Neitzert bstr act This pap er in estigates tec hnique to 57356nd an optimal set of conditions for an isothermal pro cess to extrude pro duct for giv en shap and material prop erties with minimal defects The inputs to this mo del are the pro duct geo ID: 29950 Download Pdf

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Neitzert bstr act This pap er in estigates tec hnique to 57356nd an optimal set of conditions for an isothermal pro cess to extrude pro duct for giv en shap and material prop erties with minimal defects The inputs to this mo del are the pro duct geo

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Optimal Pro cess Con trol arameters Estimation in Aluminium Extrusion for Giv en Pro duct Characteristics Kathirgamanathan and T. Neitzert bstr act |This pap er in estigates tec hnique to nd an optimal set of conditions for an isothermal pro cess to extrude pro duct for giv en shap and material prop erties with minimal defects. The inputs to this mo del are: the pro duct geometry and its mate- rial data suc as o curv and microstructure during dynamic recrystallization. This is an in erse problem and the mo del is form ulated as non-linear least- squares

minimization problem coupled with nite elemen mo del for the extrusion pro cess. It is done constructing an iterativ pro cedure using an op- timisation routine suc as TLAB ’s lsqnonlin and at eac iteration, the extrusion o is solv ed using ABA QUS First all con trol oin ts of Bezier-curv for the die surface that minimizes the redundan strain inside the deformation zone are found. Then the ini- tial billet temp erature, die temp erature and the ram sp eed that closely matc with the strain rates and temp eratures for the desired microstructure (grain size) are obtained. Keywor

ds: Extrusion; Inverse pr oblem; Par ameter es- timation; Optimization. In tro duction The metal o during extrusion through die is complex and not uniform, whic causes cross-crac king, ending, distorting and wisting of the extruded pro duct. im- pro the qualit of the orkpiece, the die cross-section la out and op erating conditions ust tak en in to ac- coun in the design of new pro duct. Computer sim ulation of extrusion using the nite elemen metho is opular option to replace the traditional trial and error metho during pro cess design. nite ele- men mo del, whic is

capable of describing the eha vior of metal o during extrusion, requires sev eral input data suc as die geometry material eha vior la ws, friction la ws and op erating conditions. In realit material data can obtained using ailable exp erimen tal data, but optimal alues of die geometry Man uscript receiv ed Marc 01, 2008. This ork as supp orted ec hnology New Zealand. Authors are with Sc ho ol of Engineer- ing, Auc kland Univ ersit of ec hnology Auc kland, New Zealand; email: pak athir@aut.ac.nz, thomas.neitzert@aut.ac.nz and op erating conditions are often not accurately kno wn and

therefore guessed. Metho ds to determine die geom- etry and op eration conditions are an imp ortan part of designing an extrusion pro cess. Sev eral articles ha een published in this area. Wi et al [6 ], used the incremen tal slab tec hnique and Bezier- curv tec hnique to nd the optim um curv ed die prole that minimizes the extrusion load for hot extrusion pro cess. Lee et al [1 used the nite elemen metho and exible olyhedron searc metho to pro duce uniform microstructure. They used Bezier-curv es to generate all ossible die proles and used plain carb on

steel for the billet. The no el concept of this researc is to determine the optimal conditions of isothermal extrusion with regards to billet temp erature, con tainer temp erature, extrusion sp eed and microstructure for non-adiabatic die prole basing it on tec hniques ailable in the literature. orw ard Problem Extrusion is thermo-mec hanical deformation pro cess in whic blo of metal (billet) is forced through the die op ening of smaller cross sectional area than that of the original billet. In this pro cess, the large deformations are mainly plastic or viscoplastic, allo wing the

elastic part to neglected. Strain is measure of deformation and at high strain rates metal o is analogous to uid o w. Therefore the material eha vior can describ ed as that of uid o w. (i) Conserv ation of mass (1) where is the densit of the material and is the elo cit ector. If the material is incompressible, densit is unc hanged and the Equation (1) is simpli- ed to (2) (ii) Conserv ation of momen tum (3) Proceedings of the World Congress on Engineering 2008 Vol II WCE 2008, July 2 - 4, 2008, London, U.K. ISBN:978-988-17012-3-7 WCE 2008

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where is the dy force er unit mass, and is Cauc stress tensor. or viscous incompressible uid (4) where is the pressure, is the unit tensor and ij ij (5) where is dened (6) here represen ts the consistency of the material, is the strain rate sensitiv index and the eectiv strain rate ij ij (7) is the strain rate tensor and it is dened (8) (iii) Conserv ation of energy c r (9) where is the sp ecic heat, denotes ther- mal conductivit is the temp erature and is the rate of heat generated er unit olume. The exact mathematical

analysis of the extrusion pro cess is ery complex and has not een fully resolv ed. But nite elemen tec hniques deliv er appro ximate umerical solutions to it. implemen ted this with the ABA QUS nite elemen program. In erse Problem The goal here is to determine the input data (param- eters) of the forw ard problem whic leads to giv en result. In this problem, the die design parameters and pro cess parameters are not kno wn, but the shap and prop erties of the nal pro duct are kno wn. assume here that the desired grain size of the material is kno wn, but the exact die

geometry initial billet temp erature and ram sp eed are unkno wn. Our aim is to estimate the exact die geometry and the pro cess parameters. This problem is kno wn mathematically as an in erse problem and can seen as an optimization problem whereb the ob jectiv function to minimize is the gap et een the exp ected result and nite elemen sim ulation results. First of all, mathematical denition of the gap et een the exp ected and sim ulation results is required as an ob jectiv function. The aim of optimization is to nd the est set of parameters that minimizes an ob jectiv

function impro ving the erformance in the direction of optimas. The aim of the approac is to nd the global minim um on giv en searc space maximizing or minimizing the ob jectiv function sub jected to the giv en constrain ts. In the case of minimization problem, the mathematical form ulation of the problem can stated as follo ws Minimize sub ject to (10) where is the ob jectiv function, ), are constrain ts function and is ector of design ariables and pro cess ariables. 3.1 Design ariables In order to determine the optimal die prole, it is essen- tial to describ an arbitrary die

prole mathematical expression and obtain the design ariables from the ex- pression. The design ariables will then optimized to get desired result. In this study Bezier curv es with v con trol oin ts are used to obtain the arbitrary die prole. The Bezier curv whic has the con trol oin ts is giv en [2 =0 +1 i;N and =0 +1 i;N (11) where 4, [0 1] and i;N !( 1)! (1 Using these con trol oin ts an innite um er of die sur- faces can created. 3.2 Pro cess ariables Theoretically the pro cess ariables whic can inuence the extrusion pro cess are the ram sp eed

initial tem- erature of the billet and the friction conditions at the billet/to ol in terface. 3.3 Imp ortan factors in the pro duct opti- mization 3.3.1 Die life It is ob vious that high extrusion loads will lead to an in tolerable amoun of ear in die. Therefore it is im- ortan to minimize the extrusion load. In addition, the Proceedings of the World Congress on Engineering 2008 Vol II WCE 2008, July 2 - 4, 2008, London, U.K. ISBN:978-988-17012-3-7 WCE 2008

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strength of die can aected ear and this ear is closely related to the load on the surface of die ca v- it By

distributing the load on the die surface ear can distributed along the surface and it leads to increases in die life. Therefore the optimization constrain for in- creases in die life ha to in the form AR =) min +1 +1 (12) where is the axial stress in the th no de, is the friction stress in the th no de, is the extrusion load, is enalt parameter and is ector of design and pro cess parameters. 3.3.2 Isothermal extrusion Isothermal extrusion is pro cess of main taining con- stan temp erature during the extrusion pro cess inside the deformation zone. Ac hieving this condition is ery im- ortan for

the pro duction of uniform, high-qualit pro d- ucts. Isothermal conditions can ac hiev ed mec hanically adjusting the sp eed of the ram, initial temp erature of billets and die and carefully designing the die for eac pro duct. Based on these considerations the optimization constrain to ac hiev isothermal extrusion can written as follo ws. min x; dt; (13) where x; is the temp erature inside the deformation zone as function of space and time, is the erage temp erature or desired temp erature inside the deforma- tion zone and is time to complete the particular extrusion pro cess. 3.3.3 Flo

balance An ev enness of exit o is an imp ortan part in the die design to oid part distortion and to able to yield an extruded part within the sp ecied tolerances. The uniform die exit elo cit can obtained adjusting the die land length. The optimization constrain to increase the ev enness of exit o is therefore min (14) where is the no dal elo cit of the th no de, is the um er of no des along the die exit and is the erage no dal elo cit along the die exit. 3.3.4 Distortion The success of an extrusion pro cess dep ends mainly on reducing the distortion and minimizing the

redundan strain in the extruded pro duct as ell as con trolling the strain rate inside the deformation zone. Literature in the eld of metal forming pro cesses suggests that redundan strain and distortion are related to eac other. Based on the theory and ailable literature the follo wing three criteria can used to minimize distortion. (i) Distortion ariation: min (15) where is the displacemen of the no de and is the erage displacemen ts of transv erse no des at the die exit. (ii) Av erage eectiv strain: min deal (16) where deal is the ideal eectiv strain and is

dened deal ln (17) for the axisymmetric extrusion. (iii) Strain rate deviation: min (18) 3.4 Ob jectiv and constrain function The selection of the ob jectiv function is asso ciated with the user’s sp ecic needs. Here our aim of the study is to extrude pro duct with giv en haracteristics. Therefore the ob jectiv function ust in olv the microstructure (grain size) of the nal pro duct. But the optimal strain rate, strain and temp erature tra jectories can obtained to get desired alue of grain size minimizing [4 (19) where is desired strain, is eigh ting factor, is the

extrusion time, is desired alue of erage grain size and the erage grain size is [4 exp (20) The ob jectiv function to determine the optimal die pro- le is the least square deviation et een the results ob- tained using Equation (19) and the nite elemen solu- tion of Equations (1-9). If sub divide the time in terv al Proceedings of the World Congress on Engineering 2008 Vol II WCE 2008, July 2 - 4, 2008, London, U.K. ISBN:978-988-17012-3-7 WCE 2008

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Input data Close enough? Estimate desired strain strain rates temperatur (MA TLAB) rite the FEM input file FEM

simulation (ABAQUS) Extraction of results and export to MA TLAB (PYTHON) Output Optimization algorithm (MA TLAB) New paramer es No Figure 1: Illustration of design pro cess. [0 in to partitions then the ob jectiv can ex- pressed min +1 min +1 min +1 (21) urther if an to consider die life and o balance then the Equations (12) and (14) can included as con- strain ts with the ob jectiv function (21). Sc hema of the design Let us no consider the tec hnical design details of the algorithm to solv Equation (21). The solution will arriv ed at through sev eral steps as sho wn in Figure 1. They

consists of: (i) The input data dene the ph ysical structure its dimensions, material prop erties and oundary con- ditions. (ii) TLAB’s lsqnonlin optimization algorithm is ap- plied to obtain the optimal alues of strain-rate, strain alues and temp erature for giv en microstruc- ture prop erties of the material. (iii) Input le mo dule creates mo del for the nite ele- men analysis. It consists of geometry of the mo del, material prop erties, con tact denitions and loading sequence etc. This is done with TLAB script whic creates an input le for the ABA QUS

pro- gram. The reason for ho osing this approac is the ossibilit of easy mo dication of the die geometry for eac iteration pro cess in the optimization routine. (iv) FEM sim ulation part executes the created input le using ABA QUS explicit. (v) Extraction mo dule retriev es results from ABA QUS ODB database. Here the in terface is set up et een TLAB ’s optimization pro cedure and ABA QUS - nite elemen sim ulation. This is ac hiev ed through the use of PYTHON scripts to retriev necessary results from the ODB database. It is sa ed in dieren le with an EX

CEL le format and then pro cessed with TLAB to obtain the results for optimization. (vi) The optimisation part is implemen ted using TLAB ’s lsqnonlin built-in function whic uses the Lev en erg-Marquardt algorithm. This approac en- sures an easy implemen tation of ultiple runs of ABA QUS within an optimization routine. Unlik the linear problem, the non-linear ob jectiv func- tion giv en Equation (21) ma ha more than one minim um. Therefore, the solution pro cess should include nding the global minim um. deal with these problems rst nd all or most of the lo cal

minima of (21) for sequence of design ariables at larger in terv al. Then pic the lo est alue of the minima. In the next step use the minim um obtained from the previous step as the starting alue to solv Equation (21). Results and discussion In order to study the optim um results in an aluminium extrusion pro cess, FE-sim ulation is carried out using MA TLAB and ABA QUS. The pro cess here is hot ex- trusion pro cess with heat transfer et een the ork piece and die also eing considered. The ork piece and die has an initial temp erature of 450 The extrusion ratio is 1.33. The friction factor at

the die-material in- terface is assumed to 0.1. The die is considered as deformable dy Ram elo cit is 25 mms The o stress-strain relationship of the ork-piece material is [7 209 122 =mm (22) Other data alues used in the sim ulation of the ork-piece are: oung’s mo dulus of 10 10 co ecien of expansion= 10 at 20 oisson’s ratio= 35, inelastic heat fraction 9, sp ecic heat 910 densit y= 2750 conductivit Proceedings of the World Congress on Engineering 2008 Vol II WCE 2008, July 2 - 4, 2008, London, U.K. ISBN:978-988-17012-3-7 WCE 2008

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204 when 225 when 300

and the alues used for the die material are oung’s mo dulus of 20 10 10 co ecien of expansion= 10 at 20 oisson’s ratio= 30, inelastic heat fraction 9, sp ecic heat 450 densit y= 7200 conductivit 204 when 225 when 300 First minimized extrusion pressure, die pressure, temp erature ariation inside the deformation zone, elo cit ariation at the die exit, distortion ariation, strain ariation and strain rate ariation at the die exit separately to study ho eac prop ert ary with the pro cess and design parameters. It is in teresting to nd that the optimal parameter alues ere not

the same (eg the parameter whic minimizes the extrusion pressure is not same as the parameter whic minimizes temp erature ariation inside the die) for all categories. Secondly compared the circumferen tial and axial stress at the die exit. It has een found that the die whic minimizes the strain rate ariation at the exit also pro duces the prole with lo est circumferen tial stress and the die whic minimizes the extrusion pressure pro duces the prole with minim um axial stress, whic can ex- ected. The review of literature [3 sho ed that the ossi- bilit of crac king increases with

increasing tensile circum- feren tial stresses and hevron(or cen tral) bursts increase with increasing axial stress. Therefore it is imp ortan and necessary to include the minimization of circumferen tial and axial stress during the optimization pro cess. The tendency to ard hevron crac king increases if plas- tic zones do not meet together inside the forming zone. If this happ ens the ydrostatic pressure at that region is zero. It can sho wn that the ydrostatic pressure is not zero inside the forming zone for an of the ab men tioned cases (whic minimizes extrusion pressure, die pressure, temp

erature ariation, strain rate ariation etc) and in certain non optimal alues (eg when extrusion pressure is maxim um) the ydrostatic pressure inside the forming zone is zero. Av erage grain size is prop ortional to exp (23) The ariation of four most optimized d= alues along the die exit with dieren pro cess conditions are sho wn in Figure 2. Most optimal (in terms of uniformit y) d= al- ues of four dieren cases and their resp ectiv die surfaces whic giv es the optim um are sho wn in Figure 3. These gures clearly sho that the grain size is more sensitiv to extrusion sp

eed and the initial temp erature than the die geometry urther the uniformit of the grain size is more inuenced the shap of the die surface than 10 11 0.22 0.24 0.26 0.28 0.3 0.32 0.34 10 11 0.22 0.24 0.26 0.28 0.3 0.32 0.34 Proportional average grain size (d/ 10 11 0.22 0.24 0.26 0.28 0.3 0.32 0.34 Nodes along the die exit data1 data2 data3 data4 data1 data2 data3 data4 data1 data2 data3 data4 Figure 2: Grain size ariation along the die exit d= (A) when mm=s 450 500 (B) when 25 mm=s 450 500 (C) when 25 mm=s 400 450 10 0.22 0.24 0.26 0.28 0.3 0.32 0.34 Nodes along the die exit a (v=6.3

mm/s, TD =450,TB =500) b(v=25 mm/s TD =450, TB =500) c(v =25 mm/s, TD =400, TB =450) d(v=6.3 mm/s,TD =450, TB =500 Proportional average grain size (d/ a (v=6.3 mm/s, TD =450,TB =500) b(v=25 mm/s TD =450, TB =500) c(v =25 mm/s, TD =400, TB =450) d(v=6.3 mm/s,TD =450, TB =500 Figure 3: Grain size ariation along the die exit d= ):(A) Optimal ales with dieren pro cess conditions, (B) Op- timal die shap with dieren pro cess conditions Proceedings of the World Congress on Engineering 2008 Vol II WCE 2008, July 2 - 4, 2008, London, U.K. ISBN:978-988-17012-3-7 WCE 2008

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initial temp erature or ram sp eed. Finally lo ok ed at the grain size ariation with arious desired grain sizes for sequence of design and pro cess parameters and found that alues of parameters (design and pro cess) exist to satisfy equations (22), (18), (16) and (17). Summary The goal of the ork presen ted here is to in estigate umerical tec hnique whic is capable of sim ultaneously estimating the optimal die prole and the pro cess param- eters suc as extrusion sp eed and initial temp eratures. The approac is based on non-linear least squares esti- mation using the desired prop erties

of the pro duct whic is extruded. The examples considered in the ab section describ ho alues of the design and pro cess parameters inu- ence the optimization pro cess. The results from these examples suggest that the prop osed tec hnique is capable of estimating the optimal alues reasonably ell. The alidit and eectiv eness of this tec hnique has to eried using an exp erimen tal metho d. will address this issue as part of the ongoing pro ject. References [1] Lee S. K., Ko D. C., Kim B. M., \Optimal die prole design for uniform microstructure in hot extruded pro

duct", International Journal of Machine ols Manufactur V40, pp.1457{1478, 2000. [2] Mathews j. H., Fink K. D., Numeric al Metho ds Using Matlab Pren tice-Hall Inc, 4-th Edition, 2004. [3] Shivpuri R., \Adv ances in umerical mo delling of man ufacturing pro cess", ansIndian Inst Metals V57, pp.345{366, 2004. [4] en ugopal S., Ro driguez ., \Strategy for the de- sign of thermomec hanical pro cesses for AISI yp 304L stainless steel using dynamic materials mo del stabilit criteria and mo del for the ev olution of mi- crostructure", Journal of Materials Scienc V39, pp.5557{5560, 2004. [5] en ugopal

S., Medina E. A., Malas I., Medeiros S., razier W. G., \Optimization of microstructure during deformation pro cessing using con trol the- ory principles", Scripta materialia V36, pp.347{353, 1997. [6] Wi A. S., Shatla M. N. Ab del-Hamid A., \An optim um-curv ed die prole for the hot forw ard ro extrusion pro cess", Journal of Materials Pr essing chnolo gy V73, pp.97{107, 1998 [7] an H., and Xia J., \An approac to the optimal de- sign of tec hnological parameters in the prole extru- sion pro cess", Scienc and chnolo gy of dvanc Materials V7, pp.127{131, 2006. Proceedings of the

World Congress on Engineering 2008 Vol II WCE 2008, July 2 - 4, 2008, London, U.K. ISBN:978-988-17012-3-7 WCE 2008

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