Nanoscale heterogeneity promotes energy dissipation in bone

Transcription

1 Nnosle heterogeneity promotes energy dissiption in one KUANGSHIN TAI 1 *,MINGDAO 1 *, SUBRA SURESH 1,2, AHMET PALAZOGLU 3 AND CHRISTINE ORTIZ 1 1 Deprtment of Mterils Siene nd Engineering, Msshusetts Institute of Tehnology, 77 Msshusetts Avenue, Cmridge, Msshusetts 2139, USA 2 Division of Biologil Engineering, Msshusetts Institute of Tehnology, 77 Msshusetts Avenue, Cmridge, Msshusetts 2139, USA 3 Deprtment of Chemil Engineering nd Mterils Siene, University of Cliforni, One Shields Avenue, Dvis, Cliforni 95616, USA *These uthors ontriuted eqully to this work e-mil: ortiz@mit.edu Pulished online: 21 My 27; doi:1.138/nmt1911 Nnomehnil heterogeneity is expeted to influene elstiity, dmge, frture nd remodelling of one. Here, the sptil distriution of nnomehnil properties of one is quntified t the length sle of individul ollgen firils. Our results show elorte ptterns of stiffness rnging from 2 to 3 GP, whih do not orrelte diretly with topogrphil fetures nd hene re ttriuted to underlying lol struturl nd ompositionl vritions. We propose new energy-dissiption mehnism rising from nnomehnil heterogeneity, whih offers mens for dutility enhnement, dmge evolution nd toughening. This hypothesis is supported y omputtionl simultions tht inorporte the nnosle experimentl results. These simultions predit tht non-uniform inelsti deformtion over lrger res nd inresed energy dissiption rising from nnosle heterogeneity led to mrkedly different iomehnil properties ompred with uniform mteril. The fundmentl onepts disovered here re pplile to rod lss of iologil mterils nd my serve s design onsidertion for iologilly inspired mterils tehnologies. Bone, like mny nturl mterils, is inherently struturlly, nd hene mehnilly, heterogeneous owing to sptil distriutions in the shpe, size nd omposition of its onstituent uilding loks. Heterogeneity lso rises from multitude of different types of iomromoleule, pores ontining fluids of vrile visosities nd numerous interfes (for exmple solid solid, solid fluid, dmge sed) 1,2. As onsequene of the hierrhil nture of one 3, mehnil heterogeneity is expeted to exist t multiple length sles. Mrosopilly, signifint vritions in mehnil properties hve een oserved for different ntomil lotions 4, s well s for regions within prtiulr ntomil lotion 5, nd re thought to develop in response to distint loding requirements 6. At the mirosopi level, instrumented indenttion hs further identified differenes in moduli nd hrdness for speifi fetures suh s treule, interstitil lmelle nd thik nd thin lmelle in osteons, whih hve een ttriuted to ollgen firil orienttion nd nisotropy, s well s vritions in minerl ontent 7,8. Heterogeneity t this length sle is lso expeted to rise from the ontinul ellulr remodelling proesses resulting in mixture of old nd new one t ny given time. For exmple, osteonl one typilly undergoes sustntil remodelling, wheres older interstitil one 9 hs higher degree of minerliztion 1, leding to inresed stiffness 8. Atomi fore mirosopy (AFM)- sed nnoindenttion hs een used to distinguish mehnilly heterogeneous mirosle regions in one tissue from genetilly modified mie with vrious levels of trnsforming growth ftor, TGF-β, whih orrelte with ompositionl heterogeneity mesured y Rmn mirospetrosopy, nd dditionlly hs een shown to possess the pility for quntifying nnosle heterogeneity 11. All of these studies hve ignited speultion on the role of heterogeneity in strin onentrtion, frture risk, dpttion nd dmge umultion 8, They lso rise importnt issues s to whether heterogeneity is dvntgeous or disdvntgeous to the mehnil funtion of one 2. Detiled nd quntittive studies of the onsequenes of heterogeneity, in prtiulr t the nnosle, on the struturl integrity nd proper funtion of the tissue re lrgely unknown. In this pper, we first quntify mehnil heterogeneity t nnometre length sles. Here, the mehnil response rises from n individul minerlized ollgen firil within the one extrellulr mtrix, whih is omposed of pltelike ronted ptite minerlites ( tens of nnometres in length nd width, nd 3 5 nm in thikness 1 ) tht permete in nd round type-i ollgen in n overlpping mnner 17, s well s smll onentrtion of 2 different types of non-ollgenous protein (<1 wt% of totl protein) 1. It hs eome inresingly evident tht the unique nnosle properties of one ply key role in its mrosopi iomehnil funtion 17 22, s this is where inelsti deformtion nd frture must first initite. Furthermore, euse osteoyti proesses hve dimensions tht re of the nnometre length sle 23, nnosle heterogeneities in mteril properties of the surrounding extrellulr mtrix would e expeted to modulte lol stresses, therey potentilly influening proesses suh s remodelling, migrtion nd dhesion. Nnoindenttion using sptilly ontrolled AFM-sed instrument ws rried out with shrp tetrhedrl silion proe tip (end rdius 15 nm, s estimted y high-resolution snning eletron mirosopy imging nd AFM imging of goldnnoprtile lirtion stndrd) in mient onditions s shown in Fig. 1 (see the Methods setion). This instrument hs fully three-dimensionl (3D) losed-loop piezoeletri system tht ensures positionl uries in three dimensions of <1 nm. Grids of indenttions over 2 μm 2 μmrewererried out on dult ovine tiil ortil one (prepred with no lohol 454 nture mterils VOL 6 JUNE 27

2 dehydrtion, emedding or hemil fixtion) oth perpendiulr nd prllel to the long one, where the seprtion distne etween the indents ws 1 nm. This vlue ws hosen to void overlp of neighouring inelsti strin/residul stress zones estimted y pproprite finite-element nlysis (FEA) simultions s detiled in the Methods setion. Inelsti deformtion is predited y FEA simultions (dt not shown) to tke ple lmost immeditely fter ontt owing to the high lol stress onentrtions due to the shrpness of the proe tip, whih is onsistent with finite residul depth oserved experimentlly on unloding (s will e shown lter on). Hene, this size of the inelsti strin/residul stress zones determines the sptil resolution of the experiments or the interindent sping, tht is, it is not limited y instrumentl ftors, only y the smple deformtion. The deformtion of the proe tip is negligile, euse the elsti modulus of silion is pproximtely one order of mgnitude greter thn tht of one. The deformtion of the instrument ws negligile (ompline Nm 1 ) ompred to the deformtion of the ntilever (spring onstnt 56.2Nm 1 ). It ws ensured tht during indenttion the lser spot ws mintined within the liner rnge of the position-sensitive photodiode. The mximum indenttion depth ws <4 nm t 5 μn mximum lod (for 9% of ll of the experimentl dt), orresponding to n elsti-zone rdius (ontining the inelstilly deformed region) of 7nm (estimted y FEA simultions; see the Supplementry Informtion). As type-i ollgen firils in these types of one smple were mesured to hve men dimeter of 15 nm (ref. 24), the deformtion indued y indenttion ws on the length sle of n individul ollgen firil. At eh position in the grid, the unloding slope of the fore-versus- urve ws used to lulte n indenttion modulus on the sis of widely used ontt mehnis model 25 (see the Methods setion). Owing to the ssumptions of this model nd the hierrhil struture of one, the extrted indenttion-modulus vlues reflet reltive mehnil stiffness nd re not solute quntittive vlues. However, the reltive sptil differenes in stiffness n e urtely ompred. Two-dimensionl (2D) ontour mps of lol stiffness were onstruted, whih enled the diret visuliztion of nnomehnil sptil heterogeneity. These mps were ompred with high-resolution.. intermittent ontt-mode topogrphi AFM imges of the sme re of the one ultrstruture tken efore indenttion. The ultrstruture of osteonl one viewed perpendiulr to the long one showed nnogrnulr morphology of dense rry of minerlites (Fig. 2) with n verge mximum lterl dimension of 64.9 ± 26.4nm, pek-to-vlley height of 11.5 ± 9.1 nm nd r.m.s. surfe roughness of 11.5 nm lulted over the entire 2 μm 2 μm sn (mesured from orresponding height imges). Distint type-i ollgen firils nd their hrteristi 67 nm periodiity were not oserved owing to the dense rry of minerlites present. However, s reported previously 24, AFM imging fter prtil deminerliztion, using one-seond eth in ethylenediminetetreti id, revels surfe of type-i ollgen firils with vrying degrees of orienttion within the smple plne, the expeted 67 nm nding nd men dimeter of 151 ± 17nm. The stiffness mp for the loding perpendiulr to the long one (osteonl) (Fig. 2) orresponding to the re imged in Fig. 2 shows distriution with vlues rnging from 2 GP (white) to 26 GP (lk) with n verged indenttion-modulus vlue of 8.55 ± 3.7 GP nd oeffiient of vrition or COV (rtio of the stndrd devition to the men) of.43. Individul fore versus depth urves orresponding to few of the positions shown in Fig. 2 re given in Fig. 2 nd show finite residul depth fter unloding rnging from 17 to 27 nm, inditing the presene of inelsti deformtion. This ~18 μm 33º 29º ~18 μm 12º 5 μm 56º 18º 18º ~9 μm Nnomehnilly mpped region 2 μm 33º 33º ~9 μm 29º ~1 μm Proe-tip end rdius <15 nm Contt rdius ~7 nm Figure 1 Shemti digrm of experiment used to quntify nnomehnil heterogeneity in one., 3D shemti digrms illustrting proe-tip geometry nd dimensions from the side nd k view (left), front view (middle) nd ottom view (right) s determined y snning eletron mirosopy., Optil mirosopy imge of n individul osteon in dult ovine ortil one showing the Hversin nl, irumferentil lmelle nd the luno-nliulr porosity., Shemti digrm of 2 μm 2 μm region proed nnomehnilly drwn to sle reltive to osteon size; the size of the indented region is lso drwn to sle. is due to the lrge stress onentrtions imprted y the very smll proe tip whih initites inelsti deformtion lmost immeditely on ontt (s verified oth experimentlly nd omputtionlly, dt not shown). A urved region of lower stiffness ppers digonlly ross the mpped region (length 1.8 μm), wheres higher-stiffness regions pper horizontlly ross the top of the mpped region (Fig. 2). Smller lolized vritions nd grdients our throughout the mp s well. Multiple mpping experiments were rried out on ones from five different nimls nd nnosle heterogeneity ws found to exist in ll smples, with COV vlues rnging from.26 to.46 (dt not shown). The fine ptterns of eh stiffness mp vried mong different nimls (dt not shown), s well s the men stiffness vlues (rnging from 8.5 ± 3.7 to14.6 ± 5. GP), s expeted. We lso investigted the effet of surfe roughness y rrying out >3, indenttion experiments over rnge of mximum depth, h mx /r.m.s. surfe roughness (where the r.m.s. surfe roughness ws mesured diretly t every position using.. intermittent ontt-mode AFM height imges). It ws found tht the COV styed within.3.4 for the entire rnge of h mx /r.m.s. surfe roughness 2 16 nd ws sttistilly independent of h mx /r.m.s. surfe roughness s ssessed y n nlysis of vrine test using the O Brien method to test for homogeneity of vrines etween the different COV groups reltive to one nother. Using this nlysis, it ws shown tht the ssumption for homosedstiity ws met (F = 1.113, p >.5), tht is, tht there ws no overll sttistilly signifint trend for the dependene of the COV on h mx /r.m.s. surfe roughness (see the Supplementry Informtion). Assuming helioidl ollgen firil rrngement 26 nd trnsversely isotropi properties of individul ollgen firils 27, vritions in ollgen firil orienttion re not expeted to ply mjor role in the oserved heterogeneity, euse the loding is perpendiulr to the ollgen firil. Other possile soures of oserved heterogeneity inlude nture mterils VOL 6 JUNE

3 Topogrphy 1 nm Modulus 1 nm GP Fore (μn) Long one Indenttion depth (nm) d Topogrphy 1 nm Modulus 1 nm e GP f Fore (μn) Long one Indenttion depth (nm) g Long one Long one 2 nm 2 nm Figure 2 The ultrstruture nd nnomehnil sptil heterogeneity of one stiffness., An.. intermittent ontt-mode AFM mplitude imge viewed perpendiulr to the long one., 2D ontour mp of modulus redued from nnoindenttion dt 25 for the re shown in with loding perpendiulr to the long one., Individul nnoindenttion urves t different lotions within the modulus mp ; moduli (GP) lue dimonds 12.5, green irles 1.5, purple tringles 5.4, rown squres 4.2. d, An.. intermittent ontt-mode AFM mplitude imge prllel to the long one. The verge mximum lterl dimension mesured from orresponding AFM height imges ws determined to e 64.9±26.4 nm, whih is onsistent with the known dimensions of minerl prtiles in one nd other lified tissues, s mesured y snning eletron mirosopy 5, trnsmission eletron mirosopy 51 nd smll-ngle X-ry sttering 52. These vlues re somewht smller thn those previously oserved on similr smples 2 owing to optimiztion of the experimentl imging protool to hieve higher resolutions. e, 2D ontour mp of modulus for the re shown in d with loding prllel to the long one. f, Individul nnoindenttion urves t different lotions within the modulus mp e; moduli (GP) lue dimonds 24.2, green irles 18.1, purple tringles 15.8, rown squres 5.7. g, 3D grphi illustrtions of the modulus mps shown in (left), where loding ws rried out perpendiulr to the long one, nd e (right), where loding rried out prllel to the long one. The vertil orresponds to stiffness, where the highest vertil topogrphy orresponds to the mximum vlue nd the lowest topogrphy orresponds to the minimum vlue. For nd e, in the 26+ in 1.5% of the dt points showed moduli > 28 GP. nno- to mirosle orgni inorgni ompositionl or struturl vritions (for exmple, different sizes, shpes nd/or sping etween nnosle onstituents). Compositionl flututions re known to exist down to 1 μm, s reveled y miro- Rmn spetrosopy 28. Indenttion-modulus dt tken with the loding prllel to the long one on the sme smple show less heterogeneity (oeffiient of vrition of.37), with vlues rnging from 5 (light yellow) to 31 (lk) GP nd inresed men stiffness ( 12.9 ± 4.8 GP) (Fig. 2d f) ompred with the perpendiulr orienttion (y ftor of 1.5). This trend is onsistent with mrosopi tissue-level dt 5 nd ollgen firil nisotropy (inresed stiffness long the firil ompred with trnsverse diretions) 27. A lrge urved region is oserved with inresed stiffness (thikness 25 nm) in the upper left hnd side of the mp, s well s the lower right hnd side (distne etween high stiffness regions 1.5 μm). Similr experiments were rried out on five different osteonl regions of the sme one smple. Nnosle mehnil heterogeneity ws oserved in ll of these mps nd showed COV vlues rnging from.37 to.43 (dt not shown). Agin, the fine 456 nture mterils VOL 6 JUNE 27

4 GP d GP 5 5 Figure 3 Quntittive nlysis of nnomehnil-property mps using disrete wvelet trnsform., Level-1 pproximtion for modulus mp given in Fig. 2 (loding perpendiulr to the long one )., Level-1 pproximtion for modulus mp given in Fig. 2e (loding prllel to the long one )., First-level detil orresponding to. d, First-level detil orresponding to. ptterns of eh stiffness mp vried mong different regions nd the men stiffness vlues rnged from 11.8±3.6 to 14.1±5.3GP. Figure 2g provides 3D visuliztion of the nnosle stiffness dt, with the vertil diretion orresponding to the stiffness mgnitude, ompring the prllel nd perpendiulr orienttion diretions. Although the COV for the ontour mp perpendiulr to the long one is greter, the differenes in pek-to-vlley modulus vlues re more pprent for the mp from indents tken prllel to the long one in the 3D visuliztions. Assuming the spirl model for ollgen firil orienttion 29, this ourrene my rise from off- loding of ollgen firils, s opposed to inplne loding in the perpendiulr orienttion. If the heterogeneity oserved (for loding prllel to the long one ) ws solely due to ollgen firil orienttion, more-or-less-ontinuous stiffness grdient would e oserved, whih does not seem to e the se here. Thik nd thin lmelle hve een mesured to e pproximtely 5 7 nd 1 μm in width, respetively 3, with the thin lmelle eing 1% softer thn the thik ones 8. No distint ptterns were oserved tht ould e onlusively ssoited with suh fetures. Hene, the flututions oserved must prtilly originte from smller-length-sle fetures. The modulus mps were nlysed quntittively using the disrete wvelet trnsform 31, where deomposition of sptil frequenies retes orser pproximtion of the imge (see the Methods setion). Detils n e ssoited with pseudofrequeny tht identifies the length sles of the ptterns removed t sptilly speifi regions. Figure 3, depits the first-level pproximtions of the originl imges orresponding to indents perpendiulr nd prllel to the long one, respetively. The detils for these orresponding mps were removed to otin the orser imge (Fig. 3,d). The detil imges show lrgely rndom fetures tht previl t the high sptil frequenies. However, they lso pture speifi ptterns ssoited with the high-modulus res. One n oserve high-modulus peks tht oinide with the lrger high-modulus domins (more pprent with the imge prllel to the one, Fig. 3d). The first level of deomposition thus seems to remove the lol high-modulus peks, leving roder regions of stiffness. The hrteristi length sle given y the pseudo(-sptil-)frequeny nlysis, identified s 156 ± 28 nm, orresponds losely to the experimentlly mesured men dimeter of n individul ollgen firil 24. The mehnil heterogeneity desried ove for the AFMsed nnoindenttion is muh more pronouned ompred with depth-sensing miroindenttion rried out on similr speimens (see the Supplementry Informtion). The COVs for mirosle nture mterils VOL 6 JUNE

5 5 μm 2 μm 2 μm Inelsti equivlent strin 3 nm.74 μm x 2 x 1 Homogeneous inelsti region Heterogeneous inelsti region Homogeneous elsti region Homogeneous inelsti region μm 3 nm 1.64 μm º rottion 9º rottion 2.9 μm 18º rottion 27º rottion 2.5 μm d Inelsti deformed re (μm 2 ) Fore (μn) Hom. mp Het. mp Hom. mp Het. mp e Normlized energy dissiption Long one Hom. mp Het. mp Hom. mp Het. mp Figure 4 FEA simultions of the effet of nnomehnil sptil heterogeneity on lrger-sle iomehnil properties., 2D nothed four-point end FEA model geometry showing dimensions nd mesh. Modulus mps for dult ovine ortil one were superimposed in 2 μm 2 μm heterogeneous region hed of the noth. The simultion ws ontrolled with the ottom two oundry onditions (sped 2 μm prt)displed 2.5 μm upwrds from the ottom wheres the two fixed regions re t the end. Four-node iliner plne-strin qudrilterl elements (CPE4) were used in this study nd the mesh ws highly refined in the heterogeneous mpped region. There were 9,424 elements in the 6 nm nothed speimen; 3,116 of these elements were within the heterogeneous region. The numer of yielded elements ws trked through the equivlent inelsti strin nd the retion fore in the vertil diretion ws mesured t the fixed points. Energy dissiption ws lulted y integrting the re etween the loding nd unloding fore urves up to 2.5 μm.,snpshots of equivlent inelsti strin for the four-point end FEA virtul simultions ompring the homogeneous to the heterogeneous se for loding up to 2.5 μm. This prtiulr dtset used heterogeneous dt with the loding perpendiulr to the long one (Fig. 2)., Equivlent inelsti strin for the four-point end FEA virtul simultions, whih inorported the heterogeneous nnoindenttion dt tken with the loding perpendiulr to the long one (Fig. 2), where the 2 μm 2 μmmpis rotted in the plne of the long one (the plne of the pper) y, 9, 18 nd 27. d, The inelstilly deformed re is plotted s funtion of fore for eh set of tests, oth perpendiulr nd prllel to the long one, ompring the homogeneous nd heterogeneous ses. The heterogeneous plots re verges of the rotted nd shifted ontour-mpped regions, indited y stndrd error rs ( shifted simultions involved moving the noth 3 nm vertilly upwrds nd downwrds; see Supplementry Informtion). e, Energy-dissiption r plots for eh set of simultions, oth perpendiulr nd prllel to the long one, ompring the homogeneous nd heterogeneous ses where the vlues re normlized to the homogeneous se. The heterogeneous plots re verges of the rotted nd shifted ontour-mpped regions, indited y stndrd error rs. Aunit thikness of 1 μm in the 2D model is used to ompute the energy dissiption. indenttions (Berkovih geometry t 5 μn mximum lod) perpendiulr nd prllel to the long one were found to e.25 nd.26, respetively (methods re provided in our previous pulition 24 ). This is mrkedly less thn the nnosle dt presented here, suggesting sle-dependent homogeniztion effet. Other studies hve reported even smller mirosle COV vlues of.7.15 (refs 7,8,32). To etter understnd how nnosle mehnil heterogeneity influenes lrger-length-sle iomehnil properties, finiteelement simultions were rried out using the experimentlly 458 nture mterils VOL 6 JUNE 27

6 Inelsti equivlent strin 1. Fore (μn) 1, Homogeneous mp (114 pj) Heterogeneous mp (162 pj) 3 nm Depth (nm) 3 nm Inelsti deformed re (μm 2 ) 1. Homogeneous mp.8 Heterogeneous mp Fore (μn) Figure 5 FEA simultions of the effet of nnomehnil sptil heterogeneity on lrger-sle ompressive loding., 2D indenttion FEA model showing snpshots of equivlent inelsti strin for n indenttion virtul simultion ompring the homogeneous (top) with the heterogeneous (ottom) se for loding up to 85 μn. This prtiulr dtset used heterogeneous dt with the loding perpendiulr to the long one (Fig. 2). A unit thikness of 1 μm in the 2D model is used to ompute the energy dissiption., The fore depth urves for the homogeneous nd heterogeneous ses., The inelstilly deformed re s funtion of fore, ompring the homogeneous nd heterogeneous ses. determined fine-sle heterogeneity nd lol mehnil-property vritions extrted from AFM-sed indenttion experiments. One previous study reported finite-element nlysis of perfetly elsti heterogeneous mirostruture on the sis of men tissuelevel experimentl dt. In tht work, the indenttion elsti modulus deresed nonlinerly with inresing COV (rtifiilly indued in the simultion). In ddition, mrked inrese in the numer of filed elements (determined y n elsti-limit-strin riterion) ws oserved with inresing heterogeneity 16.Here,we diretly inorporte the experimentlly mesured heterogeneous nnosle-stiffness mps into highly refined region of finite elements just hed of nothed four-point end mesh (Fig. 4), so s to pture the loliztion of strin in region with nnosle struturl non-uniformity. Inelstiity ws lso inorported into the four-point end mesh s follows. Inelsti deformtion ws first ssessed during individul indenttion experiments y onstruting n ymmetri 2D FEA simultion with n equivlent tip pex ngle of 23.5, tht is, the sme indenttion volume versus depth s for the AFM proe tip, tht ssumed perfet von Mises inelsti onstitutive lw, s well s modulus evluted from the unloding slope 25. For numer of foreversus-depth indenttion urves, vrious tril yield-stress (σ Y ) vlues were hosen to determine the optiml vlue t whih the simulted dt fitted the experimentl urve. Eh individul estfit vlue (with n verge σ Y vlue of 395 MP) ws then used to lulte the orresponding equivlent (elsti) limit strin t the yield point s given y ε Y = (2(1 + ν)σ Y )/3E, whereν is Poisson s rtio nd E is the elsti modulus (ν ws tken to e.25). The ε Y vlues were found to e.4 ±.2 for speimens tested prllel to the long one nd.44 ±.2 for speimens tested perpendiulr to the long one. These oservtions suggest tht onstnt-elsti-limit-strin (inelstiyield-strin) riterion is resonle first-order pproximtion. Therefore, tking the ove verge vlue of ε Y, the lol yieldstress vlue t eh position within the heterogeneous mp, σ Y, n e otined onversely using σ Y = (3E ε Y )/2(1 + ν). The energydissiptive inelsti deformtion is inorported here euse finite residul depth is oserved fter unloding of the indenttion urves (Fig. 2,f). Strin-ontrolled miro- nd mro-sle filure of one hs lso een studied experimentlly using four-point end speimens with rounded nothes 33. Dmge preeding filure ws oserved to our immeditely t the noth tip (where mximum strin ours) s opposed to further hed of the noth tip (where mximum stress ours), supporting the onstnt-filure-strin hypothesis. The tensile yield ehviour of one hs lso een hrterized y ritil strin 34. As noted in erlier work 33, the inelsti onstitutive response of one is t present not fully understood. In ompression, nnogrnulr frition nd introrgni ohesion re thought to e mjor ontriuting ftors to the inelsti resistne of one initilly 2. Other mehnisms proly ome into ply t higher strins owing to the effets of ollgen firil sher 19 nd denturtion 24, rk initition/propgtion 33 nd srifiil ond rupture 17. Although one is known to show signifint tension ompression symmetry, plstiity models involving the von Mises yield riterion re typilly used to pture, t lest pproximtely, the stress/strin fields in one undergoing inelsti deformtion. In the spirit of developing detiled quntittive trends nture mterils VOL 6 JUNE

7 f (x,y) l (x) h (x) 2 x 2 x f L (x,y) f H (x,y) l (y) h (y) l (y) h (y) 2 y 2 y 2 y 2 y f LL (x,y) f LH (x,y) f HL (x,y) f HH (x,y) Figure 6 Wvelet deomposition of 2D imge 47. f(x, y) is the 2D imge, h(x) the high-pss filter in the x diretion, l(x) the low-pss filter in the x diretion, f L (x, y) the 2D imge following the low-pss filter, f H (x, y) the 2D imge following the high-pss filter, f LL (x, y) the 2D imge following the low-pss/low-pss filter omintion, resulting in the smoothed pproximtion, f LH (x, y) the 2D imge following the low-pss/high-pss filter omintion, resulting in the horizontl detil, f HL (x, y) the 2D imge following the high-pss/low-pss filter omintion, resulting in the vertil detil, f HH (x, y) the 2D imge following the high-pss/high-pss filter omintion, resulting in the digonl detil, nd 2( )x the downsmpling in the x diretion. for reltive ssessments of the role of struturl heterogeneity from the nno- to the mirosle, we invoke, s in erlier studies 33, plstiity nlysis (desried elow) in our finite-element model, with the full reliztion tht omprehensive onstitutive models for inelsti deformtion of one hve not yet een developed. These simultions, however, re seen to pture key trends of the inelsti deformtion of one when ompred with our prllel omputtionl studies tht invoke nisotropy nd pressure sensitivity of deformtion (s noted elow). Figure 4 shows the progression of equivlent inelsti strin (defined s the time integrl of 2/3 multiplied y the mgnitude of the inelsti strin rte) during two virtul four-point end FEA simultions ( ontrolled). The first (right olumn) inorported the heterogeneously mpped dt from Fig. 2 (loding perpendiulr to the long one ). The seond (left olumn) ws ontrol, whih ws n elsti perfetly plsti simultion with purely homogeneous mehnil properties (set to the volume-verged men vlue of modulus mesured experimentlly in Fig. 2 nd the volume-verged men vlue of yield stress otined from the strin-sed yield riterion). The mesured inelsti strin in the viinity of the noth is ontinuously greter for the heterogeneous se thn the homogeneous from 1.5 to 2.5 μm. The inelsti strin develops irregulrly hed of the noth nd extends out twie s fr for the heterogeneous mteril ompred with the homogeneous one t 2.5 μm. Non-uniform noth opening nd lunting took ple in the heterogeneous se, s ompred with more uniform shpe hnge for the homogeneous se. Figure 4 shows the inelsti equivlent strin for the four-point end FEA simultions t 2.5 μm for the sme nnoindenttion dt (Fig. 2), where the heterogeneous mp hed of the noth is rotted y,9, 18 nd 27 in the plne defined y the long one (in the plne of the pper). Although the fine detils of the inelstiity pthwys re different for different rottions, ll simultions showhrteristi heterogeneousdiffuse deformtion over muh greter re ompred with the homogeneous mteril. There is less thn 1% vrition in the inelstilly deformed re for omplete rottion over 36. The inelstilly deformed re s funtion of pplied fore is given (Fig. 4d) for ses oth perpendiulr nd prllel to the long one for oth heterogeneous nd homogeneous ses. For tests perpendiulr to the long one, the inelstilly deformed re ws out three times s gret for the heterogeneous thn the homogeneous se t 39 μn mximum fore. For tests prllel to the osteonl, the inelsti deformtion re ws out twie s gret for the heterogeneous se t 61 μn mximum fore. In generl, for 2.5 μm, the inelsti strin region orresponded to n inresed pplied fore for stiffer modulus vlues, on verge. A omprison etween mps tken with different loding orienttions showed similr energy dissiption, whih ws out 2.4 times greter for the heterogeneous mp ompred with the homogeneous se (Fig. 4e). Also, s the noth dimeter ws vried etween 3 nd 12 nm, there ws slight inrese ( 1%) in the inelsti strin region for heterogeneous nd homogeneous ses in oth orienttions (dt not shown). However, the reltive rtio of inelstilly strined res etween heterogenous nd homogeneous ses remined similr. Further FEA simultions were rried out inorporting nisotropy (using elsti onstnts from ultrsoni veloity tests 35 ) nd, seprtely, pressure sensitivity 2. These dditionl simultions (see the Supplementry Informtion) showed the sme overll trends s the foregoing elstoplsti simultions, of inresed inelstilly deformed re (out threefold) nd energy dissiption (out twofold) for the heterogeneous system ompred with the homogeneous se. Therefore, lthough the speifi extent of dmge evolution nd energy dissiption ould e somewht influened y the prtiulr onstitutive model, mteril nisotropy nd pressure sensitivity, it is evident tht the overll onlusions extrted from the present omputtionl simultions hold irrespetive of the prtiulr hoie of mteril model. Another importnt issue pertins to the ury with whih the 2D plne strin model ptures the ehviour of the 3D mirostruture. Indeed, signifint quntittive differenes would e expeted in the inelsti strin distriutions predited y the 3D model s ompred with the 2D se 36,37. Nevertheless, sttistilly, inelsti strin nd stress (for exmple mximum prinipl stress) distriutions from plne-strin model re in generl quite similr to those from full 3D model, with the heterogeneity slightly underestimted y the plne-strin model 36. These results 36,37 suggest tht the plne-strin model used here should e suffiiently urte to distinguish sttistilly the reltive differene etween the heterogeneous se nd the homogeneous se. An dditionl 2D FEA nnoindenttion model ws lso uilt with ompression s the hief loding mehnism (Fig. 5). Here, similr to the ending se, the modulus mp ws superimposed onto fine mesh with 9 inluded-ngle indenttion geometry to demonstrte tht the sme trends re oserved s with the fourpoint end simultions. Figure 5 shows tht the simulted foreversus-indenttion-depth urves for the homogeneous stiffness mp show smller deformtion for given lod ompred with the heterogeneous mteril (Fig. 2) for lods lrger thn 25 μn. The energy dissiptions given y the urves were 114 nd 162 pj for the homogeneous nd heterogeneous ses, respetively (for 85 μn mximum lod). The inelstilly deformed re for the heterogeneous se ws mrkedly greter thn tht for the homogeneous se (Fig. 5), onsistent with the results otined from the four-point ending simultions. This result further suggests tht, even though heterogeneity t the miro- nd mrosle my hve deleterious effets, leding to frture nd interfil instilities 38, inelsti heterogeneity t the nnosle offers n dvntgeous mehnism for dutile energy dissiption. In summry, we propose new energy dissiption mehnism for one rising from nnomehnil heterogeneity, y reourse to experimentl results tht provide high-resolution, sptilly speifi nnomehnil tests in onjuntion with detiled omputtionl simultions of lolly inelsti deformtion. These results provide ompelling evidene for enhned dutility nd 46 nture mterils VOL 6 JUNE 27

8 energy dissiption through nnosle struturl heterogeneity. This onept is generlly pplile to rod lss of nturl mterils euse nnomehnil heterogeneity is expeted to e uiquitously present. In ddition, euse osteoytes re ontinully smpling their mehnil environment 39, deformtion spred out over greter sptil res owing to nnomehnil heterogeneity my filitte dmge detetion in the extrellulr mtrix nd improved remodelling responses. The heterogeneous nnomehnil ptterns mesured experimentlly would in turn use orresponding lol heterogeneous strins when loded mrosopilly. Suh strins re expeted to e mplified y the softer surrounding periellulr mtrix of osteoytes 4, further ffeting ellulr proesses, possily ting s nnomehnil messges. Lstly, heterogeneous lol strins re lso expeted to influene interstitil fluid flow, whih hs een shown to e ritil to the proper mintenne of metoli tivity 41. METHODS SAMPLE PREPARATION AND CHARACTERIZATION Smples of dult ovine ortil one from the proximl tiil metphysis were prepred s previously desried 24. Smple preprtion did not involve hemil tretment, ethnol dehydrtion or emedding. The minerl ontent mesured y k-sttered eletron mirosopy ws found to e 58 wt%. The.. intermittent ontt-mode AFM imging t mient temperture nd humidity ws used with 3D moleulr fore proe (MFP-3D, Asylum Reserh) on hydrted smples of osteonl one using silion ethed OMCL-AC16TS (Olympus) retngulr ntilever with tetrhedrl proe tip (ntilever spring onstnt, k 56.2Nm 1, proe-tip end rdius 15 nm, Fig. 1) in mient onditions. The tip rdius ws mesured y..-mode imging of gold nnoprtile lirtion stndrd 42. NANOINDENTATION Nnoindenttion experiments were onduted in mient onditions using the MFP-3D nd the sme proe tip s tht used for..-mode AFM imging. The inverse optil lever sensitivity nd ntilever spring onstnt lirtions were rried out s previously reported 43. Displement-ontrolled nnoindenttion ws rried out y loding t rte of 1 μms 1 up to trigger fore of 5 μn followed y unloding t the sme rte. The loding/unloding rte ws vried from.5 to 1 μms 1 nd no sttistilly signifint differenes in lulted moduli were oserved t loding rtes etween.5 nd 5 μms 1, inditing tht visoelsti effets were minimized. The tip ontt-re funtion ws lirted with smple of glyol methrylte using 6 GP modulus from previous indenttion dt tken on Hysitron Trioindentor. Vlues for elsti moduli were lulted from 95 to 2% of the initil unloding urve y fitting these dt to ontt mehnil theory for n isotropi, elsti hlf-spe 25 for eh individul urve. Due to the ssumptions of this model 44,45 (for exmple isotropy) nd the hierrhil nnostruture of one, the extrted modulus is more ppropritely defined s n indenttion modulus in tht it reflets reltive mehnil stiffness nd does not produe solute quntittive vlues. However, the reltive sptil differenes in stiffness n e urtely ompred. The 1 nm interindent sping ws hosen to e suffiiently lrge for miniml interferene with neighouring residul inelstilly deformed zone s well s residul stresses, s verified y 2D elsti inelsti FEA simultions (desried in detil in the FEA methods setion). Additionlly, experimentl histogrm distriutions of the modulus tken t muh lrger interindent sping of 1 μm showed sttistilly similr modulus distriutions (p <.5). An epoxy ontrol smple (MMster-Crr) ws tested to ensure tht the rnge of stiffness vlues oserved ws outside the instrumentl stter nd the verged modulus mesured ws determined to e 2.9±.7 with COV of.23. WAVELET DECOMPOSITION ANALYSIS Wvelet trnsform enles loliztion of oth spe nd frequeny, hieved y dilting nd trnslting finite wvelet funtion to pture different frequeny rnges nd their sptil lotions 46. The disrete wvelet trnsform represents one-dimensionl smpled dt with limited numer of deomposition levels (frequeny sles). A set of low- nd high-pss filters is used for deomposing the dt into orse pproximtion nd residul detil. Wvelets lso enle prtiulr frequeny su-nds to e isolted within n imge, whih re treted s 2D signls 47. Figure 6 depits one-level deomposition of 2D imge. Filters re pplied in the x-diretion nd the results re down-smpled y deleting every other olumn. This yields two imges of pproximtely hlf the size of the originl, one ontining high-frequeny omponents of the rows nd the other ontining low-frequeny omponents. These two imges re then eh filtered down the olumns using the orresponding filters nd down-smpling the results long the rows. This results in the smoothed-pproximtion, the horizontl-detil, the vertil-detil nd the digonl-detil suimges. The proess is repeted on the smoothed-pproximtion suimge to otin the next level of deomposition. FINITE-ELEMENT ANALYSIS The elsti ontt rdius t mximum lod (using threshold of.1 logrithmi strin) nd the size of the residul inelsti strin zone (using threshold of.1 inelsti equivlent strin) nd residul stress zone (using threshold of.1 σ Y ) fter unloding were pproximted y n elsti perfetly inelsti 2D ymmetri-indenttion FEA model using the softwre pkge ABAQUS (Pwtuket, RI, USA) fitted to individul fore depth urves. The proe-tip geometry ws inorported from diret snning eletron mirosopy visuliztion. The modulus vlue ws fixed to the vlues otined from the Oliver Phrr nlysis on the sis of the initil unloding slope 25.The yield stress ws kept s free fitting prmeter, from whih the yield strin ould e itertively determined. More detiled disussions on the omputtionl indenttion modelling nd mehnil property extrtion n e found elsewhere 48,49. A 2D plne-strin four-point end model (4 μm wide nd 5 μm high) ws uilt (Fig. 4). The experimentlly mesured modulus mps were inorported within this FEA model for oth speimens with indents perpendiulr nd prllel to the long one y ssigning speifi mteril setion to whihever elements sptilly oinided with eh prtiulr mteril region. An elsti-limit-strin-sed von Mises plstiity onstitutive lw ws used to model the energy-dissipting inelsti deformtion nd to otin lol yield-stress vlue s desried in the min text. The elements in the re immeditely to the right nd left of the 2 μm 2 μm heterogeneous region were ssigned the experimentlly mesured verge homogeneous vlue of modulus nd orresponding yield stress (on the sis of the riterion disussed previously). The elements in the ottom portion were ssigned only the purely elsti verge homogeneous modulus vlue. Reeived 25 Septemer 26; epted 16 April 27; pulished 21 My 27. Referenes 1. Weiner, S. & Wgner, H. D. The mteril one: Struture mehnil funtion reltions. Annu. Rev. Mter. Si. 28, (1998). 2. Currey, J. 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Aknowledgements The uthors thnk the MIT Deprtment of Mterils Siene nd Engineering Nnomehnil Testing Fility, The Whitker Foundtion nd the US Army through the MIT Institute for Soldier Nnotehnologies (ontrt numer DAAD-19-2-D2), nd the NIH grnt 1-R1-GM on multisle modelling for funding. M.D. nd S.S. lso knowledge prtil support from the United Sttes Army Reserh Offie nd the Joint Improvised Explosive Devies Defet Orgniztion under ontrt numer W911NF The ontent does not neessrily reflet the position of the government nd no offiil endorsement should e inferred. The uthors would lso like to thnk grphi rtist Beryl Simon for preprtion of Fig. 2g. Correspondene nd requests for mterils should e ddressed to C.O. Supplementry Informtion ompnies this pper on Competing finnil interests The uthors delre no ompeting finnil interests. Reprints nd permission informtion is ville online t nture mterils VOL 6 JUNE 27