P.-A. Eggertsen a,b, Kjell Mattiasson a,b, Mats Larsson c,d

Transcription

1 A COMPREHENISVE ANALYSIS OF BENCHMARK 4: PRE-STRAIN EFFECT ON SPRINGBACK OF D DRAW BENDING P.-A. Eggertsen a,b, Kjell Mattiasson a,b, Mats Larsson c,d a Div. of Material and Comutational Mechanics, Det. of Alied Mechanics Chalmers University of Technology, SE Göteborg, Sweden b Volvo Cars Safety Centre, SE Göteborg, Sweden c Det Press, C15-1 TMNKM, Saab Automobile, SE Trollhättan, Sweden d Deartment of Engineering Science, University West, SE Trollhättan, Sweden Abstract. In order to be able to form high strength steels with low ductility, multi-ste forming rocesses are becoming more common. Benchmark 4 of the NUMISHEET 011 conference is an attemt to imitate such a rocess. A DP780 steel sheet with 1.4 mm thickness is considered. In order to understand the re-strain effect on subsequent forming and sringback, a D draw-bending is considered. Two cases are studied: one without restrain and one with 8% re-stretching. The draw-bending model is identical to the U-bend roblem of the NUMISHEET 93 conference. The urose of the benchmark roblem is to evaluate the caability of modern FEmethods to simulate the forming and sringback of these kinds of roblems. The authors of this article have reviously made exhaustive studies on material modeling in alications to sheet metal forming and sringback roblems, [1],[],[3]. Models for kinematic hardening, anisotroic yield conditions, and elastic stiffness reduction have been investigated. Also rocedures for material characterization have been studied. The material model that mainly has been used in the current study is based on the Banabic BBC005 yield criterion, and a modified version of the Yoshida-Uemori model for cyclic hardening. This model, like a number of other models, has been imlemented as User Subroutines in LS-DYNA. The effects of various asects of material modeling will be demonstrated in connection to the current benchmark roblems. The rovided material data for the current benchmark roblem are not comlete in all resects. In order to be able to erform the current simulations, the authors have been forced to introduce a few additional assumtions. The effects of these assumtions will also be discussed. Keywords: Multi-stage forming, sringback, material modeling, re-straining PACS: Relace this text with PACS numbers; choose from this list: htt:// INTRODUCTION Multi-ste forming of sheet metal materials is a kind of rocess that is becoming more and more common in the industry. In terms of numerical simulations, very little has been done in this field. Benchmark 4 of the NUMISHEET 011 conference is an attemt to imitate such a rocess. In order to understand the re-strain effect on subsequent forming and sringback, a D draw-bending is considered. Two cases are studied: one without re-strain and one with 8% re-stretching before the draw-bending rocess. The draw-bending model is identical to the U-bend roblem of the NUMISHEET 93 conference. The urose of the benchmark roblem is to evaluate the caability of modern FE-methods to simulate the forming and sringback of these kinds of roblems. The current aer starts with a brief descrition of the material models used in the resent study. The following section describes the material arameter identification rocesses, and in the final section the analyses of the drawbending roblem are accounted for. The 8th International Conference and Worksho on Numerical Simulation of 3D Sheet Metal Forming Processes AIP Conf. Proc. 1383, (011); doi: / American Institute of Physics /$

2 MATERIAL MODELING The authors of the resent aer have earlier shown the imortance of the modeling of the material behavior for successful sringback redictions [1],[],[3]. In the revious work it is demonstrated that there are three material model constituents, which are imortant for a roer descrition of the material behavior. These are: the yield surface, the hardening law, and, finally, the modeling of the elastic stiffness degradation with lastic strain. Below follows a brief descrition on how these constituents are chosen for the resent roblem. Yield Surface To accurately account for the anisotroic behavior, the eight arameter yield criterion by Banabic and Aretz [6] is chosen: The functions Γ, Ψ and Λ are defined as Lσ Γ= ( 1 M M M σ = Γ+Ψ + Γ Ψ + Λ xx + Kσ yy ( Nσxx Pσ yy ) Ψ= + Q σxyσ 4 ( Rσxx Sσ yy ) Λ= + T σxyσ 4 where the arameters L,K,N,P,Q,R,S and T are identified from uniaxial- and equibiaxial-test data. xy xy 1 M (1) () Kinematic Hardening Descrition To account for the material behavior at cyclic loading, two different kinematic hardening laws are considered: the classical hardening law by Arnstrong and Frederick (A-F) [7] and a modified version of the Yoshida-Uemori (Y- U) [8] hardening law. The back-stress evolution of the A-F law is described by: σ = Cx αsat ε (3) where C x and sat are material arameters. The A-F hardening law is able to account for the Bauschinger effect and the transient behavior during reverse loading. However, the imortant ermanent softening effect (see Refs. [1][3][8]) is not accounted for. To also consider the ermanent softening effect, a modified version of Y-U hardening law is also used in this study. In this modified version of the Y-U model, the lastic hardening curve is rovided as inut data, instead of being a art of the arameter identification rocedure as in the original formulation. This also results in that the number of unknown material arameters is reduced from seven to four. Below follows a brief descrition of this modified version. For a full descrition the reader is referred to earlier work by the authors [1], [3]. The Y-U model includes both translation and exansion of the bounding surface, while the active yield surface only evolves kinematically. The evolution of the back-stress is exressed as: * = + (4) with 1065

3 * R * = Cx ( - ) ε Y b = k ( - ) ε Y + R (5) * where is the relative kinematic motion of the yield surface with resect to the bounding surface, is the centre of the bounding surface, Y is the size of the yield surface, C x, b and k are material arameters, and R is the isotroic hardening of the bounding surface. In the current modified version, the isotroic hardening is described as: R ( ε ) = H ( ε ) β ( ε ) (6) where H ( ε ) is the given lastic hardening curve, and β ( ε ) is the uniaxial backstress of the bounding surface. Besides the three already mentioned material arameters, the Y-U model also includes a fourth material arameter, h. This arameter affects the amount of cyclic hardening. The larger value of the arameter h the smaller the rediction of cyclic hardening. Elastic stiffness The amount of sringback during unloading deends to a great extent on the elastic stiffness of the material. It has been shown that during lastic deformation, the elastic stiffness decreases [1][3][9]. In this work, an analytical exression roosed by Yoshida et al. [8], was used to describe the elastic stiffness degradation with lastic strain: u ( sat) ( ) E = E E E e ξε (7) where E 0 is the initial Young s modulus, E sat is a value that the unloading modulus saturates towards, and is a material arameter. MATERIAL CHARACTERIZATION The material used in this study is a DP780 material with a thickness of 1.4 mm. The material characterization is based on a number of exeriments. Below follows a brief descrition of these exeriments, and on how the various material arameters are determined from these exeriments. Uniaxial Tension Tests Tension tests were erformed in the rolling, transverse and diagonal directions. The basic exerimental data from the tension test in the rolling direction are resented in TABLE 1. The uniaxial tension tests also form the basis for descrition of the anisotroic behavior of the material. Unfortunately, only the material coefficients for the Barlat Yld000 and the Hill 48 yield criteria are rovided in the benchmark material data sheet. However, the Banabic BBC005 model (used in our simulations) and the YLD000 model are identical, although their coefficients have different meanings. It has therefore been ossible to transform the YLD000 coefficients to BBC005 coefficients. The recalculated coefficients for the BBC05 criterion are resented in TABLE. The exonent M (a in Yld000) has the same meaning for both criteria and is set to 6. TABLE 1. Uniaxial tension test data Direction E [GPa] YS [MPa] UTS [MPa] R-value Poisson s ratio Rolling direction (assumed) TABLE. Material constants for the Banabic 8-arameter (BBC005) yield criterion Samle L K N P Q R S T M DP

4 The uniaxial tension test in the rolling direction also forms the basis for the lastic hardening curve. However, due to the occurrence of diffuse necking, lastic hardening data is only rovided u to 10% lastic strain. The maximum lastic strain level that can be anticiated in the sringback examles is much higher than that. This means that the lastic hardening curve has to be extended u to higher values of effective lastic strain by extraolation. This introduces a significant source of error in the simulations. It would have been desirable, if the exerimental hardening curve was extended by means of data from a bulge test or a shear test. In our simulations the hardening curve is extended by a simle extraolation of Hollomon tye: H( ε ) n = K ε (8) where H is the effective stress, and K and n are material arameters. Both these arameters are determined based on the sloe at the end of the exerimental tension test curve, such that K = 1.4 and n = An illustration of the extraolation is given in FIGURE 1a. The resulting hardening curve can be seen in FIGURE 1b, where the first art of the curve is the tension test data and the second art is based on the extraolation described above. In the numerical simulations this hardening curve is given directly in the inut data. FIGURE 1. Extraolation of the isotroic hardening curve. Resulting isotroic hardening curve with one art from uniaxial tension test in the rolling direction and one art from Hollomon extraolation. In-Plane Tension-Comression-Tension Tests In order to characterize the cyclic behavior of the material, in-lane tension-comression-tension tests were erformed. The test secimens were reared in the rolling direction, using the testing aaratus described in Kuwabara et al. [4], for different amounts of re-strains: 0.0, 0.04, 0.06, 0.08 and 0.1. The identification rocedure is erformed with the otimization code LS-OPT, such that a stress-strain relationshi from the constitutive driver becomes as close to the exerimental values as ossible. In the otimization rocedure we have chosen to use two of the exerimental stress-strain curves as target curves: the curves corresonding to 0.04 and 0.1 lastic re-strain. The reason for using two target curves is to assure that the obtained arameter set-u covers a wide range of lastic strain levels. By using only one target curve, there is a risk that the otimal solution only is suitable for that articular strain level. A detailed discussion on this matter can be found in an earlier work by the authors [5]. The obtained material arameters are rovided in TABLE 3. In FIGURE the erformance of the kinematic hardening models with the obtained material arameters is shown. TABLE 3. Kinematic hardening arameters for the two considered hardening laws Hardening law C x sat b k h A-F Y-U

5 FIGURE. Verification of redicted uniaxial tension-comression-tension data for: the modified Yoshida-Uemori kinematic hardening model and, the Armstrong-Frederick hardening law, with material arameters according to TABLE 3. The red curves reresent redicted values and the blue curves are exerimental data. Elastic Stiffness Degradation The stiffness degradation as a function of lastic strain is measured from stress-strain relationshis at five different levels of re-straining. Based on this data, the material coefficients in Eq. 7 are determined by a least square fitting of the analytical exression to the exerimental data oints. In TABLE 4 the obtained arameters are listed. An illustration of the obtained analytical curve can be seen in FIGURE 3. In the figure the exerimental data oints are illustrated as red stars. TABLE 4. Material constants describing the elastic stiffness degradation according to Eq. 7 Samle E 0 [GPa] E sat [GPa] DP FIGURE 3. Unloading modulus curve according to Eq. 7 with arameters resented in TABLE 4. The stars reresents exerimental data. SPRINGBACK The aim of this aer is to discuss benchmark roblem 4 of this conference. The benchmark considers D-draw bending of the DP780-steel described earlier in this aer. Two different cases are comared: in the first one the virgin base material is used in the draw bending rocess, and in the second case the material is re-tensioned u to 8% engineering strain. An illustration of the exerimental set-u is given in FIGURE 4a, and the dimensions of the tools are rovided in TABLE 5. The alied blank holding force is.94 kn. In FIGURE 6 and TABLE 7 the dimensions of the secimens used in the draw-bending examle are defined. Numerical Simulations The numerical simulations were erformed by means of the LS-DYNA code. For the forming simulations an exlicit time integration scheme was used. The sheet material was modeled by fully integrated quadratic shell 1068

6 elements. The forming tools were modeled by shell elements and were assumed to be comletely rigid. The friction coefficient between all contact surfaces was assumed to follow the simle Coulomb frictional law with a constant friction coefficient equal to 0.1. The imortance of the mesh size in sringback simulations has been demonstrated in revious investigations. In this study a mesh convergence analysis was carried out. The result from the mesh sensitivity study is shown in FIGURE 5 and TABLE 6. As can be seen, the mesh that is called mesh 3, 0.469x0.469 mm, seems to be the best choice. Therefore, this mesh density is used for both the considered cases. The re-straining of the material is modeled such that the nodes in one end of the material are locked in the length direction, while the nodes in the other end are given a rescribed dislacement, corresonding to an axial strain level of 8% engineering strain. After the re-straining, a sringback simulation is erformed so that the secimen becomes stress-free. The re-strained secimen is then used in a forming oeration, and a subsequent sringback simulation. For the sringback simulations imlicit time integration is used. The unloading method for the sringback simulation is the so-called one ste unloading method, which means that all the forming tools are removed before the sringback calculation takes lace. FIGURE 4. Schematic view of the tools of the -D draw bending test. Definition of the various sringback measures. TABLE 5. Dimensions for the D draw-bending test Parameters W1 W W3 W4 R1 R G1 Stroke Dimensions FIGURE 5: Illustration of the mesh sensitivity. sringback rofile, zoom-in of the secimen tis in figure. 1069

7 TABLE 6: Mesh-sizes in the mesh sensitivity study (FIGURE 5) Mesh Size [mm x mm] Mesh x Mesh x Mesh x Mesh x 0.34 FIGURE 6. Schematic view of secimens and their dimensions for: the base material case (without re-strain), the restrained case. TABLE 7. Dimensions of the secimens in FIGURE 6. Base material Pre-strained secimen (8% engineering strain) Parameters L W L 0 W 0 L gri Gri dislacement L Dimensions The sringback rofiles for the two considered kinematic hardening laws are dislayed in FIGURE 7. FIGURE 7a shows the results for the base material, while in FIGURE 7b the corresonding results for the re-strained material are dislayed. Obviously the A-F hardening model yields more sringback than the Y-U model. The reason for this can be deduced from the determination of the kinematic hardening arameters, and from the stress-strain relationshi for the A-F hardening law at high cyclic strain levels (see FIGURE b). As can be seen in FIGURE b the redicted stresses at reverse loading are significantly higher than the exerimental ones. This means that the numerical simulations of the draw-bend test, using the A-F hardening law, will also over-redict the stresses in the side-wall of the formed art. Therefore, also the bending moment will be over-redicted, and as a consequence of this also the resulting sringback. Comaring FIGURE 7a and FIGURE 7b, we note that the difference between the results for the two considered hardening laws is larger for the base material case than for the re-strained case FIGURE 7. Sringback rofiles for the base material, the re-strained material. Since the Y-U hardening model fits exerimental data best, it can be anticiated to result in the most accurate sringback redictions. The resulting sringback rofiles for the Y-U model alied to the base material and to the re-strained material, resectively, are shown in FIGURE 8a. It can be observed that the re-strained material results in considerably more sringback. The reason for this can be seen in FIGURE 8b. The figure shows the through 1070

8 thickness stress distribution at a oint in the middle of the side wall, after the forming hase, and before the sringback hase. As can be seen the stresses are much higher for the re-strained case, which in turn results in higher side-wall curl. The reason for the higher stresses in the re-strained case can easily be understood, if one considers the re-strained material just as a material with a higher hardening curve than the base material. In TABLE 8 the values of the sringback measurements illustrated in FIGURE 4b are shown for both the base material and for the re-strained material, for the two considered kinematic hardening laws. FIGURE 8. Y-U hardening model. Sringback rofiles for the base material and the re-strained material. Longitudinal stress distribution through the thickness, for the base material, and for the re-strained material, in a oint in the middle of the sidewall. TABLE 8. Sringback measurements for all considered cases Without re-strain With re-strain Hardening law 1 1 Y-U A-F CONCLUSIONS Benchmark 4 of the NUMISHEET 011 conference has been studied. It has been shown that the sringback deformation after a draw-bend oeration, such as the one resented in the benchmark, is very sensitive to the mesh size and the choice of kinematic hardening model. In this aer both the A-F hardening law and the Y-U hardening law has been used. Since the Y-U hardening model fits exerimental cyclic stress-strain data best, it can be anticiated to yield the most accurate sringback redictions. When it comes to the effect of re-straining of the material before the draw-bend oeration, it is shown that the resulting sringback for the re-strained case becomes higher than for the base material case. This is a result of the rediction of higher stresses in the side-wall after the forming stage, resulting in larger side-wall curl, for this case. It should finally once again be ointed out that the numerical results obtained in this study are highly deendent on the aearance of the hardening curve, which we, unfortunately, to a great extent have had to guess. The errors originating from this assumtion may erhas be of the same order as the differences in results obtained with the two kinematic hardening laws. REFERENCES 1. P.A. Eggertsen and K. Mattiasson, Int. J. of Mech. Sci. 51, (009).. P.A. Eggertsen and K. Mattiasson, Int. J. of Mech. Sci. 5, (010). 3. P.A Eggertsen, "Prediction of sringback in sheet metal forming with emhasis on material modeling", licentiate Thesis, Chalmers University of Technology, Kuwabara et al, Int. J. of Plast.. 5, (009). 5. P.A. Eggertsen and K. Mattiasson, Int. J. of Mtrl. Forming (011). 6. D. Banabic et al, Int. J. of Plast, 1, (005). 7. P.J. Armstrong and C.O. Frederick, G.E.G.B reort RD/B/N 731 (1966). 8. F. Yoshida and T. Uemori, Int. J. of Plast, 18, (00). 1071