152
6 Mathematical Models of Micro- and Nano-cylindrical Panels in Temperature Field
∂
∂ x
J 0
∂u
∂ x
+ kw +
1
2
∂w
∂ x
2
+ J 1
∂ϕ
∂ x
+ f −
∂ N
T
∂ x
= I 0
∂
2 u
∂t 2 + I 1
∂
2
ϕ
∂t 2 ,
(6.75)
∂
∂ x
J 1
∂u
∂ x
+ kw +
1
2
∂w
∂ x
2
+ J 2
∂ϕ
∂ x
− A 0
ϕ +
∂w
∂ x
+
+
1
4
∂
∂ x
B 0
∂ϕ
∂ x
−
∂
2 w
∂ x 2
+
C 0
2
−
∂ M
T
∂ x
= I 1
∂
2 u
∂t 2 + I 2
∂
2
ϕ
∂t 2 ,
(6.76)
∂
∂ x
A 0
ϕ +
∂w
∂ x
− k
J 0
∂u
∂ x
+ kw +
1
2
∂w
∂ x
2
+ J 1
∂ϕ
∂ x
+
+
∂
∂ x
J 0
∂u
∂ x
+ kw +
1
2
∂w
∂ x
2
+ J 1
∂ϕ
∂ x
∂w
∂ x
+
+
1
4
∂
∂ x
∂
∂ x
B 0
∂ϕ
∂ x
−
∂
2 w
∂ x 2
+ k
2 B 0
∂w
∂ x
+
+
1
2
∂C 0
∂ x
+ k N
T
−
∂
∂ x
N
T ∂w
∂ x
+ q = I 0
∂
2 w
∂t 2
(6.77)
with the corresponding boundary conditions
J 0
∂u
∂ x
+ kw +
1
2
∂w
∂ x
2
+ J 1
∂ϕ
∂ x
− N
T
x=L
x=0
= ¯
N or u|
x=L
x=0 = ¯
u, (6.78)
J 1
∂u
∂ x
+ kw +
1
2
∂w
∂ x
2
+ J 2
∂ϕ
∂ x
+
1
4
B 0
∂ϕ
∂ x
−
∂
2 w
∂ x 2
− M
T
x=L
x=0
= ¯
M
(6.79)
or
ϕ|
x=L
x=0 = ¯
ϕ,
C 0
2
+ A 0
ϕ +
∂w
∂ x
+
1
4
k
2 B 0
∂w
∂ x
+
1
4
∂
∂ x
B 0
∂ϕ
∂ x
−
∂
2 w
∂ x 2
+
+
J 0
∂u
∂ x
+ kw +
1
2
∂w
∂ x
2
+ J 1
∂ϕ
∂ x
∂w
∂ x
−N
T
∂w
∂ x
x=L
x=0
= ¯
Q
(6.80)