本文應用Biot多孔彈性理論與古典板理論，推導含飽和流體多孔彈性板的彎曲振動統御方程組。再於頻域中應用Galerkin型態有限元素法推導多孔彈性板三角形與四邊形元素的剛性矩陣及荷重向量。並藉由脈衝負荷作用與彈性邊界限制完成多孔彈性板的彎曲振動有限元素頻域分析，探討含飽和流體多孔彈性圓板的脈衝響應。
透過分析彈性板的無因次頻率係數及驗證前人發表的多孔彈性板數值與實驗結果，顯示有限元素頻域分析確可準確模擬多孔彈性圓板因材質、負荷及邊界限制等變化影響的彎曲振動行為。含飽和流體多孔彈性圓板具有流體與固體架構交互作用的動態消散特性，由多孔圓板撓度振幅的衰減可發現流體黏滯係數愈大其振幅衰減愈顯著，流體體積模數增加亦顯著提升多孔圓板的基礎自然頻率，藉由流體的改變可調整多孔彈性圓板的基礎自然頻率與撓度。最後研究應用無因次有限元素頻域分析探討各無因次參數變異於邊界固定支撐多孔彈性圓板脈衝響應的影響。結果顯示無因次固體有效密度及固體與流體的耦合有效密度分別對基礎自然頻率及撓度振幅的影響最為顯著。

In this study, Biot’s poroelastic theory and classical plate theory are applied to derive the governing equations of flexural vibrations of fluid-saturated poroelastic plates. The Galerkin type finite element approach is applied to derive the stiffness matrices and load vectors of triangular as well as quadrilateral poroelastic plate elements in the frequency domain. After applying the impulsive loadings and adjusting the elastic restraints, the finite element frequency domain analysis of flexural vibrations of poroelastic plates can thus be accomplished. The impulse responses of fluid-saturated poroelastic circular plates are explored.

Upon examining the results of the non-dimensional frequency parameters of elastic plates as well as the numerical and experimental results of poroelastic plates published by other researchers, it is validated that the finite element frequency domain analysis can obtain accurate results which are influenced by material properties, loadings and boundary restraints for the flexural vibrations of poroelastic circular plates. A fluid-saturated poroelastic circular plate can present a dynamic dissipation effect owing to the interactions of the fluid and the solid skeleton. Upon examining the reduction in deflection amplitude of poroelastic circular plates, it is found that the dissipation effect is an increasing function of fluid’s viscosity, and the fundamental natural frequency is an increasing function of the bulk modulus of the fluid. Accordingly, the fundamental natural frequencies and the deflections of poroelastic circular plates can be adjusted by changing the properties of the saturated fluid. At the end of this study, the dimensionless finite element frequency domain analysis is applied to explore the influence of dimensionless parameters on the impulse responses of clamped poroelastic circular plates. The results indicated that the value of dimensionless effective mass of the solid has a pronounced effect on the fundamental natural frequency, and the value of dimensionless mass coupling parameter between the fluid and the solid has a pronounced effect on the deflection amplitude.

目錄
中文摘要 I
英文摘要 II
目  錄 IV
圖目錄 VI
表目錄 IX
第一章 緒論 1
1.1 前言 1
1.2 研究動機 3
1.3 文獻回顧 4
1.4 研究內容 11
第二章 含飽和流體多孔彈性板理論 13
2.1 多孔彈性力學理論 13
2.2 多孔材料係數 15
2.2.1 孔洞率 15
2.2.2 多孔材料有效密度 15
2.2.3 消散係數 16
2.2.4 空氣體積模數 16
2.2.5 彈性係數與材料係數的關係 16
2.3 多孔彈性板彎曲振動統御方程組 17
2.3.1 應力、應變與位移關係 18
2.3.2 古典板理論 21
2.3.3 古典板理論─能量法 24
2.3.4 平面應力理論 28
第三章 有限元素頻域分析 30
3.1 多孔彈性板三角形元素 31
3.2 多孔彈性板四邊形元素 35
第四章 結果與討論 40
4.1 彈性板無因次頻率係數分析 40
4.2 多孔方板有限元素頻域分析 46
4.3 多孔圓板有限元素頻域分析 52
第五章 無因次參數變異分析 60
5.1 多孔彈性板無因次方程式 60
5.2 多孔圓板無因次參數變異分析 61
5.2.1 無因次彈性係數變異的影響探討 66
5.2.2 無因次有效密度變異的影響探討 68
5.2.3 無因次消散係數變異的影響探討 70
5.2.4 無因次均佈負荷與流體壓力差變異的影響探討 70
第六章 結論與未來展望 72
5.1 結論 72
5.2 未來展望 75
參考文獻 76
符號索引 82


圖目錄
圖2-1 多孔彈性圓板承受脈衝負荷示意圖 18
圖2-2 多孔彈性板彎曲後x-z平面示意圖 21
圖2-3 多孔彈性板彎曲後y-z平面示意圖 21
圖2-4 多孔彈性圓板受彈性邊界限制示意圖 25
圖3-1 多孔彈性板三角形元素及其節點位移示意圖 31
圖3-2 三角形元素面積座標系示意圖 32
圖3-3 四邊形元素自然座標系示意圖 35
圖3-4 多孔彈性板四邊形元素及其節點位移示意圖 38
圖4-1 彈性方板四邊形板元素分割示意圖 42
圖4-2 彈性方板三角形板元素分割示意圖 43
圖4-3 彈性圓板三角形板元素分割示意圖 43
圖4-4 40×40個多孔板四邊形元素使用4×4個高斯積分點與40×40個多孔板矩形元素板中心點的撓度頻率響應誤差分析 48
圖4-5 40×40×2個多孔板三角形元素使用7個高斯積分點與40×40個多孔板矩形元素板中心點的撓度頻率響應誤差分析 49
圖4-6 40×40及50×50個多孔板四邊形元素使用4×4個高斯積分點板中心點的撓度頻率響應收斂性分析 49
圖4-7 四邊固定支撐含飽和空氣G-Foam方板受0.1 Pa均佈脈衝負荷板中心點的撓度頻率響應驗證 50
圖4-8 四邊固定支撐含飽和空氣Y-Foam方板受0.1 Pa均佈脈衝負荷板中心點的撓度頻率響應驗證 50
圖4-9 四邊固定支撐含飽和空氣Coustone方板受0.1 Pa均佈脈衝負荷板中心點的撓度頻率響應驗證 51
圖4-10 2768及3616個多孔板三角形元素使用7個高斯積分點板中心點的撓度頻率響應收斂性分析 54
圖4-11 邊界固定支撐含飽和空氣G-Foam圓板受0.1 Pa均佈脈衝負荷板中心點的撓度頻率響應驗證 54
圖4-12 邊界固定支撐含飽和空氣Y-Foam圓板受0.1 Pa均佈脈衝負荷板中心點的撓度頻率響應驗證 55
圖4-13 邊界固定支撐含飽和空氣Coustone圓板受0.1 Pa均佈脈衝負荷板中心點的撓度頻率響應驗證 55
圖4-14 邊界固定及簡支撐含飽和空氣G-Foam圓板受0.1 Pa均佈脈衝負荷板中心點的撓度頻率響應 56
圖4-15 邊界固定及簡支撐含飽和空氣Y-Foam圓板受0.1 Pa均佈脈衝負荷板中心點的撓度頻率響應 57
圖4-16 邊界固定支撐含飽和流體G-Foam圓板受0.1 Pa均佈脈衝負荷板中心點的撓度頻率響應 57
圖4-17 邊界固定支撐含飽和流體Y-Foam圓板受0.1 Pa均佈脈衝負荷板中心點的撓度頻率響應 58
圖4-18 邊界固定支撐含飽和流體G-Foam圓板受0.1 Pa均佈脈衝負荷板中心點的撓度時間響應 58
圖4-19 邊界固定支撐含飽和流體Y-Foam圓板受0.1 Pa均佈脈衝負荷板中心點的撓度時間響應 59
圖5-1 無因次彈性係數 變異於多孔圓板撓度頻率響應的影響 66
圖5-2 無因次彈性係數 變異於多孔圓板撓度頻率響應的影響 67
圖5-3 無因次彈性係數 變異於多孔圓板撓度頻率響應的影響 67
圖5-4 無因次有效密度 變異於多孔圓板撓度頻率響應的影響 68
圖5-5 無因次有效密度 變異於多孔圓板撓度頻率響應的影響 69
圖5-6 無因次有效密度 變異於多孔圓板撓度頻率響應的影響 69
圖5-7 無因次消散係數 變異於多孔圓板撓度頻率響應的影響 70
圖5-8 無因次均佈負荷 變異於多孔圓板撓度頻率響應的影響 71
圖5-9 無因次流體壓力差 變異於多孔圓板撓度頻率響應的影響 71


表目錄
表2-1  多孔彈性板彎曲振動統御方程組中抗撓剛度的差異	29
表4-1  四邊固定支撐彈性方板的無因次頻率係數	44
表4-2  四邊簡支撐彈性方板的無因次頻率係數	44
表4-3  邊界固定支撐彈性圓板的無因次頻率係數	45
表4-4  邊界簡支撐彈性圓板的無因次頻率係數	45
表4-5  含飽和流體多孔彈性板材料係數[23]	48
表4-6  四邊固定支撐含飽和空氣G-Foam方板的自然頻率	51
表4-7  四邊固定支撐含飽和空氣Y-Foam方板的自然頻率	51
表4-8  四邊固定支撐含飽和空氣Coustone方板的自然頻率	52
表4-9  邊界固定支撐含飽和空氣G-Foam圓板的自然頻率	56
表4-10 邊界固定支撐含飽和空氣Y-Foam圓板的自然頻率	56
表4-11 邊界固定支撐含飽和空氣Coustone圓板的自然頻率	56
表5-1  含飽和流體多孔彈性板材料係數[23][43][44]	62
表5-2  Biot多孔彈性係數、多孔材料有效密度及消散係數	63
表5-3  含飽和流體多孔彈性板無因次參數值	64
表5-4  無因次參數值變異範圍	65

參考文獻
1. M. A. Biot, “General Theory of Three-Dimensional Consolidation”, Journal of Applied Physics, Vol. 12, No. 2, pp. 155-164, 1941.
2. M. A. Biot, “Theory of Elasticity and Consolidation for a Porous Anisotropic Solid”, Journal of Applied Physics, Vol. 26, No. 2, pp. 182-185, 1955.
3. M. A. Biot, “Theory of Propagation of Elastic Waves in a Fluid-Saturated Porous Solid. I. Low-Frequency Range”, Journal of the Acoustical Society of America, Vol. 28, No. 2, pp. 168-178, 1956.
4. M. A. Biot, “Theory of Propagation of Elastic Waves in a Fluid-Saturated Porous Solid II. Higher Frequency Range”, Journal of the Acoustical Society of America, Vol. 28, No. 2, pp. 179-191, 1956.
5. M. A. Biot and D. G. Willis, “The Elastic Coefficients of the Theory of Consolidation”, Journal of Applied Mechanics, Vol. 24, pp. 594-601, 1957.
6. M. A. Biot, “Generalized Theory of Acoustic Propagation in Porous Dissipative Media”, Journal of the Acoustical Society of America, Vol. 34, No. 5, pp. 1254-1264, 1962.
7. M. A. Biot, “Mechanics of Deformation and Acoustic Propagation in Porous Media”, Journal of Applied Physics, Vol. 33, No. 4, pp. 1482-1498, 1962.
8. M. A. Biot, “Theory of Buckling of a Porous Slab and Its Thermoelastic Analogy”, Journal of Applied Mechanics, Vol. 31, No. 2, pp. 194-198, 1964.
9. H.-S. Tsay and H. B. Kingsbury, “Influence of Inertia and Dissipative Forces on the Dynamic Response of Poroelastic Materials”, International Journal of Solids and Structures, Vol. 29, No. 5, pp. 641-652, 1992.
10. F.-H. Yeh and H.-S. Tsay, “Dynamic Behavior of a Poroelastic Slab Subjected to Uniformly Distributed Impulsive Loading”, Computers and Structures, Vol. 67, No. 4, pp. 267-277, 1998.
11. H.-S. Tsay and F.-H. Yeh, “Frequency Response Function for Prediction of Planar Cellular Plastic Foam Acoustic Behavior”, Journal of Cellular Plastics, Vol. 41, No. 2, pp. 101-131, 2005.
12. H.-S. Tsay and F.-H. Yeh, “Finite Element Frequency-Domain Acoustic Analysis of Open-Cell Plastic Foams”, Finite Elements in Analysis and Design, Vol. 42, No. 4, pp. 314-339, 2006.
13. H.-S. Tsay and F.-H. Yeh, “Three-Dimensional Finite Element Frequency Domain Acoustic Analysis of Open Cell Plastics Foams”, Journal of Cellular Plastics, Vol. 42, No. 6, pp. 487-515, 2006.
14. H.-S. Tsay and F.-H. Yeh, “The Influence of Circumferential Edge Constraint on the Acoustical Properties of Open-Cell Polyurethane Foam Samples”, Journal of the Acoustical Society of America, Vol. 119, No. 5, pp. 2804-2814, 2006.
15. H.-S. Tsay, “Analysis of Normal Incidence Absorption of Pyramidal Polyurethane Foam by Three-Dimensional Finite Element Frequency Domain Acoustical Analysis”, Journal of the Acoustical Society of America, Vol. 120, No. 5, pp. 2686-2693, 2006.
16. H.-S. Tsay and F.-H. Yeh, “Optimal Design of Corrugated Cellular Plastic Foam with Semi-Elliptical Shaped Upper Surface by Finite Element Frequency Domain Acoustic Analysis”, Journal of Cellular Plastics, Vol. 43, No. 3, pp. 197-214, 2007.
17. H.-S. Tsay and F.-H. Yeh, “Analysis of Mode Shapes of a Rigidly Backed Cylindrical Foam Using Three-Dimensional Finite Element Acoustical Analysis”, Applied Acoustics, Vol. 69, No. 9, pp. 778-788, 2008.
18. L. A. Taber, “A Theory for Transverse Deflection of Poroelastic Plates”, Journal of Applied Mechanics, Vol. 59, No. 3, pp. 628-634, 1992.
19. D. D. Theodorakopoulos and D. E. Beskos, “Flexural Vibrations of Fissured Poroelastic Plates”, Archive of Applied Mechanics, Vol. 63, No. 6, pp. 413-423, 1993.
20. D. D. Theodorakopoulos and D. E. Beskos, “Flexural Vibrations of Poroelastic Plates”, Acta Mechanica, Vol. 103, No. 1-4, pp. 191-203, 1994.
21. L. P. Li, G. Cederbaum and K. Schulgasser, “Theory of Poroelastic Plates with In-Plane Diffusion”, International Journal of Solids and Structures, Vol. 34, No. 35-36, pp. 4515-4530, 1997.
22. P. Leclaire, K. V. Horoshenkov, and A. Cummings, “Transverse Vibrations of a Thin Rectangular Porous Plate Saturated by a Fluid”, Journal of Sound and Vibration, Vol. 247, No. 1, pp. 1-18, 2001.
23. P. Leclaire, K. V. Horoshenkov, M. J. Swift and D. C. Hothersall, “The Vibrational Response of a Clamped Rectangular Porous Plate”, Journal of Sound and Vibration, Vol. 247, No. 1, pp. 19-31, 2001.
24. M. Etchessahar, S. Sahraoui and B. Brouard, “Bending Vibrations of a Rectangular Poroelastic Plate”, Comptes Rendus de l'Academie de Sciences, Vol. 329, No. 8, pp. 615-620, 2001.
25. M. Schanz and A. Busse, “Acoustic Behavior of a Poroelastic Mindlin Plate”, 17th ASCE Engineering Mechanics Conference, 2004.
26. P. H. Wen and Y. W. Liu, “The Fundamental Solution of Poroelastic Plate Saturated by Fluid and Its Applications”, International Journal for Numerical and Analytical Methods in Geomechanics, Vol. 34, No. 7, pp. 689-709, 2010.
27. P. H. Wen, “The Analytical Solutions of Incompressible Saturated Poroelastic Circular Mindlin's Plate”, Journal of Applied Mechanics, Vol. 79, No. 5, 2012.
28. J. Sladek, V. Sladek, M. Gfrerer and M. Schanz, “Mindlin Theory for the Bending of Porous Plates”, Acta Mechanica, Vol. 226, No. 6, pp. 1909-1928, 2014.
29. A. S. Rezaei and A. R. Saidi, “Exact Solution for Free Vibration of Thick Rectangular Plates Made of Porous Materials”, Composite Structures, Vol. 134, No. 15, pp. 1051-1060, 2015.
30. 蔡慧駿、葉豐輝、陳央澤、廖聖善，“應用有限元素頻域法於含飽和流體多孔薄板之彎曲振動分析”，中華民國力學學會第三十四屆全國力學會議論文集，2010。
31. 蔡慧駿、葉豐輝、林益誠，“具橫向加強肋多孔板受彈性支撐之動態反應”，中華民國力學學會第三十五屆全國力學會議論文集，2011。
32. 蔡慧駿、葉豐輝、吳鑄城，“多孔板與聲場耦合之有限元素頻域分析”，中華民國力學學會第三十六屆全國力學會議論文集，2012。
33. 蔡慧駿、葉豐輝、陳正輝，“含飽和流體多孔彈性圓板之脈衝響應”，中華民國力學學會第三十九屆全國力學會議論文集，2015。
34. G. Kirchhoff, “Über das Gleichgewicht und die Bewegung einer elastischen Scheibe”, Journal für die Reine und Angewandte Mathematik, Vol. 40, pp. 51-88, 1850.
35. E. Reissner, “The Effect of Transverse Shear Deformation on the Bending of Elastic Plates”, Journal of Applied Mechanics, Vol. 12, pp. 69-77, 1945.
36. R. D. Mindlin, “Influence of Rotatory Inertia and Shear on Flexural Motions of Isotropic, Elastic Plates”, Journal of Applied Mechanics, Vol. 18, pp. 31-38, 1951.
37. J. N. Reddy, “A Simple Higher-Order Theory for Laminated Composite Plates”, Journal of Applied Mechanics, Vol. 51, No. 4, pp. 745-752, 1984.
38. G. P. Bazeley, Y. K. Cheung, B. M. Irons and O. C. Zienkiewicz, “Triangular Elements in Plate Bending - Conforming and Non-Conforming Solutions”, Proceedings of the First Conference on Matrix Methods in Structural Mechanics, pp. 547-576, 1965.
39. A. W. Leissa, “The Free Vibration of Rectangular Plates”, Journal of Sound and Vibration, Vol. 31, No. 3, pp. 257-293, 1973.
40. C. Rajalingham, R. B. Bhat and G. D. Xistris, “Vibration of Rectangular Plates Using Plate Characteristic Functions as Shape Functions in the Rayleigh-Ritz Method”, Journal of Sound and Vibration, Vol. 193, No. 2, pp. 497-509, 1996.
41. J. R. Airey, “The Vibrations of Circular Plates and Their Relation to Bessel Functions”, Proceedings of the Physical Society of London, Vol. 23, No. 1, pp. 225-232, 1910.
42. A. W. Leissa, “Natural Frequencies of Simply Supported Circular Plates”, Journal of Sound and Vibration, Vol. 70, No. 2, pp. 221-229, 1980.
43. D. L. Johnson, T. J. Plona and H. Kojima, “Probing Porous Media with First and Second Sound. II. Acoustic Properties of Water-Saturated Porous Media”, Journal of Applied Physics, Vol. 76, No. 1, pp. 115-125, 1994.
44. B.-N. Kim, K. I. Lee and S. W. Yoon, “Phase Velocity and Attenuation of Acoustic Waves in Water-Saturated Sandy Sediment: Application of Biot's Theory”, Journal of the Korean Physical Society, Vol. 44, No. 6, pp. 1442-1448, 2004.
45. T. R. Chandrupatla and A. D. Belegundu, “Introduction to Finite Elements in Engineering”, Prentice Hall, New Jersey, 2002.
46. A. W. Leissa, “Vibration of a Simply-Supported Elliptical Plate”, Journal of Sound and Vibration, Vol. 6, No. 1, pp. 145-148, 1967.
47. Y. Narita and A. W. Leissa, “Transverse Vibration of Simply Supported Circular Plates Having Partial Elastic Constraints”, Journal of Sound and Vibration, Vol. 70, No. 1, pp. 103-116, 1980.
48. B. Singh and S. Chakraverty, “Transverse Vibration of Circular and Elliptic Plates with Quadratically Varying Thickness”, Applied Mathematical Modelling, Vol. 16, No. 5, pp. 269-274, 1992.
49. C.-F. Liu and G.-T. Chen, “A Simple Finite Element Analysis of Axisymmetric Vibration of Annular and Circular Plates”, International Journal of Mechanical Sciences, Vol. 37, No. 8, pp.861-871, 1995.
50. M. Sathyamoorthy, “Influence of Transverse Shear and Rotatory Inertia on Nonlinear Vibrations of Circular Plates”, Computers and Structures, Vol. 60, No. 4, pp. 613-618, 1996.
51. H. S. Yalcin, A. Arikoglu and I. Ozkol, “Free Vibration Analysis of Circular Plates by Differential Transformation Method”, Applied Mathematics and Computation, Vol. 212, No. 2, pp. 377-386, 2009.
52. Z. H. Zhou, K W. Wong, X. S. Xu and A. Y. T. Leung, “Natural Vibration of Circular and Annular Thin Plates by Hamiltonian Approach”, Journal of Sound and Vibration, Vol. 330, No. 5, pp. 1005-1017, 2011.
53. X. Liu and J. R. Banerjee, “Free Vibration Analysis for Plates with Arbitrary Boundary Conditions Using a Novel Spectral-Dynamic Stiffness Method”, Computers and Structures, Vol. 164, pp. 108-126, 2016.