Axisymmetric extrusion through adaptable dies—Part 2: Comparison of velocity fields

In Part 1 of this series of papers, six kinematically admissible velocity fields, along with the power terms, were developed for use in upper bound models for arbitrarily shaped dies for axisymmetric extrusion. The three base velocity fields in the deformation zone were derived:

(1) assuming proportional angles in the deformation zone,

(2) assuming proportional areas in the deformation zone, or

(3) assuming proportional distances from the centerline in the deformation zone.

In each case the base velocity was modified by an additional term comprised of two functions, each function containing pseudo-independent parameters. One function allows extra flexibility in the radial direction, and the second function allows extra flexibility in the angular direction. In Part 2, the results obtained in upper bound models for the six velocity fields for extrusion through a spherical die are compared to one another. The velocity fields are compared based upon: (a) the base velocity field, (b) the number and distribution of pseudo-independent parameters in the flexible functions, and (c) the form of the angular flexible function. A spherical extrusion die shape is used to evaluate and compare the three velocity fields. The results demonstrate that the sine-based velocity field is the best. Furthermore, a natural boundary condition exists which allows the shear surface associated with the streamlined portion of a die to energetically disappear. Part 3 uses the best velocity field to determine an adaptable die shape, which minimizes the extrusion pressure and compares the shape to the arbitrarily curved and streamlined die shape of Yang and Han.     

Axisymmetric extrusion through adaptable dies—Part 1: Flexible velocity fields and power terms

An adaptable die is one that not only produces the correct geometrical shape, but also is designed through an adaptable method to impart other desirable properties to the product or process. In this first part of a series of papers, six kinematically admissible velocity fields are developed for use in upper bound models for axisymmetric extrusion through various dies, including extrusion through adaptable dies. Three base velocity fields are presented:

(1) assuming proportional angles in the deformation zone,

(2) assuming proportional areas in the deformation zone, or

(3) assuming proportional distances from the centerline in the deformation zone.

The base velocity is modified by an additional term comprised of two functions. One function allows extra flexibility in the radial direction, and the second function allows extra flexibility in the angular direction. There are two forms of the second function, which meet the required boundary conditions. The flexibility function in the radial direction is represented by a series of Legendre polynomials, which are orthogonal over the deformation region. The power terms derived for these velocity fields for use in upper bound models are also presented.Part 2 of this series compares the results obtained in upper bound models for the six velocity fields for a spherical extrusion die. In Part 3, the use of the best velocity field for extrusion through streamlined dies is developed to determine the adaptable die shape, which minimizes the required extrusion pressure. Additionally, the adaptable die shape is compared with results from Yang and Han for arbitrarily curved and streamlined dies.     

Upper bound analysis of extrusion from square billets through circular and square/rectangular dies

Upper bound elemental technique (UBET) for prediction of extrusion pressure in three-dimensional forward extrusion process is presented. Using square/rectangular billets, the study of the effect of die land length has been extended for the evaluations of extrusion pressures to extrude sections such as circular, square and rectangular shaped sections with power of deformation due to ironing effect at the die land taken into account. The extrusion pressure contributions due to the die land evaluated theoretically for these shaped sections considered are found to increase with die land lengths for any given percentage reduction and also increase with increasing percentage die reductions at any given die land length. The effect of die land lengths on the extrusion pressures increases with increasing complexity of die openings geometry with rectangular section giving the highest extrusion pressure followed by circular with square section die opening, giving the least extrusion pressure for any given die reduction at any given die land lengths. The proper choice of die land length is imperative if excessive pressure buildup at the emergent section is to be avoided so as to maintain good quality and metallurgical structure of the extrudates. This paper was recommended for publication in revised form by Associate Editor Youngseog Lee Ajiboye, Joseph S. received his B.Eng, M.Eng, and PhD degrees in Mechanical Engineering from the University of Ilorin, Nigeria, in 1988, 1995 and 2006 res-pectively. Dr. Ajiboye is a lecturer in the Department of Mechanical Engineering, Uni-versity of Lagos, Nigeria. He is currently a Contract Research Scientist at KAIST Valufacture Institute of Mechanical Engineering, School of Mechanical, Aerospace & Systems Engineering, Korea Advanced Institute of Science and Technology, Daejeon 305 - 701, Korea. Dr. Ajiboye’s research interests include ECAE/P, determination of frictional effects in metal forming operations, upper bound and finite element in plasticity.     

On the optimal die curvature in deep drawing processes

The paper presents an attempt to increase the limit drawing ratio of deep drawing processes by searching an optimal die curvature, which minimizes the drawing load. The search done here for an optimal die curvature is based on experimental observations and followed by a detailed upper bound analysis. The analysis takes into account the non-steady character of the process (from a 2D circular plane blank into a 3D axisymmetric cup). The plastic flow along the die curvature is expressed in a toroidal coordinate system which seemingly describes more naturally a smooth velocity field along the real toroidal profile of the die. The outcome provides more closely the relationship between the energy dissipation rate and the die curvature so that a preferred die curvature is obtainable by energy minimization.Circular sheet blanks, made from aluminum and copper, were drawn through dies with different radii of curvature (with at least five repetitions at each radius) to capture the features of the optimal dies whenever exists.The main result is that under certain circumstances an optimal die curvature does exist. It depends largely on the drawing ratio and the blank/die interfacial friction, m, but appears quite insensitive to the initial thickness of the blanks. The optimal die curvature is pronounced in the cases where the frictional resistance is relatively low, otherwise it is indistinctable and remains practically undeterminable by designers.     

An investigation of the tribological interaction between die damage and billet deformation during MMC extrusion

This study deals with material flow behaviour during the extrusion process of a metal-matrix composite (MMC), and the effects of this behaviour on the damage to die flat surfaces. AA 6063 aluminium matrix composite billets reinforced with SiC particles (167 μm) were prepared using the stir-casting method for extrusion. Extrusion of the MMC billets were conducted at 500 °C with a ram speed of 2 mm s⁻¹ and an extrusion ratio of 25:1 under laboratory conditions. The extrusion die with two different channel profiles was manufactured from AISI H13 steel that was hardened, tempered and grounded. The flow patterns of the deformed billet during the MMC extrusion determine the positions of the SiC particles in the deformation zone. While some of the SiC particles flow within the deformed material, some flow at the deformed billet surface; these SiC particles play the most important role in the damage mechanism of the die-bearing surface and the geometry of the dead metal zone (DMZ). The possible damage to the die-bearing surfaces is severe at the entrance of the die bearing. On the other hand, some SiC particles are broken in this zone due to the severe deformation stress of the MMC billet.     

Axi-symmetric tube extrusion through a smooth conical or cosine die and over a conical or ogival mandrel: numerical construction of axi-symmetric slip-line fields and associated velocity fields

The numerical method of construction of axi-symmetric slip-line fields and their associated velocity fields suggested earlier for rod extrusion (Chitkara NR, Butt MA. Axi-symmetric rod extrusion through smooth conical, cosine and lat-faced dies, International Journal of Mechanical Sciences, submitted) and tube extrusion (Chitkara NR, Butt MA. Axi-synnetric tube extrusion through a flat-faced circular die: International Journal of Mechanical Sciences, 1997;39(3);341–366) is employed to construct slip-line fields and their associated velocity fields for a few cases of forward tube extrusion through smooth, rigid conical and cosine dies and over a smooth rigid conical or ogival mandrel. The computographic plots of slip-line fields and associated velocity fields are given in the form of weighted and directed velocity vectors as are the normal pressure distribution on both the die and the mandrel surfaces. The values of the non-dimensionalised mean extrusion pressures, are compared in each case with a similar case of plane strin extrusion and the results commented upon.     

On the pressure projection in elasto-hydrostatic lubrication problems—Part 2: FE approximation to observe the pressure projection

We have argued in Part 1 that the conventional Reynolds equation including the deformation of elastic disc does not obtain the pressure spike. However, as the experimental results in some papers have found the pressure projection, it can be intuitively explained as being caused by something hard in the oil film; by the pressure sensor, namely, manganin; or by the solidified lubricant. In this paper, the first of these possible causes is analyzed, for the first time, by introducing the fluid velocity variable in the direction of the oil film thickness and using the solid–fluid coupled finite-element (FE) method. It is shown by the numerical results based on Hamilton's experimental data that the pressure projection can be obtained by introducing the thickness of the pressure sensor, and that the projection profile varies with the sensor location on the disc. However, to solve the non-linearity of lubricant viscosity accurately is left for future research because the above FE analysis does not converge with the experimental data.     

On the pressure projection in elasto-hydrostatic lubrication problems—Part 1: Mechanical study of the Reynolds equation

The problems of elastic-hydrostatic lubrication (EHL) are discussed here from the mathematical and mechanical point of view. We begin with the formulation of the Reynolds equation and derive analytical solutions of the tilting plate and the disc models. We show that the so-called “Pressure spike” cannot be observed analytically and numerically by the newly developed computer codes, which is a reasonable simulation of one-dimensional flow from a mechanical point of view. Then, we discuss the possible factors that may be responsible for generating the conventional pressure spike including some unreasonable assumptions incorporated in formulating the numerical equation and in coding the computer program. A small pressure projection, however, has been experimentally observed. Therefore, some possible mechanical and mathematical explanations for it will be presented in a series of our subsequent papers.     

Scope of in-process measurement, monitoring and control techniques in machining processes—Part 3: In-process techniques for cutting processes and machine tools

In-process techniques for cutting processes and machine tools are surveyed through representative examples. Detection of cutting force and chatter vibration are especially important in cutting processes. As for machine tools, four measuring items will be important which include the driving system, the bearing and rotating systems, the temperature control system, and the monitoring system. They are the primary factors to be taken into consideration for achieving high machining accuracy.