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>
HAL Id: jpa-00247080
https://hal.archives-ouvertes.fr/jpa-00247080
Submitted on 1 Jan 1995
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J.
Phys.
France
I
(1995)
5
547-554
MAY
1995,
PAGE
547
Classification
Physics
Abstracts
81.40Np
05.40+j
62.20Mk
Short
Communication
From
Quasi.Static
to
Bouchaud
Navéos
E.
and S.
(OM),
O.N.E.R.A.
(Received
7
Quatre profils
avec
étudiés
sont
décroît
exposant
une
trie
crack
la
avec
sur
de
vitesse.
Four
same
fracture
sample
velocity.
Pour
hypothesis, 1-e-,
lengthscales
trie
measured
24
B-P.
72, 92322
Châtillon
Cedex,
France
February 1995)
correspondant à quatre vitesses de propagation de fissure
échantillonj et révèlent
l'existence
d'une longueur de coupure
des
échelles de longueur
supérieures à (, on
retrouve
un
0.84. A plus petite échelle, l'exposant
mesuré
accord
est
en
quasi-statique (QS) du front de fissure, et vaut (~s Ù 0.45.
profiles
show
lengthscales
For
At
and
Leclerc,
Division
rupture
le même
rugosité voisin de ( ct
hypothèse de propagation
recovered.
is
de la
de
Abstract.
on
Avenue
Fracture
February 1995, accepted
Résumé.
différentes
( qui
29
Rapid
trie
larger
smaller
corresponding
existence
(,
(,
trie
index
is
thon
than
roughness
of
trie
a
to
four diOEere1~t
crack
velocities
are
studied
lengthscale ( decreasmg with the
previously reported roughl~ess index ( ct 0.84
fracture
profile lits with a quasi-static (QS)
close to (~s CÎ 0.45.
crossover
fracture
surfaces
clearly established
that
be
considered
self-affine
objects.
can
as
pioneering work of Mandelbrot
experiments using various experiet ai. [l], many
mental
techniques (profilometry [2,3], microscopy and image analysis [4-10], scanning tunnel(STM) [11], electrochemistry [12], etc.) on materials as dilferent as
electron
microscopy
mg
aluminium alloys [7], rocks [3],
steels il,5,10,12],
intermetallic
compounds [8,9] or ceramics [4],
decades [7]
fracture
surfaces
exhibit scaling properties on two [2,8,9] or three
bave shown that
materials),
for
of lengthscales. At "large enough" lengthscales (trie
scale
for
metallic
micron
rapid crack propagation (~'uncontrolled fracture" ), all reported values of trie roughness index
(or Hurst exponent) ( are close to 0.8. It was suggested [7] that this value might be "universal", i-e-, independent of trie fracture mode and of trie material (see also [2]). However,
significantly smaller values are measured either at very small lengthscales (nanometers), or in
the case of slow crack
propagation. As a matter of fact, STM experiments il1] report values
of the roughness index close to 0.6 in trie case of fractured
tungstene (regular stepped region),
On the
other hand, low cycle fatigue
experiments on a
and close to 0.5 for graphite iii].
results
particularly
attractive
steel sample have led to a value of ( close to 0.6 [12]. These
are
surface in a
they report roughness indices close to the roughness of a minimum
energy
since
[13,14]. It was suggested by Chudnowsky and Kunin [15], Kardar [16],
r&ndom
environment
It
is
After
Q
Les
now
trie
Editions
de
Physique
1995
iowNALD8mYsiQu8L-T.
s,
NOS.MAY199s
21
JOURNAL
548
DE
31.0
I
N°5
mm
Fatigue
tension
in
PHYSIQUE
4.4
Profile
Fig.
Sketch
l.
1 is in
trie
trie
to
and
that
by
the
fracture
for
rapid
velocities.
shall
Pr°fl'e
4
of trie
fatigue
zone,
fracture"
and
François
by a
path
chosen
energy
crack
is
~~°fi'~
3
on
the
crack
minimised.
This
basis
in
a
of
propose
the
following
an
fracture
a
random
applicatiol~
alternative
model
environment
for
porous
should be
ductile
such
which clearly
enough
lengthscales
at
compatible with the prediction
quasi-static
propagation, might be valid
However, although our results are
in
~ ~~°fi~~
sample, showing the four profiles which have been investigated. Profile
Profile 4 corresponds
and thus corresponds to trie lower crack velocity.
for which trie crack velocity is much higher.
zone,
broken
fracture
"instantaneous
Roux
6
assumption,
small
'~quasi-static"
description,
the
overall
cannot
be
fulfilled
for
low
or
which
il?]
materials
that
of this
should
enough
model, we
lead
to
a
comparable exponent, but which should be doser to the actual crack propagation mechanism.
mechanism
is valid indeed up to a distance ( which
In this letter, we show that a quasi-static
of this new
length and the
The
existence
decreases
with increasing crack velocity.
crossover
the
result of this
analysis of the smaii iengthscales behaviour
central
are
paper.
A notched
CT sample (dimensions 12.5 x 30 x 31 mm~, see Fig. l) of the Super a2 alloy [18]
T13Al-based
first precracked in fatigue at 30 Hz, with a fixed ratio R of 0.1 (the applied
is
ioad
oscillates
between a
load P~~~ and a
load P~~~/10 at a frequency of
minimum
maximum
2750 N), in order that the total length of the crack after the fatigue test
30 Hz with Pm~~
Fracture is achieved through
uniaxial
is close to 60% of the sample length.
tension (mode I,
of the alloy is composed of a2 (ordered
microstructure
[19] for example). Note that the
see
phase) lath in a fl (disordered phase, stable at
higher than lllo °C) matrix.
temperatures
brittle
needles
both in thickness, length and
The
Plasticity of the fi-phase
orientation.
vary
shown to play an
important role in the fracture properties of the material.
The broken
was
sample is electrochemically nickel-plated (the thickness of the deposit being approximateiy
Four profiles are
obtained
by subsequently cutting and polishing the sample
100 microns).
perpendicularly to the direction of propagation of the crack (see Fig. l). Only the crack
velocity corresponding to the fatigue profile (1) could be estimated, since the crack increased
during the last part of the fatigue test, corresponding to 13000 cycles.
millimeter
by only one
This velocity is close to 2
micrometers
the sound velocity Cs
per second, 1-e-, 5 x 10~ times
material
(Cs ct 4700 m/s), whereas in the
uncontrolled
in the
fracture
the velocity is
zone,
expected to saturate at a value which is at least 0.2
0.3Cs.
with a scanning electron
These profiles are
observed
Zeiss DSM 960 at
various
microscope
magnifications
10 to 12 images were
made for each profile with magnifications ranging from
=
N°5
QUASI-STATIC
FROM
~
FRACTURE
RAPID
~$~
'
~
siope
d
$
TO
549
100
046
~i
~
~~
o
-
i
~
~'
l
Îi
~.
a
o-1
0.01
0
100
(micrometers)
r
(see Eq. (l)
for a.
definition
of zmax(r)). Averaged
expenmental
as
a
r
plotted with error bars computed from the variance of experimental
results
obtained
from
various
micrographs (at trie same or at dioEerent magnifications). Trie continuous
fine corresponds to
trie 3- (profile 1) or
2-parameter
non-hnear
fit (see Eq. (2)), with (~s fixed to trie value
0.45.
curve
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.
Fig.
zmax(r)
2.
points
function
of
are
=
=
=
=
give a better
between trie
contrast
registered
through
Delta
Kevex
in
are
a
segmentation is performed using the
system, and sent to an IBM PC 486-33,
system Synoptics Synergy Board. The obtained binary images (the weight of each point located
the profile being 1, the weight of any other point being 0) of length 703 pixels are sent to
on
workstation
their
where
statistical
various
When
properties are computed.
the profile is
a
branched
with secondary cracks, both the whole
and its
backbone
considered.
In
structure
are
the
profiles)
this letter however, only the results
backbones (including
non-branched
concerning
reported.
are
shown in
that a particularly reliable quantity to be
It was
occasions
measured
various
a
on
self-affine profile in order to
determine
height
maximum
its roughness index ( is the
average
x50
Backscattered
x3000.
to
alloy
trie
and
z~~x(r),
which
defined
is
~~'~~~~~
z~ax(r)
In
law
the
is
computed
case
of
electrons
regime'extending
of profile
the
case
order
[20]
Ml~lZ(r')lz<r'<z+r
micrograph for profiles
each
4
(rapid
over
two
fracture
to
to
grey levels
where trie image
256
~~~l~(~')l~<r'<~+r
"~
on
profile
follows
as
used in
are
Images
deposit.
nickel
three
zone),
ail the
>~co
r~
(1)
to 4.
analysed micrographs
decades, for which
the
exponent
contrary,
present
remains
close
a
power
0.8.
to
micrographs at high
propagation zone), on the
1 (slow crack
magnification also present a power law increase, but the exponent is significantly smaller, lying
generally between 0.4 and 0.6. In the four cases, z~a~(r) is averaged over the results obtained
distribution
micrographs. Error bars are estimated from the variance of the
from the various
relative to the various micrographs.
of points coming from the results
smaller
of the
relative
behaviour
The
to profile 1 (fracture in fatigue) at
curve
average
with an exponent
m0.46 (see inset of
first studied, showing a power law
distances
increase
is
theoretical
roughness index (~s of a minimum
Fig. 2), 1-e-, remarkably close to the
energy
0.4, 0.45 and 0.5 is plotted against r, and the
surface [13,14]. Then z~~~/r(Qs, with (~s
fit using the expression
three plots are fitted with the Kaleidagraph°/~ non-linear
curve
In
=
~~~~
r(QS
Similar
results
are
obtained
using
the
=
Xvgr
A + B
~(-(QS
non-linear
curve
(2)
fit.
JOURNAL
550
PHYSIQUE
DE
I
N°5
jitting of zm~~(r).
non-linear
Table I.
Fatigue fracture (profile 1, see Fig. l ): results of the
(~s assuming the ~alues o-1, 0.$5 (see Fig. 2) and 0.5, zm~~(r) is jitted according to equadetermined
according ta equation (3) and is ezpressed
tion (2). The quasi-static blob size fi is
Errer bars are only resulting from the fit.
in microns, r is the confidence ratio.
Fatigue
results
fracture:
of the
fitting
non-linear
of
zm~~(r)
(~s
A
B
(
fi
r
0.4
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
0.5
0.64 + 0.02
0.20 + 0.01
0.838
+
0.875
+
0.07
5
0.987
0.009
10
0.977
Expression (2) is the simplest to account for the asymptotic power law behaviour corresponding, at short distances, to a quasi-static fracture mode
power law with a roughness index (~s
mode
and at larger distances, to a rapid fracture
power law with a roughness index ( yet
function
certainly
complex, but, as will be seen
real
be
deterrnined.
The
is
to
more
crossover
(fatigue)
for profiles
far
from
reality,
especially
following,
tl~is
assumption
the
is
not
in
too
this
Furthermore,
and 4 (~'uncontrolled fracture" )~ which are closer to the asymptotic
cases.
length
profile
the
fracture
for
between
quasi-static
allows us to define the
(~
zone
crossover
equal, 1e.,
the length at which the two
and the rapta
fracture
asymptotic
terms
are
one
as
i
The
fit
sets
of
(~s
for
obtained
results
obtained
.
0.45 is
=
exp
in
Figure
2.
shown
summarised
are
~ç ~ç~~~
in
Table
in
iii1
Results
(3)
the
concerning
following
three
I.
high velocity cracks [7-9], as well as
favours a value of (~s close to 0.45;
Fig.
2)
power
(qs
0.5 leads to a particularly high value of (. Subsequently, the
relative to profiles 2
curves
according to equation (2), but ( is kept equal to its previously determined
to 4 are also fitted
value, while A and B are the results of the fitting procedure. Values of (~ (i
2, 3, 4) are again
determined
through equation (3). The results are
summarized
corresponding
in Table II. Fits
0.45 are shown in Figures 3 to 5 for profiles 2 to 4, respectively.
to the value (~s
As a
of fact, the actual
values of (~ are very
sensitive
matter
to the value of (~s, for which
precision is rather bad because of the too few experimental points at short distances,
the
although, in the case of profile 1, one con determine a '~short distance"
exponent close to 0.46.
Consistency
previously
with
short-distance
the
law
values
measured
(see
behaviour
inset
of ( for
of
"
=
=
On
hand,
other
the
that (~ decreases
increases.
clear
it is
correlatively, the crack velocity
fatigue (between 3 and 5 pm,
the
much
fracture
process,
in
the
Although
the
local
the
stress
intensity
micrometer
factor
region
micron
II), it could decrease below 1
crack propagation regime, for which the
Table
unstable
"quasi-static"
alternative
an
be
environment
motion
rnodelled
as
[22, 23].
of
vortex
the
for
at
crack
K
or,
fracture
in
the
of
end
velocity
is
trace
From
lines in
seems
more
appropriate
to
describe
to
the
crack
surface
propagation
minimum
front
surfaces
it was recently suggested that the fracture
velocity
V
random
propagating
in
at
a
non-zero
a
a
this
picture, and using the results of Ertas and Kardar [24] for
dirty superconductors, one finds that the high velocity fracture
matter
a
exponent is remarkably close
measured
description
during fatigue loading. As
the
the
in
higher.
ezponent,
could
see
when
Lying
of
of
fact,
litre
QUASI-STATIC
FROM
N°5
TO
FRACTURE
RAPID
551
length for profile1 ezpressed in micrometers,
determined
by
(;
rather
the
is
sensiti~e
imposed
of
~alue
ta
(QS. fi is desee
con
three-parameter
non-linear
jitting (Fig. 2), while for1= 2,3,4, (~
curue
determined
through a two-parameter
(Figs.
is
jitting
3-5), ( being kept ta the ~alue decurue
termined from the analysis of profile 1 (Fig. 2). One con note that the ~alue of ( is not ~ery
sensiti~e
ta the ~alue of (Qs.
Table
(~ is the
II.
crosso~er
equation (3). One
through a
termined
that
lengths
Crossover
(QS
(
~l
0.4
0.815
3
for
the
~2
2
0.45
0.838
5
2
0.5
0.875
10
6
various
analysed profiles
~4
~3
0.6
0.05
0.3
3
ioo
ioo
Z
$É
~
o
Û
(
.
-
#
_
'
«
0.01
0.01
r
surface
hould
propagation
anisotropic,
(~
with
ci
reported
here
a
is
also
indeed
ioo
a
to
index
0.75
measured
3.
the
in
to propagation).
a
of
certain
regime,
however,
Interestingly,
velocity
low
rack
egime
which
is rather
for
this
ioo
Gi
É
E
Î
2i
a
o-i
0.01
r
4.
close
to
the
here
problem,
where
JOURNAL
552
PHYSIQUE I
DE
N°5
ioo
Z
É
Î
~
(
~'
C
~'w
fi
0.01
r
5.
ig.
ransition,
the
elow
ould
rack
investigated
this
regime [25],
behaviour
is
to
V~~,
sting
is
et
Roux
Similar
ai.
to
within
are
(AFM)
which,
-
crack
actual
the
relationship
by
the
with
ncreasing
The
observes
crack
is
In
i
-,
due
that,
in
It
was
ranching
latter
8,9,19,22,23],
that
are
this
in
interested
to
result
the
ere,
power
a
determination
lower
limit
in
that
quantitative
of
(c
as
no
no
y", with
aw,
of
perfectly
is
the
our
limit
y
as
in
interested
in
upper
with
well
experiment as
checked
we
ontrary,
emphasize
very
lead
to
the
to
case,
rougher
an
which
urther
compatible
of
(
the
increases
with
fracture
orphology
the
urfaces,
low
but
has
which
transformations
espect:
homogeneous
for
disorder
at
urfaces
[21]
crack
of
importance
fracture
flat
nstability
the
their
the
with
shown
is
the
of
determination
racture
also
velocities
edge.
crack
(c
were
as
thon
of
can
often
Higher
erformed
-
notch
of
more
average
smaller
significantly
authors
these
fact,
initial
be
hieved).
length
crack
be
explore
and
lectrically,
In
the
describe.
(note
atomic
to
might
above,
Visilog
y.
roughness
One
one
velocity
we
regime"
0.83
could be
crack velocity
ours.
On
the
with
A
the
with
an
precise
should provide a
intensity factor K.
stress
has been
fracture surfaces
[28].
to
y
out
ointed
of
property
observations
ould
istance
to
decreases
kinetic
distance
crack
xperiments
local
the
egmented
in fatigue, for
measured
ai.
the
~'long
close
be
and
the
with
of
determination
and
nough,
eing
a
is
load
high
of
this
pinning.
studied
by
rid
get
images are
with
order
which
the
in
already
con
et
chmittbuhl
further
hand,
another
length
lation
of
pinning
the
not
is
currently
is
the
when
to
lane)
1024)
x
way
under
as
between
that
Note
are
other
the
velocity
0
confirmed. Quantitative
duly
crack
=
(
The
observations.
discontinuous:
low-velocity
as
with
velocity
be able
z
(1024
larger
which
results
previous
microscope
On
the
highly
this
urthermore,
27].
[26,
for
force
if
our
resemblance
some
is
its
front
obstacles
icrostructural
lie
with
bears
).
decreasing
ngth
also erformed
are
plated,
system. The
to
onstrained
velocity
=
((s
pendent
agreement
load
the
crack
(the
ransition
below
the
on
that
result
lengths
trapped
recently
a
in
of
ovement
be
of
the
hold
crack
short
Kardar
and
Ertas
with
3,
=
the
can
ncrease
next
ather
for
since
transition
it
the
çi
with
during fatigue
predicted
ail.
at
propagate
not
only
crack
crack velocities,
seems
it
never
een
to
referred
observed
now
on
as
that this
roughening
metallic alloys,
to
concern
secondary
velocity.
omparison
of
bath
N°5
QUASI-STATIC
FROM
disordered
the
and
homogeneous
more
materials
RAPID
STM
FRACTURE
553
lengthscales
various
at
AFM, and
real "phase diagram"
scale~ using both
micrometer
TO
from
electron
standard
or
or
the
nanometer
to
optical microscopy
certainly help to draw a
for fracture.
conclusion, it has been shown that there exists a
length ( which decreases with
crossover
increasing crack length, and correlatively~ with increasing stress intensity factor and crack
velocity, which separates two fracture regimes. At lengthscales higher than (, the previously
reported roughness index ( ce 0.8 is measured, while the small lengthscale (< () behaviour lits
with a pinning/depinning
mechanism
hypothesis, for which ( m 0.45.
Finally, it should be noted that these two fracture regimes
characterized
by two fixed roughindices and separated by a
length which depends on the crack velocity cari be
ness
crossover
misinterpreted as a unique regime
characterized
by a fractal
dimension
continuously varying
with the velocity. A similar
confusion
made in the past by considering that the roughwas
exponent would vary continuously with the fracture toughness KIC, while it was recently
ness
could
In
that
shown
it is the
length
correlation
self-affine
of the
fracture
surface
which
in
some
cases
is
of KIC (29].
function
a
Acknowledgments
experiments
Enlightening
Fracture
Thomas.
achieved
were
collaboration
in
discussions
with
Lemoine, G. Lapasset and
Thomas
and G. Lapasset
A.
with
Bouchaud,
J.-P.
M.
M.
are
gratefully acknowledged.
References
iii
Mandelbrot
[2]
Maloy K-J-,
[3]
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