Limit analysis of flow through conical converging dies

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.     

The analytical determination of friction for flow through conical converging dies

The effects of friction upon the intermediate and final distorted grids for wire drawing and/or extrusion were analytically studied for an assumed triangular velocity field. An upper-bound solution for the process was used. This solution predicted that the shape of the final and intermediate distorted grids were functions of the process geometry and of friction. Initially, combinations of reduction and semi-cone angle (α) were found for which the triangular velocity field was energetically preferred over an existing spherical velocity field. The analytical final distorted grids were then compared to experimentally obtained final distorted grids to determine the experimental friction. This was done by plotting calibration curves for distortion where friction served as the parameter and by comparing the actual distortion with the family of calibration curves.     

The strength of composite materials loaded in uniaxial tension under pressure

This paper presents a method for the evaluation of the tensile strength of composite materials made of a soft matrix with hard inclusions. The tensile strength is calculated as the force per unit area required for the onset of plastic flow in the material. In addition to the tensile strength, the severity of the core deformation and the rate of void formation are also estimated. The results are based on an upper-bound analysis with a two parameter family of velocity fields which accounts for both non-homogeneity of deformation and void formation.The model of the composite and the family of the velocity field used here have already appeared in published articles. Moreover, the application of the present method to estimate the tensile strength in cases when ductile deformation prevails (i.e. when there is no void formation) would lead to the same results as in the previous studies based on the same model and velocity fields.New here is the completeness of the analysis. All possible flow patterns, including for the first time void forming flows, were considered. This made it possible to show that the conditions required to ensure that one or another flow pattern prevails (i.e. those conditions found here and in other articles based on the same model) are not only necessary but also sufficient, at least from a theoretical point of view.The results of this paper have not yet been confirmed experimentally. Although the general trends are in the expected direction, so far no attempt has been made to find the degree of correlation between the theory and the behavior of real materials.