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.     

凹模型线对挤压过程的数值模拟和试验研究

本文用数值拟和试验研究相结合的方法分析凹模型线对挤压过程变形状态的影响。模拟计算采用刚塑性有限元法,试验研究用坐标网格法和密栅云纹技术。文章对文献中提出的圆锥凹模、余弦曲线凹模、等应变率曲线凹模、双曲线凹模、椭圆曲线凹模和最短流线凹模,以及另外新设计的正弦曲线凹模,进行数值模拟,得到了各种型线模挤压时的速度场、应力场和应变场,以及由于变形热效应引起的试件和模具温升。并用光刻网格法和密栅云纹法进行试验校核。计算结果与实验结果相互吻合良好,研究结果表明七种曲线凹模中,以新设计的正弦曲线凹模最佳,余弦曲线凹模次之,这两种型线凹模的挤压力低,而且内部应变分布均匀。     

Determination of friction coefficient encountered in large deformation processes

A new technique, namely, the open-die backward extrusion test technique, was developed as an alternative method to the ring compression test in order to quantitatively evaluate the coefficient of friction, μ, at the die/workpiece interface. This technique relates the percentage deformation in height of the specimen to the percentage increase in extruded height of the specimen. In this study, the open-die backward extrusion tests (ODBET) were simulated for different aspect ratios, (H/D), and different die geometries, (d/D), by utilizing an elastic-plastic finite element code (ANSYS) in order to obtain the friction calibration curves (FCCs). The results indicated that the extruded height is related to the friction conditions at the die/workpiece interface. Therefore, ODBET can be used to generate FCCs to determine the coefficient of friction at the die/workpiece interface in large deformation processes.     

The use of generalised deformation boundaries for the analysis of axisymmetric extrusion through curved dies

A generalised kinematically admissible velocity field is derived for axisymmetric extrusion through curved dies by employing rigid-plastic boundaries expressed in terms of arbitrarily chosen continuous functions. The corresponding upper-bound extrusion pressure is related directly to boundary functions for the plastically deforming region when the die shape, lubrication condition and material characteristics of the billet are given. The proposed method of analysis makes it possible to predict the deformation pattern as well as extrusion pressure. In computation a third-order polynomial is chosen for the die boundary and the bounding function for the plastic region is chosen to be a fourth-order polynomial. The workhardening effect is considered in the formulation. The plastic boundaries as well as stream lines are affected by various process parameters. The theory predicts the relatively faster axial flow at the center than near the die boundary for greater friction factor even with the same die shape. The effects of area reduction and die length are also discussed in relation to extrusion pressure and deformation. Experiments are carried out for steel billets at room temperature. Deformation patterns are measured for several area reductions by the photoetching technique and the extrusion pressure is measured using a load-cell. The predicted extrusion pressure is in excellent agreement with the value computed by the finite element method. The deformation patterns agree well with the experimental observation.     

混合多目标遗传算法及其在热挤压模具型腔形状优化设计中的应用

将复合形法引入遗传算法来反映决策者对多目标问题中各目标函数的偏好信息，提出一种新的结合复合形法的混合多目标遗传算法。针对热挤压模具型腔轮廓形状优化问题，结合刚-粘塑性有限元模拟和神经网络技术，利用三次样条函数插值来表达凹模型腔轮廓形状，以表面载荷沿凹模型腔轮廓表面均匀分布和挤压力最小为目标，建立了多目标优化的数学模型，对挤压模具型腔轮廓形状进行多目标优化设计，得到了最优的凹模形状。对几种不同模具凹模型腔采用MARC／AutoForge有限元软件进行数值模拟对比研究，结果表明，结合复合形法的多目标遗传优化算法是一种较好的模具型腔形状多目标优化设计方法，其优化结果是有效的和显著的。研究结果说明，通过优化型腔形状来提高模具寿命的效果十分显著。     

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.     

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。     

硬质PVC结皮发泡板的加工工艺研究

介绍了硬质PVC结皮发泡板材生产工艺流程,讨论了螺杆转速、挤出温度、挤出压力及机头、口模、冷却定型模结构等影响板材质量的主要工艺因素。采用新开发的硬质PVC板材生产设备,设置相对应的结皮发泡加工工艺参数,生产出具有优良性能的硬质PVC结皮发泡板材。     

Mechanical property changes in drawing/extrusion of hardening viscoplastic materials with damage

This paper is concerned with the mechanical property changes in drawing/extrusion operations of hardening viscoplastic materials with damage. The process model in this study includes two state variables, the hardness for strain hardening from slip dominated plastic distortion and the porosity for damage from growth of microvoids. The decrease in apparent density in strip drawing of aluminum for several die angles are computed and compared with that from experiments. Simulations of axisymmetric drawing/extrusion have shown that the accumulated porosity in drawing is much bigger than that in extrusion while the difference between the hardness distributions in these processes are insignificant. The effects of the process conditions, such as conical die angle, friction and drawing/extrusion speed, on the mechanical property changes are also examined.     

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.     

Round-to-hexagon drawing through straightly converging dies: an application of the SERR technique

The modelling of metal forming processes in recent years has brought the development of different analytical and/or numerical techniques. The SERR technique is eminently suitable for analyzing drawing/extrusion of sections having re-entrant corners. However, in its present form, it cannot be used to analyse forming processes where the deformation zone has curved boundaries, since the elementary regions will no longer remain rigid. The present study is an endeavor to remodel this technique so that it can handle round billets. The circular cross section of the round billet is approximated by a regular polygon of equal area and the number of sides of the polygon is progressively increased until convergence of the drawing stress is achieved. As a test, the drawing of hexagon section bars from round billets through the straightly converging dies is analyzed     

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。     

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。     

Die design optimization for axisymmetric hot extrusion of metal matrix composites

Strain rate sensitive materials such as Ti alloys, superplastic materials and metal matrix composites (MMCs) can be deformed only in very narrow range of strain rate. In this work, a new process design method for controlling strain rate in workpiece during hot extrusion process is proposed. In this approach, a coupled numerical approach of finite element analysis and optimization technique to optimal profiled die which yields more uniform strain rate distribution in the deforming region is applied to the hot extrusion process of MMCs. Extrusion die profiles are defined by Bezier curves, and FPS (flexible polyhedron search) method is used as optimization technique. The change of relative deviation of strain rate, the progressive development of die profiles with increase of iteration for optimization and the corresponding strain rate distributions are investigated. In addition, the die profiles by optimization scheme for different extrusion ratios are compared with those by analytical solution.     

Radial flow simulation of drawing and extrusion of rigid/hardening materials

A combination of rigorous relations (drawn from continuum plasticity) with pragmatic engineering arguments, leads to new approximate formulae for the stress fields and velocity profiles in some basic steady forming processes. The latter include drawing and extrusion in two (plane strain) and three (axisymmetric) dimensions.The theory derived here accuonts for wall friction and, more important, is not restricted by any particular choice of the strain hardening function. The applicability of the present analysis is limited, though, to dies of sufficiently small angles, taper and wall friction.Confidence in our results is gained by comparing them with a few consistent approximations obtained from available exact solutions in theoretical plasticity.The simplicity and usefulness of the new formulae is demonstrated by specifying the results for rigid/power-hardening materials, and evaluating the drawing limits associated with the occurrence of necking and separation at the exit of the die.It is also shown that the present approximate solution can be used for rapid evaluation of the stress field in drawing of rigid/perfectly-plastic composite (bimetallic) sheets and tubes.     

外径不等杯-杯型复合挤压过程的数值模拟

采用刚塑性有限元法对外径不等杯－杯型复合挤压过程进行了数值模拟,并给出网格重新划分技术和处理八节点四边形等参元奇异点的方法。数值模拟结果表明：对于外径不等杯－杯型复合挤压在晚期非稳态流动阶段有压力回升现象;两端变形程度的大小对金属流出量大小起主要作用,摩擦条件的改变对金属流出量大小影响很小。变形力和金属流出长度的计算结果与实验结果相吻合。     

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.     

A computational study of die geometry and processing conditions effects on equal channel angular extrusion of a polymer

Equal channel angular extrusion (ECAE) is an efficient process to obtain enhanced microstructures via super-plastic deformation. In view of its optimisation, it is of prime importance to assess the relationships between processing conditions and material flow. More precisely, detailed knowledge of the plastic strain distribution in the extruded material in relation to the ECAE processing variables is required. The key parameters of the ECAE process are primarily die geometry, ram speed, extrusion temperature, use of back-pressure, number of extrusion sequences and processing route (e.g. rotation of the sample between successive passes). A numerical investigation was achieved to check out the influence of these parameters on the homogeneity of plastic strain distribution in the case of a conventional thermoplastic polymer. Material parameters of a phenomenological elastic viscoplastic model were deduced from compressive deformation tests at different temperatures and strain rates on high-density polyethylene (HDPE). Recommendations on tool geometry and processing conditions can then be provided, according to the numerical results.It was found that optimum ECAE die geometry is strongly material dependent. The application of a back-pressure significantly contributes to reduce the corner gap and consequently promotes the homogeneity of the plastic strain field. A slight sensitivity of plastic strain to ram speed and friction conditions was pointed out. The extrusion temperature strongly influences the magnitude of the plastic strain and has a slight effect on its homogeneity. The number of passes has a significant effect on the magnitude of the plastic strain but has a negligible influence beyond a certain temperature. The extruded material reaches a stationary strain state after few passes. The homogeneity of the plastic strain field is strongly affected by the processing route.     

Closed form solutions for plane-strain ideal plastic deformations

A Riemann plane mapping methodology to analyze plane-strain ideal plastic flow processes such as extrusion and drawing is presented. A solution function for a generalized die profile is obtained in the form of a zeroth-order Bessel function. Explicit solutions in closed form using functional approximation are obtained for the deformation forces and stresses for a wedge-shaped die without requiring numerical evaluation of any integrals. Two limiting solutions of the wedge shaped die profile, including wall friction effects, are derived and discussed. A comparison of the method and the results with the traditional slip-line field and upper-bound-lower-bound approaches is provided.     

Portholes-equal channel angular pressing: novel technique for extrusion of 6061 aluminum alloy tube by subsize billet

Wrought 6061 Al alloy exhibits the prospective applications in the form of tube extrusions. In this study, billets of 6061 Al alloy were extruded under optimized conditions by a novel technique namely portholes-equal channel angular pressing (P-ECAP) extrusion. This technique is different from the conventional extrusion as the dimension of the product is greater than that of the billet. The extruded tube produced by the method was characterized for their microstructure as well as for their physical and mechanical properties. The tube that was fabricated using P-ECAP die showed significant refinement in microstructure with improved mechanical properties outside the seam joint portion. However, the extrusion loads using porthole die were less compared to that in the conventional method by using columniform billet because of the decrease in billet dimension or extrusion ratio. Furthermore, microstructure at seam joints of 6061 Al alloy extrusion was discussed in detail in this study. Thus, the P-ECAP technique has significant potential for substantial energy saving.