Identification of the Hardening Curve Using a Finite Element Simulation of the Bulge Test

Transcription

1 Manuscipt No.72 Abdessalem Chamekh, Hédi Bel Hadj Salah, Mohamed Amen Gahbiche, Abdelmejid Ben Amaa & Abdelwaheb Dogui Identification of the Hadening Cuve Using a Finite Element Simulation of the Bulge Test Summay: The numeical simulations ae a useful tool fo contolling and optimizing the foming pocesses. The quality of these simulations stongly depends on the efficiency of the behaviou models of the defomed mateials. In ode to identify such models, expeimental tests must be used. In this wok, the bulge test with cicula and elliptical dies ae consideed. Duing the bulge test, the displacement of the cental point and the pessue wee continuously monitoed and ecoded. The numeical simulations of this test ae caied out using ABAQUS softwae. These cuent simulations, the behaviou of the specimen ae assumed to be elastoplastic. The Hill s citeion (48) and the isotopic hadening ae consideed. An invese modelling based on the finite element computations and expeimental data is made in seveal situations. The aim of this analysis is to identify the paametes of the hadening cuve and the Lankfod s coefficients. Seveal sets of esults ae pesented and discussed. KEYWORDS: metal foming, finite element, bulge test, invese modelling, paamete identification, elastoplasticity, Hadening cuve Laboatoie de Génie Mécanique, Ecole Nationale d Ingénieus de Monasti, Monasti / Tunisie Aims and scope: Nowadays, the foming pocess of sheet metals is a significant industial activity. In ode to contol this pocess, the numeical simulation seems to be the best tool. In fact, with the ecent developments in the non-linea computational tools, we can make tests, compae and optimize these pocesses at a low cost. But the quality of such simulations stongly depends on the good knowledge and the fomulation of the behaviou of the defomed mateials. In the sheet metal foming pocess, the behaviou law is consideed as anisotopic, elastoplastic in finite defomations. The identification of such models using the expeimental tests is still difficult. Indeed, the simple tensile test is not enough to identify the behaviou of the inceasingly efined models. Thus additional infomation must be taken fom othe tests [1], [2], [3], [4]. The bulge test is consideed in this wok. It has the advantage of poceeding without fiction. Numeical simulations of the consideed test ae caied out by the finite element method using the ABAQUS softwae. In the following, we will pesent the expeimental set-up, the consideed constitutive model and the finite element analysis. With egad to the discepancy between the numeical esults and the expeimental data, an invese identification stategy is used [5][6][7]. It is applied to identify the anisotopy coefficients of the mateial and the hadening cuve using the bulge test.

2 The second pat of the pape will be then devoted to the invese modelling pesentation, the discussion of the identification esults and the conclusions. Expeimental setup: The test concens a thin steel disk, clamped along its bounday by a daw-bead between the blankholde and the die and subjected to a unifom pessue P on one face (Fig.1). e, e and h epesent, espectively, the initial thickness, the final thickness and the height of the cental point. One of the significant advantages of this test is in the fact that it held almost without fiction. The geometical data and a sketch of the expeimental dies ae shown in Fig.2 fo the cicula die and in Fig.3 fo the elliptical die. The steel disk has a adius of 66.5mm and a thickness of.8 mm. The pat exceeding the snap ing is neglected. The cicula die is a evolution shell having an extenal adius of 66.5 mm and an intenal adius of 45.5mm. The elliptical die has a 11 mm along the lage axis and a 74mm along the small axis. The two dies have a 24mm of height and a adius of 6mm fo the beaing section. Duing the bulge test, the displacement of the cental point Up and the pessue P wee continuously monitoed and ecoded. The disk is a pat of a sheet made of high-stength steel X6 C Ni18-9. The simple tensile test is done on this mateial. It povides the stain hadening cuve (Fig.4) and the Lankfod s coefficients in seveal diections (Table1). Table1: The expeimental Lankfod s coefficients Mateial model: Ou goal is to study the behaviou of the mateial constituting the sheet. It is assumed to be elastoplastic. The elastic law is supposed isotopic linea with Young s modulus E = 193 GPa and Poisson s atio =.3. The plastic model is based on the following fomulations: f (, ) ( ) ( ) c s Whee c ( ) is the equivalent yield stess of the model and ( ) is the isotopic hadening law. s The equivalent yield stess is assumed equal to the Hill s othotopic yield citeion (Hill, 1948) [8] given, in the plane stess case, as follows: c ( G H) 11 2H 11 ( F H) 2 12 Whee ij epesents the Cauchy stess tenso in the othotopic axes; F, G, H and N ae the anisotopic coefficients. The plastic flow ule is witten as follows: p c ; ; f ; f The identification of the Hill s othotopic model unde associative flow ule assumption and in the case of plane

3 stess equies a stain hadening cuve s ( ) and anisotopic coefficients (F, G, H and N). The Lankfod s coefficients ae expessed vesus the anisotopic coefficients as follows: H, 9 G H, 45 F 2 N ( F G) 2( F G) n In the following, we adopt the elation s ( ) k Finite element analysis: Tests with 3D-models with diffeent element type have been pefomed to detemine the element to be used in the simulation of the bulge test. Hee, the disk is meshed with 1619 thee-dimensional elements of type S4R. It is a fou-node doubly-cuved shell fo finite stains. The softwae (ABAQUS) uses numeical integation to calculate the cosssectional behavio of the shell elements, such as shea defomation and thickness change. The element type has six active degees of feedom: thee fo the displacement and thee fo the otation. The pessue of fluid applied to the lowe face of the sheet is supposed to be unifom accoding to a linea pofile. The peimete of the sheet undegoes an imposed zeo displacement and the contact between the die and the sheet is taken in to account (Fig.5 and Fig.6). The mateial data used in the simulation wee the stain hadening cuve s ( ) and Lankfod s coefficients (, 45 and 9 ). The fist set of simulation was pefomed with a standad stain hadening cuve given in Fig.4 (caied out by the tensile test) and the expeimental Lankfod s coefficients given in Table.1. These simulations ae called case1. Fig.7 and Fig.8 pesent the defomed shape espectively fo the cicula and the elliptical die. In a fist step, we have caied out a numeical analysis of the bulge test. As an example of esults, we wee inteested hee in the atio of the stess components and the atio of the plastic stain components PE PE ij S S ij and to the evolutions of these atios duing the test. Fig. 9 and Fig.1 pesent espectively the evolutions of the atios of stesses and stains in the pole of the specimen. We can see that these atios ae constant duing the test. We note that PE 11 is close to 1 than 11 PE S. So, the bulge test is stain S bi-axial athe than it is stess bi-axial. We looked to the same atios in the nodes a and b indicated at Fig.11. The evolutions of these atios ae given in Fig.12 and Fig.13. We note that as long as the consideed node goes away fom the pole, the atios PE 11 and 11 PE S go away fom 1. So, the bulge test is less bi-axial fa fom the pole. In the second step we ae inteested in the global esponse of the test. We can see in Fig.14 (case1) and Fig.15 (case1) that when we use a stain hadening cuve fom the simple tensile test and the expeimental Lankfod s coefficients, the calculated displacement of the cental point undeestimates the expeimental data. To exclude the possibility that the discepancies ae caused by a quite coase mesh o unealistic S

4 bounday conditions, seveal simulations have been done using fine meshes and diffeent kinds of bounday conditions [9]. Then, the geatest eason fo the gap between the expeimental data and the numeical esults seems to stem fom the choice of the constitutive paametes. So, the invese modeling is consideed in ode to obtain a bette identification of these paametes. Invese modeling: In ode to minimize the gap between the expeimental measuements of the pessue and thei numeical values vesus the displacement of the pole, we use an optimization pocedue. The aim of this pocedue is to find the values of the constitutive paametes to be used in the finite element calculation and minimizing a measuement of the gap which epesents the objective function. This optimisation pocedue is used unde MATLAB softwae and is coupled to ABAQUS/Standad code. In a fist analysis, we identify the Lankfod s coefficients, 45 and 9. This simulation is called case 2. Case3 concens the identification of the hadening paametes, k and n. The poposed model is elative to the identification of both sets of paametes (hadening and anisotopy) fom the bulge test with a cicula die in two steps. Table.2: Analyses summay, k, n, 45, 9 Case 1 Case 2 Case 3 Poposed model Identification fom the simple tensile test Identification fom the simple tensile test Identification fom the bulge test (cicula die) Identification fom the bulge test (cicula die) Expeimental Identification fom the bulge test (cicula die) Expeimental Identification fom the bulge test (cicula die) Evaluation of the esults: As it is shown on Fig.14, the identified Lankfod s coefficients using the invese modeling give a good ageement between the expeimental and the numeical (case2) pessue-displacement cuves in the case of the cicula die. We pesent in Table.3 the identified Lankfod s coefficients in seveal oientations elative to the olling diection and the expeimental ones. Table.3: Compaison of Lankfod coefficients 45 9 Expeimental values Identified values : case We note fistly that these identified values, and mostly 9, ae not close to the expeimental ones. Secondly, one of these values ( 9 ) is almost out of ange. So, we can conclude that this analysis is not sufficient. This allows us to ty to educe the discepancy between the expeimental and the numeical esults using the modification of the stain hadening cuve. This appoach was used in othe woks [1], [11]. The invese modeling is then used to identify the

5 paametes, k and n fo the stain hadening cuve s ( ). As in the case 2 befoe, we seach the paametes minimizing the eo between the measued pessue-displacement cuve and the numeical one caied out by finite element computation of the bulge test. This identification is called case 3. Duing this analysis, the expeimental values of Lankfod s coefficients ae consideed. Table.4 pesents the identified values of the hadening paametes and thei expeimental ones. Table.4: Expeimental and identified values of the hadening paametes. Paamete Unit Identified values Expeimental values MPa k MPa n The identified and the expeimental hadening cuves ae plotted on Fig.4. We can see, as expected, that thee s no ageement between these two cuves. As shown in Fig.14, and although it esults fom an invese calculation, the numeical pessue-displacement cuve (case 3) is not close to the expeimental one. So, this identification is no longe sufficient. Poposed identification: As a esult of the analyses pesented befoe, we conside the hadening cuve identified fom the bulge test (case 3). Using this cuve, we identify the Lankfod s coefficients, 45 and 9 by the invese modelling fom the bulge test again with a cicula die.. Table.5 pesents the identified Lankfod s coefficients obtained thoughout this analysis. The cuve pessuedisplacement of the pole is pesented again in Fig.14. It is indicated by poposed model. Table.5: Compaison of Lankfod s coefficients 45 9 Expeimental case Poposed model As it is shown in Table.5, the identified Lankfod s coefficients in this case ae close to expeimental coefficients than those identified using the stain hadening cuve fom the simple tensile test. Moeove, fom Fig.14, we can conclude that the esults of the poposed model ae bette than those obtained with the paametes coming fom the tensile test. Validation and accuacy: The bulge test with the elliptical die is used in ode to eveal the effectiveness of the poposed model. Fig.15 pesents the pessuedisplacement cuves elative to this test. We find the expeimental measuements and the simulations esults. Cuent these simulations, we conside the constitutive paametes elative to case1, case2 and the poposed model as defined in Table 2. It is clealy shown fom Fig.15 that the poposed model is close to the

6 expeimental esults than the othe cases. But, the gap pesists impotant. The poposed model has also, the advantage to use Lankfod s coefficients in the ange of the expeimental ones contaily to case2. Finally, as it is shown in Fig.4, the gap between the hadening cuves is vey lage. It is in the ange of the gap between the expeimental data and the simulation s esults (case1) as pesented in Fig.14. Whethe the hadening cuve is obtained using an optimisation pocedue (case3), its esult is not close to the expeimental one (Fig14) and the value of the objective function is not zeo. Then, the Lankfod s coefficients ae identified in the way that they minimize the same objective function. Seveal stategies can be used instead of the poposed one. As an example, we can identify the hadening cuve and the Lankfod s coefficients in a same optimisation. In this case, the hadening cuve depends on the Lankfod s coefficients and the poblem pesents seveal optimums. The solution pesented in case 2 is one of them. Then, we have a lage vaiability on the identified paametes; and we can not have a good accuacy. In ou poposed appoach, the option of taking the expeimental values of the Lankfod s coefficients in the step of the identification of the hadening cuve avoids patially this difficulty. Conclusions: In this wok, we ae inteested in the bulge test. Ou goal is to be able to epoduce numeically the measued global esponse of the test. We consideed hee, the cuve of the pessue vesus the displacement of the pole of the specimen. As expected, we show that the Lankfod s coefficients and the hadening paametes coming fom the simple tensile test measuements do not give a good simulation of the test. The identification of the Lankfod s coefficients fom the bulge test using an invese computation can not educe significantly the discepancy between the expeimental and the simulation esults mostly in the validation case (elliptical die). We conclude that the identification of the hadening cuve fom the bulge test, and using an invese calculation, is also necessay to obtain anisotopic paametes in the ange and in the neighbouhood of those given by the simple tensile test. Then, the identified paametes give a numeical global esponse elatively close to the measued one in the bulge test using the elliptical die. This test is used to validate the identified esults. Howeve, moe investigations ae necessay in ode to educe the gap between simulations and expeiments. We believe that the Hill s (48) citeion is not enough to descibe such a mateial behaviou. So, we will use a nonquadatic citeion, a non-associative plastic law and kinematical hadening. Refeences [1] Balat F., Feeia Duate J., Gácio J. J., Lopes A. B., Rauch E. F., Plastic Flow fo non-monotonic loading conditions fo an aluminium alloy sheet sample, Intenational Jounal of Plasticity, (23), N 19, pp [2] Bel Hadj Salah H., Khalfallah A, Znaidi A., Dogui A. & Sidooff

7 F., Constitutive paametes identification fo elastoplastic mateials in finite defomation, Jounal de Physique IV, Vol. 15 (23), pp 3-1 [3] Khalfallah A., Bel Hadj Salah H. & Dogui A., Anisotopic paamete identification using inhomogeneous tensile test, Euoean Jounal of Mechanics A/solids, Vol. 21 () N 6, pp [4] Gelin J-C. and Ghouati O. Identification of mateial paametes diectly fom metal foming pocesses, J. Mat. pocessing Tech., (1998) pp8-81. [5] Khalfallah A., Bel Hadj Salah H., Dogui A., Anisotopic paamete identification using invese method, 4th Int. Esafom. Conf. on Metal. Foming., Apil 21, Liège (Belgium), pp [6] Mahnken R., Kuhl E., Paamete identification of gadient enhanced damage models with the finite element method, Eu. J. Mech. A/Solids, Vol. 18 (1999), pp [7] Kleinemann J-P., «Identification paamétique et optimisation des pocédés de mise a fome pa poblèmes inveses», Univesité de liège, Belgique, 2, (Thesis) [8] Hill R., A theoy of yielding and plastic flow of anisotopic metals. Poc. Roy. Soc. London, A193, (1948), pp [9] Chamekh A, «Simulation numéique de l emboutissage des tôles minces», Monasti, Tunisie, (23), (Mas-Diss) [1] Pasquinelli G., Simulation of metal-foming-pocesses by the finite element method, Intenational Jounal of Plasticity, Vol. 11, (1995), pp [11] Chamekh A., Gahbiche M. A., Bel Hadj Salah H, Ben Amaa A. & Dogui A., «Etude expéimentale et numéique de l essai de gonflement hydaulique», Actes des JS23, 21- mai 23, Boj El Ami ( Tunisia), Vol 2, pp 2-7

8 Flow stess (Mpa) 74mm METAL FORMING 24 - Section: e e Fig.1: Bulge test appaatus Snap ing Snap ing 91mm R6 11mm R6 Fig.2: geometical data and a sketch of the expeimental cicula die Fig.3: geometical data and a sketch of the expeimental elliptical die ,1,2,3,4,5 Equivalent plastic stain Tensile test plastic data Optimised plastic data Fig.4: Compaison between expeiment stain hadening cuve and identified stain hadening cuve

9 Fig.5: Finite element model of bulge test with cicula die Fig.6: Finite element model of bulge test with elliptical die Fig.7 : defomed shape when using cicula die Fig.8: defomed shape when using elliptical die

10 Fig.9: Evolution of the stesses atio in the pole duing the test Fig.1: Evolution of the plastic stains atio in the pole duing the test a b Fig.11: Position of nodes a and b in the specimen Fig.12: Evolution of the stess and stain atios in the node b duing the test Fig.13: Evolution of the stess and stain atios in the node a duing the test

11 Pessue (Mpa) Pessue (Mpa) METAL FORMING 24 - Section: Expeiment Case1 Case2 Case3 Poposed_Model Displacement, H(mm) Fig.14: Pessue vesus the displacement of the cental point in diffeent cases of analysis: bulge test with cicula die Expeiment Case1 Case2 Poposed_Model Displacement, H(mm) Fig.15: Pessue vesus the displacement of the cental point in diffeent cases of analysis: bulge test with elliptical die