Page 300 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Inhomogeneities
if we assume that the plate lies symmetrically about its neutral axis and
placed at z = 0 . In order to preserve a simple algebraic structure later
on, we define the averaged plate strains by
t ---
 2
{ ε , ε , ε } =   12 3∫ { ε , ε , ε }zz (7.134a)
d
------
xx
xx
yy
xy
xy
yy
h – --- t
2
t ---
2
h ∫
{ ε , ε } =  2 t { ε , ε } z (7.134b)
d
---
yz
xz
yz
xz
–
2 ---
We now join the above relations to obtain the plate-specific constitutive
relation for an isotropic material, namely
(
M = D ε + νε ) (7.135a)
xx xx yy
(
M yy = D ε + νε ) (7.135b)
yy
xx
M = ( 1 – ν)Dε (7.135c)
xy xy
and
Q xz = µ′hR xz (7.136a)
Q yz = µ′hR yz (7.136b)
2
3
(
where the plate stiffness is D = Eh ⁄ ( 12 1 – ν )) and the modified
shear stiffness is µ′ = k µ .
M
The first z moment of the first two mechanical equations of motion, rep-
resented by (7.110), with the third equation of motion combine to give
the plate equations of motion
t
∂M xx ∂M xy --- 2
d
------------ + ------------ – Q xz = ρ t ∫ u˙˙ zz (7.137a)
x
∂x ∂y – ---
2
t
∂M xy ∂M yy --- 2
d
------------ + ------------ – Q yz = ρ t ∫ u˙˙ zz (7.137b)
y
∂x ∂y – ---
2
Semiconductors for Micro and Nanosystem Technology 297