Deep drawing simulation of α-titanium alloys using LS-Dyna

Transcription

1 Deep drawing siulation of α-titaniu alloys using LS-Dyna Sebastijan Jurendić a, b, Silvia Gaiani a, c a Akrapovič d.d., Ivančna Gorica, Slovenia b University of Ljubljana, Faculty of Mechanical Engineering, Ljubljana, Slovenia c University of Modena and Reggio Eilia, Dept. of Materials Engineering, Italy Abstract Titaniu alloys have excellent properties for their target applications; however their use is still liited by high price and forability issues. To avoid extensive on-site trials and to cut developent costs, a nuerical siulation ethod is developed for the deep drawing process of α-titaniu (hexagonal close packed) alloy sheet using LS-Dyna. The Barlat 1989 aterial odel is adopted for odelling the plastic response of the aterial and the necessary input data is exained. It is found that in order to adequately capture the plastic properties of HCP titaniu, load curves are needed both for strain hardening and to capture the strain dependency of the plastic strain ratio. A procedure for deterining the aterial input data fro the tensile test results is developed and an exeplary data set is given. To identify a suitable value of the Barlat flow potential exponent a paraetric analysis is carried out using a siulation of the Erichsen cupping test. Foring liit diagras are adopted for failure prediction, the foring liit curves are deterined using the Nakajia ethod and a siplified procedure for obtaining liiting shear strains on a tensile testing achine is presented. To confir the ethod an exaple of a deep drawn end-cap for a otorcycle exhaust uffler is presented and the siulation copared to the physical foring process with good results. Keywords: HCP titaniu; Foring siulation; Deep drawing siulation; Sheet etal; LS-Dyna; 1 Introduction With the ever widening field of application of titaniu alloys, classical anufacturing techniques are being applied to these high-tech aterials. This can pose a proble under ass production conditions, since their relevant echanical properties are quite different to those of traditional engineering aterials. Such is the case with deep drawing of heat resistant α-titaniu alloys for high-end autootive exhaust applications. The exceptional light-weight, echanical, theral and corrosion-resistant properties of the finished product outweigh the high cost of the aterial and coplications in the production process. However, difficulties associated with forability of the aterial need to be overcoe in order to establish a reliable production process. This becoes a ajor issue as the high cost of the raw aterial, along with relatively low production volues, akes extensive on-site trials uneconoical. There is a clear need for a nuerical siulation ethod in this field to optiize the process paraeters and tooling geoetry beforehand and therefore iniize the aount of costly on-site trial and error testing. A key feature of a ethod applied in an industrial environent is the ability to proptly and efficiently identify all the necessary input data, especially the aterial paraeters, with readily available tests. The Barlat 1989 aterial [1] odel is well suited in this respect, as it allows for all of its input paraeters to be derived fro the standard tensile test, which is routinely carried out as part of quality control on incoing raw aterial. The ai of this paper is to develop a robust and efficient ethod for siulating deep drawing of α- titaniu alloy sheet using LS-Dyna. The LS-Dyna ipleentation of the Barlat 1989 aterial odel is reviewed and adopted as the ost appropriate aterial odel currently available. The necessary input data for the aterial odel is exained and a aterial characterization procedure is defined. The Barlat flow potential exponent is deterined through a paraetric analysis using a siulation of the Erichsen cupping test. The α-titaniu alloy 1. ASN fro Kobe Steel is characterized by this ethod and the data is used on a deep drawing siulation of an exhaust end-cap. 8 th European LS-DYNA Users Conference, Strasbourg - May 011 1

2 General properties of α-titaniu and the Barlat 3-parater aterial odel.1 General properties of α-titaniu Titaniu alloys used in this application are all based on the α phase with a hexagonal close packed (HCP) crystal structure. A description of the hcp titaniu crystal structure and echanics of deforation is available in []. The fundaental properties that ake these aterials difficult to for and significantly coplicate the phenoenological descriptions copared to steels and aluiniu alloys are: highly anisotropic yielding high plastic strain ratios that wary greatly with orientation highly anisotropic hardening extension to R falls off significantly fro longitudinal to transverse direction asyetry in yielding (tensile vs. copressive strength differential SD) due to twinning phenoena It should be noted that the twinning deforation ode in titaniu is activated during in-plane copression [3].. Barlat 1989 aterial odel The Barlat 3-paraeter 1989 [1] aterial odel (Material 36 in LS-Dyna) is chosen for the siulations at this stage because it is based on input paraeters which have a well defined physical relevance and can be readily easured in an industrial environent on a standard tensile testing achine. It was developed priarily for BCC and FCC aterials and thus lacks the capacity to odel SD effects, the yield locus however can be varied trough the flow potential exponent to suit the titaniu yield surface and should give acceptable results at least under predoinantly tensile conditions, where twinning odes are not activated. According to the Barlat odel, the anisotropic plane stress yield criterion is defined as: Φ = a K 1 + K + a K 1 K + c K = where K 1 and K are defined as: σ Y (1) σ x + hσ y K1 = K = σ x hσ y + p τ xy a, c, h, p, are aterial paraeters defined as: () (3) a = R00 R R00 1+ R90 (4) c = a (5) h = R00 1+ R 1+ R R (6) According to the authors of the aterial odel the plastic strain ratio R for an arbitrary angle fro the rolling direction can be deterined by: Rφ = σ Φ + σ x Y Φ σ σ y φ 1 (7) where the paraeter p is calculated iteratively fro the above expression to fit the data for the uniaxial tensile test in the diagonal (45 ) direction. The LS-Dyna ipleentation [4] allows for autoatic calculation of the aterial properties fro the plastic strain ratios, which significantly siplifies the use of the odel. To account for the anisotropic hardening properties the plastic strain ratio values in the longitudinal, diagonal and transverse direction should be input as functions of equivalent plastic strain, as this will odify the yield locus with the plastic flow. 3 Input paraeter deterination 3.1 Input paraeters Fro the atheatical forulation described above, the necessary input data for the constitutive odel are: Plastic strain ratio in three directions: R 00, R 45, R 90, which are derived directly fro the tensile test with the width to thickness strain ratio easureent. To capture the anisotropic hardening the paraeters are R ε p. input as load curves ( ) Yield stress as a function of equivalent plastic strain ( p ) i σ in the rolling direction. Y ε Although it is coon practice to use the 8 th European LS-DYNA Users Conference, Strasbourg - May 011

3 power law plasticity in foring siulations, this proved inadequate in this application; a full load curve inversely identified fro the tensile test should be used instead. copleted curves are converted into a discreet function to be used as a load curve in the odel. Flow potential exponent defines the shape of the yield locus. References for this paraeter in literature are sparse, however soe research on the subject indicates that a quadratic yield locus ay be appropriate for titaniu [ref], thus = is used for initial evaluation. 3. General echanical properties of 1. ASN The properties were easured using the standard tensile test in accordance to the EN ISO 689:009 standard with the extensoeter gauges at 80. The sheet thickness is 0,9. Five saples were tested in each direction, the presented values are average vales of all the tests in their respective directions (Table 1). Table 1 General echanical properties of 1. ASN Dir. E [GPa] R p [MPa] R [MPa] A gt [%] A tot [%] ,4 3, ,7 35, ,0 34,5 3.3 Plastic strain ratio These paraeters should be deterined first as they are needed in the subsequent steps. They can be deterined directly fro the tensile test using biaxial strain easureent. The procedure for plastic strain ratio deterination is defined in the ISO standard [5]. This procedure is necessary because of the inherently large error associated with plastic strain ratio easureent [6]. Following the standard procedure, the plastic strain ratios are deterined in intervals of 1% fro initial yield to the onset of localized necking on the specien. Figure 1 shows the variation of the plastic strain ratio with plastic strain. The variation in the longitudinal direction is nearly negligible, in the diagonal and transverse directions there is a significant variation in the initial plastic region before the values stabilize. The curves are extrapolated anually fro the onset of necking to large strains using an exponential function. The Figure 1. Plastic strain ratio as a function of true plastic strain in the rolling, diagonal and transverse direction. 3.4 Yielding curve deterination n Initially the Holloon equation σ = K ε was used as the yield stress function, however the fit at higher strains is unacceptable. In contrast to steel, titaniu exhibits substantial additional elongation past R, with a fairly gradual onset of localization, which is in line with the properties expected fro its crystal structure. This phenoenon becoes even ore pronounced in the diagonal and transverse directions, where the peak force occurs at very low strains yet the aterial still achieves a oderate strain at fracture. The assuption follows that quite soe useful deforation occurs after the onset of localized thinning in etal foring applications. This property cannot be reasonably captured by any of the traditional hardening laws, thus a load curve representing yield stress as a function of equivalent plastic strain ( p ) σ is identified using an inverse Y ε procedure proposed by Koc et al. [7]. To suarize, the load curve is identified iteratively by running nuerical siulations of the tensile test and coparing the response to the test data. After each iteration the load curve is odified until an acceptable agreeent between the siulation and the tensile test is achieved. A fit within the scatter between saples of the sae batch can be achieved without difficulties; however, since the deviation of aterial properties on a single coil of sheet is usually substantial, a 8 th European LS-DYNA Users Conference, Strasbourg - May 011 3

4 perfect fit is not necessary. Figure shows the results of such an inverse procedure, the fit between the easured and calculated σ-ε curves is practically perfect over the entire range of easureent. Points on the FLD were easured using the Arais optical 3D foring analysis syste (Figure 4) to easure the in-plane strains. This syste uses two caeras to recognize a stochastic pattern spraypainted onto the speciens surface and to track the deforation in real tie. Figure 4. FLD deterination with the Arais syste. Figure. Tensile test siulation vs. the tensile test results. The final load curve for the aterial is shown in Figure 3 along with the easured true stress true strain curve and the functional approxiation using the Holloon equation. Copared to the Holloon approxiation it is soewhat steeper to support the extensive post-r deforation. The load curve is extrapolated past the breaking point, this is necessary as the aterial can reach uch greater stains under different loading conditions, such as biaxial tension. In our case the Nakajia ethod was used to deterine the foring liit curve. With this ethod the inor strain is varied using speciens with different elliptical aspect ratios, as shown in Figure 5. Figure 5. Nakajia test speciens. The tests have been perfored using three speciens for each of the seven different geoetries, extracted along the rolling and transverse direction. A total of 4 tests were perfored for the aterial. The foring liit diagras obtained is shown in Figure 6. Figure 3. The inversely identified yield curve shown against the Holloon approxiation and the tensile test results. 4 Liits of forability As the goal of the siulation is to ultiately deterine the feasibility of a given deforation process, a way of detecting aterial defects is needed. For this reason the foring liit diagra (FLD) was deterined. Figure 6. Foring liit diagra for 1. ASN. 8 th European LS-DYNA Users Conference, Strasbourg - May 011 4

5 4.1 Shear foring liit Failure of the aterial in shear is a coon occurrence in deep drawing processes. If the initial blank size is too sall the blank will shear at the die radius. This failure ode is not tested in the conventional Nakajia test, thus a different ethod has been devised. The in-plane strains are plotted in the inor-ayor strain space and a line perpendicular to the upper foring liit line is plotted through the largest negative inor strain as shown in Figure 9. A special specien shown in Figure 7 has been prepared so that it fits in the standard tensile testing achine to allow for the test to be carried out inhouse. The width at the grips is 50 and the specien allows for the use of a standard 80 extensoeter. The shape was optiized to localize the deforation in the shearing zone. Figure 9. The coplete foring liit curve with the in-plane strains fro the siulation at specien fracture. This is a soewhat siplified approach, however it yields good results as the siulation conditions directly copare to the conditions under which the foring siulations are run. Figure 7. Geoetry of the shear test specien. The specien is strained until it breaks and the total extension at the extensoeter is noted, a nuerical siulation of the test is then carried out to the sae aount of deforation. The nuerical odel of the specien is shown in Figure 8. It is odelled using three and four node shell eleents and syetry is taken into account. 5 Erichsen cupping test siulation 5.1 The Erichsen test The Erichsen cupping test is a standard test perfored on sheet etal to deterine the stretch forability of the aterial. The test is governed by the ISO 048 standard. Figure 10 shows the tooling geoetry. Figure 8. The nuerical odel for siulating the shear test. Figure 10. Scheatic of the Erichsen cupping test. 8 th European LS-DYNA Users Conference, Strasbourg - May 011 5

6 This test is particularly well suited for evaluating the aterial odel in the first stage as it strains the specien in biaxial tension. There is no in-plane copression or shear, thus avoiding the strain differential effects that pose a proble for applications using the Barlat 1989 aterial odel, so it provides a good platfor for the paraetric analysis of the paraeter. 5.3 Paraetric analysis results The results of the paraetric analysis, shown in Figure 1, indicate that = is an appropriate value. The force displaceent plot shows good agreeent between the test and the siulation, also the FLD predicts a break at around 8,7 of deflection, which is nearly identical to the test. Results of the Erichsen test for the 1. ASN aterial are shown in Table. The sheet thickness was 0,9 and three speciens were tested, the results are average values of all three tests. Table Erichsen test results for 1. ASN Maxiu load [N] Erichsen index [] 800 8,86 5. The nuerical odel The nuerical odel used for the siulations is shown in Figure 11, it is coprised of three and four node shell eleents. We assue that there is no drawing of the aterial fro under the holder, thus only the free portion on the blank is odelled and fixed around its perieter in all degrees of freedo, avoiding the need to odel the holder. The die and punch are odelled using rigid aterials (Material 0 in LS-Dyna) and friction contacts are prescribed between the tools and the blank. The penalty contact forulation proved to be inadequate around the die radius as unacceptable contact penetration occurred and instead a constraint forulation is used. Mass and tie scaling are used to cut calculation tie, adaptive reeshing was found to not provide a significant tie saving in this case, thus a denser esh was used for the entire siulation. Figure 1. Force displaceent plot of the Erichsen test siulation for different values of. 6 Siulation of a deep drawn part 6.1 The physical part A otorcycle exhaust end-cap, shown in Figure 13, was selected for the siulation. It is a probleatic shape to fro because of the tapering sides and a sall radius at the top. This particular part cannot be drawn fro the 1. ASN aterial (the exaple shown in Figure 13 is drawn fro a different titaniu alloy), the aterial fractures at the leading edge of the punch. This allows for better evaluation of the ethod as the liits of forability are surpassed. Figure 11. Nuerical odel of the Erichsen cupping test. Figure 13. A fully fored exhaust end-cap. 8 th European LS-DYNA Users Conference, Strasbourg - May 011 6

7 A series of drawing tests was carried out with the 1. ASN aterial with a sheet thickness of 0,9. Longitudinal and transverse orientations of the aterial rolling direction with regard to the longer axis of the end-cap were tested and the axiu safe drawing depth for this shape was established to be 47 (the full depth is 90 ). The thickness distribution was then easured on these saples along the line shown in Figure Results The process was evaluated for longitudinal and transverse orientation of the aterial with regard to the longer axis of the rosette. The in-plane strains at 47 of drawing depth for the longitudinal and transverse directions were plotted on the FLDs in Figures 16 and 17 respectively. They clearly show the localized deforation in the uniaxial strain region of the FLD that exceeds the foring liit curve. Those points coincide with the leading edge of the punch where fracture occurs. The siulations predict aterial fracture on the FLD at around 40 of depth for the longitudinal orientations and around 4 for the transverse orientation, which is soewhat conservative. Figure 14. The end-cap drawn to 47 with the thickness distribution easureent line. 6. The nuerical odel Figure 15 shows the nuerical odel of the endcap deep drawing process. The odel is coprised of three and four node shell eleents. As before, the tools are considered rigid and friction contacts are prescribed between the tools and the blank. Even with this larger 5 die radius, contact penetration is still an issue for the penalty forulation, and therefore the constraint forulation is used instead. Mass and tie scaling as well as adaptive reeshing were used to cut calculation tie. Figure 16. Minor ajor strain plot for the longitudinal blank orientation at 47 of punch displaceent. Figure 15. The nuerical odel of the deep drawing process. The siulations were run to the full 90 drawing depth and both blank orientations were tested. Figure 17. Minor ajor strain plot for the transverse blank orientation at 47 of punch displaceent. 8 th European LS-DYNA Users Conference, Strasbourg - May 011 7

8 The siulated thickness distributions with both blank orientations at the axiu safe drawing depth were copared to the experiental thickness distributions easured on the real parts. The zero point on the horizontal axis corresponds to the die shoulder, with the distance easured along the surface of the part. Figures 18 and 19 show the results of the coparison. The siulation results copare fairly well to the easureents; the severe discrepancy in the longitudinal direction is due to the siulation predicting a break in the rosette before this depth is achieved. between the siulation and experients, although the siulations are soewhat conservative. The outlined procedure for acquiring aterial data proved to be quite robust overall. Most of the aterial data can easily be derived fro the tensile test, one area of concern however is the inverse yield curve deterination procedure, as it requires anual alterations to the load curve after each iteration, which is unnecessarily tie consuing. This process would greatly benefit fro autoation. Although the Barlat 1989 odel proved to be adequate, there have been recent developents of bespoke HCP aterial odels [8, 9] that could be ipleented into LS-Dyna using the UMAT library, and could potentially iprove the accuracy of the ethod, however the deterination of the input paraeters for these odels would require a ore coplex procedure. 8 References [1] F. Barlat, J. Lian, Plastic behavior and Stretchability of Sheet Metals, Part 1: A Yield Function for Orthotropic Sheets Under Plane Stress Conditions, International Journal of Plasticity, 5 (1989), p Figure 17. Thickness distribution with the longitudinal blank orientation. [] G. Lutjering, J.C. Willias, Titaniu, nd edition, Springer, 007. [3] M.E. Nixon, O. Cazacu, R.A. Lebensohn, Anisotropic response of high-purity α-titaniu: Experiental characterization and constitutive odeling, International Journal of Plasticity, 6 (010), p [4] LS-Dyna Keyword Users Manual Volue V971-R4, LSTC, 009. [5] International Standard ISO 10113, International Organization for Standardization, 006. [6] ASTM Standard E , ASTM International, 000. [7] P. Koc, B. Štok, Usage of the yield curve in nuerical siulations, Journal of Mechanical Engineering, 54 (008), p Figure 18. Thickness distribution with the transverse blank orientation. 7 Conclusion Despite the initial concerns about the liitations of the Barlat 1989 aterial odel with regard to HCP aterials, the results show that it is quite adequate for siulating deep drawing processes at this level. Both the Erichsen cupping test and the otorcycle exhaust end-cap exaple show good agreeent [8] O. Cazacu, B. Plunkett, F. Barlat, Orthotropic yield criterion for hexagonal close packed etals, International Journal of Plasticity, (006), p [9] M.G. Lee, H.R. Wagoner, J.K. Lee, K. Chung, H.Y. Ki, Constitutive odeling for anisotropic/asyetric hardening behavior of agnesiu alloy sheets, International Journal of Plasticity, 008 (4), p th European LS-DYNA Users Conference, Strasbourg - May 011 8