
系統識別號 
U00262001201711381900 
論文名稱(中文) 
含角點非對稱複合材料疊層板之邊界元素分析 
論文名稱(英文) 
Boundary Element Analysis of Unsymmetric Laminates with Corners 
校院名稱 
成功大學 
系所名稱(中) 
航空太空工程學系 
系所名稱(英) 
Department of Aeronautics & Astronautics 
學年度 
105 
學期 
1 
出版年 
106 
研究生(中文) 
張瀚文 
研究生(英文) 
HanWen Chang 
學號 
P48961031 
學位類別 
博士 
語文別 
英文 
論文頁數 
113頁 
口試委員 
指導教授胡潛濱 口試委員夏育群 口試委員楊文彬 口試委員江達雲 口試委員吳光鐘 口試委員馬劍清 口試委員趙振綱 口試委員陳正宗

中文關鍵字 
邊界元素法
複合材料疊層板
耦合問題
角點不連續
孔洞
史磋公式

英文關鍵字 
Boundary Element Method
Coupled Stretchingbending Analysis
Unsymmetric Composite Laminates
Corner Discontinuities
Rectangular Hole
Complex Variable Formalism
Stroh Formalism

學科別分類 

中文摘要 
邊界元素法在現今彈性力學領域中的發展已愈趨成熟，並且呈現出更多元的面向。文獻裡，相關的研究與討論同時關注在其數學上的特性以及對於各種不同材料之彈性體的應用；另一方面，學者對於其發展要素譬如基本解的求得、奇異積分的計算，以及在角點不連續處的處理也有詳盡的討論。然而，這些研究多侷限於求解等向性或者金屬材料彈性體之二維問題、板的純彎矩問題，或是三維問題。為了符合工業設計上對於輕量化的要求和結構體中在受力方向上的結構加強需要，設計者或工程師會更傾向於採用複合材料疊層板來建立結構本體或其部件。近二十年來，以此為基礎和目標的研究多致力於正交性材料、對稱性疊層板和反對稱疊層板的應用；吾人若需考慮非對稱疊層板做為結構材料時，就需應對其更為複雜之力學行為，也就是平面內與出平面之變形/拉伸或彎矩的耦合問題。然而，此類問題在文獻中很少藉由邊界元素法得到較完善的分析和解決。
為了顧及所有不同類型疊層(對稱、反對稱、非對稱)所造成的板之不同力學行為，以及伴隨著角點存在之不連續性所產生的影響，在本研究中，藉由相應之邊界積分式和經由史磋公式推導而來之基本解，非對稱複材疊層板之邊界元分析受到了適當的建立和處理。在離散化邊界積分式的過程中，為準確地計算奇異積分以及解決在角點上因幾何不連續所衍生之線性相依方程式的問題，各種不同的計算和處理方式經過了實際的測試和驗證，並提出了奇異積分之解析顯示解以提供更準確的結果和更有效率的運算，以及四條輔助方程式用以取代相依方程式。另一方面，在傳統的邊界元分析中，節點上或靠近節點處之完整的應變與應力分量還需透過類似於後處理的方式來求得；因此在文獻裡，學者對此問題的處理採取了各種不同的方法與手段。在本研究中，由於邊界上節點之位移和曳引力結果，與稍微遠離邊界之內點上的位移、應變、曲率、合應力和彎矩力分量皆可被正確地求出與計算，這些正確的數值結果均被用以內插手法計算節點上或者鄰近邊界處之近似值。因此，在這過程裡，我們不需要再去執行包括超奇異積分與強奇異積分的計算。
綜合以上各階段的求解方式與流程，此非對稱複材疊層板之邊界元素分析可順利地提供疊層板的全域解，並且其整體的結果能保持一致的穩定性與正確性。

英文摘要 
For the boundary element analysis of the elastic bodies in the practical engineering problems, researchers in this field have explored into the many aspects of this numerical method when it is engaged with different applications and different kinds of materials. In their studies, some critical concerns had been paid attention to such as the need of an adequate fundamental solution, the singular integrals, and the corner discontinuities. All of these problems can be regarded as the “classical” topics in these days, and many applications had indeed been well treated in the literature. However, most of these studies were confined to the two dimensional problems, plate bending problems or three dimensional problems utilizing isotropic or metallic materials. In the industry, in order to meet some structural designs such as the criteria of the light weight, or the strengthening of the materials in the directions of applying force, today engineers are more willing to take advantages of the designable characteristics of the composite materials or laminates. With this understanding, over the past few decades some researches had been conducted for the types of laminates such as specially orthotropic materials, symmetric or antisymmetric laminates. Nevertheless, if an unsymmetric laminate is considered, the mechanical behaviors of the plates will become more complex in such a way that the coupling between the inplane and outofplane bending problems will be unavoidable, and this problem was seldom received a thorough solution via the use of boundary elements.
In this dissertation, to cover the complex mechanical behaviors of the composite laminates in response to all the possibility of symmetric, antisymmetric, or unsymmetric stacking sequences and the corner discontinuities of a laminated plate, the coupled stretchingbending analysis of the general composite laminates via boundary elements has been developed with the help of the associated boundary integral equation, and the fundamental solution obtained via the Green’s function written in the form of Strohlike complex variable formalism. To effectively treat the singular problem and the corner discontinuities which may result in dependent equations in the system of equations established via the discretization of the boundary integral equation, various methodologies were investigated to see their adequacies for the present application. And, the explicit solutions of the weakly and strongly singular integrals and the four auxiliary equations employed to replace the dependent equations are proposed in this study to solve the nodal displacements and tractions accurately and promptly.
Besides, similar to the needs in the traditional boundary element analysis, the postprocessing for the calculations of the complete components of the strains and stress resultants at or near the boundary nodes was also implemented and carried out in the present study. In order to obtain these results, we can make useful the already known nodal displacements through the method of finite difference, and all the other correct results of strains and stress resultants calculated via the derivatives of boundary integral equation at the points not so close to the boundary, and the use of the constitutive equation of laminates. Finally, by utilizing the moving least square method with these results, we can further approximate the solution in the vicinity of boundary nodes with good accuracy. In this process, we don’t need to tackle again the singular problems which involve hypersingular and strongly singular integrals. Hence, based on all the works required at different stages, the fulldomain solution can be obtained for the coupling analysis of composite laminated plates via boundary elements with accuracy and efficiency.

論文目次 
CONTENTS
ABSTRACT.................................................i
CONTENTS.................................................v
LIST OF TABLES.........................................vii
LIST OF FIGURES.......................................viii
NOMENCLATURE.............................................x
CHAPTER I INTRODUCTION.............................1
CHAPTER II ANALYSIS OF COMPOSITE LAMINATES..........8
2.1 Classical Lamination Theory..........................8
2.2 Strohlike Formalism................................12
CHAPTER III BOUNDARY ELEMENT ANALYSIS...............16
3.1 Boundary Integral Equation..........................16
3.2 The Fundamental Solution............................18
3.3 Boundary Element Formulation........................20
3.4 Displacements, Strains and Stresses at Internal Points..................................................26
3.5 Locations of Source Points Related to the corner..................................................27
CHAPTER IV AUXILIARY RELATIONS FOR CORNERS.........35
CHAPTER V SINGULAR INTEGRALS......................43
5.1 Numerical Integration by Finite Part Integrals......44
5.2 Explicit ClosedForm Solutions of the Singular Integrals...............................................47
5.3 Indirect Calculation of the Singular Integrals...............................................54
5.4 Discussion..........................................56
CHAPTER VI COMPLETE SOLUTION AT OR NEAR THE BOUNDARY NODES...................................................59
6.1 Strains and Stresses at Boundary Nodes..............60
6.2 Regularization for the Points Near the Boundary Nodes...................................................64
6.3 Discussion..........................................65
CHAPTER VII NUMERICAL EXAMPLES......................67
7.1 Calculation of the Singular Integrals...............67
7.2 Location of Source Points Related to Corner Nodes with Various Boundary Conditions.............................69
7.3 Calculation of the Strains and Stresses at or near the boundary nodes..........................................76
CHAPTER VIII CONCLUSIONS.............................81
REFERENCES..............................................85
TABLES..................................................90
FIGURES.................................................97
PUBLICATION LIST.......................................114

參考文獻 
Ameen, M., Boundary Element Analysis: Theory and Programming, Narosa Publishing House, India, 2001.
Barnett, D.M. and Lothe, J., “Synthesis of the sextic and the integral formalism for dislocations, Greens function and surface waves in anisotropic elastic solids,” Phys Nor, 7, pp. 1319, 1973.
Bonnet, M., Poon, H., and Mukherjee, S., “Hypersingular formulation for boundary strain evaluation in the context of a CTObased implicit BEM scheme for small strain elastoplasticity,” International Journal of Plasticity, 14(1011), pp. 10331058, 1998 .
Brebbia, C. A., Telles, J.C.F. and Wrobel, L.C., Boundary Element Techniques, SpringerVerlag Berlin, Heidelberg, 1984.
Chang, H.W. and Hwu, C., “Complete solutions at or near the boundary nodes of boundary elements for coupled stretchingbending analysis,” Engineering Analysis with Boundary Elements, 72, pp. 8999, 2016.
Chaudonneret, M., “On the discontinuity of the stress vector in the boundary integral equation method for elastic analysis,” In: Brebbia, C.A., ed., Recent advances in boundary element methods, Pentech Press, London, pp. 185194, 1978.
Chaves, E.W.V., Fernandes, G.R., and Venturini, W.S., "Plate bending boundary element formulation considering variable thickness," Engineering Analysis with Boundary Elements, 23, pp. 405–418, 1999.
Chen, H.B., Lu, D., and Schnack, E., “Regularized algorithms for the calculation of values on and near boundaries in 2D elastic BEM,” Engineering Analysis with Boundary Elements, 25, pp. 851876, 2001.
Chien, C.C., Rajiyah, H., and Atluri, S.N., “On the evaluation of Hypersingular integrals arising in the boundary element method for linear elasticity,” Computational Mechanics, 8, pp. 5770, 1991.
Chen, J. T. and Hong, H.K., “Dual boundary integral Equations at a Corner Using Contour Approach around Singularity,” Advances in Engineering Software, 21(3), pp.169178, 1994.
Chen, J. T. and Hong, H.K., “Review of dual boundary element methods with emphasis on hypersingular integrals and divergent series,” Applied Mechanics Reviews, ASME, 52(1), pp.1733, 1999.
Davis, P.J. and Rabinowitz, P., Methods of Numerical Integration, 2nd ed., Academic Press, New York, 1984.
Deng, Q, Li, C.G., Wang, S.L., Tang, H., and Zeng, H., “A new method to the treatment of corners in the BEM,” Engineering Analysis with Boundary Elements, 37, pp. 182186, 2013.
Ehao, E. and Lan, S., “Boundary stress calculation – a comparison study,” Computers & Structures, 71, pp. 7785, 1999.
ElZafrany, A. and Fadhil, S., “A modified Kirchhoff theory for boundary element analysis of thin plates resting on twoparameter foundation,” Engineering Structures,18(2), pp. 102104, 1996.
Gakwaya, A, Dhatt, G., and Cardou, A., “An implementation of stress discontinuity in the boundary element method and application to gear teeth,” Applied Mathematical Modelling, 8(5), pp. 319–327, 1984.
Gao, X.W. and Davies, T.G., “3D multiregion BEM with corners and edges,” International Journal of Solids and Structures, 37, pp. 15491560, 2000.
Gaul, L., Kogl, M., Wagner, M., Boundary Element Methods for Engineers and Scientists, SpringerVerlag, Berlin, 2003.
Gu, Y., Dong, H., Gao, H., Chen, W., and Zhang Y., “An extended exponential transformation for evaluating nearly singular integrals in general anisotropic boundary element method,” Engineering Analysis with Boundary Elements, 65, pp. 3946, 2016.
Gu, Y., Gao, H., Chen, W., and Zhang C., “A general algorithm for evaluating nearly singular integrals in anisotropic threedimensional boundary element analysis,” Computer Methods in Applied Mechanics and Engineering, 308, pp. 483498, 2016.
Guiggiani, M., Casalini, P., “Direct computation of Cauchy principal value integrals in advanced boundary elements,” International Journal for Numerical Methods in Engineering, 24, pp. 17111720, 1987.
Guiggiani, M. and Gigante, A, “A General algorithm for multidimensional Cauchy principal value Integrals in the boundary element method,” Journal of Applied Mechanics, 57, pp. 906915, 1990.
Hasebe, N. and Iida, J., “Intensity of Corner and Stress Concentration Factor,” Journal of Engineering Mechanics, 109(1), pp. 346356, 1983.
Heise, U., “Numerical properties of integral equations in which the given boundary values and the sought solutions are defined on different curves,” Computers & Structures, 8, pp. 199205, 1978.
Hwu, C., “StrohLike formalism for the coupled stretchingbending analysis of composite laminates,” International Journal of Solids and Structures, 40(1314), pp. 36813705, 2003.
Hsieh, M.C. and Hwu, C., “Explicit solutions for the coupled stretchingbending problems of holes in composite laminates,” International Journal of Solids and Structures, 40(15), pp. 39133933, 2003.
Hwu, C., “Green's function for the composite laminates with bending extension coupling,” Composite Structures, 63, pp. 283292, 2004.
Hwu, C., “Green's functions for holes/cracks in laminates with stretchingbending coupling,” ASME Journal of Applied Mechanics, 72, pp. 282289, 2005
Hwu, C. and Tan, C.Z., “InPlane/OutofPlane Concentrated Forces and Moments on Composite Laminates with Elliptical Elastic Inclusions,” International Journal of Solids and Structures, 44, pp. 65846606, 2007.
Hwu, “Some Explicit Expressions of Extended Stroh Formalism for TwoDimensional Piezoelectric Anisotropic Elasticity,” International Journal of Solids and Structures, 45, pp. 44604473, 2008.
Hwu C., “Boundary integral equations for general laminated plates with coupled stretchingbending deformation,” Proceedings of the Royal Society, Series A, 466, pp. 10271054, 2010a.
Hwu, C., Anisotropic Elastic Plates, Springer, New York, pp. 458459, 2010b.
Hwu, C., “Boundary element formulation for the coupled stretchingbending analysis of thin laminated plates,” Engineering Analysis with Boundary Elements, 36(6), pp. 10271039, 2012.
Hwu, C. and Chang H.W., “Coupled StretchingBending Analysis of Laminated Plates with Corners via Boundary Elements,” Composite Structures, 120, pp. 300314, 2015a.
Hwu, C. and Chang, H.W., “Singular integrals in boundary elements for coupled stretchingbending analysis of unsymmetric laminates,” Composite Structures, 132, pp. 933943, 2015b.
Jones, R.M., Mechanics of Composite Materials, 2nd ed., Taylor & Francis, 1999.
Kermanidis, T., “Kupradze functional equation for the torsion problem of prismatic barspart 2,” Computer Methods in Applied Mechanics and Engineering,7, pp. 249259, 1975.
Kutt, H.R., “Quadrature formulae for finitepart integrals. Report WISK 178,” The National Research Institute for Mathematical Sciences, Pretoria, 1975a.
Kutt, H.R., “On the numerical evaluation of finitepart integrals involving an algebraic singularity. Report WISK 179,” The National Research Institute for Mathematical Sciences, Pretoria, 1975b.
Kutt, H.R., “The numerical evaluation of principal value integrals by finitepart integration,” Numerische Mathematik, 24, pp. 205210, 1975c.
Lachat, J.C., and Watson, J.O., “Effective numerical treatment of boundary integral equations  a formulation for threedimensional elastostatics,” International Journal for Numerical Methods in Engineering,10, pp. 9911005, 1976
Lei, X.Y., “A new BEM approach for linear elasticity,” International Journal of Solids and Structures, 31(24), pp. 33333343, 1994.
Levin, D., “The Approximation Power of Moving Least Squares,” Mathematics of Computation, 67(224), pp. 15171531, 1998 .
Matsumoto, T., Tanaka, M., “Boundary stress calculation using regularized boundary integral equation for displacements gradients,” International Journal for Numerical Methods in Engineering, 36, pp. 783797, 1993.
Nik, A.M.N. and Tahani, M., “Analytical solutions for bending analysis of rectangular laminated plates with arbitrary lamination and boundary conditions,” Journal of Mechanical Science and Technology, 23, pp. 22532267, 2009.
Nomura, Y., Ikeda, T., and Miyazaki, N., “Stress intensity factor analysis at an interfacial corner between anisotropic bimaterials under thermal stress,” Engineering Fracture Mechanics, 76, pp. 221–235, 2009.
Reddy, J.N., Mechanics of laminated composite plates: theory and analysis, CRC Press, Inc., Boca Raton, 1997.
dos Reis, A., Albuquerque, É.L., Torsani, F.L., Palermo, L., and Sollero, P, “Computation of moments and stresses in laminated composite plates by the boundary element method,” Engineering Analysis with Boundary Elements, 35(1), pp. 105113, 2011.
Rudolphi, T.J., “An implementation of the boundary element method for zoned media with stress discontinuities,” International Journal for Numerical Methods in Engineering, 9, pp. 115, 1983.
Sokolnikoff, I.S., Mathematical Theory of Elasticity. McGraw Hill, 2nd Edition, 1956.
Sun, Y., Kagawa, Y., “Regular boundary integral solution with dual and complementary variational formulations applied to twodimensional Laplace problems,” International Journal of Numerical Modelling, 8, pp.127137, 1995.
Syngellakis, S. and Cherukunnath, N., “Boundary element analysis of symmetrically laminated plates,” Engineering Analysis with Boundary Elements, 28, pp. 1005–1016, 2004.
Telles, T.C.F., “A selfadaptive coordinate transformation for efficient numerical evaluation of general boundary element integral,” International Journal for Numerical Methods in Engineering, 24, pp. 959973, 1987.
Tsamasphyros, G. and Theocaris, P.S., “On the Convergence of Some Quadrature Rules for Cauchy PrincipalValue and FinitePart Integrals,” Computing, 31, pp. 105114, 1983.
Venturini, W.S. and Paiva, J.B., "Boundary element for plate bending analysis," Engineering Analysis with Boundary Elements, 11, pp. 18, 1993.
Wearing, J.L. and Bettahar, O., “The analysis of plate bending problems using the regular direct boundary element method,” Engineering Analysis with Boundary Elements, 16(3), pp. 261271, 1995.
Wen, P.H., Dirgantara, T., Baiz, P.M., and Aliabadi, M.H, “The boundary element method for geometrically nonlinear analyses of plates and shells,” In: Aliabadi, M.H., Wen, P.H., eds., Boundary Element Method in Engineering and Science, Imperial Collage Press, pp.149, 2011.
Williams, M.L., “Stress singularities resulting from various boundary conditions in angular corners of plate in extension,” Journal of Applied Mechanics, 19, pp. 526–528, 1952.
Zhang, Q. and Mukherjee, S., “Design sensitivity coefficients for linear elastic bodies with zones and corners by the derivative boundary element method,” Journal of Applied Mechanics, 27, pp. 983998, 1991.
Zhao, Z., “On the calculation of boundary stresses in boundary elements,” Engineering Analysis with Boundary Elements, 16(4), pp. 317322, 1995.
Zeng, ZJ, Cai, RY, “Treatment of singular integral caused by employing linear boundary elements for twodimensional elastostatic problems,” Computers & Structures, 34(6), pp. 855859, 1990.

論文全文使用權限 
同意授權校內瀏覽/列印電子全文服務，於20170125起公開。同意授權校外瀏覽/列印電子全文服務，於20170125起公開。 


