功能梯度多鐵纖維複合材料反平面剪力波之多重散射

Abstract

本研究旨在探討功能梯度多鐵纖維狀複合材料受反平面剪力波(SH波)之多重散射。其中，功能梯度材料之組成呈梯度變化，其無明顯交界面之特性可避免應力集中、增強結合強度與耐性；而多鐵性複合材料有較佳之磁電彈耦合效應，此兩類材料皆為近代材料研究之重點。 本研究將多鐵性材料反平面問題之廣義勢能控制方程式解耦成一條Helmholtz與兩條Laplace方程式，而內含物呈指數函數漸變，需利用Whittaker函數求解內含物之廣義勢能，並將母材域之勢能場分解為受剪力波入射與內含物干擾產生散射波之勢能場。首先，以波函數展開表示母材之廣義勢能，以Whittaker函數表示含物之廣義勢能，並藉由交界面條件與三角函數之正交性，建立勢能係數之轉換關係。再由多重座標系統展開，將母材域之勢能場轉換為受入射波勢能場與散射波勢能場之疊加，最後利用Graf’s addition定理、二項式展開、勢能係數轉換關係與三角函數之正交性，求解廣義勢能函數。 藉由廣義勢能函數，進而探討漸變參數 對廣義勢能之收斂性、交界面勢能、強度因子、方向場型、散射截面等物理量之影響。從鈦酸鋇(BTO)置入鈷鐵氧(CFO)進行數值模擬結果發現： 影響內含物個數與廣義勢能收斂性之關係。當 時交界面勢能隨 上升而降低，而雙內含物在特定角度會隨 有較明顯之改變。強度因子方面，特定情況下之強度因子皆隨 數值上升而降低；方向場型與散射截面之數值則隨 數值上升而有所提升。

This study is concerned with multiple scattering of the functionally graded multiferroic fibrous composites subjected to an anti-plane shear wave. The main characteristic of functionally graded materials is the tailoring of graded composition to enhance the material performance. On the other hand, the multiferroic composite materials have better magneto-electric-elastic coupling effect. Both of them are the focus of modern material research. We decouple the governing equations of potentials for the anti-plane problem of multiferroic materials into Helmholtz and Laplace equations. Following, the potentials of the matrix and the inclusions are expressed by the wave function expansion and the Whittaker function, respectively. The coefficients relationship is established by the interface conditions and the orthogonality of trigonometric functions. Further, the potentials of the matrix are also expressed by the superposition of the incident wave and the scattered waves of the multiple cylinders. Finally, the coefficients are found by the Graf’s addition theorem, binomial expansion, the interface conditions and orthogonality of trigonometric functions. Numerical examples of BTO-CFO composites are presented for several configurations to show the effect of the grading factor on the convergence of potentials, potentials of the interface, intensity factors, directivity patterns and scattering cross-sections. Numerical results show clearly that the grading factor influences all the above physical quantities. Also in particular cases, the grading factor increases when the potentials of the interface and intensity factors decrease, while the directivity patterns and scattering cross-sections increase.