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Failure model for heterogeneous structures using
structured meshes and accounting for probability aspects
Martin Hautefeuille, Sergiy Melnyk, Jean-Baptiste Colliat, Adnan Ibrahimbegovic
To cite this version:
Martin Hautefeuille, Sergiy Melnyk, Jean-Baptiste Colliat, Adnan Ibrahimbegovic. Failure model for
heterogeneous structures using structured meshes and accounting for probability aspects. Engineering
Computations, Emerald, 2009, pp.166 - 184. �10.1108/02644400910924852�. �hal-00542668�
meshes and a ounting for probability aspe ts
M. Hautefeuille, S.Melnyk, J.B. Colliatand A. Ibrahimbegovi
E ole Normale Superieure de Ca han,
LMT-Ca han, Genie Civilet Environement
61, avenue de president Wilson, 94235 Ca han, Fran e
e-mail: ailmt.ens- a han.fr, fax. +33147402240
Abstra t
Purpose-Inthisworkwedis usstheinelasti behaviorofheterogeneousstru tureswithin
theframeworkof niteelementmodelling,bytakingintotherelatedprobabilisti aspe tsof
heterogeneities.
Design/methodology/approa h- We show how to onstru t the stru tured FE mesh
representationfor the failure modelling forsu h stru tures, by using a building-blo k of a
onstant stress element whi h an ontain two di erent phases and phase interfa e. We
present all the modi ationswhi h are needed to enfor e for su h an element in order to
a ountforinelasti behaviorinea hphaseandthe orrespondinginelasti failuremodesat
thephaseinterfa e.
Findings- Wedemonstratebynumeri alexamplesthat theproposedstru tured FEmesh
approa hismu hmoreeÆ ientfromthenon-stru turedmeshrepresentation. Thisfeatureis
ofspe ialinterestforprobabilisti analysis,wherealargeamountof omputation isneeded
in order to provide the orrespondingstatisti s. Onesu h ase of probabilisti analysis is
onsideredin this work wherethe geometryof thephaseinterfa eisobtainedas theresult
oftheGibbsrandompro ess.
Originality/value- Con rms that one an make the most appropriate interpretation of
theheterogeneousstru turepropertiesbytakingintoa ountthe nedetailsoftheinternal
stru ture,alongwiththerelatedprobabilisti aspe tswiththepropersour eofrandomness,
su h astheoneaddressedhereinin termsofporosity.
Keywords: heterogeneousstru tures,failuremodes, niteelement,stru turedmesh,prob-
abilityaspe ts
Paper type: resear hpaper
1 Introdu tion
Thedomainofnumeri alanalysisforultimateloadbehaviorofCivilEngineeringstru tureleads
to many important issues, hief among them a ounting forheterogeneities of real stru tures.
For example, the stru tures built of ement-based materials, su h as on rete or mortar, an
be modelled at di erent s ales, depending on the obje tives and the physi al me hanisms to
bea ountedfor. Namely, forengineeringappli ationsand omputations atthe stru tures ale
(ma ro-s ale), the material might be onsidered as homogeneous, and its properties obtained
byusingthe key on ept of RVE (see [2 ℄,[13 ℄) to obtainphenomenologi almodelsof inelasti
behavior (e.g. see [28℄, [1 ℄, [7 ℄) The main advantage of those models is their robustness and
small omputational ost,hen e thisapproa h is widely spread. On theother hand, su h phe-
nomenologi almodelsarebasedonasetof "material"parameterswhi houghtto beidenti ed,
mainly from experiments performed with pres ribed load paths. This methodology leads to a
path, thusleadingto a non-predi tivema ro-model.
In order to over ome thismajordrawba k many authors triedto furnishmi ro-me hani al
bases to thema ros opi modelset of parameters(see [17 ℄, [15 ℄)and provideamore predi tive
model. One wayto a hieve thisgoal isto employhomogenization methodsleadingto a urate
results forlinear problems. In ase of non-linearitiessu h methods arenot providing good es-
timates for the e e tive (ma ros opi ) properties (see [6 ℄). Moreover, su h approa h does not
take intoa ount theinherentun ertaintiesatta hedto heterogeneousmaterialsandstru tures.
Consideringasmalls ale,thisvariabilitymightbeviewedfromthegeometri alpointofview
throughthe sto hasti des ription of the meso-stru ture. In thiswork we propose to ompute
thema ros opi parametersforaporousmediaaswellastheirstatisti sbytakingintoa ount
the variability of the meso-stru ture. The key point is that the material parameters at this
levelare assumed to bedeterministi ,so thatthe variabilityis onlyrelated to thesize and the
positions of the voids. In order to solve this sto hasti problem and ompute the statisti al
moments forthe responsequantities, we employthe Monte-Carlo method withina distributed
softwareenvironment. This sto hasti integration method is basedon manyevaluations of the
meso-stru turesresponsesthusleadingto atime- onsumingpro ess. Moreover,astheerror an
dire tlybeevaluatedintermsofthenumberofrealizations,itisne essaryto hoosearelatively
smalldis reteproblem,eveninthe aseof omplexmeso-stru tures. Toa hieve thiswepropose
amodelbasedonaregularmeshwhi hisnot onstrainedbythephysi alinterfa es. Thismodel
relies on lassi al CSTelements, whose kinemati s des riptionis enri hed bythe use of strain
and displa ementsdis ontinuitiesinorderto represent two phases.
Theoutlineofthispaperisasfollows;inSe tion2,wepresenttheplasti itymodelemployed
at the meso-s alelevel. Being based on regular meshes,thismodel an lead to fast omputing
ofnon-linearresponseeven for omplexmeso-stru turegeometries. InSe tion3wedes ribethe
sto hasti problem,thegeometri aldes riptionpro ess forde ningthemeso-stru tureand the
sto hasti integration method. Finally, in Se tion 4 we show and dis uss the resultsobtained
throughnumeri alsimulations.
2 Plasti ity model for failure of heterogeneous materials
Meshing is one of the major issues in modelling heterogeneous two-phase materials and fre-
quentlyleadsto undesirablyhighnumberof degrees-of-freedomanddistortedmeshes. Forthat
reason, themeshingpro essmightrequirea omplexand time- onsumingalgorithmand, more
importantly, produ ethe set ofdis rete equationswhi his poorly onditioned. In thisse tion,
we present another approa h by using stru tured (regular) meshes whi h are not onstrained
bythephysi alinterfa esbetweendi erentphases. Thekeyingredientforprovidingsu hmod-
els are eld dis ontinuitiesintrodu ed inside the elements in whi h the physi al interfa es are
present. The latter an bedeveloped as thekinemati s enhan ementswhi h belongwithin the
frameworkoftheIn ompatibleModesMethod(see[26 ℄, [11 ℄),andrequiresa dedi atedsolution
algorithm whi his illustratednext.
2.1 Plasti ity model with stru tured meshes
Intwo dimensional ontext, we onsideraheterogeneous materialforwhi hthein lusionsposi-
tionsandshapesareknown,thusleadingto xedpositionsofthedis ontinuitiesinea helement.
Figure1showsa3-nodetriangular niteelement representing twophases. Inorder to takeinto
a dis ontinuity of the strain eld and a dis ontinuity of the displa ement eld, both of them
lyingatthesame position(pres ribedbythephysi alinterfa ebetweentwophases). Thestrain
dis ontinuitypermitsthe properstrain representation oftwo di erent sets ofelasti properties
orresponding to ea h phase. The displa ement dis ontinuityleads to the possibilityto model
adebonding failureme hanismat theinterfa e. Forthelatter,two failureme hanismsare on-
sidered: one orresponding to theopening of the ra k inthe normal dire tionand the se ond
onetotheslidinginthetangentdire tion(see[22 ℄). Bothofthesedis ontinuitiesareintrodu ed
byusingtheIn ompatible Modes Method(see [26 ℄,[11 ℄ ) leadingto thesame numberof global
degrees-of-freedom.
These kinemati s enhan ements areadded on topof thestandard CSTelement (Figure1).
Thiselementisdividedintotwopartsbyintrodu inganinterfa ewhosepositionisde nedbytwo
parameters:
1
;
2
2[0;1℄. Theseparameters
i
areobtainedfromtheinterse tionofthe hosen
stru tured mesh with the in lusionspla ed within the stru ture. The orresponding values of
i
at ea h element boundaryare sharedbetween two neighboringelements. The domain e
of
the standard 3-node CST element is thus dividedinto two sub-domains e
1
and e
2
. Di erent
elasti -plasti orelasti -damagebehaviorlawsmightbe hosenforea h ofthesetwoparts, with
di erent elasti properties (see[9 ℄).
Figure1: Twophase3 node triangular element;withstressve tor ontinuity enfor ed a rossthe
interfa e.
Contrary to thedispla ement elddis ontinuity, whi his a tivateda ording to the hosen
failure riterion,thestrain elddis ontinuityisalwayspresent. Introdu ingthosedis ontinuities
requires to enhan ethekinemati s of theelement byusingtwoin ompatible modes. Thus, the
displa ements eld an be writtenasfollows:
u h
(x;t)= 3
X
a=1 N
a (x )d
a
(t)+M
I (x)
I
(t)+M
I (x)
I
(t)+M
II (x)
II
(t) (1)
This expression ontains four terms: the rst one provides onstant strain inside the element
(CST). The se ond term represents a jump in the displa ements eld in the normal dire tion
and the third a jump of displa ements eld in the tangential dire tion. Finally, the last part
provides thestrain eld dis ontinuity.
The shape fun tions M
I
(x ) for the rst in ompatible mode (Figure 2a.) orresponding to
the displa ements eld dis ontinuity for both normal and tangent dire tions (see [8℄) an be
writtenas:
M
I
(x )=H
S (x)
X
a2 e
1 N
a
(x ) (2)
where N
a
represents the normal shape fun tions of a CST element and H
S
the Heaviside
fun tionpla edat theinterfa eposition.
II
thejumpinthe strain eld (SeeFigure 2b.) an be writtenas:
M
II (x)=
8
>
>
>
>
>
>
<
>
>
>
>
>
>
:
1
(x
6 x
1 )(y
4 y
1 ) (x
4 x
1 )(y
6 y
1 )
(y y
1 )
(x
4 x
1 )+(x
6 x
1 )z
4
(x x
1 )
(y
6 y
1 )z
4 +(y
4 y
1 )
; x;y2
e
1
1
(x6 x2)(y3 y2) (x3 x2)(y6 y2)
(y y
2 )(x
3 x
2 ) (y
3 y
2
)(x x
2 )
; x;y2 e
2
(3)
The shapefun tion M
II
(x ) expressionis obtainedbyusingtheequations ofthe two planes
de ned bynodes 2,3 and 6 for one sub-domain and 1, 4 and 6 forthe se ond one (see Figure
1). These geometri onditions are suÆ ient to de ne M
II
(x) for the real displa ement eld.
Thesameshapefun tionalso satis esthepat htest ondition(e.g. see[11 ℄)whi henfor esthe
element apabilityto represent onstantstress eld.
Figure2: In ompatible modes orrespondingto displa ements a)and strain b) dis ontinuitiesof
CST element
With these resultsinhand,thestrain eld an be writtenasfollows:
"
h
(x;t) = 3
X
a=1 B
a (x)d
a
(t)+G
II (x )
II (t)
+ (n T
n)G
Ir (x)
I (t)+
1
2 h
n T
m+m T
n i
G
Ir (x )
I (t)
= Bd+G
II
II +(n
T
n)G
I
r
I +
1
2 h
n T
m+m T
n i
G
I
r
I
(4)
where B(x) are the well known strain-displa ement matrix for CST element, ontaining the
derivatives oftheelement shape fun tions(e.g. see [28 ℄),
B(x )= 2
6
4
N
1
x 0
N
2
x 0
N
3
x 0
0
N
1
y 0
N
2
y 0
N
3
y
N
1
y
N
1
x
N
2
y
N
2
x
N
3
y
N
3
x 3
7
5
(5)
and G
I
r
(x ) ontainsthederivativesof the rst in ompatiblemode
G
Ir (x)=
2
6
4
N
2
x +
N
3
x
0
0
N
2
y +
N
3
y
N
2
y +
N
3
y
N
2
x +
N
3
x 3
7
5
(6)
G
Ir
= G
Ir +G
Ir Æ
S
=
X
a2 e+
B
a
| {z }
G
Ir
+nÆ
S
|{z}
G
Ir
(7)
In (7) above, Æ
S
is the Dira delta fun tion providing the jump of displa ement eld. It is
important to notethat su h a shape fun tionought to be modi ed into G
Iv
for representation
of thevirtualstrain eld
G
I
v
=G
I
v +G
I
v Æ
S
(8)
This kind of modi ation, needed to enfor e the satisfa tion of the Pat h Test ([28 ℄), an be
obtainedbyfollowingpro edure rstproposedforamodi edversionoftheIn ompatibleModes
Method (see [11 ℄)leadingto:
G
Iv
(x ) = G
Ir (x )
1
A Z
e
G
Ir (x)d
=
X
a2 e+
B
a +
1
A Z
e
X
a2 e+
B
a d
l
S
A n
| {z }
G
I
v
+ nÆ
S
|{z}
G
I
v Æ
S
(9)
Finally, in (4), G
II
is the matrix ontaining the derivatives of the se ond shape fun tion
M
II (x):
M
II (x)
x
= (
1
(x
6 x
1 )(y
4 y
1 ) (x
4 x
1 )(y
6 y
1 )
[ (y
6 y
1 )z
4 +(y
4 y
1
)℄; x;y2 e
1
1
(x
6 x
2 )(y
3 y
2 ) (x
3 x
2 )(y
6 y
2 )
[ y
3 y
2
℄; x;y2
e
2
(10)
M
II (x )
y
= (
1
(x
6 x
1 )(y
4 y
1 ) (x
4 x
1 )(y
6 y
1 )
[(x
4 x
1 )+(x
6 x
1 )z
4
℄; x;y 2 e
1
1
(x
6 x
2 )(y
3 y
2 ) (x
3 x
2 )(y
6 y
2 )
[x
3 x
2
℄; x;y 2
e
2
(11)
2.2 Operator split solution for interfa e failure
The total system onsists of four equilibrium equations, with (12a) as the global equilibrium
equation and(12b) to(12d) are orrespondingto thelo alones. Equations(12b) to(12 ) have
to be solved only in ase of a tivation of the displa ement dis ontinuityin the normal or the
tangentialdire tion.
8
>
>
<
>
>
: A
nel
e=1
f int
f ext
=0
h ;e
I
=0
h ;e
I
=0
h e
II
=0
=) 8
>
>
>
<
>
>
>
: R
e
B T
d R
e
N T
bd=0
R
e
G ;T
Iv
d=0
R
e
G ;T
I
v
d=0
R
e
G T
II
d=0
(12)
Bythe onsistentlinearization(e.g. see[7 ℄)ofthissystemofequationsweobtaininthematrix
form
2
6
6
6
6
4 K
e
F ;e
I
r F
;e
I
r F
e
II
F ;e
T
I
r H
;e
I F
e
H F
;e
S
F ;e
T
I
r F
e T
H H
;e
I F
;e
S
F e;T
II F
;e T
S F
;e T
S H
e
II 3
7
7
7
7
5 (k)
n+1 0
B
B
d
I
I
II 1
C
C
A (k+1)
n+1
= 0
B
B
r
0
0
0 1
C
C
A (k)
n+1
(13)
K e
= R
e
B T
:C ep
:Bd
F ;e
I
r
= R
e
B T
:C ep
:(n T
n)G
Ir d
F ;e
I
I
= R
e
B T
:C ep
: 1
2 n
T
m+m T
n
G
I
r d
F e
II
= R
e
B T
:C ep
:G
II d
F ;e
T
I
r
= R
e
G ;T
I
v :C
ep
:Bd
H ;e
I
= R
e
G ;T
I
v :C
ep
:(n T
n)G
Ir d+
R
S G
;T
I
v
t
S
I d
F e
H
= R
e
G ;T
I
v : C
ep
: 1
2 n
T
m+m T
n
G
I
r d
F ;e
S
= R
e
G ;T
Iv :C
ep
:G
II d
F ;e
T
I
r
= R
e
G ;T
I
v :C
ep
:Bd
F e
T
H
= R
e
G ;T
I
v :C
ep
:(n T
n)G
Ir d
H ;e
I
= R
e
G ;T
I
v :C
ep
: 1
2 n
T
m+m T
n
G
I
r d+
R
S G
;T
I
v
t
S
I d
F ;e
S
= R
e
G ;T
Iv :C
ep
:G
II d
F e;T
II
= R
e
G T
II :C
ep
: Bd
F ;e
T
S
= R
e
G T
II :C
ep
:(n T
n)G
Ir d
F ;e
T
S
= R
e
G T
II : C
ep
: 1
2 n
T
m+m T
n
G
Ir d
H e
II
= R
e
G T
II :C
ep
:G
II d
(14)
In order to solve this system, we arry out stati ondensations (e.g see [27 ℄). The last three
equationsaresolvedatlo allevel(numeri alintegrationpoints),thusthetotalnumberofglobal
unknownsremainsthesameaswiththestandardCSTelement. Thesestati ondensationsleads
to thee e tive sti ness matrix(see [23 ℄), whi h an be writtenasfollows:
b
K e;(k)
n+1
=K e;(k)
n+1 h
F ;e
I
r F
;e
I
r F
e
II i
(k)
n+1 2
6
4 H
;e
I F
e
H F
;e
S
F e
T
H H
;e
I F
;e
S
F ;e
T
S F
;e T
S H
e
II 3
7
5 (k)
1
n+1 2
6
4 F
;e T
I
r
F ;e
T
I
r
F e;T
II 3
7
5 (k)
n+1
(15)
Finally, the global system of equations (12) is solved to obtain the updated value of the dis-
pla ement eldd (k+1)
n+1
=d (k)
n+1 +d
(k+1)
n+1
b
K (k)
n+1
d (k+1)
n+1
= r (k)
n+1
(16)
2.3 Model problem of lo alized failure
Inthisse tionwe onsiderthe onstitutivebehaviorattheinterfa einmoredetails. Asalready
mentioned,thepositionsoftheinterfa esarepres ribedinadvan ea ordingtothepositionsof
thein lusionsandwesupposethat ra ks ano uronlyattheinterfa es. Inordertoinvestigate
these interfa es behavior, we shall onsiderthe se ondequation of thesystemin(12).
Z
e
G T
Iv
d+ Z
S G
T
Iv t
S
d =0 (17)
Z
e
G T
Iv
d+G T
Iv t
S l
S
=0 (18)
lengthof theinterfa eand of thestrain eld
t
S
= 1
l
S Z
e
G T
Iv
d
= 1
l
S ngp
X
l =1 G
T
I
v : C
ep
:
"
Bd+G
II
II +(n
T
n)G
I
r
I
+ 1
2 h
n T
m+m T
n i
G
Ir
I
#
j
l w
l
(19)
wherengpisthetotalnumberofintegrationpointsinea helementandj
l andw
l
arerespe tively
theisoparametri transformation ja obianand thenumeri al integration weight.
Considering onlya nonlinearpart ofstrain at the given interfa e, we an furtherobtainan
in rement of tra tion-ve torsa ording to:
t
S
= 1
l
S npg
X
l =1 G
T
Iv :C
ep
:
"
(n T
n)G
Ir
I +
1
2 h
n T
m+m T
n i
G
Ir
I
#
j
l w
l
= 1
l
S npg
X
l =1 G
T
I
v :C
ep
:(n T
n)G
I
r
j
l w
l
| {z }
K
oef
I
1
l
S npg
X
l =1 G
T
I
v :C
ep
: 1
2 h
n T
m+m T
n i
j
l w
l
| {z }
K
oef
I
= K
oef
I +K
oef
I
(20)
In order to represent thedebonding between the two phases, we hoose a softening law of
exponential form for the ra k both in normal and in tangent dire tions. The orresponding
failure riteriahave been hosen asfollows:
D;n
n+1
=t
n+1
n (
f q
D
n+1
)0 (21)
where
q D
n+1
=(
1
s )
h
1 e b
n+1 i
(22)
The integration algorithm isbasedon a trialstate whi hlookslike:
D;n;tr
n+1
=t
n
n (
f q
D
n
) (23)
A ording to this trial state, the in rement of the softening variable depends on the plasti
multipliers
n+1
=
n +
n+1
=
n +
n+1
(24)
In order to obtainthese given parameters
n+1
, we solve the following lo al equation by using
Newton'smethod
D;n
n+1
= t
S;n+1
n (
f q
D
n+1 )
= t
s
n (
f q
D
n )
| {z }
D;n;tr
n+1
+q D
n+1 q
D
n +t
n+1
n
= D;n;tr
n+1 +(
1
s )
h
1 e b
n+1 i
e b
n
+t
S;n+1
n (25)
softening:
t
S;n+1
=t tr
S;n+1 +t
S;n+1
; q
D
n+1
=q D;tr
n+1 +q
D
n+1
(26)
Finally, the in ompatible mode parameters
I and
I
are omputed in the manner similar to
plasti strainat the interfa e:
I;n+1
= tr
I;n+1 +
I;n+1
;
I;n+1
= tr
I;n+1 +
I;n+1
(27)
Withthelo alin ompatiblemodesparameters
I;n+1 and
I;n+1
we an updatethestrain eld
inea h sub-domainof theelement. Byusingoperator-splitsolutionpro edure(15)wesolve the
globalsystemof equationsin(16).
3 Probability aspe ts of inelasti lo alized failure for heteroge-
nous materials
The mainobje tive ofthis se tionis to illustrate thepossibilitiesprovided by theuseof stru -
tured meshrepresentationand theeÆ ient omputation apabilities of theproposedmodel for
dealingwithrandomheterogeneities. Tothatend,we onsiderhereinaporousmaterial(typi al
ofmany ement-basedmaterial)atameso-s alelevel. Atthiss aleweassumethatsu hmaterial
is hara terized byatwo-phasemi rostru turewithasolidphaseanda uidphase. Theformer
willbereferred asthe "matrix" andthe latteris supposedto represent thevoids orin lusions.
Depending on the number of in lusions, their sizes and positions, the non-linear ma ros opi
response of su h a material will vary. In other words,the ma ros opi hara teristi s, su h as
Young's modulus or the yield stress, willbe in uen ed by the meso-s ale geometry. Our goal
here is to arry out numeri allythe variations of the ma ros opi hara teristi s upon the in-
lusion sizes and positions. The key point for thisstudyis that thevariabilityintrodu edinto
the model is restri ted to the spe imen geometry only, whereas the me hani al hara teristi s
of thetwo phasesareassumed to be deterministi .
To be more pre ise, thematrix phase is supposed to be a urately modelled by an elasti -
perfe tlyplasti model based uponthe Dru ker-Prager riterion (see [5 ℄). The voids arerepre-
sented bya simplelinear isotropi elasti itymodelwith very smallYoung'smodulusvalue. In
thefollowingse tionswe rstbeginto des ribe theGibbspointpro ess,leadingto therealiza-
tions of the meso-stru tures. We also show an example of one typi al mesh obtained and the
orresponding ma ros opi responseto a tension test. Then we turn to the des riptionof the
sto hasti integrationmethod whi hhasbeen hosen tonumeri allysolvethisproblemand the
orresponding Software Engineering aspe ts. Finallywe show and dis ussthe resultsobtained
forthissto hasti problem.
3.1 Meso-s ale geometry des ription
Here we des ribeboththepro ess and thehypothesis leadingto themeshing pro edure within
a re tangular domain (3:61:8 m 2
). The meso-stru ture geometry of su h domain is here
supposedto bea urately modelledbya Gibbspointpro ess. Su h point pro ess is builton a
two steps s heme. The rst one is the determination of the in lusions number a ording to a
Poissonlaw. The se ondstep onsistsinthedetermination ofthe in lusion enters oordinates
as wellas theradius forea h in lusion. While su h a Gibbs pro ess already naturally leads to
a set of non-interse ting in lusions, we applied an even more restri tive riterion, by hoosing
the minimaldistan e between the in lusions (here equal to 2mm). Moreover, in order to be
onsistent withthemeshsizeand themodelfeatures, thein lusionsradiiareboundedbetween
orrespondingstru turedmesh. We an noti ethatea hin lusionis orre tly modelledbyaset
of dis ontinuitieswithoutanymajordistortion.
Figure 3: Meso-stru ture geometry a)and orresponding meshb)
Sin ethematerialparametersare hosentobedeterministi ,thestatisti softhema ros opi
response depends on the meso-stru ture geometry only, de ned by the in lusions radius and
enterspositions. Thusthema ros opi problemissto hasti andrequiressto hasti integration
method whi his presentedinthenext se tion.
3.2 Sto hasti integration
Sin ethe positionsand thedimensionsofthein lusionsinthematrixaredes ribedbydis rete
random elds de ned by Gibbs point pro esses,we obtaina random ma ros opi behavior for
this me hani al model. A 2D random point pro ess an be de ned as a nite set of random
variables, whi h are indexed by the spatial oordinates ve tors in R 2
. As a result, the geom-
etry of our stru tureis de ned as a random eld,whi h impliesthat every solution omputed
by the me hani al model is also a random eld. For example, the stru ture displa ement at
a xed point is also a random variable. In this study, we are interested in hara terizing the
ma ros opi me hani al properties of our stru ture. To a hieve this goal, we use a global ap-
proa h whi h onsists in identifying the material properties governing the global behavior of
the stru ture. More pre isely, we aim to determine the e e tive global material properties by
the orresponding identi ationof theglobalresponse omputedbytheFiniteElement model.
Therefore, sin e the global responses (displa ement and rea tions) are random variables, the
globalmaterialpropertiesweaim toidentify,su hastheYoungmodulusortheyieldstress,are
also randomvariables.
Probabilisti hara terizationofthema ros opi me hani alproperties an beviewedasde-
s ribingtheprobabilisti lawfollowedbyea hoftheseproperties. Twoapproa hes anbedrawn
to nda probabilisti lawdes ribing a randomphenomena. The rst one, so- alledfrequentist
approa h [14 ℄, is based on statisti al tests, like the 2
test for the Gaussian probability law.
Results of these tests are error margins that evaluate how the out omes of the given random
phenomena twithrespe tto agiven probabilitylaw. The se ond,so- alledBayesianapproa h
[12 ℄,is trying to useall theavailableinformationalong withthemaximumentropy theory(see
[21 ℄, [25 ℄)inorder to providethe mostgeneralprobabilitylawfora given state of information;
thus,tofullydes ribethisprobabilitylaw,thestatisti almomentsofdi erentordershavetobe
omputed. In this work, the se ond approa h is hosen. The ma ros opi material properties
we tend to hara terize are all de nedon thepositive real line. Moreoverwe assumethat they
an be given a mean value and a nitestandard deviation. On the basi of su h information,
thelog-normal distribution,whi his fullydes ribed by its omputedmean value andstandard
deviation.
Consequently, in order to hara terize the ma ros opi me hani al properties using the
Bayesian approa h, the rst two statisti al moments of ea h of these properties have to be
omputed. The statisti al moment of anyrandomvariable is an integralof a fun tionalof this
random variable over a probabilityspa e. Hen e, an eÆ ient numeri al tool to ompute su h
integral in multi-dimensional spa e is required. Rather than high order quadrature rules like
Smolyakalgorithm[24 ℄,weusehereasimpledire tintegrationalgorithmisMonteCarlosimula-
tion[3℄. Thebasi ideaofMonte Carlosimulationisto approximatetheintegralsofafun tional
of a random variable by a weighted sum of realizations of thisrandom fun tional. Let be a
randomvariablede nedon some probabilityspa e (;B;P), whereisthespa e ofevents, B
isa-algebrabuiltonandP aprobabilitymeasure. Anyde nedmomentof an bewritten
as R
f((!))dP(!). The simple Monte Carlo algorithm onsist in approximating thisintegral
as a nite weighted sum of realizations f((!
i
)), ea h omputed at a randomly independent
hosen point !
i
in , multipliedby the orrespondingweights 1
N
(withN thegiven numberof
realizations)
Z
f((!))dP(!) 1
N N
X
i=1 f((!
i
)) (28)
For this kindof numeri al integration, the onvergen e rate an be a priori omputed thanks
to the entrallimit theorem [16℄. We an ndthe error estimate whi h is proportional to the
standarddeviation of f() over p
N, N being thenumber ofevaluations off(). Asea h real-
ization of the Gibbs pro ess is sto hasti ally independent from the others, thismethod an be
dire tlyappliedhereandfurthermoreparallelizedusinganappropriatesoftwareenvironmentto
eliminatethemaindrawba kofMonteCarloalgorithm,theslow-rate onvergen e. Inthis ase,
where no orrelation exists in the geometri al spa e, no other tools su h as Karhunen-Loeve
expansionis required(see [16 ℄,[4 ℄).
The software ar hite ture used here is based on the software omponent te hnology and
the middleware CTL [19 ℄, whi h provides the adequate network environment to enable ode
ommuni ationunder apres ribedproto ols and more generally ode oupling. The basi idea
ofsoftware omponent te hnologyis todivideasoftwareframeworkintoseveral tasksand then
to implement software omponents, ea h of them being able to arry out this parti ular task.
Existing software an be turned into a omponent by de ning an interfa e through whi h the
ommuni ationwillbe hannelled. Implementinga omponentfrom fora pre-existingprogram
onsists in oding a set of methodsthat other software an allthrough this interfa e. In the
ase of Monte Carlo simulations,twodi erent tasks an be drawn. One is to generatea Gibbs
pro ess and to transfer thisresult de ning thein lusions geometry ina stru tured mesh. The
otheris to runa omputationwiththisgivengeometry withintheme hani almodel de nedin
the rst se tion. A CTL software omponent has beenpreviouslyobtained [18 ℄ from theFEM
odeFEAP[28 ℄wherealsotheme hani almodelhasbeenimplemented. These ond omponent
in harge of thegeometry generation(the so- alled lient inFig. 4) willask forseveral runs of
theFEAP omponent at thesame timeea h usingadi erent geometry.
Further detailson the useof thisparallel framework and resultsare presented inthe following
subse tion.
FEAP
FEAP
FEAP
FEAP
FEAP CLIENT
Figure 4: Parallel software ar hite ture for Monte Carlo simulations
4 Results of the probabilisti hara terization of the two phases
material
4.1 Illustrative examples
In order to show the main features of this model, we shall rst present two simple examples.
The rst dealswith aperfe tly-plasti Dru ker-Prager matrix(see [5 ℄) inwhi h are pla ed ir-
ularvoids(Figure5). We showbythisexamplethe apabilityofourmodelto representstrain
eld dis ontinuityby hoosinga small value of Young's modulusfor the voids sub-domain. In
Figure 6a we show the stress-strain diagram omputed for elasti -perfe tly-plasti behavior of
thematrixmaterial.
Figure 5: Tension test on a square spe imen witha ir ular in lusion
In the se ond example we use the same spe imen geometry with a ir ular in lusion (see
Figure 5), but assumingthan thein lusion willhave the same Young's modulusas the matrix
and thatthe ra k an o uronlyat theinterfa ebetweentwo phases.
The post-peak behaviorat theinterfa eisrepresentedbyexponentialsoftening law,leading
to omputedstress-strain responseshown inFigure6b.
4.2 Comparison between stru tured and unstru tured mesh approa h
Inthispartwe onsiderthesamemi rostru tureasinthepreviousse tion. Wewanttoshowthe
di eren ebetween two meshes. The rst ase (Fig. 7a) presents adaptive exa t meshobtained
byusingthesoftwareGMSH,where ea h element ontains onlyone phase. Inthis ase several
E
1
, MPa 30000 30000
E
2
, MPa 30 30000
0.2 0.2
y
, MPa 20 -
f
, MPa - 1.5
u
, MPa - 4.0
u, m 0.04 0.001
Table 1: Material parameters fortwo examples
0 0,002 0,004 0,006 0,008 0,01
Strain, %
Stre ss , M Pa
0 10 20 30 40
0 5e-05 0,0001 0,00015 0,0002 0,00025
Strain, % 0
5 10 15 20
Stre ss , M Pa
Figure6: Strain-stressdiagramfora)elasti -perfe tly-plasti matrix andb)exponentialsoftening
law at the interfa e
elementsarestronglydistorted,sin ewedo notoptimizethismeshwithrespe ttotheelement
sizes,the sti nessmatrix ispoorly onditioned. The se ond ase (Fig. 7b)is ourregular mesh
whi h we use in the al ulation. In this ase, the elements an represent two phasesto model
thein lusions.
Figure 7: Adaptive mesha) and regular meshb) within lusions
Moreover, Fig. (8)shows theaxial displa ement ontour plot (with an ampli ation fa tor
of 100)and the orrespondingma ros opi axial rea tionsdispla ement urve.
Weobtainalmostthesameresponseforboth ases,butwithverydi erenttimeof al ulation
for irregularmesh as11774.68 s and forregular meshas 646.41 s. This simpleexample points
outone ofthemajoradvantageof theproposedmodelintermof omputationtimede reasing.
This point is a key point in order to ta kle ma ros opi models of heterogeneous materials
takinga ountforthemeso-stru turegeometry(forexamplethroughnumeri alhomogenization
methods).
Time = 1.00E+00
4.17E-04 8.33E-04 1.25E-03 1.67E-03 2.08E-03 2.50E-03 2.92E-03 3.33E-03 3.75E-03 4.17E-03 4.58E-03 0.00E+00
5.00E-03 DISPLACEMENT 1
Min = 0.00E+00 Max = 5.00E-03
Time = 1.00E+00 Time = 1.00E+00
4.17E-04 8.34E-04 1.25E-03 1.67E-03 2.08E-03 2.50E-03 2.92E-03 3.33E-03 3.75E-03 4.17E-03 4.58E-03 0.00E+00
5.00E-03 DISPLACEMENT 1
Min = 0.00E+00 Max = 5.00E-03 Time = 1.00E+00
Figure 8: Longitudinal displa ement ontour plot orresponding to max.load for adaptive mesh
a) and regular mesh b)
0 0,001 0,002 0,003 0,004 0,005
Displacement (cm) -20
-15 -10 -5 0
Reactions Sum (kN)
Figure9: Rea tions sumvs. displa ement urve b)
4.3 Numeri al results and dis ussion
By ombining both the deterministi problem presented above and the sto hasti numeri al
integration methoddetailedinthepreviousse tion,we performed Z =9999 integration points,
ea hofthem orrespondingtoasinglemeso-stru turerealization. Theseintegrationpointshave
beendistributed on 9 pro essors leading to a 7-day omputing time and we shall present here
thedi erent results.
The rst point to be mentioned deals with the meso-stru ture geometry, (whi h are the "in-
put" parameters a ording to the sto hasti integration method point of view). Namely, ea h
meso-stru ture realizationis builtby usinga modi ed Gibbs point pro ess with in lusionsra-
diusboundedbetween0:01mmand 0:3mm. Fig. 10 shows thevoids volume fra tion(ratio of
the voids volume versusthe total volume) histogram orresponding to the Z realizations. The
asso iatedmeanvalue is6:26%and thestandarddeviation 3:59%.
The global sto hasti integration pro ess is leading to a set of Z axial rea tion for e-
displa ement diagrams. In Fig. 10we show100 realizationssample forthisma ros opi result.
It is worth to re all again that the variabilityshown by this sample is only due to the meso-
stru ture geometry variability (the material parameters being deterministi and so onstant
along the realizations). Moreover we an note that some meso-stru tures inside this sample
have no in lusions. This behavior is dire tly linked to the Gibbs point pro ess properties, in
parti ular to thedis retePoisson law leadingto thein lusionsnumber.
InFig. 11 weshowbyusingthesetofZ ma ros opi axial rea tions-vs-displa ement urves
the estimated mean ma ros opi stress-strain urve as well as the 99:9% on den e interval.
Withthis on den eintervalbeingquitenarrow,we an on ludethatthenumberofintegration
pointsused inthe sto hasti integration method issuÆ ient to make a urate on lusions and
0 200 400 600 800 1000 1200
0 0.05 0.1 0.15 0.2 0.25 0.3
Number of realizations
Volume fraction of inclusions ( % ) Histogram of the inclusions volume fraction
-20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0
0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004 0.0045 0.005
Reactions Sum ( kN )
Displacement ( cm )
100 realizations of the reactions sum w.r.t the displacement
Figure 10: a) Histogram of the volumefra tion b) 100 realizations sample results
0 10 20 30 40 50 60 70
0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014
Mean Stress ( MPa )
Strain ( % )
Mean stress with 99.9% confidence interval w.r.t the strain
Mean stress Elastic Response 99.9% Error Bars
0 10 20 30 40 50 60 70 80
0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014
Mean Stress ( MPa )
Strain ( % )
Mean stress and standard deviation interval w.r.t the strain
Mean stress standard deviation
Figure 11: Mean stress w.r.t the strain a) withe error bars on the mean b) with standard
deviation interval
to provide good estimates of statisti al moments. The ma ros opi stress and strain " are
de nedasequivalenthomogeneous quantities,
"= u
L
x
= P
i R
i
L
y
(29)
whereL
x and L
y
arethesizeof thedomainand R
i
theaxialrea tions. Thisma ros opi mean
urve leads to the determination of an estimate for the ma ros opi mean Young's modulus
as well as to an estimate of the maximum stress mean
f
. In order to provide a mean yield
stress estimate, Fig. 12 shows the evolution of the Young's modulus mean along the ma ro-
s opi strain. We an notethat themodulusis smoothlyde reasingup to a strainlimit before
a mu h more rapidde reasebeyond thispoint. We assume thislimitto be an estimatefor the
ma ros opi yield strainorforthema ros opi yield stress aswell.
Table 2 summarizes all the statisti al ma ros opi estimates obtained from this numeri al
example.
5 Con lusion
When dealing with the FiniteElements modellingof heterogeneous stru tures,meshing is one
of themajorissues leading to distortedand bad onditionedtangent operators aswellastime-
0 10000 20000 30000 40000 50000 60000 70000 80000 90000 100000
0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014
Slope of the mean stress / strain curve ( MPa )
Strain ( % )
Evolution of the slope of the mean stress / strain curve w.r.t the strain
Figure12: Slope of the urve mean stress vs. strain w.r.t the strain
Mean Estimator 99:9% on den einterval std-dev interval
u
66:3651 MPa [ 66:3575MPa; 66:3727MPa℄ [58:2215MPa; 74:5087MPa℄
y
23:4998 MPa [ 23:4946MPa; 24:5050MPa℄ [21:5254MPa; 25:4742MPa℄
E 9:9371 GPa [9:7887GPa; 10:0855GPa℄ [ 9:2742GPa; 10:6000GPa℄
Table2: Statisti sof theout omeproperties ofMonte Carlo simulations
onsuming algorithms. In this work we rst proposed a numeri al approa h based on regular
and stru turedmeshes whi h arenot onstrainedbythephysi alinterfa es. Based on lassi al
CSTelementsweshowedhowtoenhan etheelementskinemati susingtheIn ompatibleModes
Method providing two kind of dis ontinuities. The rst one onsists in a strain dis ontinuity
inside the element in order to model the di erent elasti properties of the two phases. The
se onddis ontinuity orrespondstoadispla ementoneandallowstomodeltheinterfa efailure
(e.g. debonding) a ording to two di erent me hanisms (normal and tangential). By using a
2D numeri al omparison on a porous media with a perfe tly plasti matrix, we showed that
the omputation timeisstrongly lowerwithinthe ontext ofsu ha regularmesh.
With su h a modelling tool in hand we also presented how to take into a ount for the
variabilityofthegeometri aldes riptionat themeso-s alelevel. Thesegeometriesaremodelled
byusingmodi edGibbs pointspro esses with ir ularin lusions. Althoughthematerial prop-
erties of the two phases are assumed to be deterministi , this variability leads to a sto hasti
problem to be solved. In this work we employed the lassi al Monte-Carlo method in order
to produ ethe statisti al moments of thedesired quantities. UsingtheComponentsTemplate
Library(CTL)theFiniteElements odeFEAPweprodu ed9999 realizations. Thestatisti sof
the out ome properties exhibit quite narrow on den e intervals. These numeri al results an
thenbeviewed asma ros opi propertiesforthisporousmediawithinthe ontextof a lassi al
phenomenologi almodel.
A knowledgements
This work was supported by the Fren h Ministry of Resear h. The ollaboration with TU
Braun hweigresear hgroupofProf. HermannG.Matthies,espe iallyDr. RainerNiekampand
M. Martin Kros he is also gratefully a knowledged. AI also a knowledges the support of the
Alexandervon Humboldt Foundation.
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