Page 62 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Elastic Properties: The Stressed Uniform Lattice
represents a pure translation and α
where α
a rotation plus stretch.
o
d
Next, we consider what happens to the line element x under the defor-
mation field u
T
T
d x′ = Q′ – P′ = α + ( δ + α) • Q ( α + ( δ + α) • P)
–
o o
(2.27)
T
= d x + α • d x = d x + d u
T
We can rewrite du via the chain rule as du = ( ∇ u) • d x , thereby
making the association α = ∇ u , to find that we can now rewrite the
expression for x'd only in terms of the original quantities that we mea-
sure
T
d x' = d x + ( ∇ u) • d x (2.28)
We are interested in how the line element deforms (stretches), and for
this we form an expression for the square of its length
T
T
T
2
T
T
d ( x') = d ( x) • d x + ( [ ∇ u) • d x] • d x + d ( x) • ( [ ∇ u) • d x]
T
T
T
+ ( [ ∇ u) • d x] • ( [ ∇ u) • d x] (2.29)
T
T
T
•
= d ( x) • [ δ + ∇ u + ( ∇ u) + ( ∇ u) • ∇ u] d x
T
T
Consider the argument δ +[ ∇ u + ( ∇ u) + ( ∇ u) • ∇ u] . If the stretch is
small, which we have assumed, then we can use 1 + ≅ 1 + s , 2 ⁄
s
ε
which defines the strain by
1 T T 1 T
ε = --- ∇ +[ u ( ∇ u) + ( ∇ u) • ∇ u] ≈ --- ∇ +[ u ( ∇ u) ] (2.30)
2 2
The components of the small strain approximation may be written as
1 ∂u α ∂u β
ε αβ = --- x β + ∂ x α (2.31)
2 ∂
The small strain is therefore a symmetric rank two tensor.

Semiconductors for Micro and Nanosystem Technology 59