Fitting function representation for strain fields and its application to the optimizing process

International Journal of Minerals, Metallurgy and MaterialsVolume 21, Number 6, June 2014, Page 556DOI: 10.1007/s12613-014-0941-6Fitting function representation for strain fields and its application tothe optimizing processKun Chen', Ke-jia Liut), Li-qun Wei', Yi-tao Yang21) School of Materials Science and Engineering, Shanghai Institute of Technology, Shanghai 201418. China2) School of Materials Science and Enginering, Shanghai University, Shanghai 200072, China(Received: 22 November 2013; revised: 15 January 2014; accepted: 29 January 2014)Abstract: A ftted function method to describe the strain fields during forging was discussed to optimize the homogeneous distribution ofstrain in the axial forging zones during successive stretching. The results are verified by experiment and numerical simulation, and the devia-tions between experiment and simulation are less than 24%. Therefore, the ftted function method can be applied to optimize the stretchingprocess for large forgings. The optimal value of feed determined by the analytic method ensures that the degree of inhomogeneity in strain inthe axial ingot zone is less than 6%. This work provides a mathematic model to optimize technological parameters in stretch forging of largeingotsKeywords: forging; strain; ftting function representation; optimizationmensional plastic problems are complicated for exact ana-1. Introductionlytical solutions. In recent years, finite element simulation(FEM) has been widely applied to the forging process [4- -5],Forging has been widely employed to improve the mi-which can provide the flow state during metal forming. Thecrostructure and mechanical properties of materials. Duringstrain ditribution with specified technological parametersforging, internal voids from initial cast ingots can be elimi-can be evaluated, i.e., numerical results for strain can be ob-nated [1-2]. Thus, investigations on strain ditributions in-tained. However, the functional expression that describes theside of forgings are very important.regular pattern of strain distribution is hardly determined. So,Currently, there are three types of research methods onthe optimization can only be achieved by repeated trial solu-plastic strain: physical simulation methods (experimentaltions, modification of technological parameters based onmethods), analytic function methods, and numerical analysissimulation results, and comparisons of additional FEM re-methods. The experimental methods include grid method,sults, one by one [6- -7]. This approach is time-consumingscrew method, hardness method, compared-grain method,and inefficient, thereby has negative effect on the competi-moire method, and photoplasticity [3]. However, because oftive stance of enterprise.restrictions imposed by high temperature, the integrity ofmaterials, and differences in mechanical properties between ing the strain field [8], i.e, a functional representationmodel and prototype materials, it is difficult to determine themethod, and the optimum technological parameters werestrain distribution by experimental methods listed above. Indetermined by logical deductions from the mathematicalthe analytic function method, exact solutions are obtained byfunction. This approach could significantly reduce the timessolving the mechanical differential equations. Althoughfor numerical simulation and have a very important impactanalytic solutions can be obtained for plane deformationon the optimization of forging processes. The purpose ofproblems and axisymmetric deformation problems, three-di-this paper was to further explore this method on stretchCorresponding autho: Kun Chen◎University of Science and Technology Bejjing and Springer-Verlag Berlin Heidelberg 2014E- mail: chk119@ st.du.cn中国煤化工包SpringerYHCNM HGK. Chen et al, Fitting function representation for strain fields and its application to the optimizing process557forging.expression was determined by measuring and analyzing theThe strain distribution in a stretch-forged material is re-distribution of E in the x direction [8].lated to the tool width ratio (W/H) and the deformation de-gree [9-1]. It is well known that the forging effect canpenetrate intermally and exterally, internal voids can be ef-fectively eliminated using a value of W/H in the range of0.5- 0.8, and the size of reduction is 20%- -30% [12]. Usually,internal voids appear on the centerline of the longitudinalingot section, so the strain distribution in that section is veryimportant. However, induced by the friction and rigid zoneyIof the ingot during practical forging, the strain distributionnear the axial ingot regions is not continuously uniform, and2345678910111213141516internal voids in the small deformation region are difficult toeliminate. At a certain value of feed, a fairly uniform de-Fig. 1. Specimen divided along the symmetry axis after sin-formation along the longitudinal ingot section can be real-gle-pass forging.ized and intermal voids can be eliminated [13]. The inveti-gation shows that the required value of feed is usually ob-It is diffcult to measure the internal strain distributiontained by practical experience rather than theoretical deriva-during hot forging, and it is also difficult to obtain the strainion; e.g， in practice, a feed magnitude employed isdistribution at room temperature when the microstructure isone-third of the anvil width [14]. Banaszek and Szota [15]transformed from austenite to martensite. Therefore, in theused numerical simulation to calculate the strain distributionpresent study, the numerical distribution of strain on thewith different process parameters and used statistical analy-symmetry axis was determined by measuring the residualsis to determine an objective function, which served in thestrain of retained austenite by X-ray diffraction (XRD) [8].optimization of successive stretching. However, the result-To improve the precision of calculated results, the diffrac-ing objective function depends on the simulation and thetion peaks of retained austenite should be obtained using thestatistical calculation is rather complicated, so the approachlargest possible Bragg angle and intensity. Thus, the meas-is difcult to apply in practice. The new method proposed [8] ured interplanar distance (d) was in the (311) plan of re-can be applied to solve such problems.tained austenite at the selected points. The residual strainIn this study, first, the strain distribution was determinedcan be calculated from Gresidual= (d - do)ldo, where do is theby experiments based on specified technological parameters,interplanar distance of undeformed retained austenite.and then a functional expression of the strain field was ob-Fig. 2 shows the experimental result and fiting functiontained on the symmetry axis by mathematical function fit-representation of the residual strain distribution of retainedting. Finally, the optimization of the fiting function pro-austenite along the symmetry axis in the x-direction (& residua)vided the optimal value of feed, resulting in a homogeneous after compressive deformation. A Gaussian function is in-distribution of strain in the axial forging zones. This papertroduced to characterize the strain distribution along thepresented a novel method for performing the optimization ofsymmetry axis, as shown in Eq. (1).technological parameters for large-ingot stretch forging, andεz=βe-ax2(1)the paper demonstrated the feasibility of the proposedwhere a is the parameter related to the width of strain dis-method.tribution and β the maximum value of &. Both a and β de-pend on the processing parameters, and their values can be2. Fitting function of strain fields during forgingdetermined by the experiment as well as other methods.A specific experiment was chosen to ilustrate Eq. (1). A2.1. Upsettingcylindrical specimen (high-chromium stee) with the diame-To analyze the operation of upsetting, a specimen waster of 70 mm and the height of 84 mm was used for analysis.divided along the symmetry axis after compressive defor-The compression experiment conditions were the deforma-mation, as shown in Fig.1, and points are located and num-tion degree of 45%, the heating temperature of 1200C, andbered along the x direction of its symmetry plane. The strainthe friction facto中国煤化工deformation andin the pressing direction (<2) was denoted by ε. A functionaloil quenching, td, as shown inYHCNMHG558Int. J. Miner. Metall. Mater, Vol. 21, No.6, .Jun. 2014Fig. 1. The result obtained from ftting the experimental datasion, numerical simulation was used to obtain the strain fieldwith the Gaussian function is shown in Fig. 2, hence, a=described above. Finite element (FE) computations based on2.75 x 10-3 mm-2. The value of β can be determined fromthe software DEFORM-3D were carried out to investigatethe total deformation based on the constant volume; namely,the deformation behavior of a high- chromium steel speci-men, and the detailed true stress-true strain data obtained-0H=J', e,dr=J" e.du=jIT",-a"dr.from a Gleeble 3500 thermal mechanical simulator werewhere△H is the total deformation, &, the strain in the radialloaded into the material model. Then, the simulation resultsdirection (r), and &r the strain in the pressing direction (z).were ftted to the Gaussian function, shown in Fig. 3, whereThus, β = 0.618. The fit cofficient of determination is R2 =a= 2.59x 10- mm- and β = 0.635. Compared to the val-0.99, which means that Eq. (1) can be adopted to describeues obtained from ftting the experimental data, the devia-the strain distribution in the symmetry axis after compres-tions are less than 6%, and the coefficient of determinationsive deformation.is R2 = 0.99. It is indicated that the strain distribution can bedetermined by various methods, including XRD, numerical-0.020simulation, and so on.■Experimental-0.016Eq. (1)0.7 rFE simulation0.6 tGaussian fitting0.5 t-0.0080.4 t0.3 t-0.0040.000-60-45-30-15015304560750.1 t).0 L-50-40-30-20-100 10203040 50Fig. 2. Residual strain distribution of retained austenite alongx/ mmthe symmetry axis in the x-direction.Fig. 3. FE simulation and Gaussian ftting of the strain dis-This approach was also used to determine the proportion-tribution along the symmetry axis in the x-direction.ality coefficient K= &/ Euresidual [8]. It is difficult to measurethe true strain, but it is relatively easier to measure the cor-2.2. Stretch forgingresponding residual strain. This is because the crystallineStretch forging is characterized by local loading, localslip is the main part of the true strain, and the grain defor-pressure, and local deformation. In the deformation zone, .mation is minor. However, the measurement of grain de-the deformation is similar to that in the upsetting case de-formation, compared with that of crystalline slip, can bescribed above, but still has the difference. In this study, theconveniently performed by XRD. Moreover, the residualtrain distribution in the deformation zone during stretchstrain distribution fathfully reflects the strain distributionforging was obtained by experiment and numerical simula-after deformation. Therefore, the residual strain can be ex-tion.pressed as E_residual = βe -ax-，even if the elastic deforma-A high-chromium steel was used in the following study.tion will be recovered after the stress is eliminated, there stillA stock with the cross section of 50 mm x 50 mm and theexists some residual strain without reversion, although thelength of 100 mm was used in the experiment and numericalvalues of E-residua are far smaller than those of the true strain.simulation. The process parameters were as follows: defor-Nevertheless, there is a direct proportional relationship be-mation degree 25%, tool width ratio 0.6, heating tempera-tween &q and Eresidual as the following the equation.ture 1200C, and friction coefficient 0.3. The measuredεz=β'e-ax =Kβe ax' = KE:xresidualtheological parameters of the selected steel at deformationThus, the values of proportionality coefficient K can be ob-temperature as well as DEFORM 3D software were em-tained from the fitted Gaussian function.ployed to simulate the stretch forging process. Fig. 4 showsTo further verify the reliability of the functional expres-the schematic o中国煤化工YHCNMH GK. Chen et al, Fitting function representation for strain fields and its application to the optimizing process559Experimental and simulation results for the strain distri-bution in the axial forging zones are shown in Fig. 5. Basedon the investigation described above, the Gaussian functiongiven in Eq. (1) was adopted to fit the distributions of &.The coefficients of determination for fits are both R2 = 0.99,indicating that Eq. (1) can be used as the analytic function10 mmto accurately describe the strain fields during stretch forg-Fig. 4. Schematic of the stretch forging process.ing.-0.003(a■Experimental-0.4b)■Simulation. FittingFitting-0.002-0.3-0.2-0.001-0.10.00010 0102030 4x/ mmFig. 5. Residual strain (a) and strain (b) distribution along the symmetry axis in the x-direction.3. Optimization of successive stretching in flattively predict this inhomogeneous distribution of strain, theanvilsdegree of strain inhomogeneity (δ&) is proposed and ex-pressed by the difference between & max and E min, as shown3.1. Analytic optimization of successive stretchingin Eq. (3).The successive stretching for stock described above wasδ&(x)=&max-&min(3)chosen for analytic optimization. Once the Gaussian func-The degree of strain inhomogeneity (δ&) is a function oftion is determined, the main concern is to obtain the optimalfeed value (xo), as shown in Fig. 7(a). The minimum valuevalue of feed to ensure a homogeneous distribution of strainof 8&; (δ&, = 0.01) is obtained when the value of feed is x =in the axial forging zones.16 mm, which assures the highest degree of strain homoge-It is assumed that the value of feed for forging in succes-neity. Therefore, optimization in terms of the homogeneoussive drafts is constant and equal to X0, where x denotes thedistribution of strain in the axial forging zones near thevalue of feed in the axial direction (x). Since the stretchsymmetry center is obtained at the minimum value of δ&,forging is only carried out in the x-direction (as shown inFig. 4), the strain along the z-axis, which is denoted by 6,can be superimposed on the strain function of each draft.).3 tTaking three successive drafts for an example, En in thex-direction can be expressed as.心0.2εz= pe-a(-5)2 + βe-a(x)2 + pe-a(+x)2(27MMTwhere a= 6.50x 10- mm~ and β= 0.407 obtained fromexperiment. Fig. 6 shows & from Eq. (2) at a feed of x0= 300.0mm.60-40-200204(50Fig. 6 shows that the values of strain in the axial billet re-gions are distributed in an inhomogeneous manner, thFig. 6. Strain al中国煤化工ging zones aftermaximal strain (E; max), secondary maximal strain (ε' max ),three successive dTYHCNMH G560Int. J. Miner. Metall. Mater,, Vol. 21, No.6, Jun. 2014(a0.4 rb)).3 t.2-官0.23 0.1-0.1.0 F.1-102045003405x/ mmFig. 7. Relationships between the feed value and the degree of strain inhomogeneity: (a) δ& and.xo; (b) 8ε': and xX0A further study of Eq. (3) indicates that the minimumin the (311) plane of retained austenite at the selected points.value of δ& cannotBased on the formula for calculating residual strain, the dis-strain values always appear at the center of symmetry. Thus,tribution of Eresidwal in the axial forging zones was deter-the relationship between the secondary maximal strainmined. Then, the values of Eresdual were multiplied by the(E2 max ) and the minimal strain (& min) should also be takenproportionality coefficient K between the residual strain andinto account in view of whole strain homogeneity in the ax-total strain. Fig. 9 shows the experimental results of & dis-ial forging zones. When the difference between e'max anctribution in the axial forging zones after three successive& min is equal to zero, the whole strain homogeneity can bedrafts.achieved. So, the degree of strain inhomogeneity betweenE'max and & min can be given as .δe'(x)=e'max -ε:min(4which quantitatively expresses the degree of strain homo-广geneity for zones far away from the center of symmetry. Fig.7(b) shows the relationship between δe' and xX0. When the28456789101112 1314 15feed values are small, δε' < 0, which means that the val-ues of & min are larger than those of Ezmax. This is very un-Fig. 8. Laterally divided specimen after three successivefavorable for drawing effciency [16]. Therefore, δε' =0 drafts.should be used to characterize the optimum condition for-0.60strain homogeneity. The optimal value of feed is 19.4488mmat δε! = 0. In contrast, δ&, = 0.042 in Fig. 7(a) givesthe optimal result in terms of whole strain homogeneity in-0.45the axial forging zones.Optimization in terms of the homogeneous distribution of心-0.30strain in the axial forging zones can be performed by inte-grating two concepts. That is, the analytic optimization offeed (xo) has been achieved by combining Eq. (3) and Eq.-0.15(4).3.2. Experimental verification of optimal successive6080100120stretching in flat anvilsx1 mmfig. 9. Experimental results for the distribution of & in theTo verify the analytic optimization results, experimentsaxial forging zones after three successive drafts.were performed for three successive drafts using a feed ofX= 19.4488 mm. The deformed specimen was laterally di-Data in Fig. 9 indicate that in the axial forging zones, thevided along the symmetry plane, and points were selectedvalues of & fall within 0.24 0.29. The degree of strain in-and numbered along the longitudinal specimen section, ashomogeneity for this stretch forging process is δ = 0.05.shown in Fig. 8. Residual strain measurements were con-Comparing the中国煤化工2) with the ex-ducted by XRD; the measured interplanar distance (d) wasperimental resul2on is less than4HCNMHGK. Chen et al, Fitting function representation for strain fields and its application to the optimizing process6116%. Therefore, it is feasible to optimize successive stre-ratio 0.6, heating temperature 1200°C, and friction factor 0.3.tching in flat anvils by analytic calculation proposed in thisThe strain distribution of large ingots (denoted as。) in thestudy.axial ingot zone was determined in successive stretching..3. Simulation verification of optimal successiveThe functional expression is written as ε°=β° e" xwhere, in this forging case, a*= 1.3x 10-5 mm7 2 and β*=stretching in flat anvils0.409. For the successive stretching of large ingots, theIn addition to experimental verification described in Sec-value of feed is denoted by xX0, and then the functional ex-tion 3.2, optimization results in Section 3.1 were also veri-pression for three successive drafts can be written asfied by numerical simulation. Finite element simulation wascarried out for three successive drafts with the feed value6:=β° (=-rr(xo) = 19.4488 mm. Fig. 10 shows the simulation results ofSimilar to the development presented in Section 3, the& distribution in the axial forging zones. For the optimaloptimal results were obtained at x= 217.444 mm and δε =value of feed, the values of &n lie within 0.24 -0.295. The0.041. Thus, the ftting function representation method wasdegree of strain inhomogeneity is δ& = 0.055. Comparingsuccessfully applied to the optimization of successivethe calculated results (δε = 0.042) with the simulated resultsstretching for large ingots.(δ&z = 0.055), the calculated deviation is less than 24%，Since there are some difficulties in the actual successivewhich validates the analytic calculation. Comparing thestretching of large ingots, FE simulation for three successivesimulation results (δ& = 0.055) with the experimental resultsdrafts with the feed value X = 217.444 mm was carried out(δε = 0.05), the deviation is less than 10%, which validatesto verify the optimization results. Fig. 11 shows the simula-the numerical simulation.tion results of ε; distribution in the axial forging zones. It-0.60shows that the values of ε: fall within 0.35- -0.40 for theoptimal value of feed. The degree of strain inhomogeneityfor the stretch forging process is δε° = 0.05. Comparing-0.45the calculated result (δε° = 0.041) with the simulation re-sult (δε° = 0.05), the deviation is less than 18%.-0.30-0.5-0.15-0.4.--0.30.002(10120x/ mm-0.2Fig. 10. Simulation results for & distribution in the axialforging zones after three successive drafts.-0.14. Optimization of large ingot successive0.050010001500stretchingFig.11. Simulation results for ε distribution in the axialBased on the results above, the fitted function methodingot zone after three successive drafts.was applied to the optimization of large-ingot successivestretching. Since there are some difficulties in the actual5. Conclusionssuccessive stretching of large ingots, the valuesof aand βwere determined by computer simulation.(1) The ftting function representation method proposedThe material used in this study was still the high chro-earlier is further tested in this paper. For single-pass forging,mium steel. A stock with the cross section of 500 mm X 500the regular patterns of strain distribution in forgings can bemm and the length of 1200 mm was used to simulate therepresented by Gaussian functions. Then, the optimizationstretch forging process. Process parameters used in theof technological Darameters can be determined from thesimulation were as follows: relative draft 25%, tool widthfunction calculal中国煤化工mes needed forYHCNMHG562Int. J. Miner. Metall. Mater, Vol. 21, No.6, .Jun. 2014optimization, compared to the times required for experi-No.48-49, p. 6676.ments or FE computation.5] H.S. Valberg, Applied Metal Forming, Cambridge UniversityPress, Cambridge, 2010, p. 223.(2) The optimization of successive stretching is based on[6] X.W. Chen, J. Chen, S.Y. Zuo, and X.Y. Ruan, Research anda deduced mathematical function. For the successive stre-development of forging optimization technique based on fi-tching, the distribution of & can be expressed as the linearnite element analysis, Chin. 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Cui, Finite element simulation of stretchexperimental and simulation validation of the analytic cal-forging using a mesh condensation method, Sci. China Ser. E,culation method proposed in this study to optimize the suc-53(2010), No. 1,p. 227.cessive stretching in flat anvils.10] G. Banaszek, H. Dyja, D. Rydz, and S. Berski, Strain distri-(4) The fiting function representation method can also bebution in forging process in flat anvils with inclined lateralapplied to the successive stretching of large ingots. Usingsurfaces, [in] Proceedings of the 10th Intemational Melallur-gical and Materials Conference, Ostrava, 2001, p.15.the determined optimal value of feed, a homogeneous dis-[11] L. Li, Q. Wang, S.Q. Yu, and L.Y. 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