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7-10 MEMS: Design and Fabrication

where D is the flexural rigidity (N–m) of the metal membrane material and is expressed as:
m
E t 3
m m
D (7.3)
2
12(1 ν )
m
where E represents the Young’s modulus (Pa) of the diaphragm, t (m) is the metal diaphragm thick-
m m
ness, and ν is the metal Poisson constant. Deflection of the metal diaphragm, w , at the center resulting
m
f
from a concentrated load at the center
Fa 2
w (7.4)
f
16πD
m
where F (N) is the concentrated load or the contact force on the boss section of the metal diaphragm. The
net deflection resulting from combined loading, when the concentrated load acts in the opposite direc-
tion of the applied pressure, is determined by:

a 2 Pa 2 F
w w w (7.5)
net. p f 16D 4 π
m
Deflection of a beam fixed (clamped) on one end, “guided” on the other (guided means that the slope at
the guided point is always zero, but slight deflection is allowed and may change). Applying the Castigliano
(1966) method allows the beam deflection to be expressed as:

Fl 3
w (7.6)
b 12E I
b b

where w beam deflection (m); l length of the beam (m); E modulus of elasticity of the beam
b
b
(Pa); and I moment of inertia of the beam, expressed as:
b
bh 3
I (7.7)
b
12
where b beam width (m) and h beam thickness (m). During loading, the deflection of the beam will
be equal to that of the diaphragm; therefore, Equations (7.5) and (7.6) can be set equal to solve for the
contact force, F (which is the contact force between the diaphragm boss and the beam). Because the
radius of the diaphragm and the length of the beam are, for all intents and purposes, equal, the equation
for F simplifies to:

2
3 Pa πE bh 3
b
F (7.8)
3
4 4aE t π 3E bh 3
b
m m
However, the maximum strains (ε max ) occur at either end of the beam and have opposite signs.
Fat
ε (7.9)
max 4E I
b b
The maximum stress, σ max , at the edge of the beam can be calculated from Hook’s Law:

σ ε E (7.10)
max max b
Based on the dimensions of the bossed metal diaphragm and the beam (diaphragm thickness, t 0.2mm;
m
Young’s modulus, E 207GPa, diaphragm radius, a 4mm, beam thickness, h 305µm; beam length,
d
l 4mm, beam width, b 1.52 mm; beam Young’s modulus, E 448 GPa, the strain in the SiC beam
b
calculated from the dimensions of the metal diaphragm,was approximately 1 nanostrain/Pa ( 7µstrains/psi).

© 2006 by Taylor & Francis Group, LLC