An analytical solution for tube sinking by strain rate vector inner-product integration

A new integration method of strain rate vector inner-product is proposed in present paper. First, cylindrical coordinate effective strain rate is established and expressed in terms of two-dimensional strain rate vector for tube sinking. Then, plastic power expressed by the vector inner-product is integrated term by term. Second, sum of the integrated results yields the total power. By minimization of the power, the stress effective factor, the maximum possible reduction Ro/Rof and the optimal die angle are obtained. It is proved that the expression of power by linear integration is similar with that by traditional immediate integration. In order to avoid wrinkling in one process, the reduction should be designed according to the constant wall thickness region defined by the ratio of the outer diameter to the original thickness of the tube. Also, the solution is compared with that of Avitzur. The basic law is analyzed.