140
6 Mathematical Models of Micro- and Nano-cylindrical Panels in Temperature Field
(N , M, P) =
A
σ 11 (1, z, z
3
)d A, (Q, R) =
A
σ 13 (1, z
2
)d A,
(6.29)
where N , M, Q stand for classical axial force, moment and cutting force, whereas
P, R are higher order resultants. Besides, we introduce couples of stresses of higher
orders
(Y 12 , T 12 ) =
A
m 12 (1, z
2
)d A, (Y 23 , T 23 ) =
A
m 23 (1, z)d A.
(6.30)
Formula (6.28) with an account of (6.29), (6.30) takes the following form:
δU =
L
0
N
∂δu
∂ x
+ kδw +
∂w
∂ x
− βku
∂δw
∂ x
− βkδu
+
+ M
∂δϕ
∂ x
− α P
∂δϕ
∂ x
+
∂
2
δw
∂ x 2
+
(6.31)
+Q
δϕ +
∂δw
∂ x
− βkδu
− 3α R
δϕ +
∂δw
∂ x
− βkδu
+
+
Y 12
2
∂δϕ
∂ x
−
∂
2
δw
∂ x 2 + βk
∂δu
∂ x
−
3
2
αT 12
∂δϕ
∂ x
+
∂
2
δw
∂ x 2 − βk
∂δu
∂ x
+
+
Y 23
2
βkδϕ + k
∂δw
∂ x
− βk
2
δu
− 3αT 23
δϕ +
∂δw
∂ x
− βkδu
dx.
After integration by parts and algebraic transformations, we get
δU =
L
0
−
∂ N
∂ x
+ βk N
∂w
∂ x
− βku
+ βk Q +
βk
2
∂ Y 12
∂ x
+ kY 23
−
− 3αβk
R + T 23 −
1
2
∂ T 12
∂ x
δu −
∂ M
∂ x
− Q +
1
2
∂ Y 12
∂ x
− βkY 23
−
− α
∂ P
∂ x
+
3
2
∂ T 12
∂ x
+ 3α (R + T 23 )
δϕ −
∂ Q
∂ x
− k N +
(6.32)
+
∂
∂ x
N
∂w
∂ x
− βku
+
1
2
∂
2 Y 12
∂ x 2 + k
∂Y 23
∂ x
+