6.3 Modified Couple Stress Theory of Thermoelastic Curvilinear Panels
143
+
βk
2
(C 0 + 3αC 2 ) +
∂ 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+
−
(I 1 − α I 3 )
∂
2 u
∂t 2 + (I 2 − α I 4 )
∂
2
ϕ
∂t 2 − α (I 4 − α I 6 )
∂
2
ϕ
∂t 2 +
∂
3 w
∂ x∂t 2
+
+
1
2
(C 0 − 3α C 2 ) +
∂ M
∂ x
− Q +
1
2
∂Y 12
∂ x
− βkY 23
− α
∂ P
∂ x
+
3
2
∂ T 12
∂ x
+
+ 3α (R + T 23 )] δϕ +
−
I 0
∂
2 w
∂t 2 + α I 3
∂
3 u
∂ x ∂t 2 + α I 4
∂
3
ϕ
∂ x ∂t 2 −
−α
2 I 6
∂
4 w
∂ x 2 ∂t 2 +
∂
3
ϕ
∂ x∂t 2
+
1
2
∂
∂ x
(C 0 + 3αC 2 ) +
(6.39)
∂ Q
∂ x
− k N +
∂
∂ x
N
∂w
∂ x
− βku
+
1
2
∂
2 Y 12
∂ x 2 + k
∂Y 23
∂ x
+ α
∂
2 P
∂ x 2 −
−3α
∂ R
∂ x
+
∂ T 23
∂ x
−
3
2
∂
2 T 12
∂ x 2
+ q
δw
dxdt+
+
t 2
t 1
¯
N − N −
3
2
αβkT 12 − βk
Y 12
2
δu +
¯
M − M + α
P +
3
2
T 12
−
Y 12
2
δϕ+
+
¯
Q +
α I 3
∂
2 u
∂t 2 + α (I 4 − α I 6 )
∂
2
ϕ
∂t 2 − α
2 I 6
∂
3 w
∂ x∂t 2
−
C 0 + 3αC 2
2
−
Q +
1
2
kY 23 +
1
2
∂Y 12
∂ x
+ N
∂w
∂ x
− βku
+ α
∂ P
∂ x
− 3α
R + T 23 −
1
2
∂ T 12
∂ x
δw+
+
α ¯
P − α P −
Y 12 + 3αT 12
2
∂δw
∂ x
x=L
x=0
dt = 0.
Employing the fundamental lemma of the variational computations into equation
(6.39) with an account of arbitrariness of kinematically allowed variable δuδϕ and δw
for x ∈ [0, L] and t ∈ (t 1 , t 2 ), the following system of the Euler–Lagrange equations
is obtained for the beams:
∂ N
∂ x
+ βk N
∂w
∂ x
− βku
+ βk Q +
βk
2
∂Y 12
∂ x
+ kY 23
−