零場邊界積分方程法求解含圓形孔洞功能梯度介質的反平面問題

An anti-plane problems containing circular holes in a functionally graded material by using the null-field boundary integral equation method

湯幃丞(Wei-Chen Tang)

指導教授 ： 李家瑋

淡江大學/工學院/土木工程學系碩士班/碩士(2020年)

https://doi.org/10.6846/TKU.2020.00064

摘要

本論文使用零場邊界積分方程法（null-field boundary integral equation method）求解含圓孔洞之功能梯度材料的反平面問題，本文所採用的功能梯度材料其剪力模數為水平向指數變化，因此控制方程式並非為Laplace方程式，但可藉由變數變換將控制方程式轉換成修正型Helmholtz方程式。針對單圓邊界問題，將考慮三種邊界條件:1.圓孔洞問題、2.剛性置入物、3.Eshelby的置入物問題，其中前2個問題在無窮遠處受一剪切應力作用。因考慮圓形邊界可透過搭配退化核函數（degenerate kernel）與傅立葉級數（Fourier series）取代閉合型基本解與邊界密度，可得到其半解析解。本文更延伸至無限域中含多圓孔洞之功能梯度材料的反平面問題，藉由自適性座標系統(局部座標系統)的使用，可充分使用三角函數的正交性，因此無須數值積分的方式求解邊界弧長積。最後將本文方法的數值結果與Matlab R2019a的工具箱PDE Toolbox，也就是有限元素法（finite element method）的數值結果做對比，針對不同非均勻空間變換參數（non-homogeneous parameter）對場解的影響，除了全場位移的比較之外，也針對圓形孔洞邊界上的位移與應力集中因子（stress concentration factor）做比較，其兩種方法的結果都一致吻合，且當β=0時，也與均質的結果一致吻合。

關鍵字

功能梯度材料 ； 零場邊界積分方程法 ； Modifided Helmholtz方程式 ； 反平面力場問題 ； Eshelby圓形置入物

並列摘要

The null-field boundary integral equation method is employed to solve anti-plane problems containing circular holes of a functionally graded material (FGM). The shear modulus of the present FGM is an exponential variation. Therefore, the governing equation isn’t a typical Laplace equation. By using the change of variable, the governing equation can be transform into the modified Helmholtz equation. In this thesis, we consider three kinds of boundary condition, for the problem containing a single circular boundary. For the former two kinds, one is a traction free boundary condition and the other is rigid inclusion. Both are subject to a remote shear. The third one is an Eshelby inclusion problem. By using the degenerate kernels and Fourier series expansions, the semi-analytical solution can be obtained. We also extend the problem containing a single circular hole to multiple circular holes by using the adaptive observer system, while the orthogonality of angular function can be fully utilized. In this way, using the numerical quadrature to calculate the boundary arc length integral is free. Finally, all numerical results are compared with those results by using the finite element method (FEM). The displacement field and the stress concentration factor along the circular hole are considered to discuss the effect of the non-homogeneous parameter of materials. The results by using the present method and the FEM are consistent. For the special case of non-homogeneous, the present results are the same with the results of homogeneous case.

並列關鍵字

Functionally graded materials ； Null-field boundary integral equation method ； Modified Helmholtz equation ； Anti-plane problem ； Eshelby inclusion problem

參考文獻

[1] 王保林, 韩杰才, 张幸红, “非均匀材料力学”, 科学出版社, 北京市,(2003)

[2] 新野正之, 平井敏雄, 渡辺龙三, “倾斜机能材料―宇宙机用超耐热材料を目指して”, 日本複合材料学会志, (10), 1-8, (1987)

[3] G.I. Popov, “Axisymmetric contact problem for an elastic inhomogeneous half-space in the presence of cohesion”, Journal of Applied Mathematics and Mechanics, 37, 1109-1116, (1973)

[4] Y.Y. Yang, “Stress analysis in a joint with a functionally graded material under a thermal loading by using the Mellin transform method”, International Journal Solids and Structures, 35, 1261-1287, (1998)

[5] H. Li, J. Lambros, B.A. Cheeseman and M.H. Santare, “Experimental investigation of the quasi-static fracture of functionally graded materials”, International Journal of Solids and Structures, 37, 3715-3732, (2000)