Deep drawing simulation of dual phase steel using hardening curves and anisotropic parameters from uniaxial and biaxial tensile tests

Among the various factors, the accuracy of the predictions from the numerical simulation of sheet metal forming processes depends on the material model used to define the mechanical behavior of the blank material. The coefficients of the hardening model to define the flow curve and the plastic strain ratios are commonly determined using the uniaxial tensile tests. The advanced anisotropic yield criteria incorporate material flow behaviour and plastic strain ratio in equi-biaxial tension. In this work, deep drawing of a flat bottom cylindrical cup has been simulated using dual-phase steel (DP600) sheets. The biaxial material properties obtained by conducting hydraulic bulge test and cruciform specimen test are used in the anisotropic yield criteria in the simulations. The hardening curves are extrapolated using different hardening laws in which the coefficients are determined from the stress-strain curves obtained from both uniaxial tensile and hydraulic bulge tests. The predicted peak drawing load and thickness variation in the drawn cups are compared with the experimental results of deep drawing.


Introduction
In the automotive industry, the major focus has been towards reducing weight and minimizing exhaust gas emissions.A significant area of emphasis has been the careful selection of materials for automotive bodies and other sheet metal parts [1].High strength steels, particularly advanced high strength steels like dual-phase (DP) steel, are being used in the automotive industry for manufacturing critical parts.DP steel is a low carbon steel containing both soft ferrite and hard martensite and hence is highly suitable for meeting the dual requirements of high strength and good formability [2].Due to the complexity of the components in automotive industries, the state of stress is usually biaxial.The standard uniaxial tensile tests can not be used to determine the behaviour of materials in this condition.Material anisotropy leads to change in the deformation behaviour, subsequently influencing the predicted stress and strain distributions in sheet metal forming operations analysed through finite element analysis [3].So, choosing an appropriate material model that consists of the hardening model to define the flow curve and the yield criterion that incorporates anisotropy is crucial for accurate predictions in simulation of processes such as deep drawing.The Hill48 quadratic yield criterion [4] was the first formulation that incorporated anisotropy in the yielding behaviour of the material.Hassannejadasl et al [5] indicated that changes in the material anisotropy coefficients with respect to strain rate alters the yield surface of DP600.Lenzen et al. [6] conducted elliptical hydraulic bulge tests to get the material properties in the plane strain condition in DP600 specimens.The modified yield surface led to better predictions in the drawing behaviour of the square cups.A non-quadratic anisotropic yield function referred to as Barlat89 was proposed in which the anisotropic coefficients are obtained by conducting the uniaxial tensile tests [7].Some advanced yield criteria have been proposed by Barlat (2000) [8] and Banabic (2005) [9] to predict the yield locus more accurately.The advanced yield criteria incorporate the material properties obtained from the biaxial tensile tests also in which the specimens are subjected to equi-biaxial tension.F. Ozturk et al. [10] demonstrated the effect of different yield functions on the prediction of the limit strains of dual-phase (DP600) steel sheets in FE simulation using Marciniak-Kuczynski model.
In the present work, deep drawing of a flat bottom cylindrical cup has been simulated using dualphase (DP600) steel sheets of 1.4 mm thickness.The combined effect of yield criterion and hardening model on the predicted load and thinning in deep drawing of DP600 steel sheets has been studied.The biaxial material properties obtained by conducting hydraulic bulge test and cruciform specimen test are used to determine the coefficients in the anisotropic yield criteria used in the simulations.The hardening curves are extrapolated using different hardening laws in which the coefficients are determined from the stress-strain curves obtained from both uniaxial tensile and hydraulic bulge tests.The predicted peak load and thickness variation in the drawn cups are compared with the experimental results of deep drawing.

Uniaxial tensile test
Uniaxial tensile (UT) tests were conducted on dual phase (DP600) steel sheet specimens of 1.4 mm thickness in accordance with the ASTM E8M Standard [11] specification.The samples were prepared by blanking process on a mechanical press.The tests were conducted on a Instron make universal testing machine at a constant speed of 5 mm/min.The specimens were tested in three different orientations: parallel (0°), diagonal (45°), and perpendicular (90°) to the rolling direction of the sheet.For each orientation, three specimens were tested to determine the tensile properties.Anisotropy plays an important role in the formability of the sheet metal especially in the deep drawing process.Anisotropy of sheet metals is characterised by determination of plastic strain ratio (R) which is defined as the ratio of true strain in width to true strain in thickness before the onset of necking in a uniaxial tension test.The plastic strain ratio (R) of the DP600 sheet was determined as per the ASTM E517 [12] specification.For determining the R-value, specimens were elongated up to 15% longitudinal strain before the maximum load was reached.The R values were determined in all the three orientations as mentioned earlier in the tensile tests.

Hydraulic bulge test
The hydraulic bulge tests (HBT) were conducted using a hydraulic press with a capacity of 630 tons at LFT, FAU, Erlangen, Germany.The hydraulic bulge test setup is shown in Figure 1(a).In this test, a circular blank is deformed by fluid pressure until failure.High hydrostatic pressure acting on the sample during deformation delays the onset of necking and improves ductility.Thus, a much higher degree of uniform deformation can be achieved resulting in equivalent stress vs. equivalent strain curve up to a much larger strain than that obtained in uniaxial tensile test.
In this test, a circular blank with a diameter of 395 mm was clamped between an upper die and a lower die using a blank holding force of 3500 kN.This clamping force ensured the complete restriction of material flow from the flange region.The inner diameter of the die is 200 mm with an entry radius of 25 mm to reduce the bending effects that can occur with a smaller entry radius.The applied fluid pressure was monitored during the test using a pressure transducer.An online 3-D optical strain measurement system ARAMIS was used for measuring the major and minor strains at the pole throughout the test.

Cruciform test
The biaxial tensile tests (BCT) using cruciform specimens were performed on a biaxial tensile machine at Tata steel, R&D Jamshedpur as shown in Figure 1(b).The geometry of the biaxial test specimen used in this work is based on the design proposed as per ISO 16842:2021 standard [13].Each arm of the specimen has seven slits, 60 mm long and 0.2 mm wide at 5.6 mm interval.The slits are provided to relax the geometric constraint against the uniform deformation in 45 × 45 mm 2 gauge area.The slits were prepared through laser cutting.The specimens were cut in such a way that the rolling and transverse directions of the sheet material coincide with the specimen arms.Two load cells attached in along the two in-plane mutually perpendicular directions were used for measuring the force values in rolling direction (RD) and transverse direction (TD) directions, respectively.The test was performed in equibiaxial condition that is, with a force ratio (Fx/Fy) of 1.0.The strains along the rolling and transverse directions were measured using the optical strain measurement system (GOM).

Numerical analysis of cylindrical deep drawing
Numerical simulations of flat bottom cylindrical cup deep drawing of 1.4 mm thick DP600 steel sheets were carried out using AutoForm software to predict the thinning and the drawing load.The FE model used for the simulation is shown in Figure 2. Circular blanks of 100 mm diameter were deep drawn with a 50 mm diameter flat bottom cylindrical punch.The elasto-plastic shell (EPS-5) elements of size 1.0 mm were used for meshing the blank with five integration points.The refining mesh option with a maximum refinement level of 7 was used for better accuracy in the predicted results.A blank holding force of 30 kN was applied to prevent wrinkling during deep drawing which corresponds to a blank holding pressure of 4 MPa.A punch speed of 1 mm/s was used in the simulations.The coefficients of the Swift hardening law determined using the flow curve of UT and HBT-ISO were used to define the hardening behaviour in the simulation.The simulations were performed using different yield criteria (Mises, Barlat89, and BBC2005).An advanced yield criteria (BBC2005) using the equibiaxial properties determined using both cruciform tests (BBC2005 (CT)) and hydraulic bulge tests (BBC2005 (HBT)) were used in the simulation.

Deep drawing experiments
To validate the predicted results from the above simulations, Swift flat bottom cup deep drawing tests were conducted using circular blanks of 1.4 mm thick Dual Phase steel sheet (DP600).The setup consists of flat bottom cylindrical punch with a diameter of 50 mm, a blank holder to apply the blank holding force to prevent wrinkling, and the drawing die, designed to accommodate the increase in thickness of the sheet to avoid ironing.Blanks diameter of 100 mm (draw ratio of 2.0) were deep drawn in the experiments.Figure 2(b) illustrates the dimensions of the tools used in the deep drawing process.The tests were carried out on a 100 ton double action hydraulic press maintaining a constant punch speed of 1 mm/s.To prevent wrinkling, blank holding force was applied such that the initial blank holding pressure was 4 MPa.The cups were drawn to a depth of 35 mm, and the measurements of punch load and displacement were performed using a load cell and an LVDT, respectively.

Prediction of flow stress model
The stress strain curves obtained from the uniaxial tensile tests and the hydraulic bulge tests are shown in Figure 3(a).It clearly shows that the stress-strain curve of DP600 steel can be plotted up to a much larger plastic strain ( > 0.8) using hydraulic bulge tests when compared to the uniaxial tensile tests in which early onset of necking/instability takes place.However, the material undergoes large plastic strains during deep drawing.Therefore, if uniaxial flow curve is used to predict the hardening behavior by extrapolation to large plastic strains, the accuracy of such curves depends on the hardening law used.The difference between the flow stress approximated by different hardening laws could be significantly high at large plastic strain.In the present work, three models (Swift hardening law, Hockett-Sherby law, and Voce hardening law [14]) were used to plot the flow curves up to a large plastic strain as shown in Figure 3 The results of hydraulic bulge test were analysed, and the biaxial plastic strain ratio (Rb)HBT was determined in accordance with the circular bulge test DIN EN ISO 16808 standard [15].The equivalent biaxial stress has been determined using the membrane theory.It is valid for a small ratio of sheet thickness to bulge diameter, where bending effects can be neglected.The equivalent strain is determined using Hill's 1948 anisotropic material behavior.To extrapolate the uniaxial stress-strain curve, the biaxial stress strain curve is scaled by a factor obtained from the plastic work principle [16].The transformed equivalent stress -equivalent strain curve is referred to as HBT-ISO curve in this work.Biaxial yield stress (σb)HBT is determined using the plastic work equivalence corresponding at the plastic work dissipated in the uniaxial tensile test at the plastic strain of 0.004 in the rolling direction.The yield locus at this low plastic strain is necessary for generation of the initial yield surface.The HBT-ISO curve is already up to a large plastic strain, but it is further extrapolated upto a strain of 1.0 using the Swift hardening law (referred to as HBT-ISO-Swift).Therefore, the flow curve from uniaxial tensile tests extrapolated using Swift hardening law (UT-Swift) and the flow curve from HBT-ISO-Swift method are used in the simulation.The coefficients of the Swift hardening law for UT and HBT-ISO curves are shown in Table 1.The stress strain curves obtained from the cruciform tests along rolling and transverse direction are shown in Figure 3(b).The limit strain reached in the cruciform tests is very small compared with the tensile tests due to strain localization at cross corners and necking in arms.The true stress-true strain curve in RD and TD directions is converted into true stress-true plastic strain curves using ISO 16842:2021 standard.The biaxial plastic strain ratio (Rb)CT is calculated from the ratio of plastic strain in the transverse direction to the rolling direction.Plastic work per unit volume was calculated from the true stress -true plastic strain curves obtained from the cruciform tests.The summation of plastic work in the RD and TD directions of the cruciform specimen at the yield corresponds to the total plastic work dissipated in the uniaxial tensile test at a plastic strain of 0.004 in the rolling direction.The equivalent biaxial yield stress (σb)CT is calculated from the stresses in the RD and TD directions using the Hill48 anisotropic yield criterion.

Prediction of different yield surfaces
The ratios of yield stresses and plastic strain ratios obtained from uniaxial and biaxial tests are summarized in Table 2. Figure 4 shows the predicted yield surfaces obtained from Mises, Barlat89, and BBC2005 yield criteria.Stress ratios derived from both uniaxial and biaxial tensile tests are represented as dotted points.The figure reveals disparities between Mises and Barlat89 under both uniaxial and equibiaxial stress conditions, particularly in equibiaxial and uniaxial transverse scenarios.This discrepancy arises because Mises assumes the material to be isotropic in the material, while Barlat89 considers anisotropy but with only uniaxial properties (0, 45, 90, and 0) for defining the yield function.In equibiaxial stress conditions, both Mises and Yld89 tend to overestimate yielding compared to the experimental values.On the other hand, BBC2005 utilizes both uniaxial and biaxial material properties to define its yield function as given in Table 2.The BBC2005 yield function plotted using the biaxial properties (biaxial plastic strain ratio (Rb) and the equibiaxial flow stress ratio (σb)) obtained from the cruciform tests and the hydraulic bulge tests are represented as BBC2005 (CT) and BBC2005 (HBT).

Deep drawing simulation using different yield criteria
The predicted load-displacement curves from the numerical simulations of deep drawing are compared with the experimental results at a draw depth of 35 mm. Figure 5(a) shows the effect of different yield models on the load-displacement curves.The resistance against deformation in the flange region leads to an increase in the drawing force.The error in the prediction of maximum load is 5% on using the isotropic yield model (Mises), whereas it reduced to less than 3% when anisotropic Barlat89 yield model is used.The error in prediction further reduced to less than 1% when the advanced non quadratic BBC2005 yield function is used.Figure 5(b) shows the thickness distribution in the drawn cup from the centre of the cup to the flange.The maximum thinning is observed in the punch corner radius near to the cup bottom.The maximum thinning is predicted with an accuracy of 5% using the BBC2005 yield model.Combining the advanced yield criterion (BBC2005) with a reliable hardening model of the material up to large strains leads to a more accurate prediction peak load and thinning behaviour.

Effect of biaxial properties in deep drawing simulation
The load-displacement curves and variation of % thickness change obtained from simulations using uniaxial (UT-Swift) and biaxial (HBT-ISO-Swift) flow curves are shown in Figure 7(a) and 7(b) respectively.These curves are obtained with BBC2005 yield surface using the equibiaxial properties determined using both cruciform tests (BBC2005 (CT)) and hydraulic bulge tests (BBC2005 (HBT)).The load-displacement curves show a good agreement with the experimental results.The difference in the predicted results is negligible when the hardening laws used are UT-Swift and HBT-ISO-Swift.The error in predictions would have been higher if the uniaxial flow stress-strain was extrapolated using other flow stress models.Also, the flow curve obtained from the hydraulic bulge tests is up to a large plastic strain and hence the uncertainty related to the choice of hardening law is very low when hydraulic bulge test flow curve is used.A significant difference between the thinning behaviour predicted using the BBC2005 (CT) and BBC2005 (HBT) yield criteria is observed as shown in Figure 7(b).This could be attributed to the fact that thinning in the cup wall is due to biaxial stretching.The BBC2005 yield surface obtained using the biaxial properties from the cruciform tests shows better predictions with the experiments than the one obtained from the hydraulic bulge test.The in-plane deformation in cruciform tests leads to a more accurate determination of biaxial properties than the out-of-plane deformation in hydraulic bulge tests [17].At the location where thinning is maximum, the deviation of the predicted value using the BBC2005 (CT) from the experimental value is nearly 5%, whereas the deviation is more than 24% when BBC2005 (HBT) is used.

Conclusions
It clearly shows that the stress-strain curve of DP600 steel can be plotted up to a much larger plastic strain using hydraulic bulge tests when compared to the uniaxial tensile tests in which early onset of necking/instability takes place.The uncertainty related to the choice of hardening law decreased when hydraulic bulge test flow curve is used.The extrapolation of the stress strain curve obtained from the uniaxial tensile tests using Swift hardening law predicts similar hardening behaviour as obtained by the hydraulic bulge tests even at large plastic strains.BBC2005 yield criterion predicted the yield surface more accurately while Mises and Barlat89 overestimated yield stress in equi-biaxial tension compared to the experimental values.In deep drawing simulations, prediction of peak load, thickness variation and earing profile is closer to the experimental observations with BBC2005 yield function than with Mises, and Barlat89.BBC2005 yield surfaces were compared using the equibiaxial properties determined using both cruciform tests (CT) and hydraulic bulge tests (HBT).A significant difference between the thinning behaviour predicted using the BBC2005 (CT) and BBC2005 (HBT) yield criteria is observed.Simulation with BBC2005 yield surface obtained using the biaxial properties from the cruciform tests showed better prediction than the one obtained from the hydraulic bulge test.

Figure 1 .
Determination of biaxial flow curves by (a) hydraulic bulge test and (b) cruciform test (a) (b) Figure 2. (a) FE model used in simulation of Swift flat bottom cup test and (b) dimensions of the tools used in the deep drawing.
(a).The coefficients in different hardening laws were determined by regression analysis and minimizing the Sum of Squares with a minimum R² values of 99.9%.Using the equations of Swift hardening law, Hockett-Sherby law, and Voce hardening law, the flow curves were generated up to a true plastic strain of 1.0.(a) (b) Figure 3. (a) Extrapolation of UT-Exp curve using different hardening laws and their comparison with HBT, (b) True stress-strain curve obtained from cruciform test and its comparison with UT.

Figure 4 .
Figure 4. Predicted yield surfaces using different yield criteria

Figure 5 .
Comparison of (a) load-displacement curves and (b) variation of change in thickness in the deep drawn cup predicted using different yield models.The cup is drawn to a depth of 35mm successfully as shown in Figure6(a).An experimentally determined forming limit curve, shown in Figure6(b), is used in the FE simulation to predict the failure.The simulation also shows successful forming of the cup.The planar anisotropy present in the material leads to earing in the flange region during deep drawing.

Figure 6 .
Figure 6 (c) shows comparison of the earing profile predicted using the Mises and BBC2005 yield models.The effect of planar anisotropy is visible in the flange region clearly and it leads to earing at the top of the cup wall when drawn completely.The profile obtained from the von-mises yield model is circular due to assumption that the material is isotropic.BBC2005 model incorporates the anisotropic material behaviour and hence resulted in better prediction of the earing profile.(a) Experimentally drawn cup, (b) Simulation result of deep drawing, and (a) Comparison of the earing profile using different yield models at a draw depth of 35 mm.

Figure 7 .
Comparison of predicted (a) load-displacement curves, (b) % change in thickness in the drawn cup using BBC2005 yield criterion with experimental curves.

Table 1 .
Coefficients of swift hardening law predicted using uniaxial tensile and hydraulic bulge tests.

Table 2 .
Material properties determined to plot the BBC2005 yield surface in AutoForm.