104
J. Chen et al.
The stress-strain relation in layer e l is then expressed as
σ
l
= D
l
ε
l
,
(26)
where σ l =
¯
τ 21 ¯
σ 22 ¯
τ 23
T ;
D
l
=
⎡
⎣
G 0 0 0
0 E 0 0
0 0 G 0 /k
⎤
⎦ ,
(27)
where G 0 = E/[2(1 + ν)]; E 0 = E/
1 − ν 2
; k = 1.2.
Thus, the strain energy of layer e l can be derived as
l
=
a
l
T
H / 2
−H / 2
l / 2
−l / 2
N
l
T
D
l N
l d ¯
α 1 dα 3 a
l
/(2d ).
(28)
3.3 Imposition of the Displacement Boundary Condition
As illustrated in Fig. 7, along boundary 1–2 of element e j , the rotations
¯
θ 1 , ¯
θ 2
or
displacements (¯ u 10 , ¯
u 20 , u 30 ) in the local coordinate system ¯
α 1 ¯
α 2 α 3 are fixed. A fictitious
thin layer e b on boundary 1–2 is also introduced to impose the displacement boundary
condition. The displacement approximation in Eq. (11) is divided into two parts to
separately impose the rotation and middle surface displacement boundary conditions as
follows:
Fig. 7. Imposition of the displacement boundary condition
u
P r
¯
α
P
1 , ¯
α
P
2 , α
P
3
≈ λ
b N
r
¯
α
P
1 , ¯
α
P
2 , α
P
3
a
b
;
(29)