第五章 結論與建議
5.2 建議
利用 Ritz 法分析具應力奇異點之功能梯度板振動，由多項式函數結合 描述應力奇異性之角函數形成所需之允許函數，有效並精確的分析含裂縫 板之問題。將來可直接應用本研究所提之函數於其他具應力奇異之問題。
41
亦可將本研究所提方法之精神應用於其他理論或求解不同材料(例：壓電材 料)之問題。
42
參考文獻
Aggarwala, B. D. and Ariel, P. D. (1981) “Vibration and bending of a cracked plate”, Rozprawy Inzynierskie, 29(2), pp. 295-310.
Bachene, M., Tiberkak, R. and Rechak, S. (2009) “Vibration analysis of cracked plates using the extended finite element method”, Archive of Applied Mechanics, 79, pp. 249-262
Delale, F. and Erdogan, F. (1983) “The crack problem for a nonhomogeneous plane”, Journal of Applied Mechanics, 50, pp. 609-614.
Erdogan, F. (1985) “The crack problem for bonded nonhomogeneous materials under Antiplane shear loading”, Journal of Applied Mechanics, 64, pp.
449-456.
Ferreira A. J. M., Batra R. C., Roque C. M. C., Qian L. F. and Jorge R. M. N.
(2006) “Natural frequencies of functionally graded plates by a meshless method”, Composite Structures, 75, pp. 593-600.
43
He X. Q., Ng T. Y., Sivashanker S. and Liew K. M. (2001) “Active control of FGM plates with integrated piezoelectric sensors and actuators”, International Journal of Solids and Structures, 38, pp. 1641-1655.
Hirano, Y. and Okazaki, K. (1980) “Vibration of cracked rectangular plates”, Bulletin of the Japan Society of Mechanical Engineers, 23(179), pp.
732-740.
Hosseini-Hashemi, S. h., Heydar Roohi, G. h. and Hossein Rokni, D. T. (2010)
“Exact free vibration study of rectangular Mindlin plates with all-over
part-through open cracks”, Computer and Structures, 88, pp. 1015-1032.
Hosseini-Hashemi, S. h., Khorshidi, K., Es’haghi, M., Fadaee, M., Karimi, M.
(2011) “On the effects of coupling between in-plane and out-of-plane vibrating modes of smart functionally graded circular annular plates”, Applied Mathematical Modelling, 36(3), pp. 1132-1147.
Huang, C. S. and Leissa, A. W. (2009) “Vibrtaion analysis of rectangular plates with side cracks via the Ritz method”, Journal of Sound and Vibration, 323(3-5), pp. 974-988.
44
Huang, C. S., Leissa, A. W. and Li, R.S. (2011) “Accurate vibration analysis of thick, cracked rectangular plates”, Journal of Sound and Vibration, 330(9), pp. 2079-2093.
Khadem, S. E. and Rezaee, M. (2000) “Introduction of modified comparison functions for vibration analysis of a rectangular cracked plate”, Journal of Sound and Vibration, 236(2), pp. 245-258.
Krawczuk, M. (1993) “Natural vibrations of rectangular plates with a through crack”, Archive of Applied Mechanics, 63(7), pp. 491-504.
Lee, H. P. and Lim, S. P. (1993) “Vibration of cracked rectangular plates
including transverse shear deformation and rotary inertia”, Computers
&Structures, 49(4), pp. 715-718
Liew, K. M., Hung, K. C. and Lim, M. K. (1994) “A solution method for analysis of cracked plates under vibration”, Engineering Fracture Mechanics, 48(3), pp. 393-404.
Lynn, P. P. and Kumbasar, N. (1967) “Free vibrations of thin rectangular plates having narrow cracks with simply supported edges”, Developments in
45
Mechanics, 4, Proc. 10th Midwestern Mechanics Conference, Colorado State
University, Fort Collins, Colorado, August 21-23, pp. 911-928.
Matsunaga H. (2008) “Free vibration and stability of functionally graded plates according to a 2-D higher-order deformation theory”, Composite Structures, 82, pp. 499-512.
Neku, K. (1982) “Free vibration of a simply-supported rectangular plate with a straight through-notch”, Bulletin of the Japan Society of Mechanical Engineers, 25(199), pp. 16-23.
Noda, N. and Jin, Z. H. (1993) “Thermal stress intensity factors for a crack in a strip of a functionally gradient material”, International Journal of Solids and Structures, 30, pp. 1039-1056.
Ozturk, M. and Erdogan, F. (1992) “Diffusion problem in bonded nonhomogeneous materials with an interface cut”, International Journal of Engineering Science, 30(10), pp. 1507-1523.
Ozturk, M. and Erdogan, F. (1995) “An axisymmetric crack in bonded materials with a nonhomogeneous interfacial zone under torsion”, Journal of Applied Mechanics, 62(1), pp. 116-125.
46
Qian, G. L., Gu, S. N. and Jiang, J. S. (1991) “A finite element model of
cracked plates and application to vibration problems”, Computers and Structures, 39(5), pp. 483-487
Qian L. F., Batra R. C., Chen L. M. (2004) “Static and dynamic deformations of thick functionally graded elastic plates by using higher-order shear and normal deformable plate theory and meshless local Petrov-Galerkin method”, Composites: Part B, 35, pp. 685-697.
Reddy, J. N. (2009) “Analysis of functionally graded plates”, International Journal of Numerical Method for Engineering, 47(1-3), pp. 663-84.
Solecki, R. (1983) “Bending vibration of a simply supported rectangular plate with a crack parallel to one edge”, Engineering Fracture Mechanics, 18(6), pp. 1111-1118.
Stahl, B. and Keer, L. M. (1972) “Vibration and Stability of cracked rectangular plates”, International Journal of Solids and Structures, 8(1), pp.
69-91.
47
Vel, S. S. and Batra, R. C. (2004) “Three-dimensional exact solution for the vibration of functionally graded rectangular plates”, Journal of Sound and Vibrations, 272(3), pp. 703-730.
Yang, Y. and Shen, H. S. (2001) “Dynamic response of initially stressed functional graded rectangular thin plates”, Composite Structures, 54, pp.
497-508.
Yuan, J. and Dickinson, S. M. (1992) “The flexural vibration of rectangular plate systems approached by using artificial springs in the Rayleigh-Ritz method”, Journal of Sound and Vibration, 159(1), pp. 39-55
Zhao, X., Lee, Y. Y. and Liew, K. M. (2009) “Free vibration analysis of functionally graded plates using the element-free kp-Ritz method”, Journal of Sound and Vibration, 319, pp. 918-939.
Zhou, L. and Xiang, Y. (2011) “Vibration Analysis of Mindlin Plates with Cracks by MLS-Element Method”, Procedia Engineering, 14, pp.
1637-1644.
張明儒 (2008) ，「功能梯度厚板之應力奇異性分析」，國立交通大學土 木工程學系博士論文，黃炯憲指導。
48
李榕師 (2009) ，「利用 Ritz 法分析具有裂縫之矩形 Mindlin 板振動」，
國立交通大學土木工程學系碩士論文，黃炯憲指導。
李昱成 (2009) ，「利用 Ritz 法分析具有裂縫之矩形薄板振動」，國立交 通大學土木工程學系碩士論文，黃炯憲指導。
王凱平 (2010) ，「含邊裂縫矩形功能梯度板之三維自由振動分析」，國 立交通大學土木工程學系碩士論文，黃炯憲指導。
49
表 2.1 材料參數
Material
Properties
E(Gpa) Poisson’s
ratio (kg/m ) Aluminum(Al) 70.0 0.3 2702 Alumina( ) 380 0.3 3800
50
Polynomial Solution Size (I× J) @ Stahl and Keer
51
Polynomial Solution Size (I× J) @ Stahl and Keer
52
Polynomial Solution Size (I× J) @ Stahl and Keer
53
Polynomial Solution Size (I× J) @ Stahl and Keer
54
Polynomial Solution Size (I× J) @ Stahl and Keer
55
56
Polynomial Solution Size (I× J) @
3×3 4×4 5×5 6×6 7×7 8×8 9×9
57
表 3.2 (續上頁)
Mode No.
Crack Functions
（N）
Polynomial Solution Size (I× J) @
3×3 4×4 5×5 6×6 7×7 8×8 9×9
5
0
[18.56] [18.56] [18.56] [18.56] [18.56] [18.56] [18.56]
(18.56) (18.56) (18.56) (18.56) (18.56) (18.56) (18.56) {18.56} {18.56} {18.56} {18.56} {18.56} {18.56} {18.56}
2
[18.56] [18.56] [18.56] [18.56] [18.56] [18.56] [18.56]
(18.56) (18.56) (18.56) (18.56) (18.56) (18.56) (18.56) {18.56} {18.56} {18.56} {18.56} {18.56} {18.56} {18.56}
3
[18.56] [18.56] [18.56] [18.56] [18.56] [18.56] [18.56]
(18.56) (18.56) (18.56) (18.56) (18.56) (18.56) (18.56) {18.56} {18.56} {18.56} {18.56} {18.56} {18.56} {18.56}
4
[18.56] [18.56] [18.56] [18.56] [18.56] [18.56] [18.56]
(18.56) (18.56) (18.56) (18.56) (18.56) (18.56) (18.56) {18.56} {18.56} {18.56} {18.56} {18.56} {18.56} {18.56}
5
[18.56] [18.56] [18.56] [18.56] [18.56] [18.56] [18.56]
(18.56) (18.56) (18.56) (18.56) (18.56) (18.56) (18.56) {18.56} {18.56} {18.56} {18.56} {18.56} {18.56} {18.56}
Note: [ ]: results from N ˆ
z 3 ; ( ): results from N ˆ
z 4 ; { }: results from N ˆ
z 5
58
59
Polynomial Solution Size (I× J) @
文獻值
60
表 3.3 (續上頁)
Mode No.
Crack Functions
（N ）
Polynomial Solution Size (I× J) @
文獻值
3×3 4×4 5×5 6×6 7×7 8×8 9×9
5
0
[19.48] [19.48] [19.48] [19.48] [19.48] [19.48] [19.48]
- (19.48) (19.48) (19.48) (19.48) (19.48) (19.48) (19.48)
{19.48} {19.48} {19.48} {19.48} {19.48} {19.48} {19.48}
2
[19.48] [19.48] [19.48] [19.48] [19.48] [19.48] [19.48]
(19.48) (19.48) (19.48) (19.48) (19.48) (19.48) (19.48) {19.48} {19.48} {19.48} {19.48} {19.48} {19.48} {19.48}
3
[19.48] [19.48] [19.48] [19.48] [19.48] [19.48] [19.48]
(19.48) (19.48) (19.48) (19.48) (19.48) (19.48) (19.48) {19.48} {19.48} {19.48} {19.48} {19.48} {19.48} {19.48}
4
[19.48] [19.48] [19.48] [19.48] [19.48] [19.48] [19.48]
(19.48) (19.48) (19.48) (19.48) (19.48) (19.48) (19.48) {19.48} {19.48} {19.48} {19.48} {19.48} {19.48} {19.48}
5
[19.48] [19.48] [19.48] [19.48] [19.48] [19.48] [19.48]
(19.48) (19.48) (19.48) (19.48) (19.48) (19.48) (19.48) {19.48} {19.48} {19.48} {19.48} {19.48} {19.48} {19.48}
Note: [ ]: results from N ˆ
z 3 ; ( ): results from N ˆ
z 4 ; { }: results from N ˆ
z 5
*：Stahl and Keer (1972)之結果
+：Huang et al. (2011) 之結果
61
Polynomial Solution Size (I× J) @
文獻值
*：Stahl and Keer (1972)之結果
+：Huang et al. (2011) 之結果
62
Polynomial Solution Size (I× J) @
文獻值
*：Stahl and Keer (1972)之結果
+：Huang et al. (2011) 之結果
63
Polynomial Solution Size (I× J) @
文獻值
*：Stahl and Keer (1972)之結果
+：Huang et al. (2011) 之結果
64
Polynomial Solution Size (I× J) @
文獻值
*：Stahl and Keer (1972)之結果
+：Huang et al. (2011) 之結果
65
Polynomial Solution Size (I× J) @
文獻值
*：Stahl and Keer (1972)之結果
+：Huang et al. (2011) 之結果
66
67
Polynomial Solution Size (I× J) @
文獻值
68
表 3.5 (續上頁)
Mode No.
Crack Functions
（N）
Polynomial Solution Size (I× J) @
文獻值
3×3 4×4 5×5 6×6 7×7 8×8 9×9
5
0
[19.48] [19.48] [19.48] [19.48] [19.48] [19.48] [19.48]
- (19.48) (19.48) (19.48) (19.48) (19.48) (19.48) (19.48)
{19.48} {19.48} {19.48} {19.48} {19.48} {19.48} {19.48}
2
[19.48] [19.48] [19.48] [19.48] [19.48] [19.48] [19.48]
(19.48) (19.48) (19.48) (19.48) (19.48) (19.48) (19.48) {19.48} {19.48} {19.48} {19.48} {19.48} {19.48} {19.48}
3
[19.48] [19.48] [19.48] [19.48] [19.48] [19.48] [19.48]
(19.48) (19.48) (19.48) (19.48) (19.48) (19.48) (19.48) {19.48} {19.48} {19.48} {19.48} {19.48} {19.48} {19.48}
4
[19.48] [19.48] [19.48] [19.48] [19.48] [19.48] [19.48]
(19.48) (19.48) (19.48) (19.48) (19.48) (19.48) (19.48) {19.48} {19.48} {19.48} {19.48} {19.48} {19.48} {19.48}
5
[19.48] [19.48] [19.48] [19.48] [19.48] [19.48] [19.48]
(19.48) (19.48) (19.48) (19.48) (19.48) (19.48) (19.48) {19.48} {19.48} {19.48} {19.48} {19.48} {19.48} {19.48}
Note: [ ]: results from N ˆ
z 3 ; ( ): results from N ˆ
z 4 ; { }: results from N ˆ
z 5
*：Stahl and Keer (1972)之結果
+：Huang et al. (2011) 之結果
69
70
Polynomial Solution Size (I× J) @
3×3 4×4 5×5 6×6 7×7 8×8 9×9
71
表 3.6 (續上頁)
Mode No.
Crack Functions
（N）
Polynomial Solution Size (I× J) @
3×3 4×4 5×5 6×6 7×7 8×8 9×9
5
0
[18.56] [18.56] [18.56] [18.56] [18.56] [18.56] [18.56]
(18.56) (18.56) (18.56) (18.56) (18.56) (18.56) (18.56) {18.56} {18.56} {18.56} {18.56} {18.56} {18.56} {18.56}
2
[18.56] [18.56] [18.56] [18.56] [18.56] [18.56] [18.56]
(18.56) (18.56) (18.56) (18.56) (18.56) (18.56) (18.56) {18.56} {18.56} {18.56} {18.56} {18.56} {18.56} {18.56}
3
[18.56] [18.56] [18.56] [18.56] [18.56] [18.56] [18.56]
(18.56) (18.56) (18.56) (18.56) (18.56) (18.56) (18.56) {18.56} {18.56} {18.56} {18.56} {18.56} {18.56} {18.56}
4
[18.56] [18.56] [18.56] [18.56] [18.56] [18.56] [18.56]
(18.56) (18.56) (18.56) (18.56) (18.56) (18.56) (18.56) {18.56} {18.56} {18.56} {18.56} {18.56} {18.56} {18.56}
5
[18.56] [18.56] [18.56] [18.56] [18.56] [18.56] [18.56]
(18.56) (18.56) (18.56) (18.56) (18.56) (18.56) (18.56) {18.56} {18.56} {18.56} {18.56} {18.56} {18.56} {18.56}
Note: [ ]: results from N ˆ
z 3 ; ( ): results from N ˆ
z 4 ; { }: results from N ˆ
z 5
72
73
Polynomial Solution Size (I× J) @
3×3 4×4 5×5 6×6 7×7 8×8 9×9
74
表 3.7 (續上頁)
Mode No.
Crack Functions
（N）
Polynomial Solution Size (I× J) @
3×3 4×4 5×5 6×6 7×7 8×8 9×9
5
0
[12.64] [12.64] [12.64] [12.64] [12.64] [12.64] [12.64]
(12.64) (12.64) (12.64) (12.64) (12.64) (12.64) (12.64) {12.64} {12.64} {12.64} {12.64} {12.64} {12.64} {12.64}
2
[12.64] [12.64] [12.64] [12.64] [12.64] [12.64] [12.64]
(12.64) (12.64) (12.64) (12.64) (12.64) (12.64) (12.64) {12.64} {12.64} {12.64} {12.64} {12.64} {12.64} {12.64}
3
[12.64] [12.64] [12.64] [12.64] [12.64] [12.64] [12.64]
(12.64) (12.64) (12.64) (12.64) (12.64) (12.64) (12.64) {12.64} {12.64} {12.64} {12.64} {12.64} {12.64} {12.64}
4
[12.64] [12.64] [12.64] [12.64] [12.64] [12.64] [12.64]
(12.64) (12.64) (12.64) (12.64) (12.64) (12.64) (12.64) {12.64} {12.64} {12.64} {12.64} {12.64} {12.64} {12.64}
5
[12.64] [12.64] [12.64] [12.64] [12.64] [12.64] [12.64]
(12.64) (12.64) (12.64) (12.64) (12.64) (12.64) (12.64) {12.64} {12.64} {12.64} {12.64} {12.64} {12.64} {12.64}
Note: [ ]: results from N ˆ
z 4 ; ( ): results from N ˆ
z 5 ; { }: results from N ˆ
z 6
75
76
Polynomial Solution Size (I× J) @
3×3 4×4 5×5 6×6 7×7 8×8 9×9
77
表 3.8 (續上頁)
Mode No.
Crack Functions
（N）
Polynomial Solution Size (I× J) @
3×3 4×4 5×5 6×6 7×7 8×8 9×9
5
0
[12.64] [12.64] [12.64] [12.64] [12.64] [12.64] [12.64]
(12.64) (12.64) (12.64) (12.64) (12.64) (12.64) (12.64) {12.64} {12.64} {12.64} {12.64} {12.64} {12.64} {12.64}
2
[12.64] [12.64] [12.64] [12.64] [12.64] [12.64] [12.64]
(12.64) (12.64) (12.64) (12.64) (12.64) (12.64) (12.64) {12.64} {12.64} {12.64} {12.64} {12.64} {12.64} {12.64}
3
[12.64] [12.64] [12.64] [12.64] [12.64] [12.64] [12.64]
(12.64) (12.64) (12.64) (12.64) (12.64) (12.64) (12.64) {12.64} {12.64} {12.64} {12.64} {12.64} {12.64} {12.64}
4
[12.64] [12.64] [12.64] [12.64] [12.64] [12.64] [12.64]
(12.64) (12.64) (12.64) (12.64) (12.64) (12.64) (12.64) {12.64} {12.64} {12.64} {12.64} {12.64} {12.64} {12.64}
5
[12.64] [12.64] [12.64] [12.64] [12.64] [12.64] [12.64]
(12.64) (12.64) (12.64) (12.64) (12.64) (12.64) (12.64) {12.64} {12.64} {12.64} {12.64} {12.64} {12.64} {12.64}
Note: [ ]: results from N ˆ
z 3 ; ( ): results from N ˆ
z 4 ; { }: results from N ˆ
z 5
78
79
Polynomial Solution Size (I× J) @
文獻值
80
表 3.9 (續上頁)
Mode No.
Crack Functions
（N ）
Polynomial Solution Size (I× J) @
文獻值
3×3 4×4 5×5 6×6 7×7 8×8 9×9
5
0
[19.48] [19.48] [19.48] [19.48] [19.48] [19.48] [19.48]
- (19.48) (19.48) (19.48) (19.48) (19.48) (19.48) (19.48)
{19.48} {19.48} {19.48} {19.48} {19.48} {19.48} {19.48}
2
[19.48] [19.48] [19.48] [19.48] [19.48] [19.48] [19.48]
(19.48) (19.48) (19.48) (19.48) (19.48) (19.48) (19.48) {19.48} {19.48} {19.48} {19.48} {19.48} {19.48} {19.48}
3
[19.48] [19.48] [19.48] [19.48] [19.48] [19.48] [19.48]
(19.48) (19.48) (19.48) (19.48) (19.48) (19.48) (19.48) {19.48} {19.48} {19.48} {19.48} {19.48} {19.48} {19.48}
4
[19.48] [19.48] [19.48] [19.48] [19.48] [19.48] [19.48]
(19.48) (19.48) (19.48) (19.48) (19.48) (19.48) (19.48) {19.48} {19.48} {19.48} {19.48} {19.48} {19.48} {19.48}
5
[19.48] [19.48] [19.48] [19.48] [19.48] [19.48] [19.48]
(19.48) (19.48) (19.48) (19.48) (19.48) (19.48) (19.48) {19.48} {19.48} {19.48} {19.48} {19.48} {19.48} {19.48}
Note: [ ]: results from N ˆ
z 3 ; ( ): results from N ˆ
z 4 ; { }: results from N ˆ
z 5
*：Stahl and Keer (1972)之結果
+：Huang et al. (2011) 之結果
81
82
Polynomial Solution Size (I× J) @
3×3 4×4 5×5 6×6 7×7 8×8 9×9
83
表 3.10 (續上頁)
Mode No.
Crack Functions
（N）
Polynomial Solution Size (I× J) @
3×3 4×4 5×5 6×6 7×7 8×8 9×9
5
0
[6.377] [5.793] [5.139] [5.112] [5.077] [5.074] [5.070]
(6.324) (5.733) (5.094) (5.066) (5.033) (5.031) (5.028) {6.322} {5.731} {5.093} {5.064} {5.032} {5.030} {5.026}
2
[5.395] [5.322] [5.066] [5.055] [5.026] [5.024] [5.019]
(5.352) (5.272) (5.020) (5.009) (4.983) (4.981) (4.977) {5.351} {5.270} {5.019} {5.008} {4.982} {4.980} {4.976}
3
[5.065] [5.059] [5.030] [5.028] [5.020] [5.019] [5.017]
(5.022) (5.016) (4.987) (4.985) (4.978) (4.977) (4.975) {5.021} {5.015} {4.986} {4.984} {4.977} {4.975} {4.973}
4
[5.033] [5.030] [5.020] [5.018] [5.015] [5.013] [5.012]
(4.990) (4.986) (4.977) (4.975) (4.972) (4.971) (4.970) {4.988} {4.985} {4.976} {4.974} {4.971} {4.970} {4.968}
5
[5.017] [5.016] [5.014] [5.013] [5.010] [5.010] [5.009]
(4.975) (4.974) (4.971) (4.971) (4.968) (4.968) (4.967) {4.973} {4.972} {4.970} {4.969} {4.967} {4.967} {4.966}
Note: [ ]: results from N ˆ
z 3 ; ( ): results from N ˆ
z 4 ; { }: results from N ˆ
z 5
84
85
Polynomial Solution Size (I× J) @
3×3 4×4 5×5 6×6 7×7 8×8 9×9
86
表 3.11 (續上頁)
Mode No.
Crack Functions
（N）
Polynomial Solution Size (I× J) @
3×3 4×4 5×5 6×6 7×7 8×8 9×9
5
0
[6.377] [5.793] [5.139] [5.112] [5.077] [5.074] [5.070]
(6.324) (5.733) (5.094) (5.066) (5.033) (5.031) (5.028) {6.322} {5.731} {5.093} {5.064} {5.032} {5.030} {5.026}
2
[5.091] [5.038] [4.881] [4.872] [4.849] [4.846] [4.840]
(5.041) (4.986) (4.839) (4.829) (4.806) (4.803) (4.797) {5.040} {4.984} {4.838} {4.827} {4.804} {4.802} {4.796}
3
[4.832] [4.819] [4.799] [4.791] [4.781] [4.779] [4.777]
(4.790) (4.776) (4.757) (4.749) (4.739) (4.738) (4.736) {4.789} {4.775} {4.756} {4.748} {4.738} {4.737} {4.735}
4
[4.801] [4.789] [4.779] [4.777] [4.775] [4.774] [4.772]
(4.759) (4.747) (4.738) (4.736) (4.734) (4.733) (4.732) {4.758} {4.746} {4.737} {4.735} {4.733} {4.732} {4.730}
5
[4.789] [4.781] [4.774] [4.773] [4.772] [4.771] [4.771]
(4.748) (4.740) (4.733) (4.732) (4.731) (4.730) (4.730) {4.747} {4.739} {4.732} {4.731} {4.730} {4.729} {4.729}
Note: [ ]: results from N ˆ
z 3 ; ( ): results from N ˆ
z 4 ; { }: results from N ˆ
z 5
87
88
Polynomial Solution Size (I× J) @
3×3 4×4 5×5 6×6 7×7 8×8 9×9
89
表 3.12 (續上頁)
Mode No.
Crack Functions
（N ）
Polynomial Solution Size (I× J) @
3×3 4×4 5×5 6×6 7×7 8×8 9×9
5
0
[9.793] [8.894] [7.880] [7.838] [7.780] [7.777] [7.770]
(9.744) (8.839) (7.841) (7.798) (7.744) (7.741) (7.734) {9.744} {8.839} {7.841} {7.798} {7.744} {7.741} {7.733}
2
[8.276] [8.167] [7.769] [7.750] [7.705] [7.701] [7.694]
(8.238) (8.121) (7.728) (7.710) (7.667) (7.663) (7.657) {8.238} {8.121} {7.728} {7.710} {7.667} {7.663} {7.657}
3
[7.764] [7.756] [7.712] [7.709] [7.696] [7.694] [7.690]
(7.727) (7.718) (7.674) (7.671) (7.659) (7.657) (7.653) {7.727} {7.717} {7.673} {7.670} {7.658} {7.656} {7.653}
4
[7.715] [7.710] [7.695] [7.691] [7.686] [7.685] [7.681]
(7.677) (7.672) (7.657) (7.654) (7.650) (7.648) (7.645) {7.677} {7.672} {7.657} {7.654} {7.649} {7.647} {7.645}
5
[7.690] [7.688] [7.685] [7.684] [7.679] [7.679] [7.677]
(7.653) (7.651) (7.648) (7.647) (7.643) (7.642) (7.641) {7.653} {7.651} {7.648} {7.647} {7.643} {7.642} {7.641}
Note: [ ]: results from N ˆ
z 3 ; ( ): results from N ˆ
z 4 ; { }: results from N ˆ
z 5
90
91
Polynomial Solution Size (I× J) @
3×3 4×4 5×5 6×6 7×7 8×8 9×9
92
表 3.13 (續上頁)
Mode No.
Crack Functions
（N ）
Polynomial Solution Size (I× J) @
3×3 4×4 5×5 6×6 7×7 8×8 9×9
5
0
[9.793] [8.894] [7.880] [7.838] [7.780] [7.777] [7.770]
(9.744) (8.839) (7.841) (7.798) (7.744) (7.741) (7.734) {9.744} {8.839} {7.841} {7.798} {7.744} {7.741} {7.733}
2
[7.813] [7.732] [7.486] [7.472] [7.436] [7.432] [7.422]
(7.768) (7.685) (7.449) (7.434) (7.398) (7.394) (7.384) {7.768} {7.685} {7.448} {7.433} {7.397} {7.393} {7.384}
3
[7.409] [7.389] [7.358] [7.346] [7.329] [7.326] [7.322]
(7.372) (7.352) (7.321) (7.308) (7.293) (7.289) (7.286) {7.372} {7.351} {7.321} {7.308} {7.292} {7.289} {7.286}
4
[7.361] [7.342] [7.327] [7.323] [7.320] [7.319] [7.315]
(7.324) (7.305) (7.291) (7.287) (7.284) (7.283) (7.280) {7.324} {7.305} {7.291} {7.287} {7.284} {7.282} {7.279}
5
[7.341] [7.328] [7.318] [7.317] [7.315] [7.314] [7.313]
(7.305) (7.292) (7.282) (7.281) (7.279) (7.278) (7.277) {7.305} {7.292} {7.282} {7.280} {7.279} {7.278} {7.277}
Note: [ ]: results from N ˆ
z 3 ; ( ): results from N ˆ
z 4 ; { }: results from N ˆ
z 5
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
y
a b
x
d
r1
x
0y
0r2
1
2
(a)板上視圖
x z
陶瓷面
金屬面
2 h
2 h
(b)板側視圖
圖 2.1 具內部裂縫矩形板示意圖
138
y
a b
x
A
1
2
B
圖 2.2 具內部裂縫奇異點與連續線段示意圖
139
圖 4.1 Al/ 功能梯度材料參數 E(z)和 (z)沿厚度變化圖根據式(2.1)
140
m ˆ
x0/a,y0/b d/a Mode Number
1 2 3 4 5
0
0
(5.965) (14.88) (14.88) (23.76) (29.66)
(0.5,0.5)
0.3
(5.665) (14.58) (14.84) (23.68) (26.71)
(0.5,0.75)
(5.789) (13.67) (14.85) (23.63) (27.71)
5
0
(3.925) (9.787) (9.787) (15.62) (19.49)
(0.5,0.5)
0.3
(3.725) (9.581) (9.760) (15.56) (17.53)
(0.5,0.75)
(3.808) (8.984) (9.766) (15.53) (18.19)
圖 4.2 具不同內部裂縫簡支之均質與 Al/Al2O3 FGM 方形薄板 2D 模態圖 (h/b=0.02)
0
141
圖 4.3 具不同水平內部裂縫簡支之方形均質中厚板模態圖
b
h / d / a
1 2
Mode
3 4 5
0.1 0
(5.777) (13.81) (13.81) (19.48) (19.48)
142
圖 4.3 (續上頁)
b
h / d / a
1 2
Mode
3 4 5
0.2 0
(5.304) (9.742) (9.742) (11.65) (11.65)
143
圖 4.3 (續上頁) (
x
0/ a , y
0/ b 0 . 5 , 0 . 5
)b
h / d / a
1 2
Mode
3 4 5
0.1 0.3
(5.421) (13.22) (13.76) (19.46) (19.48)
144
圖 4.3 (續上頁)
b
h / d / a
1 2
Mode
3 4 5
0.2 0.3
(4.959) (9.728) (9.742)
(10.84) (11.60)
145
圖 4.3 (續上頁)
b
h / d / a
1 2
Mode
3 4 5
0.1 0.5
(5.068) (11.10) (13.55) (19.27) (19.48)
146
圖 4.3 (續上頁) ((
x
0/ a , y
0/ b 0 . 5 , 0 . 75
))b
h / d / a
1 2
Mode
3 4 5
0.1 0.3
(5.547) (12.48) (13.74) (18.92) (19.46)
147
圖 4.3 (續上頁)
b
h / d / a
1 2
Mode
3 4 5
0.2 0.3
(5.059) (9.465) (9.728) (10.55) (11.58)
148
圖 4.3 (續上頁)
b
h / d / a
1 2
Mode
3 4 5
0.1 0.5
(5.199) (11.28) (13.57) (16.95) (17.60)
149
b
h / d / a
1 2
Mode
3 4 5
0.1
30
0.3(5.412) (13.21) (13.74) (19.45)
(19.48)
圖 4.3 (續上頁) (
x
0/ a , y
0/ b 0 . 5 , 0 . 5
)150
b
h / d / a
1 2
Mode
3 4 5
0.1
30
0.5(5.017) (11.10)
(13.45) (19.26)
(19.48)
圖 4.3 (續上頁)
151
b
h / d / a
1 2
Mode
3 4 5
0.1 0
(3.772) (8.929) (8.929) (12.64) (12.64)
圖 4.4 具不同水平內部裂縫簡支之 Al/Al2O3 方形 FGM 中厚板模態圖(
m ˆ 5
)152
b
h / d / a
1 2
Mode
3 4 5
0.2 0
(3.406) (6.296) (6.296) (7.347) (7.347)
圖 4.4 (續上頁)
153
b
h / d / a
1 2
Mode
3 4 5
0.1 0.3 (3.537) (8.525) (8.897)
(12.61) (12.64)
圖 4.4 (續上頁) (
x
0/ a , y
0/ b 0 . 5 , 0 . 5
)154
b
h / d / a
1 2
Mode
3 4 5
0.2 0.3
(3.185) (6.274) (6.296) (6.823) (7.322)
圖 4.4 (續上頁)
155
b h / d / a
1 2
Mode
3 4 5
0.1 0.5 (3.306) (7.125) (8.764) (12.47)
(12.64)
圖 4.4 (續上頁)
156
b
h / d / a
1 2
Mode
3 4 5
0.1 0.3
(3.619) (8.068) (8.883) (12.27) (12.61)
圖 4.4 (續上頁) (
x
0/ a , y
0/ b 0 . 5 , 0 . 75
)157
b
h / d / a
1 2
Mode
3 4 5
0.2 0.3
(3.247) (6.119) (6.279) (6.656) (7.303)
圖 4.4 (續上頁)
158
b
h / d / a
1 2
Mode
3 4 5
0.1 0.5
(3.389) (7.291) (8.771) (10.74) (11.63)
圖 4.4(續上頁)
159
b
h / d / a
1 2
Mode
3 4 5
0.1
30
0.3(3.531) (8.525) (8.888) (12.62)
(12.63)
圖 4.4 (續上頁) (
x
0/ a , y
0/ b 0 . 5 , 0 . 5
)160
b
h / d / a
1 2
Mode
3 4 5
0.1
30
0.5(3.272) (7.132)
(8.700) (12.48)
(12.61)
圖 4.4 (續上頁)
161
1 2Mode
3 4 5
90
(1.042) (2.446) (6.107) (6.601)
(7.732)
圖 4.5 具不同內部裂縫懸臂之方形均質中厚板模態圖(h/b=0.1)
162
1 2Mode
3 4 5
90
(1.016)
(2.233) (3.306) (5.361) (6.828)
圖 4.5 (續上頁) (h/b=0.2)
163
d / b
1 2
Mode
3 4 5
0
0.3(1.016) (2.195) (3.221) (5.359) (6.285)
圖 4.5 (續上頁) (h/b=0.2,
x
0/ a , y
0/ b 0 . 5 , 0 . 5
)164
d / b
1 2
Mode
3 4 5
90
0.3(1.020) (2.420) (5.727) (6.427) (7.567)
圖 4.5 (續上頁) (h/b=0.1,
x
0/ a , y
0/ b 0 . 5 , 0 . 5
)165
d / b
1 2
Mode
3 4 5
90
0.3(0.9897)
(2.195) (3.220) (5.010) (6.686)
圖 4.5 (續上頁) (h/b=0.2,
x
0/ a , y
0/ b 0 . 5 , 0 . 5
)166
d / b
1 2
Mode
3 4 5
90
0.5(0.9800) (2.398) (5.185)
(6.096) (7.398)
圖 4.5 (續上頁) (h/b=0.1,
x
0/ a , y
0/ b 0 . 5 , 0 . 5
)167
d / b
1 2
Mode
3 4 5
90
0.3(0.9660) (2.425) (6.032) (6.384) (7.634)
圖 4.5 (續上頁) (h/b=0.1,
x
0/ a , y
0/ b 0 . 25 , 0 . 5
)168
d / b
1 2
Mode
3 4 5
90
0.5(0.8639) (2.401) (5.754) (5.965) (7.360)
圖 4.5 (續上頁) (h/b=0.1,
x
0/ a , y
0/ b 0 . 25 , 0 . 5
)169
d / b
1 2
Mode
3 4 5
150
0.3(1.037) (2.403) (6.049) (6.426) (7.302)
圖 4.5 (續上頁) (h/b=0.1,
x
0/ a , y
0/ b 0 . 5 , 0 . 5
)170
d / b
1 2
Mode
3 4 5
150
0.5(1.027) (2.341) (5.991) (6.092) (6.812)
圖 4.5 (續上頁) (h/b=0.1,
x
0/ a , y
0/ b 0 . 5 , 0 . 5
)171
1 2Mode
3 4 5
90
(0.6844) (1.595) (3.969) (4.287)
(5.026)
圖 4.6 具不同內部裂縫懸臂之 Al/Al2O3 方形 FGM 中厚板模態圖(
m ˆ 5
, h/b=0.1)172
1 2Mode
3 4 5
90
(0.6637)
(1.432) (2.154) (3.396) (4.347)
圖 4.6 (續上頁) (