# Tensile failure of bio-inspired lattices with different base topologies

Gabriele Simone, Riccardo Manno, Fabrice Barbe, Ivano Benedetti

► To cite this version:

Gabriele Simone, Riccardo Manno, Fabrice Barbe, Ivano Benedetti. Tensile failure of bio-inspired lattices with different base topologies. AIP Conference Proceedings, 2023, FDM 2022 - 20th International Conference on Fracture and Damage Mechanics, 5-7 septembre 2022, Malaga, Spain, 2848, pp.020019. 10.1063/5.0145012. hal-04351622

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# Tensile Failure of Bio-inspired Lattices with Different Base Topologies

Gabriele Simone,$^{1,2, a)}$ Riccardo Manno,$^{3, b)}$ Fabrice Barbe,$^{2, c)}$ and Ivano Benedetti$^{1, d)}$

$^{1)}$Department of Engineering, University of Palermo, Viale delle Scienze, Edificio 8, 90128, Palermo, Italy

$^{2)}$INSA Rouen, UNIROUEN, Groupe de Physique des Matériaux, Normandie Univ, UMR CNRS 6634, Rouen, F-76000, France

$^{3)}$Bristol Composite Institute (ACCIS), Department of Aerospace Engineering, Queen's Building, University Walk, Bristol, BS8 1TR, United Kingdom

$^{a)}$Electronic mail: gabriele.simone@community.unipa.it

$^{b)}$Electronic mail: r.manno@bristol.ac.uk

$^{c)}$Corresponding author: fabrice.barbe@insa-rouen.fr

$^{d)}$Corresponding author: ivano.benedetti@unipa.it

**Abstract.** In the last decades the use of cellular materials, either in the form of foams or lattices, has widely spread in engineering due to their specific properties, namely their high mechanical and multifunctional properties in terms of strength, stiffness, energy absorption, thermal and acoustic insulation at small weight compared to bulk materials. These features can be achieved, in the case of lattices, by designing their structure at different scales, both in two and three dimensions. Additionally, nowadays, complex desired geometries may be easily obtained thanks to consolidated and even recent technologies of production, especially Additive Manufacturing (AM) techniques. The aim of the present work is to investigate the mechanical behaviour exhibited by two-dimensional lattices with different types of engineered topologies under tensile loadings, focusing on damage evolution and eventual failure. Since past studies have shown, numerically and experimentally, the advantages of bio-inspiration from wood microstructures on honeycombs with hexagonal cells, in terms of crack deflection and therefore absorbed fracture energy, here also the performances of 2D lattices with different base topologies, namely triangular and triangular-hexagonal (aka "Kagome") cells, are evaluated. The comparison is based considering not only tensile tests on lattices with different base topologies, but also assessing, for each topology, the effect of lattice-density variations, ranging from sparse to variable density to fully dense lattices. The studies are carried out through Finite Element Analysis (FEA), using base material properties derived from curve fitting between experimental and numerical results on engineered specimens in ABS, 3D printed using Fused Filament Fabrication (FFF). Further theoretical, computational and experimental investigations are currently undergoing, towards a framework for lattice materials by design.

## INTRODUCTION

Engineering and technology progress is often focused on the development of materials able to deliver specific structural performances at the lowest possible weight. In particular, in the recent decades, considerable efforts have been devoted, especially in the aerospace and automotive fields, to the design of mechanical systems with superior energy efficiency [1], aimed at maximising the payload while minimising fuel consumption. Such developments have direct impact on transportation costs as well as on greenhouse gases emissions [2].

In this context, in recent years, considerable attention has been attracted by cellular materials, such as foams and lattices, for their multi-functional features coupled with their low density and thus weight. Cellular materials feature micro-structures made up of clusters of cells that, thanks to the presence of voids or pores between solid walls, exhibit macro-properties with interesting applications in acoustic insulation, thermal management, energy absorption [3], while maintaining relatively high strength and stiffness at very low density compared to usual bulk materials.

Among cellular solids, lattices are built by tessellating a unit cell throughout space either in two or three dimensions. The planar ones, also named "honeycombs", can be regular, semi-regular or show other configurations. Regular lattices are generated by the repetition of a regular polygon (such as a hexagon or a triangle); semi-regular lattices are assembled from elementary cells of two or more regular shapes, like the 'triangular-hexagonal' lattice also known as "Kagome" [4]. Other configurations include for example auxetic lattices, which exhibit a negative Poisson's ratio, useful in several mechanical applications [5].

Recent advancements in manufacturing, namely those related to Additive Manufacturing (AM), offer the possibility of designing the micro-structure of lattices and thus obtaining materials with tunable macro-properties [6]. Although AM is not yet suitable for massive productions in terms of times and costs, it offers some interesting advantages if compared to conventional machining. Indeed, AM technologies allow manufacturing complex parts directly from
the 3D designed models [7]; for such a reason, AM enables the realisation of architectured cellular materials with a
precisely defined geometry and size [8].

Since AM broadens the materials design space, there is the need to identify materials architectures able to take
advantage of such manufacturing freedom. In this context, Nature provides an invaluable source of inspiration for the
design of nano-, micro- or meso- cellular architectures. Many biological materials, like bones, stems and woods, show
remarkable mechanical properties despite the modest constitutive strength of their constituents: this is achieved thanks
to their material structures hierarchically arranged at different scales [9]. In recent years, several studies have shown
how bio-inspired lattices are ideal candidates for biomedical and energy absorption applications. On the other hand,
the cellular meso-structures of several kinds of wood offer a model for materials with increased fracture toughness,
which is achieved by crack deflection induced by the alternation of layers with different stiffness [10, 11].

The present work focuses on material architectural strategies addressed at increasing the fracture energy of honey-
combs; it is the continuation of previous studies [11, 12, 13] and it specifically addresses the effects of lattice-density
variations in honeycombs with hexagonal, triangular and Kagome unit cells subjected to tensile loading. Here, a first
experimental study is performed on *ad hoc* designed specimens, 3D printed in ABS, to derive the printed material’s
constitutive properties; such constitutive properties have then been used in a set of numerical tensile tests performed
on notched cellular structures with different architectures. The main aim of this work is assessing the effect of density
gradients in homogeneous lattices with different base topology towards increased fracture toughness.

TENSILE TESTS ON ABS

The present work presents a set of simulation baselines for further experimental validation. The material properties used in this context have been determined from experimental tests on 3D printed specimens manufactured via Fused Filament Fabrication (FFF) of Acrylonitrile Butadiene Styrene (ABS). The experimental campaign to determine the tensile properties of printed ABS has been performed on an *ad hoc* designed specimens consisting of three parallel walls having the same thickness as that of the lattice walls tested in the next steps. The details about the three-wall specimens geometry are reported in Fig.(1).

<figure><img src="image_2.png" /><figcaption>FIGURE 1: Geometry of the specimens for ABS tensile tests. Left: 3D model. Right: dimensions in mm.</figcaption></figure>

All the specimens have been fabricated using a Ultimaker 3 Extended machine with a 0.4mm diameter nozzle,
setting the layer height to 0.1mm and a line width - the width of the deposited filament - that allowed to have a single
filament of polymeric material along each one of the three longitudinal walls. After material relaxation, the samples
have then been tested on a 10kN capacity MTS electro-mechanical testing machine using a classical extensometer with a base length of 25 mm. The employment of the extensometer justifies the choice of specimens with at least three parallel walls instead of a single one, as in the case of conventional dogbone specimens. Indeed, given the small size of the gauge section and the relatively low stiffness of ABS, the classical extensometer would exert a non-negligible bending on the specimen if a single wall was considered; the measured stress-strain curve would then not correspond to the response of the material under uniaxial tension.

A set of eight engineering stress-strain curves has been experimentally obtained. An average behaviour of the material has been extracted from the experimental curves and used to numerically calibrate an approximate stress-strain relation, see Fig.(2), to infer the right set of material constants necessary to define the constitutive behaviour in the finite element environment. Tab.I shows the average mechanical properties extracted using the proposed approach.

<figure><img src="image_3.png" /><figcaption>FIGURE 2: Comparison between the experimental stress-strain curves, the average tensile behaviour and the approximate curve resulting from the numerical simulation for 3D printed ABS.</figcaption></figure>

<table><caption>TABLE I: Mean mechanical properties of printed ABS derived from tensile testing of three-walled specimens.</caption><thead><tr><td>Young's modulus [GPa]</td><td>Poisson's ratio</td><td>Ultimate Tensile Strength [MPa]</td><td>Strain at failure [%]</td></tr></thead><tbody><tr><td>1.9</td><td>0.34</td><td>35.77</td><td>3.94</td></tr></tbody></table>

# FINITE ELEMENT ANALYSES OF FAILURE

## Modelling

All the Finite Element Analyses (FEAs) have been set and performed in the +Abaqus/CAE+ environment. The +Abaqus/Explicit+ solver has been used as it allows for adequate study of highly nonlinear problems including damage onset and evolution. As mentioned in the previous section, the constitutive properties used in the numerical
model have been obtained simulating the tensile test of the three-walled specimen as in Ref. [12]. The numerical analyses have been carried out on a 3D model of the specimen, in which the gripping parts have been removed via Computer-Aided Design (CAD), imposing the displacements at the sample's top and bottom surfaces. The resulting geometry has been meshed using eight-node linear hexahedral elements with reduced integration (C3D8R). The approximate constitutive law used in the simulation was the von Mises elasto-plastic model coupled with ductile damage to simulate failure onset and propagation: the model is adopted for its relative simplicity of calibration with respect to the experimental curves, although different materials models are being considered for further studies.

<figure><img src="image_4.png" /><figcaption>FIGURE 3: Zoom on the three configurations of the considered base topologies. From top to bottom: hexagonal (H), triangular (T) and Kagome (K). From left to right: sparse (S), variable (V) and dense (D).</figcaption></figure>

After characterizing and calibrating the material constitutive law for employment in the computational models, different lattice geometries inspired from Ref. [11] have been considered. Their FE models have been generated in
+Abaqus/CAE+ via dedicated Python script pre-processors. Four-node plane strain elements with reduced integration have been used (CPE4R). The boundary conditions have been enforced in terms of displacement at the nodes located in correspondence to the theoretical grips. Furthermore, mass scaling has been used so to improve the computational time, ensuring however that any artificial inertial effects introduced were negligible. Three different base cell topologies have been considered: Hexagons (H), Triangles (T) and Kagome (K), as represented in Fig.(3). For each base topology, two different widths of an elementary cell have been used (where the width of a cell is its dimension along the horizontal direction), from which three types of lattices have been designed: Sparse (S), Dense (D) and Variable (V). In the variable configuration, an alternation of sparse and dense regions is introduced in the form of bands aligned with the height of the specimens. Further assigning the letter L to "lattice", the acronym HLV then, for example, refers to the lattice with hexagonal cells in the variable density configuration.

The overall sizes of all the considered lattices are comparable, in terms of width and height of the whole specimen. Specimens have all the same thickness (third dimension) of 4 mm and a unified width of cell wall of 0.65 mm (equal to those of the ABS three-walled specimens). The unit cells have been created so that the internal sides of the cells (hexagons for HL and KL, triangles for TL) present the same dimensions. In other words, for each configuration the lengths of cells' edges – the vertical (h) and the oblique (l) ones – have the same values in hexagons and triangles. The internal width of a cell in the dense configurations has been set to half the internal width of its related sparse cell.

Finally, a mid-height horizontal notch has been inserted in each honeycomb specimen. The notch lengths range approximately from 18% to 22% of the whole lattice width. Such variability from a lattice to the other is inevitable since the width of a unit cell is not necessarily the same for all lattices and since, besides, the length of the notch is necessarily equal to a discrete sum of unit cells widths.

The main parameter to describe a lattice with variable density is the ratio $R$ as defined in Eq. (1), where $l_d$ and $l_s$ are the widths of the dense and sparse regions respectively.

$$R = \frac{l_d}{l_d + l_s} \qquad (1)$$

Therefore, sparse lattices are characterized by $R = 0$ while dense ones by $R = 1$. Previous computational analyses [11] have indicated that hexagon lattices with a ratio $R = 0.6$ exhibit the highest work of fracture under tensile loading, as the crack propagates from a side of the specimen to the other. Hence, as also carried out experimentally in [12], this value of $R$ has been taken as a reference for the creation of the 2D variable-density lattices in this study. Nevertheless, due to geometrical constraints, here variable density lattices with hexagonal and Kagome cells were characterized by $R = 0.58$, while the one with triangular cells had a value of $R = 0.6$. For each base topology, three sparse-dense bands have been taken into account: this constraint has been posed by the dimensions of the FFF manufacturing plate, which will be used in further studies to validate the current work.

## Results and discussion

Nine numerical tensile tests have been performed, obtaining as output a set of curves of reaction force per unit thickness vs displacement. The comparisons between the curves related to the sparse, variable, and dense configurations, for each one of the cells' base geometries, are reported in Figs.(4–6). The results show that, as expected, dense configurations of 2D lattices exhibit the highest stiffness and ultimate tensile force, while sparse configurations tend to have the highest displacement at failure.

From all the fracture tests, the work of fracture $G$ defined as

$$G = \int_{0}^{\delta_f} F(\delta) d\delta, \qquad (2)$$

where $\delta_f$ is the imposed displacement upon failure, has been computed.

Here, the hexagonal-cell lattice with variable density seems to have a lower fracture energy if compared with the homogeneous ones, not showing the positive effect of density variations, contrary to what was reported in Ref. [11]. This could be related to a dimensional effect linked to the printing constraints: here only three alternate bands are considered and, due to the unit cell dimensions, the number of cells is sensitively lower than that studied in the mentioned reference, probably leading to non-negligible boundary effects altering the overall fracture behaviour. In the lattices with triangular cells, the curves for dense and variable configurations are comparable, without relevant advantages in the use of density variations. The opposite occurs for the Kagome lattices, where the density variation results in the desired - slight - increase in the work of fracture $G$.
<figure><img src="image_5.png" /><figcaption>FIGURE 4: Force-Displacement curves for the hexagonal lattices.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>FIGURE 5: Force-Displacement curves for the triangular lattices.</figcaption></figure>

In order to compare the performances of lattice specimens with different topology and density, taking into account their mass, which plays an important role in weight-sensitive applications, the work of fracture per unit mass for each specimen, computed by dividing *G* by their respective mass, is reported in Tab.II, together with the percentage value with respect to the dense configuration, assumed as reference.

Fig.(7) shows the crack path upon failure in the three cases of lattices with variable density. One may notice how the presence of an alternation of cell density in the form of bands can affect the crack path: most of the kinks are related to the change in density. More specifically for triangle and Kagome topologies, the dense regions are characterized by almost rectilinear crack propagation in the direction transverse to the load, which is consistent with the observations made in [11, 12].
<figure><img src="image_7.png" /><figcaption>FIGURE 6: Force-Displacement curves for the Kagome lattices.</figcaption></figure>

<table><caption>TABLE II: Work of fracture per unit mass for all the considered lattice specimens. For each case, a percentage of the work of fracture with respect to the dense configuration assumed as reference is also given.</caption><thead><tr><th>G/m (J/kg)</th><th>Sparse</th><th>Dense</th><th>Variable</th></tr></thead><tbody><tr><td>Hexagonal</td><td>316.08 (91.89%)</td><td>343.98 (100%)</td><td>252.51 (73.41%)</td></tr><tr><td>Triangular</td><td>38.04 (66.76%)</td><td>56.98 (100%)</td><td>56.24 (98.69%)</td></tr><tr><td>Kagome</td><td>58.02 (137.73%)</td><td>42.13 (100%)</td><td>59.70 (141.72%)</td></tr></tbody></table>

It is observed how the values of G/m for honeycombs with triangular and Kagome topology fall in a lower range with respect to the values for hexagonal honeycombs. The reason for this occurrence, linked to the different displacements at failure as shown in Fig.(8), could be related to the interplay between lattice geometry and parent material constitutive behaviour. This question is currently under investigation.

<figure><img src="image_8.png" /><figcaption>FIGURE 7: Crack paths in the lattice specimens with density variations. Left: HLV. Center: TLV. Right: KLV.</figcaption></figure>
<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /><figcaption>FIGURE 8: Comparison between force-displacement curves for sparse lattices with different base topology.</figcaption></figure>
CONCLUSION

In this work, numerical failure analyses have been performed to assess the fracture behaviour of two-dimensional lattices with different base topologies. First, the constitutive properties of printed ABS have been experimentally investigated and numerically fitted. Then, a set of computational tensile tests have been performed on different configurations of honeycombs: sparse, dense and with variable lattice density, with either hexagonal, triangular or Kagome unit cells.

The investigation is implemented after promising results have been computationally predicted and experimentally verified with hexagonal-cell lattices in terms of an enhancement in fracture energy when a cell density gradient is introduced in a baseline homogeneous lattice. In this work, the study is extended to the effect of cell topology. In order to prepare and guide the forthcoming experimental analyses, which will be conducted employing the different cell topologies, the analyses presented here have been conducted in the same configuration as the foreseen experiments: same building material, same dimensions of the unit cells, same number of cells, and same boundary conditions.

According to what has been found, for the considered base material and the geometries taken into account, one major conclusion can be drawn: the most favourable topology to highlight the positive effect of a cell density gradient seems to be Kagome. Therefore, here the use of density variations results advantageous in the case of semi-regular two-dimensional lattices, while it does not lead to particularly favourable mechanical behaviours when applied to regular honeycombs. Moreover, the analyses have shown that the overall dimensions of the lattices, in terms of number of cells within each band of a uniform cell density, could be an important factor in the process of increasing the work of fracture of the lattice. Further investigations are needed to assess the sources of the discrepancies between the ranges of work of fracture per unit mass for the different honeycombs analysed in this work.

ACKNOWLEDGMENTS

LABEX EMC3 (French government "Programme d'Investissement d'Avenir") is acknowledged for its financial support on GS fellowship at Groupe de Physique des Matériaux. IB acknowledges the support of the *Italian Ministry of Education, University and Research – MIUR –* through the project DEVISU, funded under the scheme PRIN-2107 – Grant 22017ZX9X4K-006.

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