1
Chapter 2
References
J.H.Weiner，Statistical mechanics of elasticity, Wiley,
1981
Green & Zerna，Theoretical elasticity, 1968
Ashby & Jones，Engineering
materials
Two Physical Origins of Elasticity
Crystals
,
silicon
15
Chapter 2.2
Two Physical Origins of Elasticity
Crystals
,
16
Chapter 2.2
Two Physical Origins of Elasticity
ionic, covalent or metallic melt at 1000-5000K
The secondary bonding
Van der Waals and hydrogen bonds melt at 100-500K
11
Chapter 2.2
Two Physical Origins of Elasticity
24
Chapter 2.3
Tensor Description of Elasticity
Voigt Symmetry
ij
ji
C ijkl kl C jikl kl kl
C ijkl C
jikl
kl lk
C ijkl kl C ijlk lk C ijlk kl kl
σ T ε , X
σ T ε

infinitesimal deformation homogeneous material
σ C : ε linear elasticity
6
Chapter 2.1
Definition of Elasticity
Hyperelasticity Two assumptions: ① The response of the elastic body only depends on its current state. ② The current state of an elastic body can be described by a tensor.
bond energy U repulse between nuclei
B r
n
Crystals
0 Coulomb attraction electron cloud interactive attraction
A r
12
m
stable zone
Chapter 2.2 inter-atomic distance r
Chapter 2
2
Definition of Elasticity
Difference between solids and fluids Mechanics of Solids, The New Encyclopedia of Britannica, 15th edition, Vol. 23, pp. 734-747, 2002, “A material is called solid rather than fluid if it can also support a substantial shearing force over the time scale of some natural process or technological application of interest.” J. R. Rice
3
Chapter 2.1
Definition of Elasticity
Elasticity

σ T F , X
Where

F
F x X
Explicit dependence on X can be eliminated for homogeneous material
4
Chapter 2.1
Definition of Elasticity
Remarks
Stress is irrelevant to the strain rate, as well as to the history of deformation. No hysteresis: the original configuration is recovered after unload.
Theory of Elasticity
Introduction Elasticity of Solids Field Equations of Elasticity Differential Formulation Prismatic Rods Plane Problems – Theory and Solutions Plane Problems – Applications Variational Formulation of Elasticity Three-dimensional Problems
NK B T
2

: the stretching ratio N : the number of links per unit volume
22
Chapter 2.2
Two Physical Origins of Elasticity
Long Chain Polymers
0
Index
Elasticity of Solids
Definition of Elasticity Two Physical Origins of Elasticity Tensor Description of Elasticity Physical Foundation of Elastic Symmetry
ij
U ij
T
ij T
T ij T
ij is constant.
entropy stress
energetic stress
U ij
T
10
Chapter 2.2
Two Physical Origins of Elasticity
Crystals
The primary bonding
generalized Hooke’s law:

ij

W ij
C ijkl
kl
8
Chapter 2.1
Two Physical Origins of Elasticity
Energetic and Entropic Stresses Helmholtz free energy: H U ST Maxwell relation:
Path independent condition
by great mathematician Green
7
Chapter 2.1
Definition of Elasticity
Hyperelasticity

ij

W
ij
linear elastic:
W
1 2
C ijkl ij kl
For a three-dimensional problem, all indices i, j, k and l may have 3 possible numbers. That gives the maximum possible combinations of indices as 34=81.
C ijkl
W
2
ij kl

W
2
kl ij
Crystals
crystal lattice sites
r n1 n 2 n 3 n 1 a 1 n 2 a 2 n 3 a 3 r n1 n 2 n 3 n i a i r ' n1 n 2 n 3 n i a i ξ
,
a composite cell
14
Chapter 2.2
S ij T
ij
ij
U ij
T
S ij

U ij
T
ij T
9
Chapter 2.2
Two Physical Origins of Elasticity
Energetic and Entropic Stresses
ij
~
Gaussian distribution
3R 3 2 W R,n exp 2 2 2 nb 2 nb
2 3
b denotes the effective length
, R R
20
Chapter 2.2
Two Physical Origins of Elasticity
Two Physical Origins of Elasticity
F
Crystals Stiffness ：
S d U dr
2 2
Fmax
(dislocation radius) attraction
d U dr
2 2
0
0
r0
rD
r
Maximum inter-atomic force occurs at rD
90

constant
18
Chapter 2.2
Two Physical Origins of Elasticity