Mechanical Response at Very Small Scale Lecture 3: The Microscopic Basis of Elasticity
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Transcript of Mechanical Response at Very Small Scale Lecture 3: The Microscopic Basis of Elasticity
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Mechanical Responseat Very Small Scale
Lecture 3:The Microscopic Basis of
Elasticity
Anne TanguyUniversity of Lyon (France)
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III. Microscopic basis of Elasticity.
A. The Cauchy-Born theory of solids (1915).
1) General expression of microscopic and continuous energy.2) The microscopic expression for Stresses.3) The microscopic expression for Elastic Moduli.
B. The coarse-grained theory for microscopic elasticity (2005).
1) Coarse-grained displacement and fluctuations2) The microscopic expression for Stresses.3) The computation of Local Elastic Moduli.
S. Alexander, Physics Reports 296,65 (1998)C. Goldenberg and I. Goldhirsch (2005)
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Microscopic expression for the local Elastic Moduli:Simple example of a cubic crystal.
On each bond:
....)(2
1).( 02
22
0000 rdr
drrr
dr
drrrr ijijij
ijijijij
EEEE
strain
stress
0
011 r
rrij
20
2
02
0
20
11
).(
4
4'
rdr
rdrr
r
f ijij
ijE
elastic modulus
30
0
2
02
011111111 .
1/'
r
r
dr
rd
rC ijij EE
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A. The Cauchy-Born Theory of Solids (1915).
ij
kj
Regular expression of the Many-particlesEnergy:
N particlesD dimensions
N.D parameters-D(D+1)/2 rigid translations and rotations
N.D –D(D+1)/2 independent distances
ij
ikjkijkji
ijji
i
r
rrrrr
E
EEE ijkij
...),,()(),,(),(
2-body interactions(Cauchy model)Ex. Lennard-Jones Foams BKS model for Silica
3-body inter.Ex. Silicon
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Expression of local forces:
Internal force exerted on atom i: )()(
)( rfr
rrf
jij
ii
E
Force of atom j on atom i:
ij
ijij
ij
ij
jiijij
ij
r
rrT
r
rrT
rrrr
rrf
)()().(
)()(
E
E
with
with
Tension of the bond (i,j)in the configuration {r}.
The equilibrium on each atom i writes:
extieq
ij
eqijeq
jij
exti
eq
jij
fr
rrT
frf
).(
0)(
thus
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Particles displacement, and strain:
ij
ui
uj
rijeq
rij
uij
uijP
uijT
rurrr
uruuu
rrurru
rurru
rruruu
rrr
eqij
eqj
eqjeq
ijjiij
eqj
eqjeq
ijeqj
eqj
eqij
eqj
eqij
eqj
eqii
jieqij
.2
.
2.
.
,
..
)(..
...,,,,
eqij
eqij
eqijeq
eqij
eqij
eqij
eqij
eqijeq
ijeqij
eqij
ijPij
Tij
Pijij
r
rrru
r
rr
r
rur
r
ruu
uuu
eqijijeq
ij
eqijijP
ij
eqij
Pij
eqij
Tij
eqij
Pijij
rrr
rru
rururur
2
)(
)(.222
2222
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First order expansion of the energy, and local stresses:
....
2
1
...2
1
....2
1
....2
1
,,
i j
eqij
eqij
eqij
ijeqi
P
i jijij
eqi
i jij
ij
ij
ij
eqi
i jij
ij
eqii
r
rrTr
uTr
ur
r
rr
ur
rr
E
E
EE
EEE
To compare with:
....::2
1:0 CdV E
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First order expansion of the energy, and local stresses:
i j
eqij
eqij
eqij
ijeqii r
rrTrr ...
.
2
1,,
EE
To compare with:
.....0
E dV
« Site stress »: )(.
2
1,,
energyr
rrTis eq
ij
eqij
eqij
ijj
Local stress: )(.
2
11,,
0 Par
rrT
Vi eq
ij
eqij
eqij
ijji
)().()( 00 isVidVriV
i
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Second order expansion of the energy, local Elastic Moduli:
.....!2
1.
2
1
),)(,(
2
lkji
klklij
iji j
ijij
eqii u
rruu
rrr
EEEE
with
ij
2Tij
)kl),(ij(ijP
klP
ijklij
2
3ij
ijij
ijklij)kl),(ij(
ijklij
ij
ij
kl
kl
klij
2
klijij
ij
ijkl
klij
klij
2
klklij
2
ij
r
u..Tu.u.
rr
r
r.r
r.u.u..
ru.u.
r
r.
r
r.
rr
u.u.r
r.
rru.u.
rru.
rr.u
E
EE
EEE
Local stiffness
bound elongation rotation
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Born-Huang approximation for local Elastic Moduli:
..r.r
r.r.r.r.
rr!2
1
u.u.rr!2
1r
)kl),(ij(eq
kleq
ij
,eqkl
,eqkl
,eqij
,eqij
klij
2
)kl),(ij(
Pkl
Pij
klij
2
iQ
E
EE
Tij=0
To compare with:
::2
1CdVQ E
)(..
....
1)( 4321)(
)
,,,,2
4321
4321 4321
43432121
4321
iiiiinrr
rrrr
rrViC iiii
iiiieq
iieq
ii
eqii
eqii
eqii
eqii
iiiii
E
(
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Born-Huang approximation for local Elastic Moduli:
nrr
rrrr
rrViC
iiiieq
iieq
ii
eqii
eqii
eqii
eqii
iiiii
..
....
1)(
)
,,,,2
4321 4321
43432121
4321
(
E
2-body contribution (central forces): (i1i2)=(i3i4) n=1/2
i
3-body contribution (angular bending): i=i1 and i=i3 or i=i4 n=2/3
i i
4-body interactions (twists): (i1i2) ≠ (i3i4) n=2/4
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Number of independent Elastic Moduli, from the microscopic expression:
Warning: CMACRO ≠ < C
MICRO (i) > (cf. lecture 4)
C=C and C=C 36 moduliC=C 21 moduli
nrr
rrrr
rrViC
iiiieq
iieq
ii
eqii
eqii
eqii
eqii
iiiii
..
....
1)(
)
,,,,2
4321 4321
43432121
4321
(
E
Additional symetries , for 2-body interactions (Cauchy model):Permutations of all indices: C=C and C=C
(Cauchy relations for 2-body interactions) 3 C + 6 C + 3 C + 3 C 15 moduli.
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B. The coarse-grained theory for microscopic elasticity
For ex.
with
and
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1) Coarse-grained displacement:
Velocity dependent
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Separate coarse-grained (continuous) response, and « fluctuations »:
)t,r(U)r(u)r(u ilin
iifluct
C. Goldenberg et I. Goldhirsch (2004)
gaussian funct. of width w continuous
Coarse-grained displacement and fluctuations:
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2) Microscopic expression for Stresses
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cf.
Note that, at this level, there is no explicit linear relation between and !!
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Use of the coarse-grained (continuous) disp. fieldfor the computation of local elastic moduli:
Gaussian with a width w ~ 2
using 3 independent deformations for a 2D system
strain
stress
2D case:
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C1 ~ 2 1 C2 ~ 2 2 C3 ~ 2 (+
2D Jennard-Jones w=5a N = 216 225 L = 483 a
Maps of local elastic moduli:
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Large scale convergence to homogeneous and isotropic elasticity:
Elastic Moduli:
Locally inhomogeneous and anisotropic.
Progressive convergence to the macroscopic moduli and homogeneous and isotropic.
Faster convergence of compressibility.
No size dependence, but no characteristic size !
~ 1/w
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1%
Departure from local Hooke’s law, for r < 5 a.Which characteristic size ?
?
At small scale w:ambigous definition of elastic moduli
(9 uncoherent equations for 6 unknowns)
Error function:S
SCEMinC ).(
Local rotations?Long-range interactions ?Role of the « fluctuations » ?
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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)