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.     

Axisymmetric extrusion through adaptable dies—Part 3: Minimum pressure streamlined die shapes

In Part 1 of this series of papers, six kinematically admissible velocity fields, as well as the power terms, were developed for use in upper bound models for arbitrarily shaped dies for axisymmetric extrusion. Part 2 compared the results obtained in upper bound models for the six velocity fields through a spherical die shape and demonstrated that the sine-based velocity field was the best. In this final part, the application of the sine-1 field to extrusion through streamlined dies is developed. By fixing the values of two additional constants in the radial flow flexibility function, the two surfaces of velocity discontinuity, which separate the deformation zone from the incoming and outgoing material, will have no shear. In effect, the analysis for streamlined dies can be modeled without the surface of velocity discontinuity power terms. The results for an arbitrarily curved streamlined die, as proposed by Yang and Han, using the sine-1 velocity field and the cylindrical velocity field from the work by Yang and Han are compared. It is found that the upper bound model using the sine-1 velocity field predicts lower values for the extrusion pressure. A method to determine a streamlined die shape is proposed. The method allows flexibility between the entrance and exit by the use of a Legendre polynomial series for representation of the die surface. The method is termed an adaptable die design. The adaptable die design method is used to determine streamlined die shapes, which will minimize the pressure required for the extrusion process.     

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.     

A generalized spherical velocity field for bi-metallic tube extrusion through dies of any shape

In this paper, an upper-bound approach is used to analyze the extrusion process of bi-metallic tubes through dies of any shape with moving cylindrical shaped mandrel. A generalized kinematically admissible velocity field using spherical coordinate system is developed to evaluate the internal power and the power dissipated on frictional, velocity discontinuity surfaces and the total power. The extrusion process is also simulated using the finite element code, ABAQUS. Analytical results are compared with the results given by experiments of other researchers and also by the finite element method. These comparisons show a good agreement.     

Axi-symmetric tube extrusion through a flat-faced circular die: Numerical construction of slip-line fields and associated velocity fields

A numerical method of construction of axi-symmetric slip-line fields and their associated velocity fields [13–15] is applied to investigate the problem of axi-symmetric tube extrusion through smooth and rough flat-faced circular dies and the salient results presented in a computographic manner. A reasonably satisfactory agreement was observed between the estimated values of the non-dimensionalised specific extrusion pressures in each case when these were compared with the analytical and experimental work of Kudo [6] and that carried out by the authors. The deficiency in the technique used, the difficulties encountered and also some specific errors noticed during the construction of the fields, are highlighted and the results commented upon.     

A combined upper bound and finite element model for prediction of velocity and temperature fields during hot rolling process

In this work, velocity field and temperature distribution during hot strip rolling are predicted, employing a combined upper bound and finite element analysis. At first, a velocity field is proposed utilizing the principle of volume constancy, and then the velocity field is modified by means of upper bound theorem. At the same time, a thermal-finite element analysis is utilized to determine temperature distribution within the metal as well as to calculate the flow stress of deforming material. The model is capable of considering the effects of different factors on temperature and velocity distributions such as rolling speed and interface heat transfer coefficient. In order to verify the predictions, the calculated time-temperature curves as well as roll forces have been compared with the measured ones and a good agreement has been observed. The main advantage of the present model is that the model requires relatively lower computational effort in comparison with the required one in standard finite element codes.     

Development of an expert system for cold forging of axisymmetric product

This paper deals with an automated computer-aided process planning and die design system by which the designer can determine operation sequences even if they have little experience in process planning and die design for axisymmetric products. An attempt is made to link programs incorporating a number of expert design rules with the process variables obtained by commercial FEM softwares, DEFORM and ANSYS, to form a useful package. The system is composed of four main modules. The process planning and the die design modules consider several factors, such as the complexities of preform geometry, punch and die profiles, specifications of available multi-former, and the availability of standard parts. They can provide a flexible process based on either the reduction in the number of forming sequences by combining the possible two processes in sequence, or the reduction of deviation of the distribution on the level of the required forming loads by controlling the forming ratios. In the die design module optimal design technique and the horizontal split of the die insert were investigated for determining appropriate dimensions of components of the multi-former die set. It is suggested that the proposed method can be beneficial for improving the tool life of the die set in practice.     

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.     

Combined rod and tube extrusion: Numerical solution of axi-symmetric slip-line fields and associated velocity fields

Numerical method of construction of axi-symmetric slip-line fields and associated velocity fields in the form of computographic plots of axial and radial velocity components, earlier given by the authors for rod extrusion and tube extrusion, was extended to the solution of two types of combined rod and tube extrusions: (i) where the rod and tube are extruded in opposite direction to each other and (ii) where, both the rod and tube move in the same axial direction as the punch movement during the extrusion process. Computographic plots of slip-line fields and associated velocity fields in the (r, z) plane are given and the deficiencies noticed in the solutions are highlighted. The results of specific punch pressures are observed to compare well with those estimated, using simple upper bound analysis and the experimental work of the authors.     

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.     

Forward extrusion through steadily rotating conical dies. Part I: experiments

Forward extrusion of a cylindrical rod from a round billet was carried out through steadily rotating conical dies. Die rotation was shown to decrease the extrusion load and impose a twist shear strain into the partially extruded billet. The material twisting occurred both inside the container and inside the convergent die. However, not all the rotary work was transferred into shearing the bulk of the material, and this led to circumferential slippage at the rotating tool/material interface. The twisting degree on the outer surface of the material is quantified by a simple measurement technique. The influences of various process parameters on the extrusion load are also studied, including die semi-cone angle, die rotating speed and lubrication condition.     

Forward extrusion through steadily rotating conical dies. Part II: theoretical analysis

Forward extrusion of a cylindrical rod from a round billet was carried out through steadily rotating conical dies. Die rotation was shown to decrease the extrusion load and impose a twist shear strain into the partially extruded billet. The material twisting occurred both inside the container and inside the convergent die. However, not all the rotary work was transferred into shearing the bulk of the material, and this led to circumferential slippage at the rotating tool/material interface. The twisting degree on the outer surface of the material is quantified by a simple measurement technique. The influences of various process parameters on the extrusion load are also studied, including die semi-cone angle, die rotating speed and lubrication condition.     

Scuffing — a review: Part 2: The mechanism of scuffing

Surface films and experimental evidence of scuffing in discs and gears are discussed, and the importance of hydrodynamic and boundary lubrication mechanisms have been analysed. The breakdown conditions of these mechanisms and thermal instability work are summarised.     

Indentation of ring-stiffened cylinders by wedge-shaped indenters—Part 2: Scale model tests

The main objective of this account was to examine the effects of scale on the deformation and penetration processes of ring-stiffened thin metal cylindrical shells with fully fixed ends under quasi-static loading by wedge-shaped indenters. To this end three sizes of geometrically similar shells of the same material with outer diameters of 157.7, 236.5 and 315.4 mm were tested under geometrically identical conditions. The shells, which incorporated flat ring-stiffeners and deep frames (bulkheads), were loaded laterally at the midsection to failure. The results demonstrate that they conform to simple scaling laws.     

Friction induced vibration for an aircraft brake system—Part 2: Non-linear dynamics

Non-linear dynamics due to friction induced vibrations in a complex aircraft brake model are investigated. This paper outlines a non-linear strategy, based on the center manifold concept and the rational in order to evaluate the non-linear dynamical behaviour of a system in the neighbourhood of a critical steady-state equilibrium point. In order to obtain time–history responses, the complete set of non-linear dynamic equations may be integrated numerically. But this procedure is both time consuming and costly to perform when parametric design studies are needed. So it is necessary to use non-linear analysis: the center manifold approach and the rational approximants are used to obtain the limit cycle of the non-linear system and to study the behaviour of the system in the unstable region. Results from these non-linear methods are compared with results obtained by integrating the full original system. These non-linear methods appear very interesting in regard to computational time and also necessitate very few computer resources.     

Stochastic model updating: Part 2—application to a set of physical structures

The application of a stochastic model updating technique using Monte-Carlo inverse propagation and multivariate multiple regression to converge a set of analytical models with randomised updating parameters upon a set of nominally identical physical structures is considered. The structure in question is a short beam manufactured from two components, one of folded steel and the other flat. The two are connected by two rows of spot-welds. The main uncertainty in the model is concerned with the spot-weld but there is also considerable manufacturing variability, principally in the radii of the folds.     

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.     

Cross-sectional deformations of rectangular hollow sections in bending: Part I — experiments

This investigation deals with deformations of individual cross-sectional members as flanges and webs in bending of rectangular hollow sections. Part I describes the experimental work, while analytical models developed to predict pre- and post-buckling deformations are presented in a paper to follow (Part II). The experimental program involved rectangular single- and double-chamber aluminium alloy AA6060 extrusions with three different wall thicknesses. The profiles were given two distinct heat treatments to obtain different hardening characteristics. Multiaxial tests were performed to determine the mechanical properties of the materials. The profiles were then bent into a number of different bend radii. Measurements of strains, curvatures, deflections and bending forces were taken. The results show that cross-sectional distortions take place from the very beginning of bending; at first in the form of a uniform sagging-like deformation along the entire length of both sides of the bend until the inner (compressive) flange buckles into several half-waves, superimposing the pre-deformation modes. The instant at which buckling occurs is found to be mainly associated with the width-to-thickness ratio of the flange and the strain hardening characteristic of the material. The magnitude of pre- and post-deformations, however, appears to be more directly related to the actual width of the flange than to its slenderness. The material stress–strain curve is shown to have an increasingly effect on the distortions of members directly sustained to buckling as bending proceeds beyond the onset of buckling, leading to severely concentrated deformations for sections made of low hardening materials. The material has less impact on sagging of the outer (tensile) flange.     

Cross-sectional deformations of rectangular hollow sections in bending: Part II — analytical models

In this part, analytical models to predict the deflection of cross-sectional members such as flanges and webs are developed. The models are based on the deformation theory of plasticity along with the energy method, using appropriate shape functions capable of including the restraining effect of adjacent members. The present method provides explicit solutions of cross-sectional deformations prior to buckling, onset of buckling, as well as post-buckling deformations at different stages of bending. The predictions show that the suck-in of the tensile flange is closely related to geometry parameters, particularly the flange width. Plastic anisotropy appears to be the most significant material parameter. The width-to-thickness ratio tends to be the governing parameter with respect to buckling of the inner (compressive) flange. Also, the strain hardening of the material has a major effect on onset of buckling as well as post buckling deformations. Upon continued bending after buckling, the wavy deformation of the inner flange develops more rapidly than the more uniform deformation of the outer (tensile) flange. For relatively compact sections, however, the deformation mode of the compressive flange resembles that of the tensile flange without any typical buckling waves. There are also obvious interactions between deformations of different members. Comparing the theoretical predictions with the experimental results presented in Part I, a reasonably good agreement was found.     

The weight vector theory of Coriolis mass flowmeters — Part 2. Boundary source of secondary vibration

The purpose of this short paper is to draw attention to a boundary source of secondary vibration in Coriolis mass flowmeters. This is important in the calculation of meter sensitivity using the weight vector theory if the effect of fluid viscosity is to be taken into account in the vibrational flow.