The role of architecture

Manipulating the architecture or microstructural geometry^{Reference Bauer, Meza, Schaedler, Schwaiger, Zheng and Valdevit15,Reference Yu, Zhou, Liang, Jiang and Wu23,Reference Yeo, Oh and Yoo24} can effectively alter the mechanical behavior of architected materials and significantly expand the material design space. Architectures are classified as truss-, plate-, and shell-based with periodic or stochastic cellular structures. In contrast to the stochastic structures, the periodic ones offer greater potential to improve the efficiency in mechanical properties due to the ordered and regular arrangement of internal pores.^{Reference Yeo, Oh and Yoo24}

For truss-based lattices, the common periodic unit cells include the octet-truss, iso-truss, 3D kagome, cuboctahedron, tetrahedron, and tetrakaidecahedron^{Reference Meza, Phlipot, Portela, Maggi, Montemayor, Comella, Kochmann and Greer25–Reference Tancogne-Dejean and Mohr28} (Figure 1a–c). These unit cells have different degrees of rigidity and nodal connectivity,^{Reference Meza, Phlipot, Portela, Maggi, Montemayor, Comella, Kochmann and Greer25} which mainly determine the mechanical response. Considering the deformation mode of a single strut, the truss-based lattice architectures are generally classified as stretching- and bending-dominated. In the ideal stretching-dominated one, each strut is subjected to uniaxial tension or compression (Figure 1a, c). However, in the real stretching-dominated lattice, the struts might experience local buckling, especially under a large strain. The bending-dominated lattice deforms via bending or buckling of single struts (Figure 1b). As a typical bending-dominated lattice, the tetrakaidecahedron lattice exhibits large deformability and good recoverability.^{Reference Meza, Phlipot, Portela, Maggi, Montemayor, Comella, Kochmann and Greer25} Under large compressive strain, it accommodates the deformation by local deformation at nodes, which makes these lattices sensitive to nodal compliance or imperfections.^{Reference Meza, Phlipot, Portela, Maggi, Montemayor, Comella, Kochmann and Greer25}

Figure 1.

Schematic illustrations of typical unit cells of truss-, plate-, and shell-based lattices. (a) Octet-truss; (b) tetrakaidecahedron; (c) iso-truss; (d) octet unit cell for plate-based lattice; (e) hybrid unit cell of cubic and octet structures for plate-based lattice; (f) triply periodic minimal surface-like unit cell for shell-based lattice; (g) structural hierarchy of architected materials. Reprinted with permission from Reference 34. © 2016 Nature Publishing Group. The arrows in (a–c) indicate the load transfer and deformation mode of struts in the truss-based lattices.

As a representative of stretching-dominated lattices, the octet-truss lattice exhibits good specific mechanical properties, but apparent anisotropies in both elastic modulus and yield strength.^{Reference Tancogne-Dejean and Mohr28} Young’s modulus in the [111] direction of the octet-truss lattice is maximum, while the compressive strength in both [111] and [001] directions are maximum over all directions.^{Reference Tancogne-Dejean and Mohr28} Topology optimization based on finite element modeling showed that the iso-truss is isotropic and has optimal stiffness compared to other lattice topologies.^{Reference Messner26} Recent experimental and computational studies^{Reference Zhang, Vyatskikh, Gao, Greer and Li27} further demonstrated that the iso-truss lattice has better flaw tolerance than the lattices with octet-truss and tetrahedron unit cells, due to its better load-bearing and load-transfer ability.

Recently, plate-based lattices, replacing the struts with plates on the close-packed planes of crystalline structures, have emerged.^{Reference Berger, Wadley and McMeeking29,Reference Tancogne-Dejean, Diamantopoulou, Gorji, Bonatti and Mohr30} Based on finite element modeling and theoretical analyses, a design map was developed for the elastically isotropic plate-based lattice with cubic symmetry^{Reference Tancogne-Dejean, Diamantopoulou, Gorji, Bonatti and Mohr30} (Figure 1d–e). The stiffness of these plate-based lattices is at least 3× higher than that of the stiffest truss-based lattices of equal mass.^{Reference Tancogne-Dejean, Diamantopoulou, Gorji, Bonatti and Mohr30} These plate-based lattices are well-distributed networks of plates, which have more efficient loading transfer than the truss-based lattices with similar geometries.^{Reference Berger, Wadley and McMeeking29,Reference Tancogne-Dejean, Diamantopoulou, Gorji, Bonatti and Mohr30} Notably, a plate-based lattice with a combination of cubic and octet topologies has achieved the Hashin–Shtrikman upper bound,^{Reference Berger, Wadley and McMeeking29} which is a theoretical limit for the modulus of isotropic porous materials.

A novel shell-based lattice has been conceived by replacing the hollow truss with a continuous and curved shell.^{Reference Nguyen, Cho and Kang31,Reference Lee, Lee, Han and Kang32} In shell-based lattices, the nodes are interconnected through smooth shells, which reduce stress concentration around the nodes. Based on topology optimization, the curved shell is indeed a triply periodic minimal surface (TPMS),^{Reference Nguyen, Cho and Kang31} which has a constant mean curvature at every point on the surface (Figure 1f). Shell-based lattices consisting of TPMS are stretching-dominated and experience coplanar stresses.^{Reference Nguyen, Cho and Kang31} The shell-based lattices with well-developed interconnections are less sensitive to flaws than truss-based lattices with similar geometries, but show only little improvement in strength, because local buckling dominates the deformation.^{Reference Nguyen, Cho and Kang31} By optimizing the shape or size of the shells, and by suppressing local buckling mechanisms, the mechanical properties of shell-based lattices can be further improved.

Inspired by hierarchical natural materials (e.g., bone, wood, and conch shell), hierarchy has been introduced into the design of architected materials.^{Reference Meza, Zelhofer, Clarke, Mateos, Kochmann and Greer33,Reference Zheng, Smith, Jackson, Moran, Cui, Chen, Ye, Fang, Rodriguez, Weisgraber and Spadaccini34} 3D hierarchical architected materials consist of self-similar or dissimilar unit cells at different hierarchical levels with feature sizes spanning several orders of magnitude^{Reference Meza, Zelhofer, Clarke, Mateos, Kochmann and Greer33,Reference Zheng, Smith, Jackson, Moran, Cui, Chen, Ye, Fang, Rodriguez, Weisgraber and Spadaccini34} (Figure 1g). Such a hierarchical design can expand the building volume without sacrificing structural integrity and improve mechanical properties by combining the benefits of microarchitectures at successive length scales. Therefore, such hierarchical structures reach into the ultralow density material property spaces (Figure 2). Because of the decoupling of strut length effective for Euler buckling (i.e., buckling of a monolithic beam under compression) and the relative density,^{Reference Bauer, Meza, Schaedler, Schwaiger, Zheng and Valdevit15} hierarchical architected materials exhibit improved buckling resistance and recoverability^{Reference Bauer, Meza, Schaedler, Schwaiger, Zheng and Valdevit15,Reference Meza, Zelhofer, Clarke, Mateos, Kochmann and Greer33,Reference Zheng, Smith, Jackson, Moran, Cui, Chen, Ye, Fang, Rodriguez, Weisgraber and Spadaccini34} facilitating linear scaling of compressive strength with relative density.

Figure 2.

Strength and recoverability of existing micro- and nanoarchitected materials. (a) Material property chart showing compressive strength versus density. Many architected materials fill the empty spaces not accessible by known materials. For graphene,^{Reference Lee, Wei, Kysar and Hone48} the tensile strength has been included. (b) Glassy carbon and different composite-beam lattices reach the highest specific strengths, while such a clear advantage is not observed for the (c) recoverability. The lattices were deformed to different strain levels, but the high-entropy alloy (HEA)-coated structures still appear superior at combining high strength and recoverability underlining the importance of materials selection and architectural design.

Exploiting material size effects

Material size effects are generally observed when extrinsic feature sizes approach the atomic scale and the continuum assumption of materials becomes invalid.^{Reference Kraft, Gruber, Mönig and Weygand3,Reference Greer and De Hosson4,Reference Gu, Wu, Zhang, Srolovitz and Greer7} We focus here on size effects in mechanical properties, the most significant of which are enhanced strength^{Reference Kraft, Gruber, Mönig and Weygand3,Reference Greer and De Hosson4,Reference Gu, Wu, Zhang, Srolovitz and Greer7} and ductility.^{Reference Chen, Jang, Guan, An, Goddard and Greer35} Although these novel nanomaterial properties are generally lost in bulk materials, careful control over nanoscale features enables them to be utilized in micro- and nanoarchitected materials.

Nanoscale strengthening is dependent upon both the constituent material system and its corresponding crystal or grain structure. In single-crystalline metals, size-effected strengthening can arise due to dislocation starvation, dislocation source truncation, and the elimination of “weakest-link” sources.^{Reference Greer and De Hosson4} In micro- and nanoarchitected materials, this is seen predominantly, for instance, in electrodeposited copper mesolattices with strengths greater than that of bulk copper, despite having 50% of the latter’s density,^{Reference Gu and Greer36} and electrodeposited nickel double gyroid structures that have strengths equal to or greater than that of bulk nickel despite having just 38% of the bulk density.^{Reference Khaderi, Scherer, Hall, Steiner, Ramamurty, Fleck and Deshpande19} Similar behaviors have been observed in nanoporous metals.^{Reference Jin, Weissmüller and Farkas37}

Nanocrystalline metals conversely experience a smaller-is-weaker trend due to the enhanced mobility of grain boundaries at free surfaces. This is observed in sputtered gold nanolattices where structures with smaller features are weaker than structures with larger features at the same overall density^{Reference Montemayor and Greer38} and, to some degree, in directly additive-manufactured pyrolyzed premetallic photopolymer nickel systems.^{Reference Vyatskikh, Delalande, Kudo, Zhang, Portela and Greer39} In ceramics and other intrinsically brittle materials, strength is traditionally governed by flaws via a “weakest link” mechanism that can be well characterized using a Weibull statistical distribution.^{Reference Jang, Meza, Greer and Greer40,Reference Bauer, Schroer, Schwaiger, Tesari, Lange, Valdevit and Kraft41} This has the effect of strengthening many nanoarchitected materials systems, for instance enabling metamaterial recoverability in nanolattices made from atomic-layer-deposited (ALD) aluminum oxide (Al₂O₃)^{Reference Meza, Zelhofer, Clarke, Mateos, Kochmann and Greer33,Reference Meza, Das and Greer42} and ultrahigh strength in lightweight nanolattices made from pyrolyzed carbon.^{Reference Zhang, Vyatskikh, Gao, Greer and Li27,Reference Bauer, Schroer, Schwaiger and Kraft43} Nanoscale strengthening also has a unique effect on the flaw tolerance of nanoarchitected materials. It has been observed that ALD Al₂O₃ nanolattices tested in tension are insensitive to notches^{Reference Montemayor, Wong, Zhang and Greer44} and pyrolyzed carbon nanolattices exhibit large strains to failure due to insensitivity to flaws of their constituent beams.^{Reference Zhang, Vyatskikh, Gao, Greer and Li27}

Ductility can be enhanced at the nanoscale due to the availability of free surfaces, and the damage state at the free surfaces often drives the extent of ductility.^{Reference Chen, Jang, Guan, An, Goddard and Greer35} This effect is most pronounced in brittle materials such as silicon, ceramics, and metallic glasses, wherein ductility is enabled when feature sizes are smaller than the intrinsic size necessary for failure nucleation (e.g., shear transformation zones in metallic glass that are ∼100 nm). The exploitation of enhanced ductility can be seen in nanolattices made from sputtered metallic glasses that suppress brittle failure when their feature sizes are reduced below ∼50 nm.^{Reference Lee, Jafary-Zadeh, Chen, Zhang and Greer45,Reference Liontas and Greer46} It is also observed in copper nanolattices with a ∼250-nm-thick coating of chemical-vapor-deposited (CVD) silicon that resists cracking during lithiation.^{Reference Xia, Di Leo, Gu and Greer47} Although the previously mentioned results show promise for the enhancement of ductility using size effects in nanoarchitected materials, it is still an underutilized phenomenon.

Local deformation and failure

The mechanical properties of architected materials depend on the local tensile, compressive, and shear loading states. The corresponding stiffness, strength, hardening/softening behavior, damage evolution, and strain limit of each can be set by the constituent material or by instabilities such as buckling, each of which can be extremely anisotropic and exhibit large tension–compression asymmetry.

Constituent material behavior is defined by atomic structure and corresponding deformation mechanisms. For example, carbon can form weakly bonded graphitic sheets (e.g., in carbon fiber having axial stiffness >10× its transverse stiffness),^{Reference Totry, Molina-Aldareguía, González and Llorca58} or it can form amorphous carbon,^{Reference Bauer, Schroer, Schwaiger and Kraft43} which is relatively isotropic in both strength and stiffness, but whose properties are affected by defect size and carbon bond configurations.^{Reference Albiez and Schwaiger59} Brittle materials undergo catastrophic failure in both tension and compression (Figure 3c), and can be orders of magnitude weaker in tension because their strength is flaw-limited.^{Reference Danzer, Supancic, Pascual and Lube60} Brittle nanomaterials have minimal tension–compression asymmetry in strength due to the smaller number of defects,^{Reference Jang, Meza, Greer and Greer40,Reference Bauer, Schroer, Schwaiger, Tesari, Lange, Valdevit and Kraft41} although the tensile failure strain is still ${{\rm{\varepsilon }}_f} \approx 5\%$ even for the strongest nanomaterials.^{Reference Zhu and Li61}

Figure 3.

Local behaviors. (a) Compression of single-crystal niobium and (b) tension of single-crystal gold showing the inherent tension–compression asymmetry due to densification versus necking. Adapted with permission from Reference 62. © 2009 The Minerals, Metals & Materials Society. (c) Tension of an Al₂O₃ beam showing brittle response.^{Reference Bauer, Schroer, Schwaiger, Tesari, Lange, Valdevit and Kraft41} (d) Compression of four-bar hollow TiN beams showing buckling response.^{Reference Meza and Greer63} (e) Catastrophic (pink) versus gradual (red) collapse of a hollow NiP cylinder due to shell buckling. Adapted with permission from Reference 71. © 2011 American Chemical Society.

Ductile materials generally have little tension–compression asymmetry, as their strength is governed by dislocation motion. For isolated ductile members, there is an emergent tension–compression asymmetry after the onset of plasticity because tension causes necking and a loss of integrity, while compression causes expansion and densification (Figure 3a–b).^{Reference Zheng, Smith, Jackson, Moran, Cui, Chen, Ye, Fang, Rodriguez, Weisgraber and Spadaccini34,Reference Greer, Kim and Burek62} Anisotropy in mechanical properties can have a significant effect on overall properties, but it has yet to be thoroughly studied or effectively utilized in tuning the mechanical properties of nanoarchitected materials.

While constituent materials properties are important, it is critical to consider the local distribution of stresses that emerge in various structures. Beams, plates, and shells can be loaded in pure tension or compression, but they often experience bending, which causes both tensile and compressive stresses. Brittle materials in bending will fail catastrophically when the tensile fracture stress is reached,^{Reference Meza, Das and Greer42,Reference Bauer, Schroer, Schwaiger and Kraft43,Reference Meza and Greer63} while ductile materials will generally form a plastic hinge and experience localized deformation.^{Reference Gu and Greer36,Reference Montemayor and Greer38,Reference Schaedler, Jacobsen, Torrents, Sorensen, Lian, Greer, Valdevit and Carter55} Bending in structural members is a less significant factor in denser structures (with >5–10% relative density), because the intersections between beams and plates at nodes and joints contribute considerably to the load-carrying ability and are much more significant sources of stress concentration.^{Reference Meza, Phlipot, Portela, Maggi, Montemayor, Comella, Kochmann and Greer25,Reference Portela, Greer and Kochmann64} Nodes are generally loaded in bending and have high stress concentrations, meaning they are often the initial point for failure.^{Reference Meza, Das and Greer42,Reference Bauer, Schroer, Schwaiger and Kraft43,Reference Bauer, Schroer, Schwaiger and Kraft65,Reference Juarez, Schroer, Schwaiger and Hodge66} Attempts to work around this issue by changing the node geometry have had minimal success,^{Reference Portela, Greer and Kochmann64} while other types of spinodal structures^{Reference Hsieh, Endo, Zhang, Bauer and Valdevit67} and minimal surface structures^{Reference Han, Lee and Kang68} have had more success by introducing architectures without any defined nodes.

Structural instabilities in lightweight architected materials have an effective material response. For example, slender beams made from brittle constituents have a brittle response in tension, but can bend or buckle in compression, thereby sustaining large deformations^{Reference Jang, Meza, Greer and Greer40,Reference Meza and Greer63} (Figure 3d). This means that instabilities can generate a considerable tension–compression anisotropy. Instabilities occur in slender beam members via beam buckling, which occurs at a critical stress ${{\rm{\sigma }}_{cr}} = {{{{\rm{\pi }}^2}EI} \over {A{{\left( {kL} \right)}^2}}}$, where E is Young’s modulus, I is the second moment of area, A is the area, L is the beam length, and k is a factor to account for boundary conditions.^{Reference Timoshenko and Gere69} Different types of lattices can create different effective boundary conditions, with more interconnected structures having a more fixed boundary ($k \approx 0.5$) and less connected structures having a more pinned boundary ($k \approx 1$).^{Reference Meza70} It is convenient to describe a slenderness ratio.

$${\rm{\lambda }} = \sqrt {A{L^2}/I}.$$

Instabilities in thin plates and shells occur via shell buckling. This type of buckling is extremely complex due to the numbers of buckling modes that can occur depending on the boundary conditions^{Reference Lian, Jang, Valdevit, Schaedler, Jacobsen, Carter and Greer71} (Figure 3e). The simplest case is for a thin hollow circular shell, for which the buckling limit is

$${{\rm{\sigma }}_{cr}} = {E \over {\sqrt {3\left( {1 - {\nu ^2}} \right)} }}\left( {{t \over r}} \right),$$

where v is the Poisson’s ratio, t is the wall thickness, and r is the beam radius.^{Reference Timoshenko and Gere69} Buckling is an inherently elastic phenomenon, but it can result in high local stresses in buckled regions that lead to plastic deformation or failure. If the members are sufficiently slender, the post-buckled stress will not cause failure and structures can recover, as is seen in micro- and nanoarchitectures made from both ductile and brittle constituents^{Reference Meza, Das and Greer42,Reference Schaedler, Jacobsen, Torrents, Sorensen, Lian, Greer, Valdevit and Carter55,Reference Maloney, Roper, Jacobsen, Carter, Valdevit and Schaedler56} (Figure 2). This is better enabled in micro- and nanoarchitected materials due to the high stresses to failure of the constituent materials. However, residual plasticity or failure will reduce recovery and cause a prebuckled path that lowers the stress necessary to reinitiate buckling.

Global deformation and failure

The architecture, applied load, and distribution of defects affect how load is transferred and correspondingly affect the emergent deformation and failure behavior of micro- and nanoarchitected materials. The compression behavior of lightweight architectures is dominated by the slenderness or aspect ratio of the constituent members. For architectures with non-slender beams or “thick” shells, their behavior will be yielding-dominated (i.e., governed by the plasticity of the constituents)^{Reference Meza70} (Figure 4a, c). Yielding-dominated architectures with ductile constituents comprise a plateau region correlated with the onset of plasticity, which often starts at nodes and joints, and an eventual densification region when members contact one another.^{Reference Meza, Phlipot, Portela, Maggi, Montemayor, Comella, Kochmann and Greer25,Reference Zheng, Smith, Jackson, Moran, Cui, Chen, Ye, Fang, Rodriguez, Weisgraber and Spadaccini34,Reference Gu and Greer36,Reference Montemayor and Greer38,Reference Vyatskikh, Delalande, Kudo, Zhang, Portela and Greer39,Reference Schaedler, Jacobsen, Torrents, Sorensen, Lian, Greer, Valdevit and Carter55,Reference Valdevit, Godfrey, Schaedler, Jacobsen and Carter72} The onset of densification occurs earlier for architectures with higher relative density.

Figure 4.

Global behaviors. (a) Transition from beam buckling to yielding-dominated behavior in a solid polymer cuboctahedron nanolattice with varying slenderness, λ, and (b) corresponding buckled state.^{Reference Meza70} (c) Transition from shell buckling to fracture-dominated behavior in a hollow Al₂O₃ cuboctahedron nanolattice with varying thickness to semi-major ellipse axis ratio, t/a, and (d) corresponding shell buckled state.^{Reference Meza70} (b, d) Scale bars = 20 μm.

Yielding-dominated architectures with brittle constituents undergo catastrophic collapse when the fracture strength of one of the members or nodes is reached.^{Reference Zhang, Vyatskikh, Gao, Greer and Li27,Reference Bauer, Schroer, Schwaiger and Kraft43,Reference Lee, Jafary-Zadeh, Chen, Zhang and Greer45,Reference Bauer, Hengsbach, Tesari, Schwaiger and Kraft51,Reference Meza and Greer63,Reference Bauer, Schroer, Schwaiger and Kraft65,Reference Eckel, Zhou, Martin, Jacobsen, Carter and Schaedler73,Reference Cui, Hensleigh, Chen and Zheng74} For architectures with slender beams or “thin” shells, their behavior is governed by buckling of the members^{Reference Meza70} (Figure 4). This buckling-dominated behavior depends on the structure and is generally ductile-like, independent of the properties of the constituent materials, making it a metamaterial behavior.^{Reference Meza, Das and Greer42,Reference Maloney, Roper, Jacobsen, Carter, Valdevit and Schaedler56} Buckling events lead to load drops that occur in a layer-by-layer manner, and the magnitude of the drops depends on the architecture.^{Reference Zheng, Deotte, Alonso, Farquar, Weisgraber, Gemberling, Lee, Fang and Spadaccini12,Reference Meza, Zelhofer, Clarke, Mateos, Kochmann and Greer33,Reference Liontas and Greer46,Reference Schaedler, Jacobsen, Torrents, Sorensen, Lian, Greer, Valdevit and Carter55,Reference Maloney, Roper, Jacobsen, Carter, Valdevit and Schaedler56,Reference Meza70,Reference Torrents, Schaedler, Jacobsen, Carter and Valdevit75,Reference Zheng, Lee, Weisgraber, Shusteff, DeOtte, Duoss, Kuntz, Biener, Ge, Jackson, Kucheyev, Fang and Spadaccini76} The absolute slenderness of the constituents determines the residual yield or fracture in the members, which affects the recoverability.^{Reference Meza, Das and Greer42,Reference Schaedler, Jacobsen, Torrents, Sorensen, Lian, Greer, Valdevit and Carter55,Reference Maloney, Roper, Jacobsen, Carter, Valdevit and Schaedler56} This means that architectures with more slender members will be more recoverable and will display a more repeatable stress–strain response.^{Reference Salari-Sharif, Schaedler and Valdevit77}

The tension behavior of micro- and nanoarchitected materials is generally dominated by stress concentrations in the architecture, although this aspect is much less studied than compression. Tension in architectures made from ductile constituent beams and shells fail progressively and experience large load drops as individual members fail.^{Reference Zheng, Deotte, Alonso, Farquar, Weisgraber, Gemberling, Lee, Fang and Spadaccini12} Tension in architectures made from brittle constituent beams and shells fail in a single catastrophic event.^{Reference Montemayor, Wong, Zhang and Greer44,Reference Bauer, Schroer, Schwaiger and Kraft65,Reference Mateos, Huang, Zhang and Greer78} In both, nodes serve as initiation points for failure because they are the source of highest stress concentration. Although there are architectures that minimize stress concentrations at nodes,^{Reference Nguyen, Cho and Kang31,Reference Hsieh, Endo, Zhang, Bauer and Valdevit67} it is unlikely they would be a significant improvement on the overall tensile response. The progression of failure in architectures in tension causes a reduction in integrity as members begin to neck or fracture entirely, so measurements must be made in a displacement-controlled setup if they are to extract meaningful information before the onset of catastrophic failure.^{Reference Zheng, Deotte, Alonso, Farquar, Weisgraber, Gemberling, Lee, Fang and Spadaccini12}

The imperfection landscape is somewhat overlooked, but it can strongly influence behaviors. Waviness in solid beams or hollow shells can significantly reduce the load-carrying capability of the individual members in compression and thereby the whole structure, particularly when the waviness is on the order of the member thickness.^{Reference Meza, Phlipot, Portela, Maggi, Montemayor, Comella, Kochmann and Greer25,Reference Meza, Zelhofer, Clarke, Mateos, Kochmann and Greer33} Misaligned nodes, however, have a minimal effect on mechanical properties,^{Reference Meza, Phlipot, Portela, Maggi, Montemayor, Comella, Kochmann and Greer25} meaning architecture is more important to overall properties. Missing members can quickly degrade the properties of the architecture beyond the strength and stiffness scaling with density,^{Reference Meza, Phlipot, Portela, Maggi, Montemayor, Comella, Kochmann and Greer25,Reference Gross, Pantidis, Bertoldi and Gerasimidis79} but the controlled distribution of missing members can potentially be used to improve deformation pathways and increase energy absorption.^{Reference Vangelatos, Komvopoulos and Grigoropoulos80} Some architectures can even be insensitive to large imperfections such as missing beams, meaning they are highly flaw-tolerant and suggesting that the complex load carrying must be fully modeled and understood to predict behavior.^{Reference Montemayor, Wong, Zhang and Greer44} This flaw tolerance is not to suggest that micro- and nanoarchitectures are tough. On the contrary, it is predicted that the toughness of lattice materials scales with the square root of the unit cell size.^{Reference O’Masta, Dong, St-Pierre, Wadley and Deshpande81} Thus, a structure with nanoscale beams would have a significantly lower toughness, although this has yet to be demonstrated experimentally for any sufficiently small (i.e., submillimeter-sized) sample.