System No. U0002-0109201513201800 含飽和流體多孔柱之步階響應 Step Response of a Fluid-Saturated Porous Column 淡江大學 機械與機電工程學系碩士班 Department of Mechanical and Electro-Mechanical Engineering 103 2 104 林玟萱 Wen-Xuan Lin 601371155 碩士 Traditional Chinese 2015-06-26 66page advisor - 葉豐輝 co-chair - 柯德祥 co-chair - 李經綸 多孔柱 位移 Biot理論 有限元素頻域分析 Porous Column Displacement Biot Theory Finite Element Frequency-Domain Analysis ```本文旨在探討含飽和流體多孔柱之步階響應分析，所應用的分析方法有二，第一種方法使用拉普拉斯轉換後之Biot動態統御方程組，給予材料參數及邊界條件，直接解析算出多孔柱在上表面中心點位移頻域函數之理論解，第二種方法乃應用有限元素頻域分析(FEFDA)，執行多孔柱之動態分析，兩種方法所求得頻域的位移解，可藉由傅立葉轉換，而得到時域的位移結果。 研究模擬首先驗證分析方法之正確性，使用Biot動態統御方程組頻域理論解與有限元素頻域分析，進行含飽和水多孔砂岩柱之動態行為分析，所得到上邊界垂直位移曲線與前人所發表之結果一致。其次透過幾何或參數之變化，利用二維有限元素頻域分析，探討含飽和流體多孔柱動態行為之影響因子。由於多孔柱孔洞中含有流體在與固體相互作用下，會具有特殊的動態消散特性，分析結果顯示流體黏滯係數越大或孔洞率提高，消散係數與位移都會相對增加，故藉由流體的變化及孔洞率的改變，可精準調整含飽和流體多孔柱的動態行為反應。``` ```This thesis aims to explore the step response of a fluid-saturated porous column. Two analysis methods are applied. The first method involves the use of the Biot’s dynamic governing equation after Laplace transformation with material parameters and boundary conditions to directly compute the theoretical solution of midpoint displacement function in frequency domain. The second method involves the Finite Element Frequency Domain Analysis (FEFDA) applied to execute porous column dynamic analysis. The displacement solution of frequency domain obtained using the two methods then utilizes Fourier transformation to obtain the time domain displacement results. In the study, the analysis methods were first verified for correctness. Through the theoretical solution and FEFDA of Biot’s dynamic governing equation in the frequency domain, the dynamic behavioral analysis was conducted on the water-saturated porous sandstone column to obtain the top boundary vertical displacement curve coincided with the results published by predecessors. Secondly, through geometric or parameter changes, two-dimensional FEFDA was adopted to explore the impact factors contributing to the dynamic behavior of a fluid-saturated porous column. Since the porous column containing fluid, through interaction with solids, display the special characteristic of dynamic dissipation, the analysis results show that the higher the viscosity coefficient of fluid or porosity, the higher the dissipation coefficient and displacement. Therefore, fluid or porosity changes can facilitate the accurate adjustment of dynamic behavioral response of a fluid-saturated porous column.``` ```目 錄 中文摘要 I 英文摘要 II 目 錄 III 圖目錄 V 表目錄 VIII 第一章 緒論 1 1.1 前言 1 1.2 研究動機與目的 2 1.3 文獻回顧 3 1.4 研究內容 5 第二章 多孔柱之統御方程組與參數簡介 7 2.1 多孔柱統御方程組 7 2.1.1 應力、應變與位移關係 8 2.1.2 應力應變函數關係 9 2.1.3 動能與消耗能量 11 2.1.4 多孔柱之統御方程組 12 2.2 多孔柱參數 14 2.2.1 孔洞係數 14 2.2.2 多孔柱與Biot彈性係數之關係 15 2.2.3 結構因子 17 2.2.4 動態消散係數 18 2.2.5 空氣之體積模數 18 第三章 有限元素頻域分析 20 3.1頻域多孔柱表面位移函數解析解 20 3.2多孔柱之二維有限元素頻域分析 27 3.2.1多孔柱二維矩形元素 27 第四章 結果與討論 31 4.1多孔柱之階響應 31 4.1.1含飽和流體多孔柱頻域位移步階響應之驗證 31 4.1.2含飽和流體多孔柱時域位移步階響應之驗證 35 4.2多孔柱參數異變之影響 42 4.2.1孔洞係數的影響 42 4.2.2動態消散係數的影響 43 4.3多孔柱二維有限元素頻域分析 44 4.3.1高度網格數目多寡之變化 44 4.3.2高度及寬度網格數目多寡之變化 46 4.3.3同固體架構流體變化之比較 49 4.3.4流體相同固體架構變化之比較 52 4.3.5孔洞率之影響 53 4.3.6受力變化之影響 54 第五章 結論與未來展望 60 5.1 結論 60 5.2 未來展望 61 參考文獻 63 圖目錄 圖3-1 上表面為可穿透表面受均勻壓力之多孔柱示意圖 25 圖3-2 多孔柱矩形元素位移示意圖 28 圖3-3 多孔柱矩形元素之直角座標系示意圖 29 圖4-1 多孔柱應用二維有限元素分析之邊界示意圖 32 圖4-2 底邊固定之含飽和水多孔砂岩柱受1Pa均勻負荷後上表面之中心點位移頻域響應圖 33 圖4-3 底邊固定之含飽和空氣多孔砂岩柱受1Pa均勻負荷後上表面之中心點位移頻域響應圖 34 圖4-4 Bettina Albers 等人發表之結果 [19] 36 圖4-5 含飽和水多孔砂岩柱受1 Pa均佈壓力之時域位移圖 36 圖4-6 Biot理論解、二維有限元素頻域分析及Bettina Albers 等人發表之結果[19]之比較 37 圖4-7 含飽和空氣多孔砂岩柱之時域位移圖 38 圖4-8 含飽和空氣多孔氧化鋁柱之時域位移 40 圖4-9 含飽和空氣多孔泡棉柱之時域位移 40 圖4-10 含飽和水多孔泡棉柱之時域位移 41 圖4-11 含飽和水多孔砂岩柱在0.1秒之時域位移 41 圖4-12 孔洞率影響動態消散係數之關係圖 42 圖4-13 含飽和水多孔砂岩柱之消散係數與頻率關係圖 43 圖4-14 含飽和空氣多孔砂岩柱之消散係數與頻率關係圖 44 圖4-15 含飽和水多孔砂岩柱高度網數目格多寡之上表面中心點位移比較 45 圖4-16 含飽和空氣多孔砂岩柱高度網格數目多寡之上表面中心點位移比較 46 圖4-17 含飽和空氣多孔泡棉柱高度及寬度網格數目多寡之上表面中心點位移比較圖 47 圖4-18 含飽和空氣多孔氧化鋁柱高度及寬度網格數目多寡之上表面中心點位移比較 48 圖4-19 含飽和流體多孔泡棉柱之上表面均勻壓力0.1Pa的中心點位移圖 50 圖4-20 含飽和流體多孔砂岩柱之上表面平均壓力0.1Pa的中心點位移圖 51 圖4-21 含飽和空氣多孔氧化鋁柱及泡棉柱之上表面平均壓力0.1Pa的中心點位移比較圖 52 圖4-22 含飽和空氣多孔氧化鋁柱變化孔洞率之位移影響 53 圖4-23 含飽和空氣多孔砂岩柱之上表面均勻壓力不同的中心點頻率位移圖 55 圖4-24 含飽和水多孔砂岩柱之上表面均勻壓力不同的中心點頻率位移圖 56 圖4-25 含飽和空氣多孔泡棉柱及氧化鋁柱之上表面均勻壓力不同的中心點頻率位移圖 57 圖4-26 含飽和空氣多孔砂岩柱之上表面均勻壓力不同之中心點時域位移圖 58 圖4-27 含飽和水多孔砂岩柱之上表面均勻壓力不同之中心點時域位移圖 58 圖4-28 底含飽和空氣多孔泡棉柱及氧化鋁柱之上表面均勻壓力不同之中心點時域位移圖 59 表目錄 表4-1 多孔砂岩柱材料參數 31 表4-2 流體參數 32 表4-3 含飽和水多孔砂岩柱上表面中心點位移數值整理之結果 37 表4-4 多孔氧化鋁及泡棉材料參數 39 表4-5 含飽和空氣多孔泡棉柱上表面中心點位移結果 47 表4-6 含飽和空氣多孔泡棉柱上表面中心點位移之相對誤差之結果 47 表4-7 含飽和空氣多孔泡棉柱上表面中心點位移結果 48``` ```參考文獻 1. Somiya, S. (2013). Handbook of advanced ceramics: Materials, applications, processing, and properties. Waltham, Massachusetts, USA: Academic Press. 2. Tsay, H.-S., & Yeh, F.-H. (2005). Frequency response function for prediction of planar cellular plastic foam acoustic behavior. Journal of Cellular Plastics, 41(2), 101-131. 3. 陳央澤（民 97）。應用頻域有限元素法於含飽和液體多孔樑之彎曲振動分析（碩士論文）。取自臺灣博碩士論文系統。（系統編號 097TKU05489005） 4. 廖聖善（民 99）。多孔陶瓷材料之製備與彎曲振動分析（碩士論文）。取自淡江大學電子學位論文服務。（系統編號 U0002-3108201010364400） 5. 吳鑄城（民 102）。加肋多孔板與聲場耦合之脈衝響應（碩士論文）。取自淡江大學電子學位論文服務。（系統編號 U0002-2602201322574100） 6. 顧庭碩（民 98）。應用 Biot 理論於海綿骨之相速度與聲響性質分析（碩士論文）。取自淡江大學電子學位論文服務。（系統編號 U0002-1002200922380700） 7. Biot, M. A. (1941). General theory of three‐dimensional consolidation. Journal of Applied Physics, 12(2), 155-164. 8. Biot, M. A. (1955). Theory of elasticity and consolidation for a porous anisotropic solid. Journal of Applied Physics, 26(2), 182-185. 9. Biot, M. A. (1956). Theory of propagation of elastic waves in a fluid‐saturated porous solid. II. higher frequency range. The Journal of the Acoustical Society of America, 28(2), 179-191. 10. Biot, M. A. (1964). Theory of buckling of a porous slab and its thermoelastic analogy. Journal of Applied Mechanics, 31(2), 194-198. 11. Allard, J. F., Depollier, C., & L’Esperance, A. (1986). Observation of the biot slow wave in a plastic foam of high flow resistance at acoustical frequencies. Journal of Applied Physics, 59(10), 3367-3370. 12. Gibson, L. J. (1985). The mechanical behaviour of cancellous bone. Journal of Biomechanics, 18(5), 317-328. 13. Biot, M. A. (1956). Theory of propagation of elastic waves in a fluid‐saturated porous solid. I. Low‐frequency range. The Journal of the Acoustical Society of America, 28(2), 168-178. 14. Zwikker, C., & Kosten, C. W. (1949). Sound absorbing materials. London, United Kingdom: Elsevier publishing company, INC. 15. Terzaghi, K. (1943). Theoretical soil mechanics. New York, USA: John Wiley and Sons, INC. 16. Craggs, A., & Hildebrandt, J. G. (1984). Effective densities and resistivities for acoustic propagation in narrow tubes. Journal of Sound and Vibration, 92(3), 321-331. 17. Craggs, A., & Hildebrandt, J. G. (1986). The normal incidence absorption coefficient of a matrix of narrow tubes with constant cross-section. Journal of Sound and Vibration, 105(1), 101-107. 18. Chandrupatla, T. R., & Belegundu, A. D. (2002). Introduction to finite elements in engineering. New Jersey, USA: Prentice Hall. 19. Albers, B., Savidis, S. A., Taşan, H. E., Estorff, O. V., & Gehlken, M. (2012). BEM and FEM results of displacements in a poroelastic column. International Journal of Applied Mathematics and Computer Science, 22(4), 883-896. 20. Eringen, A. C., & Suhubi, S.S., (1975). Elastodynamics. New York, USA: Academic Press. 21. Korsawe, J., Starke, G., Wang, W., & Kolditz, O. (2006). Finite element analysis of poro-elastic consolidation in porous media: Standard and mixed approaches. Computer Methods in Applied Mechanics and Engineering, 195(9), 1096-1115. 22. Vgenopoulou, I., & Beskos, D. E. (1992). Dynamics of saturated rocks. IV: Column and borehole problems. Journal of Engineering Mechanics, 118(9), 1795-1813. 23. Tsay, H.-S., & Kingsbury, H. B. (1992). Influence of inertia and dissipative forces on the dynamic response of poroelastic materials. International Journal of Solids and Structures, 29(5), 641-652. 24. Tsay, H.-S., & Yeh, F.-H. (2006). Finite element frequency-domain acoustic analysis of open-cell plastic foams. Finite Elements in Analysis and Design, 42(4), 314-339. 25. Tsay, H.-S., & Yeh, F.-H. (2008). Analysis of mode shapes of a rigidly backed cylindrical foam using three-dimensional finite element acoustical analysis. Applied Acoustics, 69(9), 778-788. 26. Depollier, C., Allard, J. F., & Lauriks, W. (1988). Biot theory and stress–strain equations in porous sound‐absorbing materials. The Journal of the Acoustical Society of America, 84(6), 2277-2279. 27. Stinson, M. R. (1991). The propagation of plane sound waves in narrow and wide circular tubes, and generalization to uniform tubes of arbitrary cross‐sectional shape. The Journal of the Acoustical Society of America, 89(2), 550-558. 28. Leclaire, P., Horoshenkov, K. V., & Cummings, A. (2001). Transverse vibrations of a thin rectangular porous plate saturated by a fluid. Journal of Sound and Vibration. Journal of Sound and Vibration, 247(1), 1-18. 29. Dureisseix, D., Ladevèze, P., & Schrefler, B. A. (2003). A computational strategy for multiphysics problems: Application to poroelasticity. International Journal for Numerical Methods in Engineering, 56(10), 1489-1510.``` Within Campus： I request to embargo my thesis/dissertation for 2 year(s) right after the date I submit my Authorization Approval Form.Duration for delaying release from 2 years. Outside the Campus： I grant the authorization for the public to view/print my electronic full text with royalty fee and I donate the fee to my school library as a development fund.Duration for delaying release from 2 years.