146
6 Mathematical Models of Micro- and Nano-cylindrical Panels in Temperature Field
Next, to get the resultants Q, R, expressed in terms of displacements.
We denote
(A 0 , A 2 , A 4 ) =
A
G (x, z, T (x, z)) (1, z
2
, z
4
)d A.
(6.53)
Hence, we have
Q = (A 0 − 3α A 2 )
ϕ +
∂w
∂ x
− βku
,
(6.54)
R = (A 2 − 3α A 4 )
ϕ +
∂w
∂ x
− βku
.
(6.55)
Proceeding in the analogous way, to find the higher order resultants, we employ
the following notation:
(B 0 , B 1 , B 2 , B 4 ) =
A
G (x, z, T (x, z)) l
2
(x, z, T (x, z)) (1, z, z
2
, z
4
)d A,
(6.56)
and hence
Y 12 =
1
2
B 0
∂ϕ
∂ x
−
∂
2 w
∂ x 2 + βk
∂u
∂ x
− 3α B 2
∂ϕ
∂ x
+
∂
2 w
∂ x 2 − βk
∂u
∂ x
, (6.57)
T 12 =
1
2
B 2
∂ϕ
∂ x
−
∂
2 w
∂ x 2 + βk
∂u
∂ x
− 3α B 4
∂ϕ
∂ x
+
∂
2 w
∂ x 2 − βk
∂u
∂ x
, (6.58)
Y 23 =
1
2
B 0 k
βϕ +
∂w
∂ x
− βku
− 6α B 1
ϕ +
∂w
∂ x
− βku
,
(6.59)
T 23 =
1
2
B 1 k
βϕ +
∂w
∂ x
− βku
− 6α B 2
ϕ +
∂w
∂ x
− βku
.
(6.60)
Occurrence of Y 12 , Y 23 , T 12 , T 23 in the consider model (see (6.40)–(6.46) is implied
by direct contribution of the couples of stresses of higher orders to the beam deformation energy (see (6.26), (6.30). The derived four results explicitly depend on the
length scale material parameter l, which exhibits the influence of the higher order
couples of stresses.
Finally, substitution of relations (6.50)–(6.52), (6.54), (6.55), and (6.57)–(6.60)
into equations (6.40)–(6.46) yields equation of motion with third-order shear deformation of the panel with regard to displacements u, ϕ, w.