154
6 Mathematical Models of Micro- and Nano-cylindrical Panels in Temperature Field
∂
2
∂ x 2
J 1
∂u
∂ x
+ kw +
1
2
∂w
∂ x
2
− J 2
∂
2 w
∂ x 2
−
∂
2
∂ x 2
B 0
∂
2 w
∂ x 2
−
−k
J 0
∂u
∂ x
+ kw +
1
2
∂w
∂ x
2
− J 1
∂
2 w
∂ x 2
+
+
∂
∂ x
J 0
∂u
∂ x
+ kw +
1
2
∂w
∂ x
2
− J 1
∂
2 w
∂ x 2
∂w
∂ x
+
(6.83)
+
k
2
4
∂
∂ x
B 0
∂w
∂ x
+
∂C 0
∂ x
+ k N
T
−
∂
∂ x
N
T ∂w
∂ x
−
∂
2 M
T
∂ x 2 + q =
= I 0
∂
2 w
∂t 2 + I 1
∂
3 u
∂t 2 ∂ x
− I 2
∂
4 w
∂t 2 ∂ x 2
with the corresponding boundary conditions
J 0
∂u
∂ x
+ kw +
1
2
∂w
∂ x
2
− J 1
∂
2 w
∂ x 2 − N
T
x=L
x=0
= ¯
N or u|
x=L
x=0 = ¯
u, (6.84)
J 1
∂u
∂ x
+ kw +
1
2
∂w
∂ x
2
− J 2
∂
2 w
∂ x 2 − B 0
∂
2 w
∂ x 2 − M
T
x=L
x=0
= ¯
M
(6.85)
or
∂w
∂ x
x=L
x=0
=
____
∂w
∂ x
,
∂
∂ x
J 1
∂u
∂ x
+ kw +
1
2
∂w
∂ x
2
− J 2
∂ 2 w
∂ x 2
+
1
4
k 2 B 0
∂w
∂ x
−
∂
∂ x
B 0
∂ 2 w
∂ x 2
+
+
J 0
∂u
∂ x
+ kw +
1
2
∂w
∂ x
2
− J 1
∂ 2 w
∂ x 2
∂w
∂ x
−N T
∂w
∂ x
−
∂ M T
∂ x
+
C 0
2
x=L
x=0
= ¯
Q
(6.86)
or
w|
x=L
x=0 = ¯
w.
In Sect. 6.4, the resolving equations of motion of size-dependent Sheremetev–
Pelekh beam on the basis of modified couple stress theory have been obtained. If we
take the material length parameter l = 0 in (6.56) and if we assume a homogeneous