A new formulation for three-dimensional extrusion and its application to extrusion of clover sections

The metal flow in the extrusion process is an important factor in controlling the mechanical property of the extruded products. It is, however, difficult to predict the metal flow in three-dimensional extrusion of complicated sections due to the difficulty in representing the geometry of the die surface and in expressing the corresponding velocity field. In this study a new kinematically admissible velocity field for a generalized three-dimensional flow is derived, in which the flow is bounded by the die surface expressed by an analytic function. Then, by applying the upper-bound method to the derived velocity field, the flow pattern as well as the upper-bound extrusion pressure are obtained. As a computational example, extrusion of clover sections from round billets is chosen. A new method of die surface representation is proposed by which there is a smooth transition of die contour from the die entrance to the die exit. Computation is carried out for work-hardening materials such as aluminium and steel. The analysis takes into account the effect of product shape complexity, lubrication condition and reduction of area on extrusion pressure, average effective strain and distribution of effective strains on the cross-section of the extruded product.     

A generalised CAD/CAM solution to the three-dimensional off-centric extrusion of shaped sections: analysis

In analytical method based on the upper-bound theory is proposed to investigate the three-dimensional off-centric extrusion of arbitrarily shaped sections from arbitrarily shaped billets through linearly converging and smooth curved dies. A set of generalised kinematically admissible velocity fields are derived on the assumption of Bezier-type streamlines that provide compatibility of surface representation with most CAD/CAM systems. To obtain a more realistic non-uniform metal flow, a special velocity function was also incorporated into the derived velocity fields and work hardening effect of the material considered. Based on the method presented, for a given reduction in area, material property, friction condition and off-centric positioning of the exit cross-section, predictions of the deforming grid pattern, curvature of the extruded product as well as upper-bound to the extrusion pressures may be obtained. The above procedure is highlighted and commented upon.     

A new method for bearing design in the metal extrusion of profiled sections

The International Journal of Advanced Manufacturing Technology - Bearing design is a major issue in controlling material flow and producing high-quality extruded products. Adjusting the bearing...     

Analysis of three-dimensional extrusion of sections through curved dies by conformal transformation

A new method of analysis is proposed for the extrusion of arbitrarily shaped sections through curved die profiles. A kinematically admissible velocity field is found by deriving the equation of a stream line. Conformal transformation of a unit circle onto a section is utilized in the derivation. The upper-bound method is then applied to determine the extrusion pressure for the rigid-perfectly plastic material. The redundant work relating to the velocity discontinuities at the entrance and the exit is included in the formulation. The general formulation for an arbitrary cross section is obtained by use of conformal transformation. The upper-bound pressure for extrusion through curved die profiles is computed for a complex section with a curved boundary. Two curved die profiles widely used are chosen to compare the effects of die profiles. From the derived velocity field, the upper-bound extrusion pressures are also computed for the extrusion of regular polygons and rectangles of various aspect ratios. The effects of sectional shape, die profile and interfacial friction at the die surface are discussed.     

Further investigation into extrusion of trocoidal gear sections considering three-dimensional plastic flow

In the present study, theoretical development proposed in previous work carried out in extrusion of clover sections in relation to a generalized die design method is presented in the extended scope for three-dimensional extrusion of trocoidal gear sections from round billets with experimental verification. Computations are carried out for clover and trocoidal gear sections. The CAD/CAM of the suggested dies is introduced for the experiments. Experiments are carried out for a clover section and a trocoidal gear section with eight teeth. Al 2024 aluminum billets were used as the working material. Half-cut specimens are used for flow visualization of the extrusion process. The theoretical predictions are in good agreement with the experimental results in extrusion load and metal flow.     

Application of an upper bound method to off-centric extrusion of square sections, analysis and experiments

The previously given analytical method [1], which was based on the upper-bound theory, was used to design the streamlined dies and to investigate the three dimensional off-centric extrusion of circular sections from initially circular billets through linearly converging (ruled-surface) and smooth curved (advancedsurface) dies. For a reasonably correct upper-bound to the load, a set of generalised kinematically admissible velocity fields were derived on the assumption of Bezier-type streamlines by incorporating a special velocity function that takes care of the non-uniform material flow. Based on the present method, for a given reduction in area, material property, friction condition and off-centric positioning of the exit cross-section, computations were carried out to predict the upper-bound to the extrusion pressures, the deforming grid patterns and curvature of the extruded product. Experiments were carried out for some off-centric circular sections with varying die lengths and reduction in areas. A sophisticated CAD/CAM package was used in conjunction with the CNC and EDM processes to manufacture the streamlined dies for the off-centric extrusion of circular rods. Both the solid and split-type specimens made of tellurium lead under both the lubricated and the dry conditions were used for experimental investigations in order to assess pressures, deformation modes and to visualise the deforming grid patterns. The theoretical predictions were observed to be in good agreement with the experimental.     

Formulation of a new generalized kinematically admissible velocity field with a variable axial component for the forward extrusion of shaped sections

For most three-dimensional analytical solutions proposed for the extrusion of shaped sections, the axial component of the velocity vector has been assumed to be constant at each cross section throughout the deforming zone. This shortcoming means that these velocity fields are not in accordance with the reality of the extrusion problem, and hence, the upper bounds based on such fields give high values for the extrusion pressure. To overcome this, a new formulation has been presented in this paper for which a kinematically admissible velocity field has been developed using a variable axial velocity component. For this purpose, curved surfaces of velocity discontinuities at the entry and exit have been proposed and incorporated into the formulation for the extrusion of shaped sections from circular billets. As an example, a square profiled section has been chosen for the extrusion problem. The upper bound on extrusion pressure was computed using the new formulation. It was shown that the initial velocity discontinuity surface at the entry to the deforming region was flat, and as one travels into the deforming region towards the exit section, the velocity discontinuity surface gradually became convex, having the highest convexity at the exit section. This was contrary to what has been suggested in the literature so far. The measure of convexity depends on the extrusion parameters which have been investigated in this work. Experiments were also carried out to verify the theoretical results, and good agreements were observed between the two. Comparison of the present results with similar previous works showed good improvements as well.     

An analytical solution for the spread extrusion of shaped sections

Spread extrusion could be used for manufacturing of wide profiles in the extrusion industry. In this paper a new method of design and analysis has been presented for such a problem. Special dies were designed for profiled sections such as square, rectangular, elliptical and cross shapes. These dies force the material to flow sideways and spread so as to extrude sections with wider dimensions than the initial billet or the maximum container diameter. The geometry of the deforming zone in the die was formulated and based upon that, a kinematically admissible velocity field was derived. Using this velocity, we estimated the field upper bound on extrusion power. Profile sections with different aspect ratios were investigated and the influence of other process parameters such as friction and reduction of area on the extrusion pressure were studied. Optimum die lengths for each die were calculated so as to minimize the extrusion pressure. Finite element analysis for the numerical simulation of the process was also carried out. The finite element results were also used as an aid to the design process of the extrusion dies. Dies were manufactured for different sections such as square, rectangle, and ellipse and cross shapes. Experiments were carried out to obtain data to verify the theory. Comparison of the results showed good agreement between the theoretical, numerical and experimental data. It was concluded the present method could be used to design dies for the spread extrusion of different shaped dies.     

Round-to-channel section extrusion through linearly converging die: a three-dimensional analysis

Metal flow in the extrusion process is an important factor in controlling the mechanical properties of the extruded products. It is, however, difficult to predict the metal flow in three-dimensional extrusion of sections due to the involvement of re-entrant corners. The present work is an attempt to find an upper bound solution for the extrusion of channel section from round billet through the taper die. The rigid-perfectly plastic model of the material is assumed, and the spatial elementary rigid region (SERR) technique is presented for which the kinematically admissible velocity field is found out by minimizing the plastic dissipation of power. The presented analysis allows for specification of process control parameters and their relation to extrusion load, equivalent die angle, reduction ratios and friction factor.     

An approximate method for a three-dimensional analysis of rolling

A theory, based on the extremum principle for rigid perfectly plastic materials, is given for the analysis of three-dimensional deformation in rolling. Theoretical solutions are obtained for single-pass rolling in terms of sideways spread, roll torque and the location of neutral points. The results on spread and roll torque showed excellent agreement with those found by experiments in the literature.     

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.     

A three-dimensional finite element method for plate bending

A three-dimensional finite element method is proposed for plate bending. The displacement function is assumed to have a particular form with respect to the thickness variable.The assumed displacement function is substituted into the three-dimensional potential energy functional and a two-dimensional variational problem emerges.The two-dimensional problem is treated by the finite element method and it is seen that for conforming solutions the only requirement is continuity of the basis functions.The method includes transverse shear and thickness effects and may be used for both thin and moderately thick shells.To illustrate the method and to compare it with other results, the problems of a square plate under point and distributed loads and with simply supported and clamped boundary are treated numerically.     

The presentation of three-dimensional reconstructions from serial sections

A simple and effective way is described for showing particular structures in serial sections. By using waterproof inks and acetate sheets supported on a frame immersed in water and illuminated from beneath, the information from up to sixty sections can be easily visualized. This technique is particularly useful for displaying enclosed structures and by a suitable choice of colours enables special emphasis to be given to selected components. The information can be recorded simply through stereo photography.     

A method for sectioning and handling frozen sections for scanning electron microscopy

Glutaraldehyde-fixed insects were embedded in Tissue-Tek^R and sectioned on a liquid-nitrogen-cooled stage. The sections were sandwiched between two layers of microscope lens paper and all postsectioning treatments were carried out on this sandwich, including dehydration and critical-point drying. In some cases, the sections were placed on filter membranes and lyophilized. These procedures produced intact specimens which maintained internal morphology as well as inter- and intracellular integrity without expensive or specialized equipment.     

铝型材挤压过程的一种数值模拟方法

通过优化几何模型,采用有限元法与有限体积法相结合,并在有限体积模拟阶段进行分步计算模拟的方法,成功地进行了一薄壁大挤压比铝型材挤压过程的数值模拟,获得了型材挤压过程中的材料流动速度、应力、应变和温度分布图,并对其结果进行讨论.     

A new method for three-dimensional skeleton graph analysis of porous media: application to trabecular bone microarchitecture

This paper introduces a new three-dimensional analysis of complex disordered porous media. Skeleton graph analysis is described and applied to trabecular bone images obtained by high resolution magnetic resonance imaging. This technique was developed bearing in mind topological considerations. The correspondence between vertices and branches of the skeleton graph and trabeculae is used in order to get local information on trabecular bone microarchitecture. In addition to real topological parameters, local structural information about trabeculae, such as length and volume distributions, are obtained. This method is applied to two sets of samples: six osteoporosis and six osteoarthritis bone samples. We demonstrate that skeleton graph analysis is a powerful technique to describe trabecular bone microarchitecture.     

A three-dimensional or penalty finite element method for plate bending

This paper, a sequel to an earlier paper by the same authors, uses a three dimensional or penalty function approach to obtain finite element solutions for plate bending problems. The advantage in using this approach is that the finite element subspaces need only be continuous; the disadvantages are that more functions are needed.In our previous paper, we used piecewise quadratic and linear functions, in the present paper we use piecewise cubic and quadratic functions with a resultant reduction of the discretization error. We also suggest an interpolation method or an a priori choice of the thickness to length ratio which are appropriate for thin or moderately thick plates. Numerical results are given for clamped and simply supported plates with point and distributed loads.