Measurement and modeling of simple shear deformation under load reversal: Application to advanced high strength steels

International Journal of Mechanical Sciences 98 (2015) 144–156

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Measurement and modeling of simple shear deformation under load reversal: Application to advanced high strength steels J.S. Choi a, J.W. Lee b, J.-H. Kim a, F. Barlat a,e,n, M.G. Lee c,n, D. Kim b,d a

Graduate Institute of Ferrous Technology, Pohang University of Science and Technology, Pohang 790-784, South Korea Materials deformation group, Korea Institute of Materials Science, Changwon 642-831, South Korea c Department of Materials Science and Engineering, Korea University, Seoul 136-713, South Korea d Korea University of Science and Technology, Daejeon 305-350, South Korea e Center for Mechanical Technology and Automation, Department of Mechanical Engineering, University of Aveiro, 3810-193, Portugal b

art ic l e i nf o

a b s t r a c t

Article history: Received 29 August 2014 Received in revised form 16 April 2015 Accepted 21 April 2015 Available online 29 April 2015

In this paper, the stress–strain behavior under load reversal of advanced high strength steel (AHSS) sheet samples was measured using a modified simple shear (SS) apparatus. The forward–reverse loading behavior was characterized for three different grades of AHSS; namely, DP, TRIP and TWIP steels. For comparison purpose, compression–tension (CT) tests were also carried out for the same materials. For all the cases, a typical complex anisotropic hardening behavior, including the Bauschinger effect, transient strain hardening with high rate and permanent softening, was observed during load reversal. No premature localization and sheet buckling occurred in these experiments, which have been major technical hurdles in CT tests. For example, an engineering shear strain of over 40%, which corresponds to an effective strain of roughly 0.2, at reversal was achieved for DP980 although the uniform elongation of this material in uniaxial tension is only 5%. A recently developed distortional hardening model (HAH) was employed to reproduce the SS stress–strain curves. Using the coefficients determined with these SS data, the CT behavior was predicted with the HAH model and compared with experimental results. This complementary study indicated that the constitutive model determined from the SS flow curves satisfactorily reproduced the CT hardening behavior. As an application, finite element simulations of springback were carried out for 2D draw-bending of a strip sheet. & 2015 Elsevier Ltd. All rights reserved.

Key words: Simple shear Advanced high strength steels Springback Compression–tension

1. Introduction Advanced high strength steels (AHSS) have been replacing conventional low strength steels for automotive parts because of weight saving and improved crash performance. However, it is a challenge to successfully introduce AHSS because of their inferior formability and larger springback compared to conventional steels. Especially, springback, the undesirable elastic recovery in a formed part when forming loads are removed, should be compensated to allow the assembly of a component. In order to understand and predict the mechanics of springback for AHSS, there have been numerous approaches using finite element (FE) simulations [1–8]. The FE approach can reduce the development cost and time significantly by suggesting manufacturing modifications without resorting to numerous experimental trial-and-error steps. Therefore, it is important to increase the

n

Corresponding authors. E-mail addresses: [email protected] (F. Barlat), [email protected] (M.G. Lee). http://dx.doi.org/10.1016/j.ijmecsci.2015.04.014 0020-7403/& 2015 Elsevier Ltd. All rights reserved.

accuracy of FE simulations to provide results similar to those obtained in real forming processes. For this purpose, a reliable constitutive model that properly describes the elastic–plastic response of a sheet metal is necessary. In particular, for an accurate prediction of springback, various constitutive models, which capture the complex material behavior during strain path changes, have been proposed. This is because bending and unbending with a superimposed stretch, which occur in a typical process involving load reversal and the Bauschinger effect, influence the springback behavior significantly. The Bauschinger effect, which is characterized by a lower yield stress and higher strain hardening rate upon load reversal, was successfully captured by advanced constitutive models [9–14]. In order to provide suitable coefficients for such advanced constitutive models, it is necessary to employ the non-conventional testing method. In particular, it is important to probe the material response at large strain without premature plastic flow localization and to achieve a sufficient load at reversal to effectively determine suitable coefficients for AHSS. In this sense, a conventional uniaxial tension test is not appropriate to capture the Bauschinger behavior of high strength materials because of its

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monotonic nature and its tendency to premature flow localization at low strain. Alternate testing methods have been reported, which include torsion-reverse torsion [15,16], tension–compression [17] and forward–reverse simple shear tests [18]. The disadvantage of the torsion-reverse torsion test is that, for a sheet sample, it requires bending and welding prior to testing, which are likely to change the mechanical properties of the original material. In the case of inplane tension–compression, an anti-bulking system is necessary for sheet specimens, which modifies the stress state and introduces uncertain friction effects [12,17,19]. Moreover, the strains attainable in the tension–compression test are limited due to either flow localization during tension or buckling during compression, phenomena which are a challenge in high strength steels. For example, the uniform elongation of DP980 steel measured in uniaxial tension is less than 5%, which is not sufficient for an accurate constitutive model identification and FE modeling [20]. The simple shear (SS) test has been designed to circumvent the disadvantages of the existing methods. In this test, the specimen preparation is simpler than in other methods and large homogenous strains can be achieved without plastic flow localization. Moreover, reverse loading is easy to control without major changes in the deformation mode. Due to these advantages, several researchers adopted this approach to characterize the anisotropic hardening behavior of metals after load reversal [21–27]. However, most of the previous studies pertaining to the SS test focused on softer materials such as mild steels [21,23,24], aluminum alloys [22,25,27] and polymers [28]. Only few investigations on the SS behavior of AHSS have been published [29,30]. The aim of the present work is first to introduce a modification of the SS device that allows testing for AHSS, sheets for which the tensile strength ranges from 700 to 1000 MPa (or even higher). Then, this improved SS device is used to measure the forward– reverse stress–strain responses of AHSS sheets, which involve complex hardening behavior during load reversal. The measured stress–strain curves are approximated by a recently developed anisotropic hardening model, so-called HAH [31–33]. Finally, the effectiveness of the measurement method is validated by a comparison of the predicted reverse loading behavior in compression-tension (CT) and by the finite element (FE) prediction of springback.

2. Experiment 2.1. Simple shear device The simple shear (SS) test device for sheet metals consists of two rigid grips, which translate with respect to each other along the sample longitudinal direction [28,34,35]. A rectangular sample is firmly clamped by the two grips, one of them immobile while the other translates along the x-axis (Fig. 1). The constant width h of the deformation area is maintained during the test. In Fig. 1, L and Δx are the current length of the specimen and the relative displacement of the two grips during simple shear deformation, respectively. Compared to lower strength materials, the main challenge of SS for high strength steels is to achieve a robust clamping. While a mechanical gripping approach relying on friction was successful for soft materials, the same method was insufficient in a preliminary study for materials with strengths of 780 MPa or higher. Therefore, one of the motivations of the present work was to develop an enhanced gripping system in the SS test for AHSS. The improved SS test device installed in a 500 kN MTSs universal tension–compression machine, is shown in Fig. 2. For a firm and controlled gripping of the rectangular sample, hydraulic clamping is applied on the grip ends of the specimen as shown in

145

Fig. 1. Schematic description of the SS test.

Fig. 2. The modified SS test device.

Fig. 1. Preliminary trials showed that hydraulic pressures up to 30 MPa were not sufficient to prevent the specimen from slipping during the test for steel of 780 MPa strength or higher. To prevent slipping due to this insufficient gripping pressure, the entire specimen is encapsulated within a serrated groove specially designed as shown in Fig. 3. The specimen is inserted into the groove, which is shown as the shaded area in Fig. 3(c). The size of the groove depends on the length of the shear specimen. In this study, the depth of the groove d is 3 mm and the distance h between the two bottoms or top grips is 4 mm. The thickness of the grips is 50 mm, which is large compared with the thickness of the specimen (See Fig. 3). Moreover, a mechanism forces the grips to only translate with respect to each other. Therefore, during testing, any rotational rigid motion of the SS apparatus is prevented. This groove design prohibits slipping of the specimen in a direction perpendicular to its length. For all AHSS sheets, the hydraulic clamping pressure was maintained to 30 MPa. 2.2. Materials To evaluate the performance of the improved SS device, three AHSS sheet samples provided by POSCO were considered; i.e., a 980 MPa strength (DP980) dual-phase steel, a 780 MPa transformation induced plasticity steel (TRIP780) and a 980 MPa twinninginduced plasticity steel (TWIP 980). The thicknesses of the DP980,

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Fig. 3. Grips with groove design of improved SS test device: (a) front view, (b) side view, and (c) side view after full clamping.

TRIP780 and TWIP980 sheet sample were 1.2, 1.4 and 1.4 mm, respectively. The main alloying elements of these materials, which exhibit three different microstructures, are listed in Table 1. The dual-phase steels consist of ferrite and martensite phases with specific volume fractions that control the macroscopic strength. The TRIP steel has a typical microstructure consisting of ferrite, bainite, martensite and retained austenite. The latter phase is metastable and contributes to enhance strain hardening and formability during deformation. The TWIP steel is made of a single austenitic phase, which deforms by slip and twinning. This microstructure is known to contribute to the excellent combination of strength, elongation and work hardening of this material.

Table 1 Main alloying elements (in wt%).

DP980 TRIP780 TWIP980

C

Mn

0.07 0.15 0.60

2.3 2.1 18.0

2.3. Shear stress and strain The deformed sample after SS deformation is illustrated in Fig. 4. The shear stress (τ) can be simply calculated by dividing the applied external force (F) by the whole cross-section area (A), i.e., τ¼

F F ¼ A Lt

ð1Þ

Actually, the shear stress is not uniform over the entire crosssection area of the specimen because of the end effects. During the shear test, the ends of the specimen undergo tensile and compressive stress, as schematically shown in Fig. 4. However, these end effects are minimized with a large L/h ratio which, for metallic materials, was recommended to be greater than 10 [27]. In this study, the L/h ratio is 12.5 ( ¼50 mm/4 mm). Therefore, the end effects of the specimen during the shear test are assumed to be negligible when the shear stress is calculated with Eq. (1). This assumption is rectified in later section. For the deformation measure, the Hencky strain concept is used because it is a proper measure for large deformation. The Hencky strain is the tensor of the logarithmic strain extended to threedimensional analysis [36]. It is defined using the deformation gradient tensor F, which is classically decomposed as

Fig. 4. Schematic representation of the end effects of SS sample.

case, the Hencky strain tensor H is defined using U as H ¼ ln U

ð3Þ

ð2Þ

where ln U is the logarithm of the matrix U. In this study, The Hencky strain component considered for SS is " # rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  γ 2ffi γ εH ¼ ln þ 1 þ ð4Þ 2 2

In the above relationship, R is an orthogonal tensor and U is a symmetric tensor with positive eigenvalues. The tensor R describes a pure rotation while the tensor U, called the right stretch tensor, represents the essential deformation in F. In this

as demonstrated by Onaka [36]. The important aspect of the Hencky strain component in this work is that it is equal to half of the engineering strain, γ, within a few percent error since γ r 0:5.

F ¼ RU

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147

Fig. 5. DIC images of specimen:(a) AOI (50  4 mm), (b) Hencky shear strain in AOI (50  4 mm), (c) AOI (16  4 mm), (d) Hencky shear strain in AOI (16  4 mm).

2.4. Simple shear test The dimensions of the rectangular simple shear (SS) specimen are 50  16 mm and the thickness is 1.2 or 1.4 mm. The width of shear deformation zone h is 4 mm as shown in Fig. 1 and the remaining of the specimen is used for gripping. All the tests were performed at an approximate strain rate of 2  10  3/s and at room temperature. In order to determine the strain, a VIC digital image correlation (DIC) system was employed to measure the displacement field. The VIC-3D software (www.correlatedsolutions.com) was used to analyze the strain values from optical images. The maximum area of the strain measurement by the DIC technique was 50  4 mm. Fig. 5(a) represents the specimen area of interest (AOI) before starting the test. Fig. 5(b) is an image of the specimen after deformation, which shows an inhomogeneous shear strain distribution. The magnitude of the shear strain in the whole specimen deformation zone ranges from 0.097 to 0.196 because of the specimen free end effects. In order to eliminate the region of high shear strain gradient, an appropriate AOI (16  4 mm) was selected in Fig. 5(c), in which the deformation is almost uniform (ranges from 0.152 to 0.161) as shown in Fig. 5(d). 2.5. Compression–tension test In order to compare the reversal effects in simple shear (SS) with those in another deformation mode, compression–tension (CT) tests, illustrated in Fig. 6, were also carried out for the DP980, TRIP780 and TWIP980 steel sheets with a constant strain rate of 10  3/s. An horizontal typed tension–compression device includes two solid support plates applying a constant normal force, which prevent the sheet samples from buckling during compression [37,38]. Because of the constant normal force through the thickness

Fig. 6. CT experimental apparatus installed at Korea Institute of Materials Science (KIMS).

direction, the raw stress–strain curves require corrections from frictional and biaxial stress effects as discussed in Boger et al. [19]. The CT tests were performed with a 4.9 kN normal force and the Coulomb friction coefficient 0.06, which was estimated by comparing the tensile flow curves obtained with or without the support plates. Using the Coulomb law, the frictional stress was eliminated from the raw stress–strain data. Because the normal stress was much smaller than the longitudinal stress, the biaxial stress effect was neglected.

3. Constitutive models In order to utilize the measured SS stress–strain curves in the FE simulations, an accurate constitutive model is necessary to describe the complex material responses under strain path changes. In this study, an advanced constitutive model, which consists in an anisotropic non-quadratic yield function and a distortional hardening

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approach, was considered. The main features of the constitutive models are briefly summarized in the following sections and additional details can be found elsewhere [31,32].

strain increment.

3.1. Non-quadratic anisotropic yield function: Yld2000-2d

The parameters fk are given by " #1=q 1 f k ¼ q 1 for k ¼ 1 and 2 gk

The yield function Yld2000-2d is expressed as [39]    a  a  a  1=a 1  ~0 s1  s~02  þ 2s″~2 þ s″~1  þ 2s~01 þ s″~2  ϕ¼ 2

ð5Þ

where exponent ‘a’ represents the crystal structure and s~K are the principal value of the tensor s~ defined by a linear transformation ~ ¼ L″σ. on the stress tensor σ, namely, s~0 ¼ L0 σ and s″ The linear transformation 4th order tensors L0 and L″ contain a total of 8 anisotropy coefficients α1  8 for an orthotropic rolled sheet and are written as follows 2 0 3 2 3 L11 L012 0 2α1  α1 0  0 6 0 1 0 7 6 0 5 ¼ 4  α1 2α12 0 7 L ¼ 4 L21 L22 ð6Þ 5 3 0 0 0 L66 0 0 3α7 2

3

0 L″11 L″12 6 0 7 ½L″  ¼ 4 L″21 L″22 5 0 0 L″66 2 8α5 2α3  2α6 þ 2α4 6 ¼ 4 4α3 4α5  4α4 þ α6 0

4α6 4α4  4α5 þ α3 8α4  2α6 2α3 þ 2α5 0

0

3

0 7 5 9α8

ð7Þ

The anisotropy coefficients are usually calculated using the flow stresses and r-values determined from uniaxial tension tests conducted at 0, 45 and 90 degrees from rolling, as well as the balanced biaxial flow stress (σb ) and r-value (rb ¼dεx/dεy). 3.2. Distortional anisotropic hardening HAH model The isotropic hardening model cannot capture the Bauschinger and associated effects in the flow curve upon load reversal. In contrast, the kinematic hardening model, which assumes a back stress to translate the yield surface, has been successfully employed to reproduce these phenomena for metals. Recently, the authors of the present article proposed the homogeneous yield function-based anisotropic hardening (HAH) model as an alternative to kinematic hardening [31]. This approach allows the modeling of the complex hardening response of metals with a yield surface distortion, which is controlled by the so-called microstructure deviator. A series of papers relating this distortional hardening model to the Bauschinger and cross-loading effects have been published [40,41]. Theoretical details about the HAH model and its numerical implementation can be found in the literature [31–33]. In this study, the HAH model was adopted to describe the hardening behavior measured in the SS tests. In the original version of the HAH model, the yield condition is expressed as follows       i h  b q  b q 1=q qb q b : s  h : s þ f 2 h : s þ h : s σ ðsÞ ¼ ϕq ðsÞ þf 1 h ¼ σðεÞ

ð8Þ

where s is the deviatoric stress tensor and q is a constant exponent. ϕðsÞ; called the stable component, can be any isotropic or anisotropic positively homogeneous yield function of first b which was introduced to degree. The microstructure deviator h, capture the deformation history in a continuum sense [42], controls the details of the yield surface distortion. The superscript “^” stands for a normalized value and “:” denotes 0the double dot b is assumed to product [31]. The initial microstructure deviator h 0 correspond to the stress deviator s , which leads to the first plastic

s0ij b 0 ¼ qffiffiffiffiffiffiffiffiffiffiffi h ij 8 0 0 3sij sij

ð9Þ

ð10Þ

where g k are state variables that are defined by evolution laws. b : s. The The subscript “k” is determined depending on the sign of h original HAH model has been extended with additional state variables, which allow the model to capture permanent softening and transient hardening under various loading path changes including cross-loading [32,41]. However, the original model is sufficient for the present study. The state  variables  evolve as functions of the equivalent plastic σ dε b These strain ε dε ¼ ij σ ij and depend on the sign of s : h. evolution laws were originally given by: b o0 If s : h

 dg1 k1 g4  g1 ¼ g1 dε

dg2 σ0 ¼ k2 k3  g2 dε σ dg3 ¼ k5 ðk4  g3 Þ dε   b dh 8b b b ¼ k b sþ h s:h dε 3

ð11Þ

b Z0 If s : h



dg1 σ0 ¼ k2 k3  g1 dε σ

 dg2 k1 g3  g2 ¼ g2 dε dg4 ¼ k5 ðk4  g4 Þ dε   b dh 8b b b s:h ¼k b s h dε 3

ð12Þ

The two state variables g1 and g2 control the evolution of the yield stress and hardening rate after load reversal. The two additional state variables g3 and g4 were introduced to capture permanent softening. The evolution laws for all these variables produce a distortion of the yield surface and capture the Bauschinger effect.

4. Simple shear results 4.1. Strain homogeneity in simple shear The measured SS deformation is not homogeneous over the whole specimen surface during the test, particularly, near the ends of the shear specimen. Nevertheless, the shear stress is calculated from the SS test as the ratio of the force recorded by the load cell divided by the cross-section area along the whole specimen length. In contrast, the value of average shear strain is determined using digital image correlation (DIC) in the AOI in which the deformation field is more generally uniform. Therefore, in order to check the validity of the stress–strain curve calculated from the described experimental procedure, FE simulations involving different AOIs were performed. Fig. 7(a) is a schematic illustration of a SS specimen (DP980) for FE simulations. The number of FE elements is 7000 for the entire

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149

Fig. 7. A schematic illustration of SS specimen in the FE simulation: shear stress and strain calculated from; (a) whole specimen area to calculate, and (b) a selected AOI with uniform deformation.

Fig. 9. Shear strain distribution measured from FE simulation.

Fig. 8. Shear stress–strain curves measured by the three cases (i-iii) in Fig. 7.

4.2. Experiments

specimen and for the selected AOI 2500 as shown in Fig. 7(b). The macroscopic shear stress and strain were obtained by averaging the corresponding microscopic quantities at all the integration points considered. Three cases were investigated: (i) the shear stress and strain were calculated from the whole specimen, (ii) the shear stress was calculated from the whole region but the shear strain was calculated from the AOI, and (iii) the shear stress and shear strain were calculated from the AOI. Note that the experimental stress–strain curves presented in this study are determined according to Case (ii). As shown in Fig. 8, the stress–strain curves obtained in the three cases are virtually the same indicating that the effect of inhomogeneous deformation on the flow curves is negligible. For clear validation, Fig. 9 shows the strain gradient in the shear specimen obtained from the FE simulation. It shows an inhomogeneous strain distribution, especially, near the edges of the shear specimen as shown in the DIC image of Fig. 5(b). However, except those regions, the other areas of the shear specimen experience a reasonably uniform shear deformation. Therefore, the average shear strain over the elements is nearly the same as the average shear strain over the elements of the AOI.

The stress–strain curves corresponding to the forward–reverse simple shear tests for DP980, TRIP780 and TWIP980 steel sheet samples are shown in Figs. 10–12, respectively. Three different reversal strains were considered for each material and at least triplicate tests were carried out for each condition. Since the reproducibility of the tests was excellent, only one representative curve for each condition is shown in these figures. To facilitate the comparison between the different tests, the reverse shear stress– strain curves were also rotated in the positive quadrant and represented as a function of the accumulated shear strain in Figs. 10(b), 11(b) and 12(b). The measured SS flow curves indicate that the maximum attainable strains are large enough, namely, 21%, 20% and 25% for DP980, TRIP780 and TWIP980 steels, respectively, to provide a suitable range for the analysis of strain hardening. In particular, the maximum shear strain achieved for DP980 using the current SS device is much larger than the 5% uniform elongation measured in uniaxial tension. However, in the monotonic case for DP980, TRIP780 and TWIP980 steels, a decrease of the shear stress is observed at strains of 22.8%, 20% and 26.7%, respectively. This likely happens because of the initiation of a crack resulting from the groove in which the specimen is placed (See Fig. 4). Depending on the material hardening characteristic, the strains at which the shear

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Fig. 10. SS stress–strain curves of DP980 for three prestrains 4.5%, 7.5% and 12%: (a) forward–reverse curves, (b) accumulated forward–reverse curves.

Fig. 11. SS stress–strain curves of TRIP780 for three prestrains 4.4%, 8.7% and 13.8%:(a) forward–reverse curves, (b) accumulated forward–reverse curves.

Fig. 12. SS stress–strain curves of TWIP980 for three prestrains 4.8%, 10.0% and 13.7%: (a) forward–reverse curves, and (b) accumulated forward–reverse curves.

stresses decrease are different. This decrease is a limit of the current groove and a better design must be considered in future work. In spite of these cracks, a Hencky shear strain value of 20% or more can be safely obtained for the AHSS sheets investigated in this study. Moreover, these experimental SS stress–strain curves indicate that the three materials exhibit significant amounts of Bauschinger effect and transient hardening. In other words, premature re-yielding occurs after load reversal, followed by a high hardening rate that saturates after a few percent strain. In addition, a permanent softening, which corresponds to the apparent gap between monotonic and reverse flow curves, is also observed, particularly for large reversal strains. 4.3. FE simulations One of the advantages of the HAH model is that the material coefficients for the initial yield surface, isotropic hardening and distortional hardening can be determined independently. For

example, the anisotropic function (or the so-called stable component in Eq. (8)) is characterized using the results of uniaxial tension tests in different directions and balanced biaxial tests. Then, the isotropic hardening law (or the right hand side of Eq. (8)) is determined from the best approximation of a monotonic flow curve, such as uniaxial tension, by a hardening law. Finally, distortional hardening is controlled by the state variables characterized by the non-proportional loading flow curves, namely, from the forward–reverse SS tests in this study. For isotropic hardening, the Swift hardening law was used, σ ¼ K ðε0 þ ϵÞn

ð13Þ

where K, ε0 and n are coefficients, which were determined from the best approximations of the stress–strain curve obtained either from RD uniaxial tension tests for TRIP780 and TWIP980 steels or from the hydraulic bulge experiment for DP980 steel. The latter test is particularly useful because the corresponding strain range is usually quite large [32], unlike the uniform elongation in uniaxial

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Table 2 Constitutive model parameters for DP980. Swift law Yld2000-2d HAH

K (MPa) 1235 α1 0.914 q 2

ε0 0.0002 α2 1.109 k 30

Table 4 Constitutive model parameters for TWIP980. Swift law

n 0.035 α3 1.052 k1 86

α4 1.033 k2 120

α5 1.016 k3 0.3

α6 0.821 k4 0.90

α7 1.039 k5 10

α8 1.097

Yld2000-2d HAH

K (MPa) 1506 α1 0.908 q 2

ε0 0.0021 α2 1.046 k 30

n 0.23 α3 0.623 k1 110

α4 0.950 k2 40

Yld2000-2d HAH

K (MPa) 2398 α1 0.911 q 2

ε0 0.12 α2 0.891 k 30

n 0.685 α3 0.914 k1 110

α4 0.892 k2 40

α5 0.961 k3 0.3

α6 0.649 k4 0.79

α7 0.930 k5 8

α8 1.083

hardening behavior and permanent softening, are accurately described by the HAH material model unlike the isotropic hardening model.

Table 3 Constitutive model parameters for TRIP780. Swift law

151

α5 0.958 k3 0.35

α6 0.721 k4 0.96

α7 0.955 k5 10

α8 1.283

tension of the DP980 steel, which is less than 5% and does not provide reliable hardening data for use in FE simulations. The three isotropic hardening parameters for each material are listed in Tables 2–4. For the yield function ϕ in the HAH model, the non-quadratic anisotropic yield function Yld2000-2d described in Section 3.1 was used. The corresponding eight anisotropy coefficients were calculated assuming the exponent a ¼6 for all three materials and are listed in Tables 2–4. Although for TWIP steel, the crystal structure of austenite is FCC, a yield function exponent of 6 (not 8) was selected in order to achieve a better description of plastic anisotropy, as recommended in [43]. After swift hardening and Yld2000-2d were determined, distortional hardening coefficients ki were obtained. In order to determine the best coefficients for the distortional hardening, iterative single element FE simulations of a forward reverse shear sequence was conducted. For this, the HAH constitutive model was implemented into the FE software ABAQUS/Standard using the user material subroutine UMAT as described by Lee [33,44]. A 4-node shell element with reduced integration point (S4R) was considered. The detailed boundary conditions for this test were identical at those in Yoon et al. [22]. The resulting optimized coefficients of the state variables for the three materials are listed in Tables 2–4. Note that the parameter k1 is related to the transient hardening rate during load reversal, k2 and k3 to the Bauschinger ratio, and k4 and k5 to permanent softening. Fig. 13 compares the experimental SS stress–strain curves with the HAH predictions for DP980, TRIP780 and TWIP980. For comparison purpose, the hardening curves resulting from the isotropic hardening model (ISO) and the same yield function are also presented. Note that the HAH hardening coefficients of Tables 2–4 were optimized using the forward–reverse SS stress–strain curves for the intermediate pre-strains; i.e., 7.5%, 8.7% and 10% for DP980, TRIP780 and TWIP980, respectively. Then, the flow curves for the other pre-strains were predicted using the determined hardening coefficients. Fig. 13 indicates that the HAH predicted SS stress–strain curves are in good agreement with the experimental results from the monotonic and load reversal tests. Again, note that the reference flow curves for isotropic hardening were those corresponding to equi-biaxial tension for DP980 and uniaxial tension for TRIP780 and TWIP980 steels. This indicates that the anisotropic yield function Yld2000-2d employed in this study capture the anisotropy of the three materials very well. Fig. 13 also shows that the complex hardening features such as the Bauschinger effect, transient

5. Discussion 5.1. Effect of anisotropic yield function For the CT simulations, the yield function may not influence the predicted stress–strain response when the reference hardening curve is based on uniaxial data. However, for SS simulations, the yield function may play an important role in the predicted stress– strain response. In other words, when the reference flow curve is obtained from uniaxial or biaxial tension test, the shear stress response depends on the employed yield function. In this study, three different yield functions were considered to investigate this effect; i.e., von Mises, Hill 48 [45] and Yld2000-2d. The coefficients of the Hill 48 model are listed in Table 5. Note that the HAH model coefficients were obtained from the results of the CT tests and are listed in Table 6. Fig. 14 shows the effect of the yield function on the SS curves for the three materials. In Fig. 14(a), Yld2000-2d predicts the lowest SS flow curve compared to the other two yield functions for DP980. The SS to uniaxial tension yield stress ratios are 0.537, 0.577, and 0.581 for the Yld2000-2d, von Mises and Hill 48, respectively. For TRIP780, the yield stress ratios for Yld2000-2d and von Mises are very similar; i.e., 0.572 and 0.577 and the overall simple shear flow stresses are also similar as shown in Fig. 14(b). In contrast, Hill 48 model predicted a higher ratio of 0.614. For TWIP980, the yield stress ratios for Yld2000-2d and Hill 48 have similar values of 0.596 and 0.600, respectively, but that for von Mises has a lower value of 0.577. The above simulations demonstrate that the predicted SS flow stress curves are highly dependent on the yield function if the reference flow curve is not simple shear. In addition, the more advanced anisotropic yield function Yld2000-2d leads to a better prediction of the flow curve in SS, as Figs. 13 and 14 indicate. 5.2. Comparison of constitutive parameters determined by SS and CT tests For the simulations of springback, it is important to capture the load reversal effect on strain hardening which, during sheet metal forming processes, occurs after bending and unbending at die and punch radii. For this reason, material tests accompanying load changes are important and the uniaxial tension–compression test has been frequently adopted [33]. In this section, the forward– reverse stress–strain curves determined by the two test methods, CT and shear reversal, are compared and the suitability of the SS test to provide the basic mechanical properties needed for springback predictions are assessed. For this purpose, the material parameters of the HAH model, which are listed in Table 6, were first identified based on the measured CT stress–strain curves. All distortional hardening coefficients were determined as same method as those described in

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Fig. 13. Comparison between HAH model and experimental result of forward–reverse simple shear loading curves: (a) DP980, (b) TRIP780, and (c) TWIP980.

Table 5 Hill 1948 coefficients.

DP980 TRIP780 TWIP980

F

G

H

N

0.448 0.460 0.385

0.506 0.509 0.587

0.425 0.491 0.413

1.480 1.326 1.388

Table 6 Constitutive model parameters identified from CT and SS. DP980

CT

TRIP780

CT

TWIP980

CT

k1 160 k1 150 k1 110

k3 0.35 k3 0.35 k3 0.40

k4 0.90 k4 0.78 k4 0.86

SS SS SS

k1 86 k1 110 k1 110

k3 0.30 k3 0.35 k3 0.30

k4 0.90 k4 0.96 k4 0.79

*

k2, k5 are the same for two different experiments, which are shown in Tables 2–4.

simple shear. The boundary conditions for this test were followed at those in Lee et al. [46]. Fig. 15 compares the experimental CT stress–strain curves with those predicted with the HAH model based on the CT parameters. For the purpose of comparison with a simpler model, the curves computed with the isotropic hardening model are also represented in these figures. In general, the predicted CT stress–strain curves reproduce the experimental results very well for the three materials. For TRIP780, the calculated flow curves during monotonic compression exhibit some difference with the experimental results. This is likely because the isotropic hardening coefficients were determined from the uniaxial tension stress–strain curve, which is different from compression. This asymmetric response between tension and compression is consistent with the difference in martensitic transformation kinetics observed under tension and compression [47]. The best set of optimized HAH coefficients obtained from the two test methods are compared in Table 6. Note that parameters k2

and k5 are the same. The three parameters k1, k3, and k4 control the transient hardening rate during load reversal (k1), Bauschinger ratio (k3) and permanent softening (k4). The values of k1 determined from the CT tests are larger than those from SS for the DP and TRIP steels, which means that the transient hardening rate during CT is higher than that of SS. In contrast, the values of k3 are similar, irrespective of the test method, which means that the Bauschinger ratios are similar for the three materials. For TRIP780, the value of k4 determined from the CT test data is smaller than that from SS, which is associated to a larger permanent softening. In Fig. 16, the SS stress–strain curves predicted using the HAH model with the CT-based coefficients, are compared to those from the SS experiments. In general, the predicted and experimental curves are in reasonable agreement. The calculated reverse shear flow curves of TRIP780 exhibit more deviation from the experiments compared to the other materials which is likely, as mentioned previously, related to different phase transformation kinetics in tension and compression for this material. For a more quantitative analysis, the deviation (DEV) between calculated and experimental flow stresses is defined as follows Pn DEV ð%Þ ¼

i¼1

mo YðXex i Þ  YðXi Þ YðXex i Þ

n1

 100ð%Þ

ð14Þ

mo In Eq. (14), YðXex i Þ and YðXi Þ are the experimental and HAHpredicted stress values corresponding to each strain, respectively. The DEV of each reverse flow curve for a predefined pre-strain was calculated using 20 equally spaced (n¼ 20) strain values from the load reversal. Table 7 shows that the overall relative errors are within 10%, which means that the two sets of HAH coefficients determined using the CT and SS test data are consistent. In other words, the developed SS testing procedure for forward and reverse loading can be viewed as an alternate method to estimate the hardening behavior in tension–compression (or compression–tension). This might be particularly interesting for the recently developed AHSS, which have very limited uniform elongation in uniaxial tension. However, the discrepancy observed for the TRIP steel requires

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Fig. 14. Effect of initial yield function on the predicted SS flow curves:(a) DP980, (b) TRIP780, and (c) TWIP980.

Fig. 15. Predicted CT flow curves using the HAH model:(a) DP980, (b) TRIP780, and (c) TWIP980.

additional investigations to assess the effects of transformation kinetics on the evolution of anisotropy during deformation.

6. Application to springback simulation Finally, springback simulations were conducted using the HAH model with the two sets of coefficients based on CT and

SS data (Table 6) in order to assess whether a similar deformation and springback behavior is obtained in forming simulations. The 2D draw-bending springback model proposed in the Numisheet'93 benchmark [48] was considered in Fig. 17. All the numerical conditions such as element type, number of elements, integration scheme, boundary conditions, etc., are the same as in the benchmark problem excepted for the HAH constitutive model.

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Fig. 16. Comparison of HAH predicted SS curves with CT parameters with experimental SS curves: (a) DP980, (b) TRIP780, and (c) TWIP980.

SS test HAH (CT) CT test HAH (SS)

Pre-strain (%) DEV (%) Pre-strain (%) DEV (%)

4.5 4.87 1 6.03

7.5 8.1 2 8.20

12 6.17 3 7.5

TRIP780

SS test HAH (CT) CT test HAH (SS)

Pre-strain (%) DEV (%) Pre-strain (%) DEV (%)

4.5 10.51 7 11.25

8.7 9.30 9.5 12.51

13.8 10.24

and TWIP980 steels. In particular, the profiles of the sidewall regions are almost identical for the two cases. In contrast, a larger difference was observed for TRIP780 steel, which is likely the result of the difference between tension and compression flow stresses. This analysis suggests that the SS tests can be used as an alternate method to describe the Bauschinger effect and provide the coefficients of an advanced constitutive model necessary for the prediction of springback of AHSS sheets.

TWIP980

SS test HAH (CT) CT test HAH (SS)

Pre-strain (%) DEV (%) Pre-strain (%) DEV (%)

5 2.13 4.7 1.60

10 4.82 9.5 3.45

13.7 3.78

7. Conclusions

Table 7 A standard deviation (DEV) in stress between predicted and experimental data.

DP980

Fig. 17. 2D draw bending springback model in Numisheet'93 benchmark.

Fig. 18 shows the predicted springback profiles for the three materials. As expected, the isotropic hardening model leads to the highest springback because of the large elastic recovery in the absence of the Bauschinger effect. The springback profiles predicted using the CT- and SS-based coefficients are very similar for DP980

A simple shear (SS) test device was improved to determine the flow curves of advanced high strength steel (AHSS) sheets under monotonic and reverse loading. Three AHSS sheet samples were considered, namely, DP980, TRIP780 and TWIP980. From the measured SS flow curves with load reversals, a recently proposed distortional anisotropic hardening model, so-called HAH, was identified. Comparisons of predicted flow curves and springback results using the SS-based and CT-based constitutive coefficients validate the proposed SS test method. Further specific conclusions of the present work follow, namely, 1) Complex hardening behaviors after load reversal such as the Bauschinger effect, transient hardening and permanent softening were successfully captured with the SS tests for the three AHSS sheet samples investigated. 2) The SS stress–strain curves in forward and reverse loading were satisfactorily captured by the HAH distortional hardening model. 3) The accuracy of the model in predicting the SS stress–strain curves from uniaxial or balanced biaxial test results is highly dependent on the yield function embedded in the HAH approach, namely, von-Mises, Hill 48 and Yld2000-2d in this work. The latter provided the best predicted flow curves.

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Fig. 18. Comparison of 2D draw bending springback profiles predicted by HAH model based on SS and CT parameter: (a) DP980, (b) TRIP780, and (c) TWIP980

4) For TRIP780 steel, the HAH distortional hardening model slightly underestimated the monotonic SS flow stress curve when the isotropic hardening component of the model was determined using uniaxial tension data. This effect is likely due to the difference in phase transformation kinetics between simple shear and tension, as suggested in the literature. 5) The predicted flow curves based on the SS coefficients were in reasonably good agreement with those based on CT and vice versa. This validates the SS test method in providing suitable material coefficients for a constitutive model that captures anisotropic hardening effects. 6) The 2D draw-bending springback profiles predicted using FE simulations with two different sets (SS and CT) of HAH coefficients were very similar, particularly for DP and TWIP steels. However, the differences were larger for TRIP steel, which is likely due to different phase transformation kinetics between tension and simple shear. 7) The modified SS test device can produce reliable data for very high strength steels larger than 1000 MPa. In the future, this device will be used effectively for strength higher than 1200 MPa but the groove design must be slightly modified to prevent the initiation of cracking. 8) In summary, the SS test method is a good alternative to the compression–tension (or tension–compression) test of AHSS for providing suitable material coefficients in advanced hardening model.

Acknowledgments Authors appreciate the support by POSCO and by the National Research Foundation of Korea (NRF) Grant funded by the Korea government (MSIP) (No. 2012R1A5A1048294) and (No. 2014R1A2A1A11052889). FB appreciates the support of FEDER funds through the Operational Program for Competitiveness Factors - COMPETE and National Funds through the Portuguese FCT - Foundation for Science and Technology under the projects PTDC/EMS-TEC/2404/2012. References [1] Wagoner RH, Lim H, Lee M-G. Advanced issues in springback. Int J Plast 2012.

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