含复杂孔口有限尺寸经典弹性体的边界元分析

Boundary Element Analysis of Finite Large Classical Elastomers with Complex Orifices

分析含不同孔口有限尺寸二维经典弹性体的应力集中问题的边界元。首先,基于平面应力问题的基本解,利用Betti互等定理,建立包含孔口的边界积分方程,采用二次插值函数及高斯积分离散含未知量的边界积分方程并重新组织格式。其次,对含圆形和椭圆形孔口的孔边环向应力进行数值求解,并与有限元方法的数值结果及无限大板的解析解进行对比验证。最后,分析在垂向拉伸作用下,板的尺寸、孔口的尺寸以及孔口分布对孔边应力的影响。数值实例结果表明,含单孔口的有限尺寸二维弹性板,孔口上半平面和下半平面孔边环向应力对称;多边形边数增加,孔口环向应力集中系数逐渐变小,且趋于圆形孔口。椭圆孔口长轴两个端点处的环向应力值达到最大,且孔边环向应力随着长轴与短轴比值的增加而增加;圆角四边形孔口的四个圆角处应力值达到最大,且环向应力随着孔口尺寸的增大而增大。含两个孔口的有限尺寸二维弹性板,孔口水平同轴时孔周环向应力值最大,且孔边环向应力随着孔口之间距离的增大而减小。

The boundary element analysis is conducted for the stress concentration problem of twodimensional classical elastomers with finite dimensions and different holes.Firstly,the boundary integral equation containing the hole is established by using Betti reciprocity theorem based on the fundamental solution of the plane stress problem.The boundary integral equation with the unknown quantity is discretized by quadratic interpolation function and Gaussian integral,and the discrete format is reorganized to form the linear equations with unified variables.And then,the circumferential stress at the edge of hole containing circular and elliptical is solved numerically,and the results are compared with the numerical results of finite element method and the analytical solution of infinite plate to confirm the effectiveness of the boundary element method.Finally,the influence of size of plate,size of hole and distribution of hole on side stress of hole under vertical tensile action is analyzed.The numerical results show that the circumferential stress of the upper and lower half plane of the orifice is symmetrical for the finite size two-dimensional elastic plate with a single orifice;as the number of polygonal edges increases,the circumferential stress concentration coefficient decreases gradually and tends to the circular stress concentration coefficient;the circumferential stress at the two ends of the long axis of the ellipse aperture reaches the maximum value,and the circumferential stress at the edge of the ellipse aperture increases with the increase of the ratio of the long axis to the short axis;the stress values of the four corners of a rounded quadrilateral aperture reach the maximum,and the circumferential stress increases with the increase of the aperture size;for a finite two-dimensional elastic plate with two orifices,the circumferential stress around the orifices is the largest when the orifices are horizontally coaxial,and the circumferential stress at the orifices decreases with the increase of the distance between the orifices.