Chapter 2 Elasticity and Viscoelasticity. Mechanical Testing Machine.
Mechanical Response at Very Small Scale Lecture 2: The Classical Theory of Elasticity
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Transcript of Mechanical Response at Very Small Scale Lecture 2: The Classical Theory of Elasticity
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)