Bone-inspired microarchitectures achieve enhancedfatigue lifeAshley M. Torresa,b, Adwait A. Trikanadc, Cameron A. Aubina, Floor M. Lambersa, Marysol Lunaa, Clare M. Rimnacd,Pablo Zavattieric, and Christopher J. Hernandeza,b,e,1

aSibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14853; bMeinig School of Biomedical Engineering, CornellUniversity, Ithaca, NY 14853; cLyles School of Civil Engineering, Purdue University, West Lafayette, IN 47907-2051; dDepartment of Mechanical & AerospaceEngineering, Case Western Reserve University, Cleveland, OH 44106-7222; and eHospital for Special Surgery, New York, NY 10021

Edited by Huajian Gao, Nanyang Technological University, Singapore, and approved October 24, 2019 (received for review April 7, 2019)

Microarchitectured materials achieve superior mechanical proper-ties through geometry rather than composition. Although ultra-lightweight microarchitectured materials can have high stiffnessand strength, application to durable devices will require sufficientservice life under cyclic loading. Naturally occurring materials pro-vide useful models for high-performance materials. Here, we showthat in cancellous bone, a naturally occurring lightweight micro-architectured material, resistance to fatigue failure is sensitive to amicroarchitectural trait that has negligible effects on stiffness andstrength—the proportion of material oriented transverse to appliedloads. Using models generated with additive manufacturing, weshow that small increases in the thickness of elements orientedtransverse to loading can increase fatigue life by 10 to 100 times,far exceeding what is expected from the associated change in den-sity. Transversely oriented struts enhance resistance to fatigue byacting as sacrificial elements. We show that this mechanism is alsopresent in synthetic microlattice structures, where fatigue life can bealtered by 5 to 9 times with only negligible changes in density andstiffness. The effects of microstructure on fatigue life in cancellousbone and lattice structures are described empirically by normalizingstress in traditional stress vs. life (S-N) curves by√ψ, where ψ is theproportion of material oriented transverse to load. The mechanicalperformance of cancellous bone and microarchitectured materials isenhanced by aligning structural elements with expected loading;our findings demonstrate that this strategy comes at the cost ofreduced fatigue life, with consequences to the use of microarchitec-tured materials in durable devices and to human health in thecontext of osteoporosis.

microarchitecture | osteoporosis | microarchitectured materials | bone |additive manufacturing

Microarchitectured materials can achieve high stiffness andstrength per unit mass through underlying geometry rather

than material composition (1–4). Recent developments in additivemanufacturing and lattice design software allow for rapid optimi-zation of lattice density and architecture to meet stiffness, strength,and/or energy absorption demands with low-density micro-architectures (5). Advancements in micro- and nanofabricationhave allowed for the design of microarchitectured materials froma variety of different substrates with high stiffness and strength(6–11). Resistance to fatigue failure is not as often considered inthe design of lattice microstructures. However, microarchitecturedmaterials can be susceptible to fatigue failure, because their com-plex geometry results in stress concentrations that can be an orderof magnitude greater than stresses applied to the bulk material,thereby promoting the initiation and propagation of fatigue dam-age (12–14). Balancing the needs for fatigue life with stiffness,strength, and other desired material properties is a major challengefor the use of microarchitectured materials in durable devices.Naturally occurring materials can display exceptional mechan-

ical performance and are useful models for the design of micro-architectured materials (13, 15). Bone is a biological material withhigh stiffness and strength relative to density. Whole bones consist

of an outer shell made of dense tissue known as cortical bone thatsurrounds a foam-like tissue known as cancellous bone. Cancel-lous bone consists of a network of interconnected plate-like androd-like struts called trabeculae (∼50 to 300 μm in thickness).Trabeculae in cancellous bone are preferentially aligned in thedirection of stresses generated by habitual physical activity,resulting in a transversely isotropic microstructure. Althoughmicroarchitecture is widely recognized as a contributor to themechanical performance of cancellous bone, to date, only density/porosity and fabric tensor (a measure of anisotropy) have beenshown to be major contributors to cancellous bone stiffness andstrength; all other aspects of microarchitecture provide only neg-ligible contributions (16). The effect of microarchitecture on thefatigue properties of cancellous bone is not as well studied.The stiffness and strength of cancellous bone and other cellular

solids have been studied for some time and are related to densitythrough power law relationships (4, 17). Although there are ana-lytical methods of relating the foam density to fatigue life (number ofcycles to failure, Nf) (17), the fatigue life of foams is better explainedby normalized stress vs. life (S-N) relationships of the form (18, 19)

σ

E0= ANB

f , [1]

where σ is the maximum compressive stress, E0 is the initialYoung’s modulus (alternatively, yield stress or plateau stress is

Significance

Microarchitectured materials, such as foams and lattice struc-tures, can achieve high stiffness and strength while remainingextremely lightweight. Applying high-porosity microarchitecturedmaterials to durable devices, such as vehicles, however, will re-quire the materials to also resist failure during cyclic loading.Here, we identify an aspect of microstructure in cancellous bonethat greatly influences failure under cyclic loading and show thatthe effect is generalizable to synthetic microarchitectured mate-rials. Our findings demonstrate that a common design strategy toimprove stiffness and strength of microarchitectured materialscomes at the cost of impaired service life. Our findings are usefulfor the design and application of microarchitecturedmaterials andadditionally provide insight into human health in situations ofosteoporosis.

Author contributions: A.M.T., A.A.T., F.M.L., C.M.R., P.Z., and C.J.H. designed research;A.M.T., A.A.T., C.A.A., F.M.L., and M.L. performed research; A.M.T., A.A.T., C.A.A., F.M.L.,M.L., C.M.R., P.Z., and C.J.H. analyzed data; and A.M.T., A.A.T., C.M.R., P.Z., and C.J.H.wrote the paper.

The authors declare no competing interest.

This article is a PNAS Direct Submission.

Published under the PNAS license.1To whom correspondence may be addressed. Email: cjh275@cornell.edu.

This article contains supporting information online at https://www.pnas.org/lookup/suppl/doi:10.1073/pnas.1905814116/-/DCSupplemental.

First published November 18, 2019.

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used), Nf is the number of cycles to failure, and A and B areempirical constants (in cancellous bone, A ranges from 0.0091 to0.013, and B ranges from −0.121 to −0.094 [20]). Although thenormalized S-N relationship improves prediction of fatigue life,the empirical constants differ among microstructures (cancellousbone from different animals, distinct lattice microstructures, etc.)in ways that are not yet well understood but are commonly attrib-uted to local deformation mechanisms (bending vs. stretching) orin synthetic microarchitectured materials, flaws generated duringmanufacturing (19).

Results and DiscussionTo better understand the effects of microarchitecture on fatiguelife, we determined the relationship between microstructure andfatigue damage processes in high-porosity (>90%) cancellousbone from the vertebral bodies of deceased human donors (n =44 specimens from 18 donors) (Materials and Methods). Cycliccompressive loading (0 to compressive stress) was applied in thedirection of habitual loading in vivo. Fatigue loading was sus-pended at a specified amount of cyclic loading (determined byaccumulated cyclic strain), and the resulting amount and locationof microscopic damage within the microstructure (microscopiccracks and accumulation of submicroscopic cracks; referred to as“microdamage” in the bone literature) were detected using con-trast agents (Fig. 1 A and B and SI Appendix, SupplementaryMethods and Materials and Fig. S1) (21, 22). Microarchitecturewas assessed using 3-dimensional (3D) images and analyzedusing a morphological decomposition approach that isolateseach individual strut within the structure and classifies the strutas plate like or rod like as well as determining its orientationrelative to loading (Materials and Methods and Fig. 1 C and D)(23). The amount of tissue damage caused by fatigue loadingwas correlated with maximum applied apparent strain (SI Ap-pendix, Table S1) but was not correlated with specimen density,other specimen-average measures of microstructure, or measuresof plate-like trabeculae (the primarily load-carrying elements).Surprisingly, the amount of tissue damage was reduced in speci-mens with thicker rod-like struts (Fig. 1E and SI Appendix, TableS2) (R2 = 0.76, P < 0.01). This finding was unexpected, since rod-like struts in cancellous bone are primarily oriented transverse tothe applied load, constitute on average only 20% of the solidvolume of high-porosity cancellous bone (SI Appendix, Table S3),carry only a small proportion of longitudinally oriented loads, andhave negligible effects on stiffness and strength in the longitudinaldirection (24) (transversely oriented elements have similarly smalleffects in static properties of nanolattice structures [25]).To better understand the effect of rod-like struts on fatigue

failure, we examined the distribution of tissue damage at differentpoints during the fatigue loading process. The failure of individualtrabeculae during fatigue loading occurs nonlinearly with cyclenumber and differs by trabecular type/orientation (Fig. 1F). Earlyin fatigue, failure of struts occurs primarily in rod-like trabeculae;substantial damage accumulation in plate-like trabeculae does notoccur until overt failure (Fig. 1F). The pattern of strut failure isalso related to orientation: failed rod-like trabeculae are pre-dominately transversely oriented, while failed plate-like trabeculaeare predominately oriented longitudinally (SI Appendix, Fig. S2).We attribute the pattern in failure of individual trabeculae to thedistribution of tensile stresses generated by loading: finite elementmodels indicated that apparent compressive loading results intensile stresses in rod-like trabeculae (primarily transversely ori-ented) and compressive stresses in plate-like trabeculae (primarilylongitudinally oriented) (SI Appendix, Fig. S3). These findingssuggest that, in cancellous bone, transversely oriented trabeculaeact as sacrificial elements during cyclic loading by accumulatingtissue damage and thereby, protecting the load-carrying, longitu-dinally oriented, plate-like trabeculae, the failure of which is in-dicative of final fatigue failure.

Tissue heterogeneity is also a major contributor to damageaccumulation in cancellous bone (22, 26) and is, therefore, apotential explanation for our findings in human bone tissue.To isolate the effects of microstructure from those associatedwith material heterogeneity, we generated 3D models of can-cellous bone microstructures using a high-resolution projectionstereolithography printer (Fig. 2 A and B) (M1; Carbon) (27). Can-cellous bone microstructures (Fig. 2B) were modified by addingmaterial to the surface of transverse trabeculae in 1 of 3 increments:no modification (original geometry), +20 μm on the surface (anaverage increase in rod thickness of 20 ± 5%; mean ± SD), or+60 μm on each surface (an average increase in rod thickness of45 ± 14%). Because transverse rod-like trabeculae constitute onlya small portion of the solid volume and carry only a small portion

Fig. 1. Microarchitecture influences fatigue damage accumulation in cancel-lous bone. (A) The creep fatigue curve of cancellous bone is shown with thethree phases of fatigue loading indicated. Cyclic compressive loading of can-cellous bone was stopped at different points along the creep fatigue curve (datapoints) to determine patterns of damage accumulation (fatigue life was esti-mated as in ref. 21). (Inset) Cyclic loading waveform is shown. The 3D images ofcancellous bone with (B) green indicating damage, (C) plate-like and rod-likestruts, and (D) strut orientation relative to anatomical position (longitudinal,oblique, and transverse). (E) The amount of damage in cancellous bone (dam-age volume fraction, DV/BV) was correlated with maximum applied strain, butspecimens with thicker rod-like trabeculae experienced less damage accumula-tion (R2 = 0.76, P < 0.01). Error bars indicate the SDs as determined from thelinear mixed effects model. (F) Early in fatigue life, strut failure occurs primarilyin transversely oriented rod-like struts; final mechanical failure is characterizedby widespread failure of longitudinally oriented plate-like struts.

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of longitudinal loads, thickening of rod-like struts had only a smalleffect on density (Fig. 2C) (increase of 11 ± 8%; mean ± SD) andapparent stiffness (22 ± 19% increase in longitudinal Young’smodulus) (Fig. 2D). When applied uniformly across the micro-structure, such small increases in density and stiffness cause onlysmall changes in fatigue life that are well described by the nor-malized S-N relationship (19). However, when applied only to rod-like trabeculae, fatigue life was increased by as much as 2 orders ofmagnitude (Fig. 2E) and followed entirely new normalized S-Nrelationships (SI Appendix, Fig. S4).To confirm that damage accumulation in the 3D models did

not differ considerably from that of cancellous bone, we examineddamage (microscopic cracking, constrained microcracks, etc.) inthe additively manufactured specimens after loading using aradioopaque dye penetrant. Locations of damage accumulationidentified with the dye penetrant were distributed throughout thestructure in a manner qualitatively similar to that seen in cancel-lous bone (Fig. 2F). Furthermore, printed specimens with thickerrod-like struts showed reduced damage accumulation (Fig. 2G).Hence, damage accumulation during fatigue loading is modifiedby the thickness of rod-like trabeculae in both materials. Fur-thermore, finite element models of the specimens indicated thatthe average tensile stresses in rod-like trabeculae (predominatelytransversely oriented) were greater than those in plate-like tra-beculae (predominately longitudinally oriented) (SI Appendix, Fig.S3), demonstrating that the localization of damage follows stress

distributions within the microarchitecture as seen in true cancel-lous bone. Together, these findings indicate that small increases inmass applied to transversely oriented structural components of themicrostructure can reduce tensile stresses, leading to dispropor-tionately large beneficial effects on fatigue life.To determine if our findings are generalizable to other cellular

solids and other deformation mechanisms (bending vs. stretching),we created printed models of an octet truss (1) as well as an octettruss modified to have plate-like and rod-like elements mimickingthe microstructure and anisotropy of cancellous bone (Fig. 3A).Cancellous bone microstructure shows bending-dominated be-havior, the octet truss shows a stretching-dominated deformationbehavior, and the bone-like microarchitecture displayed a com-bination of both stretching and bending deformation behaviors (SIAppendix, Fig. S5). In the bone-like microarchitectures, increasesin transverse strut thickness resulted in an increase in the fatiguelife by a factor of 8 (Fig. 3B), with only a small change in density(+4%) or longitudinal stiffness (+20%). In the octet truss, in-creases in transverse strut thickness resulted in an increase in fa-tigue life by a factor of 5 (Fig. 3B), with only minor changes indensity (+10%) or longitudinal stiffness (+14%) (Fig. 3B and SIAppendix, Table S4). In contrast, when the modified octet trussmodel was rotated 90° so that the thickened elements were verti-cally oriented and oblique to the applied loads, the fatigue life wasreduced by a factor of 9 compared with the model without thick-ened struts (Fig. 3B), demonstrating that the effect of transverse

Fig. 2. Models of cancellous bone generated using additive manufacturing show that fatigue life is sensitive to small changes in microarchitecture. (A)Digital images of human vertebral cancellous bone were edited and printed into (B) high-resolution 3D models. Increases in the thickness rod-like struts hadsmall effects on (C) density and (D) stiffness (Young’s modulus in first cycle of loading) yet resulted in (E) increases in fatigue life by as much as 2 orders ofmagnitude. (Lines connect samples derived from the same bone specimen; 2 magnitudes of normalized cyclic stress are shown. Scatter is a result of variationsin microstructure among the 5 bone samples.) (F) A microcomputed tomography image of a 3D-printed sample of cancellous bone after fatigue loading tofailure. A radioopaque dye penetrant indicated regions of accumulated damage. (Inset) Magnified view. (G) The amount of damage (damage volumefraction, DV/BV) generated by fatigue loading to failure was reduced in 3D-printed specimens with greater thickness of rod-like struts.

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elements on fatigue life is related to the proportion of materialoriented transverse to loading rather than the thickness of thetransverse struts per se. To understand the extent to which thetransversely oriented material influenced fatigue damage accu-mulation, we performed nonlinear finite element models of cyclicloading. Fatigue damage involves a local irreversible energy-dissipating process resulting in increases in inelastic dissipationenergy. Finite element models of the first 5 to 25 cycles of loadingindicate that the fatigue life of the octet and bone-like micro-architectures with and without thickened struts is closely related tothe inelastic dissipation energy per unit work (Fig. 3C). Althoughthese models are limited to the first few cycles of loading, inelasticdissipation energy and stress triaxiality stabilize by this point (SIAppendix, Fig. S5G). Hence, increases in the transverse volumefraction (ψ ; the proportion of the solid volume oriented transverseto loading) in these microarchitectured materials reduce theamount of inelastic energy dissipation and damage accumulationduring cyclic loading, just as thicker rod-like trabeculae (pre-dominately transversely oriented) experienced less damage accu-mulation in cancellous bone (Fig. 1E). Following a single overload(50% strain), both bone (28) and microarchitectured materials(25, 29) can recover a large proportion of the applied strain, aneffect attributed to elastic deformations in transversely orientedstruts (however, in our work, the energy loss coefficient during fa-tigue loading did not vary much with alterations in strut thickness[SI Appendix, Fig. S6]). Our finding here suggests that transverselyoriented struts are also important for resisting fatigue failure underintermediate and high cycle fatigue, such as that commonly ap-plied to durable devices. Together, these findings show that theeffects of transversely oriented material on fatigue life are notunique to cancellous bone but extend to synthetic microarchitecturedmaterials. That our findings regarding damage accumulation

are consistent in human bone tissue (a biological ceramic polymercomposite) as well as a polymer used in additive manufacturingfurther supports the idea that the effect is due to geometry andmay not be limited to one class of constituent material.We developed an empirical model to characterize the rela-

tionship between fatigue life (Nf), applied cyclic normalized stress(noted as σ/E0), and transverse volume fraction (ψ) for bone,bone-like microarchitectures, and octet trusses generated withadditive manufacturing. Surprisingly, the regression models iden-tified a predictive equation only slightly different from Eq. 1:

σ

E0

1ffiffiffiffiψ

p = ANBf , [2]

in which A and B are empirical constants (A = 51,903, B = −0.14,R2 = 0.82) (regression coefficients are in SI Appendix, Table S5).Although there remains unexplained variance in Eq. 2 (likely dueto differences in tensile stresses among the bone, bone-like, and octetmicrostructures), this modification to the normalized S-N relationshipaccounts for the differences in normalized S-N relationships amongthe microstructures and provides a simple means of considering fa-tigue life during the design/selection of microarchitectured materials.In ultralightweight microarchitectured materials, lattice struc-

tures in which struts are aligned with expected loading andunderloaded struts are removed can achieve more efficiency interms of specific stiffness, strength, and energy absorption (25, 27).However, our findings demonstrate that such a strategy can comeat the cost of large reductions in fatigue life due to reductions inthe proportion of material oriented transverse to loading (as aresult of reductions in the thickness and/or number of transverselyoriented struts) (SI Appendix, Fig. S7). Future applications ofhigh-porosity microarchitectured materials to durable products,

Fig. 3. Transverse volume influences fatigue life in repeating cellular solids. (A) Images of the bone-inspired microstructure and an octet truss are shown.(Scale bar: 5 mm.) (B) The fatigue life of microarchitectured materials printed as designed or with rod-like struts thickened (colored) is shown. Thickeningtransverse struts increases fatigue life, while thickening vertically oriented struts reduces fatigue life (specific stiffness, E0/ρ is also shown). (C) Fatigue life ofthe lattice structures is related to the inelastic dissipation energy per unit work determined from finite element models. (D) Fatigue life for 3D-printedspecimens at different applied cyclic loading (σ/E0, noted in microstrains) is shown with lines indicating regression model fits (Eq. 2) (R2 = 0.82). Symbolsindicate bone (●), bone-like microstructure (▲), and octets (♦).

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such as vehicles, will require balancing performance in terms ofstiffness, strength, and energy absorption with costs associatedwith replacement and repair due to fatigue failure.Our findings indicate an effect of microstructure on fatigue;

however, material properties can also influence the accumulation ofmicroscopic damage due to fatigue loading. In particular, materialheterogeneity (designed or naturally occurring) has the potential toinfluence fatigue failure. Furthermore, as strut size approaches thatof the critical size for flaw insensitivity, there can be a large increasein the strength of a microarchitectured material (the so-called“smaller is stronger” concept) (8, 9, 11, 30). It is possible that suchapproaches could also be used to improve fatigue life, althoughlittle is known about the fatigue properties of such materials.Our findings are also relevant to osteoporosis-related fractures in

the elderly. Osteoporosis is characterized by degradations in can-cellous bone microstructure illustrated by drastic reductions in thenumber and robustness of transversely oriented trabeculae (31).However, the stiffness, strength, and energy absorption of cancel-lous bone are overwhelmingly determined by longitudinally orientedtrabeculae, and the mechanical importance of transversely orientedtrabeculae is traditionally viewed as minor and limited to resistingoccasional off-axis loading (32, 33) or providing simple support forthe more mechanically relevant longitudinally oriented trabeculae(31). Our findings challenge this view by showing that transverselyoriented trabeculae are key determinants of fatigue performanceunder longitudinal loading. While many osteoporosis-related frac-tures in humans are caused by a single overload sensitive to strength(fall from standing height, heavy lifting, etc.), the most commontype of osteoporosis-related fracture, vertebral fractures in thespine, frequently occurs in the absence of a discrete loading event,implicating a contribution of fatigue damage (34). Preferential lossof transversely oriented trabeculae during aging, therefore, causesreductions in the fatigue life of cancellous bone that are dispro-portionately larger than the reductions in stiffness and strength,potentially explaining the long-held but poorly described relation-ship between cancellous bone microstructure and clinical fractures.

Materials and MethodsMechanical Characterization and Damage Assessment in Cancellous Bone. Thisstudy examined human vertebral cancellous bone (n = 44 cancellous bonespecimens from 10 male and 8 female donors 62 to 92 y of age) as reportedin 2 prior studies (21, 22). Human tissue was received from a nonprofit hu-man tissue bank (National Disease Research Interchange) and is thereforeexempt from institutional review board review. Cylindrical specimens 8 mmin diameter and 27 mm in length were oriented in the cranial–caudal di-rection and were press fit into end caps to avoid artifacts during mechanicaltesting (SI Appendix, Supplemental Materials and Methods).

Fatigue loading was applied using a 4-Hz haversine waveform cyclicallybetween 0 N and a compressive load corresponding to σ = E0 × 0.0035mm/mm,where σ is stress and E0 is the initial Young’s modulus of the specimen. Fatigueloading of the remaining specimens was applied at room temperatureand stopped before failure (defined as 4% apparent strain [22]) at a pre-determined magnitude of cyclic strain. After loading, specimens were carefullyremoved from the testing device and bulk stained with contrast agents thatidentify microscopic tissue damage (microdamage in the bone literature), and3D images were collected. Microscopic damage detected with this methodol-ogy includes damage 5 to 10 μm in size, including both individual microscopiccracks as well as aggregation of submicroscopic cracks. Hence, individualsubmicroscopic cracks generated at lower levels of structural hierarchy (forexample, those caused by individual lamellae, individual mineral crystals, etc.)are only detected in aggregate. More details on microdamage detection are inSI Appendix, Supplementary Methods and Materials and refs. 21 and 22.

Cancellous Bone Microarchitecture. Microarchitecture was assessed usingspecimen average measurements [BoneJ; http://bonej.org/ (35); 21-μm

isotropic voxel microcomputed tomography images collected prior to fatigueloading] (SI Appendix, Table S6) as well as morphological analysis of each in-dividual trabecula (SI Appendix, Table S7). Morphological analysis involvedclassifying each trabecula as rod like or plate like based on Digital TopologicalAnalysis (Fig. 1C) (ITS software; Columbia University). Failure of cancellousbone has been referred to as accumulation of failed trabeculae (28, 36). Here,we consider failure of a trabecula to occur when at least 10% of the volume ofthe trabecula includes microdamage (37). Variation in this definition of failuredid not influence conclusions.

Additive Manufacturing of Modified Cancellous Bone Microstructures. Micro-computed tomography images of cancellous bone samples (21, 22) that hadbeen collected before mechanical loading were modified digitally by addingmaterial to surfaces. A high-resolution stereolithography system (M1; Carbon)was used to generate 3D models of the modified and unmodified micro-structures from a urethane methacrylate polymer resin (UMA 90; Carbon;E = 2 GPa) (27) at 1.5 times isotropic magnification (12-mm diameter, ∼30-mmlength). The accuracy of the printed geometries was confirmed using micro-computed tomography images (SI Appendix, Tables S7 and S8). Models generatedthrough additive manufacturing were submitted to cyclic fatigue loading from0 to a normalized initial compressive stress magnitude σ/E0 corresponding to 9,500,6,500, or 4,500 μe until failure (4% applied strain). A total of 45 models of can-cellous bone microstructure were used for this experiment (5 distinct microstruc-tures × 3 distinct rod thicknesses × 3 different normalized stress magnitudes).Damage in polymer specimens was identified using a radio opaque dye penetrant.

Bone-like microarchitectures derived from the octet truss were designed byelongating the longitudinal axis (the longitudinal axis) andaddingplates to achieveporosity, transverse volume fraction, longitudinal volume fraction, and platevolume fraction similar to the cancellous bone specimens (SI Appendix, Fig. S5).

Finite Element Modeling. Finite element modeling was used to characterizethe stress distributions in bone and lattice structures generated using additivemanufacturing. Models were meshed with elastic perfectly plastic brick el-ements (1.5 million per model), and quasistatic analyses were carried out forcyclic compressive loading. Inelastic dissipation energy was defined as thatportion of the internal strain energy that is dissipated by rate-independentand rate-dependent plastic deformation calculated as

Ep =Zt

0

�ZVσc : _«pldV

�dτ,

where Ep is the inelastic dissipation energy, σc is the stress derived from theconstitutive equation, and _«pl is the plastic strain rate. These properties areintegrated in time and over the volume of the specimen.

To characterize the primary deformation mechanism (bending dominatedor stretching dominated) within eachmicrostructure, the stress triaxiality wasdetermined at each point within the model. Stress triaxiality is defined as

Triaxiality =−pq,

where p is the hydrostatic stress defined as positive when stress is com-pressive and q is the von Mises stress. By this definition, points under uniaxialtension would have a triaxiality of +0.33, and points under pure uniaxialcompression would have a triaxiality value of −0.33 (SI Appendix, Fig. S5).

Data Sharing.All data, documentation, and custom code used in this work areavailable from the corresponding author on reasonable request.

ACKNOWLEDGMENTS. We thank Rob Shepherd for comments. This workwas supported in part by National Institute of Arthritis and Musculoskeletaland Skin Diseases, NIH Grants AR057362 and AR073454; NSF GraduateResearch Diversity Supplement Grant 1068260, Graduate Research Fellowship(to A.M.T.), and Faculty Early Development Program (CAREER) Award 1254864;a Cornell–Colman Fellowship (to A.M.T.); and the Wilbur J. Austin Professor ofEngineering Chair (C.M.R.). Imaging data were performed in the CornellBiotechnology Resource Center-Imaging Facility (NIH Grant S10OD012287).The content of the work is solely the responsibility of the authors and doesnot necessarily represent the official views of the funding agencies.

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