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Phys 460 Lecture 7
(
pdf version - 6 slides/page
)
Monday, September 18, 2006
Lecturer: Richard Martin
No new homework today
Reading:
Kittel, Chapt. 3
Elasticity
Outline
From last times:
Crystal Structures, Reciprocal Lattice
Binding Mechanisms
Fundamental difference between solids and liquids
Volume deformations (dilation) - similar
Shear deformations fundamentally different
Key ideas
What is determined purely by symmetry?
What depends upon the particular element or compound?
Strain
A position in the unstrained crystal
r
= x
x
+ y
y
+ z
z
is displaced to
r'
= x
x'
+ y
y'
+ z
z'
Displacement
Delta r
= x (
x'-x
) + y (
y' - y
) + z (
z'- z
)
Generalized:
Delta r
(
r
) = u(
r
)
x
+ v(
r
)
y
+ w(
r
)
z
To linear order, strain (tensor) is defined by change of lengths and angles
Leads to 6 independent variables: e
i
where i = 1,6
Stress (tensor) = Force (vector) per unit area (vector direction = normal)
Also 6 independent variables: sigma
i
where = 1,6
Linear elastic constants, compliances, energy
sigma
i
= Sum
j
C
ij
e
j
, (i,j = 1,6) (We will describe elasticity in terms of elastic constants C
ij
)
e
i
= Sum
j
S
ij
sigma
j
, (i,j = 1,6) (Some references use compliances S
ij
- not used here)
E = (1/2) Sum
ij
e
i
C
ij
e
j
Volume dilation: Bulk Modulus B, Compressibility K = 1/B
B = - V dP/dV = V d
2
E/dV
2
, i.e., curvature of E(V)
Cubic Crystals
3 independent quantities: C
11
, C
12
, C
44
B = (1/3)(C
11
+ 2 C
12
)
Shears: (1/2)(C
11
- C
12
) and C
44
Elastic Waves
(omega)
2
= (C / mass per unit length) k
2
or omega = s k, where s = sound velocity
Example for cubic crystals
More later
Young's modulus, Poisson ratio
See problem and figure in Kittel
Yield in a crystalo
The breaking point in a crystal happens when planes separate or slip
Determined by dislocations
See Kittel Ch 20 - not discussed further here
Email question/comments/corrections to
rmartin@uiuc.edu