Journal of Mechanics of Materials and Structures vol. 5 (2010 ... - MSP

834 ELIA EFRAIM AND MOSHE EISENBERGER
Appendix: Entries of the coefficient matrices K (0) , K (1) , and K (2) in Equation (14)
Only nonzero entries are listed. We set � = 1/φ0.
K (0)
11 = ω2 (ρ/E)(1 − µ 2 )R 2 p R3 0 h + R p R0(R ′ p h′ + R ′′ p h)�2 µ − 1
2 (1 − µ)(R2 p R0κh + R 3 0 hn2 )
K (0)
12
= 1
2 R2 0 n�R′ p h(µ − 3) + R2 0 n�R ph ′ µ
K (0)
13 = R2p R0h ′ �(1 + µ) K (0) 1
14 = 6ω2 (ρ/E)(1 − µ 2 )R 2 p R2 0h3 + 1
2 (1 − µ)R2p R2 0κh K (0)
21
1
= 2 R2 0n�R′ 1
ph(µ − 3) − 2 R2 0n�(1 − µ)R ph ′
K (0)
22 = ω2 (1 − µ 2 )(ρ/E)R 2 p R3 1
0h − 2 (1 − µ)R p R ′′ p R0h� 2 µ − 1
2 (1 − µ)R p R ′ p R0h ′ � 2 − R 3 0hn2 − 1
2 (1 − µ)R2p R0κh − 1
2 (1 − µ)R′2 p
R0h� 2
K (0)
23 = −(1+µ)R p R 2 1
0hn− 2 (1−µ)R p R 2 (0) 1
0κhn K 25 = 6ω2 (ρ/E)(1−µ 2 )R 2 p R2 0h3 + 1
2 (1−µ)R2p R2 0κh K (0)
31 = −(1 + µ)R p R ′ p R0�h − 1
2 (1 − µ)R p R ′ p R0κh� − 1
2 (1 − µ)R2 p R0κh ′ �
K (0)
32
K (0)
34
K (0)
41
K (0)
44
K (0)
45
K (0)
52
K (0)
55
= K (0)
23
= 1
2 (1 − µ)� R 2 p R2 0 κh′ + R p R ′ p R2 0 �κh�
= K (0)
14
K (0)
33 = ω2 (ρ/E)(1 − µ 2 )R 2 p R3 1
0h − 2 (1 − µ)R3 0κhn2 − 2(1 + µ)R 2 p R0h
K (0)
35
= 1
12 ω2 (ρ/E)(1 − µ 2 )R 2 p R3 0 h3 − 1
24 (1 − µ)R3 0 h3 n 2 − 1
12 R′2 p R0h 3 � 2 (1 + µ)
= 1
4 nµR p R 2 0 h′ h 2 � − 1
24 R′ p R2 0 n�h3 (3 − µ)
= K (0)
25
K (0)
53
= K (0)
35
= 1
2 (1 − µ)R p R 3 0 κhn
− 1
2 (1 − µ)R2p R3 1
0κh + 12 R p R ′′ p R0h 3 � 2 µ + 1
4 R p R ′ p R0h ′ h 2 � 2 µ
K (0)
54
= − 1
8 (1 − µ)n R p R 2 0 h′ h 2 � − 1
24 R′ p R2 0 n�h3 (3 − µ)
= 1
12 ω2 (ρ/E)(1 − µ 2 )R 2 p R3 0 h3 + 1
24 (1 − µ)R p R ′′ p R0h 3 � 2 + 1
8 (1 − µ)R p R ′ p R0h ′ h 2 � 2 µ
K (1)
11 = R p R ′ p R0h� 2 + R 2 p R0h ′ � 2
K (1)
12
K (1)
21
K (1)
31
K (1)
33
K (1)
43
− 1
2 (1 − µ)R2p R3 1
0κh − 12 R3 0n2h 3 − 1
12 R2p ′ R0h 3 � 2 (1 − µ)
1
= 2 (1 + µ)R p R 2 (1)
0hn� K 13 = (1 + µ)R2p R0�h + 1
2 (1 − µ)R2p R0K �h
= −K (1)
12
= −K (1)
13
K (1)
22
= 1
2 (1 − µ)R p R ′ p R0h� 2 + 1
2 (1 − µ)R2 p R0h ′ � 2
1
= 2 (1 − µ)(R2p R0κh ′ � 2 + R p R ′ p R2 0�2κh) K (1) 1
34 = 2 (1 − µ)R2p R2 0κh� = K (1)
34
K (1)
44 = �2 R0
K (1)
54
= −K (1)
45
� 1
12 R p R ′ p h3 + 1
4 R2 p h′ h 2�
K (1)
55
K (1)
45
= 1
24 (1 + µ)R p R 2 0 �h3 n
= 1
8 (1 − µ)2�2 R 2 p R0h ′ h 2 + 1
24 (1 − µ)R p R ′ p R0h 3 � 2