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Two-dimensional s aling properties of experimental fra ture surfa es 1 1 1 L. Ponson, D. Bonamy, and E. Bou haud 1 Fra ture Group, Servi e de Physique et Chimie des Surfa es et Interfa es, DSM/DRECAM/SPCSI, CEA Sa lay, F-91191 Gif sur Yvette, Fran e 6 (Dated: February 5, 2008) 0 0 The self-a(cid:30)ne properties of post-mortem fra ture surfa es in sili a glass and aluminum alloy 2 were investigated through the 2D height-height orrelation fun tion. They are observed to exhibit anisotropy. The roughness, dynami and growth exponents are determined and shown to be the n sameforthetwomaterials,irrespe tiveofthe ra kvelo ity.Theseexponentsare onje turedtobe a universal. J 5 PACSnumbers:62.20.Mk,46.50.+a,68.35.Ct ] h (a) Understanding the physi al aspe ts of fra ture in het- c e erogeneous materials still presents a major hallenge. m Sin e the pioneering work of Mandelbrot [1℄, a large 2 amount of studies have shown that ra k surfa e rough- - m 0 t ening exhibits some universal s aling features although n−2 a 1000 t it results from a broad variety of material spe i(cid:28) pro- 1000 s esses o urring at the mi rostru ture s ale (see Ref. [2℄ 800 800 at. for a review). Fra ture surfa es were found to be self- z (nm6)00400 400 600x (nm) m a(cid:30)ne over a wide range of length s ales. In other words, 200 200 ∆h(∆r)=<[h(r+ - ∆thre)h−eighh(tr-)h]2eig>h1t/ 2orrelationfun tion 0 0 d r ∆ ohm∼pu(t∆edr)aHlong a giHven dire tion is (b) n found to s ale as where refers to the o c HurHst≈ex0p.8onent. The roughness exponent was found to [ be , weakly dependent on the nature of the ma- 200 terialandonthefailuremode[3℄.Thisquantitywasthen µm 0 1 onje tured to be universal. −200 v 500 400 7 Sin etheearly90s,alargeamountoftheoreti alstud- 400 300 8 ies suggesteds enariosto explaintheseexperimentalob- 300 200 0 servations. They an be lassi(cid:28)ed into two main at- z (µm) 200100 100 x (µm) 1 egories: (i) per olation-based models where the ra k 0 0 0 propagation is assumed to result from a damage oales- 6 en epro ess[4,5℄;(ii)elasti stringmodelsthat onsider 0 FIG. 1: Topographi image of fra ture surfa e of pure sili a / the ra kfrontasanelasti linepropagatingthroughran- glass (a) and aluminum alloy (b). x is the dire tion of ra k t a domly distributed mi rostru tural obsta les [6, 7℄. The propagation. z is parallel to theinitial ra kfront. m fra turesurfa e orrespondsthentothetra eleftbehind this ra k front. - d All these models lead to self a(cid:30)ne fra ture surfa es ever, their s aling properties are not isotropi as usually n o withvariousexponents.However,noneofthem hasbeen believed but require the use of a tw∆oh-(d∆i~mr)en=sio<na[lh((2~rD+) c able to predi t the measured value of the roughness ex- ∆h~erig)h−t-hhe(~rig)h]2t > o1r/r2elation fun tion v: ponent. The main di(cid:27)eren e between the predi tions of ~r for a omplete des ription. This 2D these two ategories of theoreti al des riptions is that i des ription involves two independent s aling exponents X models(i)leadtoisotropi fra turesurfa eswhilemodels whi h orrespondtotheHurstexponentsmeasuredalong r (ii),wherethedire tionoffrontpropagation learlyplays the ra k propagation dire tion and the perpendi ular a a spe i(cid:28) role, predi t anisotropi surfa es. The analysis one,the ra kfrontdire tion.Theyarefoundtovaryin- of su h anisotropy on experimental examples is the en- signi(cid:28) antlyforthetwomaterialsandfromslowtorapid tral point of this paper. ra k growth. Su h observations are interpreted within We investigate the 2D s aling properties of fra ture the framework of elasti line models driven in a random surfa es in sili a glass and metalli alloy, representative medium. of brittle and du tile materials respe tively. Those were Experimental setup. - Sili a glass and a metalli al- broken using various fra ture tests (stress- orrosion and loy are hosen as the ar hetypes of brittle and du tile dynami loading).Fra turesurfa esobservedforallthese materials respe tively. materials/failure modes are shown to be self-a(cid:30)ne, in Fra tureofsili aisperformedonDCDC5(×d5o×ub2l5em lemav3- agreement with results reported in the literature. How- agedrilled ompression)parallelepipedi ( ) 2 samples under stress orrosion in mode I (see ref. [8℄ tudes but also in the Hurst exponen0t.s8.3A±lo0n.0g5the ra k for details). After a transient dynami regime, the ra k front0,.7th5e±e0x.0p3onentsarefound to be forsili a propagates at slow velo ity through the spe imen under and formetalli alloy,i.e.fairly onsisteζnt≃w0it.h8 stress orrosion.This velo ity is measured by imagingin the "universal" value of the roughness exponent realtimethe ra ktippropagationwithanAtomi For e widely reported in the literature [3℄. In the ra k growth Mi ros ope (AFM). In the stress orrosion regime, the dire tion,theHu0r.s6t3e±xp0o.0n4entis0f.o5u8n±d0t.o03besigni(cid:28) antly ra k growth velo ity an be varied by adjusting prop- smaller, loseto and forsili aglass erlythe ompressiveloadappliedtotheDCDCspe imen andthealuminumalloyrespe tively.Inallthefollowing, [8℄. The proto ol is then the following: (i) a large load is the Hurst exponentmeasured alongthe z-axis,the ra k appliedtorea hahighvelo ity;(ii)theloadisde reased frontdire tion,and the x-axis,the ra kpropagationdi- ζ β to a value lower than the pres ribed one; (iii) the load re tion, will be r(eaf)erred to as and respe tively. is in reased again up to the value that orresponds to the pres ribed velo ity and maintained onstant. This 101 pro edure10a−ll6ows1u0s−t1o1get −va1rious ra k velo ities rang- ing from to m.s orrespondingto zones on 100 (a) the post-mortem fra ture surfa es whi h are learly sep- arated by visible arrest marks. The topography of these 2) m10−1 fra turesurfa esisthenmeasuredthroughAFMwithan n in-plan5enamndout0-o.1f-npmlaneresolutionsestimatedoftheor- 2 ( derof and respe tively.Toensurethatthere h)10−2 ∆ is no bias due to the s anning dire tion of the AFM tip, ( Along the crack propagation direction ea h image is s anned in two perpendi ular dire tions 10−3 Along the crack and the analyses presented hereafter are performed on front direction b1o×th1iµmmages. These images represent a square (cid:28)eld of (1024 by 1024 pixels). 10−4 100 102 104 Fra ture surfa es of the ommer ial 7475 aluminum ∆x or ∆z (nm) alloy were obtained from CT ( ompa t tension) spe - imens whi h were (cid:28)rst pre ra ked in fatigue and then 104 brokenthroughuniaxialmodeItension.The ra kvelo - (b) ity varies during the fra ture pro ess, but has not been measured. In the tensile zone, the fra ture surfa e has 103 been observed with a s anning ele tron mi ros ope at 2) m twotiltangles.Ahighresolutionelevationmaphasbeen µ produ edfromthestereopairusingthe ross- orrelation 2 (102 ) based surfa e re onstru tion te hnique des ribed in [9℄. h ∆ rTeh etarneg uolnasrtr(cid:28)ue ldteodfi5m65ag×e4o0f5tµhme t(o5p1o2gbryap5h1y2rpeipxreelss)e.nTtshae (101 Aprloonpga gthaetio cnr adcirke c t io n Along the crack i2n−-pl3anµemand out-of-plane resolutions are of the order of front direction . 100 Experimental results. - A typi al snapshot of sili a 100 101 102 103 glass (resp. metalli alloy) fra ture surfa e is presented ∆x or ∆z (µm) in Fig. 1a (resp. Fig.1b). In both ases, the referen e frame (x, y ,z) is hosen so that axis x and z are respe tively parallel to the dire tion of ra k propagation FIG. 2: Height-height orrelation fun tion al ulated along and to the ra k front. The in-plane and out-of-plane the propagation dire tion and the ra k front dire tion on a hara teristi length s ales are respe tively of the order fra tu−r1e1surf−a 1e of sili a glass obtained with a ra k velo ity nm nm 10 of 50 and 1 for the sili a glass, and of the order of m.s (a)andaluminumalloy(b).Thestraightlines µm µm of 100 and 30 for the aluminum alloy. In or- are power law (cid:28)ts (see textfor details). dertoinvestigatethes alingpropertiesofthesesurfa es, ∆h(∆z) =< t[hh(ez1+D∆hze,igxh)t−-hhei(gzh,tx) ]o2r>re1l/a2tion fun tiozns The observation of two di(cid:27)erent s aling behaviors in z,x ∆h(∆x) =< [h(z,x+∆x)−ha(lozn,xg)t]2he>1/d2ire tion, anxd two di(cid:27)erent dire tions of the studied fra ture surfa es z,x suggestsanew approa hbased on theanalysisof the2D along the dire tion were omputed. Theyarerepresentedin Fig.2a he∆ighht(-∆hezi,g∆htx) o=rr<el[aht(iozn+fu∆nz ,txio+n d∆ex(cid:28))n−edha(sx:,z)]2 >1/2 z,x (resp. Fig.2b) for sili a glass (resp. metalli alloy). For . both materials, the pro(cid:28)les were found to be self-a(cid:30)ne This fun tion ontains informations on the s aling inbothdire tions.Moreover,these urvesindi atea lear properties of a surfa e in all dire tions. The variations ∆h∆x anisotropyofthefra turesurfa es.Thisanisotropyisre- of the orrelation fun tions are plotted as a fun - ∆z (cid:29)e tednotonly in the orrelationlengthsand theampli- tion of in the insets of Figure 3a and 3b for sili a 3 (a) 100 ra k growth velo ity over the whole range from ultra- 100 slow stress orrosion fra ture (pi ometer per se ond) to ζ m) rapid failure (some metersper se ond). The ratioof to n β h (∆x is given in the fourth olumn of Table I. Itzis=wζo/rtβh ∆ β to note that the exponent z ful(cid:28)lls the relation . ∆x 10−1 The sameexponentshavealsobeen observedon fra ture /∆x 1∆00z (nm1)01 ∆∆xx==01.n5mn m surfa esofmortarandwood[10℄.Theyaretherefore on- ∆h ∆x=1.5nm je tured to be universal. 10−1 ∆x=2.5nm ∆x=3.5nm ∆x=5.5nm ζ β ζ/β (a) ∆∆xx==81n2mnm ± ± ±z ∆x=19nm sili a glass 0.77 ± 0.03 0.61 ± 0.04 1.30 ± 0.15 1.26 ∆x=28nm metalli alloy 0.75 0.03 0.58 0.03 1.26 0.07 1.29 10−2 100 102 ∆z/∆x1/z TABLEI:S alingexponentsmeasuredonfra turesurfa esof (a) ζ β z ζ/β sili aglassandmetalli alloy. , , and arerespe tively 102 interpreted as the roughnesszexponent, the growth exponent 102 and the dynami exponent while the fourth olumn on- ζ β tains the ratio of to . Error bars represent an interval of h (µm)∆x101 on(cid:28)den eof 95 %. β ∆ ∆x101 100 ∆h/∆x 100 ∆z1 (0µ1m) 102 ∆∆∆xxx===123µµµmmm expDliosr eudsstihoen.2-DTsh aelienxgpeprriompeenrttsiersepoofrftread tiunrethsisurlfeat teers ∆x=6µm of two di(cid:27)erent materials. Three main on lusions an ∆x=9µm ∆x=16µm be drawn: (i) 1D pro(cid:28)les s anned parallel to the ra k 100 (b) ∆∆xx==2463µµmm front dire tion and to the dire tion of ra k propagation ∆x=73µm both exhibit self a(cid:30)ne s aling properties, but those are ∆x=120µm hara terized by two di(cid:27)erent Hurst exponents referred 10−2 10−1 100 101 102 ζ β ∆z/∆x1/z to as and respe tively; (ii) the 2D height-height orrelation fun tion is shown to ollapse on a single urve (Eq. 1) when appropriatlyres aled.Thiss alinginvolves ζ β z three exponents , and ; (iii) These three exponents FIG.3:Theinsetsshowthe2Dheight-height orrelationfun - ∆h∆x(∆z) ∆x ∆z areindependent of both the material onsideredand the tions orrespondingtodi(cid:27)erentvaluesof vs for a fra tur−e11surfa −e1of sili a glass obtained with a ra kve- ra k growth velo ity overthe explored range. 10 lo ity of m.s (a) and aluminum alloy (b). The data These on lusions enables a dis ussion on the vari- ollapse wasobtainedusingEq.1withexponentsreportedin ous ompeting models developed to apture the s al- Tab. I. ing properties of fra ture surfa es [4, 5, 6, 7℄. The anisotropy learly eviden ed in the s aling properties of fra ture surfa es annot be aptured by stati models glass and aluminum alβloy fraz ture surfa es respe tively. like per olation-based models [5℄. On the other hand, Foradequatevaluesof and ,it anbeseeninthemain these results are reminis ent to what is observed in ki- graphs of that same (cid:28)gure that a very good ollapse of neti roughening models [11℄. These models onsider the the urves an be ob∆txa1in/zed by∆noxrβmalizing the abs issa time evolution of an elasti manifold driven in a rahn(dzo,mt) andtheordinateby and respe tively.There- medium. The roughness development of the line h(z,t = 0) = 0 sulting master urveis hara terizedby aplateau regioζn starting from an initially straight line and followed by a power law variation with exponent . is then hara terized by a 1D height-height orrelation ∆h(∆z,t) In other words: fun tion that s ales as [11℄: ∆h(∆z,∆x)=∆xβf(∆z/∆x1/z) ∆h(∆z,t)=tβg(∆z/t1/z) where f(u)∼(cid:26)1uζ if u≪≫11 (1) where g(u)∼(cid:26)u1ζ if u≪≫11 (2) if u if u β z ζ β z The exponents and whi h optimize the ollapse, where , and refer to the roughness,growth and dy- ζ andthe exponentdeterminedby(cid:28)ttingthelarges ales nami exponents respe tively. Signature of this rough- regimeexhibited bythemaster urvesaζre≃lis0t.e7d5inβT≃a0b.l6e nening s aling an also be found in the steady state I. Thze t≃hre1e.2e5xponentsarefound to be , regime rea hed at long times when the roughness be- and , independent of the material and of the omestimeinvariant.Inthisregime,the2Dheight-height 4 ∆h(∆z,t) orrelationfun tion isexpe tedtos aleas[12℄: propagatingin a linear elasti solid, the restoring elasti for es are long range rather than lo al [17℄. Correspond- ∆h(∆z,∆t)=∆tβf(∆z/∆t1/z) ing elasti line models predi t logarithmζi ≃ 0o.r7r5elations [7℄, whi h is signi(cid:28) antly di(cid:27)erent from as re- where f(u)∼(cid:26)1uζ if u≪≫11 (3) appoprtleiedditnotthhiesiζnpta≃epref0ar.. 3i9Lalet ruas knportzoeb≃tlhema0t.7lte5haedssatomaermouogdhe-l if u ness exponent [18℄ and ζ ≃ 0.6[15, 1z9℄,≃wh1.i2le experiments [20℄ reported values and . whi h is exa tly the s aling (1) followed by the exper- t These experimental values are mu h loser to the ones imental surfa es after time has been repla ed by o- x expe ted in elasti line models with short rangζe≃ela0s.t6i3 ordinate measured along the ra k propagation dire - intera tions, that predi t roughness exponents tion. This provides a rather strong argument in favour [21℄. Understanding the origin of the intera tion s reen- of models like [6, 7℄ that des ribe the fra ture surfa e as ingin ra k problems providesasigni(cid:28) ant hallengefor thejuxtapositionofthesu essive ra kfrontpositions- future investigation. modelledasapseudoelasti line-movingthroughmate- Finally,it isworthtomention thatthes alingproper- rials with randomly distributeζd≃lo0 .a7l5toughβne≃ss0..I6n this ties exhibited by fra ture surfa es may have interesting s enario,the hurst exponents and mea- expertise appli ations. It allows indeed to determine the suredalongthe ra kfrontdire tionand thedire tionof dire tion of ra k propagationfrom the analysis of post- ra k propagation respe tively oin ide with the rough- mortem fra ture surfa es and, thus to re onstru t the nessexponentandthegrowthexponentasde(cid:28)nedwithin history of the pro esses that have lead to the failure of the framework of elasti string models [11℄. Let us note z the stru ture. moreoverthatinsu hmodels,thedynami exponent is ζ β z =ζ/β expe ted to be related to and through [13℄. z = 1.25 This leads to a value of in perfe t agreement A knowledgments with the value measured experimentally. ζ β In elasti line models, the set of exponents , and z depends only on the dimensionality [11, 14℄, the range We thank H. Auradou, J. P. Bou haud, E. Bou h- of the elasti intera tion [14, 15℄ and, to some extent, on binder, C. Guillot, J. P. Hulin, G. Mourot, S. Morel, I. the line velo ity [16℄. It has been shown that for ra k Pro a ia and S. Sela for enlightening dis ussions. [1℄ B.B. Mandelbrot,D.E. Passoja, andA. J. Paullay,Na- surfa e growth (Cambridge UniversityPress, 1995). ture308, 721 (1984). [12℄ M. Kardar, Phys.Reports 301, 85 (1998). [2℄ E. Bou haud, J. Phys.Condens. Matter 9, 4319 (1997). [13℄ F. Family and T. Vi sek, Dynami s of fra tal surfa es [3℄ E.Bou haud,G.Lapasset,andJ.Planès,Europhys.Lett. 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