利用三維彈性理論配合Ritz法分析具內部裂縫矩形功能梯度材料板之振動問題

Abstract

本研究利用3D彈性理論配合Ritz法分析含內部裂縫矩形功能梯度板自由振動；提出能準確描述裂縫尖端奇異性並滿足跨越裂縫位移不連續行為之裂縫函數配合滿足邊界條件之多項式函數當允許函數。透過收斂性分析，並與文獻中具裂縫均質板的結果比較，驗證所發展電腦程式的正確性與所提解之準確性。本文所考慮的功能梯度板是由金屬鋁(Al)及陶瓷(Al2O3)組成，使用簡單的冪級數與體積分數來描述材料的性質。本文探討不同材料組成、不同邊界條件、不同板厚及不同裂縫長度、角度與位置對板的振動行為之影響，並繪有中平面及三維變形之模態圖；此些結果均首見於文獻。

This study examines the three-dimensional (3-D) vibrations of functionally graded material (FGM) rectangular plates having internal cracks. We employ 3-D theory of elasticity and a variational Ritz methodology, and propose new hybrid series of mathematically complete orthogonal polynomials and crack functions as the assumed displacement fields to enhance the convergence modeling of the stress singular behavior of the crack terminus edge front in a rectangular FGM plate. The proposed admissible hybrid series properly describe the 3-D stress singularities at the terminus edge front of the crack, allowing for displacement discontinuities across the crack sufficient to explain the most general 3-D “mixed modes” of local crack-edge deformation and stress fields typically seen in fracture mechanics. The correctness and validity of the proposed approach are confirmed through comprehensive convergence studies and comparisons with published results for homogeneous and FGM rectangular plates with internal cracks based on various plate theories. The FGM plates under consideration are composed of aluminum and ceramic (Al/Al2O3). The locally effective material properties are estimated by a simple power law and the effects of the volume fraction on the frequencies are investigated. Frequency data, mode shapes and nodal patterns for FGM rectangular plates with different boundary conditions and thickness-width ratios, and having internal cracks with varying crack length ratios, crack positions, and crack inclination angles are reported herein for the first time in the published literature.