An Anisotropic Hardening Model for Springback Prediction

Transcription

1 An Anisotropic Harening Moel for Springback Preiction Danielle Zeng an Z. Ceric Xia Scientific Research Laboratories For Motor Company Dearborn, MI 48 Abstract. As more Avance High-Strength Steels (AHSS are heavily use for automotive boy structures an closures panels, accurate springback preiction for these components becomes more challenging because of their rapi harening characteristics an ability to sustain even higher stresses. In this paper, a moifie Mroz harening moel is propose to capture realistic Bauschinger effect at reverse loaing, such as when material passes through ie raii or rawbea uring sheet metal forming process. his moel accounts for material anisotropic yiel surface an nonlinear isotropic/kinematic harening behavior. Material tension/compression test ata are use to accurately represent Bauschinger effect. he effectiveness of the moel is emonstrate by comparison of numerical an experimental springback results for a DP6 straight U-channel test. INRODUCION Springback is an important issue in the sheet metal part an ie esign. In orer to obtain a final part shape matching its esign intent, elimination or correction of springback has to be mae uring esign stages. Otherwise it will prouce imensional eviations an result in assembly ifficulties. raitionally, ie esign engineers take great effort on trial-an-error to either reuce or compensate for springback. A number of ie re-cuts are often neee uring tryout in orer to obtain a imensionally accurate part. his approach is not only time consuming but also costly. Because of weight reuction efforts to meet fuel economy pressure an increase vehicle safety requirement, the use of Avance High Strength Steel (AHSS sheets such as ual-phase an RIP alloys becomes more wiesprea. However, their higher yiel strength an rapi work harening make springback more ifficult to anticipate in avance an harer to control in tryout. In recent years, researchers an stamping engineers start to make ie compensation by using computer simulations before cutting physical ies [] []. In such a process, springback analysis is carrie out to obtain irections an magnitues for ie compensation. Accoringly accurate springback preiction becomes a key to the success of any ie compensation whose algorithm relies on numerically preicte springback. Unfortunately accurate springback preiction remains a critical challenge for both FEA coe evelopers an en users. As we know, springback is ue to the release of resiual stresses accumulate uring the forming stage. here are lots of factors affecting the stress preiction accuracy in numerical simulations, such as time integration schemes (imicit vs. exicit, element formulations, contact algorithms, material moels etc. In this paper, the attention is focuse on the material moeling aspect with the aim to evelop a more realistic constitutive relationship which hopefully represents material eformation behavior more accurately. Isotropic harening is the simest an most popular harening moel use toay. It offers reasonable approximation for monotonic loaing cases an is easy to imement. However, when materials experience unloaing an reverse loaing, their yiel stresses in reverse loaing is usually lower than those in the case of monotonic loaing (as shown in Figure. his phenomenon is generally calle Bauchinger effect. A purely kinematic harening moel was first introuce by Prager an Ziegler where the yiel surface translates in the stress space as material yiels. However this sime moel ignores harening effects in other stress irections. In sheet metal forming process, materials experience very comicate eformation. Cyclic bening an unbening occur when the sheet passes through a rawbea or a ie raius. In orer to represent the cyclic behavior more realistically, harening rules combining both isotropic harening an kinematic harening are later evelope by various researchers [-4]. Noticeably among them is 4

2 the multi-yiel surface moel propose by Mroz [5-6] which successfully moels the non-linear harening behavior an the smooth transition from elastic to astic eformation. his moel introuces the concept of a fiel of work-harening mouli to moel the nonlinear harening behavior instea of a single moulus use in most of other kinematic moels. Chu [7] generalize Mroz s iscrete multie yiel surface concept into a continuous fiel of yiel surfaces, an later ang [8] an ang etc. [9] use the moel to analyze sheet metal formability an springback. MODIFIED MROZ MODEL he soli line OABCDEF in Figure shows the stress-strain relationship of a sime uniaxial loaing, unloaing an reverse loaing behavior accoring to the original Mroz moel. he moel was further evelope by Chu [7] where the iscrete multie yiel surfaces were generalize into a continuous fiel of yiel surfaces an is more suitable for FEA imementations. Intereste reaers shoul fin etails in [7]. Figure Illustration of Bauschinger effect he Morz's moel has several avantages over other more comex harening moels. It captures anisotropic harening behavior nicely uring reverse loaing an reuces to isotropic harening for monotonic loaing. One of its most appealing characteristics for sheet metal forming analysis an springback in particular is that the formulation oes not require any aitional experimental tests beyon stanar tensile curves. In fact there are no extra parameters neee to fit the moel. However, the moel also assumes that the material elastic unloaing an reverse loaing region is constant an twice as the initial yiel stress. While this might be true for some low yiel alloys, experimental evience suggests it is not the case in general, an Han etc. [] foun that the amount of Bauschinger effect actually epens on the harening magnitue before the loaing is reverse. Accoringly a moifie Mroz moel is propose in this paper to take into account this variable Bauschinger effect. he paper is organize as follows. First, the new moel incorporating more accurate Bauschinger effect is escribe. An then its constitutive integration algorithm is erive an imemente in commercial FEA coe. Last, the propose moel is apie to a sime test case for springback preiction. Experiments are conucte an results are compare with numerical preiction to emonstrate the new moel's apicability. Figure Uniaxial stress-strain curves in Mroz moel It can be seen from Figure that line CD represents the material elastic unloaing. Point D is the reverse yiel starting point which represents the compressive yiel strength when the material starts to unloa at point C. In Mroz moel, it assumes that the elastic region is constant an represente by the initial yiel stress. he elastic range uring reverse loaing equals to twice of the initial yiel strength. hus when the material starts to unloa at higher strength, lower compressive yiel strength is obtaine. As shown in Figure, the soli pink line is the reversal of compressive yiel strength curve at the corresponing reverse point accoring to Mroz moel for a typical DP6 material. It can be seen that as the tensile strength increases, the corresponing compression strength ecreases. For the material with higher work harening, it is conceivable that the reverse yiel point will go to positive region, i.e., the moel preicts that the material might go into reverse astic loaing when the material is release from uniaxial tension even there is no external forces apie. his is contraictory to what was observe in real material compression testing []. Figure 4 gives the compressive yiel strength at various reversal points obtaine from compression tests for ifferent graes of steels. he ot shows that when the loa reverses at higher strength point, the compressive yiel strength is higher or at least in the same level. Accoring to the testing ata, the 4

3 compressive yiel strength curve shoul follow the otte pink line instea of the soli line in Figure as preicte by Mroz moel. Clearly the Mroz moel as it is can not aequately represent material eformation behavior at reverse loaing. rue stress (Mpa.5% Reverse Yiel Stength (MPa tensile Mroz compression testing rue strain Figure Comparison of material stress-strain curves Uncertainty is ± stanar eviation Strength at Reversal Point (MPa LC HSLA DP AKDQ DDQ DQSK IF-Rephos HS44 DP5 DP6 DP8 Figure 4 Yiel strength (.5% offset on reversal as a function of steel strength before reversal (Courtesy of C. Van yne In this paper, the Mroz moel is moifie to aress its rawback as iscusse above while preserving its original formulation. In the moifie moel, material eformation follows the same rule as that of Mroz moel at monotonic loaing. However, when the unloaing is initiate, the elastic region is no long constant. he size of the elastic region is etermine through reverse compression test. hus, in Figure, instea of line CD, the elastic unloaing curve becomes CD', where D'is the compressive yiel strength at the reverse point C an can be etermine through stanar compression test. Base on this moification, the size of the elastic region enote by σ B is a function of equivalent astic strain ε an can be expresse as: σ B ( σ Y + σ c σ y = ( where σ Y is the flow stress an σ c is the compressive yiel stress at corresponing reversal point. σ B can be rewritten in the following generalize form: σ B = cσ + ( c σ Y ( where σ is the initial yiel stress of the material, an c is a material parameter reflecting the Bauschinger effect an is consiere to be a function of the effective astic strain. It is easy to note that isotropic harening is a special case of this moel when c (thus σ B = σ Y, an the moel is reuce to the original Mroz moel when c is taken to be (c, with σ B = σ. In its most general form, c is a function of ε, where it can be expresse as: σ y σ c when σ y σ c ( ε = σ σ ( y when σ y = σ FORMULAION AND CONSIUIVE INEGRAION OF HE MODIFIED MROZ MODEL he material constitutive relationship an its integration for numerical imementation will be erive in this section. Since most sheet metals exhibit ifferent yiel strengths along ifferent irections, Hill's anisotropic yiel criterion is use to account the material anisotropic behavior. For the case of combine isotropic-kinematic harening, the astic yiel criterion can be expresse as []: f ( ( P( Y ( ε (4 = σ, σ, σ, σ, σ, σ is the stress vector in most general D stress state, where { } { α, α, α, α, α α } =, is the yiel surface center, ( Y ε is the flow stress which is a function of equivalent astic strain ε, an P is a 6x6 anisotropic asticity matrix in general D case an can be expresse in terms of the anisotropic r values in three orientations as: 4

4 r r + + r r r ( r9 + r r + ( + ( + r9 r r9 r r + r r9 P = r + r9 ( r + r9 ( r + ( r45 + ( r + r 9 r9 ( r + (5 Accoring to Hooke's law, the stress vector can be obtaine through elastic strain as: el = Dt+ (6 where D is the material elastic moulus matrix. Accoring to incremental astic flow theory, all the variables at time step t+ can be calculate once the eformation history at last time step t is known. hus el the elastic strain t+ can be expresse as: el el = t + (7 Here el t is the elastic strain at time t, an are the total strain increment an astic strain increment vectors, respectively. he associate flow rule gives: f = t+ (8 where is the equivalent astic strain increment. Also, uring the astic loaing, ( f (9 where * is usually terme as stress preictor: * el = D ( t + ( he stress vector can be re-written as: = M (4 ( * t+ t+ where M = [ I + DP] Y ε (5 he position of active yiel surface center at time t+ can be etermine accoring to the rule of Mroz moel. As shown in Figure 5, F t is the active yiel surface with center O at time t. F I is the inactive yiel surface with center O I in memory an tangent to surface F t at point P. F t+ is the active yiel surface with center O'at time step t+. β is a unit vector representing the moving irection of the center of current yiel surface an can be expresse as: = (6 ( I I t t P ( I t hen the amount of active yiel surface center movement from time t to t+ can be expresse as: = Y (7 where Y is the increment of the active yiel surface raius from time t to t+ an is a function of. Substituting Equation (4 into (9, Equation (9 becomes hus, from Equations (4 an (9, we have f = P( Y t+ herefore t+ ( f * * ( [ ( ] [ ( ] ( ε = P M Y ε = M (8 t+ t+ is the only unknown in this non-linear equation an can be solve numerically. In this stuy, Newton- Raphson metho is use to solve the equation. ε = P( Y t+ ( Substituting Equations (7 an ( into (6, we get: * ε = DP( t+ Y t+ ( Figure 5 Movement of the active yiel surface 44

5 It is also necessary to obtain material stiffness matrix if imicit time integration FEA is aopte. It can be expresse as, after some algebra: = D + Y f f f ( + ' (9 Figure 7 Springback of DP6 U channel NUMERICAL EXAMPLE Base on the formulation erive above, a user subroutine was evelope to compute the elasticastic constitutive equation, an has been imemente in imicit commercial software ABAQUS/Stanar. he user subroutine can be use for D soli, ane stress shell, D ane stress an ane strain elements. A straight U channel raw test was conucte to test the ability of the propose moel in preicting springback. Figure 6 shows the geometry of tooling set up. he upper ie raius is mm an the lower ie raius is 6mm. he punch with is 5mm with a.5mm raius. he raius of the rawbea is 6mm. he material properties of DP6 are liste in able. he tensile curve, which is obtaine from uniaxial tensile test is shown in Figure 8. In the physical testing, the blank was wrappe by a teflon sheet to reuce the friction. In the numerical simulation,.8 is use as the friction coefficient. able DP6 material properties Material E ν R R 45 R 9 σ y DP6 GPa MPa rue Stress (Mpa tensile Mroz compression testing mo_mroz rue astic strain Figure 8 DP6 uniaxial tensile curve an compressive yiel strength on reversal as a function of true strain before reversal Figure 6 U channel test set up A variety of tests were conucte with variations of materials, blank holer forces, an with an without rawbeas. he case presente in this paper is a.5 mm DP6 blank with rawbea an blank holer force of KN. he blank size is 4mm x mm. he raw epth is 7mm. Figure 7 shows the picture of testing piece in this loaing case after springback. It can be seen that DP6 has very big springback both in the wall open an sie wall curl. he material compression properties use for simulation in ifferent moels are also shown in Figure 8. he soli pink an green curves are the compression yiel stress at corresponing reversal points use by Mroz moel an the moifie Mroz moel, respectively. In this exame, the compressive yiel stress use in moifie Mroz moel is obtaine by offsetting the testing reverse yiel curve (from Figure 4 own to the point that compression yiel stress is the same as the tensile yiel stress at zero strain reverse point. In this case, the c value in Equation ( is about.5. he testing curve is not irectly use in this exame since the compression yiel stress was obtaine by a.5% offset in the original testing ata an generally.% offset is use. 45

6 Figure 9 shows the comparison of springback among the testing result an the preictions by using ifferent material moels. he anisotropic yiel function was accounte in all three simulation material moels. In Mroz moel an moifie Mroz moel, the combine isotropic-kinematic harening rule was use an followe the reverse compression yiel curves shown in Figure 8. It is shown in Figure 9 that the preiction by Mroz moel gives largest eviation from testing ata while that by moifie Mroz moel goes between the isotropic harening moel an Mroz moel, an it is closest to the testing result. his inicates that the Bauschinger effect is actually not as big as assume by the original Mroz moel. testing isotropic mroz mo_mroz Figure 9 Comparison of springback results DISCUSSIONS A moifie Mroz moel is presente in this paper to preict the springback for Avance High Strength Steel. he moel combines the material isotropic harening an kinematic harening, an also consiere the anisotropic yiel criterion. he moeling of material reverse loaing behavior is incorporate to aequately represent actual Bauschinger effect uner cyclic loaing case. he springback preiction by the propose moel for the DP6 U channel correlates with testing ata very well. Work is uner way to apy the new moel to comex prouction parts where Bauschinger effect is expecte to have a larger influence on springback than the sime D exame presente in this paper. In aition, more reverse compression test ata are neee to valiate the propose moel. ACKNOWLEDGEMEN he authors woul like to thank Mr. Craig Miller for conucting the U channel raw testing an analyzing the testing ata, an Prof. C. Van yne of Colorao School of Mines for proviing compression testing ata in Figure 4. REFERENCES. L. Wu, 'ooling mesh generation technique for iterative FEM ie surface esign algorithm to compensate for springback in sheet metal stamping', Engineering Computations, Vol. 4, No.6, 997, Lumin Geng, homas Oetjens an Chung-Yeh Sa, 'Springback preiction with LS-DYNA an ie face compensation of Aluminum hoo inner', SAE J. L. Chaboche, 'ime-inepenent constitutive theories for cyclic asticity', International Journal of Plasticity, Vol., No., 986, K. Chung, M. Lee, D. Kim, C. Kim, M. L. Wenner, F. Barlat, 'Spring-back evaluation of automotive sheets base on isotropic-kinematic harening laws an nonquaratic anisotropic yiel functions, Part I: theory an formulation', International Journal an Plasticity, Vol., 5, Z. Mroz, 'On the escription of anisotropic work harening', J. Mech. Phys. Solis, Vol. 5, 967, Akhtar S. Khan an Sujian Huang, Continuum heory of Plasticity, Awiley-Interscience Publication, C.C. Chu, 'A three-imensional moel of anisotropic harening in metals an its apication to the analysis of sheet metal formability', J. Mech. Phys. Solis, Vol., No., 984, S.C. ang, 'An anisotropic harening rule for the analysis of sheet metal forming operations', Avance echnology of Plasticity, Vol., 99, S.C. ang, Z.C. Xia an F. Ren, 'Apication of the raial return metho to compute stress increments from Mroz's harening rule', J. Engr. Mat. ech., Vol.,, K. Han, C. J. Van yne an B.S. Levy, 'Bauschinger effect response of automotive sheet steel', SAE Z.C. Xia, 'A General Anisotropic Plasticity Moel for Sheet Metals', For echnical report 46