The defect sensitivity of brittle truss-based metamaterials

Architected strut-based metamaterials fabricated via three-dimensional printing exhibit a wide range of geometric and material heterogeneity, including variations in strut size, surface roughness, embedded micro-cracks and disconnected struts. The locations and severity of these defects are highly variable; combined with the complexity of the structure itself, it is exceedingly diﬃcult to identify critical defects that limit performance and inform qualiﬁcation protocols. To address this challenge, we consider the impact of distributions of defects for various strut-based lattice topologies. The role of defects is analyzed using Weibull distributions of strut failure strains. We relate the statistical distributions of strut properties to macroscopic stress-strain performance, using high through-put ﬁnite element predictions in thousands of virtual tests. Increasing the prevalence of defects decreases macroscopic strength; however, this has the complementary eﬀect of introducing apparent ductility, i.e. load capacity even after the onset of strut failures. Hence, there is a trade-oﬀ between achieving high strength and gradual loss of stiﬀness that is desirable for thermal loading or lattice cores. Predictions of average strength as a function of Weibull modulus provide knock-down factors relative to the defect-free strength. In turn, these clearly identify quantitative processing targets to mitigate the impact of defects.


Introduction
Architected strut-based metamaterials (or lattices) offer structured porosity that enables high specific stiffness and strength, efficient internal cooling, high surface areas for catalysis and functionally graded properties.[1][2][3][4] On the one hand, three-dimensional (3D) printing has dramatically expanded the range of materials and topologies that can be utilized [5]; on the other, 3D printing often introduces a wide range of geometric and material heterogeneity.This is especially true for millimeter scale features that define strut-based lattices and make it difficult to ensure performance [5,6].These defects include deviations in strut diameters [7], strut waviness [8,9] surface roughness [10,11], local variations in mechanical properties [12] or missing features (e.g.struts or nodal connections) [13][14][15][16][17][18][19].For a more exhaustive review of possible defects and imperfections and their individual and combined influence on effective properties of truss lattices, the interested reader is referred to [6].
The definition of performance-limiting critical defects is challenging, if not impossible.Potentially relevant defects are typically widely distributed spatially, and their impact is strongly dependent on their location within the lattice.Disqualification of a printed lattice simply because of the existence of any severe flaw is highly conservative, since the presence of such a flaw in one specific location may not significantly impact performance.Further, the impact of a multitude of small defects may be greater than those of isolated large defects.In light of these considerations, a broader understanding of the interaction of flaw populations with lattice topologies is needed to guide qualification, set processing targets through the identification of tolerable flaw populations, and design new defect-tolerant metamaterials.
Direct characterization of lattices with explicit defect representation is a conceptually straightforward approach to quantifying interactions between defects and topology.For instance, one can quickly quantify the role of stochastic strut thickness [7], strut waviness [7], surface roughness [10], etc. by constructing either beam-based [7,20] or full 3D models [21] and conducting parametric studies with finite element analysis.Rigorous connections between such features and component performance naturally require that defect populations agree with those obtained by a given printing process.Full computed tomography (CT) https://doi.org/10.1016/j.matdes.2024.112776Received 10 December 2023; Received in revised form 12 February 2024; Accepted 17 February 2024 scans are a powerful tool to characterize such populations [7].Ultimately, one must establish quantitative links between statistical defect distributions and the statistics of macroscopic performance.This presents several challenges; first, one must ensure that the CT scan captures salient defects that strongly impact performance.This is trivial with regards to missing struts or variations in strut size, but can be quite challenging for microcracks or embedded brittle phases.Second, one must generalize geometrical or material features such that one can generate new instances of defect populations that are consistent with such measurements.For example, one must come up with a parametric approach to generate surface roughness that generates features consistent with measurements.As not all surface features impact the overall response, this raises difficult questions about which features to include.Third and finally, one must be able to conduct a large number of simulations, each with a different incarnation of a defect population, to ensure worst-case scenarios are identified.
The need for large sample sizes is evidenced by the fact that additively manufactured materials can exhibit extreme property variations [22].Ideally, a very large number of structures, each with a unique distribution of defects, would be analyzed such that one can quantify the probability of encountering low performance.This requires extensive computational resources; there is no guarantee every unit cell contains critical defects, requiring a large number of cells, and further, actual components can be strongly influenced by boundary conditions where homogenization approaches are problematic [19].Nevertheless, despite a limited number of simulations compared to those presented here, work on the role of defects in lattices comprising elastic-plastic materials has generated significant insight into scenarios where struts yielding is the principal concern [7].
In this work, we use an alternative approach to quantify the role of defects in lattices with highly limited ductility, with the specific aim of quantifying the relationship between statistical defect distributions and the statistics of macroscopic performance.A key focus is on whether or not different strut topologies exhibit different defect sensitivity, assuming the same statistical distribution of defects.First, we adopt an implicit definition of defects; individual struts are assumed to have a fracture strain governed by sampling from a Weibull distribution, as illustrated in Fig. 1.The Weibull parameter () serves as a measure of the extent of critical defects distributed throughout the structure.While any statistical distribution could be used, the Weibull distribution is known to capture the statistics of many brittle materials.Fig. 1b displays the Weibull modulus and scale parameter of a set of additively manufactured ceramics.The majority of this data set has been published in [23] and was included for completeness at this point.Note that in the experimental results, the failure strength was measured, while in the simulations, strain defines strut failure.Since the simulations in this work assume a perfectly elastic-brittle material, failure strength and strain are directly related through the elastic modulus.More information can be found in Appendix A. Statistical distributions of fracture strain can be related to the size and spatial distribution of crack-like flaws in a structure [24,25], or the confluence of surface roughness and brittle phases.Second, we utilize a highly efficient algorithm for predicting sequential strut failures (see Fig. 1), such that thousands of samples can be analyzed in short order to compile complete statistical descriptions of response.
This work advances the understanding of the role of defects in brittle lattices in several critical aspects; first, we provide quantitative insight regarding the number of virtual tests needed to extract macroscopic failure probabilities as a function of defect distribution.Interestingly, while the material properties are governed by Weibull statistics, macroscopic strength exhibits a qualitatively different distribution.Second, we illustrate that for a single defect distribution, different lattice topologies exhibit different peak strengths and post-peak behaviors, which has important implications for designing flaw-tolerant metamaterials.Third, the results provide knock-down factors and failure probabilities for common lattices with defects (as compared to pristine (flawless) designs).Beyond being of immediate use to designers, these results provide processing targets by identifying acceptable levels of defects.That is, testing on strut-like features to obtain the Weibull parameter for a given process can be directly translated to full component performance.
The remainder of this paper is organized as follows: Section 2 outlines the model and high through-put simulation approach.Section 3 illustrates the relationship between defect statistics and the statistics of macroscopic characteristics.The implications of drawing conclusions based on limited testing are illustrated by presenting boot-strap statistics, and the influence of sample size (i.e. the number of unit cells).Section 4 provides a discussion of the results and the implications for the design and characterization of defect-tolerant lattices, with comparisons to prior work on ductile lattices [7,26].General conclusions and recommendations for future work are briefly outlined in Section 5.

Material and failure model
The material properties of the struts are assumed to be linear elastic and perfectly brittle, with failure defined as complete fracture of the strut.As illustrated in Fig. 1a, each strut is assigned a random failure strain  f which is sampled from a two-parameter Weibull distribution (see Fig. 1b) with the following probability density function: where  is the Weibull modulus and   is the scale parameter.In the context of strut fracture, the scale parameter corresponds to the characteristic fracture strain of the material.The mean of the distribution is given by   Γ(1 + 1∕), where Γ is the Gamma distribution, such that in the limit of high Weibull modulus ,   represents a deterministic fracture strain.We set   = 0.01 throughout the entire study, although results are presented in a normalized form that implies the absolute value used in the simulations is immaterial.By sampling the strut strength this way, statistical variations in the constituent material's strength as well as the influence of geometric imperfections are lumped together into one single probability of failure, and the reference volume of the Weibull distribution is implicitly assumed to be on the order of the strut size.It should be noted that 3D printed strut properties, i.e. geometry and microstructure, can be orientation-dependent, particularly for ductile materials [27]; as such, it may be worthwhile for future efforts to consider multiple property distributions to account for such effects.That said, in one of the few studies of brittle lattices [19], the influence of the build direction was rather negligible and motivates the present study of a single property distribution.Strut fracture occurs when the maximum tensile strain in a strut, computed from the finite element model, exceeds the failure strain assigned to that strut; the compressive strength is assumed to be infinite, such that failure only occurs when the peak strut tensile strain exceeds the assigned strength.The failure strain for each strut is assumed to be uniform throughout the entire strut; failure occurs at the location of peak tensile strain within the strut.

Computational framework
We use an in-house finite element code to simulate displacementcontrolled uniaxial tensile tests on lattices with the dimensions  ×  × 2 , where  is the number of unit cells along an edge and the long edge represents the tensile axis.At the bottom face, all displacements are fixed to be zero.At the top face, a displacement in the loading direction is prescribed, while all other degrees of freedom (including rotations) are constrained to be zero.The basic setup is illustrated in Fig. 1a.The rigid boundary condition on top and bottom of the lattice are one realistic scenario that could occur, e.g., when truss lattices are used as a core in sandwich structures.Naturally, while many other Probability density plot of a Weibull distribution with different modulus  (top) and measured Weibull modulus and strength parameter of six ceramic material systems (bottom).The graph displaying the experimental data was adapted from [23] and expanded by an additional set of measurements.(c) Schematic showing the algorithm for simulating sequential fracture of struts.(d) Exemplary normalized effective stress-strain curve.Note that the stress-strain curve shown here does not belong to the exact example shown in panel (a), but is randomly chosen from a sample of 500 similar lattices.boundary conditions may be of interest, we limit ourselves to this specific case to keep the scope of the study manageable.
The code uses a 2-node Timoshenko beam element, assuming small displacements and rotations.Alternatively, one could use higher order elements to further reduce computational cost; here, the 2-node element is chosen simply as a matter of convenience, i.e. to utilize existing code.The nodal degrees of freedom in the local coordinate system are denoted by  = (, , , , , ) T .The definition of the elemental stiffness matrix is given in Appendix B. Using a geometrically linear model is valid if failure is driven by fracture of struts instead of buckling, and if rotations are small.The former is a reasonable assumption if the relative density is ten percent and above [19].To investigate whether large rotations might occur within the order of effective strains relevant to the paper (about one percent), we have performed additional elastic calculations using the commercial software Abaqus.As a test case we chose a simple cubic body-centered cubic (SC-BCC) lattice topology at  = 5 and a relative density ρ = 0.1.We implemented a pristine lattice, a pre-cracked lattice to simulate highly localized failure at high Weibull modulus, and a lattice where we randomly removed 1000 struts to simu-late non-localized damage at low Weibull modulus.For all test cases, the differences in effective stiffness between the small and large displacement solutions are so small (≲ 3%) that they are hardly distinguishable from numerical noise.The maximum occurring nodal rotations in the geometrically nonlinear solutions are well within the limits of the small angle approximation (< 5 degrees).As such, we limit our investigations to the two relative densities of ρ = 0.1 and ρ = 0.3; nodal volumes and boundaries are taken into account when computing the relative density of a lattice (see Appendix C).We assume that shear strains are negligible in comparison to stretching and bending.Naturally, one may include shear strains and rigorously compute principal strains; for the lattices considered here, differences in macroscopic response were negligible for several dozen comparison cases.The peak tensile strain in the strut is computed as: where  and   are the strut radius and the element length.The first term reflects axial stretch of the strut, while the latter term is the maximum bending strain.The bending strain contribution in eqn.( 2) is (2) reflects the peak strain anywhere in the element, taking into account arbitrary bending directions.Hence, the failure condition is that  t ≥  f , where  f is sampled from the Weibull distribution given as equation (1).From basic numerical tests, we find that eight elements per strut are sufficient to assure mesh convergence when struts are assigned a uniform fracture strain (see Appendix D).
To efficiently generate the large data sets presented in the subsequent sections, we implemented the following algorithm to simulate strut failure: 1. Apply a small displacement that does not cause any strut failures and find the element that is closest to failure by computing the ratio  t ∕ f for all struts; the strut with the maximum value will be the next strut to fail, since all strut strains will increase linearly with macroscopic displacement.
2. Using the value of  t ∕ f , scale the macroscopic load and displacement from the previous step to compute the load and displacement associated with the next strut failure.3. Subtract that elemental stiffness matrix from the global system.Recompute the displacement solution for the current value of applied displacement, the elemental peak strains, and the total load on the structure after strut removal.Naturally, eliminating the stiffness of the failed strut implies that the strains in other struts increase even at fixed displacement, while the total load will drop.If the macroscopic load is sufficiently small, terminate the analysis due to complete loss of structural integrity.4. If after a strut is removed (by subtracting its elemental stiffness) max( t ∕ f ) ≥ 1, return to Step 3, removing the element stiffness for the strut with the highest overload.This is essentially a cascade of sequential strut failures at fixed macroscopic displacement.
In contrast to other methods, failed elements are not deleted from the data structure, which avoids the computationally expensive step of reformulating the global stiffness matrix from scratch at each strut failure.As a practical matter, it is necessary to introduce additional termination conditions (e.g. after a prescribed number of strut failures or displacement increments are completed), simply to avoid significant computation times associated with the final stages of rupture where strut-by-strut failures yield negligible changes.For the cases considered here, we terminated simulations after 5000 strut failures or 1000 displacement increments; for larger structures or different boundary conditions, these might require adjustments.
Our code was implemented in the Wolfram Language, where subroutines critical to performance are compiled into C-code.The simulations were run using Wolfram Mathematica 13.2.Table 1 contains some benchmark examples for the simple cubic lattice.The other lattice topologies take significantly longer to compute (some on the order of multiple days) since they contain dramatically more struts.Generally, cases with higher Weibull modulus run considerably faster, since the response strut failures quickly localize and fewer strut failures are needed for complete loss of load capacity.Cases with lower Weibull modulus exhibit widespread, de-localized strut failures that require many steps (each associated with a single strut failure).

Scope of parametric study & reported results
We report the response of the lattices using macroscopic loads that are scaled by the results obtained for an identical (yet defect-free) structure with the deterministic failure strain set to   .That is, we define σ as the macroscopic axial force divided by the value obtained for the deterministic structure.Since the area is fixed for both structures, the ratio of macroscopic forces is identical to the ratio of effective stress acting on the lattice, i.e. σ = ∕  =  ∕  , where   and   are the macroscopic stress/force on the defect-free lattice.Further, we define ε as the ratio of macroscopic strain on the lattice (defined  M = ∕(2), where  is the size of a unit cell and  is the displacement of the top platen) and the characteristic failure strain   that defines the Weibull distribution; that is ε =  M ∕  .The stress-strain response of a lattice is thus presented as σ =  ( ε), as shown in Fig. 1d.In the following, σS refers to the peak stress obtained in a simulation, i.e. the ultimate strength of the lattice as compared to that obtained for a defect-free structure.This choice of normalization emphasizes the defect-sensitivity of a given topology by providing the knock-down factor in strength for a given topology.While the specific choice of elastic modulus only influences the absolute values of strength, which can be linearly scaled to any modulus, the choice of Poisson's ratio (or shear modulus) can influence the relative roles of bending and shear deformation.For the cases considered here, shear deformation is negligible due to the relatively low relative densities considered.To confirm this, we changed the default ratio from 0.3 to zero and compared some select test cases; the results were virtually indistinguishable.Naturally, for struts with smaller aspect ratios, or different strut topologies that emphasize shear, the role of Poisson's ratio is likely to be more pronounced.In order to make cross-comparisons of raw strength for two different topologies with different defect distributions, one must use the scaling factors presented in Table 2.Note that the macroscopic failure strains for the defect-free structures are also listed; these can be smaller than the deterministic failure strain  0 due to strain concentrations in struts located adjacent to the rigid 'grips'.Note that for FCC at ρ = 0.1,  M is greater than  0 .
We investigate the defect sensitivity of three different lattice topologies: simple cubic (SC), face-centered cubic (FCC, also known as the octet truss) and simple cubic body-centered cubic (SC-BCC).Schematics of the topologies are rendered in Fig. 1c.The dominant deformation mode of the lattice, i.e. bending or stretching, depends on the coordination number , i.e. the number of struts connected at a single node [28].In an infinite medium under arbitrary loading conditions, SC ( = 6) is bend dominated, FCC ( = 12) is stretch dominated [29], and SC-BCC ( = 8, 14 for the center and corner nodes, respectively) is an intermediate case.However, under uni-axial tension, the dominating deformation is stretching in all three lattice topologies (at least prior to the first strut failure).Subsequently, each fixed set of parameters { , , , ρ}, where  , , , ρ are the lattice topology, the number Table 2 Absolute ultimate strength and strain at failure for defect-free lattices with  = 5.For simplicity, the Young's modulus is set to be  = 100 unit force per unit area.The strut radius is normalized to the lattice constant . of unit cells in the lattice, the Weibull modulus and the relative density, is called an instance.Sample quantiles, e.g. the median, are estimated using linear interpolation.Throughout the study, the minimum sample size is at least 500; i.e., for each instance, the simulation is repeated at least 500 times, each time re-sampling the strut failure strains.This makes sure that the sample size does not influence the results.Note that the chosen sample size only ensures convergence for simple statistical quantities such as estimated quantiles (see Appendix E).Precise estimation of distributions of the ultimate strength would probably require even larger samples.This will be shown in section 3.

Illustrative macroscopic stress-strain response and spatial distributions of strut failures
In this section, we illustrate representative macroscopic stress-strain responses and the spatial distribution of failed struts, simply to provide context for the more relevant discussion of macroscopic property distributions.Fig. 2 shows typical stress-strain curves of the SC lattice at relative density ρ = 0.1 and different Weibull modulus .The light and dark blue curves represent the cases that resulted in the minimum and the maximum strength out of 500 simulations, respectively.The insets visualize the fractured struts at the final increment.Videos displaying complete failure sequences are available for download as explained in Appendix F. Results for other lattices are qualitatively similar and are provided in Appendix G.
For low Weibull moduli seen in Fig. 2a, the minimum and the maximum stress-strain responses exhibit apparent ductility, in that the onset of strut failures does not coincide with a complete loss of load capacity.(Apparent ductility is more rigorously defined and discussed in the following section on statistical property distributions.)At both extremes (minimum and maximum peak strength), fractured struts are spread across the whole structure and damage is not localized.Tracking the damage evolution step by step confirms this observation.The case of  = 5 (Fig. 2b) exhibits similar behavior as  = 3, although the apparent ductility is less pronounced for the case with maximum strength, and the spread of the failed struts across the structure is slightly smaller.However, there is no obvious region of damage localization.
In the case of  = 8, (Fig. 2c), there is a shift in behavior.The minimum strength response (light blue) still exhibits apparent ductility, while the maximum strength response is perfectly brittle, i.e. the failure of the whole structure is defined by the first strut failure and the load immediately drops to zero.This shift is also reflected by the location of fractured struts.In the minimum strength case, the fractured struts are This observation becomes more obvious when examining the whole failure sequence (see Appendix F).
In the case of  = 20 (Fig. 2d), both the minimum and maximum strength curves exhibit perfectly brittle behavior, wherein the structure breaks along a single plane once the first strut is fractured.For the minimum strength curve, fracture occurs right at the boundary, which is not observed in any other case in Fig. 2.These results reflect a general trend for all lattice topologies, in that the first few strut failures lead to immediate localization for  ≥ 20, such that the macroscopic strain associated with the onset of strut failures is virtually identical to the macroscopic strain associated with peak strength.As will be shown, however, this is not to imply the macroscopic properties are deterministic; the probability of having a weak strut (defect) in highly stressed struts near boundaries leads to a distribution of strengths even at high Weibull modulus.

Macroscopic strength distributions and apparent ductility
Quantitative measures of the key features of the stress-strain response shown in the previous section, such as ultimate strength σS , were compiled for each lattice topology and each strut property distribution from 500 simulations.Here, we report such results for  = 5, i.e. a sample with 5x5 unit cells in cross-section and 10 unit cells in the loading direction.Fig. 3 shows kernel smoothed histograms of the normalized ultimate strength σS obtained for all three lattice topolo- gies, where Figs.3a, 3b and 3c correspond to the SC, SC-BCC and FCC lattice, respectively.Different Weibull moduli are shown as different colors for  = 3, 5, 8, 20, and the relative density is indicated by line type (solid for a relative density of ρ = 0.1 and dashed for ρ = 0.3).Some of the histograms exhibit multiple inflection points, which we expect would vanish at much larger sample sizes based on observations from additional cases run for the SC lattice.
A general trend that is present in all three lattice topologies is that the ultimate strength significantly decreases with decreasing Weibull modulus.The effect is the strongest in the SC lattice, where the ultimate strength can drop to less than thirty percent of the strength of a defect-free lattice.For the SC lattice, the width of the distribution increases with increasing Weibull modulus, while it stays roughly constant for the SC-BCC and FCC lattices.The shapes of the distributions do not visibly change with relative density.The results for higher relative density indicate the defect-sensitivity is slightly lower since the strengths are a slightly higher fraction of those obtained for defect-free lattices.That said, the role of relative density is rather small; the biggest difference is for the FCC lattice at  = 20, where the median value is different by about five percentage points.
A fundamental question is whether the macroscopic lattice strength exhibits the same type of statistical distribution as the individual strut fracture strains, i.e. a Weibull distribution.From visually comparing the results from Fig. 3 to the graphs shown in Fig. 1b it appears that, except for SC at  = 20, none of the macroscopic strength distributions clearly match a Weibull distribution.We used Mathematica's built-in distribution fitting algorithm [30] to fit different distributions to the data in Fig. 3.The fitting process is highly non-linear, and its results should be treated with caution, since even at a sample size of 500 the distributions might not be fully resolved in all cases.Nevertheless, computational fits to the macroscopic strength distributions indicate that the only instance where a two-parameter Weibull distribution is clearly better than alternative distributions is the SC lattice for  = 20.In this case, the Weibull modulus of the fitted macroscopic strength distribution is also close to 20.For all other instances, the data either seems to be described better by a normal distribution or the sample size is too small for a definitive conclusion.The latter is the case especially for the FCC lattice.For more information, refer to Appendix H. Fig. 4a shows an exemplary normalized stress-strain curve (SC, ρ = 0.1,  = 3).Although the intrinsic material response is perfectly brittle, the lattice exhibits significant load capacity after the onset of strut failures.While this behavior is largely irrelevant for load control on an isolated lattice structure, it has important implications for displacement control, notably thermal loading, and the response of structures with integrated lattice features, such as sandwich cores.To (d) Face-centered cubic (FCC).For all Panels  = 5.
quantify this effect, we introduce the metric of an apparent ductility εd , which we define to be the range of macroscopic lattice strain within which the lattice retains in between 90% and 50% of its initial stiffness.
Naturally, the exact range of stiffness used in this definition is arbitrary and ultimately should be defined in terms of stiffness changes important to a given application.For instance, if a part is under load control, counting the number of strut failures until ultimate strength is reached might be a reasonable metric.In this case, the FCC performs well since it can lose a lot of struts without losing much stiffness (see the fine jagged lines of the stress-strain curves given in Appendix G).In fact, even the defect-free FCC lattice reaches ultimate strength only after 21 struts have failed.Yet in practice, it might be difficult to capture the small stiffness changes in the FCC lattice through in-situ measurements before catastrophic failure occurs.Figs.4b -4d plot the apparent ductility versus the normalized ultimate strength for the SC, the SC-BCC and the FCC lattice.The colored data points are grouped by Weibull modulus.Shaded regions are convex hulls drawn around the individual groups of data points.The black stars indicate the median values of each group.Note that for SC at  = 8, the median ductility is almost zero, although a visually significant number of points are located at non-zero values.Overall, with increasing Weibull modulus, the apparent ductility decreases while the ultimate strength increases.This trade-off between apparent ductility and strength is commonly seen in conventional ductile materials.The SC-BCC lattice is the best performer in terms of apparent ductility, while FCC is by far the worst, exhibiting essentially zero ductility for a Weibull modulus of  ≥ 5. Importantly, even with its much greater apparent ductility, the SC-BCC lattice retains almost as much strength as the FCC lattice.
Figs. 5a and 5b summarize how the sample median of normalized peak strength and apparent ductility behave as a function of topology, relative density and Weibull modulus.Relative density is marked by solid and dashed lines ( ρ = 0.1, 0.3, respectively).The lattice topology is indicated by color and marker types (SC: Blue diamonds, SC-BCC: Orange squares, FCC: Green circles).The trade-off between strength and ductility has important implications for both process and component design; these are discussed in Section 4. While relative density seems to have little influence on the peak strength, the apparent ductility in the SC-BCC lattice significantly decreases with increasing relative density.
As mentioned previously, a sample size of 500 might not be enough for all instances to accurately describe the exact shape of the distributions of effective properties.Simple statistical estimators, such as the median or other sample quantiles, behave much more favorably.To determine at which sample size quantiles converge, we sub-sampled  data points without replacements from the original sample and used a simple bootstrap algorithm (see e.g.[31]) to determine 95%-confidence intervals for the 5%, 50% and 95% quantiles of the normalized ultimate strength (Fig. 5c) and the apparent ductility (Fig. 5d).The three black lines correspond to the quantiles estimated from the full sample of  = 500.The colored regions represent the confidence intervals obtained from bootstrapping.The examples given in Figs.5c and 5d are the worst-case scenarios.They correspond to the normalized ultimate strength of SC at ρ = 0.1 and  = 20, and the apparent ductility of SC-BCC at ρ = 0.1 and  = 3.Similar graphs for other instances of interest are given in Appendix E. Sample sizes smaller than  = 20 are too small to yield accurate estimates since the confidence intervals of the different quantiles overlap.At  ≥ 300, the sample size is converged.At  = 500, all of the obtained confidence intervals for the normalized ultimate strength are smaller than 2.5 percentage points.

Impact of specimen size
The previous section only considered lattices of a fixed size of  = 5, i.e. 5 × 5 × 10 unit cells.Here, we examine the scaling in terms of the impact of the number of cells (or lattice specimen size) on the ultimate strength distribution.There are generally two ways such a scaling can be examined.The first method is to fix the total dimensions of the lattice and to decrease the size of the unit cell, thereby increasing  .The second method is to fix the size of the unit cell, such that the lattice dimensions increase with  .Conceptually, both scenarios can occur in application.For deterministic properties and quasi-static linear elastic deformation, both approaches yield equivalent results that are easily related.The same is true for the simulations in this paper, as the stochastic properties are sampled on a strut basis without reference to a physical length scale.We use the second method, i.e. we increase the physical size of the specimen while keeping the cell size and strut size fixed.
Note that, in experiments, the two approaches will generally yield fundamentally different results.The rationale is as follows: assume we can fabricate a lattice without geometric imperfections, but the constituent material's strength follows a Weibull distribution that has been empirically derived from bulk material of an arbitrary reference volume.Using the first method for scaling and maintaining fixed lattice specimen size, while increasing the number of cells, yields physically smaller feature sizes while the reference volume of the Weibull distribution stays the same.This can result in a significant size effect [32,33] that has been exploited to produce nano-scale ceramic lattices with extreme strength [34].Our current analyses do not capture this effect since the reference volume is always assumed to be the same as the strut size.We leave for future work the analysis of lattices where the feature size changes in relation to the Weibull reference volume.
Fig. 6 shows the median ultimate strength relative to a unit cell ( = 1) at the same relative density and the same Weibull modulus.
That is, we normalize the median strength obtained for any lattice specimen, σ S , by the corresponding result for a single unit cell, σ1 S , keeping relative density and property distributions fixed.The different colors indicate different Weibull modulus, with blue being  = 3 and black being the deterministic lattice (loosely speaking  = ∞).Overall, we observe a decrease in strength with increasing  .
In the SC case (Fig. 6a), the scale effect shows a strong dependence on the variability of the struts.The smaller the Weibull modulus, the larger the effect.While the decrease in strength is rather small for the deterministic lattice (about 10%), it is quite large (more than 50%) at  = 3.In contrast, in the SC-BCC case (6b), the decay of strength with  is substantial irrespective of Weibull modulus.Most importantly, even the deterministic lattice shows a strong scale effect.The FCC lattice (6c) shows the overall lowest scale effect.Although the deterministic (black) curve intersects the curves of  = 3, 5, 8, the deterministic lattices strength is still the highest, yet it shows the strongest decline with respect to  = 1.It is not clear whether there is a lower bound for the strength in all cases.We have performed simulations up to  = 10 with the deterministic lattices (see inset in Fig. 6c).While the SC lattice appears to asymptotically approach a lower bound, the SC-BCC and the FCC lattices, do not exhibit an asymptotic behavior within the range of  = 10.In the stochastic domain, the computational cost quickly becomes prohibitive for  > 5 if sample sizes greater than 100 are considered.Note that size effects associated with multi-axial stress states, which in the case of this study are present at the rigid boundaries, can exceed the scale of  = 10 substantially in the presence of severe nonlinearities [19].Clearly, open questions remain regarding the presence of a continuum limit and the roles of strut topology and stress state.
Furthermore, it should be recognized that the overall component geometry and boundary conditions will play an important role with regards to size effects.The relative volume fraction of cells connected to solid components (or a test rig) has a significant impact on the distribution of strut stresses throughout the structure; similarly, the presence of bending or other loading states that do not produce nominally uniform stresses (at the cell level) will naturally influence size-effects.The present geometry/loading was chosen simply as a first step (one that is arguably easier to study experimentally), and it is likely necessary to repeat study with macroscopic specimens that represent canonical forms associated with a given application.

Discussion
We have simulated the fracture of more than thirty thousand lattices to assess their defect sensitivity under uni-axial tension.Generating the data was facilitated by an efficient finite-element algorithm that treats the fracture of a lattice as a series of linear elastic problems.While fitting distributions to the data precisely would require even more simulations, the sample sizes of 500 are large enough to accurately determine basic statistical estimators, e.g. the median ultimate strength.This is evidenced by the tight confidence intervals we computed by applying a simple bootstrap algorithm.Overall, the tensile strength of architected lattices is less affected by variability in the strut strength for higher coordination numbers.This outcome is plausible, as a similar trend has been observed in the purely linear elastic regime [35].The loss of effective strength at realistic Weibull moduli is significant.For instance, the Weibull modulus of additively manufactured ceramics commonly falls into a range of 3 <  < 12 [23,[36][37][38].In the case of  = 5 and  = 5, the median strength of the simple cubic lattice drops down to below 40% of its nominal value, while the median strength of single struts only drops to roughly 93% at  = 5.A possible explanation for the amplified defect sensitivity of a lattice versus a single strut is that, since the reference volume of the Weibull distribution is assumed to be the strut size, building lattices consisting of many struts increases the sampling volume, leading to bigger knock-down factors.This interpretation is supported by the decrease in tensile strength when the number of unit cells in a lattice is increased (see Fig. 6).Further, it should be noted that bending deformation triggered by local strut failures can significantly elevate stresses relative to those prior to the onset of failure [39].
Overall, this size scaling effect is the opposite of what has been observed for ductile materials and confirms what was postulated by White et al. [26].It raises the important question of how much ductility in the struts is needed for a lattice to fall into the more favorable ductile scaling category, where a larger sampling volume decreases the defect sensitivity [26].The answer to this question should be addressed in future work as it is relevant to set targets for the development of additive manufacturing processes.
Relative density appears to have little influence on this behavior.An increase in relative density can have at least two effects.Thicker struts reduce bending deformation and increase shear deformation in the struts and the influence of nodes becomes increasingly important with increasing relative density [40].Bending influences are captured by the present analysis, while shear deformation was neglected in the failure analysis.It can be easily included in future analysis since the elements use Timoshenko beam theory.The influence of nodes (strut intersections), however, is not captured by our model.Whether increased nodal volumes at higher relative densities have a big impact remains to be investigated by future work.
The propagation of damage through the lattices depends on the topology and Weibull modulus.In general, we observe a shift from a regime with apparent ductility without damage localization at low Weibull modulus ( = 3) to a behavior close to a perfectly elasticbrittle solid with highly localized damage at high Weibull modulus ( = 20).This trend is the most pronounced for the simple cubic lattice (SC, Fig. 2).Qualitatively, it is in good agreement with the results obtained by Quintana et al. [41], where the authors found that in 2D lattices, fracture of a pre-cracked lattice is only dominated by the K-Field around the crack-tip if the Weibull modulus is greater than four.Otherwise, the failure of the lattice is dominated by the randomness of strut strength in the entire lattice.
In practice, the intuitive target for a manufacturing process would be less variability (higher Weibull modulus) in the struts, since it yields lattices with greater ultimate tensile strength.However, in this case, the lattice fractures without warning.Hence, in a situation where the compliance of the lattice can be measured in-situ, it might be beneficial to sacrifice strength and to choose a configuration with at least a small regime of apparent ductility so that a part can be replaced before it fails catastrophically.
Whether damage localizes or not also bears important implications when aiming at the optimization of lattices for specific load cases.For instance, for the loading conditions used throughout this study, deterministic lattices will fail right at the rigid body connections due to the stress concentrations that arise from the constrained degrees of freedom.Similarly, at very high Weibull modulus ( ≥ 20), the lower bound of tensile strength is defined by the boundary (see e.g.Fig. 2d).In this case, reinforcing the material at the boundaries to deal with the stress concentrations, e.g. by introducing a gradient in relative density that puts more material towards the boundaries and less towards the center, will likely improve the effective tensile strength of the lattice.However, if the manufacturing process is highly variable, producing lattices with low Weibull modulus, and redistributing mass towards stress concentrations might have no or even adverse effects since failure is not necessarily localized at the boundary.In general terms, this means that when lattices are being optimized, it is essential to know whether the combination of stress concentration, topology and Weibull modulus leads to a localization of damage.
Finally, it should be emphasized that the present study is focused primarily on understanding fundamental relationships between defect distributions, lattice topology, and macroscopic performance for brittle materials.Naturally, there are many types of defects that warrant additional study, including heterogeneity in elastic modulus, strut size, orientation (i.e.slight misalignments), etc. Further, the present study is focused exclusively on purely brittle failures and is therefore limited to ceramics and polymers/metals with highly limited ductility (such as refractory alloys).Still further, the present analysis assumes that defects are always present at the location within the strut with the highest strain; this is entirely plausible for surface defects common to 3D printing yet may miss key mechanisms when defects are embedded near the strut centerline (e.g.interior grain boundary crack arising during processing).To establish rigorous connections between heterogeneity and macroscopic performance, care should be taken to define consistent linkages between defect type and relevant material properties.
While the specific assumptions utilized in this paper may not be applicable to all component geometries and materials, the present approach offers new opportunities to advance both processing and component design.For example, consider novel ceramic lattices for catalysis scaffolds used in rocket thrusters, fabricated with ink-jet binding followed by sintering.The spatial distribution of partially sintered particles can conceptually be linked to strength distributions, by considering the fracture strength of interparticle necks.Using the present approach to quantify links between the distribution of such defects and the macroscopic performance could achieve two important ends; first, for a fixed set of processing parameters, one could identify allowable load scenarios that would ensure scaffold integrity.Second, one could identify acceptable distributions of defects (partially sintered particles) that would then serve as processing targets to validate fabrication protocols.That is, the present approach provides a conceptual framework to define "defect-tolerant" structures for specific materials, processing routes and components.

Conclusion
A novel finite-element algorithm allows one to simulate the failure of thousands of finite sized lattices within days on a laptop or low-end desktop PC.Rigorous statistical analysis shows that the effective tensile strength of brittle truss lattices is decreased by (i) a decrease in effective Weibull modulus of struts, (ii) an increased number of unit cells in the lattice, (iii) a decrease of nodal connectivity.At low strut Weibull modulus, damage is not localized, and global stress-strain curves exhibit regimes with stable damage evolution.The higher the Weibull modulus of the struts, the more damage localization occurs and global failure becomes more instantaneous.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.where , , ,   are the Young's modulus, the shear modulus, the strut radius and the element length, respectively.Assuming isotropic material properties, the shear modulus is defined by the Young's modulus and the Poisson's ratio given in the main paper.

Appendix C. Relative density
To compute the relative density accurately, the treatment of the boundary must be taken into account, meaning it is relevant whether the cross section of the struts on the free surfaces has the shape of a half or a full circle.We use full circles.Hence, the bounding volume of a specimen is computed according to where  and  are the lattice constant and the strut radius, respectively.This implies that for finite sized lattices, to achieve the same relative density, the strut radius has to be adjusted if  is varied.Furthermore, the contribution of the nodes must be accounted for.The effective length of a strut  connecting two nodes is given by   = ||      Fig. 16 again follows the concept of Fig. 2, but for the FCC lattice.As for the other two lattice topologies, we observe a higher degree of damage localization with decreasing Weibull modulus.There are, how-ever, two striking differences.First, the FCC lattice reaches the ultimate strength at significantly higher strains than the SC and the SC-BCC lattices.Secondly, the load drops to (almost) zero right after the ultimate

Fig. 1 .
Fig. 1.(a) Schematic rendering of a lattice with  = 5 with boundary conditions.Each strut is assigned a random failure strain which is sampled from a Weibull distribution.In gray are the blueprints of three lattice topologies: Simple cubic (SC), simple cubic body-centered cubic (SC-BCC), face-centered cubic (FCC).(b)

Fig. 4 .
Fig. 4. Apparent ductility εd versus normalized ultimate strength σS at relative density ρ = 0.1.(a) Concept of apparent ductility, which is defined as the range of engineering strain where the remaining effective stiffness falls in between fifty and ninety percent.(a,b,c) Apparent ductility versus ultimate strength.Shaded regions are convex hulls around 2D point clouds.The black stars indicate the median values.(b) Simple cubic (SC).(c) Simple cubic body-centered cubic (SC-BCC).

Patrick Ziemke :Fig. 7 .
Fig. 7. Probability of failure of different ceramic material systems fabricated by DIW.This figure was adapted from [23] and expanded by data measured from Al 2 O 3 ∕Al 2 O 3 .review & editing.Matthew R. Begley: Writing -review & editing, Supervision, Software, Project administration, Methodology, Funding acquisition, Conceptualization.

Fig. 8 .
Fig. 8. Median of normalized ultimate strength σS versus number of finite ele- ments per strut for a simple cubic lattice (SC) with relative density ρ = 0.1.

Fig. 10 .
Fig. 10.Normalized ultimate strength σS versus sub-sample size  for the simple cubic (SC) lattice.Left column (a,c) ρ = 0.1 and  = 8, 20.Right column (b,d) ρ = 0.3 and  = 8, 20.The three black lines in each graph represent the 5%, 50% and 95% quantiles calculated from a sample of 500 individual simulations.The colored "scatter bands" represent the 95% confidence intervals determined via bootstrap for a sub-sample size  .

Table 1
Timings of the algorithm for the simple cubic lattice.Each row corresponds to 500 simulations at Weibull modulus  = 3 and relative density ρ = 0.1.The average number of

Table 3
Naming of video files visualizing failure sequences.