1. The Classical Theory of Elasticity
Mechanical Response
at Very Small Scale
Lecture 2:
The Classical Theory of Elasticity
Anne Tanguy
University of Lyon (France)

2. 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.
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.

3. What is a « continuous » medium?
Two close elements evolve in a similar way.
In particular: conservation of proximity.
« Field » = physical quantity averaged
over a volume element.
= continuous function of space.
Hypothesis in practice, to be checked.
At this scale, forces are short range (surface forces between volume elements)

4. 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 d >> grain size (~ mm)
liquids d >> mean free path
disordered materials d >> ???

5. Al polycristal
(Electron Back Scattering Diffraction)
Dendritic growth in Al:
Cu polycristal : cold lamination (70%)/ annealing.
TiO metallic foams, prepared with
different aging, and different tensioactif agent:
Si3N4
SiC dense

6. Classical elasticity: displacement field

7. Examples of linearized strain tensors:
Traction:
Shear:
Hydrostatic Pressure:
Units: %. Order of magnitude: elasticity OK if e<0.1% (metal)
e<1% (polymer, amorphous)
L+u
L-v L u

8. Local stresses:
antisymmetric
symmetric.
General expression for the internal rate of work:
Rigid motion
Rigid rotation
models the internal forces (Pa)

9. Equations of motion: acceleration internal forces external forces
(volume)
(at the boundaries)
with
, for any subsystem.
Equilibrium equation:
Boundary conditions:

10. Order of magnitude: MPa =106 Pa
Local stresses:
Force per unit surface
exerted along the x-direction,
on the face normal to the direction y.
Expression of forces:
vector normal
surface
Units: Pa (1atm = 105 Pa)
Order of magnitude: MPa =106 Pa

11. Examples of stress tensors:
Traction:
Shear:
Hydrostatic Pressure: F S u
By definition, pressure

12. The Landau expansion of the Mechanical Energy and the Elastic Moduli:
Expression of the rate of work of internal forces:
Mechanical Energy:
It means that
per unit volume

13. The Landau expansion of the Mechanical Energy and the Elastic Moduli:
General expansion of the Mechanical Energy, per unit volume:
No dependence in (translational invariance)
No dependence in (rotational invariance)
Thus Hoole’s Law
ut tensio sic vis
21 Elastic Moduli Cabgd
in the most general 3D case.

14. Symmetries of the tensor of Elastic Moduli:
General symmetries:
+ Specific symmetries of the crystal:
Operator of symmetry
Example of an isotropic and homogeneous material:
Units: J.m-3 , or Pa.
Order of Magnitude: -1<n ≈ 0.33<0.5 and E ≈ Gpa ≈ sY/10-3

15. Voigt notation:

16. Examples of elastic moduli in homogeneous and isotropic sys:
Traction:
Shear:
Hydrostatic Pressure: F
E, Young modulus
n, Poisson ratio u
m, shear modulus P
c, compressibility.

17. 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

18. 3 moduli 6 (5) moduli (3 equivalent axis)
(rotational invariance around an axis)

19. 6 moduli

20. 6 moduli
(2 equivalent symmetry axis)

21. 9 moduli 13 moduli 21 moduli (2 orthogonal symmetry planes)
(1 plane of symmetry)
21 moduli

22. I. Disordered Materials
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, (2003)