Limit Analysis of Plane-Strain Extrusions (Metal Forming).

Abstract

The accomplishment of this dissertation work consists of a general limit analysis algorithm for plane-strain problems, two finite element models to approximate the continuum, a minimization procedure to approach the limit solutions, and several new solutions of plane-strain extrusion problems as well as some classical solutions reproduced by the new method for the purpose of verification. The algorithm provides a powerful tool for analyzing this important class of problems frequently encountered in structural mechanics and manufacturing processes. The theoretical foundation of the algorithm is a duality theorem which equates the least upper bound to the greatest lower bound. Therefore, approaching the extremal point of the functionals from either direction can solve the same problem. We choose the more convenient method of minimizing the upper-bound functional. Unlike the classical upper-bound methods which assume the kinematic function of the flow field, the algorithm automatically converges to the correct mode (a correct mode if it is not unique) by an iterative scheme. The advantage of this general method is most evident for problems with complex geometry and boundary conditions for which the correct deformation field will be difficult to assume. Limit solutions of a continuum is known to admit certain discontinuous behavior (such as slip b and s or velocity discontinuity). The finite element analysis must account for the contribution in the functional from these discontinuities. Both conforming and non-conforming finite elements are used. Each has certain advantages and disadvantages. The extrusion problems with square dies and wedge-shaped dies are studied. In addition, the friction effect between die face and the extrudate is taken into consideration. The results obtained are compared whenever possible with the classical solutions which are in good agreement. Several new solutions with a wide geometric and friction parameter variation have shown trends that match observed experimental data. The power of the algorithm for solving general realistic problems in plane-strain extrusion is satisfactorily demonstrated.