(3.238.235.155) 您好!臺灣時間:2021/05/16 06:36
字體大小: 字級放大   字級縮小   預設字形  
回查詢結果

詳目顯示:::

我願授權國圖
: 
twitterline
研究生:許仲磊
研究生(外文):Chung-LeiHsu
論文名稱:異向性磁電彈固體之邊界元素分析
論文名稱(外文):Boundary Element Analysis for Anisotropic Magneto-Electro-Elastic Solids
指導教授:胡潛濱夏育群夏育群引用關係
指導教授(外文):Chyanbin HwuYui-Chuin Shiah
學位類別:博士
校院名稱:國立成功大學
系所名稱:航空太空工程學系
學門:工程學門
學類:機械工程學類
論文種類:學術論文
論文出版年:2020
畢業學年度:108
語文別:英文
論文頁數:101
中文關鍵詞:三維異向性彈性力學異向性熱彈問題磁電彈材料史磋公式雷冬-史磋公式多值函數分支切割
外文關鍵詞:Anisotropic thermos-elasticity3D generally anisotropic elasticitymagneto-electro-elastic materialsStroh formalismboundary element methodRadon-Stroh formalismmulti-valued functionBranch cut
相關次數:
  • 被引用被引用:0
  • 點閱點閱:31
  • 評分評分:
  • 下載下載:0
  • 收藏至我的研究室書目清單書目收藏:0
本研究建立了可處理異向性彈性、壓電、壓磁和磁電彈材料的邊界元素法的理論與程式,並執行完整的二維和三維問題的靜態分析。其中包含二維和三維的基本解(格林函數)、計算邊界節點上的位移和曳引力的系統方程式、計算邊界節點上的應力和應變,以及內部點的位移、應力和應變。本文主要利用史磋公式來推導基本解和建立相關邊界元素法的程式。在史磋公式中,對壓電、壓磁和磁電彈材料皆使用與異向性彈性材料相同的數學式,只需根據材料性質改變相關矩陣和向量的維度即可。因為這個優點,用史磋公式建立邊界元程式在程式撰寫時具有相當大的優勢。
在二維邊界元素法中使用了傳統的(無限域問題)和針對孔洞問題的兩種建立在史磋公式中的格林函數。為了正確地使用含孔洞問題的格林函數,本文提出了修正分支切割的方法,用以解決計算此函數的多值問題。經過修正後,邊界元素法若採用含孔洞問題的基本解比起無限域問題的基本解,在孔洞上或孔洞附近區域的解會更加精確。涉及熱效應的二維問題中,熱問題的基本解可以經由複數型的無限域基本解推導而來。此外,複數型的無限域基本解可以透過史磋恆等式把複變數轉換成實變數。若採用實數型的基本解,可以避免複變數對數函數的多值問題和其在邊界積分式中為了修正連續性而使用的額外線積分。因此,完全不再需要提供任何邊界以外的域內溫度資料。成功地提供一個真正的邊界元素法來處理熱問題。此外,文中也介紹了生熱率非零的狀況。
三維實數型與複數型的格林函數是透過結合史磋公式和雷冬轉換(雷冬-史磋公式)的理論來獲得,第一次的完整提供了包含位移、曳引力和它們的一階導數的推導。為了比較兩者之間的效率,文中把計算兩種型態的三維格林函數的步驟一一列出。結果顯示,使用複數型的格林函數在整體邊界元素法的計算效率上比起實數型的格林函數來的更優。然而,單純計算格林函數的話,結果卻是相反的。
文章的最後,列舉了一些數值範例分別與解析解和市售的有限元素法軟體比較以驗證文中公式的正確性。
In this dissertation, we develop both two- and three-dimensional boundary element method (BEM) for generally anisotropic magneto-electro-elastic (MEE) solids. The system of algebraic equations to calculate the nodal displacements and nodal tractions at the boundary of two-dimensional (2D) and three-dimensional (3D) solids are formulated. The relations for calculating strains and stresses at the boundary nodes as well as at the interior points are also derived. The Green’s functions in our computer codes are derived by using Stroh formalism. In Stroh formalism, the solutions for the solids with these different material types, such as elastic, piezoelectric and MEE, all bear exactly the same mathematical forms distinguished by the contents and dimensions of the related matrices and vectors. This extension is usually called expanded Stroh formalism, and it provides a big advantage for computer programming.
In 2D analysis, we consider not only general mechanical problems but also the problems with thermal effects. The available Green’s functions for an infinite plate and for an infinite plate with a central hole/crack are extended to MEE plate. To use the Green’s function containing a hole/crack in BEM, a suitable branch cut setting is provided for the multi-valued complex logarithmic functions with the mapped variable to get the correct principal value from MATLAB. For 2D thermo-MEE, the complex variable Green’s function for an infinite plate is used to derive the thermal fundamental solution for 2D thermo-MEE analysis. By using the identities of Stroh formalism, the complex form solution can be converted into real form. With the real form fundamental solution, the trouble induced by the multi-valued complex logarithmic function is circumvented and the extra line integral appeared in the thermal analysis of BEM can be eliminated. Thus, the temperature information inside the domain required by the extra line integral can be avoided, and a truly boundary element method for thermo-MEE analysis is achieved. The influence of heat source represented by the heat generation rate is also considered in our formulation.
The 3D Green’s functions of displacements and tractions in both complex and real forms, together with their first derivatives, are derived completely by Radon-Stroh formalism, which combines expanded Stroh formalism and 2D Radon transform. The calculation processes of two types of three-dimensional Green’s functions are listed step by step for comparison of efficiency. We found that the 3D BEM with complex form solution is more efficient than that with real form solution, although the opposite performance was observed for a single point calculation of Green’s function.
To verify the correctness of our solutions, some numerical examples with simple geometries involving MEE, piezoelectric, piezomagnetic and elastic material properties are demonstrated.
摘要 i
ABSTRACT xi
LIST OF TABLES xv
LIST OF FIGURES xvi
NOMENCLATURE xviii
CHAPTER I. INTRODUCTION 1
CHAPTER II. ANISOTROPIC MAGNETO-ELECTRO-ELASTIC SOLIDS 6
2.1 Three dimensional constitutive laws 6
2.1.1 Isothermal condition 6
2.1.2 Thermal effect 9
2.2 Two dimensional constitutive laws 11
2.2.1 Isothermal condition 11
2.2.2 Thermal effect 14
2.3 Specialization 17
2.3.1 Anisotropic piezoelectric/piezomagnetic solids 17
2.3.2 Anisotropic elastic solids 20
CHAPTER III. COMPLEX VARIABLE FORMALISMS 22
3.1 Stroh formalism for 2D anisotropic elasticity 22
3.2 Expanded Stroh formalism 23
3.3 Radon-Stroh formalism for 3D anisotropic elasticity 26
CHAPTER IV. GREEN’S FUNCTION 32
4.1 Two-dimensional deformation 32
4.1.1 An infinite plate 32
4.1.2 An infinite plate containing a hole or crack 33
4.1.3 Evaluation of multi-valued logarithmic functions 34
4.2 Three-dimensional deformation 38
4.2.1 An infinite solid 38
4.2.2 Evaluations of Green's function and its derivatives 39
CHAPTER V. BOUNDARY ELEMENT METHOD 44
5.1 Two-dimensional deformation 44
5.1.1 Isothermal condition 44
5.1.2 Thermal effect 50
5.2 Three-dimensional deformation 57
5.2.1 Anisotropic elastic solids 57
5.2.2 Anisotropic MEE solids 62
CHAPTER VI. NUMERICAL EXAMPLES 63
6.1 Two-dimensional analyses 63
6.1.1 Isothermal condition 63
6.1.2 Thermal environment 64
6.2 Three-dimensional analyses 67
6.2.1 3D Green’s function 67
6.2.2 Examples for 3D-BEM analysis 69
CHAPTER VII. CONCLUSIONS 73
REFERENCES 75
TABLES 79
FIGURES 84
PUBLICATION LIST 101
[1]S.G. Lekhnitskii, Anisotropic plates, Gordon and Breach, New York, 1968.
[2]T.C.T. Ting, Anisotropic elasticity: theory and applications, Oxford University Press, New York, 1996.
[3]C. Hwu, Anisotropic elastic plates, Springer, London, 2010.
[4]D.M. Barnett, The precise evaluation of derivatives of the anisotropic elastic Green’s functions, Phys. Status Solidi B, 49 (1972) 741–748.
[5]T.C.T. Ting, V.G. Lee, The three-dimensional elastostic Green’s function for general anisotropic linear elastic solid, Q. J. Mech. Appl. Math. 50 (1997) 407–426.
[6]K.C. Wu, Generalization of the Stroh formalism to 3-Dimensional anisotropic elasticity, J. elast. 51 (1998) 213–225.
[7]F.C Buroni, M. Denda, Radon-Stroh formalism for 3D theory of anisotropic elasticity, In Proceedings of the 15th International Conference on Boundary Element & Meshless Techniques, BETEQ, V. Mallardo & M. H. Aliabadi, (Florence) 2014.
[8]Y.C. Shiah, C.L. Tan, C.Y. Wang, Efficient computation of the Green's function and its derivatives for three-dimensional anisotropic elasticity in BEM analysis, Eng. Anal. Bound. Elem. 12 (2012) 1746–1755.
[9]E. Pan, W. Chen, Static Green's functions in anisotropic media, Cambridge University Press, 2015.
[10]L. Xie, C. Hwu, C. Zhang, Advanced methods for calculating Green’s function and its derivatives for three-dimensional anisotropic elastic solids, Int. J. Solids Struct, 80 (2016). 261–273.
[11]V.G. Lee, Explicit expression of derivatives of elastic Green’s functions for general anisotropic materials, Mech Res Commun, 30 (2003) 241–249.
[12]F.C. Buroni, J.E. Ortiz, A. Sáez, Multiple pole residue approach for 3D BEM analysis of mathematical degenerate and non-degenerate materials, Int. J. Numer. Meth. Engng. 86 (2011) 1125–1143.
[13]K. Malén, A unified six-dimensional treatment of elastic Green’s functions and dislocations, Phys. Stat. Sol. (b) 44 (1971) 661–672.
[14]G. Nakamura, K. Tanuma, A formula for the fundamental solution of anisotropic elasticity. Q. J. Mech. Appl. Math. 50 (1997) 179–194.
[15]E. Pan, F. Tonon, Three-dimensional Green’s functions in anisotropic piezoelectric solids, Int. J. Solids Struct. 37 (2000) 943–958.
[16]E. Pan, Three-dimensional Green’s functions in anisotropic magneto-electro-elastic bimaterails. Z. Angew. Math. Phys 53 (2002) 815–838.
[17]Q.-H. Qin, Green's function and boundary elements of multifield materials, Elsevier Science & Technology Books, 2007.
[18] M. Denda, C.-Y. Wang, 3D BEM for the general piezoelectric solids, Comput. Methods Appl. Mech. Engrg. 198 (2009) 2950–2963.
[19]L. Xie, C. Zhang, C. Hwu, J. Sladek, V. Sladek, On two accurate methods for computing 3D Green’s function and its first and second derivatives in piezoelectricity, Eng. Analysis Bound. Elem. 61 (2015) 183–193.
[22]F.C.Buroni, A. Saez, Three-dimensional Green’s function and its derivative for materials with general anisotropic magneto-electro-elastic coupling, Proc. R. Soc. A Math. Phys. Eng. Sci. 466 (2010), 515–537.
[21]L. Xie, C. Zhang, C. Hwu, E. Pan, On novel explicit expressions of Green’s function and its derivatives for magnetoelectroelastic materials, Eur. J. Mech. A-Solids 60 (2016) 134–144.
[22]I. Pasternak, R. Pasternak, V. Pasternak, H. Sulym, Boundary element analysis of 3D cracks in anisotropic thermomagnetoelectroelastic solids, Eng. Anal. Bound. Elem. 74 (2017) 70–78.
[23]S.R. Deans, The Radon transform and some of its applications, John Wiley & Sons, New York, 1983.
[24]V. Sladek, J. Sladek, Boundary integral equation method in two-dimensional thermoelasticity, Eng. Anal. 1 (1984) 135–148.
[25]V. Sladek, J. Sladek, Boundary integral equation method in thermoelasticity part III: uncoupled thermoelasticity, Appl. Math. Modell. 8 (1984) 413–418.
[26]V. Sladek, J. Sladek, I. Markechova, 1990 Boundary element method analysis of stationary thermoelasticity problems in non-homogeneous media, Int. J. Numer. Methods Eng. 30 (1990) 505–516.
[27]D. Nardini, C.A. Brebbia, A new approach to free vibration analysis using boundary elements, Appl. Math. Modell. 7 (1982) 157–162.
[28]A.J. Nowak, C.A. Brebbia, A new approach for transforming BEM domain integrals to the boundary, Eng. Anal. Bound. Elem. 6 (1990) 164–167.
[29]A. Deb, P.K. Banerjee, BEM for general anisotropic 2D elasticity using particular integrals, Commun. Appl. Num. Method 6 (1990) 111–19.
[30]F.J. Rizzo, D.J. Shippy, An advanced boundary integral equation method for three-dimensional thermoelasticity, Int. J. Numer. Methods Eng. 11 (1977) 1753–1768.
[31]Y.-C. Shiah, C.-L. Tan, Exact boundary integral transformation of the thermoelastic domain integral in BEM for general 2D anisotropic elasticity, Comput. Mech. 23 (1999) 87–96.
[32]Y.-C. Shiah, C.-L. Hsu, C. Hwu, Direct volume-to-surface integral transformation for 2D BEM analysis of anisotropic thermoelasticity, CMES 102 (2014) 257–270.
[33]X.-W. Gao, The radial integration method for evaluation of domain integrals with boundary-only discretization, Eng. Anal. Bound. Elem. 26 (2002) 905–916.
[34]M. Cui, B.-B. Xu, W.-Z. Feng, Y. Zhang, X.-W. Gao, H.-F. Peng, A radial integration boundary element method for solving transient heat conduction problems with heat sources and variable thermal conductivity, Numer. Heat. Transfer Part B 73 (2018) 1–18.
[35]Y.-C. Shiah, C.-L. Hsu, C. Hwu, Analysis of 2D anisotropic thermoelasticity involving constant volume heat source by directly transformed boundary integral equation. Eng. Anal. Bound. Elem. 93 (2018) 44–52.
[36]C. Hwu, W.-R. Chen, D.-S. Ro, Green’s Function of Anisotropic Elastic Solids with Piezoelectric or Magneto-Electro-Elastic Inclusions, Int. J. Fracture 215 (2019) 91-103.
[37]A.K. Soh, J.X. Liu, On the constitutive equations of magnetoelectroelastic solids, J. Intell. Mater. Syst. Struc.16 (2005) 44–52.
[38]W. Nowacki, Thermo elasticity, Pergamon Press, New York, 1962
[39]I.M. Gel'fand, M.I. Graev, N. Ya. Vilenkin, Generalized functions, Vol. 5, Academic Press, New York, 1966.
[40]C.A. Brebbia, J.C.F. Telles, L.C. Wrobel, Boundary element techniques: theory and applications in engineering, Springer-Verlag, Berlin, 1984.
[41]L. Gaul, M. Kogl, M. Wagner, Boundary element methods for engineers and scientists, Springer-Verlag, Berlin, 2003.
[42]G. Beer, I. Smith, C. Duenser, The boundary element method with programming for engineers and scientists, Springer-Verlag, Vienna, 2008.
[43]J. C. Lachat, J. O. Watson, Effective numerical treatment of boundary integral equations: A formulation for three-dimensional elastostatics, Int. J. Numer. Meth. Engng. 10 (1976) 991–1005.
[44]J. Aboudi, Micromechanical analysis of fully coupled electro-magneto-thermo-elastic multiphase composites, Smart Mater. Struct. 10 (2001) 867–877.
[45]P. Kondaiah, K. Shankar, N. Ganesan, Studies on magneto-electro-elastic cantilever beam under thermal environment, Coupled Systems Mechanics, 1 (2012) 205–217.
[46]J. Sladek, V. Sladek, P. Solek, Ch. Zhang, Fracture analysis in continuously nonhomogeneous magneto-electro-elastic solids under a thermal load by the MLPG, Int. J. Solids. Struct. 47 (2010) 1381–1391.
[47]R. Ansari, R. Gholami, H. Rouhi, Size-dependent nonlinear forced vibration analysis of magneto-electro-thermo-elastic Timoshenko nanobeams based upon the nonlocal elasticity theory, Compos. Struct. 126 (2015) 216–226.
[48]Y. Ootao, Y. Tanigawa, Transient analysis of multilayered magneto-electro-thermoelastic strip due to nonuniform heat supply, Compos. Struct. 68 (2005) 471–480.
[49]K.S. Challagulla, A.V. Georgiades, Micromechanical analysis of magneto-electro-thermo- elastic composite materials with applications to multilayered structures Int. J. Eng. Sci. 49 (2011) 85–104.
QRCODE
 
 
 
 
 
                                                                                                                                                                                                                                                                                                                                                                                                               
第一頁 上一頁 下一頁 最後一頁 top