Page 66 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Elastic Properties: The Stressed Uniform Lattice
1
1
B =
--- C
12
3 ( 11 + 2C ) = ---- (2.42)
K
Isotropic materials have the same structure as (2.40) but with only two
independent constants. In the literature, however, no less that three nota-
E
tions are common. Often engineers use the Young modulus and Pois-
µ
λ
ν
son ratio , whereas we find the Lamé constants and often used in
the mechanics literature. The relations between the systems are summa-
rized by:
C 11 – C 12
C 12 = λ C 44 = ----------------------- = µ 
2 

Eν 
C = λ + 2µ C = --------------------------------------
11 12 ( 1 + ν) 12ν) 
(
–

(
E E 1 – ν) 
C = -------------------- C = --------------------------------------
44 11  (2.43)
(
(
21 + ν) ( 1 + ν) 12ν)
–

(
µ 3λ + 2µ) λ 
E = ---------------------------- ν = --------------------- 
(
λ + µ 2 λ + µ) 
C ( 3C 11 – 4C ) C 11 – 2C 44 
44
44
E = ------------------------------------------- ν = ------------------------------- 
(
C – C 2 C – C ) 
11 44 11 44
The α -Quartz crystal has the following elastic coefficient matrix struc-
ture (for coefficient values, consult Table 2.1):
C C C C
11 12 12 14
C C C – C
12 11 12 14
C C C
C = 12 12 11 (2.44)
Quartz
C 14 – C 14 C 44
C 44
C 44
Semiconductors for Micro and Nanosystem Technology 63