D is the bending stiffness of the beam. G is the first moment of the cross-sectional
area of one of the ceramic layers about the x 1 axis. The transvers shear force Q in the
beam is defined by the integration of T 32 over a cross section:
Q ¼
Z
T 32 dx 1 dx 2 :
ð6:82Þ
In elementary or classical bending without shear deformation, the constitutive
relation for Q is not derived from its definition in Eq. (6.82). Instead, it will be
provided by the moment-shear force relationship below. From the equation of
motion of the differential element of the beam in Fig. 6.9 in the x 2 direction and
its moment equation, we obtain
Q ,3 þ f 2 x 3 , t
ð
Þ ¼ 2b ρ
1
ð Þ h þ ρ
2
ð Þ c
€ u 2 ,
M ,3 À Q ¼ 0,
ð6:83Þ
where f 2 (x 3 , t) is the transverse load per unit length of the beam. In Eq. (6.83) 2 , the
rotatory inertia is neglected and Eq. (6.83) 2 is the moment-shear force relationship
needed. From Eq. (6.83) 2 and Eq. (6.80), we have
Q ¼ ÀDu 2,333 þ 2Ge
1
ð Þ
33 φ ,33 :
ð6:84Þ
The total electric displacement over a cross section of the beam is
b
D 3 ¼
Z
D 3 dx 1 dx 2
¼ À2Ge
1
ð Þ
33 u 2,33 À b εφ ,3 ,
ð6:85Þ
where Eqs. (6.77) and (6.79) have been used, and
b ε ¼ ε
1
ð Þ
33 A
1
ð Þ
þ ε
2
ð Þ
33 A
2
ð Þ ,
A
1
ð Þ
¼ 2bh, A
2
ð Þ
¼ 2bc:
ð6:86Þ
Similar to the derivation of Eq. (6.83), by considering the differential element in
Fig. 6.9 under electrical loads, the charge equation of electrostatics can be written as
b
D 3,3 ¼ q Δp À Δn
ð
Þ A
2
ð Þ
:
ð6:87Þ
For the one-dimensional currents, we use the linearized constitutive relations with
uniform doping:
6.3 Bending of Beams with e 33
157