Constitutive Modelling of Superplastic AA-5083

Transcription

1 TECHNISCHE MECHANIK, 3, -5, (01, 1-6 submtted: September 19, 011 Consttutve Modellng of Superplastc AA-5083 G. Gulano In ths study a fast procedure for determnng the constants of superplastc 5083 Al alloy at 73 K s analyzed. To evaluate the dsplacement and the thckness evolutons at the dome apex of the metal sheet bulge formng experments at constant pressure are performed. The dsplacement-tme curves at the constant pressures of 0.30, 0.35, 0.40, 0.45 and 0.50 MPa are obtaned by laser measurements whereas many bulgng tests at the same gas pressures stopped up at prefxed ntervals of tme were carred out to evaluate the thckness at the dome apex. Drect measurements of the thckness were obtaned usng a centesmal mcrometer. It s known that an equbaxal stress state s present at the dome apex of the metal sheet. In contrast, at other locatons, a nonequbaxal stress state prevals. Due to the nonequbaxal stress state, the thckest locaton occurs at the sheet perphery and the thnnest locaton at the dome apex. Therefore, by usng the fnte element method, t s possble to accurately analyse the superplastc formng process n the real stress state condtons. The materal behavour s modelled by the power law relatonshp between the effectve flow stress, the effectve stran and the effectve stran-rate. In the power law relatonshp the stran-rate senstvty ndex, m, s a constant, n and K are varables that are functons of the effectve stran-rate. Expermental results allowed to easly determne the value of m. Moreover, by means of some numercal smulatons t was possble to determne the values of n and K and assocate them wth specfc stran-rates. The results of comparson of the numercal smulatons of a bulge formng process wth the expermental tests have shown good agreement and they ndcate that the materal constants are relable. 1 Introducton Ths paper descrbes the characterzaton of AA5083 alumnum sheet by usng an nverse analyss technque n order to mnmze the dfference between numercal data and expermental measurements. Usng the commercal fnte element software MSC.Marc, the bulgng process at constant pressure s smulated. A rgd-plastc flow formulaton s appled to the superplastc formng analyss (Zenkewcz, The mesh s composed of fournode, soparametrc elements used for axsymmetrc applcatons. As ths element uses blnear nterpolaton functons, the strans tend to be constant throughout the element. The stffness of ths element s formed usng four-pont Gaussan ntegraton. The element has two coordnates n the global z- and r-drecton and two degrees of freedom for node (Msc.Marc, 005. An ntegraton scheme whch mposes a constant dlatatonal stran constrant on the element s used. The deformable body correspondng to the metal sheet s dscretzed wth two layers of 18 contnuum elements wth 4 nodes. The de s consdered as a perfectly rgd body. Constant pressure s appled as a dstrbuted load. Because of the symmetry of the geometry, the load and the constrant condtons, half of the cross-secton of the sheet metal s analysed. It s necessary to lock the movement of the nodes along the axs of symmetry n a drecton that s orthogonal of the axs of symmetry, to avod penetraton of the adjacent elements. Moreover, t s necessary to mpose constrant condtons on the perphery of the sheet n order to smulate the acton of a blank holder. Fgure 1a shows the shape of the de, the fnte element mesh of the sheet and the dstrbuted load over the edges of the sheet. The materal behavour s modelled by the power law relatonshp between the effectve flow stress, s, the effectve stran, e, and the effectve stran-rate, e &, by usng the followng equaton n m s = Ke e& (1 where the stran-rate senstvty ndex, m, s a constant, n and K are varables that are functons of the effectve stran-rate. The propertes of the materal and the pressure value, appled to the top part of the sheet metal, are ncluded usng a subroutne. 1

2 Materal and Formng Tests Alumnum alloy, AA5083 n the form of a 1mm thck sheet s characterzed by usng bulge formng tests at the constant pressures of 0.30, 0.35, 0.40, 0.45 and 0.50 MPa. The bulgng tests are carred out at temperature of 73 K (450 C. A crcular 79-mm-dameter specmen s nterposed between two heated steel des. The upper de has a crcular geometry wth an aperture radus of 30.0mm and a de entry radus of.0mm. The lower de works as a blankholder. From the lower de a pressure gas acts on one sde of the sheet forcng t to expand nto the upper de cavty. The ar pressure s obtaned by a compressor and t s regulated by a proportonal valve. A pressure transducer s nserted on the ar njecton lne to ensure that the pressure nsde the de s equal to the one set for the bulgng test. The specmen temperature s ndrectly controlled through the temperature of the heatng bands wrapped around the des and connected to an electrcal feeder. The temperature of the bands s measured and checked by a LabVIEW program. Fgure 1b shows the heatng bands wrapped around the des. Further detals about the formng system are reported n (Govnco, Gulano and Testa, 010. (a (b Fgure 1: a Schematc vew of the de and sheet mesh and b formng des for bulgng test The objects of the expermental actvty are to evaluate the dsplacement and the thckness evolutons at the dome apex of the metal sheet. The dsplacement-tme curves are obtaned by laser measurements whereas many bulgng tests at the same gas pressures stopped up at prefxed ntervals of tme were carred out to evaluate the thckness at the dome apex as showed n (Gulano, 010 for a magnesum-based alloy. Drect measurements of the thckness are obtaned usng a centesmal mcrometer. formng pressure (MPa formng tme to H=1 (s average stran-rate (s x x x x x10-3 Table 1: Expermental data Fgure shows the expermental trends of the dsplacement-tme curves for constant gas pressures of 0.30, 0.35, 0.40, 0.45 and 0.50 MPa, respectvely. Fgure 3 shows the dome apex thckness evoluton at the same pressures consdered n Fg.: t s possble to consder the relatonshp [1-(s/s 0 ] versus [H /(1+H ] as lnear. In ths fgure H s the normalzed polar dsplacement (defned as the rato of the dsplacement and the de radus, s 0 and s are the ntal and the current thckness at the dome apex. Table 1 summarzes data from the expermental tests. The average stran-rate s e t, where e s the effectve stran at the dome apex for H=1 and t s the formng tme.

3 Fgure : Dsplacement-tme curves n superplastc formng process at constant pressure 3 Characterzaton of the Superplastc 5083 Al Alloy The model proposed to descrbe the behavour of the 5083 Al alloy, relates the flow stress to the stran and the stran rate, accordng to the equaton (1. The materal constants were obtaned by usng the expermental bulgng tests at constant pressure and dfferent numercal smulatons. Under a gven constant pressure, the dsplacements at the dome apex are measured by a laser devce and recorded as a functon of tme. In order to obtan materal constants, the bulgng test data at fve dfferent pressure values were used: for each sngle pressure value, the tme step taken for the sheet to pass through the normalzed polar dsplacement H=0 to H=1 was measured. Fgure 3: Dome apex thckness evoluton n superplastc formng process at constant pressure 3

4 Fgure 4: Pressure-tme expermental data at H=1 Fgure 4 shows the pressure-tme expermental data at H=1. It s possble to note that -b t = ap ( where t s the formng tme at the polar dsplacement H=1, p s the pressure and a and b are constants. Therefore, the stran rate senstvty ndex, m, can be determned by adoptng the expresson presented n (Gulano, Franchtt, 007; Enkeev, Kruglov, 1995; Gulano, 009 ln( p1 p m = (3 ln t t ( 1 where t 1 and t are formng tmes at constant pressures p 1 and p, respectvely. In ths paper, from t-p expermental relatonshp eq. (, m may be defned from the followng expresson ( ln p m = - (4 ( lnt Then, from eq. (, one can obtan that m = 1 (5 b 0,00 0,010 n 0,000-8,0-7,6-7, -6,8-6,4-6,0-0,010 ln(stran-rate -0,00 Fgure 5: Relatonshp between n and the stran-rate 4

5 Moreover, by means of some numercal smulatons t was possble to determne the values of n and K and assocate them wth specfc stran-rates. The frst numercal phase requres the bulge formng smulatons at the same pressure consdered n the expermental actvty: m s well-known through eq. (5, to K s assgned an arbtrary value and n s made to vary n a sutable range. The numercal value of n s calculated mnmzng the functon Q by N ( ln ( lnt ( lnt N ( e ( e ( e Ê t - ˆ Ê - ˆ Q( n = Á + Á (6 Ë Ë where ( ln t N and ( e N represent the values of the normalzed tme and the equvalent stran obtaned from the numercal smulaton for a fxed value of n whereas ( ln t and ( e, nstead, are the normalzed tme and the equvalent stran obtaned expermentally. The parameter t or normalzed tme to be defned as follows t t H t (7 = =1 H =0.5 t H= 0. where t and H= 1 5 constant pressure-formng process. are, respectvely, the formng tmes taken to reach the confguratons H=1 and H=0.5 n a Fgure 5 shows that t s possble to consder the relatonshp = ln( e& n as lnear. The last phase of the procedure to determne the materal constants requres the value of the constant K to be determned: for each sngle pressure value, m s well-known through eq. (5, n s known by mnmzng the functon Q(n, and K s made to vary n a sutable range. The numercal value of K s calculated by mnmzng the functon F gven by where ( N ( th = 1 ( th = 1 ( t Ê ( Á - K = Á Ë H = 1 ˆ F (8 N H 1 t = and ( t H = 1 are the formng tmes necessary to obtan the same dome geometry at constant pressure (H=1 from the numercal smulaton and from the expermental tests, respectvely. Fgure 6 shows that a thrd-order polynomal functon can gve an adequate relatonshp between - ln( e& K K ,0-7,6-7, -6,8-6,4-6,0 ln(stran-rate Fgure 6: Relatonshp between K and the stran-rate 4 FEM Results and Dscussons Usng the commercal fnte element software MSC.Marc, the bulgng process at the constant pressures of 0.30, 0.35, 0.40, 0.45 and 0.50 MPa has been smulated. The materal behavour s modelled by the power law relatonshp between the effectve flow stress, the effectve stran and the effectve stran-rate accordng to eq. (1. 5

6 In the stran rate nterval of 4.1x x10-3 s -1 whch corresponds to the regon of the superplastc behavour for alumnum alloy AA5083, t s assumed n FEM calculatons that n = Aln ( e& + B (9 K = C ( [ ( ] [ ( ] C1 ln e & + C ln e& + C3 ln e& (10 where A, B, C are constants. The values for the constants are: A=-0.01, B=-0.148, C 0 =36.80, C 1 =5.816, C = and C 3 = For each sngle numercal smulaton, the dsplacement-tme curve and the [1-(s/s 0 ]-[H /(1+H ] curve have been obtaned. All the numercal analyses show that t s possble to consder the relatonshp [1-(s/s 0 ] versus [H /(1+H ] as lnear. In the range 0 H 1, the dstance between numercal calculatons and expermental measurements was evaluated. Ths dstance s expressed as t = tnum - t t (11 ( exp 100 ( s - s 100 % exp s num (1 % = exp sexp where t s the formng tme, s s the current thckness at the dome apex and the subscrpts num and exp ndcate the numercal and expermental values, respectvely. In all the bulgng tests, t% and s% were found to be less than 10%. 5 Conclusons Constant pressure superplastc bulgng tests of AA5083 alumnum sheet at 73 K are consdered n the present paper. The dsplacement-tme curve and the thckness evoluton at the dome apex of the metal sheet are evaluated. Commercal fnte element software s used to determne the constants of superplastc alloy. The results of comparson between the numercal smulatons of bulge formng process and the expermental tests show good agreement and they ndcate that the materal constants are relable. References Zenkewcz, O.C.: Numercal Analyss of Formng Processes. Wley, New York, (1984. MSC.Marc, User s Manual, Vol. B, (005. Govnco, G.; Gulano, G.; Testa, G.: Formng apparatus to nvestgate the effect of temperature on the superplastc behavor of alloys. NUMIFORM 010, Pohang, Republc of Korea, June, (010. Gulano, G.: Modellng of superplastc formng of AZ31 magnesum alloy. AMPT 010, Pars, France, 4-7 October, (010. Gulano, G.; Franchtt, S.: On the evaluaton of superplastc characterstcs usng the fnte element method. Int. J. Mach. Tool Manufact., 47, (007, Enkeev, F. U.; Kruglov, A.A.: An analyss of the superplastc formng of a thn crcular daphragm. Int. J. Mech. Sc., 37, (1995, Gulano, G.: On the constants of the superplastc magnesum-based AZ31 alloy. ICMM, Dortmund, Germany, September, (009. Address: Dr. Gllo Gulano, DMSAT, Unversty of Cassno, Cassno (FR ITALY. emal: gulano@uncas.t. 6