Table Of ContentTwo-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 re-
spe
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
or-
relation 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.
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ity [16℄. It has been shown that for
ra
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