Optimization of die profile for improving die life in the hot extrusion process

Journal of Materials Processing Technology 142(2003)659–664Optimization of die profile for improving die lifein the hot extrusion processZou Lin ∗,Xia Juchen,Wang Xinyun,Hu GuoanState Key Lab of Plastic Forming Simulation and Die &Mold Technology,HuaZhong University of Science &Technology,1037LuoYu RD,Wuhan 430074,PR ChinaReceived 2December 2002;accepted 10April 2003AbstractAn improved method to obtain the optimal die profile which yields more uniform surface-load distribution on die profile surface via integrating finite element analysis is applied to the hot extrusion process.The die profile of the hot extrusion is represented by a cubic–spline curve.Updated sequential quadratic programming (SQP)method is adopted to accomplish the optimum calculation for unsteady metal form-ing based on the rigid–viscoplastic finite element method (FEM).The analysis for unsteady metal forming has been performed to simulate the process on the die curve acquired from optimum calculation.The above approach can improve the die life in hot extrusion processes.©2003Elsevier B.V .All rights reserved.Keywords:Hot extrusion;Die profile;Optimum design;Rigid–viscoplastic FEM1.IntroductionA large number of studies on the optimization of die pro-file via integrating the classical mathematical methods of op-timization with the finite element method (FEM)have been carried out to improve the product quality and production efficiency.Ales and Boris [1]studied tool design optimiza-tion in steady-state extrusion processes by imposing two op-timization objectives separately.Wifiet al.[2]proposed an incremental slab method to obtain the extrusion pressure of the hot forward rod extrusion process for arbitrarily curved dies.Lee et al.[3]studied the optimal die profile that can yield more uniform microstructure of a hot extrusion prod-uct.They all defined the die profiles by Bezier-curves and adopted different optimization methods.Die life is an important factor in the productivity and the quality of the finished products in hot extrusion processes.In general,the service lives of tools in metal-forming processes are to a large extent limited by wear,fatigue fracture,plas-tic deformation,etc.[4].In high-temperature metal-forming processes,wear is the predominant factor in the operating lives of tools by thermal load and the contact with the hot metal during the deformation nge and co-workers [5]reported that tool lives in high temperature forming pro-cesses are affected by wear in over 70%of cases.Therefore,∗Corresponding author.Tel./fax:+86-27-87543491.E-mail address:l.zou@ (Z.Lin).the improvement of die life in hot extrusion processes is the primary concern to raise productivity.In this paper,for the purpose of improving die life,an optimization scheme coupled with rigid–viscoplastic finite element analysis for the determination of optimum die pro-file which yields more uniform surface-load distribution on die profile surface is suggested.In the hot extrusion process,as the working surface of die is subjected to re-peated thermal–mechanical loading,and the outlet of a die is provided with a high load and an intolerable amount of wear,die life is reduced.The wear in the hot extrusion process can be optimized by proper design of die geom-etry.Starting from an initial die profile,an optimized de-sign solution based on non-linear optimization procedures is found.By performing the optimal design,the level of damage can be minimized,and an optimal die profile that gives the longer die life can be found.Finally the validity of the proposed die profile was verified by comparisons between the results of simulations and the results of real experiment.2.Optimal design problem formulationIt has been proved that the geometry of a die used in hot steady-state processes is extremely important [6].Usu-ally,the die profile is chosen by past experiences.Since an optimized die design solution depends directly on the given0924-0136/$–see front matter ©2003Elsevier B.V .All rights reserved.doi:10.1016/S0924-0136(03)00686-1660Z.Lin et al./Journal of Materials Processing Technology142(2003)659–664 process parameters,an optimal design could be performedto solve this problem.2.1.Design variables selectionIn order to determine an optimal die profile,there is theneed to describe an arbitrary die profile by a mathemati-cal model and derive independent design variables from themodel.The design variables will then be optimized so asto result in an optimal die profile.The cubic–spline curvewith12sub-intervals is applied to describe the shape in thisstudy.This cubic–spline curve may be expressed asy(t)=3h i(x i+1−t)2−2h i(x i+1−t)3y i+3h2i(t−x i)2−2h3i(t−x i)3y i+1+h i1h2i(x i+1−t)2−1h3i(x i+1−t)3y i−h i1h2i(t−x i)2−1h3i(t−x i)3y i+1i=0,1,...,n(1)Since description(1)includes a non-linear variation of the die radius y i,it can be used for analyzing streamline.We can define the die profile with a cubic–spline curve.In the present work,13control points(x0,x1,...,x12)are used.y i is the die radius at axial position,and y i is equal to f(x i)=r i=(x k)i+1and x n−1(n=13)stand for the inlet and outlet die coordinates in axial direction,respectively.x i< t<x i+1and h i=x i+1−x i.The boundary conditions can be set as:y 0=f (x0)=0,and y n+1=f (x n−1)=1.The profile of extrusion die,between point x0and point x12,will be changed during the optimization procedure.2.2.Objective and constraint functions definitionThe main problem in the optimization procedure is to find an objective function.In this study,the objective is to improve the die life in the hot extrusion process andfind such a function for streamline die.From the tool lifetime prospect rather than product quality,we are mainly focused on the determination of process parameters and conditions to minimize the extrusion load and stresses at tool–workpiece interfaces usingfinite element analysis.As we know,too high an extrusion load will lead to an intolerable amount of wear in a die,which is the major cause of die failure.Thus to minimize,the extrusion load can be considered as one of the objective functions.In addition,the strength of a die can lose effectiveness primarily by wear, including abrasive wear,adhesive wear,and fatigued wear. This wear is closely related to the load on the surface of a die cavity.Therefore,more uniform distributedsurface Fig.1.Schematic drawing of optimum curve,r i(i=0,1,...,12)design variable.load,that leads to more uniform distributed die wear,can enhance the life span of a die.To obtain the more uniform surface-load distribution on the die profile surface could be considered as another objective function.In the actual work,we choose the extrusion load as a constraint.In this case,the mathematical formulation of the optimal die profile can be written asminn−1i[(σi−σi+1)2+(τi−τi+1)2]s.t.p0−p≥0,r i+1−r i≥0,i=0,1,...,12 whereσi is the die axial stress in the i th node,τi the die friction stress in the i th node,p the total extrusion load,p0 the given extrusion load,and r i the design variable.Fig.1shows the schematic drawing of optimum die curve.3.Design procedure of optimal die profileIn order to design a die profile to impose a more uniform surface-load distribution on the die profile surface as possi-ble in the deforming region during the hot extrusion process, the above-mentioned optimization problem can be solved by using one of the mathematical programming methods[7].In the present investigation,the updated SQP,which is effective in solving non-linear optimization problem,is performed. The optimization problem is transformed to a quadratic pro-gramming sub-problem with a quadratic approximation to the Lagrange function.Fig.2briefly shows theflow diagram of searching algo-rithm for optimization procedure.This procedure includes a complicate simulation using rigid–viscoplasticfinite ele-ment analysis.The algorithm of the optimization procedure for objective function is as follows:Stage1Assume the initial design variables and define ini-tial die profile;set two small numbers that define theZ.Lin et al./Journal of Materials Processing Technology 142(2003)659–664661Fig.2.Flowchart of the optimization procedure.permissible constraint violation and convergence parameter,respectively.The Hessian is accordingly initialized by identity matrix.Stage 2Calculate the objective and constraint functions atthe current design iterate x i and calculate the gra-dients of the objective functions and the potentially active constraint functions.Calculate the maximumconstraint violation.If the constrains are violated,update the Hessian of the Lagrange function.Stage 3Construct the quadratic sub-problem and solve itfor the search direction and Lagrange multipliers.Table 1The optimal result Node Variable Value (mm)0r 012.00001r 110.65232r 29.814623r 38.566954r 47.919275r 57.058576r 6 6.210007r 7 5.649758r 8 5.236859r 9 4.8626110r 10 4.6320411r 11 4.4541312r 124.40000Stage 4Check for the convergence criteria.If the termina-tion criteria is satis fied,then stop.Otherwise con-tinue.Stage 5The design variables are updated by optimal de-sign technique,save the current penalty parameter,Fig.3.Schematic drawing of different die pro files:(1)the optimal pro file;(2)die pro file generated by tracing backward the forming path;(3)conical pro files (α=45◦).662Z.Lin et al./Journal of Materials Processing Technology 142(2003)659–664and go to Stage 2,where another iteration loop is started.When the procedure was performed,we obtained the op-timum result.The results of performed optimization was written in Table 1and displayed in Fig.3.The optimum die pro file generated by the optimization scheme is compared in Fig.3with the die pro file gen-erated theoretically by tracing backward the forming path [8].It is clear from the figure that both methods generate nearly the same die pro files for the same extrusion condi-tions.There is a slight difference between the two die pro-files in the die inlet with the optimum die pro file being closer axes,the reason of this behavior is that the FEM is a discrete numerical method,the increase of curvature in the die inlet conduced to the increase of discrete error.The optimal approach can produce an optimal die pro file that have continuous first derivatives at the die inlet and exit,causing the pro file to be smoothly merged with the con-tainer at the inlet and with the die land at the exit.Thus,the agreement between the theoretical and calculated results issatisfactory.(a)(b)(c)Fig.4.Deformed mesh at different increments for the optimal curved die pro file:(a)20th increment;(b)50th increment;(c)70thincrement.Fig.5.Distribution of effective strain rate for the optimal curved die pro file:(a)20th increment;(b)50th increment;(c)70th increment.4.Results and discussion4.1.Results and discussion of FE-simulationIn order to validate the optimum results,a rigid –visco-plastic FE-simulation of hot extrusion is carried out.For the present investigation,the process is a hot extrusion process with heat transfer between the workpiece and dies and environment and the material properties are also temperature-dependent,consequently to a given tempera-ture,we have taken account of the effects of temperature on the FE-simulations.The workpiece material employed in the simulation is 21-4N and is assumed as a rigid –viscoplastic material.The workpiece has an initial temperature of T =1000◦C.Di-ameter of the die cavity is 24mm.The extrusion ratio is 7.44.The friction factor at the die –material interface is as-sumed to be 0.2.The die and the punch are considered as rigid bodies with a constant temperature.The temperature of the die and of the punch are 300and 250◦C,respectively.Ram velocity is 25mm/s.Heat transfer of workpiece is de-fined as follows:heat transfer coef ficient during medium isZ.Lin et al./Journal of Materials Processing Technology 142(2003)659–664663parison of forming velocity for conical and optimum dies:(1)conical die (α=45◦);(2)optimumdie.parison of friction force for conical and optimal dies.0.0029N/(s mm K);heat transfer coef ficient during extrusion is 4.0N/(s mm K);the temperature of surrounding medium is constrained to 20◦C.The material constant and strain rate sensitivity are 23.23and 0.0820,respectively.Therefore,the flow stress –strain relationship of the workpiece material can be modeled as σMN /m 2=23.23(¯ε)0.0820In Figs.4and 5,for the optimal die pro file in the hot extru-sion,the result of the simulation is **shown graphically.Inparison of reaction force for conical and optimaldies.parison of normal stress for conical and optimaldies.parison of shear stress for conical and optimal dies.Fig.6,it shows clearly that the equivalent forming velocitydistribution developed with the optimal die pro file is more uniform than that developed with the conical die pro file.Moreover,under the same extrusion conditions,Figs.7–10demonstrate the comparison of stress and force along the tool –workpiece interface between the optimal and conical die pro files,which was calculated by the FEM.Those fig-ures illustrate clearly that the stress and force developedatFig.11.Schematic drawing of different working procedure of the exper-imental hot extrusion.664Z.Lin et al./Journal of Materials Processing Technology 142(2003)659–664Fig.12.The extruded products in different working procedure.the tool –workpiece interface of the optimal die pro file are lower than those of the conical die pro file.The resulting forming-load is smaller,and makes the die wear more even.Thus,the die life of the optimal die pro file is expected to be longer than that of the conical die pro file.4.2.Results and discussion of experimentFig.11shows the different working procedure of the ex-perimental hot extrusion.The material of the workpiece in this study is 21-4N and the material of dies is 3Cr 2W 8V .The initial workpiece is Ø24×H 25mm and is heated up to 1000◦C.The die and punch are given an initial temper-ature of 300◦C.The 6.3MN friction screw press is used in this experiment.The extruded products in different working procedure in this study are shown in Fig.12.Fig.13shows comparison of forming-load versus stroke curve for thesim-parison of forming-load vs.stroke curve for the simulation and experiment in the hot extrusion process.ulation and experiment in the hot extrusion process.The ex-perimental load is larger than computed load,the reason is that the cooling of workpiece during transmission of work-piece from heater to die do not been taken into account.Test shows that metal flow and forming-load in the FE-simulation have good agreement with the real experimental results.In the practical hot extrusion of valves,the production of a die that used the conical die pro file is about 1000–1200pieces,but if adopting the optimal die pro file which is de-veloped in the study,the production of a die is increased to about 3000–3500pieces.The adoption of optimum shape has improved the life span of the die more than twice in the hot extrusion of valves.5.ConclusionIn our research,an improved optimal design method for improving die life in hot extrusion processes has been pro-posed.It provides an effective approach for optimum design of the die curve.An algorithm based on a gradient method and a rigid –viscoplastic FEM is presented for process op-timizations.Die pro file was expressed by a cubic –spline curve,and updated SQP method was applied as an optimiza-tion technique.On the basis of the above-mentioned work,the mate-rial model and the equation of rigid –viscoplastic FEM have been built.A mathematical model of the die pro file in the hot extrusion process is also presented.By an updated SQP method,an optimum die pro file is obtained.The application of this method for optimization of die pro file for the hot extrusion of valves has been demonstrated.The idea that is put forward can be used to guide the optimum design for all types of extrusion processes.References[1]M.Ales,S.Boris,Tool design optimization in extrusion process,Comput.Struct.68(1998)283–293.[2]A.S.Wi fi,M.N.Shatla,A.Abdel-Hamid,An optimum-curved diepro file for hot forward rod extrusion process,J.Mater.Process.Tech-nol.73(1998)97–107.[3]S.K.Lee,D.C.Ko,B.M.Kim,Optimal die pro file design for uniformmicrostructure in hot extrusion product,Int.J.Mach.Tool Manuf.40(2000)1457–1478.[4]J.H.Kang,I.W.Park,J.S.Jae,S.S.Kang,A study on a die wear modelconsidering thermal softening.I.Construction of the wear model,J.Mater.Process.Technol.96(1999)53–58.[5]L.Cser,M.Geiger,nge,Tool life 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