Page 64 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Elastic Properties: The Stressed Uniform Lattice
a consequence, the tensor E must be symmetric with respect to its first
and second pairs of indices as well, because it now inherits the symmetry
of the symmetric strain. Thus we obtain the familiar expression for the
elastic energy as
h 1 T
--- [
U = – 2∫ ε :ER():ε] Vd
V
(2.35)
N
1  T 1  3
d
= – 2∫ ε : ------- ∑ ( R – R )DR() R –( R ) :ε R
--- 
j
j
2V
V  j = 1 
Of the possible 81 independent components for the fourth rank tensor E,
only 21 remain. We can organize these into a 6 × 6 matrix C if the fol-
lowing six index pair associations are made:
E () ≡ C () ij → m kl → n
ij kl()
m n()
11 → 1 22 → 2 33 →  (2.36)
3

23 → 4 31 → 5 12 → 6 
In order to obtain analog relations using the reduced index formalism, we
have to specify the transformation of the stress and strain as well. For the
stress we can use the same index mapping as in (2.36)

σ ≡ s ij → m
ij m
11 → 1 22 → 2 33 →  (2.37)
3

23 → 4 31 → 5 12 → 6 

For the strain, however, we need to combine the index mapping with a
scaling
e = ε 11 e = ε 22 e = ε 33
3
2
1
(2.38)
e = 2ε 23 e = 2ε 31 e = 2ε 12
4
6
5
In this way, we retain the algebraic form for the energy terms

Semiconductors for Micro and Nanosystem Technology 61