Self-Organized Criticality in a Model for Fracture on Fibrous Materials

HAL Id: jpa-00247125

https://hal.archives-ouvertes.fr/jpa-00247125

Submitted on 1 Jan 1995

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Self-Organized Criticality in a Model for Fracture on Fibrous Materials

A. Bernardes, J. Moreira

To cite this version:

A. Bernardes, J. Moreira. Self-Organized Criticality in a Model for Fracture on Fibrous Materials.

Journal de Physique I, EDP Sciences, 1995, 5 (9), pp.1135-1141. �10.1051/jp1:1995187�. �jpa-00247125�

Classification Pllysics Abstracts

62.20M 05.40 02.50

Self.Organized Criticality in

_a

Model for Fracture

_on

Fibrous Materials

A.T. Bemardes (~,*) ^{and J-G- Moreira} (~)

(~) Departamento de Fisica, Instituto de Ciências Exatas _e Biolôgicas, Universidade Federal de Ouro Preto, Campus do Morro do Cruzeiro, 35410-000 Ouro Preto, MG, Brazil

(~) Departamento de Fisica, Instituto de Ciênc1as Exatas, Universidade Federal de Minas Gerais, C.P. 702, 30161-970 Belo Horizonte, MG, Brazil

(Received

17 May 1995, ^received in final form 27 May 1995, accepted 2 June 1995)

Abstract. We investigate ^trie existence of self-orgamzed criticality _{on a} model to describe trie failure process in fibrous materials. This model _consists of ideutical parallel libers which _are pulled with constant velocity. The rupture probability of each liber depeuds _ou the elastic euergy of the liber and _on the number of unbroken neighboring libers. In trie brittle-ductile transition,

we have observed

a power law behaviour

ou trie uumber of cracks _versus its _size. This indicates that, in this _regiou, trie model shows

self-orgauized

cnticality aud the fracture pattem _con be

a

fractal. The power law exponent is not universal, but depends _on the temperature ^{and traction} velocity.

1. Introduction

Fractal pattems can be found in

severalobjects

in _nature. The formation of these pattems are

related to

dynamical

_processes, which _{are an} important ^{field of}

investigation. By

the end of last decade

Bak, Tang

and Wiesenfeld

Iii

put forth _a frame from which _{we can} understand the

formation of fractal pattems: a system evolves

spontaneously

to a cntical state,

independent

of the initial

conditions,

and stays there in _a

dynamical equilibrium.

In that _{state one can} find time and

spatial

correlations between ail parts ^{of the} system. ^The

"sandpile

model" is the first model used to

study

the

problem

of

self-organized cnticality. Next,

the existence of self-

organized criticality

was studied in several branches of

science,

such _as

earthquakes

_[2],

growth

models (3], ^erosion (4], ^{cloud formation} (Si,

biological

evolution (6],

fragmentation

_(7], etc. One

signature

^{of this} behaviour is the _power law between the

frequency

of _an event and the size of the event, which _can

explain

the existence of the

self-similarity

that _we observe in fractal

abjects.

Thus, the correlations between all parts ^of the system, present on critical state, are

responsible

for the

geometric self-similarity

observed. As

pointed

eut

by

Mandelbrot in 1984, fracture is _a class of

phenomena

in which _we observe the formation of fractal pattems _{(8] and} two

(*) Presently at Institut für Theoretische Physik, Universitàt _zu KôIn, 50923 KôIn, Germany

© Les Editions de Phvsique 1995

l136 JOURNAL DE PHYSIQUE I N°9

conjectures

_anse: the fractal dimension of trie pattem _is universal (9] or the fractal dimension and the

toughness (defined

_as the energy needed to break the

material)

_are related (8,

loi.

These

are

important questions

that remain open

Ill].

In this _{paper we propose a} model for fracture which allows _us to

gain insight

^{of the}

phenornena

of the fracture _{as an event} of

self-organized criticality.

We find that the _power law exponent ^is

dependent

_on trie extemal parameters

(temperature

and traction

velocity).

We

study

_a liber bundle rnodel to sirnulate the failure _process of fibrous

matenals,

where the libers of the material _are

joined together by

_a

homogeneous

matrix. Dur model considers the amount of elastic energy into the

material,

the

spread

of _a local

crack,

and the fusion of cracks

as the

breaking

mechanism. In

addition,

_we

adopt

the cascade

(inspired

in the avalanches _on the sand

pile model)

of

breaking libers,

_as the mechanism to form trie cracks into the liber bundle. Dur attention is focused _on computational simulation when _an uniaxial force

(parallel

to the

libers) pulls

the bundle

just

at its failure. In _a recent paper (12] we bave shown that this model in

il +1)-dimensions

describes the

principal

^features observed in _a

breaking experiment:

brittle fracture _occurs at low temperatures and ductile fracture _occurs at

high

_ones. We have also studied the

growth

of the

toughness

_as ^the temperature _increases. ^In that _{paper we} have observed features that allow _us to expect a power law behaviour between the

frequency

of cracks and the size of cracks.

However,

due to the one-dimensional character which

impedes

the formation of

big cracks,

_we could _not infer the existence of

self-organized criticality.

In the present _{paper we}

investigate extensively

the

problem

in _a

large

number of

samples

in

(2+1)-

dimensions with various sizes. We also observe the fundamental features which describe the characterization of the above mentioned failure _process. As _we shall _see

below,

_our results allow _{us to} condude

alfirmatively

_on the existence of

self-organized cnticality.

2. Mortel and Discussion

Dur model consists of

No Parallel libers,

each _one with the _same elastic constant K. These libers _are fixed in

parallel plates

where _one

plate

is fixed and the other is

pulled

with constant

velocity

_v

by

_a force in the direction

along

to the libers. At each time step T the amount of deformation of the unbroken libers _is Az

= v x _T

(in

_our units _T

=

1).

We _assume that _an

isolated liber has _a

purely

linear elastic behaviour with _a

breaking probability

which grows up with the deformation _z of the

liber, being equal

to

unity

at _a critical deformation _z~. We shall

assume that the

probability

of rupture ^{of the liber1} is

given by

Pilz)

=

j~

)

~~

exPi)lô~

¹⁾¹

^il)

Here _n~ is the number of unbroken

neighbour

libers of the ith

liber,

à

=

z/z~

is the strain of the

material,

t ₌

kBTle~

is the normalized temperature, ^{where T is the absolute} temperature,

e~ =

Kz)/2

is the critical elastic energy and kB is the Boltzmann _constant. The

exponential

term _is a

typical

form of the

probability

in fracture

models,

based _on the absolute reaction rate

theory

for chemical _process (13]. ^The normalization factor

1/(n~ +1)

guarantees ^that the deformation of _one isolated liber _can not exceed _z~, 1-e-, an isolated liber breaks when

z - z~. The

dependence

of the

breaking probability

_on the number of unbroken

neighbour

libers simulates the existence of _an interaction between the

libers,

that

is,

the distribution of the load between

neighbouring

libers.

Thus,

the

breaking probability

of _a liber increases if _a

neighbour

^liber breaks. With this

neighbour dependent

factor the deformation of _{a non-} isolated liber _can now exceed _z~. These features _are in agreement ^with experiments in composite materials

(14].

In each time step of _our simulation the system ^is

pulled by Az,

and _we

randomly

choose

Nq

_{= q} x No libers that _can be

broken,

where 0 < q < represents the percentage ^{of libers. The}

breaking probability

_is evaluated and then

compared

with _a random number in the interval

(0,1).

If the random number is less than the

breaking probability,

the liber breaks. The load

spreads

to the

neighbour

libers and the

breaking probability

of them increases due to the

decreasing

of _n~. This

procedure

describes the

propagation

of the crack

through

the liber

bundle.

Then,

the _same steps are done for _one of the

neighbouring

libers. Note that if the liber

breaks,

_a cascade beg1~ls. The cascade stops in _a

given

liber when the test of the

probability

does not allow its rupture or when another crack in the bundle is found. The

propagation

of the crack is done in all directions

perpendicular

to the force

applied

_on the system. When the cascade _process stops, another liber in the N~ set is chosen and ail steps

already

^described are

repeated.

After the

N~ trials,

we

pull

the system to _{a new}

displacement

Az and the

breaking procedure begins again.

The simulation continues until the system ruptures, that is, when no

more unbroken libers exist.

We have

performed

calculations with

No

_" L _x L libers distributed in _a square lattice for diflerent temperatures ^and traction velocities and _we obtained ail the

possible

fracture

regimes:

brittle,

brittle-ductile transition and ductile. The brittle fracture is characterized when the system ^breaks

abruptly,

the system is

pulled

until

suddenly

_one crack propagate

causing

the failure of the

material;

the rupture

profile

is flat. In the other

hard,

in ductile fracture _{one cari} observe _a formation of several small cracks

during

the

pulling

process and the rupture ^{surface is}

extremely rough.

The calculations _are done with _z~

= 1, K ₌ and _{q =} 0.001.

Figure

1 shows the

log-log diagrams

of the

frequency

of the cracks H _versus the size of the cracks

S,

defined

as the number of broken libers in _a cascade _process, for various temperatures t and traction velocities _v, obtained from _a set of100

samples

with L

= 100. As

pointed

out above the brittle process is characterized

by

the existence of _one

big

crack and _a few cracks of small sizes. A

remarkable feature is the presence of cracks with sizes _near the system size, as one can see in the

diagram

at the left-bottom _corner of the

figure ii

₌ o-1; v =

0.05).

The _curve of H _versus S has two maxima: at S

= 1 and _at the characteristic size of the system. As the temperature ^is increased

(or

the

velocity

is

decreased)

the size of the cracks becomes smaller. In the

opposite

corner, on the

right-top,

_we have _many cracks of small sizes which represent a ductile fracture.

These cracks appear in diflerent parts ^{of the} material _so that the

shape

of the _curve H _vs. S

changes.

One _{can see} the hmit _case in the

right-top diagram Ii

= 2.0; v =

0.0025). There,

we

only

observe the existence of cracks of small sizes

(maximum

in the order of100 broken libers pet

crack).

Dur main interest is in the

diagrams

in the

principal diagonal

of the

figure,

that is, in the

pairs Ii

=

o-1;

_{v =}

0.0025), (t

=

I.o;

_{v =}

0.01),

and

Ii

₌ 2.0; v =

0.05),

which _are in the brittle-ductile transition. Note that at the

beginmng

H _seems to go down

linearly

_so that

we can assume a power law

H +~ S~° 12)

In order to

verify this,

_we have

performed

calculations with diflerent

L, namely

L _"=

75,

100, 150, 250, 500, 750 and 1,000, with q = O.col.

Figure

2 shows the

plot

of H _versus S for the set of pairs above mentioned for system ^size 1, 000 _x 1,000. From _a linear fit of the

diagram

^{for the}

pair Ii

= o-1; _v

=

0.0025)

with S in the interval 10 < S < 200 _we obtained the exponent a = 1.59. The _same behaviour is observed for the other _pairs chosen _on the

brittle-ductile transition

region,

but _now with dilferent exponents.

However,

this power Iaw

was measured _on the

beginning

of the _curves.

Thus,

_to

perform

_an accurate

analysis

of these curves, we calculated the exponent o

using

dilferent

length Saales,

which is shown in

Figure

3.

By increasing

the

length

scale _{one can see in} this

figure

that the exponent o seems to stabilize

l138 JOURNAL DE PHYSIQUE I N°9

td _10~

'..~

10~

"°.,

~ 10°

""~

^~

t0°

~ t0°

~ ~

10~ ~ t0~ # tÉ Ù

s s s

tÎ lk lk

'°':,

"",

~

lÎ

'~i

^~

lÎ ^~

~

~Q°1 ~...

lÎ

~

lo~ tJ~ Ci lo~ 1

s s s

iÎ _io~ iÎ

H H

"~ H _"&

10°

~~

le _10° "

~t'.

~Î~ _'~~.

10~ "~«. i«~ io~

s s s

Fig. l. Log-log diagrarns of frequency of trie cracks H _vers~s crack size S for different temperatures and velocities. Trie siJnulations _were performed in 100 saJnples of L

= 100 libers for each pair of parameters. The figures at trie bottoJn _were calculated for t

= o-1, those in trie Jniddle for t ₌ I.o and those at trie top for t ₌ 2.0. On trie right, the figures _were calculated for _u

= 0.0025, in the center for

u = 0.01 and _on trie left for

u = 0.05.

for the pairs

Ii

= o-1; ~ =

0.0025)

and

Ii

₌ 2.0; _{~ =}

0.05)

and also that _a converge ^to diiferent values: _a

+~ 1.58 and _a

+~ 1.67,

respectively.

We obtained these two diiserent exponents but nevertheless for the

pair Ii

= I.o; ~ =

0.01)

_we observe _a

quite

distinct behaviour. For short

length

scales _{o seems} to be stable around _a

+~ 1.53.

However,

for

large length

scales it

junlps

to _a

higher

value. Front _a

analysis

for diiferent system sizes for this

pair

of externat parameters we observe the _same behaviour. In this _case the system seems to be

quasi-critical.

We observe the

conlpetition

between two diiferent

regimes

and the fracture tends to be _a ductile

one. Whereas _we observed for the other pairs _a

converging

value of _o exponent

(for

diiferent externat

parameters),

_{we cari} conclude that ail

length

scales _are present at the _saule time _so that the system is in _a critical state, 1-e-, that this model shows

self-organized criticality,

except for internlediate t and _~.

To _see how these diiferent

lenght

scales _appear in fracture _we

performed

_a s1nlulation with L ₌

10~,

_{q =}

10~~, Nq

= 1

(one

liber is chosen at each time step to enable _us to _see the crack

formation at diiferent t1nle

steps),

t

= o-1 and _~

= 2.5 _x 10~~

Figure

4a shows the _stress

tW.I vW.W25

io~ *~.

~ lo~

~~-i

lo~~

o ~

io la

s

ml.0 vW.01 m2.0 vm0.05

lo~

.. lo~

"~

g

~i

H H

lo'~

lo'~

~~-3 ^-3

lo~ lo~ ^~~

lo°

_{lo~ lo~}

S S

Fig. 2. Log-log diagrams of frequency of the cracks H _versus crack size S for the diiferent set of externat parameters in the principal diagonal of Figure 1. The pairs of temperatures ^{and velocities}

are shown _m the plots. The simulations _were perforJned in 1,000 samples of L

= 1, 000 libers for each pair of parameters.

a

(= NKz/No)

_vs. strain à

(= z/z~) diagram

for this simulation with 437,533 data

pairs.

A detail is shown in

Figure ^4b,

^with

2,000 points

around the maximum of this stress-stram

diagranl.

The variations of _{a are} of ail

lengths

and

correspond

to crack formation.

Thus,

in that

diagram

the small

(big)

variations of _{a are} related to the formation of snlall

(big)

cracks.

Following

the

assumption

that

self-organized criticality

is

responsible

for the formation of fractal pattems, the non-universal _a exportent suggests that _we do not have _a universal fractal

dimension of the fracture pattern, ⁱⁿ contrast ta what has been

conjectured by

_some authors [9].

This non-universal fractal d1nlension is in agreement ^{with the}

experimental

evidence; the fracture surface is flat for the

extremely

^{brittle fracture}

(the

d1nlension of the surface is close to the Euclidean

dimension),

and becomes

rougher

with

increasing

temperature

(for

_a constant

traction

velocity).

At

high

temperatures ^the surface is

extremely rougir

which _{means one} has

a fractal dimension that

depends

_on the externat conditions. We _can also

conjecture

that trie

toughness

^has a

positive

correlation with trie fractal dimension of trie surface. Trie

toughness

increases with the temperature ^{and is}

proportional

to the surface

generated

in the fracture process

(the

classical

Griflith/Irwing assumption).

Hence, the fracture surface increases with

temperature, and _we conclude that trie diiferent _a exponents ^obtained in _our simulations

are related _ta the _existence of diiferent fractal dimensions obtained with diiserent externat conditions.

In conclusion _we have studied _a model for fracture _on fibrous nlaterials in

(2+1)-dimensions.

1140 JOURNAL DE PHYSIQUE I N°9

1.75

1=Î'ÎÎ~~ t",

~~ ^{v=0.05 '}

*

i~lu-_

~D---D~~,,'~

^~'îJ

1.65

,1'

^'~

~

,P "'~""a"'

~~

1.60

,' ,/~', _Î

,"° _ci" "'o-""° _{ii~ ,11"°}

1.55

a'

_,"

_b"

,j'~~~~~~~~~"',,

_~

~, »---+

i.50 ^'

io ioo looo

L(scale)

Fig. 3. Asymptotic behaviour of the exportent _a ^{for different} length scales for different values of temperature and traction velocity, corresponding to the pairs of Figure 2.

1.0

0.9880

~

0.5 0.9878

0.9876

~~0.0

_{0.5 1.0 1.023 1.025 1.027}

à

a) b)

Fig. 4. a) The global stress-stram diagraJn for t ₌ o-1 and _u

= 0.0000025 with No

= 10~ and

Nq ₌ 1;

b)

_a detail with 2,000 points of this diagraJn around of this maximum.

We have simulated lattices with No _" L _x L libers distributed in _a square lattice for diiserent temperatures ^{and traction} velocities. Due ta the

simplicity

of _our model it _was

possible

ta

perform

extensive calculations that allow _us ta observe

self-organized criticality

in the brittle- ductile transition _region. The observed _power law exponent is

non-universal,

since it

depends

on the externat conditions. This fact pernlit _us to

conjecture

that the fractal d1nlension of the fracture _is also non-universal. In order to check _our

assunlptions,

_we

plan

to

study

_a model for fracture _{on a} lattice of Springs,

using

the _same ideas of this

work,

that is, the

probability

similar to the

equation (1)

and the

propagation

of the cracks _{as a} cascade. The

study

in _a lattice of

springs

will allows _us to obtain trie fracture

profile

hence the fractal dimension of the pattern. ^The

problem

in this kind of lattice is the

computational

cost. These ideas _can also be used to

explain

the results obtained in

fragmentation I?i,

^{where the} exportent

depends

_on the

geometric

form of the

sample. Along

this direction, _we ^will

study

mortels with No

" L _x

PL,

with _p

#

1, and _we

hope

to obtain _a

g-dependent

exponent.

Acknowledgments

We thank Joào Florêncio

Jr.,

Jaiserson

Kamphorst

Leal da Silva and Joâo Antônio Plascak for

helpful

criticism of the

manuscript.

One of _us

(ATB) acknowledges

^{the kind}

hospitality

of the

Departamento

de Fisica of UFMG. We also

acknowledge

the Centra de

Supercomputaçào

of

UFRGS,

for the _use of the

Cray-YMP2

_computer, where part of _our calculations _were made.

Finally,

_we thank

CNPq

and FAPEMIG for financial support.

References

iii

Bak P., Tang C. and Wiesenfeld K., Phys. Rev. Lett. 59

(1987)

381; Phys. Rev. A 38

(1988)

364.

[2] Carlson J. and Langer J., Phys. Rev. Lett. 62

(1989)

2632.

[3] Alstr@rn P., Trunfio P. and Stanley H-E-, Randorn Fluctuations and Pattern Growth, H-E- Stanley and N. Ostrowsky, Eds.

(Kluwer,

Dordrecht, 1988).

[4] Takayasu H. and Inaoka H., Phys. Rev. Lett. 68

(1992)

966.

[5] Nagel K. and Raschke E., Physica A 182

(1992)

519.

[6] Bak P. and Sneppen K., Phys. Rev. Lett. 71

(1992)

4083.

[7] Oddershede L., Dimon P., and Bohr J., Phys. Rev. Lett. 71

(1993)

3107.

[8] ^Mandelbrot B-B-, Passoja D.E. and Paullay A.J., Nature 308

(1984)

721.

[9] Màl@y K-J-, Hansen A. and Hinrichsen E-L-, Phys. Rev. Lett. 68

(1992)

213; Milman V.Y-, Blurnenfeld R., Stelrnashenko N.A. and Bali R.C., Phys. ^Rev. Lett. 71

(1993)

204; Hansen A., Hinnchsen E.L., Mal@y K-J- and Roux S., Phys. Rev. Lett. 71

(1993)

205.

[loi

Lung C.W. and Mu Z-Q-> Phys. Rev. B 38

(1988)

11781.

iii]

Meakin P., Phys. Rep. 235

(1993)

189.

[12] ^{Bernardes A.T. and Moreira} J.G., Phys. Rev. B 49

(1994)

15035.

[13] ^For a review, see: Meakin P-, Statistical Models for the Fracture of Disordered Media, H-J-

Herrmann and S. Roux, Eds.

(North-Holland,

Amsterdam,

1990).

[14] ^Grove R-A-, ASM International Engmeered Matenals Handbook, Vol VI Composites

(1989)

p. 167; Rosen B-N- and Dow N-F-, ibid., _p. 175.