144
6 Mathematical Models of Micro- and Nano-cylindrical Panels in Temperature Field
− 3αβk
R + T 23 −
1
2
∂ T 12
∂ x
+
βk
2
(C 0 + 3αC 2 ) + f =
(6.40)
= I 0
∂
2 u
∂t 2 + (I 1 − α I 3 )
∂
2
ϕ
∂t 2 − α I 3
∂
3 w
∂ x∂t 2 ,
∂ M
∂ x
− Q + 3α (R + T 23 ) +
1
2
∂Y 12
∂ x
− βkY 23
−
− α
∂ P
∂ x
+
3
2
∂ T 12
∂ x
+
1
2
(C 0 − 3αC 2 ) =
(6.41)
= (I 1 − α I 3 )
∂
2 u
∂t 2 + (I 2 − α I 4 )
∂
2
ϕ
∂t 2 − α (I 4 − α I 6 )
∂
2
ϕ
∂t 2 +
∂
3 w
∂ x∂t 2
,
∂ Q
∂ x
− k N +
∂
∂ x
N
∂w
∂ x
− βku
+
1
2
∂
2 Y 12
∂ x 2 + k
∂Y 23
∂ x
+ α
∂
2 P
∂ x 2 −
− 3α
∂ R
∂ x
+
∂ T 23
∂ x
−
3
2
∂
2 T 12
∂ x 2
+
1
2
∂
∂ x
(C 0 + 3αC 2 ) + q =
(6.42)
= I 0
∂
2 w
∂t 2 + α I 3
∂
3 u
∂ x∂t 2 + α I 4
∂
3
ϕ
∂ x∂t 2 − α
2 I 6
∂
4 w
∂ x 2 ∂t 2 +
∂
3
ϕ
∂ x∂t 2
,
and the corresponding boundary conditions
N + βk
Y 12 + 3αT 12
2
x=L
x=0
= ¯
N
or
u|
x=L
x=0 = ¯
u,
(6.43)
M − α P +
Y 12 − 3αT 12
2
x=L
x=0
= ¯
M or ϕ|
x=L
x=0 = ¯
ϕ,
(6.44)
α
2 I 6
∂
3 w
∂ x∂t 2 − α I 3
∂
2 u
∂t 2 − α (I 4 − α I 6 )
∂
2
ϕ
∂t 2 +
C 0 + 3αC 2
2
+
+
Q +
1
2
kY 23 +
1
2
∂Y 12
∂ x
+ N
∂w
∂ x
− βku
+ α
∂ P
∂ x
− 3α
R + T 23 −
1
2
∂ T 12
∂ x
x=L
x=0
= ¯
Q
(6.45)
or
w|
x=L
x=0 = ¯
w,