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# (a) CAD image of the octet-truss lattice constructed by the 3D packing of unit cell. (b) A unit cell of the nanolattice and its geometric parameters studied in this paper.

## Contexts in source publication

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... it is currently unknown how the stiffening and strengthening that has been attributed to surface effects in individual nanowires ( Park et al., 2009;Weinberger and Cai, 2012) impacts the mechanics of nanolattices. The objective of this work is to explore the mechanical response of octet-truss nanolattices, illustrated in the CAD model depicted in Fig. 1, under uniaxial compression using classical molecular dynamics (MD) simulations. The nanolattices are constructed with single crystal FCC metal (Cu) nanowires as the individual structural element, and because the structures have a nodal connectivity of 12, its mechanical response is expected to be stretching-dominated (Deshpande et ...

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... schematic model of the unit cell of the octet-truss nanolattice is shown in Fig. 1(b) together with the coordinate system. The relative density is computed by calculating the volumes of regions occupied by material in the CAD model, and scaling this by the unit cell volume, the octet-truss nanolattice relative density is í µí¼ given ...

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... t and l are the width and length of a strut defined in Fig. 1, respectively. It can be shown that Eq. (1) reduces to the relative density expression for an octet-truss lattice, if the node size is neglected (Deshpande et al., ...

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... yield point was chosen as the stress at the first peak after the linear elastic loading region. The yield strengths of the nanolattices are plotted as a function of relative density in Fig. 10 with unit cell size of 40 nm for strut size of 2.2-6 nm, and unit cell size of 30 nm for strut size of 4.3-5.6 nm. As is shown in Fig. 10, sizedependent strengthening is revealed by comparing the nanolattice yield strength against the bulk yield strength. In the paper of Gu et al. (Gu and Greer, 2015), bulk yield strength of Cu was ...

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... yield point was chosen as the stress at the first peak after the linear elastic loading region. The yield strengths of the nanolattices are plotted as a function of relative density in Fig. 10 with unit cell size of 40 nm for strut size of 2.2-6 nm, and unit cell size of 30 nm for strut size of 4.3-5.6 nm. As is shown in Fig. 10, sizedependent strengthening is revealed by comparing the nanolattice yield strength against the bulk yield strength. In the paper of Gu et al. (Gu and Greer, 2015), bulk yield strength of Cu was calculated to be 133 MPa, and the maximum yield strength of meso-lattices is about 332 MPa with relative density of 0.8 ( Gu and Greer, ...

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... yield strength. In the paper of Gu et al. (Gu and Greer, 2015), bulk yield strength of Cu was calculated to be 133 MPa, and the maximum yield strength of meso-lattices is about 332 MPa with relative density of 0.8 ( Gu and Greer, 2015). For <100> nanolattices, the maximum yield strength is about 586 MPa with relative density of 0.39. As shown in Fig. 12, comparison of the yield strengths of the nanolattices and the electroplated Cu thin film reveals that nanolattices with í µí¼ > 0.1 are stronger than the bulk, with the high-density <100> nanolattices ( having í µí¼~0.26) the same strength as the densest meso-lattices ( that were studied í µí¼~0.8) experimentally ( Gu and Greer, ...

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... 2015). In comparing the MD results for yield strength with the analytic theory predictions summarized in Tab. 1, we find that bending is the key contributor to enable the octet truss model to match the MD simulation results. Specifically, the bending term shifts the octet truss yield stress curve down as compared to the octet truss result in Fig. 10 because bending is a first order effect that scales as according to Eq. (16). The í µí±¡/í µí±' yield strength of struts not considering surface effects is 7.11 GPa, which is the compressive strength of <110> nanowires with width of 4 nm (the median width of ...

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... bending alone leads to a slight underprediction of the yield strength, additionally incorporating nodal volume effects (theory prediction 2 in Fig. 10) shifts the yield stress curve up close to the MD results. A detailed analysis demonstrates that, however, theory prediction 2, which accounts for both bending and nodal volume, deviates from the MD yield stress curve at low relative densities, or equivalently struts with small widths. This is due to the size-dependent compressive ...

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... widths. This is due to the size-dependent compressive strength of <110> nanowires, which increases with increasing nanowire width; as discussed with regards to the nanolattice modulus, surface effects had the largest effect on the low relative density nanolattices, which also manifests itself here with regards to the yield strength, as shown in Fig. 11. By incorporating surface effects in conjunction with bending and nodal volume via theory prediction 1, we are able to find good agreement with the MD simulations in Fig. 10 and 11. Thus, bending, surface effects and nodal volume contribute in decreasing order to the strength of ...

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... modulus, surface effects had the largest effect on the low relative density nanolattices, which also manifests itself here with regards to the yield strength, as shown in Fig. 11. By incorporating surface effects in conjunction with bending and nodal volume via theory prediction 1, we are able to find good agreement with the MD simulations in Fig. 10 and 11. Thus, bending, surface effects and nodal volume contribute in decreasing order to the strength of ...

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... further refinement in the ranges that the various mechanisms (bending, nodal volume, surface effects) dominate can be obtained by considering different nanolattice relative densities, as seen in Fig. 10. For low relative densities (under 5.5%), surface effects have a significant effect on the yield strength, due to the fact that the effects of bending and nodal volume are small as shown in Eq. (18), which can be simplified ...

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... the case of very small t/l (or very low relative density). í µí°¾ í µí± í µí°µ ≈ 1 This can also be observed in comparing theory predictions 1 and 2 in Fig. 10, where for low relative densities theory prediction 1, which includes surface effects, better matches the MD simulation ...

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... the relative density increases, bending effects gradually become dominant, so the green points in the middle relative density regime (between 10% and 20%) closely match theory prediction 3, which considers only bending, as shown in Fig. 10. For high relative densities (over 25%), bending and nodal volume both contribute because of enhanced constraints at the nodes and the increase of nodal volume, while surface effects become negligible. Thus, all blue points are closer to theory prediction 2, which considers both nodal volume and bending. While nodal volume was most ...

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... 2009;Park et al., 2006;Weinberger and Cai, 2012). Compared to individual nanowires, however, we find that surface effects have a smaller influence on nanolattices. This is because surface effects compete with bending and nodal volume effects, all of which depend nonlinearly with the nanolattice relative density as discussed above, and shown in Fig. 10. Thus, while surface effects do improve the accuracy of the theoretical models, particularly at lower relative densities, they are not the dominant effect in governing the mechanics of nanolattices. Fig. 11 Yield strength of nanolattices as a function of strut width (dashed lines) and length of unit cell (solid lines) from theory ...

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... with bending and nodal volume effects, all of which depend nonlinearly with the nanolattice relative density as discussed above, and shown in Fig. 10. Thus, while surface effects do improve the accuracy of the theoretical models, particularly at lower relative densities, they are not the dominant effect in governing the mechanics of nanolattices. Fig. 11 Yield strength of nanolattices as a function of strut width (dashed lines) and length of unit cell (solid lines) from theory prediction 1 and MD simulations (solid ...

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... compare the yield strength of the Cu nanolattices with other, macroscale nanolattices and other metallic engineering materials, we plot them together on the material property chart in Fig. 12, which shows the density dependent yield strength for existing materials (Ashby, 2011) and other space filling lattice structures, including alumina nanolattices ( Meza et al., 2014), Ti-6Al-4V octet-truss lattice( Dong et al., 2015) and Cu meso-lattices ( Gu and Greer, 2015). The strengths of both foams and lattices scale with those ...

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... shown in Fig. 12, Cu nanolattices provide higher strength at a lower density compared with Cu meso-lattices, and are about an order of magnitude stronger for the same relative density, which marks a new entry in the high-strength lightweight material parameter space. Compared with meso-lattices, Cu nanolattices have smaller length scales and lower ...

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... This result is close to MD simulations, which means bending effects decrease the scaling exponent. Nevertheless, octet-truss nanolattices are mainly stretching-dominated and are simultaneously lightweight and strong, while outperforming bulk copper and copper meso-lattices by reaching a previously unexplored property range in the Ashby plot in Fig. ...

## Citations

... For this reason, many research works have focused on ways to improve its performances. For instance, [7][8][9] studied octet-truss globally but also locally, revealing stress concentrations at the node level. To overcome this issue, fillets can be added at the node level. ...

The development of additive manufacturing (AM) for the fabrication of metallic parts allows structures to be directly manufactured from 3D models. The electron beam melting (EBM) technology is an example of AM technologies that enable the manufacturing of new designs and sophisticated geometries. The process is particularly well suited for the fabrication of lattice structures. Octet-truss lattice structure has been a subject for research during the past 10 years. The potentials that it possesses attract enough interest for manufacturers to use it during the production of parts. Besides being lightweight, the structure could provide solid mechanical properties. However, researchers always encounter the same issue regarding this particular structure. During finite element analysis (FEA) simulation, stress concentration tends to appear at the strut intersection. This is due to the sharp edges that have very small surface area, thus provoking the presence of singularities. In this respect, the proposed solution is to integrate rounded joints or fillets at the strut intersection. However, adding fillet entails a mass increase of octet-truss structures. To avoid this mass increase related to these fillets, it is necessary to reduce the size of octet-truss struts. This research work studies the influence of fillets onto the mechanical properties of structures with identical mass. To do so, a set of 15 octet-truss structures are designed with various fillet sizes and strut sizes and compared. Whereas some of them have thick struts and small fillets, others have smaller struts and bigger fillets. The main technical issue in this study remains the design of fillets for octet-truss structures. These latter can indeed be created for up to 12 struts that converge to the same point. Once designed, these octet-truss structures are fabricated by EBM technology and undergo static compression testing. Mechanical properties of each structure are finally determined. Results show that for the same relative density, octet-truss with fillets degrades the mechanical characteristics of the whole structure. This study shows that the strength/mass ratio is better for a structure without fillets.

... In this section, analytical models for calculating the mechanical properties, i.e. compressive modulus and initial yield strength, of the {BCC} truss structure are reviewed. Relative density ρ (He et al., 2017) is an important structural characteristic of the truss structure, and can be ...

Additive manufacturing (AM) routes have brought abundant geometric defects into as-fabricated lattice materials. Researchers take it for granted that any defects in lattice struts would decrease the mechanical properties of lattice structures, which has also been a rule followed by all engineering designers. In this paper, novel design strategies of actively utilizing defects to improve the mechanical properties of BCC lattice structures are proposed. Firstly, effects of non-periodic microstructure, waving struts and missing struts on the mechanical properties (i.e. compressive modulus and initial yield strength) of BCC lattice structures are investigated through finite element analysis. Simulation results indicate that mechanical properties of the BCC lattice structures exhibited certain sensitivity to the defects. Then, strategies to enhance the performance of BCC lattice patterns by utilizing the imperfections actively are also proposed. Finally, some typical lattice specimens are fabricated and experimental tests are also conducted to validate the accuracy of design strategies. We believe that the proposed new strategy could markedly expand the design space for the development of future materials by actively utilizing the geometric defects.

... For this reason, many research works have focused on ways to improve its performances. For instance, [7,8,9] studied octet-truss globally but also locally, revealing stress concentrations at nodes level. To overcome this issue, fillets can be added at nodes level, however this entails a mass increase of octet-truss structure. ...

The development of Additive Manufacturing (AM) for the fabrication of metallic parts allows structures to be directly manufactured from 3D models. The Electron Beam Melting (EBM) technology is an example of AM technologies that enables the manufacturing of new designs and sophisticated geometries. The process is particularly well suited for the fabrication of lattice structures. Octet-truss lattice structure has been a subject for research during the past 10 years. The potentials that it possesses attract enough interest for manufacturers to use it during the production of parts. Besides being lightweighted, the structure could provide solid mechanical properties. However, researchers always encounter the same issue regarding this particular structure. During Finite Element Analysis (FEA) simulation, stress concentration tends to appear at the struts intersection. This is due to the sharp edges that have very small surface area, thus provoking the presence of singularities. In this respect, the proposed solution is to integrate rounded-joints or fillets at the struts intersection. However, adding fillet entails a mass increase of octet-truss structures. To avoid this mass increase related to these fillets, it is necessary to reduce the size of octet-truss struts. This research work studies the influence of fillets onto the mechanical properties of structures with identical mass. To do so, a set of 15 octet-truss structures are designed with various fillet sizes and strut sizes and compared. Whereas some of them have thick struts and small fillets, others have smaller struts and bigger fillets. The main technical issue in this study remains the design of fillets for octet-truss structures. These latter can indeed be created for up to 12 struts that converge to the same point. Once designed, these octet-truss structures are fabricated by EBM technology and undergo static compression testing. Mechanical properties of each structure are finally determined. Results show that for the same relative density, octet-truss with fillets degrades the mechanical characteristics of the whole structures. This study shows that the strength/mass ratio is better for a structure without fillets. this result can be used in lightweighted applications.

... For any structure of dimensionality 2 or greater, a proper accounting of large deformations requires geometric nonlinearities that greatly complicate the network connectivity [17]. The elements can also be subject to nonlinear material response such as strain hardening [18] or nodal density dependent effects [19]. These nonlinearities disrupt the algebraic nature of the Fourier space solutions and thus represent significant obstacles that the methods used here cannot immediately surpass. ...

Truss structures at macro-scale are common in a number of engineering applications and are now being increasingly used at the micro-scale to construct metamaterials. In analyzing the properties of a given truss structure, it is often necessary to understand how stress waves propagate through the system and/or its dynamic modes under time dependent loading so as to allow for maximally efficient use of space and material. This can be a computationally challenging task for particularly large or complex structures, with current methods requiring fine spatial discretization or evaluations of sizable matrices. Here we present a spectral method to compute the dynamics of trusses inspired by results from fluid flow networks. Our model accounts for the full dynamics of linearly elastic truss elements via a network Laplacian; a matrix object which couples the motions of the structure joints. We show that this method is equivalent to the continuum limit of linear finite element methods as well as capable of reproducing natural frequencies and modes determined by more complex and computationally costlier methods.

... But this is a research field in continuous growth, which implies that many works are treating the details of the infill of those CAD designs with metamaterials. Mainly with the numerical characterization of the mechanical properties with virtual testing of different configurations and shapes [7,8,9,3] ...

Metamaterials are gaining interest with the emergence of the additive manufacturing technologies. Those implies a capability to perform local mechanical optimizations of the designs, as well as versatile external geometries of the components. The software presented is capable to fill a complex design in a .stl format with a given metamaterial, achieving a graded stiffness. The software developed is also configured to create the final design in different formats to perform a virtual test of the element in a commercial FEM, to edit the design in a CAD software, and to export it in a 3D printable format.

... Next, we derive the stretching-shear-bending coupled mechanics model for the σ * of diamond lattices. Note that deformation by shear does not contribute to the yielding of a lattice [28,29,40,44,93]. Thus, Table 2 Bending-stretching-shear coupled mechanical property models for various lattices with cylindrical strutselastic modulus. ...

... ρ RD , which are consistent with [16,93]. Table 2 (for E * ) and Table 3 (for σ * ) detail these stretching-shearbending coupled mechanics models, which consider not only the complex deformation mechanisms but also the lattice topology. ...

The Gibson-Ashby (G-A) model has been instrumental in the design of additively manufactured (AM-ed) metal lattice materials or mechanical metamaterials. The first part of this work reviews the proposition and formulation of the G-A model and emphasizes that the G-A model is only applicable to low-density lattice materials with strut length-to-diameter ratios greater than 5. The second part evaluates the applicability of the G-A model to AM-ed metal lattice materials and reveals the fundamental disconnections between them. The third part assesses the
deformation mechanisms of AM-ed metal lattices in relation to their strut length-to-diameter ratios and identifies
that AM-ed metal lattices deform by concurrent bending, stretching, and shear, rather than just stretching or
bending considered by the G-A model. Consequently, mechanical property models coupling stretching, bending and shear deformation mechanisms are developed for various lattice materials, which show high congruence with experimental data. The last part discusses new insights obtained from these remedies into the design of strong and stiff metal lattices. In particular, we recommend that the use of inclined struts be avoided.

... The deformation process also depends on the exact dimensions and the material choice. The buckling of the strut is known to occur in slender struts [103,104]. Tancogne-Dejean et al. computationally demonstrated that the Oct structures with high strut aspect ratio are prone to buckling under compressive stress [104]. They identified that there was no buckling deformation in the Oct structures with strut aspect ratio lower than 5. ...

Additive manufacturing of pure copper (Cu) via laser-powder bed fusion (L-PBF) is challenging due to the low energy absorptivity under infra-red laser. As a result, 3-dimensional architectures, known for excellent load-bearing and energy absorption capabilities, have not been fabricated in pure Cu, so far. This study, for the first time, Cu lattice structures are fabricated through laser-powder bed fusion (L-PBF) with green laser (λ = 515 nm). Structural and microstructural analysis confirm that the lattice structures consist of well-defined unit-cells and show dense microstructure. The deformation behavior is investigated under a wide range of strain rates from ∼0.001 /s to ∼1000 /s. The stress–strain curves exhibit a smooth and continuous deformation without any post-yield softening, which can be attributed to the intrinsic mechanical properties of Cu. Correlated with post-mortem microscopy examination, the rate-dependent deformation behavior of pure Cu lattice structures is investigated and rationalized. The current work suggests that the complex Cu architectures can be fabricated by L-PBF with green laser and are suitable for dynamic loading applications.

... For this reason, many research works have focused on ways to improve its performances. For instance, [7,8,9] studied octet-truss globally but also locally, revealing stress concentrations at nodes level. To overcome this issue, fillets can be added at nodes level, however this entails a mass increase of octet-truss structure. ...

The development of Additive Manufacturing (AM) for the fabrication of metallic parts allows structures to be directly manufactured from 3D models. The Electron Beam Melting (EBM) technology is an example of AM technologies that enables the manufacturing of new designs. The process is particularly well suited for the fabrication of lattice structures. Octet-truss lattice structure has been a subject for research in recent years. Besides being lightweighted, the structure could provide solid mechanical properties. However, researchers always encounter the same issue regarding this particular structure. During Finite Element Analysis (FEA) simulation, stress concentration tends to appear at the struts intersection. This is due to the sharp edges, thus provoking the presence of singularities. In this respect, the proposed solution is to integrate fillets at the struts intersection. However, adding fillet entails a mass increase of octet-truss structures. To avoid this mass increase related to these fillets, it is necessary to reduce the size of octet-truss struts. This research work studies the influence of fillets onto the mechanical properties of structures with identical mass. To do so, a set of 15 octet-truss structures are designed with various fillet sizes and strut sizes and compared. Whereas some of them have thick struts and small fillets, others have smaller struts and bigger fillets. The main technical issue in this study remains the design of fillets for octet-truss structures. These latter can indeed be created for up to 12 struts that converge to the same point. Once designed, these octet-truss structures are fabricated by EBM technology and undergo static compression testing. Mechanical properties of each structure are finally determined. Results show that for the same relative density, octet-truss with fillets degrades the mechanical characteristics of the whole structures.

... Conventional stretch-dominated lattices such as Face-Centered Cubic (FCC) [12], Octahedron [24], Kagome [25] and Octet [26] have been widely investigated for their high specific stiffness and specific strength and outstanding energy-absorbing. Deshpande et al. [27] studied the effective mechanical properties of octet lattice from the perspective of theoretical modeling, experimental validation and numerical simulations. ...

... Deshpande et al. [27] studied the effective mechanical properties of octet lattice from the perspective of theoretical modeling, experimental validation and numerical simulations. Subsequently, researchers explored the mechanical properties of octet lattices made of polymeric materials [28,29], copper [26], 316L stainless steel [30] and titanium alloys [31]. These studies also confirmed that octet structures have excellent mechanical properties and energy-absorbing performance, and are considered as a promising substitute for conventional metallic foams [32]. ...

... However, they usually suffer from low stiffness and low strength, which limit their applications. Up to now, few lattices can achieve a combination of high stiffness, high energy absorption and stable stress response, and most of the previously reported structures, such as BCC [37], Octet [26], FCC [12] lattice, possess only one or two aspects of the three. Consequently, it is always intriguing to design new lattice topologies that can simultaneously enhance the above three aspects of properties. ...

... Based on the previous analyses of the mechanics of lattice materials, lattices with a stretching-dominated deformation behavior are more desirable in the design for additive manufacturing as they offer superior mechanical properties at the same relative density when compared to the bending-dominated topologies. Therefore, stretching-dominated lattices such as the Octet-Truss lattice have been a matter of extensive research [82][83][84][85][86][87][88][89]. However, the substantial nodal interconnections of stretching-dominated strut-based topologies introduce several manufacturing and functioning limitations. ...

Cellular lattices with architectural intricacy or metamaterials have gained a substantial amount of attention in the past decade due to the recent advances in additive manufacturing methods. The lattice topology controls its physical and mechanical properties; therefore, the main challenge is selecting the appropriate lattice topology for a desired function and application. In this work, we comprehensively study the topology–property relationship of three classes of polymer metamaterials based on triply periodic minimal surfaces (TPMS) of sheet/shell and ligament types, and other types of well-known strut-based lattices. The study uses a holistic approach of designing, additive manufacturing, microstructural characterization, and compressive uniaxial mechanical testing of these polymer lattices that are 3D-printed using the laser powder bed fusion technique known as selective laser sintering (SLS). In total, 55 lattices with different topologies and relative densities were 3D-printed and tested. Printing quality was assessed using scanning electron microscopy and micro-computed tomography. The extracted mechanical properties of elastic modulus, yield strength, plateau strength, and energy absorption are thoroughly compared between the different lattice architectures. The results show that all the investigated ligament-based TPMS polymer lattices exhibit bending-dominated elastic and plastic behavior, indicating that they are suitable candidates for energy absorbing applications. The sheet-based TPMS polymer lattices, similarly to the well-known Octet-Truss lattice, exhibited an elastic stretching-dominated mode of deformation and proved to have exceptional stiffness as compared to all other topologies, especially at low relative densities. However, the sheet-based TPMS polymer lattices exhibited a bending-dominated plastic behavior which is mainly driven by manufacturing defects. Overall, however, sheet-based TPMS polymer lattices exhibited the best mechanical properties, followed by strut-based lattices and finally by ligament-based TPMS lattices. Finally, it is depicted that at high relative densities, the mechanical properties of lattices of various architectures tend to converge, which implies that the topological effect is more significant at low relative densities. Generally, this study provides important insights about the selection of polymer mechanical metamaterials for various applications, and shows the superiority of TPMS-based polymer metamaterials as compared to several other classes of polymer mechanical metamaterials.