Mechanical Responseat Very Small Scale

Lecture 4:Elasticity of Disordered

Materials

Anne TanguyUniversity of Lyon (France)

IV. Elasticity of disordered Materials.

1) General equations of motion for a disordered material

2) Rigorous bounds for the elastic moduli.

3) Examples.

Ping Sheng « Introduction to wave scattering, Localization, and Mesoscopic Phenomena » (1995)

B.A. DiDonna and T. Lubensky « Non-affine correlations in Random elastic Media » (2005)C. Maloney « Correlations in the Elastic Response of Dense Random Packings » (2006)

Salvatore Torquato « Random Heterogeneous Materials » Springer ed. (2002)

General equations:

extt fuuzyxCdivrtu

rrCrr

dVrrCrrr

))(2

1:),,((

)(:)()()(

)(:)(:)(2

1)(:)(

2

0

0

2

E

In case of homogeneous strain: cstr

dVrC

VCanddVr

VwithCV HHHH E )(

1)(

1::

2

1:. 0

But in general

Heff

affinenon

CC

rrurru

rr

.)(.

'

Inhomogeneous strain field:

Inhomogeneousresponse, rotationaldisplacements in thenon-affine part.

A.Tanguy et al. (2002,2004,2005)

A.Lemaître et C. Maloney (2004,2006)

J.R. Williams et at. (1997)G. Debrégeas et al. (2001)S. Roux et al. (2002)E. Kolb et coll. (2003)Weeks et al. (2006)

A. Tanguy et coll. Phys. Rev. B (2002), J.P. Wittmer et coll. Europhys. Lett. (2002), A. Tanguy et coll. App. Surf. Sc. (2004)F. Léonforte et coll. Phys. Rev. B (2004), F. Léonforte et coll. Phys. Rev. B (2005), F. Léonforte et coll. Phys. Rev. Lett. (2006),

A. Tanguy et coll. (2006), C. Goldenberg et coll. (2007), M. Tsamados et coll. (2007), M. Tsamasos et coll. (2009).

Atomic displacements

Example of a lennard-Jones glass:

aaeffeff ,,

other examples of inhomogeneous strain

G.Debrégeas, A.Kabla, J.-M. di Méglio (2001,2003)

F.Radjai, S.Roux (2002)E.Kolb et al. (2003)J.R. Williams et al. (1997)

emulsions, colloids, … Weeks et al. (2006)

foams Granular materials

Dynamical Heterogeneities[Keys, Abate, Glotzer, DJDurian (preprint, 2007)]

Large distribution of local Elastic Moduli:

C1 ~ 2 1 C2 ~ 2 2 C3 ~ 2 (+

Large distribution of Elastic Moduli:

Cartes de modules élastiques locaux:

2D Jennard-Jones N = 216 225 L = 483

WCCC

1

Lennard-Jones glass: homogeneous and then isotropic W>20a

General bounds for the Effective Elastic Moduli:

General bounds for the effective macroscopic elastic moduli of an inhomogeneous solid.

Example of fibers in a matrix:

VV

EVV

EE

SSSESE

SSF

SESE

SESE

SSFFF

E

mm

ffL

Lmf

mmff

mf

LL

Lmmff

mmmfff

mmffmfL

Lmf

LLL

..

...

...

....

..

.

EL,T effective Young modulusEf Fiber’s Young modulusEm Young modulus of matrix

Voigt (1889)

Reuss (1929) Vf/V

EL

ET

E

m

m

f

f

T

mm

ff

T

mmmfffTT

mftotal

Tmf

TTT

EVV

EVV

E

VV

VV

VVV

E

1.

1.

1

..

..2

1..

2

1..

2

1

.

EEE

General bounds for the effective macroscopic elastic moduli of an inhomogeneous solid.

Quadratic part of the local elastic energy:

)(:)()()()(

)(:)(:)(2

1

)(:)(:)(2

1:

0

1

0

rrCrrr

rrCr

rrCr

Q

QQ

Q

EE

with

Effective Stiffness Tensor:

)(:)( rCr effQ

Preliminary results:

Qeff

QQQ

QQ

effQ

Q

QQ

CV

CV

V

CV

CV

rdivrdiv

rrr

rrr

dVrV

::2

::2

:2

::2

::2

0))('(0))((

0)(')(')(

0)(')(')(

)(1

11

E

E

then

Voigt Bound (1889)

)(

::::

rCC

CC

eff

eff

for any deformation at equilibrium,homogeneously applied at the boundaries.

with equality only if

Reuss Bound (1929)

QQeff

QQQeff

Q

rCC

CC

)(

::::

1

11

-1

for any deformation at equilibrium,homogeneously applied at the boundaries.

with equality only if

CCC eff

1

1

Other Bounds:

...::

::::::

...::

::::::

'

'

1

1

1

2

1

1

11

11

2

1

1

1

C

CCC

C

CCC

rrr

rrr

Q

QQQeff

Q

eff

n

kk

QQ

n

kkwith

then

Ex. Exact kth order perturbative solution(n=2 Hashin and Shtrikman, 1963)

Examples:

N. Teyssier-Doyen et al. (2007)

Voigt

Reuss

Example 2: Lennard-Jones glass

Progressive convergence to the macroscopic moduli and homogeneous and isotropic medium at large scale.

Faster convergence of compressibility (homogenesous density)

effCwC

Loca

l Ela

stic

Mod

uli:

~ 1/w

M. T

sam

ados

et a

l. (2

009)

Example of an Anisotropic Material:

Wood for Musical Instruments

Elastic ModuliYoung’s Moduli:

EL>>ER ~ ET

Holographic Interferometry, Hutchins (1971)

E// ≈ 11,6 GPa E┴ ≈ 0,716 GPa ≈ 0.39 t.m-3

Simplified expresison of the Eigenmodes of an Harmonic Table:

Parallel to the Fibres:

Perpendicular to the Fibers:

Large variety of resonant Frequencies

Looking for a Material with Analogous Anisotropy: E// / E┴ ≈ 16.

E// ≈ f.Vf + rm.(1-Vf) PRFC with Vf ≈ 13%E┴ ≈ 1/ (Vf/f + (1-Vf)/m) then E// = 53 GPa

Mass Density:PRFC = 1,25 t.m-3

Comparing the Eigenfrequencies imposes:a thickness dPRFC = 0.75 x dwood ≈ 2.52 mm

Then the Total Mass of the Harmonic Table is very largeMPRFC ≈ 2.69 x Mwood !!!

Choice of a sandwich material, allowing for the same mass.

Eigenfrequencies are comparable to those of PRFC,with the following choice for the thicknesses:

d1 ≈ 0.63 d2

d2 ≈ 0.66 dwood

C. Besnainou (LAM, Paris)« sandwich » material

… convenient also for lutes:Consequences: llight, stable, humidity-resistant, less damping,

Wood

Unidirectional Carbon Fiberglued in epoxy

Acrylic Foam

Plaster Mouldin a Vacuum Bag,Heated at 140°C.

Heating with Silicone Rubbers. Heating Ramp < 1/2h.

…cellos, and string basses « COSI »

Solidity and stability, especially against humidity, With the help of composite materials with Carbon Fibers.

Richness of tone?

End

Bibliography:I. Disordered MaterialsK. Binder and W. Kob « Glassy Materials and disordered solids » (WS, 2005)S. R. Elliott « Physics of amorphous materials » (Wiley, 1989)II. Classical continuum theory of elasticityJ. Salençon « Handbook of Continuum Mechanics » (Springer, 2001)L. Landau and E. Lifchitz « Théorie de l’élasticité ».III. Microscopic basis of ElasticityS. Alexander Physics Reports 296,65 (1998)C. Goldenberg and I. Goldhirsch « Handbook of Theoretical and Computational Nanotechnology » Reith ed. (American scientific, 2005)IV. Elasticity of Disordered MaterialsB.A. DiDonna and T. Lubensky « Non-affine correlations in Random elastic Media » (2005)C. Maloney « Correlations in the Elastic Response of Dense Random Packings » (2006)Salvatore Torquato « Random Heterogeneous Materials » Springer ed. (2002)V. Sound propagation Ping Sheng « Introduction to wave scattering, Localization, and Mesoscopic Phenomena » (Academic Press 1995)V. Gurevich, D. Parshin and H. Schober Physical review B 67, 094203 (2003)