壓電材料迴轉體與楔形板幾何引致電彈奇異性之探討

Abstract

壓電材料已被廣泛應用於製造各種感測器、半導體、致動器、諧振器、振盪器與顯 示器，並在智慧結構製程中扮演著舉足輕重的角色。通盤的了解因幾何形狀所引致的電 彈奇異性對於進行壓電元件的優化設計與分析其可能的破壞行為是非常有價值的；因為 應力奇異點常是破壞之起點。另外，若需以數值方法精確的分析具應力奇異性之複雜問 題時，則該數值解通常須能準確模擬該應力奇異行為，亦即須事先了解該奇異點漸進解 之特性(至少須知道奇異階數)。 本計畫擬利用二年時間，直接利用三維壓電彈性理論(3-D peizoelasticity theory)，不 做廣義平面應變或軸對稱變形之假設，建立壓電材料迴轉體與楔形板由於幾何形狀所引 致電彈奇異性之漸近解。考慮壓電材料為transversely isotropic material，其極化軸不與 描述物體幾何之主軸(例如迴轉體之迴轉軸或楔形板之面外垂直軸)平行；探討極化軸方 向對奇異階數之影響。本研究利用特徵函數展開法(Eigenfunction expansion approach)並 結合級數解法(power series technique)對以位移(mechanical displacement)及電勢(electric potential)表示之平衡方程式與馬克斯威爾方程式求解。本研究所推導之解將與文獻之結 果(例如考慮軸對稱變形迴轉體之電彈奇異性)進行比較以驗證本文所提出方法所得解 之正確性。本研究將對單一壓電材料(PZT-4 或PZT-5H)、雙壓電材料(PZT-4/PZT-5H)或 壓電/各向同性彈性材料(PZT-4/Al or PZT-5H/Al)進行分析，並對極化方向、材料類型與 邊界條件對奇異性階數所造成的影響作通盤地研究。

Piezoelectric materials have been extensively adopted to manufacture various sensors, conductors, actuators, resonators, oscillators and monitors, and have an important role in smart structures. A comprehensive understanding of the electroelastic singularities that are induced by geometry is valuable in optimizing the design of piezoelectric components, and analyzing their failure. An accurate numerical analysis of problems that involve stress singularities depends on knowledge of such stress singularity behaviors. The main purpose of this two-year project is to establish asymptotic solutions for studying the geometrically induced electroelastic singularities in piezoelectric bodies of revolution and wedges based on 3-D peizoelasticity theory with no further assumption such as axisymmetric deformation or generalized plain strain that is made in the literature. A piezoelectric material is considered as transversely isotropic material, and its direction of polarization is not parallel to the axis of revolution for a body of revolution or the normal of the mid-plane of a wedge. An eigenfunction expansion approach is combined with a power series solution technique to establish the asymptotic solutions for geometrically induced electroelastic singularities in piezoelectric bodies of revolution and wedges. The asymptotic solutions are obtained by directly solving the three-dimensional equilibrium and Maxwell’s equations in terms of displacement components and electric potential. The correctness of the proposed solution is confirmed by comparing the present results with the published results obtained based on the assumption of axisymmetric deformation or generalized plain strain. The numerical results related to singularity orders will be shown in graphical form for bodies of revolution and wedges that comprise a single material (PZT-4 or PZT-5H) or bonded piezo/piezo (PZT-4/PZT-5H) or piezo/isotropic elastic (PZT-4/Al or PZT-5H/Al) materials. These results are going to be published for the first times in literature.