From Quasi-Static to Rapid Fracture

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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

Abstracts

62.20Mk 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 Division

Leclerc,

B-P. 72, 92322 Châtillon

Cedex,

France

(Received

7

February 1995, accepted

24

February 1995)

Résumé.

Quatre profils

de rupture

correspondant

à quatre ^vitesses de

propagation

de fissure différentes sont étudiés _sur le même

échantillonj

et révèlent l'existence d'une

longueur

de coupure

( qui

décroît _avec la vitesse. Pour des échelles de

longueur supérieures

à (, _on retrouve _un

exposant ^de

rugosité

voisin de

(

_ct 0.84. A

plus petite échelle, l'exposant

mesuré est _en accord

avec une

hypothèse

de

propagation quasi-statique (QS)

du front de

fissure,

et vaut

(~s

_Ù 0.45.

Abstract. Four fracture

profiles corresponding

to four diOEere1~t crack velocities _are studied

on trie _same

sample

and show trie existence of _{a crossover}

lengthscale ( decreasmg

with the crack

velocity.

For

lengthscales larger

thon

(,

trie previously

reported roughl~ess

index ( ct 0.84 is recovered. At

lengthscales

smaller than

(,

trie fracture

profile

lits with

a

quasi-static (QS) hypothesis,

1-e-, trie measured

roughness

index is close to

(~s

_CÎ 0.45.

It is _now

clearly

established that fracture surfaces _can be considered _as self-affine

objects.

After trie

pioneering

work of Mandelbrot et ai.

[l],

_many

experiments using

various

experi-

mental

techniques (profilometry [2,3],

_microscopy and

image analysis [4-10], scanning tunnel-

mg electron microscopy

(STM) _[11], electrochemistry _[12], etc.)

_on materials _as dilferent _as

steels

il,5,10,12],

aluminium

alloys _[7],

rocks

[3],

intermetallic

compounds [8,9]

or ceramics

[4],

bave shown that fracture surfaces exhibit

scaling properties

_on two

[2,8,9]

_or three decades _[7]

of

lengthscales.

At

"large enough" lengthscales (trie

_micron scale for metallic

materials),

for

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 "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 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),

and close to 0.5 for

graphite iii].

On the other

hand,

low

cycle fatigue experiments

on a

steel

sample

have led to _a value of

(

close _to 0.6

[12].

^{These results} are

particularly

attractive

since

they

report

roughness

indices close to the

roughness

^of a minimum energy surface _{in a}

r&ndom environment

[13,14].

It _was

suggested by Chudnowsky

and Kunin

[15],

^Kardar

[16],

Q

Les Editions de

Physique

1995 iowNALD8mYsiQu8L-T. s, NOS.MAY199s 21

548 JOURNAL DE

PHYSIQUE

I N°5

31.0 _mm

in tension Fatigue

4.4 6

Profile

4

Pr°fl'e 3 ~~°fi'~

^~

~~°fi~~ applicatiol~

Fig.

l. Sketch of trie broken

sample, showing

the four

profiles

which have been

investigated.

Profile 1 is in trie

fatigue

fracture zone, and thus

corresponds

to trie lower crack

velocity.

Profile 4

corresponds

to trie "instantaneous fracture" _zone, for which trie crack

velocity

_is much

higher.

and

by

Roux and

François

_on the basis of _a fracture model for _porous ductile materials

il?]

that the

path

chosen

by

_a crack in _a random environment should be such that the overall fracture energy ^is minimised. This

quasi-static assumption,

which

clearly

cannot be fulfilled for

rapid

crack

propagation, might

be valid at small

enough lengthscales

_or for low

enough

velocities.

However, although

_our results _are

compatible

with the

prediction

of this

model,

_we shall _propose in the

following

_an alternative

'~quasi-static" description,

which should lead to a

comparable exponent,

^{but which should be doser} to the actual crack

propagation

mechanism.

In this

letter,

_we show that _a

quasi-static

mechanism is valid indeed _up to _a distance

(

which decreases with

increasing

crack

velocity.

The existence of this _{new crossover}

length

and the

analysis

of the smaii

iengthscales

behaviour _are the central result of this _paper.

A notched CT

sample (dimensions

12.5 _x 30 x 31

mm~,

_see

Fig. l)

of the

Super

_a2

alloy _[18]

T13Al-based

_is first

precracked

in

fatigue

at 30

Hz,

with _a fixed ratio R of 0.1

(the applied

ioad oscillates between _a maximum load

P~~~

and _a minimum load

P~~~/10

at _a

frequency

of 30 Hz with

Pm~~

= 2750

N),

in order that the total

length

^{of the crack after} the

fatigue

test is close to

60%

of the

sample length.

Fracture is achieved

through

uniaxial tension

(mode I,

see

[19]

^for

example).

Note that the microstructure of the

alloy

is

composed

of _a2

(ordered phase)

lath in _a

fl (disordered phase,

stable at temperatures

higher

than lllo

°C)

matrix.

The brittle needles _vary both in

thickness, length

and orientation.

Plasticity

of the

fi-phase

was shown to

play

_an

important

role in the fracture

properties

of the material. The broken

sample

is

electrochemically nickel-plated (the

thickness of the

deposit being approximateiy

100

microns).

Four

profiles

are obtained

by subsequently cutting

and

polishing

the

sample 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

by only

one millimeter

during

^the last part of the

fatigue

test,

corresponding

to 13000

cycles.

This

velocity

is close _to 2 micrometers per

second,

1-e-, 5 _x

10~

times the sound

velocity Cs

in the material

(Cs

_ct 4700

m/s),

whereas in the uncontrolled fracture _zone, the

velocity

is

expected

to saturate at _a value which is at least 0.2

0.3Cs.

These

profiles

_are observed with _a

scanning

electron _{microscope Zeiss DSM 960 at various}

magnifications

10 to 12

images

_were made for each

profile

with

magnifications ranging

^from

100

~

'

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 of

zmax(r)). Averaged expenmental points

are

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 _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 trie

alloy

and trie nickel

deposit. Images

in 256 grey levels _are

registered through

_a Kevex Delta

system,

and sent to _an IBM PC

486-33,

where trie

image segmentation

_is

performed using

the

system Synoptics Synergy

Board. The obtained

binary images (the weight

of each

point

^located

on the

profile being

1, the

weight

of _any other

point being 0)

^of

length

703

pixels

_are sent to a workstation where their various statistical

properties

_are

computed.

When the

profile

is branched with

secondary cracks,

both the whole _structure and its backbone _are considered. In this letter

however, only

the results _concerning the backbones

(including

non-branched

profiles)

are

reported.

It _was shown _{in various} occasions that _a

particularly

reliable

quantity

to be measured _{on a} self-affine

profile

_in order to determine its

roughness

index

(

is the average ^maximum

height z~~x(r),

which is defined _as follows [20]

~~'~~~~~ "~

~~~l~(~')l~<r'<~+r Ml~lZ(r')lz<r'<z+r

_>~co

^r~ ₍₁₎ z~ax(r)

is

computed

on each

micrograph

for

profiles

to 4.

In the _case of

profile

4

(rapid

^fracture

zone),

ail the

analysed micrographs present

a power law

regime'extending

_over two to three

decades,

for which the

exponent

remains close _to 0.8.

In the _case of

profile

1

(slow

crack

propagation zone),

_on the

contrary, micrographs

at

high 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

from the various

micrographs.

Error bars _are estimated from the variance of the distribution of

points coming

from the results relative _to the various

micrographs.

The behaviour of the average curve relative to

profile

1

(fracture

_in

fatigue)

at smaller distances _is first

studied, showing

a power law _increase with _an

exponent

^m0.46

(see

inset of

Fig. 2),

_1-e-,

remarkably

close to the theoretical

roughness

index

(~s

of _a minimum energy

surface [13,14].

Then

z~~~/r(Qs,

with

(~s

=

0.4,

0.45 and 0.5 is

plotted against

_r, and the three

plots

_are fitted with the

Kaleidagraph°/~

non-linear _curve fit

using

the

expression

~~~~

= A _{+ B}

~(-(QS (2)

r(QS

Similar results _are obtained _using the

Xvgr

non-linear _curve fit.

550 JOURNAL DE

PHYSIQUE

I N°5

Table I.

Fatigue fracture (profile 1,

_see

Fig.

l

):

results

of

the non-linear

jitting of zm~~(r).

(~s assuming

the ~alues

o-1, 0.$5 (see Fig. 2)

^and

0.5, zm~~(r)

is

jitted according

to equa- tion

(2).

The

quasi-static

blob size

fi

is determined

according

ta

equation (3)

and is

ezpressed

in

microns,

r is the

confidence

ratio. Errer bars _are

only resulting from

the

fit.

Fatigue

^{fracture: results of the non-linear}

fitting

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 0.838 + 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 the}

^simplest

to account for the

asymptotic

power law behaviour

correspond- ing,

at short

distances,

to a

quasi-static

fracture mode _power law with _a

roughness

index

(~s

and at

larger distances,

to a

rapid

fracture mode _power law with _a

roughness

index

( yet

to be deterrnined. The real _crossover function is

certainly

_more

complex, but,

_as will be _seen in the

following,

tl~is

assumption

is not too far from

reality, especially

for

profiles (fatigue)

and 4

(~'uncontrolled

fracture" _)~ which _are closer to the

asymptotic

_cases.

Furthermore,

this allows _us to define the _crossover

length

_(~ for

profile

between the

quasi-static

fracture _zone and the

rapta

fracture _{one as} the

length

at which the two

asymptotic

terms are

equal, 1e.,

i

_{. exp}

~ç _~ç~~~

in

iii1 ₍₃₎

The fit obtained for

(~s

₌ 0.45 is shown in

Figure

2. Results concerning the three

following

sets of results obtained _are summarised in Table I.

Consistency

with

previously

measured values of

(

for

high velocity

cracks

[7-9],

as well _as the short-distance _power law behaviour

(see

inset of

Fig. 2)

favours _a value of

(~s

close to

0.45;

(qs

_" 0.5 leads _{to a}

particularly high

value of

(. Subsequently,

the _curves relative to

profiles

2

to 4 _are also fitted

according

to

equation (2),

but

(

is

kept equal

to its

previously

determined

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 in Table II. Fits

corresponding

to the value

(~s

₌ 0.45 _are shown in

Figures

3 to 5 for

profiles

2 to

4, respectively.

As _a matter of

fact,

the actual values of _{(~ are very} sensitive to the value of

(~s,

^{for which}

the

precision

_is rather bad because of the too few

experimental _points

at short

distances, although,

in the _case of

profile 1,

_{one con} determine _a '~short distance"

exponent

close to 0.46.

On the other

hand,

it is clear that (~ decreases when the local stress

intensity

factor K _or,

correlatively,

the crack

velocity

increases.

Lying

in the micrometer region for fracture in

fatigue (between

3 and 5 _{pm, see} Table

II),

it could decrease below 1 micron at the end of the fracture _{process, in} the unstable crack

propagation regime,

for which the crack

velocity

is

much

higher.

Although

the

"quasi-static"

measured

exponent

is

remarkably

close _to the minimum

surface

ezponent, an alternative

description

_{seems more}

appropriate

to describe crack front

propagation

during fatigue loading.

As _a matter of

fact,

it _was

recently suggested

that the fracture surfaces could be rnodelled _as the trace of _a litre

propagating

at a non-zero

velocity

V in a random environment

[22, 23].

From this picture, ^and

using

the results of Ertas and Kardar

[24]

^for the motion of _vortex lines in

dirty superconductors,

_one finds that the

high velocity

fracture

Table II. (~ ^is the _crosso~er

length for profile1 ezpressed

in

micrometers,

determined

by equation (3).

^One _{con see} that

(;

is rather sensiti~e ta the

imposed

~alue

of _(QS. fi

is de-

termined

through

_a

three-parameter

non-linear _curue

jitting (Fig. 2),

while

for1= 2,3,4,

_(~

is determined

through

_a

two-parameter

_curue

jitting (Figs. 3-5), ( being kept

_ta the ~alue de- 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.

Crossover

lengths

for the various

analysed profiles

(QS ( ~l

_~2 ~3 ~4

0.4 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 _a

certain

regime,

which

is _rather close to the

reported

_here indeed measured to propagation). Interestingly, however, here

is

also

a

low

velocity

egime

for this problem,

where

ioo ioo

Gi É

E Î

2i

a

0.01 o-i

r

4.

552 JOURNAL DE

PHYSIQUE

I N°5

ioo

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 _result

that

((s = ). urthermore, this low-velocity

behaviour

is predicted

to

hold below a _velocity ^{pendent ngth}

decreasing

as

(

V~~,

with

çi

= 3, in agreement

with

_our

observations.

The

of ^a

during

fatigue sting for ather short

crack lengths

^bears some resemblance with

the

^pinning

transition

since the

ovement of the front

is

highly

discontinuous: when the load

is

it can be trapped on icrostructuralobstacles if

its

velocity is not high

nough,

and

the next ^ncrease

of

_the load to be able

to

_get rid

of this

_pinning. A

ransition (the crack is onstrained to lie _within the z

=

^{0 lane) is}

currently

^eing

studied by

Roux

et

ai.

[26, 27].

Similar are

also

erformed

plated,

for

whichlarger

(1024

_x 1024)

images

areegmented with

the

Visilog

system.

The previous

results are

duly confirmed. Quantitative observations

with

an _atomic

force

_microscope

(AFM)

are _under

way

in

order

to explore _more of

On

the other

_hand, further _xperiments in fatigue, for

which

_the

cracklength and average

crack _velocity -

which,

as

already

ointed

out

_above, might be

significantly

smaller

thon the

actual

crack -

_con

be measured

lectrically,

should

provide ^a

precise

determination of

the relationship

between

_and the local

stress

intensity

factor

_K.

Note

that

another kinetic _property of fracture

surfaces

has _been

by chmittbuhl et

ai.

[28].

In

fact, these authors were_interested

in the

lation

length

with the

distance y

to

the initial _notch - as in our no quantitative

determination

of

the _crack velocity

ould

be ^erformed -, i

with the upper

limit

_(c

of

~'long _istance

regime" we

describe.

(c is

shown

to with y as a power aw, y",with

close

^to

0.83

_(note that,

in

their experiment as well as

ere, no determination

of

the crack velocity

could be

hieved). It

was

checked

that _this

result

_is

perfectly

^compatible

with 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 importance

of

disorder in that espect: for

homogeneous

one _{often observes}

very

flat

fracture

urfaces at _low crack velocities, _referred

to as

Higher crack velocities _lead to rougher fracture _urfaces, but it

seems

now _{that this} roughening

is

due to an

nstability

[21]

which

has

never ^een observed on metallicalloys,

to

edge.

In the _latter

case,

^urther crack orphology

transformations only concern

secondary

crack

ranching 8,9,19,22,23], which

increases with the

crack

velocity. omparison of

bath

disordered and _more

homogeneous

materials at various

lengthscales

from the _{nanometer to}

the _micrometer

scale~ using

both STM _or

AFM,

and standard electron

or

optical microscopy

could

certainly help

to draw _a real

"phase diagram"

for fracture.

In

conclusion,

it has been shown that there exists _{a crossover}

length (

which decreases with

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

rough-

ness indices and

separated by

_{a crossover}

length

which

depends

_on the crack

velocity

_cari be

misinterpreted

_{as a}

unique regime

characterized

by

_a fractal dimension

continuously

_varying

with the

velocity.

A similar confusion _was made _in the past

by considering

that the

rough-

ness

exponent

would vary

continuously

with the fracture

toughness KIC,

while it _was

recently

shown that it is the correlation

length

of the self-affine fracture surface which in _{some cases} is

a 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

_are

gratefully acknowledged.

References

iii

Mandelbrot B-B-,

Passoja

D.E. and

Paullay

A.J., Nature

(London)

308

(1984)

721.

[2] Maloy

K-J-,

Hansen

A.,

Hinnchsen E.L. and Roux S.,

Phys.

Reu. Lett. 68

(1992)

213.

[3] Schmittbum

J.,

Gentier S. and Roux

S., Geophys.

Lent. 20

(1993)

639.

[4]

Mecholsky

J-J-, Passoja D.E. and

Feinberg-Ringel K-S-,

J. Am. Geram. Soc. 72

(1989)

60.

[5] ^Mu

Z-Q-

al~d

Lung C.W.,

J.

Phys.

D

:

Appl. Phys.

21

(1988)

848.

[6] ^Dauskardt

R-H-,

Haubensak F. and Ritchie R-O-, Acia Meiall. Mater. 38

(1990)

143.

I?i Bouchaud

E., Lapasset

G. and Planès

J., Europhys.

LeU. 13

(1990)

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