Supplementary Materials

Strength and Stiffness Measurement

The data obtained from nanolattice compression experiments performed in this work has a wide range of stress-strain responses, and as such, it is necessary to formulate a consistent method to measure meaningful Young’s modulus (E) and yield strength (σ y) data. In every sample tested, the stress-strain data was comprised of a toe region, a linear region, and a failure region. The toe region is a non-linear segment of data at the beginning of loading, and here it arises from slight misalignments and imperfections between the sample and the indenter axis. For each sample, a subset of stress-strain data was taken starting at the beginning of loading and going to the onset of failure (shown in blue). The maximum slope of this data subset is measured and taken to be the Young’s modulus E. This is done to mitigate the effect of the toe region on the stiffness measurement. In polymer samples, a line with slope E is taken with a 0.02 % strain offset from the obtained Young’s modulus fit, and the intersection of this line and the stress-strain data is taken to be the yield strength σ y (Figure S1A). In hollow Al2O3 samples, the yield strength is taken to be the peak stress before failure (Figure S1B).

Figure S1: Yield Strength and Young’s Modulus Measurement.The Young’s modulus and yield strength measurement for representative data sets of A) polymer and B) hollow Al2O3 nanolattices.

Rigidity Formulation

A structure comprised of bars connected by pin-jointed links is defined to be ‘rigid’ if any shape change of the structure requires an increase in the strain energy. Rigid structures can be statically determinate or statically indeterminate. Structures that are non-rigid are kinematically indeterminate because they can change shape without any applied load in their members. To obtain a mathematically rigorous definition of rigidity, it is possible to formulate an equilibrium matrix that relates the forces at the nodes to the displacement of the bars, as was shown by Pellegrino and Calladine in 1986 (1). In this analysis, the force at each node can be related to the tension in each bar, and the displacement of each node can be related to the elongation of each bar as:

f =At , (S1)e=Bd , (S2)

where, f ∈Rdj−k is a vector that represents the force in dimension d at node j, t∈ Rb is a vector of the tension (or compression) in each bar b, and A∈Rdj− k× b is the matrix that represents a force balance between the force at the nodes and in the bars. Analogously, d∈Rdj−k is a vector representing the nodal displacements, e∈ Rb is a vector of bar elongations, and B∈ Rb × dj−k is the matrix relating them.

The equilibrium matrices A and B can be used to determine the number of mechanisms and states of self-stress. Specifically, the null space of A can be shown to be the state of self-stress of the structure, and the null space of B can be shown to be the number of structural mechanisms. It is possible to show that A=BT, so the rank of A is equal to the rank of B. Therefore, for r A=rank ( A ), the number of mechanisms and states of self-stress can be defined to be

m=(3 j−k )−r A , (S3)s=b−r A . (S4)

The rigidity of simple 3D regular polyhedra has been studied in some detail (2), and it has been shown that octahedra and tetrahedra are fully rigid geometries. An octet-truss is composed of a close packing of octahedrons and tetrahedrons, and can be shown to be a rigid topology. A cuboctahedron is composed of a periodic arrangement of octahedra, and while it is not fully rigid, the octahedral sub-units are rigid. It is therefore defined to be a periodically rigid topology. A 3D Kagome lattice similarly composed of a periodic arrangement of tetrahedra and is therefore also periodically rigid. A tetrakaidecahedron, also commonly known as a Kelvin foam, is a non-rigid topology.

Analytical Calculation of Relative Density

It is possible to derive an analytical expression for the relative density of a circular beam lattice with N beams of radius R and length L that form a cubic unit cell with width W . For small tube radius to length (R/ L) ratios, higher-order nodal interference terms can be ignored, but as R/ L increases the nodal interference has a greater effect on the relative density and the equation will be inaccurate. The interference area between circular tubes at a node is a cylindrical wedge, the volume of which can be

found to be V=23

h R2. For two beams that intersect at an angle θ, the height of the cylindrical wedge is

h=R cot (θ/2 ), meaning nodal interferences will scale as R3. Taking C to be a constant accounting for the sum of nodal interferences, the relative density can be expressed as

ρ solid=Nπ R2 L−C R3

W 3 . (S5)

For hollow structures, the correction for the hollow beams and nodes must be included. Taking

the wall thickness to be t , the corrections for a hollow cylinder and sphere are f ( tR )=2( t

R )−( tR )

2

and

g( tR )=3( t

R )−3( tR )

2

−( tR )

3

, respectively. Factoring these into the relative density equation, we

obtain the modified equation

ρhollow=Nπ R2 Lf ( t

R )−C R3 g( tR )

W 3 . (S6)

Consider the example unit cell of an octahedron (Figure S2). There are 12 beams in the structure at 6 identical nodes with beams that meet at 60 ° and 90 ° angles with respect to each other. The unit cell is cubic, and we take it to have width L. All beams therefore have a length of L/√2. The hollow correction is C≅ 61.82, which accounts for the intersections at the 6 corners, the two types of angles, and the number of intersections at each corner. We define the relative density of the solid and hollow beam lattices, respectively, to be

ρ solid=12 π R2( L

√2 )−C R3

L3 ,(S7)

ρhollow=12 π R2( L

√2 ) f ( tR )−C R3 g ( t

R )L3 .

(S8)

The equation for solid beam lattices is cubic with respect to R and the equation for hollow lattices is cubic with respect to t . It can be found that for some hollow lattice geometries with a certain

range of t / L, there is a critical radius where the relative density is at a maximum. This can be found through differentiating the hollow relative density equation and occurs at

Rc=4 √2 πL+Ct

2C. (S9)

The maximum possible radius of the structure before multiple beams begin to overlap is Rmax=L/2√6, so this critical radius may be outside of the possible range of relative densities (

t <R<L/2√6).

Figure S2: Octahedron Unit Cell Relative DensityAnalytical models for the relative density of an octahedron unit cell. A) shows the relative density for a unit cell with solid beams, and B) shows the relative density for a unit cell with hollow beams both with and without corrections for nodal interferences.

Analytical Stiffness Model for Rigid and Non-Rigid Lattices

We present a derivation of simplified analytical expressions of the stiffness of rigid and non-rigid lattice topologies. As shown in Figure S3, 3D non-rigid and rigid lattice topologies are reduced to their 2D analogs and then further reduced using symmetry to representative one- and two-beam models, respectively. The stiffness of these simplified one- and two-beam models is then generalized to non-rigid and rigid lattice topologies. This analysis considers lattices comprised of beams with uniform solid circular cross-sections, radius R, length L, and a constituent Young’s modulus E s. All beams are subject to a combination of stretching and bending, and beams are modeled using Euler-Bernoulli theory (3) for slender elastic beams. Nodes are all taken to be welded joints. Additional effects such as nodal compliance, torsion or shearing of beams, and imperfections like misalignment or waviness are ignored, as in classical theories for slender beams.

The stiffness S of any material is defined as the amount of deflection δ subject to a given load F, expressed as S=F /δ . The effective stiffness of the simplified non-rigid and rigid beam models shown in Figure S3C and S3F respectively are taken to be the deflection in the vertical direction under an applied load. The diagonal beams are taken to have length L, cross-sectional area A, area moment of inertia I , constituent material stiffness E s, and are oriented at an angle θ with respect to horizontal. The axial stiffness of the horizontal beam in the rigid-beam model is taken to be a factor α different than the diagonal beam; this factor is intended to account for differences in length and/or cross-sectional area. The stiffness of these one- and two-beam model systems is found to be, respectively,

Snon−rigid=E s

L3

12 Icos2 (θ )+ L

Asin 2 (θ )

, (

S10)

Srigid=E s(1+α (cos2 (θ )+ A L2

12 Isin2 (θ )))

L3

12 I(cos2 (θ )+α )+ L

Asin2 (θ )

.(

S11)

In the simple case where θ=π /4 and α=1, these equations reduce to

Snon−rigid=2 E

L3

12 I+ L

A

, (

S12)

Srigid=E (3+ A L2

12 I )L3

4 I+ L

A

.(

S13)

The stiffness of the non-rigid beam model can be identified as a Reuss model construction of a bending and stretching beam in series, where the axial stiffness of a beam is Saxial=E s A /L and the

stiffness of an Euler-Bernoulli beam in bending is Sbend=C E s I / L3 and C is a constant depending on the boundary conditions. The stiffness of the rigid beam model is similar to the non-rigid model but has an additional term in the numerator. This additional term arises because the horizontal beam provides a reaction force proportional to the ratio between the stiffnesses of the two beams, which can be expressed as F react∝Fapplied (Shorz /Sbend+Shorz/ Saxial). In the extreme case where α → ∞, which represents a diagonal beam fixed on a rigid surface, this equation reduces to the Voigt model construction of a bending and stretching beam as

Srigid , α →∞=Es( 12 IL3 cos2 (θ )+ A

Lsin2 (θ )).

(

S14)

In a regular lattice, such as those shown in Figure S3A and S3D, the structure can be approximated as a repeating periodic arrangement of the simple reduced non-rigid and rigid beam models. The stiffness of the lattice can then be taken to be proportional to the stiffness of the reduced model systems, and the stiffness of non-rigid and rigid lattices can be expressed as

Snon−rigid=E s

A1L3

I+ A2

LA

, (

S15)

Srigid=E s(1+B1

A L2

I )B2

L3

I+B3

LA

.(

S16)

Here, A1, A2, B1, B2, and B3 are constants that depend on the number of diagonal and horizontal beams in a structure, the angle θ, and the horizontal beam stiffness α . The effective (structural) Young’s modulus E of a lattice can then be identified as the ratio of stress (force normalized by the effective area Aeff ) and strain (displacement normalized by the effective height H eff ), so that

E=SH eff

Aeff. The height and area of a unit cell with beams of length L are approximately H eff L and

Aeff L2, giving E ≈ S / L. Taking a lattice constructed from circular beams, the area and area moment of

inertia are A s=π R2 and I s=π4

R4, respectively. From this, the expression for Young’s modulus of a

rigid and non-rigid lattice are, respectively,

Enon−rigid=E s

A1( RL )

−4

+ A2( RL )

−2 , (

S17)

Erigid=E s(1+B1( R

L )−2)

B2( RL )

−4

+B3(RL )

−2 .(

S18)

In the extreme case of very slender beams (i.e. R / L≪1), these equations approach

Enon−rigid ≈ A1 E s( RL )

4

and Erigid ≈ B1 E s( RL )

2

, which are the previously derived analytical models for a

bending- and stretching-dominated solid, respectively (4). Neither the proposed models nor the previous analytical models account for shear in the beams or stiffness of the nodes, both of which are likely to be dominant factors affecting the stiffness of the lattices. Despite this, the proposed rigid and non-rigid stiffness models serve to explain deviations from power-law scaling at high R/ L-ratios by accounting for both bending and stretching in the beams. Fitting the Euler-Bernoulli beam simulations with their respective rigid and non-rigid models yielded a correlation coefficient of R2=1.000 for all cases (Figure S4).

Figure S3: Non-Rigid and Rigid Model SystemsA) Model non-rigid lattice topology. B) 2D analog of the non-rigid 3D lattice in A. C) Simple one-beam model constructed using symmetry of the 2D lattice in B and assuming rigid nodes. D) Model rigid lattice topology (octet-truss). E) 2D analog of the rigid 3D lattice in D. F) Simple two-beam model constructed using symmetry of the 2D lattice in E and assuming rigid nodes.

Experimental Error and Coefficient of Determination

The fabrication of hollow nanolattices requires a number of processing steps, and as such, they are subject to some experimental variability. For this reason, there were between three to five experiments performed on each set of hollow nanolattice structural parameters to ensure consistency. The results from these experiments were averaged to produce the data points shown in Figure 2. Each of the geometries had different ranges of standard deviations for strength and modulus. The range of standard deviations for each hollow nanolattice topology are shown in Table S1. These are calculated by taking one standard deviation and normalizing it by its respective average strength or modulus value. These values are not presented in the figures because most of them are sufficiently small that they are obscured by the data markers. From these results, it is apparent that the stiffness is much more sensitive to experimental variability than strength. This stiffness sensitivity to imperfections is consistent with other results in the manuscript showing the deviation of hollow nanolattice stiffness from results predicted using FEM. The strength is limited primarily by bending stress concentrations near hollow nodes, and is therefore less sensitive to experimental variabilities like waviness.

Topology Modulus Std. Dev. Range

Modulus Std. Dev. Average

Strength Std. Dev. Range

Strength Std. Dev. Average

Octet-Truss 10.2% – 32.8% 23.7% 1.4% –31.5% 6.5%Cuboctahedron 3.7% –26.9 % 16.3% 1.7% – 18.2% 6.5%

3D Kagome 4.0% – 25.5% 9.4% 2.2% – 20.7% 8.0%Tetrakaidecahedron 1.5% – 20.8% 9.3% 0.7% – 17.3% 5.6%

Table S1. Range of Standard Deviations for Hollow Al2O3 Nanolattices

As discussed in the manuscript, the experimental results do not follow a true power law scaling, but they appear to follow one in the range of relative densities investigated experimentally. It is only upon further investigation aided by numerical analysis that the true nature of the scaling relationship is revealed. This apparent power law scaling can easily lead to analysis that overlooks many finer details of the mechanics. To demonstrate the agreement of a power-law fit to the data, coefficients of determination (R2) values are provided in Table S2 for each of the strength and stiffness fits presented in Table 1 (and shown in Figure 2). These R2 values generally illustrate the strong degree of agreement of a power-law fit with the experimental data.

TopologyPolymer Modulus

Fit R2 ValuePolymer Strength

Fit R2 ValueAl2O3 Modulus

Fit R2 ValueAl2O3 Strength

Fit R2 ValueOctet-truss 0.953 0.972 0.964 0.979

Cuboctahedron 0.964 0.971 0.962 0.9783D Kagome 0.924 0.931 0.967 0.987

Tetrakaidecahedron 0.973 0.944 0.991 0.986Table S2. Power Law Fit R2 Values

Euler- Bernoulli and Timoshenko Beam Simulations

In order to account for stretching and bending of struts in rigid and non-rigid topologies, initial finite element analyses were performed with three-dimensional two-node Euler-Bernoulli and Timoshenko beam elements in an in-house finite element code. The beam elements are equipped with six degrees of freedom per node and include a small-strain assumption. A linear elastic material model was used, and each strut was refined into an appropriate number of elements to ensure mesh-independent results. For the octahedron, cuboctahedron, and tetrakaidecahedron topologies, the full 5x5x5 unit cell lattice was modeled. For the 3D Kagome, the full 6x6x3 unit cell was modeled. All nodes in the bottom face of the unit cell were fixed (translations and rotations), and the top nodes were only constrained in the z-direction; a 1% compressive strain was applied to extract the effective vertical stiffness.

The beam models for the cuboctahedron and octet-truss lattices matched the theoretical scaling of the form E∝(R/ L)b at low (R/ L) ratios, with b = 2.003. Both of these models show a deviation from this theoretical scaling at higher ( R/ L ) ratios, although the deviation was not as significant as that observed in the Abaqus models. The beam model for the 3D Kagome lattice fails to match the theoretical scaling at low density and underpredicts the Abaqus results. Larger 3D Kagome lattices (18x18x9 unit cells) were created and, as the number of unit cells was increased, the stiffness at low densities increased and approached the Abaqus results. However, because of the finite size of the lattice, there are intrinsic mechanisms that are introduced from the free boundary, meaning that the beam models are fundamentally unable to match the stiffness of the periodic (infinite) lattice. The non-rigid tetrakaidecahedron models initially matched the theory with a scaling of b ≈ 4, but showed a similar deviation from the power-law fit at higher (R/ L) ratios. Since the beam models inherently ignore the effect of the intersection volume at nodes, the models in Figure S4 diverge from the refined Abaqus simulations at higher R/ L ratios for all topologies except the 3D Kagome, which are observed to converge at higher R/ L ratios. This convergence is likely due to the different orientations of the nodes in the 3D Kagome lattice and to the relatively high sensitivity of the 3D Kagome lattice to the boundary conditions.

The same beam models were performed for the hollow geometries, but they did not agree with the fully resolved Abaqus simulations in the same manner as the solid geometry simulations. Both the Timoshenko and Euler-Bernoulli beams significantly overpredicted the stiffness of all four topologies across all relative densities. This is attributed to the compliant hollow nodes that are not correctly modeled in the simple slender-beam simulations.

Figure S4. Solid Euler-Bernoulli and Timoshenko beam simulations vs. full-detail Abaqus simulations: a) octet unit cell, b) cuboctahedron unit cell, c) 3D Kagome, and d) tetrakaidecahedron unit cell. The black trend line denotes the scaling exponent b at low relative densities, calculated using the lowest three relative density simulations. The red fit lines correspond to the Erigid and Enon-rigid models, which yielded a correlation coefficient of R2=1.000 in all cases. For the 3D Kagome unit cell (c), a strong dependence on boundary conditions was observed, meaning that a much larger lattice (18×18×9) was needed to approach the full-detail simulations. The Erigid Model line for the 3D Kagome is fit to the data for the 18x18x9 sample.

Figure S5. Hollow Euler-Bernoulli and Timoshenko beam simulations vs. full-detail Abaqus simulations: A) hollow octet unit cell, b) hollow cuboctahedron unit cell, c) 3D Kagome, and d) hollow tetrakaidecahedron unit cell.

The Effect of Waviness on Hollow Beam Stiffness

Finite element simulations of hollow elliptical beams were performed to obtain a qualitative understanding of how waviness of the beams introduced in the manufacturing process affects the axial and bending stiffness. A sinusoidal perturbation with a magnitude of 30 nm and a wavelength of 1.5 μm was introduced to a long, thin-walled beam (L=9μm ,t=10 nm) and a short, thick-walled beam (L=4.5 μm , t=60nm) representing a typical beam in a low- and high-relative-density lattice, respectively. Both elliptical beams had a minor axis of 0.4 μm and a major axis of 1.6 μm.

The finite element simulations were executed using linear perturbation analysis in Abaqus and the S3R 3-node shell element. Constraints were applied to both sides of the beam to simulate clamped-clamped boundary conditions. The change in axial and bending stiffness were calculated by comparing the reaction forces of the perturbed beam to 1 nm displacements in the axial, minor axis, and major axis directions to the reaction forces of the unperturbed beam. The stiffnesses of the wavy beams as a percentage of the straight beams are given in Table S3. It is clear from the finite element data that the wavy perturbation causes a decrease in both axial and bending stiffness for hollow beams. The data also shows that long, thin-walled beams are significantly more sensitive than shorter, thicker beams.

Length (μm) Thickness (nm) Axial Stiffness Bending Stiffness (Minor Axis)

Bending Stiffness (Major Axis)

9.0 10 46.4% 44.9% 64.7%4.5 60 75.4% 81.9% 89.3%

Table S3. Axial and bending stiffness of wavy hollow beams as a percent of the unperturbed beam stiffness.

The Effect of Angle Changes on Strength and Stiffness

Three variants of hollow 3D Kagome lattices were tested to study the effect of changing the angles of the beams on the strength and stiffness scaling. These three had tetrahedron face-edge angles of 45 °, 54.7 ° (regular tetrahedron), and 65 °, and were designed with their respective compression or elongation axis in the z-direction. CAD models of the three unit cells are shown in Figure S6.

The strength and stiffness of a structure with varying beam angles has been studied both analytically and experimentally (5–7), and it has been found that for a structure with a beam angle of θ from the plane perpendicular to the compression axis, they will follow a scaling relation of

E=B E s ρmsin 4 (θ ) .(

S19)

σ y=C σ ys ρn sin2 (θ ) .(

S20)

From these equations, there should be no deviation of the scaling behavior with changes in angle but instead only a change in the magnitude of the scaling data. The effect of the angle change should also be more amplified in the stiffness data than the strength data. For the angle range tested here, these equations predict the 45 ° 3D Kagome samples to be 1.8 times more compliant and 1.3 times weaker than the regular samples, and the 65 ° samples should be 1.5 times stiffer and 1.2 times stronger than the regular samples.

These predictions are very well matched by the experimental results found here. There were some small variations between the scaling coefficients of the three Kagome angle samples of m=1.46−1.65 for the modulus scaling and n=1.45−1.77 for the strength scaling, but they generally followed the same scaling law behavior. The 65 ° samples were stiffer and stronger than the regular samples on average, and the 45 ° samples were weaker and more compliant on average. As seen in Figure S6, the increase or decrease in the stiffness data with angle changes was more pronounced than the change in strength. At very low densities, the 65 ° sample strength began to converge to that of the regular samples, but that may be due to an increased effect of imperfections.

Figure S6: Strength and stiffness vs. density of 3D Kagome lattices with varying angle . Logarithmic plots of A) the Young’s modulus vs. relative density and B) yield strength vs. relative density of all 3D Kagome samples tested in this work.

The Role of Imperfections - Missing Beams:

To study the role of imperfections in nanolattices, polymer octet-truss samples were made with beams that were removed randomly from the structure (16.7%, 33.3%, and 50% removed), randomly only from the diagonal beams (25%, 50%, and 75% removed), and randomly only from the horizontal beams (50% and 100% removed). Examples of some of these are found in Figure S8A. All beams were made to have a connectivity of at least 2 to prevent any dangling members; this becomes more important when removing large percentages of beams. All of the samples were made with an 8µm unit cell and with beams of slenderness λ ≈ 46. In an octet-truss structure, the horizontal beams make up 1/3 of the sample and diagonal beams make up 2/3, so removing 100% of the horizontal beams equates to removing 33.3% of the total beams in the structure. Equivalent percentages of different types of beams were removed from the sample to study their respective role in governing the mechanical properties. In the analytical formulation of an octet-truss structure, the strength and stiffness are governed by uniaxial tension and compression of the horizontal and diagonal beams respectively, and by systematically removing horizontal and diagonal beams it is possible to study their relative contribution to the mechanical properties.

As seen in Figure S7, a removal of any of the beams leads to a rapid decrease in both the strength and the stiffness of the structure, and the removal of identical percentages of different types of beams elicits a very similar mechanical response. The full octet-truss sample (0% beams removed) is observed to undergo a single large buckling failure event. Periodically removing larger numbers of the beams results in a gradually smaller initial burst with either periodic buckling oscillations or a plateauing behavior afterward. Removing 50% of the beams in the structure results in the elimination of any bursting behavior, and the subsequent deformation is ductile-like with a plateau in the stress. Removing 50% of the beams in an octet-truss sample results in a structure with the same number of beams as a cuboctahedron; the mechanical properties of the missing beam samples are much lower than they would be in an equivalently dense cuboctahedron, which are observed to be similar to that of an octet-truss with the same slenderness of the beams. The absence of burst behavior in the octet-truss samples with 50% of the beams removed suggests that local bending is dominating the deformation and precluding the occurrence of buckling. This likely occurs due to poor load transfer in the missing beam samples caused by the stochastic nature of the structure.

Figure S7: Stress-strain data of polymer octet-trusses with missing beams .Stress-strain plots of the three classes of octet-truss structures with missing beams. A) Randomly missing, B) missing diagonal beams and C) missing horizontal beams.

Plotting the strength and modulus against the relative density of each of the missing beam samples reveals the relative influence of the horizontal and diagonal beams on the mechanical performance. As can be seen in Figure S8, for samples with a high percentage of removed beams, removing the diagonally oriented beams has a more significant effect on the mechanical properties, but for low percentages of beams removed, the relative influence of removing beams from diagonal or horizontal regions is much less significant. Diagonal beams are more closely aligned with the compression axis and are responsible for the load transfer to horizontal beams. The octet-truss structure is highly redundant, so when only a few beams are removed there are still sufficient diagonal beams to effectively transfer the load. When a large percentage of the diagonal beams are removed, there is poor load transfer vertically in the structure and bending begins to dominate the deformation. This also suggests that structures with less redundancy, like the tetrakaidecahedron, are much more sensitive to missing-beam imperfections.

The strength and stiffness scaling trends with relative density for the missing-beam samples range between n=3.0−3.8 for the strength scaling and m=3.1−4.3 for the stiffness scaling, meaning the mechanical performance of structures with randomly removed bars degrades much more quickly than that of periodic structures.

Figure S8: Strength and modulus of polymer octet-trusses with missing beams . A) Yield strength vs. relative density and B) Young’s modulus vs. relative density of polymer octet-truss samples with beams that have been randomly removed from the entire structure, only the diagonal beams, and only the horizontal beams. Trend lines are fit to show the scaling behavior.

The Role of Imperfections - Offset Nodes:

To further study the role of imperfections in nanolattices, polymer octet-truss samples were made with randomly offset nodes. The magnitude of the offset was controlled by a uniform random distribution between 0 and a prescribed maximum; the prescribed maximum offsets were made in 0.5µm increments from 0 to 4µm. Examples of these are shown in Figure S10. The direction of the offsets were made to be random in all except the topmost and bottommost nodes of the samples, which were only allowed to displace in the x and y directions to mitigate the effect of misalignments with the indenter tip during compression. Samples were made with an 8µm unit cell and the same beam dimension as in the missing-beam samples. The exact slenderness of the beams will vary in the samples because the offset of the nodes results in variable beam lengths, but the average slenderness in each sample will remain comparable. The connectivity and relative density are effectively identical across different offset node samples. While there have been no analytical formulations on the role of offset-node imperfections on 3D octet-truss structures, modeling has been done previously and predicted little to no change in the stiffness and a sigmoid decrease before a plateau in the strength (8).

As seen in Figure S9 and S10Figure , increasing the degree of the offset in the samples results in a gradual decrease in the yield strength but causes no significant deviation in the stiffness. The lower offset samples fail via a single large buckling event that manifests as a strain burst. As the degree of offset increases, the magnitude of the initial burst event decreases and samples experience additional smaller buckling events with some post-burst densification. If samples were made with larger degrees of offset nodes the burst behavior may gradually be eliminated, but the maximum offset in the samples tested here is 71 % of the beam length, so increasing the offset further may result in a change in the topology from an unforeseen intersection of beams or an elimination of some beams entirely due to intersecting nodes.

Figure S9: Stress-strain data of offset-node octet-trusses. Representative stress vs. strain plots of polymer octet-trusses with varying degrees of nodal offsets.

Plotting the yield strength and Young’s modulus against the relative offset (max offset/beam length), it is more apparent that the modulus is almost entirely unaffected by the offset, and the variations in stiffness are statistically insignificant. The strength experiences a sigmoidal decrease, with

initially very little change, then a more significant decrease and finally a slight plateau toward higher offsets. The strength of the sample with the largest maximum offset is 22% lower than the strength of the sample with the least amount of offset. Stiffness is a purely elastic property; from these results it is apparent that it is primarily affected by the topology, connectivity, and relative density of the samples, and it is relatively insensitive to nodal-offset imperfections. The stiffness of a lattice structure is governed primarily by beam bending, compression, and tension, and the average contribution from each of these is not as affected by changes in nodal positions. The strength is more sensitive to stress concentrations and will be reduced if even a small number of beams in the sample are subject to higher stresses from additional bending. Failure is also observed to localize to whole layers in samples that are buckling-dominated, so if local beams have higher slenderness it may result in the earlier initiation of buckling, thereby reducing the strength of the whole sample.

Figure S10: Strength and modulus of offset-node polymer octet-trusses. A) Yield strength vs. relative nodal offset and B) Young’s modulus vs. relative nodal offset of polymer octet-truss samples

with offset nodes.

Macroscopic Octet-Truss Experiments

To support the generality of the results presented in this work, macroscopic octet-truss lattices have been fabricated out of PR-48 resin using an Ember 3D Printer (Autodesk, Inc.) and compressed using an Instron mechanical testing machine. These polymer “macro”-lattices have ~2.65mm-long beams with 450-850µm-diameter circular cross-sections, 3.75mm unit cells, and have overall dimensions of 19x19x19mm and 27x27x27mm. Figure S11 shows the Young’s modulus and yield strength as a function of relative density for polymer octet-truss nanolattices and polymer octet-truss macro-lattices. This plot reveals that the nano- and macro-lattices have nearly identical scaling coefficients of m=1.77 and m=1.91 for stiffness and of n=1.88 and n=1.86 for strength respectively. The slight offset in the strength and stiffness between the two lattice systems is a result of the slightly weaker properties of the PR-48 polymer (macro-lattices) compared to the IP-Dip polymer (nanolattices) and the circular beams in the macro-lattices compared with the elliptical ones in the nanolattices. The properties of the PR-48 polymer have been determined through uniaxial compression experiments on a 15mm diameter 30mm tall cylinder in an Instron testing machine, revealing a constituent stiffness of E=1.30 ±0.03 GPa and strength of σ y=33.0 ±1.0 MPa. These results directly support the applicability of the conclusions drawn in this manuscript to lattices of any absolute dimensions within the relevant ranges of R/ L.

Figure S11: Polymer Octet-Truss Nanolattice and Macro-lattice. A) Plot of Young’s modulus vs relative density for both macro- and nanolattices. Inset shows a representative image of the macroscopic polymer lattice. B) Plot of yield strength vs relative density for both macro- and nanolattices.

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