Wave reflection and transmission in composite beams containing ...

ARTICLE IN PRESS
W.-C. Yuan et al. / Journal of Sound and Vibration 313 (2008) 676–695 683
derived:
k 2 A ð1Þ
55 ð ^w; xx þ c 1 ; x Þ¼I ð1Þ
1 €^w þ p, (27a)
D ð1Þ
11 c 1; xx k 2 A ð1Þ
55 ð ^w ;x þ c 1 Þ¼I ð1Þ
2 € c 1 , (27b)
k 2 A ð2Þ
55 ð ^w; xx þ c 2 ; x Þ¼I ð2Þ
1 €^w p, (27c)
D ð2Þ
11 c 2; xx k 2 A ð2Þ
55 ð ^w ;x þ c 2 Þ¼I ð2Þ € 2
c 2 . (27d)
The contact pressure p can be eliminated by combining Eqs. (27a) and (27c), to obtain
k 2 ðA 55 ^w; xx þ A ð1Þ
55 c 1; x þ A ð2Þ
55 c 2; x Þ¼I 1
€^w. (28)
If the displacements are introduced as
^w ¼ ^W e ið ^kx otÞ ; c 1 ¼ C 1 e ið ^kx otÞ ; c 2 ¼ C 2 e ið ^kx otÞ . (29)
Then by substitution of Eq. (29) into Eqs. (28), (27b), and (27d), the dispersion relation can be determined
as follows:
82
k 2 A 55 ^k2 ik 2 A ð1Þ ^k 55
ik 2 A ð2Þ ^k
3 2
39
55 I det ik 2 A ð1Þ ^k 55
D ð1Þ ^k 1 0 0
><
2 11
þ k 2 A ð1Þ
6
55
0 7
4
ik 2 A ð2Þ ^k 55
0 D ð2Þ ^k
5 o2 0 I ð1Þ
>=
6 2
0 7
4
5 ¼ 0. (30)
>:
2 11
þ k 2 A ð2Þ 0 0 I ð2Þ
2
>;
55
The results give rise to three flexural wave modes. Loosely these modes can be related to the
fundamental flexural mode of the un-delaminated portion of the beam, A 0 (0)d , and the other two
related to the A 1 mode in the upper and lower sub-beams, A 1 (1)d and A 1 (2)d , respectively. Two cut-off
frequencies can be readily proved to be identical to those in the open delaminating case. When the frequencies
approach infinity, according to Eqs. (30) and (17), the group velocities of the three flexural modes can be
exactly derived
c ð0Þd
g0
¼ kc s ; c ð1Þd
g1
¼ c ð2Þd
g1
¼ c l as o !1. (31)
The general solution of the positive-going transmitted flexural waves in the delaminated region can be
conveniently written as
^w ¼ a t e i ^k 0 x þ b ð1Þ
t e i ^k 1 x þ b ð2Þ
t e i ^k 2 x , (32a)
c n ¼ G ðnÞ
0 a t e i ^k 0 x þ G ðnÞ
1 bð1Þ t
e i ^k 1 x þ G ðnÞ
2 bð2Þ t e i ^k 2 x ðn ¼ 1; 2Þ, (32b)
where G ðnÞ
j ¼ði ^k j =½ðoq ðnÞ Þ=ðkc ðnÞ
s ÞŠ 2 ½q ðnÞ ð ^k j =kÞðc ðnÞ
l
=c ðnÞ
s ÞŠ 2 1Þ, (n ¼ 1,2 and j ¼ 0,1,2).
The extensional waves in the two sub-beams are the same as in the open delamination case. There are a total
of eight amplitudes (a r , b r , c r , a t , b (1) t , b (2) t , c (1) t , c (2) t ) which can be determined from the following continuity
and equilibrium conditions at x ¼ 0.
w 0 ¼ ^w; c 0 ¼ c 1 ¼ c 2 ; u 1 ¼ u 0 þ h 2
2 c 0; u 2 ¼ u 0
h 1
2 c 0,
N 0 ¼ N 1 þ N 2 ; M 0 ¼ M 1 þ M 2 þ h 2
2 N 1
h 1
2 N 2; V 0 ¼ V 1 þ V 2 . (33)