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From Quasi-Static to Rapid Fracture
E. Bouchaud, S. Navéos
To cite this version:
E. Bouchaud, S. Navéos. From Quasi-Static to Rapid Fracture. Journal de Physique I, EDP Sciences,
1995, 5 (5), pp.547-554. �10.1051/jp1:1995150�. �jpa-00247080�
Classification
Physics
Abstracts62.20Mk 05.40+j
81.40Np
Short Communication
From Quasi.Static to Rapid Fracture
E. Bouchaud and S. Navéos
O.N.E.R.A.
(OM),
29 Avenue de la DivisionLeclerc,
B-P. 72, 92322 ChâtillonCedex,
France(Received
7February 1995, accepted
24February 1995)
Résumé.
Quatre profils
de rupturecorrespondant
à quatre vitesses depropagation
de fissure différentes sont étudiés sur le mêmeéchantillonj
et révèlent l'existence d'unelongueur
de coupure( qui
décroît avec la vitesse. Pour des échelles delongueur supérieures
à (, on retrouve unexposant de
rugosité
voisin de(
ct 0.84. Aplus petite échelle, l'exposant
mesuré est en accordavec une
hypothèse
depropagation quasi-statique (QS)
du front defissure,
et vaut(~s
Ù 0.45.Abstract. Four fracture
profiles corresponding
to four diOEere1~t crack velocities are studiedon trie same
sample
and show trie existence of a crossoverlengthscale ( decreasmg
with the crackvelocity.
Forlengthscales larger
thon(,
trie previouslyreported roughl~ess
index ( ct 0.84 is recovered. Atlengthscales
smaller than(,
trie fractureprofile
lits witha
quasi-static (QS) hypothesis,
1-e-, trie measuredroughness
index is close to(~s
CÎ 0.45.It is now
clearly
established that fracture surfaces can be considered as self-affineobjects.
After trie
pioneering
work of Mandelbrot et ai.[l],
manyexperiments using
variousexperi-
mentaltechniques (profilometry [2,3],
microscopy andimage analysis [4-10], scanning tunnel-
mg electron microscopy(STM) [11], electrochemistry [12], etc.)
on materials as dilferent assteels
il,5,10,12],
aluminiumalloys [7],
rocks[3],
intermetalliccompounds [8,9]
or ceramics[4],
bave shown that fracture surfaces exhibitscaling properties
on two[2,8,9]
or three decades [7]of
lengthscales.
At"large enough" lengthscales (trie
micron scale for metallicmaterials),
forrapid
crackpropagation (~'uncontrolled
fracture"),
allreported
values of trieroughness
index(or
Hurstexponent) (
are close to 0.8. It wassuggested
[7] that this valuemight
be "uni-versal",
i-e-,independent
of trie fracture mode and of trie material(see
also[2]). However, significantly
smaller values are measured either at very smalllengthscales (nanometers),
or in the case of slow crackpropagation.
As a matter offact,
STMexperiments il1] report
values of theroughness
index close to 0.6 in trie case of fracturedtungstene (regular stepped region),
and close to 0.5 for
graphite iii].
On the otherhand,
lowcycle fatigue experiments
on asteel
sample
have led to a value of(
close to 0.6[12].
These results areparticularly
attractivesince
they
reportroughness
indices close to theroughness
of a minimum energy surface in ar&ndom environment
[13,14].
It wassuggested by Chudnowsky
and Kunin[15],
Kardar[16],
Q
Les Editions dePhysique
1995 iowNALD8mYsiQu8L-T. s, NOS.MAY199s 21548 JOURNAL DE
PHYSIQUE
I N°531.0 mm
in tension Fatigue
4.4 6
Profile
4Pr°fl'e 3 ~~°fi'~
~~~°fi~~ applicatiol~
Fig.
l. Sketch of trie brokensample, showing
the fourprofiles
which have beeninvestigated.
Profile 1 is in triefatigue
fracture zone, and thuscorresponds
to trie lower crackvelocity.
Profile 4corresponds
to trie "instantaneous fracture" zone, for which trie crack
velocity
is muchhigher.
and
by
Roux andFrançois
on the basis of a fracture model for porous ductile materialsil?]
that the
path
chosenby
a crack in a random environment should be such that the overall fracture energy is minimised. Thisquasi-static assumption,
whichclearly
cannot be fulfilled forrapid
crackpropagation, might
be valid at smallenough lengthscales
or for lowenough
velocities.
However, although
our results arecompatible
with theprediction
of thismodel,
we shall propose in thefollowing
an alternative'~quasi-static" description,
which should lead to acomparable exponent,
but which should be doser to the actual crackpropagation
mechanism.In this
letter,
we show that aquasi-static
mechanism is valid indeed up to a distance(
which decreases withincreasing
crackvelocity.
The existence of this new crossoverlength
and theanalysis
of the smaiiiengthscales
behaviour are the central result of this paper.A notched CT
sample (dimensions
12.5 x 30 x 31mm~,
seeFig. l)
of theSuper
a2alloy [18]
T13Al-based
is firstprecracked
infatigue
at 30Hz,
with a fixed ratio R of 0.1(the applied
ioad oscillates between a maximum load
P~~~
and a minimum loadP~~~/10
at afrequency
of 30 Hz withPm~~
= 2750N),
in order that the totallength
of the crack after thefatigue
test is close to60%
of thesample length.
Fracture is achievedthrough
uniaxial tension(mode I,
see
[19]
forexample).
Note that the microstructure of thealloy
iscomposed
of a2(ordered phase)
lath in afl (disordered phase,
stable at temperatureshigher
than lllo°C)
matrix.The brittle needles vary both in
thickness, length
and orientation.Plasticity
of thefi-phase
was shown to
play
animportant
role in the fractureproperties
of the material. The brokensample
iselectrochemically nickel-plated (the
thickness of thedeposit being approximateiy
100
microns).
Fourprofiles
are obtainedby subsequently cutting
andpolishing
thesample perpendicularly
to the direction ofpropagation
of the crack(see Fig. l). Only
the crackvelocity corresponding
to thefatigue profile (1)
could beestimated,
since the crack increasedby only
one millimeterduring
the last part of thefatigue
test,corresponding
to 13000cycles.
This
velocity
is close to 2 micrometers persecond,
1-e-, 5 x10~
times the soundvelocity Cs
in the material(Cs
ct 4700m/s),
whereas in the uncontrolled fracture zone, thevelocity
isexpected
to saturate at a value which is at least 0.20.3Cs.
These
profiles
are observed with ascanning
electron microscope Zeiss DSM 960 at variousmagnifications
10 to 12images
were made for eachprofile
withmagnifications ranging
from100
~
'
siope 046
~ d
$ ~i
$~
~ ~~
o -
~
i
~
~' l
Îi
~.
a
o-1 0
0.01 100
r (micrometers)
Fig.
2.zmax(r)
as a function of r(see Eq. (l)
for a. definition ofzmax(r)). Averaged expenmental points
areplotted
with error bars computed from the variance ofexperimental
results obtained fromvarious
micrographs (at
trie same or at dioEerentmagnifications).
Trie continuous finecorresponds
to trie 3-(profile
1) or 2-parameter non-hnear curve fit(see Eq. (2)),
with(~s
fixed to trie value 0.45.Profile 1: A
= 0.56 + 0.02; B
= 0.28 + 0.01; ( = 0.838 + 0.007;
fi
CÎ 5 ~lm;zmax(r
=fi
Ù 2.2 ~lm.x50 to x3000. Backscattered electrons are used in order to
give
a better contrast between triealloy
and trie nickeldeposit. Images
in 256 grey levels areregistered through
a Kevex Deltasystem,
and sent to an IBM PC486-33,
where trieimage segmentation
isperformed using
thesystem Synoptics Synergy
Board. The obtainedbinary images (the weight
of eachpoint
locatedon the
profile being
1, theweight
of any otherpoint being 0)
oflength
703pixels
are sent to a workstation where their various statisticalproperties
arecomputed.
When theprofile
is branched withsecondary cracks,
both the whole structure and its backbone are considered. In this letterhowever, only
the results concerning the backbones(including
non-branchedprofiles)
are
reported.
It was shown in various occasions that a
particularly
reliablequantity
to be measured on a self-affineprofile
in order to determine itsroughness
index(
is the average maximumheight z~~x(r),
which is defined as follows [20]~~'~~~~~ "~
~~~l~(~')l~<r'<~+r Ml~lZ(r')lz<r'<z+r
>~cor~ (1) z~ax(r)
iscomputed
on eachmicrograph
forprofiles
to 4.In the case of
profile
4(rapid
fracturezone),
ail theanalysed micrographs present
a power lawregime'extending
over two to threedecades,
for which theexponent
remains close to 0.8.In the case of
profile
1(slow
crackpropagation zone),
on thecontrary, micrographs
athigh magnification
alsopresent
a power lawincrease,
but theexponent
issignificantly smaller, lying generally
between 0.4 and 0.6. In the four cases,z~a~(r)
isaveraged
over the results obtainedfrom the various
micrographs.
Error bars are estimated from the variance of the distribution ofpoints coming
from the results relative to the variousmicrographs.
The behaviour of the average curve relative to
profile
1(fracture
infatigue)
at smaller distances is firststudied, showing
a power law increase with anexponent
m0.46(see
inset ofFig. 2),
1-e-,remarkably
close to the theoreticalroughness
index(~s
of a minimum energysurface [13,14].
Thenz~~~/r(Qs,
with(~s
=
0.4,
0.45 and 0.5 isplotted against
r, and the threeplots
are fitted with theKaleidagraph°/~
non-linear curve fitusing
theexpression
~~~~
= A + B
~(-(QS (2)
r(QS
Similar results are obtained using the
Xvgr
non-linear curve fit.550 JOURNAL DE
PHYSIQUE
I N°5Table I.
Fatigue fracture (profile 1,
seeFig.
l):
resultsof
the non-linearjitting of zm~~(r).
(~s assuming
the ~alueso-1, 0.$5 (see Fig. 2)
and0.5, zm~~(r)
isjitted according
to equa- tion(2).
Thequasi-static
blob sizefi
is determinedaccording
taequation (3)
and isezpressed
in
microns,
r is theconfidence
ratio. Errer bars areonly resulting from
thefit.
Fatigue
fracture: results of the non-linearfitting
ofzm~~(r)
(~s
A B( fi
r0.4 0.51+0.02 0.34+0.01 0.815+0.005 3 0.992
0.45 0.56 + 0.02 0.28 + 0.01 0.838 + 0.07 5 0.987
0.5 0.64 + 0.02 0.20 + 0.01 0.875 + 0.009 10 0.977
Expression (2)
is thesimplest
to account for theasymptotic
power law behaviourcorrespond- ing,
at shortdistances,
to aquasi-static
fracture mode power law with aroughness
index(~s
and at
larger distances,
to arapid
fracture mode power law with aroughness
index( yet
to be deterrnined. The real crossover function is
certainly
morecomplex, but,
as will be seen in thefollowing,
tl~isassumption
is not too far fromreality, especially
forprofiles (fatigue)
and 4
(~'uncontrolled
fracture" )~ which are closer to theasymptotic
cases.Furthermore,
this allows us to define the crossoverlength
(~ forprofile
between thequasi-static
fracture zone and therapta
fracture one as thelength
at which the twoasymptotic
terms areequal, 1e.,
i
. exp~ç ~ç~~~
in
iii1 (3)
The fit obtained for
(~s
= 0.45 is shown inFigure
2. Results concerning the threefollowing
sets of results obtained are summarised in Table I.
Consistency
withpreviously
measured values of(
forhigh velocity
cracks[7-9],
as well as the short-distance power law behaviour(see
inset ofFig. 2)
favours a value of(~s
close to0.45;
(qs
" 0.5 leads to aparticularly high
value of(. Subsequently,
the curves relative toprofiles
2to 4 are also fitted
according
toequation (2),
but(
iskept equal
to itspreviously
determinedvalue,
while A and B are the results of thefitting procedure.
Values of (~(i
=
2, 3, 4)
areagain
determinedthrough equation (3).
The results are summarized in Table II. Fitscorresponding
to the value
(~s
= 0.45 are shown inFigures
3 to 5 forprofiles
2 to4, respectively.
As a matter of
fact,
the actual values of (~ are very sensitive to the value of(~s,
for whichthe
precision
is rather bad because of the too fewexperimental points
at shortdistances, although,
in the case ofprofile 1,
one con determine a '~short distance"exponent
close to 0.46.On the other
hand,
it is clear that (~ decreases when the local stressintensity
factor K or,correlatively,
the crackvelocity
increases.Lying
in the micrometer region for fracture infatigue (between
3 and 5 pm, see TableII),
it could decrease below 1 micron at the end of the fracture process, in the unstable crackpropagation regime,
for which the crackvelocity
ismuch
higher.
Although
the"quasi-static"
measuredexponent
isremarkably
close to the minimumsurface
ezponent, an alternative
description
seems moreappropriate
to describe crack frontpropagation
during fatigue loading.
As a matter offact,
it wasrecently suggested
that the fracture surfaces could be rnodelled as the trace of a litrepropagating
at a non-zerovelocity
V in a random environment[22, 23].
From this picture, andusing
the results of Ertas and Kardar[24]
for the motion of vortex lines indirty superconductors,
one finds that thehigh velocity
fractureTable II. (~ is the crosso~er
length for profile1 ezpressed
inmicrometers,
determinedby equation (3).
One con see that(;
is rather sensiti~e ta theimposed
~alueof (QS. fi
is de-termined
through
athree-parameter
non-linear curuejitting (Fig. 2),
whilefor1= 2,3,4,
(~is determined
through
atwo-parameter
curuejitting (Figs. 3-5), ( being kept
ta the ~alue de- terminedfrom
theanalysis of profile
1(Fig. 2).
One con note that the ~alueof (
is not ~ery sensiti~e ta the ~alueof (Qs.
Crossover
lengths
for the variousanalysed profiles
(QS ( ~l
~2 ~3 ~40.4 0.815 3 2 0.6 0.05
0.45 0.838 5 2 0.3
0.5 0.875 10 6 3
ioo ioo
Z
$É
~ o
Û( .
-
_
#
'«
0.01
0.01 r
3.
surface hould anisotropic,
with a index to the of rack
propagation (~ ci
0.75
in acertain
regime,which
is rather close to thereported
here indeed measured to propagation). Interestingly, however, hereis
alsoa
lowvelocity
egimefor this problem,
whereioo ioo
Gi É
E Î
2i
a
0.01 o-i
r
4.
552 JOURNAL DE
PHYSIQUE
I N°5ioo
Z
ÎÉ
~
(
~'
C
~'w fi
0.01
r
ig. 5.
ransition, elow the rack ould
not propagateat ail. Ertas and Kardar recently
investigated
this regime [25], with the resultthat
((s = ). urthermore, this low-velocitybehaviour
is predictedto
hold below a velocity pendent ngthdecreasing
as(
V~~,with
çi
= 3, in agreementwith
ourobservations.
The
of a
during
fatigue sting for ather shortcrack lengths
bears some resemblance withthe
pinningtransition
since the
ovement of the frontis
highlydiscontinuous: when the load
is
it can be trapped on icrostructuralobstacles if
its
velocity is not highnough,
and
the next ncrease
of
the load to be ableto
get ridof this
pinning. Aransition (the crack is onstrained to lie within the z
=
0 lane) iscurrently
eingstudied by
Roux
et
ai.[26, 27].
Similar are
also
erformedplated,
for
whichlarger(1024
x 1024)images
areegmented withthe
Visilogsystem.
The previousresults are
duly confirmed. Quantitative observations
with
an atomicforce
microscope(AFM)
are underway
inorder
to explore more ofOn
the other
hand, further xperiments in fatigue, forwhich
thecracklength and average
crack velocity -
which,
asalready
ointedout
above, might besignificantly
smaller
thon theactual
crack -
conbe measured
lectrically,should
provide aprecise
determination ofthe relationship
between
and the localstress
intensityfactor
K.Note
that
another kinetic property of fracturesurfaces
has beenby chmittbuhl et
ai.
[28].In
fact, these authors wereinterestedin the
lation
length
with the
distance y
to
the initial notch - as in our no quantitativedetermination
of
the crack velocityould
be erformed -, i
with the upper
limit
(cof
~'long istance
regime" we
describe.
(c isshown
to with y as a power aw, y",with
close
to0.83
(note that,in
their experiment as well as
ere, no determination
of
the crack velocity
could be
hieved). Itwas
checked
that this
result
isperfectly
compatiblewith ours.
On
the ontrary,we
are interested in the lower limit (of the
decreases with ncreasing y.
The roughness
of
racture
One
can
also emphasize the importanceof
disorder in that espect: for
homogeneous
one often observes
very
flatfracture
urfaces at low crack velocities, referredto as
Higher crack velocities lead to rougher fracture urfaces, but it
seems
now that this rougheningis
due to annstability
[21]
whichhas
never een observed on metallicalloys,to
edge.
In the latter
case,
urther crack orphologytransformations only concern
secondary
crack
ranching 8,9,19,22,23], whichincreases with the
crack
velocity. omparison ofbath
disordered and more
homogeneous
materials at variouslengthscales
from the nanometer tothe micrometer
scale~ using
both STM orAFM,
and standard electronor
optical microscopy
couldcertainly help
to draw a real"phase diagram"
for fracture.In
conclusion,
it has been shown that there exists a crossoverlength (
which decreases withincreasing
cracklength,
andcorrelatively~
withincreasing
stressintensity
factor and crackvelocity,
whichseparates
two fractureregimes.
Atlengthscales higher
than(,
thepreviously reported roughness
index(
ce 0.8 ismeasured,
while the smalllengthscale (< ()
behaviour lits with apinning/depinning
mechanismhypothesis,
for which(
m 0.45.Finally,
it should be noted that these two fractureregimes
characterizedby
two fixedrough-
ness indices and
separated by
a crossoverlength
whichdepends
on the crackvelocity
cari bemisinterpreted
as aunique regime
characterizedby
a fractal dimensioncontinuously
varyingwith the
velocity.
A similar confusion was made in the pastby considering
that therough-
ness
exponent
would varycontinuously
with the fracturetoughness KIC,
while it wasrecently
shown that it is the correlation
length
of the self-affine fracture surface which in some cases isa function of KIC
(29].
Acknowledgments
Fracture
experiments
were achieved in collaboration with A.Lemoine,
G.Lapasset
and M.Thomas.
Enlightening
discussions with J.-P.Bouchaud,
M. Thomas and G.Lapasset
aregratefully acknowledged.
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