含邊裂縫矩形功能梯度板之三維自由振動分析

Abstract

此研究分析含邊裂縫矩形功能梯度板之三維自由振動，利用3D 彈性理 論與Ritz 法；並提出一組新的允許函數，其包含完備正交多項式函數和裂 縫函數。所提的裂縫函數滿足跨越裂縫位移不連續之現象，並能正確描述 裂縫尖端局部變形及應力場。為驗證所提解及所發展電腦程式的正確性， 進行收斂性分析，並與文獻中具裂縫均質板及功能梯度板的結果比較。本 文所探討的功能梯度板是由金屬鋁及陶瓷(Al2O3 或ZrO2)組成的，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、不同厚度、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；此些結果均尚未見於文獻。

This study examines the three-dimensional (3-D) vibrations of rectangular plates of functionally graded material (FGM) having side cracks. Employing 3-D theory of elasticity and a variational Ritz methodology, new hybrid series of mathematically complete orthogonal polynomials and crack functions as the assumed displacement fields are proposed 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 (1/)rϑ3-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 vibration analysis are confirmed through comprehensive convergence studies and comparisons with published results for homogeneous and FGM rectangular plates with side cracks based on various plate theories. Two types of FGM plates, Al/Al2O3 and Al/ZrO2, are included in the study. 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 having side cracks with varying crack length ratios, crack positions, and crack inclination angles are reported herein for the first time in the published literature.