Numerical study of the temperature and porosity effects on the fracture propagation in a 2D network of elastic bonds

0.73 MB·English

by  Harold Auradou

EPJ manuscript No.
(will be inserted by the editor)
5 Numerical study of the temperature and porosity effects on the
0
0 fracture propagation in a 2D network of elastic bonds.
2

n
a Harold Auradou1,2, Maria Zei1,3, and Elisabeth Bouchaud1
J
1 Commissariat `a l’Energie Atomique, DSM/DRECAM/Service de Physique et Chimie des Surfaces et Interfaces, Bˆat. 462,
4
F-91191, Gif-sur-Yvettecedex, FRANCE.

2 Laboratoire Fluide,Automatiqueet Syst`emesThermiques, UMRNo. 7608, CNRS,Universit´eParis 6and 11, Bˆatiment 502,
]
h Universit´eParis Sud,91405 Orsay Cedex, France.
p 3 Laboratoire d’´etude des Milieux Nanom´etriques, Universit´e d’Evry, Bˆatiment des Sciences, rue du p`ere Jarlan, 91025 Evry
- Cedex.
s
s
a
February 2, 2008
l
c
.
s Abstract. This article reports results concerning the fracture of a 2d triangular lattice of atoms linked
c by springs. The lattice is submitted to controlled strain tests and the influence of both porosity and
i
s temperature on failure is investigated. The porosity is found on one hand to decrease the stiffness of the
y materialbutontheotherhanditincreasesthedeformationsustainedpriortofailure.Temperatureisshown
h to control the ductility due to the presence of cavities that grow and merge. The rough surfaces resulting
p from thepropagation of thecrack exhibit self-affine properties with aroughness exponentζ =0.59±0.07
[ overarangeoflengthscales whichincreaseswithtemperature.Largecavitiesalsohaveroughwallswhich

arefound tobefractal withadimension, D,which evolveswith thedistancefrom thecracktip.Forlarge
1
v distances, D is found tobe close to 1.5, and close to1.0 for cavities just before their coalescence with the
4 main crack.
1
0 PACS. 62.20.Mk Fatigue,brittleness,fracture,andcracks62.20.FeDeformationandplasticity(including
1 yield,ductility,andsuperplasticity) 81.40.Np Fatigue, corrosion fatigue, embrittlement,cracking, fracture
0 and failure 05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion 68.35.Ct
5 Interface structure and roughness
0
/
s
c 1 Introduction around the site. The site, which will be called “atom” in
i
s the following, undergoes displacements under the action
y Many materials such as cement or rocks have mechan- of the local forces acting on it. The third model, which
h ical properties which are greatly influenced by the pres- will not be considered here, is the beam model [6] which
p
enceofpre-existingdefectssuchasmicrocracksandmicro- containsfullbondbendingelasticity.Inthiscasethe elas-
:
v porosity due to their elaboration process [1]. In order to tic energy ofthe beam is the sum of the elongation,shear
i understandthemechanicalbehaviorofsuchmaterials,dif- andflexuralenergies.Thiscontrastswiththespringmodel
X
ferent numerical modelling are developed. for which only the elongation of the bonds leads to their
ar The most classical approach consists in using discretiza- failure.
tion schemes for the continuum description. The favorite For all networkapproaches,bonds are supposed to model
scheme in fracture and damage mechanics is the finite el- thematerialatamesoscopiclevel,andtheaimistoinves-
ement method. Yet network models constitute an alter- tigatetheinterrelationbetweendisorderandpropertiesof
native scheme which has been developed in order to sim- thenetwork-suchasfracturestressorstrain,anddamage
ulate the effect of heterogenities on the fracture process. spreading. The surprising result is that properties of the
Network models can be classified in three categories.The network are related to the system size by scaling laws in-
firstgroupiscomposedofscalar models,whichexploitthe volving non trivial exponents, independent of the precise
similarity between the failure of a heterogeneousmaterial distribution,andofthemicroscopicaspectsofthe consid-
submitted to an external load and the breakdown of an ered model [7].
array of randomly distributed fuses [3,4]. The results can Scaling is also observed on rough fracture surfaces for a
be used as a very interesting guideline, but in order to large variety of materials [8,9] (from rocks [10] to wood
compare theory with experiments, it is inevitable to con- [11]throughmetallicalloys[12]andglasses[13,14]),which
sider the vectorialnature of elasticity. In the central force can be described as self-affine structures. Self-affinity [15]
model [5], the bonds are springs which can freely rotate means that a profile extracted from such a surface, de-
2 Harold Auradou et al.: Title Suppressed Dueto Excessive Length
scribedbyaheightsdistributionz(x),wherexisaCarte- particle can be written as follows:
sian coordinate along the profile, remains statistically in-
variant under the scale transformation z(λx) = λζz(x), f =F(r r ) ri−rj (1)
where the Hurst or roughness exponent ζ characterizes ij | i− j| ri rj
| − |
the roughness of the surface. The fluctuation of the sur-
face heights over a length L is given by σ (L)=ℓ(L/ℓ)ζ. where is the modulus and F(u) is a scalar function
z |·|
Here ℓis the topothesy,definedas the horizontaldistance defining the force law. Here, we have chosenF(u) to be a
overwhichfluctuationsinheighthaveaRMSslopeofone linear function of the distance u between atoms: F(u) =
[10].For3dfracturesurfaces,experimentalvaluesofζ are α(u d)(harmonicpotential).Theparametersarefixed
− −
found to be close to 0.8, for most materials [10,11,12,13, insuchawaythatu=distheequilibriumpositionandα
14],withtheexceptionofsomematerialsdisplayinginter- is the spring constant. In order to explicitly eliminate the
granularfractures,suchassandstonerocks,whereζ 0.5 irrelevantparameters,wesuitablyrescalethespatialvari-
[16]. The exponent 0.5 was also measured on glasse∼s [13] ables as well time: in this way, both α and d can be fixed
andonmetallicalloys[17,18]atlengthscalessmallerthan to unity in allthat follows.Accordingly,allthe quantities
thelengthscalesatwhichthe0.8exponentisobserved[8, defined in this paper are dimensionless.
9]. In this work the size of the network is kept constant and
Experiments conducted on two-dimensional samples re- ismade of68608triangularbonds.Due to the orientation
ported somewhat smallest self-affine exponent; 0.6 0.1 of the lattice with respect to the network (see Fig.1), its
for paper [19] and 0.68 0.04 for the fracture of w±ood, sizes in unit of atoms distance d is 886.8 for length, and
when the crack propagat±es along the fibers[20]. 201 for width.
In this paper, we present simulations of a mode I macro As far as the sample is concerned, c denotes the fraction
crackinitiatedbyanotchgrowinginabidimensionalporous of initially missing bonds. c = 0 thus corresponds to a
material. The model is precisely described in Section II. perfectlyhomogeneousmedium.Notethatc=cp 0.653
≡
Theinitialporesaredefinedasregionsofthesamplewhere correspondstotheordinarypercolationthreshold:forc>
bonds are missing. In our model, the temperature of the cp so manybonds aremissing thatthe lattice is no longer
networkiscontrolledanditseffectonthemacroscopicme- macroscopically connected [21]. Moreover, in the case of
chanicalbehaviourof the systemis studied inSectionIII. central forces, there is a second threshold, the so-called
Inthissection,the stress-straincurvescorrespondingto a rigidity-percolation threshold (cr =0.3398 [22,23]) above
samples withno disorderandwith 30%porosityarecom- whichthelatticealthoughconnectedhaszeroYoungmod-
pared for two values of the temperature. It is shown that ulus. In what follows, the fraction of missing bonds is set
under an increasing strain, the pores will grow into cavi- to c=0.3.
ties, and merge with each other and with the main crack. A triangular notch of sides 50 atoms is carved at the left
Fracturehenceproceedsbyvoidsgrowthandcoalescence. sideofthe latticeto actasastressconcentratorandforce
The sizeandthe densityofthe cavitiesis influence bythe a main crack to propagate from the notch tip, along the
temperature.Atlowtemperature,thestressconcentration xdirection(seeFig.1).Thelattice isthensubmittedtoa
due to the initial notch dominate the junction of cavities controlledstrainwhichactsverticallyalongtheupperand
which are most likely collinear and located in its vicinity lower sides of the sample, to which fixed boundary con-
while at high temperature, the cavities spread over the ditions are imposed, while free boundary conditions are
whole materials. This has strong consequence on the fail- chosen along the right and left borders.
ure mechanisum: at low temperature, the material frac- The application of an external strain which gradually in-
tures in a brittle way, while at high temperature, it ex- creases by small steps of size δǫ = 0.000725 results in
hibits a ductile behaviour. Once the porous samples are a deformation of each spring, hence into atoms motion.
broken, we study the resulting rough profiles, which are, Between two successive increases of the strain, the new
like for real cracks, self-affine with a roughness exponent positions of the atoms are computed. The first step of
ζ = 0.59 0.07 that is independent of temperature. The thecalculationconsistsindetermining,foreachatom,the
resultsof±the analysisofthe morphologyofboth the frac- force applied by its neighbours is computed, and the var-
ture profiles and the cavities during their growth prior to ious components are added to get the total force acting
failure are presented in Section IV. Finally, Section V is on the considered atom. Newton’s equation [24] is then
devoted to discussion. solvedfor each atom i (coordinates ri(t) ; velocity r˙i(t)).
For this purpose, we use the leap-frog algorithm [24,25],
which is a modified version of the Verlet algorithm. This
algorithm uses positions and accelerations at time t and
2 The model positions at time t+δt to predict the positions at time
t+δt, where δt is the integration step, set to the value
10−2. This step is repeated N times before a new δǫ in-
Themodelconsistsina2dtriangularlatticewithnearest- crease of the strain is imposed.
∗
neighbour interactions (see Fig.1) that break as soon as A bond breaks when it reaches a critical length d which
themutualdistancebecomeslargerthanaprescribedthresh-is set to the uniform value 1.1. The fracture of a bond
old. More precisely, by noting r the position of the ith transforms its potential energy into kinetic energy, which
i
“atom”, the force f due to the interaction with the jth travelsall overthe lattice. A local dissipation, i.e. a force
ij
Harold Auradou et al.: Title Suppressed Dueto Excessive Length 3
term γr˙ , is added along the left and right boundaries of crack propagation increases with the temperature, as
i
−
(Fig. 1) where we expect the coupling with the external shownin Tab. 1. At low temperature, a sharp decrease of
worldtobemoreefficientinremovingkineticenergyfrom the stress is observed after the critical strain is reached.
the medium. In the present work, complete damping, i.e. The strain-stress curve is more rounded for a larger tem-
γ =1 is imposed. perature.Thiseffectreflectsthepresenceofdamageahead
Acloselookattheamountofkineticenergypresentinthe of the crack tip, as can been seen in Figures 3 and 4. It
system prior to any strain increase reveals fluctuation of is clear from these figures, that a temperature increase
the order of 10% with an average, E constant over the results in an increase of the number of damage cavities.
h i
wholerangeofstrainincludingtheloadingandthefailure This can be seen on the dynamics of bond failures: as
partsofthetest.Theparameterthatcontrolsthe amount shown in Fig. 5, bonds start breaking at a lower strain
∗
ofkineticenergypresentinthenetworkisthenumberN of when the temperature is increased. For T =8, the num-
iterations used to determine atomic positions. A decrease ber of brokenbonds as a function of strainalmost follows
inN resultsinanincreaseoftheamountofkineticenergy a step function, and increases abruptly when the crack
remaining in the network. The latter is used to define a starts to propagate. This distribution broadens when the
reduced equivalent temperature temperature is increased up to 80, showing that some of
the bonds are broken before the main crack propagation.
∗ E Despite thischangeintheshape ofthe distributionofthe
T = h i (2)
ǫ number of broken bonds, which has a strong influence on
c
the macroscopic mechanical property of the network, the
where ǫc = 0.5α(d d∗)2 is the energy needed to break total number of broken bonds changes only slightly, from
−
a single bond (under our conditions ǫc = 0.005). The re- 209 for T∗ = 8 to 241 for T∗ =80, which only represents
duced equivalent temperature can be seen as the number approximately 0.15% of the springs.
ofbonds that the remaining kinetic energymightbreak if The other striking difference occurring when the temper-
it was not diluted in the network. atureis increasedis anincreaseinthe verticalshiftofthe
In the present work, two different values of N are used: stress-strain curve. In fact, a linear fit of the data indi-
N =105 andN =106,whichleadrespectivelytoreduced cates that the strain-stress curves do not pass through 0.
∗ ∗
equivalent temperatures T =80 and T =8. Before dis- This indicates that an excess of stress is present within
cussing the quantitative results concerning the structure the material.This quantity is independent ofthe disorder
of damage and the roughness of the fracture profiles, let andevolvesfrom2.10−4 forT∗ =8to2.10−3 forT∗ =80.
us here briefly illustrate the phenomenology that can be Notethatstressisappliedviatheforcesactingonthesur-
∗
observed for the two values of T . face atoms: a positive stress excess thus indicates a force
acting from the bulk toward the outside and comes from
the energy flux going from the network, at temperature
3 Macroscopic mechanical properties T∗,to its ”cold”sideswhere complete damping ofthe en-
ergy is imposed. Moreover,as for a perfect gas, the stress
Let us first examine the stress response at the two differ- acting on the sides is proportionalto temperature.
ent reduced equivalent temperatures. Figure 2 shows the
variation of the stress as a function of strain for two sets This section points out that changes in the network
∗ ∗
of simulations performed at T =80 and T =8, and for porosityandtemperaturegreatlyinfluenceitsmacroscopic
twodifferentmaterials.Thefirstmaterialisinitiallyintact properties.Theporosityeasesthecreationofdamagecav-
(c = 0), meaning that no bonds were removed. From the ities, the density of which is shown to be dependent on
second one, 30% of the springs were removed at random temperature. At low temperature, the cavities are more
(c=0.3). likelyaheadthecracktip;intheregionwherethestressis
After a first stage where the system gets easily deformed concentrated.Whenthetemperaturerises,cavitiesspread
e.g. for strains smaller than 0.005 , the stress-strain overthenetworkandthecrackpropagatesinthedamaged
− −
curves all exhibit a linear behaviour. The stiffness de- material by meandering from one cavity to another. This
creases when bonds are removed, from 2.5 0.2 for the phenomenon has a strong effect on the maximum strains
±
intactmaterialto 0.70 0.04when c=0.3.This decrease thatcanbesustainedbythestructure.Thedeviationfrom
±
does not seem to be temperature-dependent. the main direction of propagationresults, after failure, in
Thereis anothermajordifference betweenthe behaviours roughfracture profiles.The nextsectionis devotedto the
of the two materials. For a given temperature, (see Tab. analysis of their statistical properties.
1), the initially damagedmaterialbreaksata lowerstress
butsustainsahigherdeformation.Thisisatypical“quasi-
brittle” behaviour, where toughening in an intrinsically
brittle material is the result of damage created ahead of 4 Self-affine properties of the fracture lines
the crack tip, which screens out the external field under-
gone by the main crack. Aftereachmechanicaltest,thepositionsofatomsbelong-
Let us now focus on the effect of the temperature. While ing to the two fracture lines are recorded.Figure 6 shows
the stiffness is only a function of the density of remain- the four profiles obtained from the two tests performed
∗
ing springs,the maximumstrainreachedbefore the onset at T = 8 and 80 on the porous material. In the past
4 Harold Auradou et al.: Title Suppressed Dueto Excessive Length
years, various methods have been developed to measure resultingpostmortemprofileshaveaself-affineroughness
the roughness exponent of self-affine structures. In this characterizedbyanexponentζ =0.59 0.07,independent
±
paper, two independent methods are considered namely ofthe temperature.Yet, the temperature has a strong in-
the average wavelet coefficient (AWC) analysis [26] and fluence on the crossover length which separates the self-
the min-max method [15]. affine regime observed at small scales and the euclidean
In the case of the AWC analysis the one-dimensional line behavior displayed at large scales. At low temperature,
z(x) is transformed into the wavelet domain as the growth of cavities is a consequence of the disordered
structure of the sample, in a region close enough to the
1 ∗ main crack tip for the stress to be high enough. Cavities
W (a,b)= ψ (x)z(x)dx, (3)
[y] √aZ a,b nucleatefrommissingatomsinthisregion,andtheprocess
zone remainsin the vicinity of the cracktip. The fracture
where ψ (x) is obtained from the analyzing wavelet ψ profiles which result from the coalescence of the macro
a,b
(in our case a Daubechies wavelet [27]), via rescaling and crack with the cavities have thus an amplitude which is
translation, ψ (x) = ψ((x b)/a). The AWC measures limited by the lateral extension of the process zone.
a,b
−
theaverage“energy”presentintheprofileatagivenscale, Ontheotherhand,whenthetemperaturerises,theexcess
defined as the arithmetic average of W (a,b)2 over all ofstress due to the undamped kinetic energy(see Section
[y]
| |
possible locations b, and for a statistically self-affine pro- 3) becomes non negligible compared to the stress created
file with exponent ζ, it scales as: W (a,b)2 a2ζ+1. by the notch, and cavities are created everywhere in the
[y] b
h| | i ∼
For the second method, the profile of length L is divided lattice. In this case, the macro crack meanders through
intowindowsofwidthr.Thelineartrendofthelineisthen thewholenetwork,andtheamplitudeofthepostmortem
subtracted from the profile for each window. The differ- profiles is larger.
ence ∆z(r) between the maximum and minimum height Recentlyexperimentalandnumericalobservationsofcrack
are computed on each window and then averaged other propagationindamagedmaterialssuggestedtheexistence
all possible windows. For a self-affine profile, a power law of two self-affine domains [9]. At the scale of the cavity,
behavior is expected : the surface is characterized by an exponent ζ 0.5 while
∼
a larger exponent, ζ 0.8, is observed at the scale of the
∆z(r) rζ (4) ”superstructure” resu∼lting from the coalescence of these
h i∝
cavities. The next section is devoted to the quantitative
For both methods, the self-affine scaling invariance will analysis of the morphology of a single cavity.
be revealedby data alignedalonga straightline ona log-
log plot, with a slope which provides an estimate of ζ.
Figures 7 and 8 shows log-log plots of the results of the 5 Structure of the damage zone
AWC and the min-max methods respectively,for the four
profiles considered. A self-affine domain can be defined in Figure 4 shows clearly that the morphology of the crack
each case and a self-affine exponent can be measured. In profiles is influenced by the presence of cavities. In order
thecaseofthewaveletanalysis(Figure7),ζ isfoundclose todescribequantitativelytheirevolution,wefocusourat-
∗ ∗
to 0.60 0.07 for T = 80 and to 0.55 0.1 for T = 8. tention on one of the largest cavities. Figure 9 shows the
± ±
For the min-max method (Figure 8), a linear fit indicates positionsofatomsbelongingtotheexternalcontourofthe
∗
that ζ = 0.65 0.1 for T = 80 and ζ = 0.55 0.02 for cavityforthree differentvaluesofthe strain,duringcrack
∗ ± ±
T =8.The self-affine exponent characterizingthe geom- propagation. Note that the total number of atoms, 1600,
etry of the profiles may appear to depend slightly on the belongingtothiscontourremainsunchangedthroughthese
temperature,with aslightincreasewhenthe temperature three stages, and that the first contour (stage (1)) is al-
rises from 8 to 80. However, the scaling domain is quite ready the result of the coalescenceof smaller cavities.We
restricted(especiallywhentheAWCmethodisused),and clearly see on this figure that as the crack tip gets closer
the difference lies within error bars. When averaged over tothecavity,thelatterismoreopenandelongated.More-
theimposedtemperature,theself-affineexponentisfound over, when the distance from the crack tip is important,
to be close to 0.59 0.07.The difference in the lowercut- the contourshowsmeanders,the importance ofwhichde-
±
off revealedby the two methods may be attributed to the creases as the crack tip gets closer. In order to describe
presenceofoverhangsontheprofiles(seeFigure6),which the tortuosity of the contours and their possible scale in-
are not included in the AWC description, as discussed variance properties, the average mass method has been
in [28]. Contrary to the value of the exponent, the self- selected [29].
affinecorrelationlength,definedastheuppercutoffofthe This method is very similar to the box counting method
power-lawdomain, appears to be temperature-dependant and consists in computing the number of atoms, N(r) lo-
∗
and is found to be close to 100 atoms spacing for T =8, cated within a circle of radius r with its center located
∗
while forT =80itoverpassesthe systemsize(886inter- on one of the atoms of the contour. The average of N(r)
atomic spacing). over all possible circle centers provides N(r) . Figure
h i
10 displays the evolution of log ( N(r) ) with respect
10
h i
In this section, we have pointed out that despite the to log (r), for the various contours. For a fractal con-
10
ductility enhancement observed on the macroscopic me- tour, N(r) shouldincreasewithr followingapowerlaw,
chanical properties when the temperature is raised, the N(r)h riD, where D is the fractal dimension. For a
h i ∝
Harold Auradou et al.: Title Suppressed Dueto Excessive Length 5
smooth, Euclidean line, D = 1, while for a line filling 6 Discussion
completely the plane, D =2.
Let us first focus on the behaviour of the contour of the Withintheframeworkofabidimensionalnumericalmodel,
cavityatstages(1)and(2).Forthesetwostages,thecav- we haveexaminedcrackpropagationand damagespread-
ity displays a fractal geometry over a domain of length ing in a porous material at two temperatures. We have
scales spanning from the atom spacing, d, up to approxi- shown that damage develops more at high temperature,
mately 50d.Its fractaldimensiondecreasesfromD =1.5 whichresults into adecreaseofthe fracturestrengthand,
(stage (1)) to 1.35 (stage (2)). For stage (3), the aver- correlatively,intoanincreaseinductility.This increasein
age mass displays a more complex behaviour: for length the elongation at failure results from a screening of the
scales smaller than 50 d, the contour has a fractal dimen- externalstressby damage.No plasticityis requiredinthe
sionof1butforlargerlengthscales,thefractaldimension model, which only involves bonds breaking and atom re-
seems to increase. In order to understand this behaviour, arrangements on a local scale. This behaviour is similar
we have analyzed separately the left and ride sides of the to the one observed in quasi-brittle materials [11]-[30].
cavity normal to the external load (Figure 9). As shown The crackmorphologyexhibits inbothcasesthe same
in Figure 10, the two sides are characterized by a fractal self-affine roughness, with an exponent close to 0.6 which
dimension D =1 and no abrupt change is detected. agrees with measurement performed on 2d materials [19,
The analysis of the contour of the cavity indicates that it 20]. The structure of damage at high temperature is also
isindeedfractal,withafractaldimensionwhichdecreases examined. Damage cavities are shown to be fractal, with
continuouslywhenthedistancebetweencracktipandcav- a fractal dimension which decreases from 1.5 to 1 prior
ity decreases.For large distances,the fractaldimension is to coalescence with the main crack. This change in the
found close to 1.5, but just before junction between the fractal dimension is due to the increase of the local stress
cavity and the main crack, the contour of the cavity has generated by the closer vicinity of the crack tip during
a fractal dimension of 1. propagation. This increase results in the coalescence of
It must be remembered however that the fractal dimen- small cavities and in atomic rearrangements of atoms on
sion of a self affine function is not uniquely defined: it thecavityfrontwhichcanbe interpretedasapartial”de-
strongly depends on the range of length scales considered pinning”. A similar change in the morphology of cavities
as well as on the method used. As pointed out in the in- with their size wasactually observedrecently in [31]. The
troduction,theheightfluctuationsofaself-affineprofileis scaling properties of the resulting crack is, in fine, due to
characterized by two parameters: the self-affine exponent the relative positions of damage cavities with respect to
ζ andthetopothesylwhichisthescaleatwhichtheslope eachotherratherthantotheirstructure,sincetheyareno
of the profile is of the order of unity. Above l, the fractal more fractal when they join the main crack and become
dimensionisequalto1fora2dprofiles.Atsmallerlength partofit.Furtherstudiesoftheinter-correlationsofdam-
scales,thedimensionwilldependonthemethodusedand age cavities for 2- and 3-dimensional systems should lead
is D = 2 ζ for the average mass method. Because of toabetter understandingofthe stillmysteriousmorphol-
−
the factthatinthe presentworkthe topothesyofthe two ogy of fracture surfaces.
sides of the cavity is less than the atom spacing the aver-
agemassmethodisnotanappropriatemethodtoanalyse
WeareindebtedtoA.Politi,whoisattheoriginofthemodel
possible self-affine nature of the sides of the cavity.
usedhere.ManythanksalsotoR.Kalia,J-PHulin,D.Bonamy
As mentioned in Sec. 4, a more appropriate tool to de-
andC.Guillot fortheirscientificsupport,andtoY.Meurdes-
scribe the self-affine nature of the profiles is the AWC
oif and P. Kloos (CEA-Saclay ComputerScience Division) for
method. The latter has been applied to the two sides of
their technical support in the parallel simulations. HA is sup-
thecavity(SeeFig.7),theydisplayaself-affinecharacter-
portedbytheCNRSandANDRAthroughtheGdRFORPRO
isticwithanexponent,ζ 0.6,closetothevalueobtained (contribution No. XXXX) and the EHDRA (European Hot
∼
for the fracture profiles over length scales ranging from 6 DryRock Association) and PNRH programs.
to 50 atoms spacing.
Thissectionwasdevotedtotheanalyseofthemorphology
of a single cavity. Previous works suggested that cavities
have rough walls with a self affine geometry character-
ized by a self affine exponent close to 0.5 [9]. The dam-
agecavitiesobtainedwithourmodelizationisfoundtobe
self-affinewithanexponentclosetotheonewhichcharac-
terizesthepostmortemsurfacei.e.0.6.Yet,theselfaffine
regime is observedover a narrowrange of scale (less than
one decade), making difficult any conclusion. Neverthe-
less, the structure of the walls of the cavity was found to
besensitivetothepresenceofthecracktip.Thiscontrast
with the scenario suggested in [9] in which the surfaces
created by the failure have a ”quenched” geometry.
6 Harold Auradou et al.: Title Suppressed Dueto Excessive Length
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man Lectures on Physics, Vol. 1, Addison-Wesley, 1963,
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Chapter9 (“Newton’s Laws of Dynamics”).
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Harold Auradou et al.: Title Suppressed Dueto Excessive Length 7
0.03
0.02
σ
0.01
0
0 0.01 0.02 0.03 0.04
ε
Fig. 2. Stress-strain evolution of the system. Circles and
squares correspond respectively to c = 0.3 and c = 0 (non-
porousmaterial).Filledandemptysymbolscorrespondrespec-
tively to T∗ =8 and 80.
Fig. 4. Breakdown under tension of a network kept at the
temperature T∗ = 80. From bottom to top the strain is re-
spectively ǫ=0.0217, 0.0231, 0.0225 and 0.0289.
0.02
0.015
σ 0.01
0.005
0
0.015 0.02 0.025 0.03
ε
Fig. 5. Filled andunfilledsymbols correspond respectivelyto
T∗ = 8 and 80. Circles show the stress-strain variation for a
Fig. 3. Breakdown of the atoms network under strain test
performedatthetemperatureT∗ =8.Frombottomtotopthe porosity c=0.3. Diamonds show thenumberof broken bonds
verticaly normalised to fit theplot.
strain is respectively ǫ=0.0223, 0.0224, 0.0225 and 0.0282.
8 Harold Auradou et al.: Title Suppressed Dueto Excessive Length
200 2.0
100 1.5
>
∆z (r) 1.0
0 <
g 10
o
l
0.5
−100
0.0
−200 0 1 2 3
0 200 400 600 800 log (r)
10
Fig. 6. Fracturesprofilesobtainedafterthebreakdownofthe
Fig. 8. log (h∆z(r)i) as function of log (r). Circles and
lattice with c = 0.3. The two top curves correspond to the 10 10
tporpofialnesdabroetftoormTp∗r=ofi8le.s obtained for T∗ =80. The two lowest spqruofiarleessashftoewr ftahileurreesautltTo∗f=the80anwahlyilseedoifamthoendtospananddtrbiaonttgolems
areobtainedwhen topand bottom failure profilesobtainedat
T∗ = 8 are considered. These results where shifted verticaly
6
for convinience. The dotted line has a slope of 0.6 .
5 800
4
>)b
2b)
a,3
W( 700
<
g(102
o
l
1
600
0
0 1 2 3
log (a)
10
Fig. 7. log h|W (a,b)|2i as function of log (a) where a is
10 [y] b 10
inunitofd.Circlesandsquaresshowtheresultoftheanalyse 500
ofthetopandbottomprofilesafterfailureatT∗ =80whiledi- 0 50 100 150 200 250 300
amondsandcrossesareobtainedwhentopandbottomfailure Fig. 9. From theleft to theright,pictures of thesame cavity
profilesobtainedatT∗ =8areconsidered.Thefilled triangles at stages (1), (2) and (3) of the test performed on the porous
correspondtotheaverageoftheanalyseofthetwosidesofthe material at temperature 80. The distance from the crack tip
cavity at stage (3) displays in Fig. 9. The dotted lines has a is respectively : 393, 359 and 355 atoms spacing. The boxes
slope of 2ζ +1 = 2.2 corresponding to a self-affine exponent indicate theleft and right sides of the cavity at stage (3).
ζ =0.6. These results where shifted verticaly for convinience.
Harold Auradou et al.: Title Suppressed Dueto Excessive Length 9
3.5
3
2.5
M(r))
g(10
lo 2
1.5
1
0.5 1 1.5 2 2.5
log (r)
10
Fig. 10. Log log representation of the average mass M(r)
measured within circle of radius r applied to the three cav-
ities displayed in figure 9. Cirlcle, squares and diamonds are
respectively for cavities 1, 2 and 3. Triangles up and left are
for the right and left sides of the cavity (3) displayed in Fig.
9. The long dashed, the dot dashed and the dahed lines have
respectively a slope of 1.5, 1.35 and 1.