272
7 Mathematical Models of Functionally Graded Beams in Temperature Field
The full moment computed in relations to the averaged line of the third layer
follows:
ˆ
M = D
γ
∂α
∂ x
−
∂
2 w
∂ x 2
+
h
2
c 13 N + M
h
,
(7.128)
where M
h stands for an additional moment due to the size dependent additives yielded
by the occurred size effects. The latter is governed by the formula
M
h
= D
−1
3γ 3 t
2
3
1 + ν 3
l 3
h 3
2 ∂α
∂ x
−
−
6γ 1 t
2
1
1 + ν 1
l 1
h 1
2
+
6γ 3 t
2
3
1 + ν 3
l 3
h 3
2
+
6γ 2 t
2
2
1 + ν 2
l 2
h 2
2
∂
2 w
∂ x 2
.
(7.129)
On this step, we introduce a new moment ˆ
H , which does not have analogy in
theory of homogeneous beams, and which defines the transversal shear in the middle
layer. It is called the shear moment and it is defined in the following way:
∧
H = M 3 + cN 1 − cN 2
(7.130)
or equivalently
∧
H =
h
2
c 12 N + Dγ
γ
1 − ϑ
∂α
∂ x
−
∂
2 w
∂ x 2
+ H
h
(7.131)
where
H
h
=
D
2
3γ 3 t
2
3
1 + ν 3
l 3
h 3
2 ∂α
∂ x
−
6γ 3 t
2
3
1 + ν 3
l 3
h 3
2 ∂
2 w
∂ x 2
(7.132)
stands for the size dependent additive term to the shear moment.
In the previous formulas, the following notation has been introduced:
= c 33 − 3c
2
13 ; γ = (c 23 − 3c 12 c 13 )
−1
;
ϑ = 1 − γ
c 23 − 3c 12 c 13
c 22 − 3c
2
12
,
(7.133)
where the explicit form of the coefficients c i j read
c 12 = t 3 (γ 1 − γ 2 ) ; c 13 = γ 1 (t 1 + t 3 ) − γ 2 (t 2 + t 3 ) ;
c 22 = t
2
3 (3γ 1 + 3γ 2 + γ 3 ) ;
c 23 = 3γ 1 t 3 (t 1 + t 3 ) + 3γ 2 t 3 (t 2 + t 3 ) + γ 3 t
2
3 ;
c 33 = γ 1
4t
2
1 + 6t 1 t 3 + 3t
2
3
+ γ 2
4t
2
2 + 6t 2 t 3 + 3t
2
3
+ γ 3 t
2
3 .
(7.134)