Limit analysis of flow through conical converging dies

Abstract

A variety of mathematical models may be used to analyse plastic deformation during a metal-forming process. One of these methods—limit analysis—places the estimate of required power between an upper bound and a lower bound. The upper and lower bound analyses are designed so that the actual power or forming stress requirement is less than that predicted by the upper bound and greater than that predicted by the lower bound. Finding a lower upper bound and a higher lower bound reduces the uncertainty of the actual power requirement. Upper and lower bounds will permit the determination of such quantities as required forces, limitations on the process, optimal die design, flow patterns, and prediction and prevention of defects. Fundamental to the development of both upper bound and lower bound solutions is the division of the body into zones. For each of the zones there is written either a velocity field (upper bound) or a stress field (lower bound). A better choice of zones and fields brings the calculated values closer to actual values. In the present work, both upper and lower bound solutions are presented for axisymmetric flow through conical converging dies. For the upper bound triangular velocity fields have been solved and compared to previously published work on spherical velocity fields. It is found that each type provides a lower solution over a part of the range of process variables. A previously published lower bound solution for axisymmetric flow is refined.