應用MLS-Ritz法於具應力奇異點問題之靜態與動態分析

Abstract

MLS-Ritz法是利用移動式最小平方差法(Moving Least-square method, MLS)建構傳統Ritz法的允許函數；MLS-Ritz法以低階之多項式函數為基底，可改善傳統Ritz法因病態矩陣發生而造成數值計算上的困難。本研究利用薄板理論、平面應力理論及三維彈性理論進行具裂縫或V形缺口板之靜態和動態分析；此外，並提出可準確描述V形缺口及裂縫尖端應力奇異行為和裂縫處位移或斜率不連續現象之增益型基底函數(enriched basis function)。利用該補強式基底函數，可依應力強度因子之定義直接求得該因子，而不需以等高線積分法(contour integration techniques)之間接方式求得。本研究展示多個案例之收斂性分析並與文獻或有限元素軟體ABAQUS做比較，以驗證所用方法之正確性與效果；並探討參數(增益型基底函數中多項式之階數、權函數之支撐域以及節點之數目)對於精確度之影響。 為了補足文獻數據的不足，本研究在動態方面列表具邊緣與內部裂縫之古典板以及具V形缺口、邊緣與內部裂縫之三維功能梯度材料(functionally Graded Materials, FGM)板之自然振動頻率；在靜態方面列表具V形缺口、邊緣與內部裂縫平面應力以及具邊緣裂縫之三維FGM板之應力強度因子。

MLS-Ritz method is the Ritz method with admissible functions constructed by the moving least-square (MLS) method. MLS-Ritz method often uses low-order polynomial basis functions and overcomes the numerical difficulties due to ill-conditioned matrix that often occurs in the conventional Ritz method. This study performs static and dynamic analyses for cracked or V-notched plates by using the classical plate theory, plane stress theory and three-dimensional elasticity theory and proposes the enriched basis functions that appropriately describe stress singularity behaviors at the tip of a crack or at the vertex of a V-notch and show displacement and slope discontinuities across a crack. Using such enriched basis functions enables to directly compute stress intensity factors based on their definitions without the aids of any contour integration techniques. The correctness and effectiveness of the proposed approach are validated through comprehensive convergence studies and comparisons of the present numerical results with the published ones or those obtained from commercial finite element package ABAQUS. Parametric studies are also carried out to show the effects of parameters such as the order of polynomials in the enriched basis functions, the support domain of weighting function, and the distribution of nodes on the accuracy of numerical results. To fill the gap in the published data, this study presents natural frequencies and mode shapes of cracked rectangular, skewed, or circular plates based on the classical plate theory, and functionally graded materials (FGM) cantilevered skewed plates with V-notches, side cracks or internal cracks using three-dimensional elasticity theory. This study also shows the stress intensity factors of cracked or V-notched plane stress plates and FGM plates with side cracks.