Diapositive 1

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Mechanical Response at Very Small Scale Lecture
4 Elasticity of Disordered Materials Anne
Tanguy University of Lyon (France)
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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)
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Inhomogeneous strain field
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Example of a lennard-Jones glass
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
Inhomogeneous response, rotational displacements
in the non-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)

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other examples of inhomogeneous strain
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Dynamical Heterogeneities

• Keys, Abate, Glotzer, DJDurian (preprint, 2007)

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Large distribution of local Elastic Moduli
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Lennard-Jones glass homogeneous and then
isotropic Wgt20a
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General bounds for the Effective Elastic Moduli
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General bounds for the effective macroscopic
elastic moduli of an inhomogeneous solid.
Example of fibers in a matrix
Reuss (1929)
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General bounds for the effective macroscopic
elastic moduli of an inhomogeneous solid.
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Preliminary results
then
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Voigt Bound (1889)
for any deformation at equilibrium, homogeneously
applied at the boundaries.
with equality only if
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Reuss Bound (1929)
for any deformation at equilibrium, homogeneously
applied at the boundaries.
with equality only if
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Other Bounds
with
Ex. Exact kth order perturbative solution (n2
Hashin and Shtrikman, 1963)
then
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Examples
N. Teyssier-Doyen et al. (2007)
Voigt
Reuss
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Example 2 Lennard-Jones glass
Progressive convergence to the macroscopic moduli
l and m, homogeneous and isotropic medium at
large scale. Faster convergence of
compressibility (homogenesous density)
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Example of an Anisotropic Material Wood for
Musical Instruments
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Holographic Interferometry, Hutchins (1971)
Simplified expresison of the Eigenmodes of an
Harmonic Table
E// 11,6 GPa E- 0,716 GPa r 0.39 t.m-3
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Looking for a Material with Analogous Anisotropy
E// / E- 16.
E// rf.Vf rm.(1-Vf) PRFC with Vf 13 E-
1/ (Vf/rf (1-Vf)/rm) then E// 53 GPa Mass
Density rPRFC 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 large MPRFC 2.69 x
Mwood !!!
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C. Besnainou (LAM, Paris)  sandwich  material
Plaster Mould in a Vacuum Bag, Heated at 140C.
Heating with Silicone Rubbers. Heating Ramp lt
1/2h.
Wood
Acrylic Foam
Unidirectional Carbon Fiber glued in epoxy
convenient also for lutes
Consequences llight, stable, humidity-resistant,
less damping,
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cellos, and string basses  COSI 
Solidity and stability, especially against
humidity, With the help of composite materials
with Carbon Fibers. Richness of tone?
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End
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Bibliography I. Disordered Materials K. 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 elasticity J. Salençon
 Handbook of Continuum Mechanics  (Springer,
2001) L. Landau and E. Lifchitz  Théorie de
lélasticité . III. Microscopic basis of
Elasticity S. 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 Materials 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) 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)