多鐵纖維複合材料反平面剪力波之多重散射

Abstract

本研究探討含多內含物無限域多鐵性纖維複合材料受反平面剪力波(SH波)之動態反應，旨在運用多極座標展開與T矩陣法的技巧，計算內含物間互相干擾所產生的多重散射場域，並求取動態應力強度因子、方向場型與散射截面等物理量，用以比較上述物理量在動態情況下，相較於靜態或只含單一內含物之情況。 本研究將多鐵性材料之反平面控制方程式解耦成一條Helmholtz與兩條Laplace方程式，並將母材域之勢能場分成入射場和散射場，其中入射場為假設內含物不存在時，母材域受SH波入射之勢能場；而散射場則為各內含物所造成。 首先分別將母材域與內含物域之勢能場以各內含物圓心進行波函數展開，並利用交界面條件和正、餘弦函數之正交性，建立勢能場間的轉換關係T矩陣。得到T矩陣後母材域再以多座標系統展開，並將母材域勢能場換成入射場和各內含物散射場的疊加，最後利用T矩陣、Graf加法定理、二項式定理和正、餘弦函數之正交性，得到用以求解待定係數的聯立線性方程組，進而得知勢能場及各物理量。

We propose a theoretical framework to investigate the multiple scattering of a multiferroic fibrous composite subjected to an anti-plane shear wave. The composite consists of piezoelectric (piezomagnetic) cylinders in a piezomagnetic (piezoelectric) matrix. These inclusions are arbitrarily distributed, and may have different sizes or material properties. The governing equations are decoupled into one Helmholtz and two Laplace equations. Wave function expansions formalism is adopted to expand the potentials into series with respect to each center of the inclusion. The T-matrix method, Graf’s addition theorem and Binomial expansion are used to connect these expansions. The admissible potentials for the inclusions and matrix are derived and calculated with sufficient accuracy for the composite. We show that the solution field is governed by a linear set of coupled algebraic equations with an infinite number of unknowns. Numerical results are presented for the dynamic intensity factors, directivity patterns, and scattering cross-section for different configurations.