Only one of Eq. (6.154) is independent. The substitution of Eqs. (6.145), (6.146),
(6.147), (6.148), (6.149), and (6.150) into Eqs. (6.151), (6.152), (6.153), and (6.154)
yields a system of linear equations for the undetermined constants. These equations
are solved on a computer using MATLAB. For a complete description of the
bending deformation, we introduce the bending curvature
K ¼
u 2,33
1 þ u 2,3
ð Þ
2
h
i 3=2 ffi u 2,33 ,
ð6:155Þ
which has been approximated according the usual small deformation assumption in
mechanics of materials.
As a numerical example, consider a beam with L ¼ 20 Â 10
À6 m, b ¼ 50 Â 10
À9 m,
c ¼ 15 Â 10
À9 m, and h ¼ 10 Â 10
À9 m. Let
p
0
0 ¼ n
00
0 ¼ a 0 þ λa 0 ,
n
0
0 ¼ p
00
0 ¼ 0:7a 0 À λa 0 ,
a 0 ¼ 10
23 m
À3
:
ð6:156Þ
When λ assumes different values, we have different doping levels.
Figure 6.16 shows the distributions of p and n along the beam near the junction.
They have become smooth distributions because of diffusion (compare to
Fig. 6.15b).
Figure 6.17 shows the bending deformation associated with the free PN junction.
The coupling to bending offers the possibility of constructing new piezotronic
devices operating with bending of composite beams.
For heterogeneous junctions we consider a relatively simple case by reversing the
poling directions in the ceramics layers of the left half of the beam in Fig. 6.15a.
When the poling direction is reversed, the relevant piezoelectric constants change
their signs. The deflection curve and bending curvature associated with such a
heterogeneous junction are shown in Fig. 6.18 for different doping levels. Compared
to the corresponding fields of a homogeneous junction, the deflection curve and the
bending curvature change their signs across the heterogeneous junction. The electric
fields are qualitatively similar to those of the homogeneous junction and therefore
are not shown.
172
6 Composite Structures
(6.147), (6.148), (6.149), and (6.150) into Eqs. (6.151), (6.152), (6.153), and (6.154)
yields a system of linear equations for the undetermined constants. These equations
are solved on a computer using MATLAB. For a complete description of the
bending deformation, we introduce the bending curvature
K ¼
u 2,33
1 þ u 2,3
ð Þ
2
h
i 3=2 ffi u 2,33 ,
ð6:155Þ
which has been approximated according the usual small deformation assumption in
mechanics of materials.
As a numerical example, consider a beam with L ¼ 20 Â 10
À6 m, b ¼ 50 Â 10
À9 m,
c ¼ 15 Â 10
À9 m, and h ¼ 10 Â 10
À9 m. Let
p
0
0 ¼ n
00
0 ¼ a 0 þ λa 0 ,
n
0
0 ¼ p
00
0 ¼ 0:7a 0 À λa 0 ,
a 0 ¼ 10
23 m
À3
:
ð6:156Þ
When λ assumes different values, we have different doping levels.
Figure 6.16 shows the distributions of p and n along the beam near the junction.
They have become smooth distributions because of diffusion (compare to
Fig. 6.15b).
Figure 6.17 shows the bending deformation associated with the free PN junction.
The coupling to bending offers the possibility of constructing new piezotronic
devices operating with bending of composite beams.
For heterogeneous junctions we consider a relatively simple case by reversing the
poling directions in the ceramics layers of the left half of the beam in Fig. 6.15a.
When the poling direction is reversed, the relevant piezoelectric constants change
their signs. The deflection curve and bending curvature associated with such a
heterogeneous junction are shown in Fig. 6.18 for different doping levels. Compared
to the corresponding fields of a homogeneous junction, the deflection curve and the
bending curvature change their signs across the heterogeneous junction. The electric
fields are qualitatively similar to those of the homogeneous junction and therefore
are not shown.
172
6 Composite Structures