5.3 Zero- and First-Order Equations
In this section we truncate the hierarchy of two-dimensional equations in the
previous section to zero- and first-order equations. These low-order equations are
not ready to be used for analyzing plate extension or bending problems. Instead, they
will be used to derive equations for extension or bending in subsequent sections with
further modifications of constitutive relations. For a complete description of zeroand first-order strains, electric fields, and carrier concentration perturbations, we
truncate the expansions in the previous section as follows:
u j x 1 , x 2 , x 3 , t
ð
Þffiu
0
ð Þ
j x 1 , x 2 , t
ð
Þ
þx 3 u
1
ð Þ
j x 1 , x 2 , t
ð
Þþx
2
3 u
2
ð Þ
j x 1 , x 2 , t
ð
Þ,
φ x 1 , x 2 , x 3 , t
ð
Þffiφ
0
ð Þ x 1 , x 2 , t
ð
Þ
þx 3 φ
1
ð Þ x 1 , x 2 , t
ð
Þþx
2
3 φ
2
ð Þ x 1 , x 2 , t
ð
Þ,
ð5:22Þ
Δp x 1 , x 2 , x 3 , t
ð
Þffip
0
ð Þ x 1 , x 2 , t
ð
Þ
þx 3 p
1
ð Þ x 1 , x 2 , t
ð
Þþx
2
3 p
2
ð Þ x 1 , x 2 , t
ð
Þ,
Δn x 1 , x 2 , x 3 , t
ð
Þffin
0
ð Þ x 1 , x 2 , t
ð
Þ
þx 3 n
1
ð Þ x 1 , x 2 , t
ð
Þþx
2
3 n
2
ð Þ x 1 , x 2 , t
ð
Þ,
ð5:23Þ
where u
0
ð Þ
a (a ¼ 1 and 2) are the extensional displacements, u
1
ð Þ
a the fundamental
thickness-shear displacements, u
2
ð Þ
a the second-order or symmetric thickness-shear
displacements, u
0
ð Þ
3
the bending or flexural displacement, u
1
ð Þ
3
the fundamental
thickness-stretch displacement, and u
2
ð Þ
3 the second-order thickness-stretch displacement. The corresponding strains, electric field and carrier concentration gradients are
S ij ffi S
0
ð Þ
ij þ x 3 S
1
ð Þ
ij , E i ffi E
0
ð Þ
i þ x 3 E
1
ð Þ
i ,
P i ffi P
0
ð Þ
i þ x 3 P
1
ð Þ
i , N i ffi N
0
ð Þ
i þ x 3 N
1
ð Þ
i ,
ð5:24Þ
where
S
0
ð Þ
11 ¼ u
0
ð Þ
1,1 , S
0
ð Þ
22 ¼ u
0
ð Þ
2,2 , S
0
ð Þ
33 ¼ u
1
ð Þ
3
2S
0
ð Þ
23 ¼ u
0
ð Þ
3,2 þ u
1
ð Þ
2 , 2S
0
ð Þ
31 ¼ u
0
ð Þ
3,1 þ u
1
ð Þ
1 , 2S
0
ð Þ
12 ¼ u
0
ð Þ
1,2 þ u
0
ð Þ
2,1 ,
ð5:25Þ
S
1
ð Þ
11 ¼ u
1
ð Þ
1,1 , S
1
ð Þ
22 ¼ u
1
ð Þ
2,2 , S
1
ð Þ
33 ¼ 2u
2
ð Þ
3 ,
2S
1
ð Þ
23 ¼ u
1
ð Þ
3,2 þ 2u
2
ð Þ
2 , 2S
2
ð Þ
31 ¼ u
1
ð Þ
3,1 þ 2u
2
ð Þ
1 , 2S
1
ð Þ
12 ¼ u
1
ð Þ
1,2 þ u
1
ð Þ
2,1 ,
ð5:26Þ
E
0
ð Þ
1 ¼ Àφ
0
ð Þ
,1 , E
0
ð Þ
2 ¼ Àφ
0
ð Þ
,2 , E
0
ð Þ
3 ¼ Àφ
1
ð Þ ,
ð5:27Þ
5.3 Zero- and First-Order Equations
117
In this section we truncate the hierarchy of two-dimensional equations in the
previous section to zero- and first-order equations. These low-order equations are
not ready to be used for analyzing plate extension or bending problems. Instead, they
will be used to derive equations for extension or bending in subsequent sections with
further modifications of constitutive relations. For a complete description of zeroand first-order strains, electric fields, and carrier concentration perturbations, we
truncate the expansions in the previous section as follows:
u j x 1 , x 2 , x 3 , t
ð
Þffiu
0
ð Þ
j x 1 , x 2 , t
ð
Þ
þx 3 u
1
ð Þ
j x 1 , x 2 , t
ð
Þþx
2
3 u
2
ð Þ
j x 1 , x 2 , t
ð
Þ,
φ x 1 , x 2 , x 3 , t
ð
Þffiφ
0
ð Þ x 1 , x 2 , t
ð
Þ
þx 3 φ
1
ð Þ x 1 , x 2 , t
ð
Þþx
2
3 φ
2
ð Þ x 1 , x 2 , t
ð
Þ,
ð5:22Þ
Δp x 1 , x 2 , x 3 , t
ð
Þffip
0
ð Þ x 1 , x 2 , t
ð
Þ
þx 3 p
1
ð Þ x 1 , x 2 , t
ð
Þþx
2
3 p
2
ð Þ x 1 , x 2 , t
ð
Þ,
Δn x 1 , x 2 , x 3 , t
ð
Þffin
0
ð Þ x 1 , x 2 , t
ð
Þ
þx 3 n
1
ð Þ x 1 , x 2 , t
ð
Þþx
2
3 n
2
ð Þ x 1 , x 2 , t
ð
Þ,
ð5:23Þ
where u
0
ð Þ
a (a ¼ 1 and 2) are the extensional displacements, u
1
ð Þ
a the fundamental
thickness-shear displacements, u
2
ð Þ
a the second-order or symmetric thickness-shear
displacements, u
0
ð Þ
3
the bending or flexural displacement, u
1
ð Þ
3
the fundamental
thickness-stretch displacement, and u
2
ð Þ
3 the second-order thickness-stretch displacement. The corresponding strains, electric field and carrier concentration gradients are
S ij ffi S
0
ð Þ
ij þ x 3 S
1
ð Þ
ij , E i ffi E
0
ð Þ
i þ x 3 E
1
ð Þ
i ,
P i ffi P
0
ð Þ
i þ x 3 P
1
ð Þ
i , N i ffi N
0
ð Þ
i þ x 3 N
1
ð Þ
i ,
ð5:24Þ
where
S
0
ð Þ
11 ¼ u
0
ð Þ
1,1 , S
0
ð Þ
22 ¼ u
0
ð Þ
2,2 , S
0
ð Þ
33 ¼ u
1
ð Þ
3
2S
0
ð Þ
23 ¼ u
0
ð Þ
3,2 þ u
1
ð Þ
2 , 2S
0
ð Þ
31 ¼ u
0
ð Þ
3,1 þ u
1
ð Þ
1 , 2S
0
ð Þ
12 ¼ u
0
ð Þ
1,2 þ u
0
ð Þ
2,1 ,
ð5:25Þ
S
1
ð Þ
11 ¼ u
1
ð Þ
1,1 , S
1
ð Þ
22 ¼ u
1
ð Þ
2,2 , S
1
ð Þ
33 ¼ 2u
2
ð Þ
3 ,
2S
1
ð Þ
23 ¼ u
1
ð Þ
3,2 þ 2u
2
ð Þ
2 , 2S
2
ð Þ
31 ¼ u
1
ð Þ
3,1 þ 2u
2
ð Þ
1 , 2S
1
ð Þ
12 ¼ u
1
ð Þ
1,2 þ u
1
ð Þ
2,1 ,
ð5:26Þ
E
0
ð Þ
1 ¼ Àφ
0
ð Þ
,1 , E
0
ð Þ
2 ¼ Àφ
0
ð Þ
,2 , E
0
ð Þ
3 ¼ Àφ
1
ð Þ ,
ð5:27Þ
5.3 Zero- and First-Order Equations
117