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。     

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。     

铝型材非稳态挤压有限体积法模拟关键技术研究与应用

采用Euler网格和有限体积法(FVM),建立了铝型材非稳态挤压过程的数值模拟模型,研究了相应的关键技术。利用SIMPLE算法迭代计算得到速度场和压力场,继而计算获得等效应变速率、温度等物理场量;根据上述场量,结合铝型材热挤压的本构方程,进行动力粘度的迭代与刷新;利用VOF(volumeoffluid)方法追踪材料的流动前沿;利用“移动网格技术”解决挤压杆移动所造成的边界移动问题;利用时间步长的自动调整保证模拟的稳定性和高效性。开发了铝型材非稳态挤压有限体积法数值模拟程序,并对典型铝型材挤压过程进行了模拟,验证了所建立的铝型材非稳态挤压过程有限体积法模型及其关键技术的可行性。     

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 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.     

An approximate analysis for studying the plane strain deformation of strain rate sensitive materials

The paper presents a numerical method for analyzing the plane strain deformation of rate sensitive materials. A rate of energy functional is introduced which is thought to take adequate account of the strain rate sensitivity of the material. In the numerical technique the functional is minimized with respect to a kinematically admissible velocity field and used in a discretized form in a finite element analysis.To serve as an illustration the frictionless, plane-strain, side extrusion process was considered. To simulate actual side extrusion processes friction was incorporated into the analysis by assuming a constant fraction, α, of the current shear stress of the material.Data were available from some preliminary experiments on the side extrusion of a superplastic tin-lead alloy. The theoretically predicted forming pressure, taking α = 0·3, showed reasonably good agreement with the experimental values.     

ADAPTIVE RIGID-PLASTIC MESHLESS GALERKIN METHOD AND ITS APPLICATION IN ANALYSIS OF EXTRUSION PROCESSES

Under the hypothesis of the rigid-plastic material,specific efforts are placed on the deve-lopments of the key simulation techniques of the meshless Galerkin method because of the complexity of the deformation process as well as the generality and atomization of the simulation procedures for non-steady state large deformation plastic processes,therefore,an adaptive rigid meshless Galerkin method is developed. The influence domain control method is used in the least square approximation by dynamic evaluation of the magnitude of the influence domain and the effective control of the amount and the positions of the points in the least square approximation in order to improve approxi-mation precision. The amount of the Gauss integration points in the discrete domain is maintained in a considerable magnitude in order to ensure the integration precision in the discrete domain. The length of the frictional boundary of the plastic deformation process may be getting longer when its deforma-tion is getting severe. Thus,the densities of the boundary points of some places get lower. The adap-tive boundary points setting method is employed to improve the approximation precision of the boundary points and enhance the constraint of the boundary condition by adaptive control of boun-dary point density. Some typical extrusion processes are analyzed,detail simulation results such as the deformation field,velocity field,effective strain field,effective strain rate field,the volume loss curve and load-stroke curve are obtained. The effectiveness of the method developed is demonstrated and the precision of the meshless simulation is proved by overall comparison with the results obtained by using the commercial software deform.     

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.     

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。     

Effect of lubrication on the stress distribution in an extruded material

The visioplasticity method is used to find the complete strain, strain rate and stress distribution in the deformation zone, according to the deformation grid lines marked on the surface of the workpiece. From the experimental data (the values of the flow function for the extrusion), the velocity, strain rate and stress fields can be calculated by the finite-difference method from the stream function, equilibrium and plasticity equations. In this article, stress components distribution in forward extruded specimens of copper alloy are analysed using the visioplasticity method. Comparisons are made between stress distribution of the specimens extruded with three different coefficients of friction. The results are shown in the form of diagrams.     

On the dead-zone formation and limit analysis in axially symmetric extrusion

For axisymmetric direct and indirect extrusion, a kinematically admissible velocity field based on a simple radial flow field combined with the asymptotic behaviour required of a real velocity field near a velocity discontinuity surface is proposed. The influence of the extrusion ratio on the shape of a dead zone and the extrusion pressure is investigated. The result obtained for the upper bound on the extrusion pressure is compared with other solutions. It is shown that using the asymptotic velocity field slightly improves the prediction of the extrusion pressure in comparison with the other solutions based on radial flow. The main advantages of the approach proposed are that it is applicable to a class of processes and that it accounts for the behaviour of a real velocity field in the vicinity of velocity discontinuity surfaces/maximum friction surfaces where various physical effects like local heating, recrystalization, and transformations occur.     

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.     

有限体积法在三维非稳态铝合金挤压过程模拟中的应用

研究基于Euler网格的有限体积法(Finite volume method,FVM)模拟三维大变形非稳态铝型材挤压过程的基本理论和关键技术,应用C语言编制程序求解速度场、温度场等物理场量,并利用运动界面追踪技术--流体体积(Volume of fluid,VOF)方法捕捉材料流动前沿.采用"移动的网格"处理计算区域边界的移动问题,实现对真实物理过程的数值模拟,并将模拟结果与比较成熟的商品化有限元软件DEFORM-3D的模拟结果进行对比,表明所建立的有限体积法模拟模型有效、可靠.结果对比表明,FvM模型的模拟结果更符合实际,证明有限体积法比有限元法更适合大变形挤压过程模拟.     

An analysis of axisymmetric hydrostatic bulging by the upper-bound method

An upper-bound solution for the analysis of hydrostatic bulging is found by minimizing the plastic energy consumption rate divided by the fluid volume change rate. The corresponding upper-bound pressure is found through optimization with respect to some parameters given in the assumed velocity field. Computations are carried out for axisymmetric hydrostatic bulging of isotropic work-hardening sheet metal. The computed results are then compared with those by the finite-element method and by the finite-difference method as well as the reported experimental results. The comparison shows that the present method renders excellent predictions in pressure and strain distributions with much reduced computational time, which are still in close agreement with those obtained by the numerical methods.     

Effects of extrusion variables on temperature distribution in axisymmetric extrusion process

A numerical method was developed to simulate the non-steady-state temperature distributions during forward extrusion process. The velocity, strain rates, and strain fields within the deformation zones during extrusion were obtained, using upper bound method of analysis to obtain internal heat generations coupled to the necessary heat transfer conduction equations. The computer program written in C++ language essentially simulates the extrusion process and takes into account extrusion variables such as material properties, friction conditions, extrusion velocity, extrusion ratio, die preheat temperature, billet height, percentage reduction in area, and die land length. The effects of billet height and percentage reduction in area on the temperature distributions within the dead metal zone give good agreements with experimental results. It is found that the higher the billet's heights and higher the percentages reduction in areas, the higher the temperature rises during the extrusion process. The die land zone shows increasing temperature rise with increasing friction coefficient, while increasing friction coefficient has no effect on the dead zone temperature. Also, increasing speed of deformation shows an increasing dead zone temperature rise than a more gradual die land temperature rise. It can be stated that the extrusion temperature increases proportionally to the increase of the container temperature.     

A numerical method for calculating temperature distributions in metal working processes

A numerical (finite element) method for calculating temperatures in metal working which largely overcomes the limitations of previous methods is described. Its application is demonstrated using an experimentally obtained flow field (streamlines) for plane strain extrusion to determine the values of strain, strain-rate and velocity needed in the temperature calculations. Account is taken of the dependence of the work material's flow stress on temperature (and strain and strain-rate) and although, to reduce computing time, the thermal properties are assumed constant it should be possible to also allow these to be dependent on temperature.     

Application of Schwartz-Christoffel conformal mapping and complex variables to the symmetric cornered die extrusion of steady state creep materials

The two-dimensional equilibrium equation for steady creep materials described by the power law is derived in terms of the stream function, the velocity potential function and the complex potential function. The solutions of the equilibrium equation show that the potential function is independent of the rate hardening exponent (m). The solutions are used to investigate the symmetric cornered die extrusion problem by applying the Schwartz-Christoffel conformal transformation. The horizontal and vertical velocities, the strain rate and the stress components are found for various rate hardening exponents. The analysis results illustrate that the stress components are dependent on the rate hardening exponent; however, the velocity and strain rate components are independent of the rate hardening exponent. Additionally, the constant dilatational stress (isobaric) line is found to be a constant velocity potential function.     

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。     

An analytical formulation as an alternative to FEM software giving strain and stress distributions for the three-dimensional solution of extrusion problems

A new improved analytical method based on the upper bound theorem is presented for the solution of extrusion problems. This method has been formulated in such a way as to eliminate the deficiencies in the previous works in order to enable it as a powerful analytical tool as an alternative to finite element software. The proposed formulation is a general method of solution, which could be used for the analysis of many bulk forming processes. However, in this paper, the forward extrusion of a square section from a round billet has been presented as an example. Kinematically admissible velocity fields were computed which gave a more physically realistic material flow patterns as compared to previous works. The distribution of strain and stress as well as the results for the effect of process parameters on the extrusion load and die geometry was also given. These results were compared with FEM data to observe the accuracy and effectiveness of the present method. For the extrusion of square sections from round billets, detailed analysis of strain distribution on the exit section was carried out using the components of the power due to internal deformation, interface friction, and velocity discontinuities at the entry and exit surfaces. Comparison with experimental data was made, and verification of the theoretical results was carried out. The improvement of the results computed using the present method was shown by comparison with the previous works.     

Modeling and simulation of polymer melts flow in the extrusion process of plastic profile with metal insert

The extrusion technology of plastic profile with metal insert is recently an advanced plastic processing method whose products keeps rising today for their excellent performance. However, the related fundamental research on polymer forming mechanism in the extrusion process of plastic profile with metal insert is lagging behind. With the development of computational fluid dynamics (CFD) theory, numerical method becomes an effective way to investigate such complex material forming problems as in the polymer extrusion process. In the present study, the mathematical model for three-dimensional non-isothermal viscous flow of the polymer melts obeying a Carreau model is developed based on the CFD theory. The Williams–Landel–Ferry equation is employed to involve the temperature dependence of material parameters. A decoupled numerical algorithm based on the penalty finite element method is conducted to predict the rheological behaviors of polymer melts within the complex flow channel. The streamline upwind/Petrov–Galerkin scheme is employed to improve the computational stability for the calculation of temperature field. Based on the theoretical model, the essential flow characteristics of polymer melts in the extrusion process of plastic profile with metal insert is investigated. The distributions of principal field variables like flow velocity, melt temperature, flow stress and pressure drop are predicted. The effects of die structure parameters including the intake angle and the distribution section length upon the melts flow patterns are further discussed. The variations of melt rheological properties versus different processing conditions like the volume flow rate and the metal insert moving velocity are also investigated. Some advice on practical processing operations of the extrusion process of plastic profile with metal insert is accordingly put forward based on the numerical results.