mechanical response at very small scale lecture 3: the microscopic basis of elasticity anne tanguy...
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Mechanical Responseat Very Small Scale
Lecture 3:The Microscopic Basis of
Elasticity
Anne TanguyUniversity of Lyon (France)
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)
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
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
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
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
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
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
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
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
(
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
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.
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:
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:
C1 ~ 2 1 C2 ~ 2 2 C3 ~ 2 (+
2D Jennard-Jones w=5a N = 216 225 L = 483 a
Maps of local elastic moduli:
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
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 » ?
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)