Alghero Summer School

Elastic MetamaterialsFrom Theory to Applications

22-29 May 2016

LIA Coss&Vita

Alghero Summer School

Inner resonance mediaTheory and Experiments

26 May 2016

Claude BOUTIN

ENTPE / LGCB - CeLyA- CNRS 5513

This short course aims at

Explaining the principles behind the inner resonance phenomenon Deriving and discussing the unconventional features …

… From several examples enabling to derive some design rules, ... and give experimental evidence on prototypes.

This will be done

By means of the asymptotic homogenization method

KeywordsHomogenisation, Metamaterials, Inner resonance

Objectives

3

Dynamic phenomenaat micro and macro scales

”Co-dynamics" regimeonly possible in heterogeneous media

Unconventional effective parametersProperties impossible to reachwith classical materials.

Inner resonance media ?

4

Usu

al

med

iaIn

ner r

eson

ance

m

edia

λ/2π> lVER

Diff

ract

ion

λ/2π< lVER

ω

An « old » Idea

Maxwell J.C. 1869 ; L. Rayleigh, 1899

PACAM XI - 2010

Question

How to design Generalized media ?

Non - Cauchy media non local in space

« Meta materials » non local in time

At the leading order

Elastic composites

And /or Other physics

Which method ?

PACAM XI - 2010

Method

Theoritical approach - Elasticity

Theory Ab initio ? Second gradient - Cosserat - Micromorphic

Mostly in statics

Non explicite link with the microstructure

A posteriori estimates of Non - Cauchy effects

Homogenization

Explicite link with the microstructure (continous or discrete)

Direct estimate of Non local effect

Dynamics vs Statics

Explicite macro wave lenght vs. Geometry and boundary conditions

Part I GeneralitiesIntroduction to homogenization

Classical and enriched elasto-dynamics of composites

Part II – III - IV Inner resonance media

Elastic media

Porous media

Resonant surface

Part V High Frequency Modulation

Recent advances in elasto–dynamics

Conclusion

Au menu

Part I GeneralitiesIntroduction to homogenization

Classical and enriched elasto-dynamics of composites

Part II – III - IV Inner resonance media

Elastic composites and reticulated media

Porous media

Resonant surface

Part V High Frequency Modulation

Recent advances in elasto –dynamics

Conclusion

Au menu

1 -Introduction to Homogenization

11

From local to global description

Key Issue

Escape from the detailed descriptionwhile

Keep the qualitative and quantitative features

Scale separation is necessary, … and sufficient (Auriault, 80)

1 - Long wavelength Λ = 2πL Λ L

2 - Medium = {ERV} ERV [l] l/L = ε << 1

12

Intuitive approach

13

Λ

λ Λ

Λ

l0/Λ = ε << 1 l/Λ = ε < 1

l/λ = Ο(1) L/Λ ≥ Ο(1)

l

Ll

l0

Guide line

Scale separation l/Λ = ε << 1 Macro - Continuum physics

ERV <==> Particle not too small (representative)not too large (infinimum % L)

Global description arises from the local physics

ERV physics Condensed in Nature of the macro-descriptionMacro-parameters Relevant information % L

Homogenization method [Sanchez-Palencia, 80], [Auriault, 80]

Rigorous mathematical approach of the two requirements

ERV Ω−Periodic media

Scale separation ε = l/L << 1Asymptotic expansionsTwo scales method

14

Homogenization method

Two-scale variables

Macro : x/L Micro : x/l = x/(Lε) x y = x/ε

Two-scale expansions

è ∂ --> ∂x + ε-1∂y

è y-Ω periodic expansionsq(x,y) = q0(x,y) + εq1(x,y) + ε2q2(x,y) + …

Resolution

Rescaled Equations EQx,y(Expansions(x,y)) = 0∀ ε −−> 0 Σ εq EQq

x,y (-) = 0 Separate ε power EQq

x,y (-) = 0 Series of local problems (y-periodic)Macro description (x)

15

2 - Elasto-dynamic of composites

16

Classification based on λ /ERV

Large scale dynamics Large scale variations

Local statics Local dynamics

1 3

Large scale dynamics

« Local dynamics

2Co-dynamics » regime

Inner resonance

Global descriptions in dynamics

Wea

kly

Con

trast

edλ /ERV <1λ /ERV > 1

Long Wavelengths----

Weakly Contrasted Composites

Wea

kly

Con

tras

ted

λ /VER <1

Hig

hly

Con

tras

ted

λ /VER > 1

Scale separation

Long wavelength Λ

Λ >> l ERV [l]

ε = 2π l/Λ <<1 Macro dynamics

è Quasi static local regime

Scale separation for U0

Elasto-dynamics

σ = a(y):e (u) div[σ] + ω2ρu = 0

E(u) + ω2ρ(y) u = 0 E(u) = div(a:e (u)) a(y) ; ρ(y) Ω-periodic

Two space variables x , y = ε-1x

E è Exy(u) = ε-2 Ey-2(u) + ε-1E-1(u) + E0(u)

Ey-2(u) = divy[a(y):e y(u)]

E-1(u) = divy[a(y):e x(u)] + divx[c(y):e y(u)]

E0(u) = divx[a(y):e x(u)] ; ω2ρ(y) u Macro dynamics

Homogenization

Governing equations (x,y)

Dominant order

è Local quasi-statics

Homogenization - 1

ΩC

ΩR

Γ

Local fields at next order

è

Macroscopic description at leading order

Mean balance

Divergence theorem + periodicity

è Macro conventional elasto-dynamics

Homogenization - 2

ΩC

ΩR

Γ

« Weak » dynamic local regime (Boutin, Auriault, IJES. 1993)ε = 2π l/Λ < 1 : Correctors

U(x) = U0(x) + U1(x) + U2(x) + U3(x) +… Ui(x) = εiui(x)

Classic Elastodynamics + Homogenized diffracted sources

div(C0 : e(U0)) + <ρ>ω2 U0 = 0

div(C0 : e (U1)) + <ρ>ω2 U1 = - [ div(C1...∇e(U0)) + ω2 R1e(U0))] ε

div(C0 : e (U2)) + <ρ>ω2 U2 = - [ div(C1...∇e(U1)) + ω2 R1e(U1))] ε2

- [div(C2....∇∇e(U0)) + ω2 R2∇e(U0))]

div(C0 : e (U3)) + <ρ>ω2 U3 = - [ div(C1...∇e(U2)) + ω2 R1e(U2))] ε3

- [div(C2....∇∇e(U1)) + ω2 R2∇e(U1))]- [div(C3.....∇∇∇e(U0)) + ω2 R3∇∇e(U0))]

Enriched Elastodynamics Non local ElasticityNon local MassRayleigh Scattering

Macro Description with Correctors

WeaklyContrasted

λ / ERV <1λ /ERV > 1

ClassicElasto-dynamic

Domain of validity - Enriched Elastodynamics