158
6 Mathematical Models of Micro- and Nano-cylindrical Panels in Temperature Field
− 3α
B 2
∂ϕ
∂ x
−
∂
2 w
∂ x 2
− 3α B 4
∂ϕ
∂ x
+
∂
2 w
∂ x 2
+
+α P
T
− M
T
x=1
x=0
= ¯
M or
ϕ|
x=1
x=0 = ¯
ϕ,
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
+
+ ( A 0 − 3α A 2 )
ϕ +
∂ w
∂ x
+
1
4
k
B 0 k
∂w
∂ x
− 6α B 1
ϕ +
∂w
∂ x
+
+
1
4
∂
∂ x
B 0
∂ϕ
∂ x
−
∂
2 w
∂ x 2
− 3α B 2
∂ϕ
∂ x
+
∂
2 w
∂ x 2
+
(6.94)
+
J 0
∂u
∂ x
+ kw +
1
2
∂w
∂ x
2
+ (J 1 − α J 3 )
∂ϕ
∂ x
− α J 3
∂
2 w
∂ x 2
∂w
∂ x
+
+α
∂
∂ x
J 3
∂u
∂ x
+ kw +
1
2
∂w
∂ x
2
+ (J 4 − α J 6 )
∂ϕ
∂ x
− α J 6
∂
2 w
∂ x 2
−
−3α (A 2 − 3α A 4 )
ϕ +
∂ w
∂ x
−
3
2
α
B 1 k
∂w
∂ x
− 6α B 2
ϕ +
∂w
∂ x
+
+
3
4
α
∂
∂ x
B 2
∂ϕ
∂ x
−
∂ 2 w
∂ x 2
− 3α B 4
∂ϕ
∂ x
+
∂ 2 w
∂ x 2
− N
T
∂w
∂ x
− α
∂ P T
∂ x
x=1
x=0
= ¯
Q
or
w|
x=1
x=0 = ¯
w
∂ϕ
∂ x
−
∂
2 w
∂ x 2
x=1
x=0
= 0
or
∂w
∂ x
x=1
x=0
=
____
∂w
∂ x
.
(6.95)
The initial conditions are as follows:
w(x, 0) = ψ 1 (x);
∂w (x, t)
∂t
t=0
= ψ 2 (x),
u(x, 0) = ψ 3 (x) ;
∂u (x, t)
∂t
t=0
= ψ 4 (x),
ϕ(x, 0) = ψ 5 (x);
∂ ϕ (x, t)
∂t
t=0
= ψ 6 (x).
(6.96)