Numerical simulation of sheet metal formability tests

Transcription

1 Numerical imulation of heet metal formability tet Ricardo Martin a a Intituto Superior Técnico, UniveridadedeLiboa, Liboa, Portugal ricardo.pedra.martin@tecnico.uliboa.pt Abtract One of the challenge faced by the cientific and engineering manufacturing community i the prediction of fracture via computational procedure. The firt tudie of formability defined the beginning of localization a the Forming Limit Curve (FLC, nonethele a certain material experienced fracture without necking and inherent ambiguity of the necking tage, the formability limit ha been readdreed with the development of a new limit uch a the Fracture Forming Limit (FFL. In addition, recent development have propoed new diagram for the repreentation of thee limit uch a the triaxiality plane that i le or not enitive to the train path. Thi thei embrace the tudy of necking and fracture formability limit in the triaxiality plane. For that, experimental reult from the aluminium alloy AA1050 H111 obtained via tenile, Nakazima, hemipherical dome and bulge tet were compared with numerical imulation of the ame tet. The imulation of heet forming conidered normal aniotropy and ductile damage criterion to determine the fracture intant. The numerical reult obtained were firt validated via the force and diplacement comparion with experimental reult, then the analyi focued the prediction of the localization and fracture by the evolution of the train path in the triaxiality plane. The compilation of the reult from all the tet defined a region of necking rather than the ingle intant repreented by the FLC. Thi confirmed the ambiguity aforementioned for the FLC and the jutified the preference for the FFL. Keyword: formability limit, necking, fracture, triaxiality, heet metal forming, numerical imulation 1. Introduction The determination of material formability i a major concern for both indutry and academia. In recent year, the development and application of new material to heet forming and the demand of thickne reduction from the project deign, require a careful and time conuming characterization of material propertie and formability limit. The material formability limit define the maximum level of deformation achievable in a forming operation without the occurrence of necking or fracture [1]. The repreentation and ue of thee material data i ubject of divergent opinion among reearcher and manufacturing expert. For example, in indutry project deign claim the ue of thickne reduction on failure tracking while reearcher prefer the ue of forming limit diagram. Thi paper emphaie the ue of numerical imulation to complement the experiment previouly done to characterize the mechanical propertie and formability limit of the aluminium alloy AA1050 H111. The tet conidered are tenile, Nakazima, hemipherical dome and bulge tet becaue cover the train path from the uniaxial deformation to biaxial expanion [2]. Ductile fracture mechanim were alo explored to model fracture and the formability reult are analyed in a diagram le or not enitive to train path variation [3, 4]: the triaxiality plane with the effective train. The analyi of the train in the heet plane, known a Forming Limit Diagram (FLD, i widely ued ince 1965 when the experimental procedure for meauring the train were propoed by Keeler and Goodwin [5, 6]. Their 1

2 reearch wa intereted in the locu of localized deformation where the train path tart it tranition to plane train deformation. Nowaday the overall locu for necking in the FLD i commonly deignated a the Forming Limit Curve (FLC. After localization the deformation proceed under plane train mode until the occurrence of fracture. Embury [7] uperimpoed the fracture intant in the FLD obtaining the Fracture Forming Limit (FFL. A fracture i a material property and therefore proce independent, Atkin [8] preent a theory of contant thickne in fracture which can be repreented in the FLD by a traith line with lope -1. A the FLD wa proven to be dependent on the train path, recent development led to the repreentation of the forming limit into a tre baed diagram that being le or not enitive to the deformation path [9], which make it more appropriate ( for the identification of fracture. Thi diagram repreented the effective train (ε a a function of the triaxiality, which i the ratio of hydrotatic tre ( m = ( /3, where 1, 2 and 3 are the principal tree with the effective tre ( defined in Equation 2. Another reaon thi repreentation i it uitable implementation in numerical analyi. Several failure model are now available in commercial oftware for metal forming [10] which were calibrated with experiment by a mixture of bulk and heet forming data. Thi generalization of the formulation for both bulk and heet i quetionable a there i a lack of experimental validation with heet forming reult. Thi paper complement thi analyi by the comparion of experiment with numerical reult. Section 2 introduce the material propertie determined via experiment and the procedure to analye the reult in the triaxiality plane. 2. Theoretical work 2.1. Material propertie The mechanical characterization of the aluminium AA1050 H111 wa performed via tenile teting. The flow rule that characterize it behaviour i defined by the Ludwik-Hollomon equation: = 140ε 4 (1 The effective value for tre and train can be calculated via an appropriate yield criterion for heet forming. The Hill 48 wa elected which conider normal aniotropy and plane tre aumption ( 3 = 0. Therefore the effective tre and train are defined by Equation 2 and 3, repectively. = r 2 ( 1 2 (2 1 + r ε = 1 + r ( ε ε r ( ε 1 ε 2 (3 1 + r where ε 1, ε 2 and ε 3 are the firt, econd and third in-plane principal train [1]. The formability limit of the aluminium AA1050 H111, preented in Figure 1(a are compoed by a bell-haped curve (FLC that indicate the tart of localization and the fracture line (FFL which being a traight line can be defined by Equation 4. ε ε 2 = 1.34 ( Tranformation into triaxiality Thee formability limit can be tranformed into the triaxiality plane with the effective train via a imple coordinate tranformation. To tart it i neceary to define the tre ratio (α and the train ratio (β by Equation 5 and 6, repectively. α = 2 1 (5 2

3 Major True Strain Effective Strain β = ε 2 ε 1 = dε 2 dε 1 (6 Thee ratio are related with each other if one ue the Hill 48. With mathematical rearrangement the tre ratio can be expreed a a function of the train ratio and the aniotropic coefficient a follow [11] α = (1 + r β + r (1 + r + r β ( To evaluate the tre triaxiality directly from the meaurement of train i intereting to rearrange thi ratio conidering plane tre aumption ( 3 = 0 which i valid for heet forming. Then combining the triaxiality with Equation 7 come: m = 1 + β ( r β + β2 The effective train determination require to calculate the third principal train that can be found by the condition of volume contancy (ε 1 + ε 2 + ε 3 = 0 and finally replaced in Equation ( 3. Thee tool are ufficient to make the coordinate tranformation from the meaured point (ε 2, ε 1 to, ε. Applying thiethodology to the FLC reult in the lower full line depicted in Figure 1(b. The evolution of thi curve how low effective train value for lower value of triaxiality and a the triaxiality increae necking i expected to happen at higher value of effective train. In a limit cae, for biaxial deformation mode, the localized deformation can occur very cloe to fracture or in theory failure i poible without necking [12]. (7 1.6 FFL 2.5 Uniaxial Plane train Biaxial FFL =0.33 =0.55 = FLC 0.5 FLC Minor True Strain (a Pricipal train plane [13] Mean Stre / Effective Stre (b Triaxiality plane Tenile Tet Circular Bulge Tet Elliptical Bulge Tet Nakazima Tet Hemipherical Dome Tet Figure 1: Forming limit diagram repreented in two different coordinate ytem. The marker repreent necking and the olid marker refer to failure by fracture The theoretical prediction of contant thickne in fracture [8] allow a different procedure to( tranform ( the FFL into the triaxiality plane, which require to define the effective train a a function of triaxiality ε = f. Combining the volume contancy condition, the train ratio (Equation 6 and the effective train (Equation 3 with mathematical re-arrangement of the term give the following reult: ε ε 1 = 1 + r 1 + r β + β2 (9 3

4 Alo the triaxiality (Equation 8 can be re-arranged a follow: 1 + r β + β2 = β m (10 Now, combining the Equation 9 and 10 reult a function valid for the fracture limit a follow: dε = K 1, where K 1 = 1 + r m 3 (ε 1 + ε 2 (11 Thi function for the fracture limit (Equation 11 repreent an hyperbole and the contant value K 1 can be evaluated for three different deformation mode available: uniaxial, plane train and biaxial (ee dahed grey line in Figure 1(b. Thee line delineate the lower and upper boundary where fracture i likely to occur. However, the FFL ha a lope different than the theoretical prediction (ee Equation 4 and therefore for a better fitting to the experiment, the deformation mode aforementioned were combined to define the FFL depicted in full line. For the triaxiality value correponding to unixaial, plane train and biaxial deformation the different pointarked with a pentagon (ee Figure 1(b were elected to model the fracture line. Thi tranformation in term of tre triaxiality can now be implemented to analye the evolution of field variable obtained from the numerical imulation Modelling condition The modelling aumption for each formability tet are addreed with indication of tool component, blank characteritic and boundary condition impoed to model the tet procedure. The tet undertudy formed three ditinguih group characterized by the tool ued or procedure imilaritie. The tenile tet procedure can be modelled with nodeotion, therefore there i no need to define extra tool. The pecimen i repreented in Figure 2 with the boundary condition conidered, namely fixed node in the left grip and velocity impoed in the right. After meh convergence tudie it wa elected the blank with minimum element length (L e of mm. v Figure 2: Boundary condition applied to the tenile pecimen. All the exitent node inide each rectangle have the ame boundary condition and the arrow indicate contant velocity on the right houlder The econd group (ee Figure 3 i compoed by the Nakazima and hemipherical dome tet becaue they hare the ame tool component (die, punch and blank holder. However, they differ in the pecimen, while the hemipherical dome tet ue circular blank (with L e = mm the Nakazima blank are trip with different width (ee Figure3(b where the minimum element length have mm. The tet condition imply fixed die, force impoition in the blank holder (50 kn wa ued and punch motion controlled by velocity. To conclude the tet remain the circular and elliptical bulge tet. The tet procedure i the ame a well a the circular pecimen ued with L e = 1.46 mm. Nonethele the tool component (die and blank holder are different: circular and elliptical hape a depicted in Figure 4(a and 4(b, repectively. The boundary condition for thee tet are fixed node of the die, blank holder force of 200 kn, fixed blank node in the region of the draw bead to analytically model thi component and finally to promote deformation, preure impoition in the blank in the internal region of the die a repreented by the blue vector in Figure 4(c. Section 3 tart with the validation of reult via force diplacement analyi followed the repreentation in the tre triaxiality plane with the methodology here preented. 4

5 VP D=165mm FB w=66, 77, 88, 99, 104, 112 mm Punch Blank holder Die Blank (a (b Figure 3: Tool component (a for the hemipherical dome and Nakazima tet with the chematic drawing of balnk (b for the Nakazima tet D =100 (a Circular die b =64 a =100 (b Elliptical die (c Load application in the blank Figure 4: Schematic model of the die ued in the circular and elliptical bulge tet (a,b and repreentation of the load applied in the blank with blue vector (c 3. Reult and dicuion 3.1. Modelling validation The validation procedure for the numerical reult obtained for the formability tet conited on analying the force evolution with the diplacement. The tenile tet (ee Figure 5 how good agreement between the imulation and experiment with an error lower than 2 percent for the maximum force value. Figure 6 preent the force diplacement curve for the hemipherical dome tet with imulation reult both neglecting friction effect between the punch and the blank and conidering a Coulomb friction coefficient of 0.1. In the numerical reult one can ee that the tet with friction achieved the larget force with an error lower than 15 percent in comparion with the maximum force value for the experiment. Thi value i within an acceptable range conidering that experimental procedure are expoed to everal ource of error both ytematic and random that will be dicued later. The Nakazima tet analyed in the validation of reult are the w66, w99 and w112 (ee Figure 7. The reult how a good compliance in trend and force value for the Nakazima with maller width but the error increae with the width of the blank. The divergence between the numerical and experimental reult for the Nakazima tetay be due to experimental difficultie that will be addreed. For example, thickne and width variation in the blank dimenion and mialignment of the pecimen in the etup of the experiment. In order to predict the influence of thee error they were modelled and the reult analyed. For the Nakazima w112 tet geometrical defect in the blank, namely a width and thickne reduction of 2 percent howed a maximum decreae in the force of 7 percent for the width change. The impact of a vertical miplacement wa tudied for the tet w66 and w112. The maller width howed a neglecting variation in the force value and it kept the good compliance with reult. For the wider blank the cae changed. Figure 8 how that increaing the vertical mialignment promote the reduction of the force. A mialignment of 6 mm that till cover the entire punch area i ufficient to drop the force to half of it initial value. 5

6 Punch load (kn Punch load (kn Punch load (kn Punch load (kn Force (kn Punch load (kn (with friction (friction-le Diplacement (mm Figure 5: Force evolution with diplacement for the tenile tet Punch depth (mm Figure 6: Punch load evolution with punch diplacement for the hemipherical dome tet with and without friction When the final hape of the pecimen (depicted in Figure 8 i compared with the reultant from the experiment the hypothei of mialignment i confirmed. Alo, if one combine blank mialignment with defect in thickne and width dimenion the force would be nicely predicted Punch depth (mm (a Width = 66 mm Punch depth (mm (b Width = 99 mm Figure 7: Punch load evolution with punch diplacement for the Nakazima tet Punch depth (mm (c Width = 112 mm mm 2mm 4mm 6mm dy=0mm dy=2mm dy=4mm dy=6mm Blank Die dy Punch Punch depth (mm Figure 8: Punch load evolution with punch diplacement for the Nakazima w112 tet for different etting. Blank ymmetrically placed and vertically mialigned by 2, 4 and 6 mm The cope of thi thei i analying the formability which i done via the interpretation of tree and train and they how a low dependence on the force reult. Therefore the force analyi wa conidered well calibrated and the tet with no defect were elected to proceed with the tudy of triaxiality evolution during the forming proce Forming analyi in the triaxiality plane For the field variable analyi wa elected the firt element deleted in each imulation which correpond to the crack opening. The value of tre triaxiality and effective train were calculated repectively by Equation 2 6

7 Effective Strain and 3 uing the tre and train tenor obtained from the oftware. Figure 9 combine the reult from the imulation for all the formability tet and preent the experimental reult of necking with marker and the fracture limit by olid marker. The evolution( of the curve for the tenile tet can be divided in three ditinguih tage. The firt i the uniaxial ( deformation = 0.33 followed by a tranition region and ending under plane train mode = The imulation reult are in accordance with the theoretical prediction for the tenile tet but the experiment depite howing the ame trend, the value for fracture intant are lightly different. Thi i related with meaurement error that can occur for obtaining the marker. The hemipherical dome tet with friction wa elected for thi analyi becaue it ( repreent a good fitting to failure under plane train deformation. Thi tet tart under biaxial deformation = 0.64, followed by a tranition tage at ε = 0.7. Then, the plane train mode i achieved FFL =0.33 Plane train =0.55 =0.64 TenileHTet CircularHBulgeHTet EllipticalHBulgeHTet NakazimaHTet HemiphericalHDomeHTet :HTenileHTet :HCircularHBulgeHTet :HEllipticalHBulgeHTet :HNakazimaHTet :HHemiphericalHDomeHTet FLC Mean Stre / Effective Stre Figure 9: FLD. The olid marker refer to failure by fracture The Nakazima ( tet hare the ame trend between each other. Depite having triaxiality value higher than plane train > 0.55, for the Nakazima tet the firt tage of imulation i not linear and vertical a it wa verified for the other imulation. After thi tage the deformation tabilize to plane train mode and the end of tranition for the Nakazima tet increae with the width of the pecimen reaching the maximum for the hemipherical dome tet. The circular bulge tet (ee A in Figure 10 require a peronalied interpretation of it reult becaue it have a contant tre triaxiality of biaxial expanion until failure (ee the vertical line where m = It mean that thi tet did not uffered localization and therefore the condition of fracture under plane train deformation wa not verified. Thi i not a urpriingly reult becaue there i no continuum theory to explain the tart of necking for thi tet [11]. Localization may occur due to the non-uniformity in the blank thickne [14] which wa artificially modelled adding thickne defect in the ome element of the blank. There were imulated four different defect identified from B to E in Figure 10. 7

8 Effective Strain FFL Plane train =0.55 =0.64 B: t=0.99 mm C: t=0.97 mm D: t=0.90 mm 1.0 D C A B E 0.5 FLC Mean Stre / Effective Stre E: t=0.99 mm Figure 10: FLD for the circular bulge tet for different blank: A i the blank 1 mm thick and the other cae with everal thickne reduction (B to D have a thickne reduction in the element indicated in the right-upper cheme and cae E in the random element painted in the left-lower drawing. The olid marker refer to failure by fracture The reult A, B and E how no difference in the tre triaxiality meaning that necking wa not likely to occur and therefore the imperfection were not ufficient to induce localization. If the thickne reduction in the pole i larger than 3 percent the tre triaxiality tart to change (ee cae C and D. It i a good reult to how the trend in analyi and confirm the theory of Marciniak et al. [14]. Thee reult ugget that necking for the bulge tet in experiment occur before fracture due to defect exitent in the blank and not by the etting of the tet which difficult the determination of FLC that will be dicued next. When analying the intant of localization for all the tet one can ee a tranition region becaue the material take time to adapt it deformation mode to plane train. Therefore the FLC which wa previouly defined via experimental tet and tranformed into thi coordinate ytem doe not have an exact definition. If thi reult i extrapolated to the claical FLD plot the variation to plane train would occur at a lower rate than the harp variation identified in Figure 1(a. In fact, one could not predict thi evolution without imulation becaue the procedure to determine the FLC i baed on a ingle point calculated for each pecimen with train value meaured after failure of the pecimen. 4. Concluion and future work Thi paper preent a coupled analyi of imulation with experiment for heet formability tet of the aluminium AA1050 H111. The formability tet analyed were tenile, Nakazima, hemipherical dome and bulge tet becaue they cover deformation mode from uniaxial deformation to biaxial expanion, where ductile fracture mechanim were verified. The numerical reult were validated via the force diplacement comparion. A good compliance wa found for the tenile and hemipherical dome tet but the Nakazima howed ome dicrepancie. Thee difference were jutified by difficultie in experimental etting uch aialignment a the imulation of thee difficultie howed. The formability analyi wa carried on in the triaxiality plane with the effective train. In thi plane the tart of necking and fracture were identified via the analyi of the train path variation during the tet. For all the tet except the circular bulge, the localization and fracture under plane train wa verified. The circular bulge tet wa ubjected the tudy of necking introducing defect in thickne of the pecimen which proved to be ufficient to promote localization. The reult alo howed that the tranition to plane train wa not intantaneouly and therefore the FLC hould be replaced by a region of tranition than a ingle line. 8

9 The model ued to find fracture, baed on ductile damage, preent good agreement with the experimental point for the range of triaxiality analyed. Thi work embraced the tudy of the FFL which characterize failure under mode I of fracture mechanic. Remain to be developed the tudy of in-plane hear fracture (mode II uing the methodology ued to tranform the FFL preented in thi article, to the hear fracture forming limit (SFFL [11]. The hear fracture can be achieved in experiment and numerically uch a in-plane hear, in-plane torion and rectangular tamping [15]. Thi would complete the triaxiality plane once lower value of triaxiality are achieved. Reference [1] Rodrigue, J. and Martin, P., Tecnologia mecânica: tecnologia da deformação plática Vol. 1, Ecolar Editora, [2] Martin, P., Montanari, L., Critino, V., and Silva, M., Formability and imulative tet in modern heet metal forming education, in Modern Mechanical Engineering, Material Forming, Machining and Tribology, page , Springer Berlin Heidelberg, [3] Kikuma, T. and Nakazima, K., Effect of deforming condition and mechanical propertie on the tretchforming limit of teel heet, in Iron and Steel Intitute of Japan, 11, page , [4] Laukoni, J. and Ghoh, A., Metallurgical Tranaction A 9 ( [5] Keeler, S. P., Circular grid ytem - a valuable aid for evaluating heet metal formability, Technical report, SAE Technical Paper, [6] Goodwin, G. M., Application of train analyi to heet metal forming problem in the pre hop, Technical report, SAE Technical Paper, [7] Embury, J. D. and Duncan, J. L., Annual Review of Material Science 11 ( [8] Atkin, A., Fracture Reearch in Retropect, Balkema, Rotterdam ( [9] Bao, Y. and Wierzbicki, T., Journal of Engineering Material and Technology 126 ( [10] Wierzbicki, T., Bao, Y., Lee, Y.-W., and Bai, Y., International Journal of Mechanical Science 47 ( [11] Martin, P., Bay, N., Tekkaya, A., and Atkin, A., International Journal of Mechanical Science 83 ( [12] Silva, M., Skjødt, M., Atkin, A., Bay, N., and Martin, P., The Journal of Strain Analyi for Engineering Deign 43 ( [13] Teodora, A., Determinação do limite de enformabilidadede chapa metálica, Mater thei, Intituto Superior Técnico, [14] Marciniak, Z. and Kuczyńki, K., International Journal of Mechanical Science 9 ( [15] Iik, K., Silva, M., Tekkaya, A., and Martin, P., Journal of Material Proceing Technology 214 (