A generalized method for analysis of three-dimensional extrusion of arbitrarily-shaped sections

Despite increasing demand for and application of three-dimensional extrusion of various sections through continuous dies, so far very little work has been done by systematic and general analysis to predict the plastic flow properly. For effective die design, efficient design method and the related method of theoretical analysis are required for extrusion of complicated sections. In generalized three-dimensional extrusion of sections through continuous dies, a new method of die surface representation, using blending function and Fourier series expansion, is proposed by which smooth transitions of die contour from the die entrance to the die exit are obtained. The flow patterns as well as the upper-bound extrusion pressures are obtained on the basis of the derived velocity field. The effects of area reduction, product shape complexity, die length and frictional condition are discussed in relation to extrusion pressure, the distorted grid pattern and distribution of the final effective strain on the cross-section of the extruded billet. As computational examples for arbitrarily-shaped products rounded rectangles and ellipses are chosen for the extruded sections. Experiments are carried out for aluminum alloys at room temperature for a rounded square section and an elliptic section. In order to visualize the plastic flow, the grid-marking technique is employed. The theoretical predictions both in extrusion load and deformed pattern are in good agreement with the experimental results.     

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.     

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 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 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.     

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.     

聚合物挤出成型模流平衡系数修正

挤出模具的合理设计对于获得挤出产品所需的外形和精确的尺寸非常重要。由模流不平衡引起的,伴随着挤出胀大产生的不均匀的几何形变,是挤出模具设计的难点所在。传统的模流平衡系数需要对截面进行子区域划分,但是对于截面形状较为复杂的异型材制品,子区域划分具有较强的不确定性。对传统的聚合物挤出成型的模流平衡系数进行修正,提出更有利于数值计算和理论研究的修正的模流平衡系数,并用数值方法对其进行验证。修正后的模流平衡系数消除了数值分析过程中挤出速度和模具出口截面积对模流平衡判断的影响,在产品形状一定时,模流平衡系数只和材料本构相关,因此可以比较准确地反映模具出口的模流平衡情况。修正后的模流平衡系数主要应用于逆向挤出和优化设计求解口模形状。     

冷冲压钢球组合模具过盈组装挤压力的计算

依据材料的塑性变形原理,详细地计算了组合模具装配时所需要的挤压力大小,进而阐述了验算有关丝杠螺母副设计时参数选择正确与否所遵循的原理。     

Localization of plastic deformation along grain boundaries in a hardening material

The deformation characteristics of ductile polycrystalline materials at elevated temperatures were studied numerically by considering a square segment of material subjected to different stress modes. The initial polycrystalline microstructure was generated numerically. The micromechanical modeling of the deformation was performed using the material point method. The constitutive behavior was taken to be isotropic, elastic–plastic with linear hardening. To effectively simulate the deformation at high homologous temperatures, the grain boundary region was assumed to be a layer of finite thickness, bearing a lower yield strength than the interior of the grain. Complex deformation patterns were observed. The deformation is dominated by the formation of zones of concentrated instantaneous deformation along grain boundaries. These zones form paths that vary with time, and depend on the microstructure and macroscopic loading mode. The macroscopic stress, as well as the microscopic stresses at various locations within the grain boundary and grain interior, were seen to correlate with the instantaneous deformation pattern throughout the deformation history. For the sample analyzed, with a relatively small macroscopic strain of 0.01, the grain boundary region experienced plastic strains as large as 0.20, a result which provided a strong indication that failure will initiate in the grain boundaries. Implications of this modeling study to actual creep deformation and failure are discussed.     

Theoretical analysis of extrusion of rectangular, hexagonal and octagonal composite clad rods

Composite clad rods with non-axisymmetric cross-sectional areas are commercially important owing to their extensive industrial applications such as in electrodes, conductors and chemical devices. For instance, in the processing of superconductor wire, increasing the packing density involves closely packing superconductor rods which have a hexagonal cross-section next to each other inside the high purity copper tube. During extrusion process, non-uniform deformation tends to occur because the core and sleeve of a composite clad rod is usually composed of materials with different mechanical properties. The first recognized paper on round-to-square drawing/extrusion with variable corner radius was presented by Boer et al. in 1979. In this study, we present a model based on upper-bound theorem to analyze the extrusion of composite clad rods with non-axisymmetric cross-section. Velocity fields for both core and sleeve are generated with the assistance of a product's cross-sectional profile functions. Products with rectangular, hexagonal and octagonal sections are chosen as the study objects. Also discussed herein are numerical results for various process variables such as semi-die angle, reduction of area, frictional condition of die, and product shape complexity. According to these results, the extrusion pressure and product dimensional change are closely related to the process variables.     

Analysis of axisymmetric extrusion of rods by the method of weighted residuals using body-fitted coordinate transformation

The method of weighted residuals with the use of the finite difference method based on a coordinate transformation to nonorthogonal curvilinear coordinate system to fit the boundary of an arbitrary shape is proposed to determine the stresses and strains in axisymmetric extrusion of rods through continuous dies. By using the body-fitted coordinate transformation, the finite difference expressions at and adjacent to the boundary do not need any interpolation between boundaries and interior grid points. The errors defined for governing equations and boundary conditions are minimized by introducing the method of weighted residuals. The work-hardening effect is incorporated by integrating the effective strain rate along the flow line. As computational examples, four types of dies with different reductions of area are chosen. Experiments are carried out for steel and aluminum alloys at room temperature. The computed results are compared with the experimental results as well as with those by the finite element method. The extrusion load and the flow pattern computed by the present method are in excellent agreement with the experimental results and those by the finite element method.     

An analysis for extrusion of helical shapes from round billets

A method of analysis is proposed for three-dimensional extrusion of a helical shape from a round billet. It is reported that a helical shape can be made by hot extrusion through a square die. In this paper, it is suggested that a helical shape be effectively cold extruded through a continuous die with appropriate lubrication. The extrusion of helical shapes can find practical application in some useful products. However, the analytical method regarding this kind of extrusion has not been attempted so far.A kinematically admissible velocity field is derived for the extrusion model where a round billet is extruded into a twisted helical section with a long elliptic cross section. The axis of the cross section is rotating during extrusion. By assuming proper stream surfaces, the velocity field is obtained by deriving the equation of a stream line. Then, an upperbound solution is formulated for the rigid-perfectly plastic material. Computation for the upperbound pressure is carried out for various process variables such as reduction of area, friction, rotation of axis, aspect ratio of a product, die length and overall die profiles.     

Three-dimensional analysis of round-to-angle section extrusion through straight converging die

The increasing interest in the modeling of metal-forming processes in recent years has brought the development of different analytical and/or numerical technique. However, due to the complexity nature of the problem, most of the attempts are made with plain strain assumptions. Among the different techniques used, the upper bound method is a convenient tool for evaluating the rate of work in processes involving predominantly plastic deformation of rigid/perfectly plastic material. The present study is an endeavor to remodel and apply the spatial elementary rigid region technique for analyzing extrusion of angle-section bars from round billets through the linearly converging die. Optimized values of the nondimensional average extrusion pressure at various area reductions have been computed and compared with experimental results. It is observed that the proposed technique can be used effectively with adequate accuracy to predict the optimal die geometry which requires a minimal forming stress at different reduction of areas and friction conditions.     

精密金属异型挤压塑性成形及模腔优化共形解析

针对异型材挤压塑性成形及模腔建模理论的焦点课题,借助近代复变共形映射理论和金属塑性成形理论成果,将三维金属异型材挤压塑性流动问题转化为二维轴对称成形问题,求解了金属异型塑性流动的流函数、速度场及应变速度场等数学解析模型,建立了金属异型挤压塑性流动通用的三维解析方法。应用能量极值原理,求解了异型材挤压成形及优化模腔工程建模的理论课题。     

Stress profiles at contact surface in ring compression test

A perfectly plastic material has been employed as a model material in simulation to analyze numerically the ring compression process, especially to examine the deformation patterns along the die/workpiece interface, which is strongly related to the frictional condition at the contact boundary. The main objective is to provide the deformation characteristics in detail in ring compression, especially at the tool/workpiece interface. The surface flow patterns at the contact boundary in ring compression are summarized and analyzed in terms of surface expansion, surface expansion velocity, pressure distributions exerted on the die surface, relative sliding velocity between die and workpiece, and sliding distance along the die surface. Movement of neutral positions and folding phenomenon are also investigated to see the effect on the deformation patterns at the interface, that is, geometrical change, which is important to measure the frictional condition at the interface using calibration curves. Finite element (FE) simulation using rigid-plastic finite element code has been performed for analysis. The results of this study reveal that surface expansion as well as other surface flow patterns, such as sliding velocity and so on, shows different and distinctive characteristics between low and high frictional conditions at the interface. This is directly related to the movement of neutral positions and folding, which affects the sensitivity of dimensional changes to tribological conditions at the interface.     

Rate-dependent transient extrusion

Plane strain slip line field solutions are developed for deformation at the edge of a cylindrical billet when this is backward extruded to form a thin-walled cup. The extrusion pressure for a perfectly plastic material is then derived assuming homogeneous compression in the centre of the billet. A geometric factor, derived from the slip line field, is identified which permits the extrusion pressure to be determined for a nonlinear viscous material.Theoretical results agree well with recent experimental data on back extrusion of highly rate-sensitive superplastic alloys. It is shown that an apparent steady state exists in the process and that extrusion pressure increases with ram speed with a rate index approximately equal to that of the material. The punch profile shape has a greater effect on extrusion pressure for viscous materials than for perfectly plastic ones. The method of analysis appears to have a generality beyond the particular process considered and may be used to optimize extrusion equipment designs.     

铝合金轴固定件热挤压成形的有限体积法模拟研究

由于变形剧烈，复杂铝型材挤压成形有限元模拟会因网格不断重划分而精度欠佳。文中基于可以有效避免网格重划分难题的有限体积法，对铝合金门轴固定产品的热挤压过程进行数值模拟，详细分析挤压成形中各个阶段金属流动情况以及应力、应变、温度、速度等场量的分布变化情况。棒料进人模口至完全流出工作带这段时间是型材挤压最为困难的阶段，材料在工作带处的应力、应变最大，温度最高，因而对模具工作带处造成的磨损也最为严重。进人到最终稳定挤压阶段时挤压方向金属流速计算值与理论挤出速度吻合很好。模拟结果表明所用有限体积法是有效的，可以为铝型材挤压的模具设计与工艺参数的选择提供理论指导。     

基于流函数法的铝型材挤压导流模合理设计

采用模拟试验法和理论分析法研究了导流模内型材挤压的流动机理;提出了确定导流腔最小深度的理论依据;用基于流函数的上限分析法确定了实心薄壁铝型材挤压变形的流线方程、动可容速度场、应变速率场及上限功率;讨论了变形程度等参数对导流模最小理论深度的影响,得出实心薄壁铝型材导流腔深度的理论最小值为导流腔半宽的0.7～0.8倍。理论结果得到了试验证实,为导流模合理设计提供了可靠的依据。     

弹塑性有限元法分析不同型线凹模静液挤压时的应力和应变状态

静液挤压时,如凹模的型线不同,变形区的应力和流动状态是不同的。目前,静液挤压时,广泛采用圆锥模,但在挤压脆性金属时,产品经常会出现缺陷。为了研究产生缺陷的原因,应用弹塑性有限元法计算和分析了等应变型线模、余弦型线模、最短流线模、椭圆型线模及圆锥模等五种不同型线凹模在静液挤压时的应力和应变状态。结果表明,等应变型线凹模挤压时的应力和应变分布最为均匀,挤压力和塑变区内的拉应力最小,适用于挤压脆性金属。计算结果与实验比较,符合很好。计算结果对设计挤压凹模有很大的参考价值。     

A study on radial flow field for extrusion through conical dies

A generalized expression for the radial flow field for extrusion through a conical die is suggested. The upper bound to the extrusion pressure for a rigid-perfectly plastic material is obtained. Other energy methods which include solutions for work-hardening and composite billets are also obtained for the generalized radial flow field. The results are used to analyse the hydrostatic extrusion of Al, Cu and Al-Cu composite billets. The extrusion pressure and hardness distribution of the product were measured in experiments and they are compared with theoretical results.