Mechanical Responseat Very Small Scale

Lecture 2:The Classical Theory of

Elasticity

Anne TanguyUniversity of Lyon (France)

II. The classical Theory of continuum Elasticity.

The mechanical behaviour of a classical solid can be entirely described by a single continuous field:The displacement field u(r) of the volume elements constituying the system.

0

0

0

z

y

x1

1

1

z

y

x

01

01

01

zzu

yyu

xxu

z

y

x

What is a « continuous » medium?

1) Two close elements evolve in a similar way.

2) In particular: conservation of proximity.

« Field » = physical quantity averaged over a volume element.

= continuous function of space.

3) Hypothesis in practice, to be checked.

At this scale, forces are short range (surface forces between volume elements)

In general, it is valid at scales >> characteristic scale in the microstructure.

Examples: crystals d >> interatomic distance (~ Å )

polycrystals d >> grain size (~nm ~m)

regular packing of grains d >> grain size (~ mm)

liquids d >> mean free path

disordered materials d >> ???

Al polycristal (Electron Back Scattering Diffraction)

Cu polycristal : cold lamination (70%)/ annealing.

Si3N4 SiC dense

Dendritic growth in Al:

TiO2 metallic foams, prepared with different aging, and different tensioactif agent:

Z

Y

Xz

y

x

Zzu

Yyu

Xxu

z

y

x

Classical elasticity: displacement field ru

WeVWVwvwW

vV

e

..2.. then and

if

: tensor strain Lagrange-Green

"distorsionangular " )21)(21(

2),cos( then0. if

extansion"unit "

WWVV

VW

VV

ee

ewvWV

eV

Vv

tensor"spin local"

2

1

tensor".strain (local) linearized" 2

1

tensor"strain Lagrange-Green" .2

1

uu

uu

uuuue

t

t

tt

V

Wv

w

Examples of linearized strain tensors:

Traction:

Shear:

Hydrostatic Pressure:

Units: %. Order of magnitude: elasticity OK if <0.1% (metal) <1% (polymer, amorphous)

L

Lu

L

vLv

y

x

00

00

00

L+u

L-vu

002

0002

00

Lu

Lu

Lv

L

vLv

z

y

x

00

00

00

Local stresses:

Rigid motion

Rigid rotation

0 A

0

dxdydztrdtrVol

P ),(:),(

,

P

t

dxdydztrVtrttrVtrAVol

),(:),(),().,(

antisymmetric symmetric.

General expression for the internal rate of work:

models the internal forces (Pa)

0dSV).n.T(dVV)).af.(div(

)V.(divdiv.VrV:t,r

dSrV.t,rTdVrV.t,rf.t,rdVrV:t,rdVrV.t,ra.t,r

^^t

^t

^^

^^^^

Equations of motion:

acceleration internal forces external forces(volume)

external forces(at the boundaries)

with

, for any subsystem.

Equilibrium equation:

Boundary conditions:

t,rTn.t,r:Sr

t,ra.t,rt,rf.t,rt,rdiv:Vr

Local stresses:

zzzyzx

yzyyyx

xzxyxx

Force per unit surface

exerted along the x-direction,

on the face normal to the direction y.

Expression of forces: dSnF .

vector normalsurface

Units: Pa (1atm = 105 Pa)

Order of magnitude: MPa =106 Pa

Examples of stress tensors:

Traction:

Shear:

Hydrostatic Pressure:

SF

00

000

000

u

00

000

00

SF

SF

P

P

P

00

00

00

F

S

3

)(trP By definition, pressure

The Landau expansion of the Mechanical Energy and the Elastic Moduli:

Expression of the rate of work of internal forces:

partby nintegratioafter .-

eunit volumper .

, t

t

u

rt

W

Mechanical Energy:

.,

E

It means that

E

per unit volume

The Landau expansion of the Mechanical Energy and the Elastic Moduli:

General expansion of the Mechanical Energy, per unit volume:

....::2

1:0 CE

Thus Hoole’s Lawut tensio sic vis

21 Elastic Moduli C

in the most general 3D case

...:0

CE

No dependence in (translational invariance)

No dependence in (rotational invariance)

u

Symmetries of the tensor of Elastic Moduli:

Example of an isotropic and homogeneous material:

Units: J.m-3 , or Pa.

Order of Magnitude: -1<≈ 0.33<0.5 and E ≈ Gpa ≈ Y/10-3

)..(:.. 11 SSSS C+ Specific symmetries of the crystal:

S Operator of symmetry

ItrEE

Itr

1

or 2

)

)

)

E

(CC

(CC

(CCGeneral symmetries:

Voigt notation:

6)12(

5)31(

4)23(

3)33(

2)22(

1)11(

Examples of elastic moduli in homogeneous and isotropic sys:

Traction:

Shear:

Hydrostatic Pressure:

Lu

Lv

Lu

ESF

.

.

u

Lu

SF

.

Etr

tr

VV

P

)21(3

23

3

)(

)(.3

.1

F

E, Young modulus

, Poisson ratio

, shear modulus

, compressibility.

P

Examples of anisotropic materials (crystals)

FCC3 moduli

C11 C12 C44

HCP5 moduliC11 C12 C13 C33 C44 C66=(C11-C12)/2

Co: HC FCC T=450°C

3 moduli(3 equivalent axis)

6 (5) moduli(rotational invariance around an axis)

6 moduli

6 moduli(2 equivalent symmetry axis)

9 moduli(2 orthogonal symmetry planes)

13 moduli(1 plane of symmetry)

21 moduli

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)