Homogeniza*on from the viewpoint of the periodic mediaPrinciples and contribu*ons

D. CaillerieGrenoble INP - UJF - CNRSSols Solides Structures et Risques

What is meant by homogeniza*on ? or upscalling or micro macro methods

2

To define for a finely heterogeneous medium, ”macroscopic” representative quantitiesand equations that enable to determine them

The keyword is separation of scales

Contents

1.Periodic media -­‐ Why?2.1D bar3.Double scale asympto8c expansion4.Self equilibrium problem on the cell of periodicity5.Mul8physics6.Related situa8ons -­‐ quasi periodic media: large strain, elastoplas8city, ...7.Intermediate conclusions and remarks8.Parallel with experiments -­‐ VER9.Granular materials (8me permiKed)

3

Periodic media

4

Media with periodically distributed heterogenei8es

L

Y2

Y1

e =

L→ 0

→

5

Finely periodic media -­‐ asympto*c method

4

Periodic media ? Why ?

6

Because there are natural or manufactured periodic media

Mainly because the method which has been widely used is very robust, it has many interes8ng features and can bring interes8ng persepec8ves to homogeniza8on in general

L

eL e = 1nc , nc number of cells k1 k2

L2L1

0 =dN

dx+ f

N = ke (x)du

dx

ke (x) = kx

e

, k (y) =

k1 0 < y < L1

k2 L1 < y < L

u (0) = 0 , N (L) = F

1D periodic bar

This problem can be easily solved

7

f = 0

1D periodic bar

8

9

1D periodic bar Convergence to the homogenized solution

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x = 0.51D periodic bar Zoom X Nc around

N

dN

dx= 0

eL

N = ke (x)du

dx

u (x + eL)− u (x) = εMeL

εM =N

eL

x+eL

x

1ke (ξ)

dξ =1L

L1

k1+

L2

k2

N

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1D periodic bar -­‐ Heuris*c homogeniza*on

At the small scale the normal stress is almost constant:

and the variation of the displacement on a length equal to the period

L

eL e = 1nc , nc number of cells k1 k2

L2L1

corresponds to the macroscopic strain:

The integration of the constitutive equation yields:

Which is the equivalent macroscopic constitutive equation

1D periodic bar -­‐ Case of

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f = −2

e→ 0

Y

C (y)

2D Elas*c periodic medium

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divσe + f = 0 in Ω

σe = C

x

e

(ue) in Ω

+ boundary conditions on ∂Ω

The 4th order tensor isY periodic

u(k)

x,

x

e

u(k) (x, y)

ue (x) = u(0)

x,

x

e

+ eu(1)

x,

x

e

+ e2u(2)

x,

x

e

+ · · ·

Double scale asympto*c expansion

Slow x and fast y variables

are almost eY periodicThe are Y periodic so the

14

15

u(0) y

u(0) (x)

ue (x) = u(0) (x) + eu(1)

x,

x

e

ue (x) = u(0) (x) + eu(1)

x,

x

e

+ e2u(2)

x,

x

e

+ · · ·

Double scale asympto*c expansion

It turns out that does not depend on

is the macroscopic displacement field

and, up to higher terms, the displacement

is equal to the macroscopic one + a small correction presenting fast variations

ue = u(0) (x) + eu(1)

x,

x

e

+ e2u(2)

x,

x

e

+ · · ·

∇ue = ∇xu(0) +∇yu(1) + e∇xu(1) +∇yu(2)

+ · · ·

(ue) = xu(0)

+ y

u(1)

+ e

x

u(1)

+ y

u(2)

+ · · ·

σe = σ(0)

x,

x

e

+ eσ(1)

x,

x

e

+ e2σ(2)

x,

x

e

+ · · ·

σ(k) (x, y)

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Double scale asympto*c expansion

are Y periodic

σe = σ(0)

x,

x

e

+ eσ(1)

x,

x

e

+ e2σ(2)

x,

x

e

+ · · ·

divσe =1edivyσ(0) + divxσ(0) + divyσ(1) + e (· · · )

1edivyσ(0) + divxσ(0) + divyσ(1) + e (· · · ) + f = 0

divyσ(0) = 0

divxσ(0) + divyσ(1) + f = 0

divσe + f = 0

divxσ(0)

+ f = 0

σ(0)

=

1|Y |

Yσ(0) (x, y) dy

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Expansion of the equilibrium equa*on

The balance equation expands into:

which, by identification of the terms of same power, yields:

Macroscopic balance equation

Macroscopic mean stress

divyσ(0) = 0 in Y

σ(0) = C (y)x

u(0)

+ y

u(1)

in Y

+ periodic boundary conditions on ∂Y

xu(0)

σ(0)

xu(0)

−→ u(1) and σ(0) −→

σ(0)

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Self equilibrium problem

u(1)σ(0)The unknowns are and

the datum is the macroscopic strain

The solving of this problem yields u(1) and σ(0)

and by averaging, the macroscopic stress

Which defines the equivalent macroscopic constitutive relation:

u(1) (y) = xu(0)

.y + u(1) (x, y)

∇yu(1) = xu(0)

+∇yu(1)

u(1)

divyσ(0) = 0 in Y

σ(0) = C (y) yu(1)

in Y

u(1)y + Y i

− u(1) (y) = x

u(0)

.Y i on Γi , i = 1, 2

σ(0)

: x

u(0)

=

σ(0) : y

u(1)

Y2y+Y1y

Y1

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Self equilibrium problem -­‐ second form

then:

the problem for reads:

Hill’s lemma

x

e = /L

y ↔ ξ = ey

u(1)ξ

= eu(1)

ξ

e

= e

x

u(0)

.ξ

e+ u(1)

x,

ξ

e

σ(0)ξ

= σ(0)

x,

ξ

e

u(1) σ(0)

Y ex

divξσ(0) = 0 in Y ex

σ(0) = C

ξ

e

ξ

u(1)

in Y e

x

u(1)ξ + eY i

− u(1)

ξ

= xu(0)

.eY i on γi

x , i = 1, 2

eY1

ξ+eY1ξeY2

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Self equilibrium problem -­‐ third form on the real cell

Homothety from the expanded cell Y to the real cell located by

The small parameter is somehow arbitrary

Change of unknowns

and are solutions of the problem set on the real cell Cx

Remak: heuristic method for the bar

Quasi periodic media

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ApplicationsGeometrically quasiperiodic mediaElastoplastic periodic mediaLarge strain framework

Mul*physics and mul* parameter modellings

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Darcy’s lawBiot’s modelling of the consolidationLarge strainsViscosity of the fluid

Medium reinforced by slender fibers

e2

Periodic plates or beams

n =ν1, ν2, n

un = u(0)eν1, eν2

+ eu(1)n

eν1, eν2

+ · · ·

Con*nuous modelling of discrete stuctures

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Graphene

Muscle Paper

Intermediate conclusions and remarks

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The homogenization method of periodic media is robust in the sense that,

starting from the description of the macroscopic scale, it enables to define

the equivalent fields at the macroscopic scale and the equations that govern

them and that with the only hypothesis of the double scale expansion.

The approximation of the real displacement field is the macroscopic

field + a small corrector presenting variations at the micro scale

The macroscopic constitutive equation is given by the solving of

a self equilibrium problem on the elementary cell

The macroscopic constitutive equation is invariant under a change of

scale in the self equilibrium problem

Hill’s lemma is satisfied

Limits of the methods

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Chapitre 7. Exemples numériques

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

0.2

0.4

0.6

0.8

1

1.2

1.4

P12

!

(1)

(3)(4)

(5)

(2)

(2) ! = 0.208

(5) ! = 0.328

(4) ! = 0.234

(3) ! = 0.215

FIG. 7.17 – Rupture non-périodique : Courbes de comportement macroscopique, contrainte (MPa) en

fonction du facteur de chargement !, pour les paramètres de la loi cohésive "n = "t = 0.01 mm, Tmax = 1

MPa, et le coefficient frottement µ0 = 0.5, pour différents VERs a 1, 4 ou 9 cellules périodiques. Les

points sur la courbe correspondent à : (1) le premier incrément pour lequel au niveau d’interface il y a

des points situés dans la zone d’adoucissement, (2) instant de bifurcation (critère de Rice est satisfait),

(3) VER à 9 cellules fissuré.

7.4.2 VER contenant plusieurs cellules

Considérons maintenant que le VER est composé de plusieurs cellules de périodicité. Nous prouvons

l’existence de nouvelles solutions qui ne peuvent pas être obtenues par la reproduction périodique des

solutions obtenues sur la cellule unitaire.

Dans les Figs. 7.17 et 7.18 on représente la contrainte globale de cisaillement en fonction du facteur de

chargement !, pour un VER composé d’une, 4 ou 9 cellules périodiques. Le même critère de convergence

est utilisé pour les solutions numériques. On recherche la même hiérarchie pour les différents moments

d’instabilités qu’avant.

Des calculs numériques sont faits pour les mêmes paramètres sauf la valeur du coefficient de frottement.

Les résultats sont montrés dans les Figs. 7.17 et 7.18.

On remarque que la présence du frottement induit un nouveau mode de rupture qui ne respecte plus la

périodicité, en revanche pour les interfaces sans frottement, la rupture reste toujours périodique. On peut

voir facilement que pour le mode de rupture périodique, la réponse globale est indépendante du nombre

des cellules choisi. Quand la rupture ne respecte plus la périodicité, étant dépendante du nombre des

cellules considérées dans le VER, le comportement homogénéisé passe alors dans un régime adoucissant.

Donc, pour une taille de grain donnée, ceci montre une dépendance de la réponse globale sur la taille du

VER.

142

From G. Bilbie’s PhD Thesis

Loss of scale separation

wave propagation

assumed macroscopic quantities that varies at the microscale

Bifurcation

Parallel with experiments -­‐ Representa*ve volume element

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Hill's definiton of the RVE (JMPS 1963)

This phrase will be used when referring to a sample that (a) is structurally entirely typical of the whole mixture on average, and (b) contains a sufficient number of inclusions for the apparent overall moduli to be effectively independent of the surface values of traction and displacement, so long as these values are “macroscopically uniform”. That is, they fluctuate about a mean with a wavelength small compared with the dimensions of the sample, and the effects of such fluctuations become insignificant within a few wavelengths of the surface. The contribution of this surface layer to any average can be made negligible by taking the sample large enough.

Triaxial, biaxial, œdometric tests are designed to be “homogeneous” test

The solving of the self equilibrium problem on the elementary cellis a numerical experiment analogous to the physical one

eY1

eY2fn'/m

fm'/n

nʼm

mʼn

Granular materials

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xm= xm + eF.Y 1

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Thank you for your attention