6.3 Modified Couple Stress Theory of Thermoelastic Curvilinear Panels
141
+α
∂
2 P
∂ x 2 − 3α
∂ R
∂ x
+
∂ T 23
∂ x
−
1
2
∂
2 T 12
∂ x 2
δ w
dx +
+
N +
3
2
αβkT 12 + βk
Y 12
2
δu
x=L
x=0
+
M − α
P +
3
2
T 12
+
Y 12
2
δϕ
x=L
x=0
+
+
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
x=L
x=0
+
α P +
3
2
αT 12 +
Y 12
2
∂δw
∂ x
x=L
x=0
.
Let us find variation of the kinetic beam energy. Let ρ (x, z, T ) denote the density
of beam material. Its kinetic energy is as follows:
K =
1
2
L
0
A
ρ (x, z, T )
∂ u
z
∂ t
2
+
∂ w
∂ t
2
d Adx .
(6.33)
Let us introduce the following resultants with regard to the beam cross section
(I 0 , I 1 , I 2 , I 3 , I 4 , I 5 , I 6 ) =
A
ρ(x, z, T )(1, z, z
2
, z
3
, z
4
, z
5
, z
6
)dz.
(6.34)
Now, the first variation of the kinetic energy with an account of (6.12) takes the
following form:
δ K = −
L
0
I 0
∂
2 u
∂t 2 + (I 1 − α I 3 )
∂
2
ϕ
∂t 2 − α I 3
∂
3 w
∂ x∂t 2
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
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
δw
dx+
+
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
δw
x=L
x=0
.
(6.35)
The work of external forces carried out in the deformed elastic body under
hypotheses of the modified couple stress theory is defined by the following formula:
[46]