Dominique Jeulin, dominique.jeulin@mines-paristech.fr
On the statistical RVE of some long range random media
It is usual to estimate the variance of local averages by means of the integral range (or integral of the centered
normalized correlation function). This procedure is followed to estimate the representative volume elements
(RVE) of the microstructure for numerical homogenization of effective properties on simulations. Long fibers
or stratified media show very long range correlations, inducing an infinite integral range. This media can be
simulated by models of Boolean random varieties. For these models non standard scaling power laws exist
for the decrease of the variance of local averages, like the volume fraction, with the volume of domains K
2 (K) of the local fraction Z = µn (A∩K) (µ (K)
. In Rn , the asymptotic expression of the variance DZ
n
µn (K)
being the Lebesgue measure of K) of a Boolean model built on isotropic Poisson varieties of dimension k
(k = 0, 1, ..., n − 1) Vk , is expressed by [1]:
2
DZ
(K) = p(1 − p)
n−k
Ak
µn (K)
n−k
n
The exponent γ = n is equal to 23 for Boolean fibers in 3D, and 31 for Boolean strata in 3D. When
working in 2D, the scaling exponent of Boolean fibers is equal to 12 . This reflects a much lower decrease of
fluctuations with respect to the volume of observations, as compared to media with a finite integral range, where
γ = 1. As a result, very large domains have to be simulated to obtain statistical RVE of the microstructure
for numerical homogenization. This behaviour was checked in the case of the estimation of elastic moduli and
thermal conductivity of simulations of various fiber systems.
References
[1] Jeulin D. (2011) Variance scaling of Boolean varieties, Paris School of Mines internal report N/10/11/MM
(2011), hal-00618967, version 1.
On the Statistical RVE of some long range Random
Media
Dominique Jeulin
Centre de Morphologie Mathématique & Centre des Matériaux P.M. Fourt, Mines ParisTech
July 24, 2014
Mines ParisTech (Institute)
CMDS 13, Salt Lake City, Utah, USA
July 24, 2014
1 / 48
Context
In some media, long …bre or extended strata networks, involving very long
range of correlations .
Needs to model and to simulate such media
Slow convergence of the RVE
Mines ParisTech (Institute)
CMDS 13, Salt Lake City, Utah, USA
July 24, 2014
2 / 48
Context
Thermisorel …brous network (X-ray microtomography and simulation)
600 x 600 x 360 voxels ; 5.6 x 5.6 x 3.37 mm3 ; Resolution US2B: 9.36
µm/voxel
Peyrega C., Jeulin D., Delisée C., Malvestio J. (2009) 3D
morphological modelling of a random …brous network. Image Analysis
and Stereology.
Mines ParisTech (Institute)
CMDS 13, Salt Lake City, Utah, USA
July 24, 2014
3 / 48
Outline
Variations of the Boolean random closed sets model to intoduce in…nite
ranges and consequences on the RVE
Principle of random structure modeling
Reminder on the Boolean model
Long range random sets: Boolean varieties
Fluctuations and probabilistic RVE
Mines ParisTech (Institute)
CMDS 13, Salt Lake City, Utah, USA
July 24, 2014
4 / 48
Principle of random structure
modeling
Mines ParisTech (Institute)
CMDS 13, Salt Lake City, Utah, USA
July 24, 2014
5 / 48
Characterization of a random structure: Main Criteria
Morphological criteria
Size
Shape
Distribution in space (Clustering, Scales, Anisotropy)
Connectivity
Probabilistic criteria
Probability laws (n points, supK )
Moments
Mines ParisTech (Institute)
CMDS 13, Salt Lake City, Utah, USA
July 24, 2014
6 / 48
Characterization of a random set
Models derived from the theory of Random Sets by G.
MATHERON
For a random closed set A (RACS), characterization by the
CHOQUET capacity T (K ) de…ned on the compact sets K
T (K ) = P fK \ A 6 = ∅ g = 1
P fK
Ac g = 1
Q (K )
In the Euclidean space R n , CHOQUET capacity and dilation
operation
T ( Kx ) = P f Kx \ A 6 = ∅ g = P f x 2 A
Mines ParisTech (Institute)
CMDS 13, Salt Lake City, Utah, USA
Ǩ g
July 24, 2014
7 / 48
Binary Morphology (Fe-Ag)
Mines ParisTech (Institute)
CMDS 13, Salt Lake City, Utah, USA
July 24, 2014
8 / 48
Basic Operations of Mathematical Morphology
Dilation by hexagon
(2)
Mines ParisTech (Institute)
CMDS 13, Salt Lake City, Utah, USA
Erosion by hexagon
(2)
July 24, 2014
9 / 48
Calculation of the CHOQUET capacity
For a given model, the functional T is obtained:
by theoretical calculation
by estimation
on simulations
on real structures (possible estimation of the parameters from the
"experimental" T , and tests of the validity of assumptions).
Mines ParisTech (Institute)
CMDS 13, Salt Lake City, Utah, USA
July 24, 2014
10 / 48
Reminder on the Boolean model
Mines ParisTech (Institute)
CMDS 13, Salt Lake City, Utah, USA
July 24, 2014
11 / 48
Poisson point process
Non homogeneous Poisson point process in R n with a regionalized
intensity θ (x ) (x 2 R n ; θ 0): the numbers N (Ki ) are independent
random variables for any family of disjoint compact sets Ki
N (K ) is a Poisson random variable with parameter θ (K ) :
θ (K ) =
Z
K
θ (dx )
Pn (K ) = P fN (K ) = ng =
Mines ParisTech (Institute)
θ (K )n
exp( θ (K ))
n!
CMDS 13, Salt Lake City, Utah, USA
July 24, 2014
12 / 48
Boolean Model
The Boolean model (G. Matheron, 1967) is obtained by implantation
of random primary grains A0(with possible overlaps) on Poisson points
xk with the intensity θ:
A = [xk Ax0 k
Any shape (convex or non convex, and even non connected) can be
used for the grain A0
Mines ParisTech (Institute)
CMDS 13, Salt Lake City, Utah, USA
July 24, 2014
13 / 48
Boolean Model
Fe Ag
Mines ParisTech (Institute)
Boolean model of spheres
(0.5)
CMDS 13, Salt Lake City, Utah, USA
July 24, 2014
14 / 48
Boolean model of spheres
Mines ParisTech (Institute)
CMDS 13, Salt Lake City, Utah, USA
July 24, 2014
15 / 48
Boolean Model
Choquet capacity, with
q = P f x 2 Ac g
T (K ) = P fA \ K 6 = ∅ g = 1
= q
µ(A 0 Ǩ )
µ (A 0 )
P fK
Ac g = 1
exp( θµ(A0
Ǩ )
Ex: contact distribution (ball), covariance, 3-points statistics,. . .
Percolation threshold obtained from simulations: 0:2895 +- 0:0005
for spheres with a single diameter
Estimation of the percolation threshold of the complementary random
set of a boolean model of spheres (constant radius): 0.0540 0.005
Percolation threshold estimated analytically from the zeroes of
NV GV (p ) (Euler characteristic, or connectivity number) for a single
convex grain
Mines ParisTech (Institute)
CMDS 13, Salt Lake City, Utah, USA
July 24, 2014
16 / 48
Change of scale in random media (Physics-Texture)
Prediction of E¤ective Properties by numerical simulations
Digital materials
Input of 3D images
Real images (confocal, microtomography)
Simulations from a random model
Use of a computational code (Finite Elements, Fast Fourier Transform,
PDE numerical solver)
Homogenization
Representative Volume Element (RVE)?
Willot F., Jeulin D. (2011) Elastic and electrical behavior of some
random multiscale highly-contrasted composites. International Journal
of Multiscale Computational Engineering, Vol. 9 (3), pp. 305-326.
Mines ParisTech (Institute)
CMDS 13, Salt Lake City, Utah, USA
July 24, 2014
17 / 48
Long range random sets: Boolean
varieties
Mines ParisTech (Institute)
CMDS 13, Salt Lake City, Utah, USA
July 24, 2014
18 / 48
Reminder on Poisson varieties
Poisson point process fxi (ω )g, with intensity θ k (d ω ) on the varieties of
dimension (n k) containing the origin O, and with orientation ω.
On every point xi (ω ) is given a variety with dimension k, Vk (ω )xi ,
orthogonal to the direction ω.
By construction, Vk = [xi (ω ) Vk (ω )xi .
For instance in R 3 can be built a network of Poisson hyperplanes Πα
(orthogonal to the lines Dω containing the origin) or a network of Poisson
lines in every plane Πω containing the origin.
Matheron G. (1975) Random sets and integral geometry.
Mines ParisTech (Institute)
CMDS 13, Salt Lake City, Utah, USA
July 24, 2014
19 / 48
Reminder on Poisson varieties
Theorem
The number of varieties of dimension k hit by a compact set K is a
Poisson variable, with parameter θ (K ):
θ (K ) =
Z
θ k (d ω )
Z
K (ω )
θn
k (dx )
=
Z
θ k (d ω ) θ n
k (K ( ω ))
(1)
where K (ω ) is the orthogonal projection of K on the orthogonal space to
Vk (ω ), Vk ? (ω ). For the stationary case,
θ (K ) =
Mines ParisTech (Institute)
Z
θ k (d ω ) µn
k (K ( ω ))
CMDS 13, Salt Lake City, Utah, USA
(2)
July 24, 2014
20 / 48
Reminder on Poisson varieties
Theorem
The Choquet capacity T (K ) = P fK \ Vk 6= ?g of the varieties of
dimension k is given by
T (K ) = 1
exp
Z
θ k (d ω )
Z
K (ω )
θn
k (dx )
(3)
In the stationary case, the Choquet capacity is
T (K ) = 1
Mines ParisTech (Institute)
exp
Z
θ k (d ω ) µn
CMDS 13, Salt Lake City, Utah, USA
k (K ( ω ))
(4)
July 24, 2014
21 / 48
Reminder on Poisson varieties
Theorem
We consider now the isotropic (θ k being constant) and stationary case,
and a convex set K . Due to the symmetry of the isotropic version, we can
consider θ k (d ω ) = θ k d ω as de…ned on the half unit sphere (in R k +1 ) of
the directions of the varieties Vk (ω ). The number of varieties of dimension
k hit by a compact set K is a Poisson variable, with parameter θ (K ) with:
θ (K ) = θ k
Z
µn
k (K ( ω ))
d ω = θk
bn
k bk +1
bn
k +1
Wk ( K )
2
(5)
π k /2
)
k
Γ (1 + )
2
4
(b1 = 2, b2 = π, b3 = π), and Wk (K ) is the Minkowski’s functional of
3
K , homogeneous and of degree n k
where bk is the volume of the unit ball in R k (bk =
Mines ParisTech (Institute)
CMDS 13, Salt Lake City, Utah, USA
July 24, 2014
22 / 48
Reminder on Poisson varieties
The following examples are useful for applications:
When k = n 1, the varieties are Poisson planes in R n ; in that
case, θ (K ) = θ n 1 nWn 1 (K ) = θ n 1 A(K ), where A(K ) is the norm
of K (average projected length over orientations).
In the plane R 2 are obtained the Poisson lines, with θ (K ) = θL(K ),
L being the perimeter.
In the three-dimensional space are obtained Poisson lines for k = 1
π
and Poisson planes for k = 2. For Poisson lines, θ (K ) = θS (K )
4
and for Poisson planes, θ (K ) = θM (K ), where S and M are the
surface area and the integral of the mean curvature.
Mines ParisTech (Institute)
CMDS 13, Salt Lake City, Utah, USA
July 24, 2014
23 / 48
Boolean random varieties
De…nition
A Boolean model with primary grain A0 is built on Poisson linear varieties
in two steps: i) we start from a network Vk ; ii) every variety Vk α is dilated
by an independent realization of the primary grain A0 . The Boolean RACS
A is given by
A = [ α Vk α A 0
By construction, this model induces on every variety Vk ? (ω ) orthogonal to
Vk (ω ) a standard Boolean model with dimension n k and with primary
grain A0 (ω ) and with intensity θ (ω )d ω.
Jeulin D. (1991) Modèles de Fonctions Aléatoires multivariables. Sci.
Terre; 30: 225-256.
Mines ParisTech (Institute)
CMDS 13, Salt Lake City, Utah, USA
July 24, 2014
24 / 48
Boolean random varieties
The Choquet capacity of this model immediately follows, after averaging
over the directions ω; it can also be deduced from Eq. (4), after replacing
K by A0 Ǩ and averaging.
Theorem
The Choquet capacity of the Boolean model built on Poisson linear
varieties of dimension k is given by
T (K ) = 1
Z
exp
θ k (d ω ) µn
k (A
0
(ω )
Ǩ (ω ))
(6)
For isotropic varieties, the Choquet capacity of Boolean varities is given by
T (K ) = 1
Mines ParisTech (Institute)
exp
θk
bn
k bk +1
bn
k +1
W k (A0
2
CMDS 13, Salt Lake City, Utah, USA
Ǩ )
July 24, 2014
(7)
25 / 48
Boolean random varieties
Particular cases of Eq. (6) when K = fx g (giving the probability
Z
q = P fx 2 Ac g = exp
θ k (d ω ) µn
k (A
0 ( ω ))
and when
K = fx, x + hg, giving the covariance of Ac , Q (h) :
2
Q (h) = q exp
Z
θ k ( d ω ) Kn
k ( ω,
!!
h . u (ω ))
(8)
where Kn k (ω, h) = µn k (A0 (ω ) \ A0 h (ω )) and !
u (ω ) is the unit
vector with the direction ω.
For a compact primary grain A0 , there exists for any h an angular sector
where Kn k (ω, h) 6= 0, so that the covariance generally does not reach its
sill, at least in the isotropic case, and the integral range is in…nite.
Jeulin D. (1991) Modèles de Fonctions Aléatoires multivariables. Sci.
Terre; 30: 225-256.
Mines ParisTech (Institute)
CMDS 13, Salt Lake City, Utah, USA
July 24, 2014
26 / 48
Boolean model on Poisson lines in 2D
Simulation of a 2D Boolean model built on isotropic Poisson lines
(alternatively 2D section of a 3D Boolean model of Poisson planes)
Mines ParisTech (Institute)
CMDS 13, Salt Lake City, Utah, USA
July 24, 2014
27 / 48
Boolean model on Poisson lines in 2D
For an isotropic lines network (…gure 1), and if A0
from equation (7):
T (K ) = 1
exp
θ L(A0
Ǩ is convex, we have,
Ǩ )
(9)
If A0 Ǩ is not convex, the integral of projected lengths over a line with
the orientation varying between 0 and π must be taken. If A0 and K are
convex sets, we have L(A0 Ǩ ) = L(A0 ) + L(K ). In the isotropic case
and using for A0 a random disc with a random radius R (with expectation
R) and for K is a disc with radius r , equation 9 becomes:
T (r ) = 1
exp
2πθ (R + r )
T (0) = P fx 2 Ag = 1
exp
2πθR
which can be used to estimate θ and R , and to validate the model.
Mines ParisTech (Institute)
CMDS 13, Salt Lake City, Utah, USA
July 24, 2014
28 / 48
Boolean model on Poisson planes
A Boolean model built on Poisson planes generates a structure with strata.
On isotropic Poisson planes, we have for a convex set A0 Ǩ by
application of equation (7):
T (K ) = 1
exp
θ M (A0
Ǩ )
(10)
When A0 and K are convex sets, M (A0 Ǩ ) = M (A0 ) + M (K ). If A0
is not convex, T (K ) is expressed as a function of the length l of the
projection over the lines
D by
Z ω
T (K ) = 1
exp
θ
2πster
l (A0 ( ω )
Ǩ
Ǩ (ω )) d ω .
If A0 is a random sphere with a random radius R (with expectation R)
and K is a sphere with radius r , equation 10 becomes:
T (r ) = 1
exp
4πθ (R + r )
T (0) = P fx 2 Ag = 1
exp
4πθR
which can be used to estimate θ and R , and to validate the model.
Mines ParisTech (Institute)
CMDS 13, Salt Lake City, Utah, USA
July 24, 2014
29 / 48
Boolean model on Poisson planes and extremal properties
Two-components microstructure obtained by an in…nite superposition of
random sets made of dilated isotropic Poisson planes with a large
separation of scales: extremal physical e¤ective properties (like electric
conductivity, or elastic properties)
when the highest conductivity is attributed to the dilated planes, the
e¤ective conductivity is the upper Hashin-Shtrikman bound
when a¤ecting the lower conductivity to the dilated planes, the
e¤ective conductivity is equal to the lower Hashin-Shtrikman bound
Jeulin D. (2001) Random Structure Models for Homogenization and
Fracture Statistics, in Mechanics of Random and Multiscale
Microstructures, ed. by D. Jeulin and M. Ostoja-Starzewski (CISM
Lecture Notes N 430, Springer Verlag), pp. 33-91.
Mines ParisTech (Institute)
CMDS 13, Salt Lake City, Utah, USA
July 24, 2014
30 / 48
Boolean model on Poisson lines in 3D
Isotropic Poison …bres
Faessel M., Jeulin D. (2011) 3D Multiscale Vectorial Simulation of
Random Models, Proc. ICS 13, Beijing.
Schladitz K., Peters S., Reinel-Bitzer D., Wiegmann A., Ohser J.,
Design of acoustic trim based on geometric modeling and ‡ow
simulation for non-woven, Computational Materials Science 38 (2006)
56–66.
Mines ParisTech (Institute)
CMDS 13, Salt Lake City, Utah, USA
July 24, 2014
31 / 48
Boolean model on Poisson lines in 3D
A Boolean model built on Poisson lines generates a …ber network, with
possible overlaps of …bers. On isotropic Poisson lines, for a convex set
A0 Ǩ
π
(11)
T (K ) = 1 exp
θ S (A0 Ǩ )
4
If A0 Ǩ is not convex, T (K ) is expressed as a function of the area A of
the projection over the planes Πω by
T (K ) = 1
exp
θ
Z
4πster
A (A0 ( ω )
Ǩ (ω )) d ω
(12)
If A0 is a random sphere with a random radius R (with expectation R and
second moment E (R 2 )) and K is a sphere with radius r , equation 11
becomes:
T (r ) = 1
exp
π 2 θ (E (R 2 ) + 2r R + r 2 )
T (0) = P fx 2 Ag = 1
exp
π 2 θE (R 2 )
which can be used to estimate θ, E (R 2 ) and R , and to validate the model.
Mines ParisTech (Institute)
CMDS 13, Salt Lake City, Utah, USA
July 24, 2014
32 / 48
Fluctuations and probabilistic RVE
Mines ParisTech (Institute)
CMDS 13, Salt Lake City, Utah, USA
July 24, 2014
33 / 48
Fluctuations and RVE
Consider ‡uctuations of average values over di¤erent realizations of a
random medium inside the domain B with the volume V . In Geostatistics,
it is well known that for an ergodic stationary random function Z (x ), with
mathematical expectation E (Z ), one can compute the variance DZ2 (V ) of
its average value Z̄ (V ) over the volume V as a function of the central
covariance function C (h) of Z (x ) by :
DZ2 (V ) =
1
V2
Z Z
B
C (x
y ) dxdy ,
(13)
B
where
C (h) = E f(Z (x )
Mines ParisTech (Institute)
E (Z )) (Z (x + h)
CMDS 13, Salt Lake City, Utah, USA
E (Z ))g
July 24, 2014
34 / 48
Fluctuations and RVE of the volume fraction
For a large specimen (with V
A3 ), equation (13) can be expressed to
the …rst order in 1/V as a function of the integral range in the space R 3 ,
A3 , by
DZ2 (V ) = DZ2
with A3 =
1
DZ2
A3
,
V
Z
R3
(14)
C (h) dh,
(15)
where DZ2 is the point variance of Z (x ) (here estimated on simulations)
and A3 is the integral range of the random function Z (x ), de…ned when
the integral in equations (13) and (15) is …nite.
When Z (x ) is the indicator function of the random set A, (14) provides
the variance of the local volume fraction (in 3D) as a function of the point
variance DZ2 = p (1 p ), p being the probability for a point x to belong to
the random set A.
Mines ParisTech (Institute)
CMDS 13, Salt Lake City, Utah, USA
July 24, 2014
35 / 48
Fluctuations and RVE of e¤ective properties
Asymptotic scaling law (14) valid for any additive variable Z over the
region of interest B
Estimation of the e¤ective elasticity or permittivity tensors from
simulations, from calculation of the spatial average stress hσi and
strain hεi (elastic case) or electric displacement hD i and electrical
…eld hE i
For given boundary conditions, local moduli obtained from the
estimations of a scalar, namely the average in the domain B of the
stress, strain, electric displacement, or electric …eld.
The variance of the local e¤ective property follows the equation (14) for a
…nite integral range A3 of the relevant …eld
Theoretical covariance of the …elds (σ or ε) usually not available; integral
range A3 estimated by working with realizations of Z (x ) on domains B
with an increasing volume V (or in the present case considering
subdomains of large simulations, with a wide range of sizes), and …tting
the obtained variance according to the expression (14).
Mines ParisTech (Institute)
CMDS 13, Salt Lake City, Utah, USA
July 24, 2014
36 / 48
Practical determination of the size of the RVE
Size of a RVE de…ned for a physical property Z , a contrast, and a given
precision in the estimation of the e¤ective properties depending on the
number n of available realizations.
From that a standard statistical approach, absolute error eabs and relative
error erela on the mean value obtained with n independent realizations of
volume V , deduced from the 95% interval of con…dence by:
eabs =
2DZ (V )
eabs
2DZ (V )
p
p .
; erela =
=
Z
n
Z n
(16)
Size of the RVE de…ned as the volume for which for instance n = 1
realization (with an ergodicity assumption on the microstructure) is
necessary to estimate the mean property Z with a relative error (for
instance erela = 1%), provided we know the variance DZ2 (V ) from the
asymptotic scaling law (14)
Mines ParisTech (Institute)
CMDS 13, Salt Lake City, Utah, USA
July 24, 2014
37 / 48
Practical determination of the size of the RVE
Alternatively, operation on smaller volumes (provided no bias is introduced
by the boundary conditions), and consider n realizations to obtain the
same relative error.
Methodology initially developped for the elastic properties and thermal
conductivity of a Voronoï mosaic [1], and wideley used.
Kanit T., Forest S., Galliet I., Mounoury V., Jeulin D. (2003)
Determination of the size of the representative volume element for
random composites: statistical and numerical approach, International
Journal of solids and structures, Vol. 40, pp. 3647-3679.
Mines ParisTech (Institute)
CMDS 13, Salt Lake City, Utah, USA
July 24, 2014
38 / 48
Scaling of the variance of the Boolean random varieties
Consider a convex domain K in R n , with Lebesgue measure µn (K ).
Theorem
µ (A \K )
In R n , the variance DZ2 (K ) of the local fraction Z = nµ (K ) of a Boolean
n
model built on isotropic Poisson varieties of dimension k
(k = 0, 1, ..., n 1) Vk , is asymptotically expressed by
DZ2 (K )
= p (1
p)
Ak
µn (K )
n k
n
the scaling exponent being γ = n n k . As particular cases, Poisson points
(k = 0) give the standard Boolean model with a …nite integral range and
γ = 1, Poisson lines (k = 1) generate Poisson …bers with γ = n n 1 , and
Poisson hyperplanes (k = n 1) provide Poisson strata with γ = n1 .
Poisson strata: very slow decrease of the variance with the volume of the
sample K , with γ = n1 .
Mines ParisTech (Institute)
CMDS 13, Salt Lake City, Utah, USA
July 24, 2014
39 / 48
Scaling of the variance of the Boolean random varieties
For isotropic Boolean …bers in 2D, the scaling exponent is γ =
1
2
For isotropic Boolean …bers in 3D, γ = 23 . Exponent recovered
from numerical simulations for the volume fraction [1]
For isotropic Boolean strata in 3D, γ = 13 . This decrease of the
variance with size is much slower than for a …nite integral range
Dirrenberger J., Forest S., Jeulin D. Towards gigantic RVE sizes for
3D stochastic …brous networks, Int. J. Structures, 51 (2014) 359-376.
Mines ParisTech (Institute)
CMDS 13, Salt Lake City, Utah, USA
July 24, 2014
40 / 48
Scaling of the variance of e¤ective properties of isotropic
Boolean …bers in 3D
For elastic properties and for conductivity of 3D random …ber
networks by …nite elements [1] or by FFT [2], γ = 23 provides an
empirical scaling law of the variance (expected, as a result of a high
correlation between the elastic or thermal …elds and the indicator
function of the random set A for a strong contrast of properties)
For …bers with a …nite length, intermediary situation [2], and scaling
coe¢ cient 23 6 γ 6 1, depending on the size of the specimen (γ ' 32
for small specimens, and γ ' 1 for large samples)
Dirrenberger J., Forest S., Jeulin D. Towards gigantic RVE sizes for
3D stochastic …brous networks, Int. J. Structures, 51 (2014) 359-376.
Altendorf H., Jeulin D., Willot F. In‡uence of …ber geometry on the
macroscopic elastic and thermal properties, Int. J. Structures (2014),
accepted .
Mines ParisTech (Institute)
CMDS 13, Salt Lake City, Utah, USA
July 24, 2014
41 / 48
Scaling of the variance of e¤ective properties of isotropic
Boolean …bers in 3D
Realization of Boolean …bers (Vv = 0.16)
Mines ParisTech (Institute)
CMDS 13, Salt Lake City, Utah, USA
July 24, 2014
42 / 48
Scaling of the variance of e¤ective properties of isotropic
Boolean …bers in 3D
local volume fraction
Thermal conductivity
Bulk modulus
Shear modulus
Mines ParisTech (Institute)
CMDS 13, Salt Lake City, Utah, USA
July 24, 2014
43 / 48
Boolean …bers with a …nite length: FFT estimated …elds
Isotropic …bers
Transverse isotropic …bers
σm maps under isotropic compression
Mines ParisTech (Institute)
CMDS 13, Salt Lake City, Utah, USA
July 24, 2014
44 / 48
Isotropic Boolean …bers with a …nite length: variance
scaling
Volume fraction
Mines ParisTech (Institute)
CMDS 13, Salt Lake City, Utah, USA
July 24, 2014
45 / 48
Isotropic Boolean …bers with a …nite length: variance
scaling
Bulk modulus
Mines ParisTech (Institute)
CMDS 13, Salt Lake City, Utah, USA
July 24, 2014
46 / 48
Conclusion
Boolean random varieties generate random media with in…nite range
correlations, with non standard theoretical scaling laws of the variance
of the local volume fraction with the volume of domains K , out of
reach of a standard statistical approach
We have theoretically shown that on a large scale, a the variance of
the local volume fraction decreases with power laws of the volume
of K (with exponent γ = 23 for Boolean …bers in 3D, 13 for Boolean
strata in 3D, and 12 for Boolean …bers in 2D).
The decrease of the variance with the scale is much slower, as
compared to situations with a …nite integral range, like the standard
Boolean model, and larger RVE are expected for the estimation of the
volume fraction
Mines ParisTech (Institute)
CMDS 13, Salt Lake City, Utah, USA
July 24, 2014
47 / 48
Conclusion
These scaling laws were shown to hold when using numerical
simulations to predict the e¤ective properties of such random media
Due to strong localization of …elds for a nonlinear behavior, long
range correlations of the …elds are expected, and scaling laws similar
to the dilated Poisson hyperplanes (with a scaling exponent close to
1
3 ) will be recovered, involving a slow convergence towards the
e¤ective properties by numerical simulations. This point remains to
be investigated.
The obtained scaling laws can help to design optimal sampling
schemes with respect to minimizing the variance of estimation
Jeulin D. (2011) Variance scaling of Boolean varieties. N10/11/MM,
hal-00618967, version 1.
Mines ParisTech (Institute)
CMDS 13, Salt Lake City, Utah, USA
July 24, 2014
48 / 48