一個用於分析含裂縫之有限異向彈性板的新邊界元素法

Abstract

本文利用Wu and Hsiao(2015)所提出的邊界積分方程式，建立一個新的邊界元素法，以分析含裂縫之有限異向彈性板受彎矩或剪力負載的問題。Wu and Hsiao(2015)所提的邊界積分方程式是以旋轉角梯度為參數，本文則在裂縫上進一步引入旋轉角梯度的差值，即旋錯密度，並據以建立含裂縫有限板的邊界積分方程式。之後將邊界積分方程式離散化，將邊界離散成Y個元素項數，而在裂縫上的邊界積分方程式可利用高斯-謝比雪夫積分法(Gauss-Chebyshev integration formulas)離散化，將裂縫離散成N點，表示成矩陣形式，再利用裂縫尖端閉合的條件來彌補不足的方程式，建立聯立方程式來求解。 為驗證本法的有效性，本文的算例涵蓋含單裂縫、雙裂縫、三裂縫、多裂縫之有限幾何板，分別受彎矩、剪力的情況，並考慮等向性、正交性、單層異向性、雙層異向性等材料。所得結果與現有文獻比較顯示，使用本新的邊界元素法，以少數的元素個數即可精確的求得各裂縫尖端的應力強度因子。

Based on the boundary integral equation proposed by Wu and Hsiao (2015), a new boundary integral equation is established for the analysis of anisotropic elastic plates under bending or transverse shear loading. The boundary integral equation contains gradients of rotation angle on the exterior boundary as parameters and uses the differences between angles of rotation gradients of the crack faces which are called disclination densities which establish the boundary integral equation for finite plate with cracks. The method of the boundary integral equation discretization is to divide into several parts of line in surrounding boundary. The parameters is constant on each line. Using Gauss-Chebyshev integration formulas and integral equation on cracks to express the disclination density in specific integral points. Crack closure conditions are used to provide additional equations. To verify the effectiveness of the proposed method, numerical examples provided include finite plates containing one or more cracks under bending and transverse shear loading; Isotropic, orthotropic, monolayer anisotropic and bilayer anisotropic materials are considered. Comparison of the numerical results with those in the literature shows that the proposed method yields accurate values of the stress intensity factors at each crack tip with few elements for the exterior boundary and integration points on the crack lines.