关键词: 扁带拉拔, 柱坐标, 应变速率矢量, 内积, 逐项积分, 解析解

Abstract: A new integration method is proposed to simplify the strain rate vector inner product by the mean value theorem in a cylindrical coordinate system. The equivalent strain rate of the strip passing through a wedge-shaped die during drawing/extrusion for plane deformation is first expressed in terms of two-dimensional vector. Then, the strain rate ratio function and direction cosine of the vector are determined by the integral mean value theorem. Finally, the termwise integration and summation of the inner product are done to give an upper-bound analytical solution to the stress state factor nσ and optimal die angle αopt. An example is given to compare the stress state factors thus solved under conditions of different α and m values with those by Avitzur's elliptic integral. The results show that if α = 15° and the friction factor m varies, the error of the value of drawing force by this analytical solution relative to that by the numerical solution resulting from elliptic integral isn't higher than 0.05%, and the absolute error of the stress state factor resulting from elliptic integral ξ(α) isn't higher than 0.002. In addition, the ultimate pass reduction Ε increases with increasing αopt and decreasing m.