The DART-Europe E-theses Portal

HAL Id: tel-00833177

https://tel.archives-ouvertes.fr/tel-00833177

Submitted on 12 Jun 2013

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

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

Modélisation mathématique de quelques problèmes de mécanique par homogénéisation

Nuttawat Sontichai

To cite this version:

Nuttawat Sontichai. Modélisation mathématique de quelques problèmes de mécanique par homogénéi- sation. Mécanique des solides [physics.class-ph]. Université Montpellier II - Sciences et Techniques du Languedoc, 2010. Français. �tel-00833177�

pour obtenir le grade de

DOCTEUR DE L’UNIVERSITE MONTPELLIER 2 Discipline : M´ecanique

Ecole Doctorale : Information, Syst`emes et Structures pr´esent´ee et soutenue publiquement

par

Nuttawat SONTICHAI Le 15 d´ecembre 2010

Titre :

MODELISATION MATHEMATIQUE DE QUELQUES PROBLEMES DE MECANIQUE PAR HOMOGENEISATION

————–

JURY

M. Amnuay KANANTAI Professeur Pr´esident

M. Christian LICHT Directeur de Recherche Directeur de Th`ese M. Somsak ORANKITJAROEN Lecturer Directeur de Th`ese

M. Fr´ed´eric LEBON Professeur Rapporteur

M. Somjot PLUBTIENG Professeur Rapporteur

M. Lo¨ıc DARIDON Professeur Examinateur

M. Boriboon NOVAPRATHEEP Lecturer Examinateur

M. Thibaut WELLER Charg´e de Recherche Examinateur

M. Andr´e CHRYSOCHOOS Professeur Invit´e

Mme Yongwimon LENBURY Professeur Invit´ee

i

ACKNOWLEDGEMENTS

I would like to deeply thank the various people who, during the several years in which this endeavor lasted, provided me with useful and helpful assistance. Without their care and consideration, the Ph.D. thesis would likely not have matured.

First of all, thank you my advisor Dr. Somsak Orankitjaroen for all advices and helps. He has supported me throughout my thesis with his patience and knowledge whilst allowing me the B.216 room to work in my own way. One simply could not wish for a better or friendlier advisor.

I owe my deepest gratitude to Professor Christian Licht, who gave an opportunity for me to do research for one year at Laboratoire de M´ecanique et G´enie Civil and Universit´e Montpellier 2, France. I am grateful for his various lectures about homogenization and Γ-convergence which are the main topics for this thesis. Without his support, I am sure that I would not have been able to achieve so much.

I would like to thank my co-advisor, Dr. Boriboon Novaprateep for his invaluable advice and his catering, especially Brazil food.

Many thanks are also to Thibaut Weller for his assistance during studying in Montpellier and for helping on the enrollment and exam appointment.

Thank all lectures in the Department of Mathematics of Mahidol Uni- versity and Department of Mathematics of King Mongkut’s University of Tech- nology North Bangkok. Thanks back to all friends who have cheered me up in several hard time.

I also would like to acknowledge Department of Mathematics of Chi- angmai University and the Ministry Staff Development Project of the Ministry of University Affairs for their scholarships that allow me to be here, and to my parents who always support me no matter how far away I live.

And finally never enough thanks to one who doesn’t want to be named but she knows who she is and so do I.

Nuttawat Sontichai

MATHEMATICAL MODELING OF SOME MECHANICS PROBLEMS BY HOMOGENIZATION.

NUTTAWAT SONTICHAI Ph.D. (MECHANICS)

THESIS ADVISORY COMMITTEE : CHRISTIAN LICHT, Ph.D., SOM- SAK ORANKITJAROEN, Ph.D.,

ABSTRACT

We proposed, by homogenization, some models for effective behaviors of structures made of two materials with a high contrast of stiffness (or permit- tivity) and of size in the occupied domains. Because each steady-state problem considered here may be formulated in terms of minimization, we used the method of variational convergence. The first two parts dealt with fibered structures: first in a non-linear scalar case and the next in a linear vector case. Whilst aiming to generalize the results of Bellieud–Bouchitt´e and Gruais, we proposed a different strategy. The third study concerned 2-D modeling of some thin flat masonry where the mortar occupied a rather thin domain and is far softer than the bricks.

A variant devoted to 3-D linearly elastic stratified media was the final part of this work.

KEY WORDS : HOMOGENIZATION, VARIATIONAL CONVERGENCE, ELASTIC MATERIALS

?? pages

iii

CONTENTS

ACKNOWLEDGEMENTS i

ABSTRACT (ENGLISH) ii

LIST OF FIGURES v

REFERENCES xv

1 Introduction 1

REFERENCES 6

2 Mathematical Modeling of Fiber Reinforced Structures by Ho-

mogenization 8

2.1 Setting the Problem . . . 8

2.2 An Alternative Strategy . . . 11

2.3 Conclusions and Remarks . . . 18

REFERENCES 20 3 A Remark on the Homogenization of a Microfibered Linearly Elastic Material 21 3.1 Setting the Problem . . . 21

3.2 A Different Approach . . . 24

3.3 Conclusions and Remarks . . . 30

3.4 Appendix : The Vector Capacitary Problem . . . 31

REFERENCES 33 4 A Simplified 2-Dimensional Model for Some Elastic Masonries 34 4.1 Setting the Problem . . . 34

4.2 The Asymptotic Model . . . 38

4.2.1 An Auxiliary Problem . . . 38

4.2.2 The Convergence Result . . . 39

4.2.3 Mechanical Interpretation, a Proposal of Model . . . 42

REFERENCES 44

5 Modeling of Some Stratified Media 45

5.1 Setting the Problem . . . 45

5.2 Asymptotic Modeling . . . 46

5.2.1 A Convergence Result . . . 47

5.2.2 Mechanical Interpretation, Proposal of Model . . . 48

5.3 A 1-Dimensional Case . . . 48

6 Conclusion and Discussion 51

REFERENCES 54

v

LIST OF FIGURES

2.1 The domain Ω = ω×(0, L) occupied by a composite material . . 9 2.2 The circular cross section of the fiber Y_εⁱ ⊂ω . . . 9 3.1 The fibered structure . . . 21 4.1 The unit cell ˆY, S, ˆM_l and ˆB_l . . . 34

R´ ESUM´ E DE LA TH` ESE

Dans cette ´etude nous avons propos´e par homog´en´eisation quelques mod`eles math´ematiques de comportement effectif de structures faites de deux phases avec un fort constraste de rigidit´e (ou conductivit´e) et de taille des domaines occup´es. Parce que chaque probl`eme (de statique) consid´er´e dans ce travail peut ˆetre ´ecrit comme un probl`eme de minimisation, la m´ethode utilis´ee est celle de la convergence variationnelle (propri´et´e de compacit´e pour les suites d’´energie uni- form´ement born´ees, bornes inf´erieures et sup´erieures de l’´energie de ces suites).

D’abord, nous ´etudions une structure renforc´ee par des fibres dans le cas scalaire (´electricit´e ou thermique stationnaire). Le domaine occup´e par la structure est Ω := ω×(0, L) avec L >0 et ω un domaine de R². Pour chaque ε >0, nous consid´erons une distribution p´eriodique de cellules (Y_εⁱ)_i∈Iε telle que Y_εⁱ := (εi1, εi2) + (−ε/2, ε/2)², et Iε := {i∈Z² |Y_εⁱ ⊂ω}. Soient (D_rⁱ_ε)i∈Iε la famille de disques de R² centr´es en ˆxⁱ_ε := (εi₁, εi₂) de rayonr_ε ¿ε, T_εⁱ :=Dⁱ_rε× (0, L) et Tε := ∪i∈IεT_εⁱ. L’ensemble de minces cylindres parall`eles Tε repr´esente les fibres. Le coefficient de conductivit´ea_ε est

aε(x) =

(1, si x∈Ω\T_ε, λ_ε, sinon.

Nous supposons r_ε→0, r_ε

ε →0, λ_ε →+∞, k_ε:=λ_εr_ε²

ε² →k, k≥0 quand ε→0.

La donn´ee au bord Γ₀ u₀ est Lipschitz, pendant que le chargement (f, g) ∈ L^p0(Ω)×L^p0(Γ₁), p⁰ =p/(p−1), Γ₁ =∂Ω\Γ₀.

Ce probl`eme peut ˆetre ´ecrit sous forme de probl`eme de minimisation : (P_ε) min©

F_ε(w)−L(w)|w∈W_Γ^1,p₀ (Ω)ª ,

vii

o`u

W_Γ^1,p₀ (Ω) :=©

w∈W^1,p(Ω)|w=u₀ sur Γ₀ª , F_ε(w) :=

Z

Ω

a_εφ_p(∇w)dx, φp(ξ) := 1

p|ξ|^p, ∀ξ ∈Rⁿ, n = 1,2,3, L(w) :=

Z

Ω

f w dx+ Z

Γ1

gw dH².

L’ ´etude d’ homog´en´eisation consiste en l’examen du comportement de la suite de solutions (u_ε) quand ε tend vers z´ero. Dans [4], il a ´et´e montr´e que la limite de (P_ε) est :

min©

Φ(u, v)−L(u)|(u, v)∈(L^p(Ω))²ª , o`u

Φ(u, v) =











 R

Ωφp(∇u)dx+^kπ_p R

Ω

¯¯

¯_∂x^∂v₃

¯¯

¯^p dx+^2πγ_p R

Ω|v−u|^pdx, si

((u, v)∈W_Γ^1,p₀ (Ω)×L^p(ω, W^1,p(0, L)), v =u₀ sur Γ₀∩(ω₀∪ω_L),

+∞ sinon, et

[0,+∞]3γ =





limε→0ε⁻²|logrε|⁻¹, sip= 2, lim_ε→0

¯¯

¯^2−p_p−1

¯¯

¯^p−1r_ε^2−pε⁻², sip6= 2.

Nous sommes concern´es par l’extension de ce r´esultat pour des sections de fibres plus g´en´erales et des densit´es d’´energie plus g´en´erales que φ_p. Notre but est de fournir une autre preuve que nous esp´erons plus apte `a traiter de tels cas plus r´ealistes. Les ´etapes de la preuve de [4] consistent `a ´etablir successivement :

(i) une propri´et´e de compacit´e pour les suites (u_ε) telles que F_ε(u_ε)< C, (ii) une in´egalit´e de borne inf´erieure pour les suites (F_ε(u_ε)),

(iii) une in´egalit´e de borne sup´erieure pour les suites (F_ε(u_ε)).

Ici, nous remplacons les ´etapes (ii) et (iii) par

(ii⁰) une ´egalit´e de borne sup´erieure de la suite (Fε(uε)).

(iii⁰) une in´egalit´e de borne inf´erieure de la suite (F_ε(u_ε)), qui essentiellement utilise une in´egalit´e sous-diff´erentielle.

La pr´ec´edente analyse peut ˆetre ais´ement ´etendue au cas o`u φp est replac´ee par n’importe quelle fonction convexe qui satisfait :

∃M > 0, ∃r ∈(1, p) ; |W(ξ)−φ_p(ξ)| ≤M|ξ|^r ∀ξ∈R³, la densit´e associ´ee `a Φ(u, v) devient :

W(∇u) + 2πγ|v−u|^p+W µ ∂v

∂x₃

¶ .

En fait, les arguments cl´es de notre analyse sont l’identification deγ en termes de solutions de probl`emes capacitaires et d’utiliser la p-positive homog´en´eit´e et la convexit´e deφ_p et le fait que φ_p(ξ)≥φ_p(ξ₃),ξ = (ξ₁, ξ₂, ξ₃). Donc, il est facile de deviner ce que pourrait ˆetreφ(u, v), quand φ_p est remplac´ee par n’importe quelle fonction strictement convexe et quand les sections des fibres sont des domaines deR² ´etoil´es r´eguliers. Nous esp´erons que notre strat´egie sera capable de r´eduire et de surmonter les difficult´es techniques mises en jeu.

Ensuite, nous ´etudions le comportement effectif d’une structure cylin- drique micro-fibr´ee comme au chapitre pr´ec´edent, faite d’un mat´eriau isotrope lin´eairement ´elastique entourant une distribution p´eriodique de fibres isotropes lin´eairement ´elastiques tr`es fines et de tr`es grande rigidit´e.

Les coefficients de Lam´e λε,µε de la structure sont tels que λ_ε(x) =

(λ₀ >0, six∈Ω\T_ε

λ_ε1, six∈T_ε , µ_ε(x) =

(µ₀ >0, si x∈Ω\T_ε µ_ε1, si x∈T_ε.

La structure est fix´ee sur la partie Γ₀ := ω × {0, L} de la fronti`ere ∂Ω de Ω, soumise `a des forces volumiques de densit´e f et surfaciques de densit´eg sur Γ₁ :=∂Ω\Γ₀. Il est bien connu que siλ_ε1,µ_ε1 >0,f ∈L²(Ω;R³),g ∈L²(Γ₁;R³), la d´etermination d’un ´equilibre conduit au probl`eme de minimisation

(Pε) min©

Fε(w)−L(w)|w∈H_Γ¹₀(Ω;R³)ª , o`u

Fε(w) :=

Z

Ω

Wε(e(w))dx, L(w) :=

Z

Ω

f ·w dx− Z

Γ1

g·w ds,

ix

et

W_ε(e) := 1

2λ_εtr²e+µ_ε|e|², ∀e∈S³ l’espace des matrices sym´etriques 3×3, H_Γ¹₀(Ω;R³) ={v ∈H¹(Ω;R³)|v = 0 sur Γ₀},

qui a une solution unique ¯u_ε.

Pour d´eterminer le comportement effectif de la structure microfibr´ee, nous ´etudions le comportement asymptotique de ¯u_ε quandε tend vers z´ero. Soit

kε :=µε1

|T_ε|

|Ω|, lε := λ_ε1 µ_ε1, et supposons que, quand ε→0,

r_ε →0, r_ε

ε →0, λ_ε1 →+∞, µ_ε1 →+∞,

k_ε →k∈[0,+∞], r²_εk_ε →κ∈[0,+∞], l_ε→l ∈[0,+∞), (ε²|lnr_ε|)⁻¹ →γ ∈[0,+∞).

Soit 1_Tε la fonction caract´eristique deT_ε etM_b(Ω;R³) l’espace des mesures sur Ω born´ees `a valeurs dans R³. Il a ´et´e prouv´e dans [5] que, quand ε tend vers z´ero,

¯

u_ε converge faiblement dans H¹(Ω;R³) vers ¯u et que ¯v_ε := _|T^|Ω|

ε|u¯_ε1_Tε converge

* faible dans M_b(Ω;R³) vers un ´el´ement ¯v de L²(Ω;R³) solution de (P^eff) min©

Φ(u, v)−L(u)|(u, v)∈L²(Ω;R³)²ª , avec

Φ(u, v) =





































 Z

Ω

W₀(e(u))dx

+µ₀πγ Z

Ω

(v−u)^T





χ+1

χ 0 0

0 ^χ+1_χ 0

0 0 1



(v−u)dx

+1 2

3l+ 2 2(l+ 1)k

Z

Ω

¯¯

¯¯∂v₃

∂x₃

¯¯

¯¯

2

dx +1

2

3l+ 2 2(l+ 1)

κ 4

Z

Ω

¯¯

¯¯∂²v₁

∂x²₃

¯¯

¯¯

2

+

¯¯

¯¯∂²v₂

∂x²₃

¯¯

¯¯

2

dx,

si (u, v)∈ D,

+∞, sinon,

et

W0(e(u)) := 1

2λ0tr²e(u) +µ0|e(u)|², χ:= λ₀+ 3µ₀ λ₀+µ₀ , D:=H_Γ¹₀(Ω;R³)×

½

v ∈L²(ω, H₀²(0, L;R³))

¯¯

¯¯ ∂v₁

∂x3

= ∂v₂

∂x3

= 0 sur Γ₀

¾ .

Donc, le comportement effectif de la structure microfibr´ee est celle d’un milieu continu ´elastique g´en´eralis´e mettant en jeu une variable d’´etat suppl´ementaire et ses deux premi`eres d´eriv´ees. Cette variable d’´etat suppl´ementaire rend compte de la microstructure dans la mesure o`u elle d´ecrit le comportement asymptotique du d´eplacement dans les fibres judicieusement mis `a l’´echelle. Notre objectif ´etait de comprendre ce r´esultat plus profond´ement dans un cadre plus g´en´eral, quant `a la g´eom´etrie des sections des fibres et au comportement de la matrice et des fibres.

N´eanmoins, nous ne sommes, `a nouveau, que parvenus `a donner une autre preuve du r´esultat de [5] en ´etudiant directement la convergence variationelle (comme dans le cas scalaire) de F_ε et avons divis´e notre preuve en trois ´etapes :

1. une propri´et´e de compacit´e pour toute suite (u_ε) telle queF_ε(u_ε) soit uni- form´ement born´ee,

2. une borne sup´erieure atteinte pour une suite (F_ε(u_ε)), 3. une in´egalit´e de borne inf´erieure pour une suite (F_ε(u_ε)).

L’ingr´edient essentiel est la construction de champs oscillants appropri´es four- nissant la “meilleure” borne sup´erieure. Donc, il semble possible avec des sec- tions de fibres plus g´en´erales (r_ε4avec∂4suffisamment r´eguli`ere) et une densit´e d’´energie de d´eformation quadratique plus g´en´erale W<sub>M</sub> pour la matrice, dans la mesure o`u les solutions θ^α_ε des probl`emes capacitaires

min









 Z

(−ε,ε)²

W_M(e(ϕ))dxˆ

¯¯

¯¯

¯¯

¯¯

¯

ϕ∈H¹((−ε, ε)²;R³),

ϕ(ˆx) =e^α sur r_ε4,{e^α}_α=1,2,3 base deR³, ϕ(ˆx) = 0 sur (−ε, ε)²\D(0, R_ε).











sont telles que :

i) ∃w_M^cap ∈S³ telle que (w^cap_M )_αβ = lim_ε→0ε⁻²R

(−ε,ε)²w_M(e(θ_ε^α), e(θ^β_ε))dx,ˆ ii) lim_ε→0R

εi+rε∂4W_M⁰ (e(θ_ε^α))·(u_ε−u¯¯_ε)dl

= lim_ε→0R

∂D_Rεⁱ W_M⁰ (e(θ^α_ε))·(u_ε−u¯_ε)dl= 0.

Dans l’´etude suivante, nous appliquons la th´eorie de l’homog´en´eisation au comportement statique de ma¸conneries lin´eairement ´elastiques minces et plates.

xi

Soit a un nombre r´eel positif, H ∈ aN, et h un nombre r´eel positif petit. Le domaine ocup´e par la ma¸connerie mince consid´er´ee ici est Ω_h := ω× (−h, h) o`u ω := (0, L)×(0, H). Ce mur est constitu´e d’une distribution p´eriodique de briques reli´ees l’une `a l’autre par un mortier adh´esif occupant un domaine tr`es mince. Plus pr´ecis´ement si ˆY := (0,1)×(0, a),

S:={yˆ∈Yˆ |y₂ =a/4 ou 3a/4;

y₁ = 1/4, et 0< y₂ < a/4 ou 3a/4< y₂ < a;

y₁ =b eta/4< y₂ <3a/4, o`u b∈(0,3/4]}, Mˆ_l:={xˆ∈Yˆ |dist(ˆx, S)< l}, l <min{b, a/4},

Bˆ_l:= ˆY \Mˆ_l, ε=L/n, n∈N,

Yˆ_εⁱ :=εi+εY ,ˆ i∈I_ε :={i∈Z² |Yˆ_εⁱ ⊂ω}, Mˆ_lεⁱ =εi+εMˆ_l, Mˆ_lε =∪_i∈IεMˆ_lεⁱ,

Bˆ_lεⁱ =εi+εBˆ_l, Bˆ_lε=∪_i∈IεBˆ_lεⁱ ,

alors B_lεh := ˆB_lε×(−h, h) et M_lεh:= ˆM_lε×(−h, h) sont les domaines repective- ment occup´es par les briques et le mortier. Les briques sont faites d’un mat´eriau homog`ene lin´eairement ´elastique dont la densit´e d’´energie W est une fonction quadratique strictement convexe v´erifiant

∃α, β >0 ; α|e|² ≤W(e)≤β|e|² ∀e∈S³,

l’espace des matrices 3×3 sym´etriques. Le mortier est supposs´e constitu´e d’un mat´eriau isotrope lin´eairement ´elastique de densit´e d’´energie W_λµ dont les coef- ficients de Lam´eλ etµ sont suppos´es ˆetre bien plus faibles queα etβ. Enfin, le mur est encastr´e le long de Γ_0h = γ₀×(−h, h), ∂Ω_h, γ₀ ´etant une partie de ∂ω de longueur positive, et est soumis `a des forces volumiques de densit´e f^h.

Pour trouver les configurations d’´equilibre du mur, on est conduit au probl`eme

(P) min

½ Z

Blεh

W(e(u)(x))dx+ Z

Mlεh

Wλµ(e(u)(x))dx− Z

Ωh

f^h(x)·u(x)dx

¯¯

¯u∈H_Γ¹_0h(Ω_h;R³)

¾

o`u (

H_Γ¹_0h(Ωh;R³) := {v ∈H¹(Ωh;R³)|v = 0 sur Γ0h au sens des traces}, e(v) := ¹₂(∇v+∇v^T).

Supposant f^h dans L²(Ω_h;R³), ce probl`eme clairement a une solution unique, mais, `a cause des faibles valeeurs de (ε, l, λ, µ, h), l’obtention d’une approximation num´erique peut ˆetre difficile. Donc, il est int´eressant de proposer un mod`ele simplifi´e mais suffisamment pr´ecis. Une premi`ere tentative [6] est de remplacer (P) par un probl`eme bidimensionnel appropri´e pos´e sur la surface moyenne du mur qui peut s’´ecrire :

( ˆP_ˆs) min

½ Z

Bˆlε

Wˆ(e(u)(ˆx))dˆx+ Z

Mˆlε

Wˆ_λ⁰_µ(e(u)(ˆx))dˆx− Z

ω

f(ˆˆx)·u(ˆx)dˆx

¯¯

¯u∈H_γ¹₀(ω;R²)

¾ , o`u

• Wˆ, d´eduite deW, est une fonction strictment convexe v´erifant

∃α,ˆ β >ˆ 0 ; α|e|ˆ ² ≤Wˆ(e)≤β|e|ˆ ² ∀e∈S²,

• Wˆ_λ⁰_µ(e) = ^λ₂⁰(e₁₁ +e₂₂)² +µ|e|², λ⁰ = _λ+2µ^2λµ , qui correspond `a W<sub>λµ</sub> sous l’hypoth`ese des contraintes planes,

• fˆest d´eduite de f^h,

• sˆ:= (ε, l, λ⁰, µ).

Supposant encore que ˆf ∈ L²(ω;R²), le probl`eme ( ˆP_ˆs) a une unique solution ˆ

u_ˆs, mais, `a cause des faibles valeurs de ˆs, obtenir une approximation num´erique reste difficile. Aussi, prenant en compte ces faibles valeurs, nous proposerons un mod`ele simplifi´e mais suffisamment pr´ecis en ´etudiant le comportement asymp- totique de ( ˆP_s) quand ˆs tend vers z´ero.

En fait, ( ˆP_s) ressemble `a un probl`eme d’homog´en´eisation p´eriodique puisque la g´eom´etrie et les propri´et´es m´ecaniques de la microstructure sont εYˆ- p´eriodiques, mais la g´eom´etrie met en jeu un param`etre suppl´ementaire l tandis

xiii

que la densit´e de d´eformation du mortier met en jeu deux autresλ⁰,µ. Si (l, λ⁰, µ)

´etaient fixes, la densit´e effective de la structure h´et´erog`ene serait W_lλ^eff0_µ(E) := min

½ 1

|Yˆ| µZ

Bˆl

Wˆ(E+e(v)(y))dy+ Z

Mˆl

Wˆ_λ⁰_µ(E+e(v)(y))dy

¶

¯¯

¯v ∈H_per¹ ( ˆY;R²)

¾

o`u

H_per¹ ( ˆY;R²) :={v ∈H¹( ˆY;R²)|les traces dev sur les cˆot´es oppos´es de ˆY sont ´egales}.

Le comportement asymptotique de W_lλ^eff0µ quand (l, λ⁰, µ)→0 est un probl`eme de mod´elisation de jonctions ´elastiques souples. Avec l’hypoth`ese suppl´ementaire

∃λ¯⁰,µ¯∈(0,+∞) ; λ⁰ ∼2¯λ⁰l, µ∼2¯µl,

qui correspond au cas le plus int´eresant du point de vue m´ecanique, les arguments de [7] et [1] montrent que

(l,λlim⁰,µ)→0W_lλ^eff0_µ(E) = min

½ 1

|Yˆ| µZ

Yˆ\S

Wˆ(E +e(v)(y))dy+ Z

S

W¯λ¯⁰µ¯([v](y))dl

¶

¯¯

¯v ∈H_per¹ ( ˆY \S;R²)

¾ , o`u

H_per¹ ( ˆY \S;R²) :={v ∈H¹( ˆY \S;R²)| les traces dev

sur les cˆot´es oppos´es de ˆY sont ´egales}, W¯¯λ⁰µ¯(v) := ˆWλ¯⁰µ¯(v⊗_sn) ∀v ∈R², a⊗_sb:= 1

2(a⊗b+b⊗a),

n est une normale unitaire `a S, [v] est la diff´erence, prise dans le sens den, des traces de v sur S.

Dans la suite, nous prouvons que cette limite est la densit´e effective d’´energie de deformation du mur en d´eterminant le comportement asymptotique de la fonctionnelle ´energie de d´eformation totale F_ˆs mise en jeu par ( ˆP_s) quand ˆ

s→0 avec la condition suppl´ementaire pr´ec´edente.

On ´etablit la convergence variationnelle, pour la convergence faible * dans BD(ω), de F_sˆ vers une fonctionnelle de densit´e W^eff donn´ee par la pr´ec´edente limite de W_l,λ^eff0,µ.

En fait, nous n’avons pas r´eussi `a prouver que les points d’accumulation des suites d’´energies uniform´ement born´ees appartiennent `aH¹(ω;R²) et v´erifient une condition homog`ene de Dirichlet sur γ₀. Donc, on ne peut ([2], [3]) affirmer que ˆu_s converge ( * faible dans BD(ω)) vers l’unique solution de

( ¯P) min

½ Z

ω

W^eff(e(v))dx− Z

ω

fˆ·v dx

¯¯

¯¯v ∈H_γ¹₀(ω;R²)

¾ .

Le probl`eme ( ¯P) d´ecrit l’´equilibre d’un corps ´elastique bidimensionel homog`ene occupantω comme configuration de r´ef´erence, de densit´e d’´energieW^eff, encastr´e le long deγ₀ et soumis `a des forces de densit´e ˆf. Nous affirmons seulement qu’un candidat raisonnable comme densit´e d’´energie de d´eformation de l’assemblage briques mortier est W<sup>eff</sup>. Ainsi, notre mod`ele est plus simple que le mod`ele de d´epart puisqu’il met en jeu un corps homog`ene et aussi suffisamment pr´ecis `a cause du r´esultat de convergence. Ceci fut a ´et´e obtenu en [6] par des arguments plutˆot heuristiques.

Le mat´eriau homog`ene ´equivalent est plus faible que le mat´eriau con- stituant les briques. Ceci est le prix `a payer dˆu `a la difficult´e `a faire un mur homog`ene. La r´esistance effective du mur serait plus grande que celle des briques si du mortier tr`es rigide ´etait utilis´e. Il serait int´eressant de consid´erer les cas o`u λ⁰, µ´etaient d’ordre de grandeur 1/l.

Enfin, nous avons proc´ed´e `a une extension bidimensionnelle en con- sid´erant des milieux stratifi´es constitu´es `a partir de deux constituants lin´eairement

´elastiques, le constituant le plus faible occupant des couches beaucoup plus minces que l’autre. Il est `a noter que dans ce cas les domaines occup´es par chaque phase ne sont pas connexes.

Comme pr´ec´edemment, nous ne sommes parvenus qu’`a prouver la con- vergence variationnelle (vis `a vis de la convergence * faible dansBD) des ´energies de d´eformations totales.

Nous conjecturons qu’il y a aussi convergence des minimiseurs pour cer- taines g´eom´etries. C’est ce que nous avons observ´e sur un exemple unidimensionel o`u l’on peut faire des calculs analytiques : la *faible limite des minimiseurs est aussi une limite uniforme et satisfait aux conditions aux limites de Dirichlet.

xv

REFERENCES

[1] A. Aitmoussa, Mod´elisation et ´etude des Singularit´es de Contraintes d’un Joint Coll´e tr`es mince, Ph.D. Thesis, University Montpellier 2, France, 1989.

[2] H. Attouch, Variational Convergence for Functions and Operators, Appli- cable Mathematics Series, Pitman Advanced Publishing Program, Boston, 1985.

[3] H. Attouch, G. Buttazzo, G. Michaille, Variational Analysis in Sobolev and BV Spaces : Application to PDEs and Optimization, MPS-SIAM Book Series on Optimization, 2005.

[4] M. Bellieud, G. Bouchitt´e, Homogenization of Elliptic Problems in a Fiber Reinforced Structure. Non local effects, Ann. Scuola Norm. Sup. CI Sci. (4) 26 (1998) p.407–432.

[5] M. Bellieud, I. Gruais, Homogenization of an Elastic Material Reinforced by Very Stiff or Heavy Fibers. Non Local Effects. Memory Effects, J. Math.

Pures Appl. 84 (2005), 55–96.

[6] A. Cecchi, K. Sab,A Multi-Parameter Homogennization Study for Modeling Elastic Masonry, European Jounal of Mechanics A/Solids, 21 (2002), p.249–

268.

[7] C. Licht, G. Michaille, A Modelling of Elastic Adhesive Bounded Joints, Advances in Mathematical Sciences and Applications, 7 (1997), no 2, p.711–

740.

CHAPTER I INTRODUCTION

Mechanics of materials is the study of behavior of forces inside and outside the solid bodies. Formulas developed in mechanics of materials relate stresses to internal forces and moments. The study of mechanics materials is very useful for helping to choose the kind of materials with appropriated using in product, industrial and electrical designs.

In this thesis, we consider some composite materials consisting of short or long, continuous or discontinuous and one or in multiple directions fibers em- bedded in a matrix. Such materials offer advantages over conventional isotropic structural materials such as steel, aluminum, and other types of metal. These advantages include high strength, low weight as well as good fatigue and cor- rosion resistance. In addition, by changing the arrangements of the fibers, the properties of the material can be tailored to meet the requirements of a specific design.

The excellent properties of composites are achieved by the favorable characteristics of the two major constituents, namely the fiber and the matrix. In low-performance composites provide some stiffening but very little strengthening.

They usually are in the form of particles, short or chopped fiber. The load is mainly carried by the matrix. In high-performance composites, continuous fibers, provide the desirable stiffness and strength, whereas the matrix provides protection and support for the fibers as well as helps redistribute the load from broken to adjacent intact fibers.

Composite materials are heterogeneous materials obtained by mixing several phases or constituent materials on a very fine (microscopic) scale. How- ever, one is usually interested only in the large scale (macroscopic) properties of such a composite. Therefore, the main problem with composite materials is to determine their effective properties without determining their fine scale structure.

Nuttawat Sontichai Introduction / 2

Composite materials can be studied from a number of different view- points each of which requires a different kind of expertise. On a scale that is large with respect to the fiber diameter, the fiber and matrix properties may be averaged, and the material may be treated as homogeneous. This assumption, commonly employed in macromechanical analysis of composites, the design steps from micromechanics (which takes into account the fiber and matrix properties) through macromechanics (which treats the properties of the composite) to struc- tural analysis. Hence, the material is considered to be quasi-homogeneous, which implies that the properties are taken to be the same at every point. These prop- erties are not the same as the properties of either the fiber or the matrix but are a combination of the properties of the constituents.

Mathematicians have been interested in composite materials since the 1970’s. Their first main contribution in this field was to give a theoretical basis for the notion of effective properties of a composite material. Indeed, homogenization theory permits one to properly define a composite material as a limit, in the sense of homogenization, of a sequence of increasingly finer mixtures of the constituent phases. Effective properties are now defines as homogenized coefficients. The application of homogenization to the modeling of composite materials has became a popular subject in applied mathematics. The physical ideas of homogenization have a very long history going back at least to Maxwell [1], Poisson [2] and Rayleigh [3]. The homogenization method in optimal design was initiated by Murat and Tartar in the late 1970’s. The first relevant references are the works of Murat and Tartar [4], [5], [6] and [7].

Our primary technique, in this work, is the homogenization of com- posite materials [8]. The procedures are mainly based on the definition of a local surrounding of a macroscopic material point with volume and the boundary.

This volume represents a characteristic part of the material which is sufficient to describe its structure and behavior, respectively. This implies that the size of heterogeneities at the microscopic level has to be one scale smaller than the size of the volume of the macroscopic point. Homogenization has many poten- tial applications, but we consider it only as a tool for deriving macroscopic or

effective properties of microscopically heterogeneous media. As such it provides a firm basis to the notion of composite material obtained by mixing, on a very fine scale, several phase components.

Although homogenization is not restricted to periodic problems, it has been used often for the asymptotic analysis of periodic structures. Indeed, in many fields of science and mechanics one has to solve boundary value problems in periodic media. Quite often the size of the period is small compared to the size of a sample of the medium, and, is often denoted by ε. An asymptotic analysis, as ε goes to zero, allows one to reduce the complexity of the problem.

Starting from a microscopic description of a problem, one seeks a macroscopic, or effective model. This process of making an asymptotic analysis in order to derive an averaged formulation is precisely the goal of homogenization.

In this thesis we focus our attention on minimization problems and apply the convergence theory for sequences of functions, variational convergence.

This convergence may be regarded as the weak convergence, which guarantees that an approach to the limit existence in the corresponding minimization prob- lems. Therefore this concept of convergence has natural applications in all branches of optimization theory.

In the framework of this thesis, the second chapter focuses on the study of fiber reinforced structures by considering the scalar case. Here we consider the homogenization of the quasi-linear elliptic problem of degree p, 1 < p < +∞. Bellieud and Bouchitt´e [9] already succeed in this problem.

Here we present another proof that we expect to be more suitable to treat more general geometrical and physical cases. Because the energy density is assumed to be positively homogeneous of degree p, the suitable Sobolev space that we use here is the W^1,p. The main theorem is the variational convergence problem.

The steps of our proof are the compactness properties, upper bound equality and lower bound inequality, respectively. Here the upper bound equality is to be proved before the lower bound inequality. Because we have to use the results in the upper bound for complete the proof in the lower bound by subdifferential inequality.

Nuttawat Sontichai Introduction / 4

The third chapter is nearly devoted like in the previous chapter, but it focuses in vector case and is considered in a homogenization of a linearly elliptic boundary value problem of elasticity. The isotropic elastic material is reinforced by a periodic distribution of very thin parallel fibers in which the Lam´e coefficients are assumed to have high values. Bellieud and Gruais [10] proved that the macroscopic behavior is the one of a generalized continuum medium involving an additional state variable accounting for the microstructure. Here we propose a proof of this result by studying the variational convergence of the energy functional. The processes of the proof are likely in the second chapter, but this one is more complicated. We have to choose the suitable capacitary problem for furnishing the result.

The fourth chapter is concerned with the homogenization method for some elastic masonries. Elastic masonry is a heterogeneous medium which shows an anisotropic and inhomogeneous nature. In particular, the inhomogeneity is due to its composite materials, mortar and bricks, which have very different me- chanical properties. The anisotropy is due to the different masonry patterns since the mechanical response is affected by the geometrical arrangement of the com- ponents. The unquestionable importance of a lot of real masonry estate requires researcher particular attention for this kind of structures. Therefore, in order to design an efficient response for repairing existing masonry structures, a large number of theoretical studies, experimental laboratory activities and computa- tional procedures have been proposed in scientific literature [11], [12] and [13];

moreover due to the low values of parameters, obtaining numerical approxima- tions may be difficult. Thus, it is of interest to propose a simplified but accurate enough model. This work is illustrated by results of 2-dimensional model. A first attempt [11] is to replace 3-dimensional problem by a suitable 2-dimensional problem set in the cross section of the wall. The techniques of bonding [12], [13]

and homogenization are used for deriving the effective behavior.

Finally, the fifth chapter is devoted to stratified medium models, it extends the problem of the fourth chapter but we consider a 3-dimensional case.

We mostly focus on a stratified medium made from two homogeneous linearly

constituents, one being far weaker than the other. Moreover, as in the case of mortar in the fourth chapter, the weak constituent occupies layers far thinner than the thin layers occupied by the stronger material. This chapter also states the example in 1-dimensional case, where closed-form formulas are available, for studying the results and comparing with the result in 3-dimensional case that corresponding or not.

In the sequel, we do not point out open problems, but it is clear that the range of applications covered by this thesis, although very important. Of course, there are many other types of homogenization problems that have not yet been attacked. It is our hope that this thesis can serve as a basis for further developments in new directions.

Nuttawat Sontichai References / 6

REFERENCES

[1] J. C. Maxwell,A Treatise on Electricity and Magnetism, 3rd Ed., Clarendon Press, Oxford, 1881.

[2] S. Poisson, Second M´emoire sur la Th´eorie du Magn´etisme, Mem. Acad, France, 5 (1822).

[3] J. W. Rayleigh, On the Influence of Obstacles Arranged in Rectangular Order Upon the Properties of a Medium, Phil. Mag., 32 (1892), p.481–491.

[4] F. Murat, L. Tartar, Calcul des Variations et Homog´en´eisation, Les M´ethodes de l’Homog´en´eisation Th´eorie et Applications en Physique, Coll.

Dir. Etudes et Recherches EDF, 57, Eyrolles, Paris, 1985, p.319–369.

[5] F. Murat, L. Tartar, Optimality Conditions and Homogenization, in Non- linear Variational Problems, A. Marino et al. eds., Pitman, Boston, 1985, p.1–8.

[6] L. Tartar, Probl`emes de Contrˆole des Coefficients dans des ´equations aux D´eriv´ees Partielles, Control Theory, Numrical Methods and Computer Sys- tems Modelling, International Symposium, Rocquencourt, June 1974, A.

Bensoussan, J. L. Lions eds., Lecture Notes in Econnomy and Mathemati- cal Systems, Springer Verlag, 107 (1975), p.420–426.

[7] L. Tartar, Estimaiton de Coefficients Homog´en´eis´es, Computing meth- ods in applied sciences and engineering, Third International Symposium, December 1977, R. Glowinski, J. L. Lions eds., Lecture Notes in Math, Springer Verlag, 704 (1979), p.364–373.

[8] H. Attouch, Variational Convergence for Functions and Operators, Appli- cable Mathematics Series, Pitman Advanced Publishing Program, Boston, 1985.

[9] M. Bellieud, G. Bouchitt´e,Homogenization of Elliptic Problems in a Fiber Reinforced Structure. Non local effects, Ann. Scuola Norm. Sup. CI Sci. (4) 26 (1998), p.407-432.

[10] M. Bellieud, I. Gruais, Homogenization of an Elastic Material Reinforced by Very Stiff or Heavy Fibers. Non Local Effects. Memory Effects, J. Math.

Pures Appl. 84 (2005), 55–96.

[11] A. Cecchi, K. Sab, A Multi-Parameter Homogennization Study for Model- ing Elastic Masonry, European Jounal of Mechanics A/Solids, 21 (2002), p.249–268.

[12] C. Licht, G. Michaille, A Modelling of Elastic Adhesive Bounded Joints, Advances in Mathematical Sciences and Applications, 7 (1997), no 2, p.711–

740.

[13] A. Aitmoussa, Mod´elisation et ´etude des Singularit´es de Contraintes d’un Joint Coll´e tr`es mince, Ph.D. Thesis, University Montpellier 2, France, 1989.

Nuttawat Sontichai Mathematical Modeling of Fiber Reinforced... / 8

CHAPTER II

MATHEMATICAL MODELING OF FIBER REINFORCED STRUCTURES BY

HOMOGENIZATION

2.1 Setting the Problem

Let p ∈ (1,+∞), we consider the homogenization of the elliptic

problem 









−divσε =f on Ω, σε =aε|∇uε|^p−2∇uε,

uε =u0 on Γ0, σε·n=g on Γ1,

(2.1)

where Ω :=ω×(0, L) with L >0 andω is a bounded domain ofR² with smooth boundary and containing the origin of coordinates. The homogenization study of (2.1) consists in examining the behavior of the sequence of the solution (u_ε) asε tends to zero. The permittivity coefficienta_ε isε-periodic and satisfies a uniform lower bound, Γ₀ is an open subset of∂Ω with Hausdorff measure H²(Γ₀) strictly positive, Γ₁ = ∂Ω\Γ₀, and n is the unit exterior normal on ∂Ω. The boundary data u₀ is Lipschitz, while (f, g)∈L^p0(Ω)×L^p0(Γ₁),p⁰ =p/(p−1).

The problem (2.1) is related to the minimization problem (P_ε) min©

F_ε(w)−L(w)|w∈W_Γ^1,p₀ (Ω)ª , where

W_Γ^1,p₀ (Ω) :=©

w∈W^1,p(Ω)|w=u₀ on Γ₀ª , F_ε(w) :=

Z

Ω

a_εφ_p(∇w)dx, φp(ξ) := 1

p|ξ|^p, ∀ξ∈Rⁿ, n= 1,2,3, L(w) :=

Z

Ω

f w dx+ Z

Γ1

gw dH². (2.2)

We are interested in the asymptotic behavior of (P_ε) as ε → 0. We present another proof of a study of Bellieud and Bouchitt´e [1] that we expect to be more suitable to treat more general geometrical and physical cases.

The bases of the cylindrical domain Ω are denoted by ω₀ :=ω× {0}

and ω_L := ω × {L}. For each ε, we consider a periodic distribution of cells (Y_εⁱ)_i∈Iε such that Y_εⁱ := (εi₁, εi₂) + (−ε/2, ε/2)², and I_ε := {i∈Z² |Y_εⁱ ⊂ω}.

Let (D_rⁱ_ε)_i∈Iε be the family of disks of R² centered at ˆxⁱ_ε := (εi₁, εi₂) of radius r_ε ¿ε,T_εⁱ :=Dⁱ_rε×(0, L) andT_ε :=∪_i∈IεT_εⁱ. The set of thin parallel cylinders T_ε represents the fibers (see Figure 2.1 and Figure 2.2). The conductivity coefficient a_ε is

a_ε(x) =

(1, if x∈Ω\T_ε, λ_ε, otherwise.

We make the assumptions r_ε→0, r_ε

ε →0, λ_ε→+∞, k_ε :=λ_εr_ε²

ε² →k, k ≥0 as ε→0. (2.3)

Ω

HH HH H Y

ω₀

6

?

L

©©

©©

¼

ω_L

©©©©©* HHHH

HHHj

T_ε

Figure 2.1: The domain Ω =ω×(0, L) occupied by a composite material

¾ ε ^-

&%

'$

&%

'$

&%

'$

Dⁱ_r

Y_εⁱ

?

r

Figure 2.2: The circular cross section of the fiberY_εⁱ ⊂ω

Nuttawat Sontichai Mathematical Modeling of Fiber Reinforced... / 10

In [1], it was shown that the asymptotic limit of (P_ε) is (P^eff) min©

Φ(u, v)−L(u)|(u, v)∈(L^p(Ω))²ª , where

Φ(u, v) =











 R

Ωφ_p(∇u)dx+ ^kπ_p R

Ω

¯¯

¯_∂x^∂v₃

¯¯

¯^p dx+ ^2πγ_p R

Ω|v−u|^pdx, if

((u, v)∈W_Γ^1,p₀ (Ω)×L^p(ω, W^1,p(0, L)), v =u₀ on Γ₀∩(ω₀∪ω_L),

+∞ otherwise,

(2.4)

and γ ∈[0,+∞],

γ =





lim_ε→0ε⁻²|logr_ε|⁻¹, if p= 2, lim_ε→0

¯¯

¯^2−p_p−1

¯¯

¯^p−1r_ε^2−pε⁻², if p6= 2. (2.5) Here, the boundary data u₀ is assumed to be Lipschitz in order to ensure that the infimum value of problem (P_ε) remains finite as ε→0.In case k= +∞, we add further assumption

k_εr_ε →0, asε→0. (2.6)

The conditions

k > 0 and {γ >0 or ω₀ ⊂Γ₀ or ω_L⊂Γ₀} (2.7) guarantee that the functional Φ is coercive in W^1,p(Ω)×L^p(ω, W^1,p(0, L)).

We are concerned with the extension of this result to more general cross sections of the fibers and more general energy density thanφp. The aim of this paper is therefore to provide another proof that we expect to be more suitable to treat such general cases. The steps of the proof in [1] are to successively establish :

(i) a compactness property of the sequence (u_ε) such that F_ε(u_ε)< C, (ii) a lower bound inequality of the sequence (F_ε(u_ε)),

(iii) an upper bound inequality of the sequence (Fε(uε)).

Here we replace the steps (ii) and (iii) by

(ii⁰) an upper bound equality of the sequence (F_ε(u_ε)),

(iii⁰) a lower bound inequality of the sequence (F_ε(u_ε)) which essentially uses a subdiffenrential inequality.

2.2 An Alternative Strategy

It consists, under (2.3), (2.5), (2.6) and (2.7), in proving the following three propositions. In the sequel, the symbols →,*and *^∗ stand for the strong convergence, the weak convergence and the weak star convergence, respectively.

As usual, the letter C denotes various constants and for all ξ= (ξ₁, ξ₂, ξ₃) inR³, ξˆstands for (ξ₁, ξ₂).

Proposition 2.1 (compactness property) Let (u_ε) be a sequence such that supF_ε(u_ε) is finite. Then (u_ε) is strongly relatively compact in L^p(Ω) and (v_ε), given by v_ε := _|T^|Ω|

ε|1_Tεu_ε, is bounded in L¹(Ω) and, up to a subsequence, (v_ε) weakly* converges in the space of bounded measures M_b(Ω) to an element v of L^p(Ω).

Proposition 2.2 (upper bound equality) For all(u, v)in(L^p(Ω))², such that Φ(u, v)< +∞, there exists a sequence (uε) such that uε → u in L^p(Ω), vε ∗

* v in M_b(Ω) and

limε→0Fε(uε) = Φ(u, v).

Proposition 2.3 (lower bound inequality) For all u in L^p(Ω) and for all sequences (u_ε) such that u_ε→u in L^p(Ω), v_ε* v^∗ in M_b(Ω), one has :

lim inf

ε→0 Fε(uε)≥Φ(u, v).

The proofs of these propositions are presented in the following sections.

Proof of Proposition 2.1 :

Compactness property was already proved in [1].

2 Proof of Proposition 2.2 :

Our sole contribution is to prove that we can replace inequality by equality, for that we use the same approximation u⁰_ε of u as in [1]

u⁰_ε = (1−θε)u+θεwε.