148
6 Mathematical Models of Micro- and Nano-cylindrical Panels in Temperature Field
Boundary conditions (6.43)–(6.46) are also simplified. This requires to take β = 0
in equations (6.50)–(6.52), (6.54), (6.55) and (6.57)–(6.60), and one gets
N |
x=L
x=0 = ¯
N
or
u|
x=L
x=0 = ¯
u
(6.64)
M − α P +
Y 12 − 3αT 12
2
x=L
x=0
= ¯
M or ϕ|
x=L
x=0 = ¯
ϕ,
(6.65)
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
+ α
∂ P
∂ x
− 3α
R + T 23 −
1
2
∂ T 12
∂ x
x=L
x=0
= ¯
Q
(6.66)
or
w|
x=L
x=0 = ¯
w
α P +
Y 12 + 3αT 12
2
x=L
x=0
= α ¯
P or
∂w
∂ x
x=L
x=0
=
____
∂w
∂ x
.
(6.67)
Let us rewrite Eqs. (6.61)–(6.63) in displacements:
∂
∂ x
J 0
∂u
∂ x
+ kw +
1
2
∂w
∂ x
2
+ (J 1 − α J 3 )
∂ϕ
∂ x
− α J 3
∂
2 w
∂ x 2
+
+ f −
∂ N
T
∂ x
= I 0
∂
2 u
∂t 2 + (I 1 − α I 3 )
∂
2
ϕ
∂t 2 − α I 3
∂
3 w
∂ x∂t 2 ,
(6.68)
∂
∂ x
J 1
∂u
∂ x
+ kw +
1
2
∂w
∂ x
2
+ (J 2 − α J 4 )
∂ϕ
∂ x
− α J 4
∂ 2 w
∂ x 2
− (A 0 − 3α A 2 )
ϕ +
∂w
∂ x
+
+3α
(A 2 − 3α A 4 )
ϕ +
∂w
∂ x
+
1
2
(B 1 k − 6α B 2 )
∂w
∂ x
− 6α B 2 ϕ
+
+
1
4
∂
∂ x
B 0
∂ϕ
∂ x
−
∂
2 w
∂ x 2
− 3α B 2
∂ϕ
∂ x
+
∂
2 w
∂ x 2
−
(6.69)
−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 +