Page 138 - MEMS Mechanical Sensors

6.4 Diaphragm-Based Pressure Sensors 127

3 Pa 2   r 2  

σ =±  3 ) ν 1 −  (6.17)
( +
 

r
2
8 h   a  
2
3
3 Pa 2 ( + ) ν
σ =± (6.18)
8 h
r max 2
The tangential stress, σ , at distance r from the center of the diaphragm is given
t
by (6.19). The maximum tangential stress occurs at the diaphragm center and is
equal to the radial stress given by (6.18):
3 P
σ =± [a 2 ( +3 ) ν − r 2 ( +1 3 ν )] (6.19)
t
8 h 2
6.4.2 Medium Deflection Diaphragm Analysis
The operation of diaphragms at deflections beyond 30% of thickness as covered in
Section 5.4.2 may be required in certain designs. In such a case, both tensile and
bending stresses must be considered. The characteristic equation, assuming the
material remains within the elastic limit, in such a case is given by [3]

16 Eh 3 7 −ν Eh
3
y
P = () y + () (6.20)
( −ν
( −ν
31 2 ) a 4 31 2 ) a 4
3
This may be written as a cubic equation form P = cy + dy , where
16 Eh 3 7 −ν Eh 3
c = and d = (6.21)
( −ν
( −ν
31 2 ) a 4 31 2 ) a 4
These represent the linear and nonlinear terms of the characteristic equation.

6.4.3 Membrane Analysis
Membranes can be considered as very thin diaphragms with large deflection
(y /h>5) [3]. In theory, a membrane has no flexural rigidity and experiences tensile
0
stress, but no bending stress. The characteristic equation for a membrane is given by
[4]

Pa 4 y 3 0
= 286 (6.22)
.
Eh 4 h 3
Radial stress in a membrane at radius r is given by (6.23); the maximum stress
occurs at the diaphragm center and, assuming ν = 0.3, is given by (6.24). Tangential
stress is given by (6.25). Maximum tangential stress occurs at the center of the mem-
brane and is equal to the maximum radial stress.
Ey  3 − ν r 2 
2
σ = 0  −  (6.23)


r 2 2
4 a 1 − ν a 