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.

B. The coarse-grained theory for microscopic elasticity

For ex.

with

and

1) Coarse-grained displacement:

Velocity dependent

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:

2) Microscopic expression for Stresses

cf.

Note that, at this level, there is no explicit linear relation between and !!

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)