Mechanical Response at Very Small Scale Lecture 4: Elasticity of Disordered Materials
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Transcript of Mechanical Response at Very Small Scale Lecture 4: Elasticity of Disordered Materials
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Mechanical Responseat Very Small Scale
Lecture 4:Elasticity of Disordered
Materials
Anne TanguyUniversity 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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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
.)(.
'
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Inhomogeneous strain field:
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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 ,,
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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
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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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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
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Lennard-Jones glass: homogeneous and then isotropic W>20a
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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:
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
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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
Q
EE
with
Effective Stiffness Tensor:
)(:)( rCr effQ
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Preliminary results:
Qeff
QQQ
effQ
Q
CV
CV
V
CV
CV
rdivrdiv
rrr
rrr
dVrV
::2
::2
:2
::2
::2
0))('(0))((
0)(')(')(
0)(')(')(
)(1
11
E
E
then
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Voigt Bound (1889)
)(
::::
rCC
CC
eff
eff
for any deformation at equilibrium,homogeneously applied at the boundaries.
with equality only if
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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
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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
n
kkwith
then
Ex. Exact kth order perturbative solution(n=2 Hashin and Shtrikman, 1963)
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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 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)
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Example of an Anisotropic Material:
Wood for Musical Instruments
Elastic ModuliYoung’s Moduli:
EL>>ER ~ ET
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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
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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
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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.
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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 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)