零場邊界積分方程法求解含圓形孔洞功能梯度介質引致的SH波散問題

SH-wave scattering by a circular hole in a functionally graded material using the null-field boundary integral equation method

施奕廷(Yi-Ting Shih)

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(2019年)

https://doi.org/10.6846/TKU.2019.00440

摘要

本論文使用零場場邊界積分方程法（null-field BIEM）求解含單一圓孔洞之功能梯度材料中水平剪力波（SH-wave）的散射問題，本文所採用的功能梯度材料參數呈指數變化，因此控制方程式並為典型的Helmholtz方程式，藉由使用變數變換將控制方程式轉換成Helmholtz方程式，且曳引力為零的Neumann邊界則轉換成Robin邊界。如此操作即可利用零場邊界積分方程法求解水平剪力波在功能梯度材料中的散射問題，搭配退化核函數（degenerate kernel）與傅立葉級數（Fourier series）取代基本解（fundamental solution）與邊界密度（boundary densities）可得到半解析解；本研究更延伸至水平剪力波在半無限域中含單一圓孔洞之功能梯度材料的散射問題，藉由引入映射法將半平面含單一圓孔洞問題轉換成全平面含兩個相等圓孔洞問題，其中也的另一關鍵則是剪力模數函數與材料密度函數的映射是關鍵；最後將本文究方法之數值結果與傳統邊界元素法（boundary element method BEM）使用常數元素的數值結果做對比，其結果都一致吻合，除位移場的比較之外，也對圓形孔洞邊界上的動態應力集中因子（dynamic stress concentration factor）做比較，針對不同非均勻空間變換參數（non-homogeneous parameter）對其場量的影響。

關鍵字

SH波的散射 ； 功能梯度材料 ； 邊界元素法 ； 零場邊界積分方程法

並列摘要

In this thesis, the problem of SH-wave scattering by a circular hole buried in infinite functionally graded materials (FGM) is solved by using the null-field boundary integral equation method (null-field BIEM). For the considered FGM, the patterns of the shear modulus and the density are the form of exponential variation. Therefore, the governing equation for the time-harmonic motion is not a typical Helmholtz equation. By using the change of variables, the original governing equation can be transformed into the Helmholtz equation. The Neumann boundary condition due to the traction free condition is transformed into the Robin boundary condition. Therefore, the null-field BIEM can be straightforward employed to solve the problem of SH-wave scattering in the FGM. Using the degenerate kernel and the Fourier series to substitute for the closed-form fundamental solution and boundary densities, the semi-analytical solution can be obtained. In addition, the problem of SH-wave scattering by a circular hole buried in semi-infinite FGM is also considered. By using the image method, the semi-infinite plane problem containing a circular hole is transformed into the infinite plane problem containing two identical circular holes. The other key point is that the functions of the shear modulus and the density are also imaged. Finally, all numerical results are compared well with those of numerical results by using the conventional boundary element method (BEM) with the constant element scheme. Not only the displacement field of the whole domain but also the dynamic stress concentration factor along the circular hole is presented. The effect of the non-homogeneous parameter of materials is also considered.

並列關鍵字

SH-wave scattering ； Functionally graded materials ； Boundary element method ； Null-field boundary integral equation method

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