三維電磁彈性迴轉體與楔形體奇異性分析

Abstract

由於電磁彈性(magneto-electro-elastic, MEE) 或壓電(piezoelectric) 材料擁有將機械能與電磁能以及機械能與電能相互轉換的特性，因此被廣泛應用於高科技之電子設備。為了對於智能元件的優化設計以及進行相關破壞分析，則需通盤了解由幾何形狀所引致之電磁彈奇異性行為。本研究主要目標為建立電磁彈性迴轉體與楔形體之三維電磁彈奇異性漸近解。透過特徵函數展開法配合級數解直接求解於以位移、電勢與磁勢表示的三維力平衡與馬克斯威爾(Maxwell) 方程組之漸近解。假設電磁彈性材料為橫向等向性(transversely isotropic) 材料，其極化方向毋須平行於迴轉體之旋轉軸或是垂直於楔形體之中平面(mid-plane)，如此將造成面外與面內之位移、電場與磁場彼此複雜之耦合關係，導致求解之困難度。在已發表的文獻中對於迴轉體或楔形體之分析，大多基於平面應變之假設，而本研究並未做任何之簡化與假設以建立迴轉體或楔形體於尖角處之漸近解。根據漸近解，將探討有關於極化方向、幾何形狀、材料種類與邊界條件對於電磁彈性與壓電迴轉體與楔型體奇異性階數之影響。其中迴轉體與楔形體可為單一電磁彈性材料(BaTiO3-CoFe2O4) 或壓電材料(PZT-4 與PZT-6B) 以及電磁彈性/各向同向性彈性材料(BaTiO3-CoFe2O4/Si)、電磁彈性/壓電材料(BaTiO3-CoFe2O4/PZT-4 或BaTiO3-CoFe2O4/PZT-6B)、雙電磁彈性材料(BaTiO3-CoFe2O4(V(1)I = 50%)/BaTiO3-CoFe2O4(V(2)I = 20%))、壓電/各向同向性彈性材料(PZT-4/Si 或PZT-6B/Si)或雙壓電材料(PZT-4/PZT-6B)。

Magneto-electro-elastic (MEE) materials and piezoelectric material are able to exchange mechanical, electric, and magnetic forms of energy among themselves and have been widely used in electronic devices. A comprehensive understanding of the magneto-electro-elastic singularities induced by geometry is valuable in optimizing the design of MEE components and analyzing their failures. The main purpose of this research is to establish three-dimensional asymptotic solutions for magneto-electro-elastic singularities in bodies of revolution and wedges. The solutions are obtained by combining an eigenfunction expansion approach with the power series solution method to solve three-dimensional equilibrium equations and Maxwell’s equations in terms of mechanical displacement components and electric and magnetic potentials. Assume the MEE material is transversely isotropic, its polarization direction is not necessarily parallel to the axis of revolution of a body of revolution or normal to the mid-plane of a wedge. Therefore, the in-plane components of displacement, electric, and magnetic fields are generally coupled with their out-of-plane components, which causes difficulties in finding the solution. The analyses of bodies of revolution and wedges in the published literature are based on generalized plane strain assumption. However, the proposed method makes no simplification or assumption to construct the asymptotic solutions at vertex of bodies of revolution and wedges. The developed solutions are further employed to examine the effects of the direction of polarization, the configuration, the material components and boundary conditions on the orders of the MEE singularities in bodies of revolution and wedges that comprise a single MEE material (BaTiO3-CoFe2O4) or a single piezoelectric material (PZT-4 and PZT-6B) and bonded MEE/isotropic elastic (BaTiO3-CoFe2O4/Si), MEE/piezoelectric (BaTiO3-CoFe2O4/PZT-4 or BaTiO3-CoFe2O4/PZT-6B), MEE/MEE(BaTiO3-CoFe2O4(V(1)I = 50%)/BaTiO3-CoFe2O4(V(2)I = 20%)), piezoelectric/isotropic elastic (PZT-4/Si 或PZT-6B/Si), or piezoelectric/piezoelectric (PZT-4/PZT-6B) materials.