular lattice has six. Lastly, the fracture sites in a triangular lattice are close to the crack tip (typically within in a distance ℓ) [19], and we observed the same for tessellations A and B, see Fig. 2a,b. In contrast, demi-regular lattice C behaves similarly to a kagome lattice since their frac- ture toughness scales with the square-root of relative density ( d= 0.5), see Fig. 3 and [19]. The high fracture toughness of the kagome lattice is attributed to elastic crack tip blunting, characterised by bands of bending deformation near the crack tip [19]. The same phenomenon is observed in demi-regular lattice C, see Fig. 4c. The fracture sites in a kagome lattice are located at a distance of ≈3ℓfrom the crack tip [19], and tessellation C exhibit the same feature, see Fig 2c. Finally, both topologies have a similar nodal connectivity: topology C has four or five bars per joint, whereas the kagome lattice has four. 8 Based on the similarities detailed above, it is justified to compare the performances of demi-regular lattices A and B to those of a triangular lattice; and those of tessellation C to a kagome lattice. Their constants DIandDIIare compared in Table 1, and their fracture envelopes are contrasted in Fig. 5. Here, the envelopes for triangular and kagome lattices were obtained following the same modelling approach detailed above and their orientations are given in Fig. 1. The performances of demi-regular lattice C are very similar to those of a kagome lattice; they have practically the same fracture toughness, see Table 1, and fracture envelope under mixed-mode loading, see Fig. 5b. Otherwise, demi-regular lattices A and B are both tougher than a triangular lattice. Tessellations A and B have a similar fracture toughness, which is roughly 15% tougher than a triangular lattice under mode I, and approximately 30% tougher in mode II, see Table 1. Demi-regular lattices A and B also outperform the triangular lattice under mixed-mode loading, see Fig. 5a. Quantitatively, the area contained within the fracture envelope of tessellation A is 43% larger than that of a triangular lattice. In summary, we showed that three previously unexplored demi-regular lattices have a remarkably high fracture toughness. Demi-regular lattices A and B have a fracture toughness that scales linearly with relative density and are tougher than a triangular lattice under mode I, mode II, and mixed-mode loading conditions. Previously, the kagome lattice was the only known topology with a fracture toughness that scales with the square-root of relative density. We discovered that demi-regular lattice C exhibit the same scaling, and has a fracture toughness equal to that of a kagome lattice. While these results are step forward in the development of tougher lightweight materials, future work is needed to evaluate the imperfection sensitivity of these three demi-regular lattices. The fracture toughness of the kagome lattice is highly imperfection sensitive [21], but we anticipate that demi-regular lattice C, with a slightly higher nodal connectivity, will be more tolerant to imperfections. Acknowledgements This work was supported by the Academy of Finland (grant 322007). 9 Supplementary material Supplementary material related to this article can be found online.