极坐标系下扇形截面杆扭转问题的有限差分法

THE FINITE DIFFERENCE METHOD ON TORSION PROBLEM OF PRISMATIC BARS WITH SECTOR CROSS-SECTIONS UNDER THE POLAR COORDINATE SYSTEM

导出了极坐标系下扇形截面杆扭转问题的差分格式;结合Mathcad编程用逐次超松弛迭代法求出了扭转应力函数差分值;采用复化二维辛普生求积公式计算了抗扭刚度;求出了相应的扭转应力的差分解;给出了张角为2π时裂纹尖端附近的第三型应力强度因子.

The difference scheme of torsion problem of a bar with a sector cross-section under the polar coordinate system is derived. With Mathcad software, using the Successive over Relaxation Method, the difference values of the torsion stress function are determined; with the compound two-dimensional Simpson integral formula, the torsion stiffness of the sectoral cross-section is obtained; consequently the difference solution to the torsion stress is obtained. When the field angle is 2π, the third stress intensity factor at the crack tip is computed.