Transcript Diapositive 1

Mechanical Response
at Very Small Scale
Lecture 2:
The Classical Theory of
Elasticity
Anne Tanguy
University 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.
x1
x0
y0
z0
y1
z1
ux  x1  x0
u y  y1  y0
uz  z1  z0
1) What is a « continuous » medium?
2) The local strains.
3) The description of local forces (stress).
4) The Landau expansion of the Mechanical Energy
and the Elastic Moduli.
J. Salençon « Handbook of Continuum Mechanics » Springer ed. (2001)
Landau « Elasticity » Mir ed.
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 ~mm)
regular packing of grains
liquids
d >> grain size (~ mm)
d >> mean free path
disordered materials
d >> ???
Al polycristal
(Electron Back Scattering Diffraction)
Dendritic growth in Al:
Cu polycristal : cold lamination (70%)/ annealing.
TiO2 metallic foams, prepared with
different aging, and different tensioactif agent:
Si3N4
SiC dense
Classical elasticity: displacement field ur 
x
y
w
X
z
Y
v
W
Z
V
Green - Lagrangestrain tensor e :
if V  v
and W  w

thenv.w  V .W  2V .e.W
v V
 eVV " unit extansion"
V
 if V .W  0 then cos(v, w) 


2eVW
(1  2eVV )(1  2eWW )
" angular distorsion"
1
t
t
u u u.u " Green - Lagrangestrain tensor"
2
1
  u t u " linearized(local)strain tensor".
2
1
t
  u u " local spin tensor"
2
e




ux  x  X
uy  y  Y
uz  z  Z
Examples of linearized strain tensors:
  vx

 L
  0


 0

Traction:
L
L+u
Shear:
u
L-v
Hydrostatic Pressure:

 0

  0
 u
 2L

0
0
0
u 

2L 
0 
0 


0

0

u

L
0
 vy
L
  vx

 L
  0


 0

0
0
 vy
L
0

0 

0 

 vz 

L 
Units: %. Order of magnitude: elasticity OK if <0.1% (metal)
<1% (polymer, amorphous)
Local stresses:
General expression for the internal rate of work:
P
 A (r , t ).V(r , t )  t (r , t ) : V(r , t )dx dy dz
V ol
t   
 antisymmetric ,
Rigid motion
 A0
Rigid rotation  
P
 symmetric.
0
   (r , t ) : d(r , t ) dxdydz
V ol

models the internal forces (Pa)
Equations of motion:
  r, t .a r, t .Vr dV     r, t  : Vr dV    r, t .f r, t .Vr dV   Tr, t .Vr dS
^
^
^
internal forces
acceleration
external forces
(volume)
^
external forces
(at the boundaries)
with
 r , t  : Vr   V.div   div ( .V)
^
^
^
t
^
^
 (div    .(f  a )).VdV   (T   .n).VdS  0
t
, for any subsystem.
Equilibrium equation:
r  V :
div r , t    r , t .f r , t    r , t .a r , t 
Boundary conditions:
r  S :
 r , t .n  T r , t 
Local stresses:
Force per unit surface
exerted along the x-direction,
  xx  xy  xz 


    yx  yy  yz 




zy
zz 
 zx
Expression of forces:
on the face normal to the direction y.
F   .n dS
surface
vector normal
Units: Pa (1atm = 105 Pa)
Order of magnitude: MPa =106 Pa
Examples of stress tensors:
F
Traction:
S
Shear:
u

0

 0
F
S

Hydrostatic Pressure:
tr ( )
By definition, pressure P  
3
0 0 0 


  0 0 0 

F
0 0

S

F

S
0 0
0 0 

0
0 
P 0


   0 P 0 
 0
0  P 

The Landau expansion of the Mechanical Energy
and the Elastic Moduli:
Expression of the rate of work of internal forces:
   u
W
 
.
per unit volume
t
r t
 
 
 -    .
t
 ,
Mechanical Energy:
after integration by part
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:
1
E   :    : C :   ....
2
0
No dependence in
u
(translational invariance)
No dependence in

(rotational invariance)
Thus
E

  0  C :   ...

Hoole’s Law
ut tensio sic vis
21 Elastic Moduli Cg
in the most general 3D case.
Symmetries of the tensor of Elastic Moduli:
General symmetries: C g  C g
(      )
C g  C  g
( g    g )
C g  Cg  
(   
E
)
 
1
1
+ Specific symmetries of the crystal: S. .S  C : (S. .S )
S Operator of symmetry
Example of an isotropic and homogeneous material:
  2m   tr I
or
1 n
n

  tr I
E
E
Units: J.m-3 , or Pa.
Order of Magnitude: -1<n ≈ 0.33<0.5 and E ≈ Gpa ≈ Y/10-3
Voigt notation:
(11)  1
( 22)  2
(33)  3
( 23)  4
(31)  5
(12)  6
Examples of elastic moduli in homogeneous and isotropic sys:
F
Traction:
F
u
E, Young modulus
 E.
S
L
v
u
n .
L
L
Shear:
u
F
u
 m.
S
L
Hydrostatic Pressure:
P
n, Poisson ratio
m, shear modulus
1 V
.
 V
tr( )
3
3(1  2n )
  3.


tr( ) 3  2m
E
P
, compressibility.
Examples of anisotropic materials (crystals)
FCC
3 moduli
C11 C12 C44
HCP
5 moduli
C11 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 Materials
K. 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 elasticity
J. Salençon « Handbook of Continuum Mechanics » (Springer, 2001)
L. Landau and E. Lifchitz « Théorie de l’élasticité ».
III. Microscopic basis of Elasticity
S. 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 Materials
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)
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)