Implementation and application of a temperature ...

International Journal of Plasticity 75 (2015) 121e140

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International Journal of Plasticity journal homepage: www.elsevier.com/locate/ijplas

Implementation and application of a temperature-dependent Chaboche model C. Zhou a, Z. Chen b, J.W. Lee c, d, M.G. Lee e, **, R.H. Wagoner b, * a

School of Mechanical & Automotive Engineering, 381 Wushan Road, South China University of Technology, 510641, China Department of Materials Science and Engineering, 2041 College Road, Ohio State University, Columbus, OH 43210, USA c Graduate Institute of Ferrous Technology, Pohang University of Science and Technology, Pohang 790-784, Republic of Korea d Materials Deformation Department, Korea Institute of Materials Science, Changwon 642-831, Republic of Korea e Department of Materials Science and Engineering, Korea University, Seoul 136-713, Republic of Korea b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 11 August 2014 Received in revised form 3 February 2015 Available online 13 April 2015

The literature shows that shear fracture of advanced high strength steels (AHSS) is affected by strain hardening at large strain, as well as the temperature dependence of flow stress and strain hardening. The role of non-isotropic hardening, such as would be expected to be important in reverse strain paths as encountered during draw-bend testing or drawing sheet metal into forming dies, has been difficult to assess without a practical constitutive model combining temperature-dependence and non-isotropic hardening capabilities. Such a model has been developed and implemented in Abaqus Standard via the UMAT subroutine. In order to apply and test the constitutive implementation, the material model was fit using alternate parameter-identification procedures starting from compression-tension (CT) data: 1) fit directly from reverse-path, CT data, and 2) fit indirectly, by combining the direct CT data plus extrapolated data at larger strains where the extrapolation uses verified large-strain monotonic hardening character. The resulting material models were used to simulate draw-bend fracture (DBF) tests for six AHSS. The results show that the indirect method improves the predictions of shear fracture significantly, allowing accurate predictions. It was also shown that the influence of non-isotropic hardening aspects are not critical to accurate predictions as long as the high-strain strain hardening is reproduced accurately. These results suggest a practical and effective method for extending measured tensile hardening to otherwise unattainable strains based on the constant ratio (a in the H/V model) of power-law and saturation-stress strain hardening at a given temperature. The success of this approach suggests that a is a material constant (describing the fundamental strain-hardening character) that depends on temperature but is unaffected by the details of transient hardening following abrupt path changes. Furthermore, the essentially transient nature of hardening following path changes is supported. © 2015 Elsevier Ltd. All rights reserved.

Keywords: Constitutive behaviour Fracture Thermomechanical processes Numerical algorithms Advanced High Strength Steel (AHSS)

1. Introduction Advanced high strength steels (AHSS) are increasingly being used in automotive body and structural parts because they offer opportunities for reducing vehicle weight while improving safety performance (Kuziak et al., 2008). However, * Corresponding author. Tel.: þ1 614 292 2079. ** Corresponding author. Tel.: þ 82 010 7108 8736. E-mail addresses: [email protected] (M.G. Lee), [email protected] (R.H. Wagoner). http://dx.doi.org/10.1016/j.ijplas.2015.03.002 0749-6419/© 2015 Elsevier Ltd. All rights reserved.

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C. Zhou et al. / International Journal of Plasticity 75 (2015) 121e140

springback and bending-affected formability remain significant impediments to their adoption (Walp et al., 2006; Chen et al., 2008; Wagoner et al., 2009a, 2009b). Shear fracture is a bending-dominated failure mode that has proven difficult to predict using standard industrial finite element procedures, testing methods, and forming limit diagrams (FLD) (Sriram et al., 2003; Stoughton et al., 2006; Huang et al., 2008; Sung et al., 2012). Shear fractures were initially reported to occur with little or no obvious thinning near the fracture (Wagoner, 2006).1 While fracture with no local thinning implies brittle behavior, the recent scientific literature, as outlined below, finds little support for that initial observation. Shear fracture is reproduced in the laboratory by reproducing the characteristic drawing, stretching, and bending over a tight radius. Several experimental techniques have been used to accomplish this (Damborg et al., 1997, 1998; Damborg, 1998; Hudgins et al., 2010; Walp et al., 2006; Luo and Wierzbicki, 2010). More recently, a specialized technique using controlled displacement rates on both sides of the bending region was introduced. Called the draw-bend fracture (“DBF”) test, it has been applied to shear fracture of AHSS (Wagoner et al., 2009a, 2009b; Kim et al., 2011; Sung et al., 2012). Such techniques have been shown to produce fracture at the maximum draw load (Yoshida et al., 2005; Hudgins et al., 2010), within 1 mm of additional draw distance (Kim et al., 2011). Using DBF tests, the roles of temperature on hardening (Kim et al., 2011; Sung et al., 2012) and the strain hardening at large strains (Lee et al., 2013) were shown to be dominant material factors affecting shear fracture, formability, as well as the process variable friction coefficient (Kim et al., 2014). Because the maximum strain rate is approximately 10/s in industrial sheet metal forming operations (Sung et al., 2012), deformation-induced heating for AHSS leads to a significant temperature rise, 100e120 C, in the bending region, thus promoting strain localization there (Kim et al., 2011; Sung et al., 2012). A new combined Hollomon (1945) and Voce (1948) constitutive form for temperature-dependent strain hardening, or “H/ V model”, was shown to reproduce the large-strain hardening behavior and the temperature sensitivity of many AHSS (Sung et al., 2009, 2010). Recently, H/V models were shown to predict strain hardening accurately at strains up to 6 times larger than tensile measurements (Smith et al., 2014). When applied to shear fracture via DBF test simulations, the formability was predicted much more accurately than with standard isothermal constitutive models based on Hollomon or Voce forms (Kim et al., 2011). In spite of the proven accuracy of H/V laws, H/V-based simulations of the DBF test still over-predicted the formability of AHSS in many cases, although such comparisons were much improved over isothermal simulations, with average errors of 2e8% vs. 26e48%, respectively (Kim et al., 2011). One obvious possibility for the remaining source of the discrepancy between simulated and measured DBF formability lies with the reverse-path hardening behavior of AHSS, which differs significantly from isotropic hardening for AHSS (Sun, 2011; Piao et al., 2012a; Sun and Wagoner, 2013). The reduction of flow stress after strain reversals would be expected to promote early localization and fracture by reducing the maximum draw stress and thus causing early instability. Such stress reversals are present for many elements of material that are drawn over the die radius. Unfortunately, it is not a simple matter to test the role of non-isotropic hardening (i.e. reproducing the reverse-path results) for DBF applications because of the known importance of small temperature excursions (typically up to 120 C) that occur because of deformation-induced heating at typical industrial strain rates. Thus, a temperature-dependent version of non-isotropic hardening is required. AHSS typically exhibit strong effects of stress reversals, including three characteristics: the Bauschinger effect (i.e. reduced yield strength compared with previous flow stress), transient behavior immediately after re-yielding (rapid strain hardening, typically for strains of ~0.03 after the path reversal), and long-term or “permanent” softening (sometimes seen over the entire strain range available from experiments, but always diminishing at larger strains) (Geng, 2000; Geng and Wagoner, 2002; Sun, 2011; Sun and Wagoner, 2013). The third of these, if present, was expected to be most important for shear fracture, because shear fracture is sensitive to flow stress and hardening at strains of approximately 0.5 (Lee et al., 2013; Luo and Wierzbicki, 2010). In addition, some steels show “overshoot”, whereby the flow stress temporarily exceeds the expected isotropic-hardening flow stress. This effect is not considered in the current work. Various plastic hardening models have been proposed to reproduce the three reverse hardening characteristics outlined above (Geng and Wagoner, 2002; Chun et al., 2002a, 2002b; Yoshida and Uemori, 2002, 2003; Chung et al., 2005; Lee et al., 2007; Sun, 2011; Sun and Wagoner, 2013). The role of constitutive complexities in the accuracy of springback simulations has been reviewed (Wagoner, 2004; Wagoner et al., 2006, 2013). The standard Chaboche model implemented with multiple nonlinear back-stress components can adequately describe the first two effects, i.e. Bauschinger effect and the transient behavior with a saturation towards the monotonic hardening curve (Chaboche, 1986, 1989; Ohno and Wang, 1993; AbdelKarim and Ohno, 2000). However, a linear term can also be added to a standard Chaboche model in order to reproduce the permanent softening (Chung et al., 2005; Lee et al., 2007; Sun, 2011; Sun and Wagoner, 2013). In particular, Sun and Wagoner (2011, 2013) showed that the reverse-path hardening of AHSS can be reproduced accurately utilizing a modified Chaboche-type model with 2 nonlinear terms and 1 linear term. In principle, the challenge for testing the hypothesis that non-isotropic hardening is needed for predicting shear fracture is to apply an accurate temperature-dependent isotropic hardening model in conjunction with a treatment of non-isotropic hardening. Such formulations have been presented (Chaboche, 1986, 2008; Ohno et al., 1989; McDowell, 1992; Mücke and

1 For those interested in seeing practical examples of shear fracture, including photos of fractured automotive parts, the (Wagoner, 2006) reference may be consulted.


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Bernhardi, 2006; Berisha et al., 2010), typically fit to isothermal tensile tests at various temperatures. However, such formulations do not follow what is now known to be the essential H/V form of strain hardening at large strain. Because the proper form of large-strain strain hardening is so critical to shear fracture and it is difficult to measure directly the large-strain strain-stress curves after reverse loading, another approach is needed. Developing and testing such a method is the object of this paper. In the current work, three constitutive models are compared for several AHSS; for DP780 a fourth model was also employed: Model 0: This model is similar to Sung et al. (2012) and Kim et al. (2011), adapted for a broader and different set of AHSS alloys. Such models have been shown to reproduce the features of shear fracture but have generally overestimated the formability significantly. Model 0 is isotropic (von Mises yield) and has isotropic hardening with strain, strain-rate and temperature dimensions. The uniaxial strain hardening parameters are taken directly from Smith et al. (2014), based on proportional-path tensile tests. For five materials tested the strain hardening form follows the H/V model introduced of Sung et al. (2010); for one material a Swift form is adopted. Model 0 is similar to that used by Kim et al. (2011) to analyze DBF tests for shear fracture. Model 1: The same as Model 0 except with anisotropic plastic yield according to the Barlat'91 (Barlat et al., 1991) yield function. The Barlat'91 parameters were only measured for DP 780, so Model 1 was only for this alloy. Nevertheless, in view of the large improvement of using the Barlat'91 yield surface, it was applied to Models 2 and 3 for all AHSS using the parameters measured for DP 780 even though this will unavoidably introduce some approximation. Model 2: Chaboche Model e A special Chaboche model (Chaboche, 1986) with 1 linear and 1 nonlinear term, and modified for temperature dependence, fit directly to tension tests and compression-tension (CT) tests. Barlat'91 yield is used, with parameters determined for DP780. During fitting, this model was not constrained to extrapolate to the known large-strain stress-strain curves under proportional loading (from Models 0 and 1). Model 3: H/V-Chaboche e A distinct Chaboche-type model consisting of Model 1 (isotropic hardening model) with two nonlinear Chaboche terms, modified for temperature dependence, fit to CT data with a special fitting procedure to produce large-strain, proportional-path hardening according to Model 1 and reverse-path transient behavior according to the Chaboche framework. Barlat'91 yield is used, with parameters determined for DP780. Model 0 has been used for past simulations; it is particularized by the developments of Smith et al., 2014. Model 1 is identical to Model 0 except for the yield surface change. Models 2 and 3 are derived in the next section of this paper; they differ in form and in the fitting procedures used to particularize them from CT data while using common proportional-path data (Smith et al., 2014). All four models, not being normally available in commercial FE programs, were constructed and implemented in Abaqus Standard version 6.10 via UMAT. They were used to simulate DBF experiments of 6 AHSS representing a wide range of mechanical forming behavior. The results for the three models were compared with DBF experiments to assess the accuracy for application to shear fracture prediction. 2. Models 0 and 1: isotropic hardening Models 0 and 1 use thermo-visco-plastic material constitutive models similar to that introduced by Sung et al. (2010) and used for DBF thermo-mechanical simulations by Kim et al. (2011). However, two changes were made to the Sung-Kim models to improve their fidelity for new alloys based on additional testing. The thermo-visco-plastic parameters for Models 0 and 1 are taken directly from Tables 2 and 3 of Smith et al. (2014), which are not reproduced here in the interest of brevity. The resulting hardening forms have been verified at strains several times larger than the tensile ones used to generate them by balanced biaxial bulge testing (Smith et al., 2014; Lee et al., 2013) at room and elevated temperatures (corresponding to temperatures generated by deformation at high rates in the DBF test). The bulge experiments were also conducted in some cases after pre-strains to verify the similar high-strain behavior after a strain path change approximating draw-bend conditions. The second modification, for Model 1, adopts the Barlat Yield'91 yield function (Barlat et al., 1991). For the current work, a small-strain, in-plane balanced biaxial tests (Lee et al., 2013) and standard tensile tests were used to determine the parameters for DP780, as presented in Table 1. Because the Barlat'91 parameters were only available for one of the current alloys, Model 1 only exists for DP780. Comparing Models 0 and 1 for DP 780 isolates the change of yield function from the other constitutive effects. As will be seen later, the yield surface shape is important for DBF formability but not for plane-strain shear fractures as encountered in industry. The details of the procedure for determining the Barlat parameters are available in Lee et al. (2013). 3. Chaboche model with temperature dimension In an isothermal context, a combined kinematic and isotropic model can be described as follows:

f ¼ ks XkH R sy 0

(1)

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Table 1 Mechanical properties of AHSS derived from tensile testing at Posco Steel (Posco). (Friction coefficients from testing at OSU.) Material

t (mm)

YS (Mpa)

UTS (Mpa)

eu (%)

et (%)

n

rRD

r45

rTD

r

rp

m

DP 590 DP 780 DP 980 TRIP 780 TWIP 980 CP 1180

1.42 1.20 1.21 1.42 1.40 1.20

372 527 800 489 543 933

606 830 1030 804 977 1199

15.3 13 4.8 20.4 49.2 4.8

25.7 20.2 10.3 27.1 51 9.1

0.16 0.14 0.06 0.24 0.34 0.06

1.18 0.79 0.78 0.86 0.67 0.82

0.93 1.00 1.04 0.86 1.01 0.92

1.28 0.94 0.87 1.12 1.25 0.85

1.08 0.93 0.93 0.92 0.99 0.88

0.29 0.14 0.22 0.14 0.05 0.09

0.05 0.07 0.03 0.06 0.1 0.1

Key:

sy e 0.2% offset yield strength. sUTS e ultimate tensile strength. eu e uniform elongation (engineering strain at maximum tensile load). et e total elongation (engineering strain at fracture). n e strain hardening index. r e plastic anisotropy ratio1. r e normal R-value. rp e planar R-value. m e Coloumb friction coefficient, Teflon-metal in tensile testing (measured at OSU). Notes: Samples tested had a gage length of 50 mm and a 1% taper. Strain rate ¼ 103/s before material yielding, 0.008/s after material yielding. Strain hardening index calculated at an engineering strain of 4% for DP 980 and CP 1180 and at a strain of 5% for all other steels. r-value calculated over the strain range from 4% to the uniform elongation for DP 980 and CP 1180 and over the strain range from 6% to the uniform elongation for all other steels (Lee, 2011). All properties are for RD only, except R-values.

Table 2 Anisotropy coefficients for the Barlat'91 model for DP780 and 5 other alloys. Barlat'91

a

a

6

1.04

c

b 1.04

g

f 1.05

0.96

h 0.99

1.05

Table 3 Model 2 Chaboche coefficients for DP 780 fitted to tensile and 4% and 8% pre compression CT test data with one linear and one non-linear backstress components, according to Step 1 for Model 2.

DP 780

Temp. ( C)

C1

r1

C2

R0 (MPa)

Rs (MPa)

b

Std. Error (MPa)

Std. Error (%)

25 50 75 100

18043 20856 23581 27704

107 126 136 153

415 491 589 578

491 443 415 370

181 175 150 161

7 13 15 17

17.8 13.8 15.2 16.9

2.2 1.7 2 2.2

Key: Relative Std. Error eStd. Error/average stress.

where s is the stress tensor, sy is the initial yield stress, X is the backstress to denote the translation of yield surface and R is the size of the yield surface, which expands uniformly by isotropic hardening. Isotropic or anisotropic yield criteria can be used. In the current work, the yield surface was defined using Hill's 1948 anisotropic yield criterion (Hill, 1950) as

ks XkH ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðs XÞ : H : ðs XÞ

(2)

where H is a fourth rank tensor related to R-Values. The back stress vector X can be composed of m components Xi such that each term evolves according to its own law (Chaboche, 2008):

X¼

m X

Xi

(3)

i¼1

In the original Chaboche (1986) model, each component evolves as proposed by Armstrong and Frederick (1966) 2:

dXi ¼ ci dεp gi Xi dp

(4)

2 Note that the original equation, Eq. (23) presented in Chaboche 2008, contained a factor of 2/3 in the term containing C that has not been retained here. This difference represents a simple change of the magnitude of C to represent the same material behavior.


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where dεp is the plastic strain increment, dp is the increment of equivalent plastic strain, a scalar defined according to the form of the yield function, and ci and gi are constants. The first term on the right of Eq. (4) was proposed by Prager (1949) to introduce linear kinematic hardening. The second term, a “return” term, introduces non-linear evolution such that the flow stress for strains following a non-proportional path change asymptotically approaches the isotropic-hardening value. By superimposing several backstress components via Eq. (4), the Chaboche model can describe the Bauschinger effect and transient behavior near re-yielding and at larger strains, but not a permanent stress offset. In order to add a linear backstress component, i.e. one that allows for permanent softening (Chaboche, 1986; Sun, 2011), the return (second) term in Eq. (4) is deleted from one of the backstress components, say the ith one (i.e. setting gi to zero):

dXi ¼ ci dεp

(5)

The Chaboche model uses a Voce-type saturation function for the isotropic part of the hardening response:

dR ¼ bðRs RÞdp

(6)

where R and Rs are the current and limiting sizes of the yield function respectively, and b is a material constant. The size of yield surface R increases with the equivalent plastic strain and saturates at Rs. The coefficients within the Chaboche model can be determined by tensile and compression-tension (CT) tests. For uniaxial loading, two equations for fitting Chaboche coefficients can be obtained by integrating Equations (3) and (4):

X¼

3 c 3 c v þ X0 v egðpp0 Þ 2 g 2 g

(7)

where v ¼ ±1 gives the direction of flow, p is the equivalent plastic strain, p0 and X0 are the pre-strain and initial reverse flow stress. The isotropic hardening component is represented as follows,

R ¼ R0 þ Rs 1 ebp

(8)

where Ro is yield stress and (Ro þ Rs) is the saturation stress at large strain. Temperature can affect the Chaboche model in two ways: the size of the yield surface (isotropic hardening) and the evolution of backstress (non-isotropic hardening) (Chaboche, 2008). The evolution of the size and position of yield surface with temperature T are described as

dR ¼ bðTÞ½Rs ðTÞ Rdp dXi ¼ ci ðTÞdεp gi ðTÞXi dp þ

(9) 1 vci ðTÞ Xi dT ci ðTÞ vT

(10)

Compared with Equation (4), a temperature-dependent backstress evolution form is introduced in Equation (10) to obtain a Chaboche form independent of temperature-driven microstructural evolution (Chaboche, 1993; Ohno and Wang, 1993; McDowell, 1992). The parameters for isotropic hardening and kinematic hardening thus can be defined as linear functions of temperature in Equations (9) and (10). A UMAT user subroutine implementing the Chaboche temperature-dependent model was constructed within the commercial finite element code ABAQUS Standard 6.10 and used for shear fracture prediction as described in Section 5. 4. Procedures for fitting the Chaboche temperature-dependent model with and without H/V-type hardening While retaining the basic constitutive structure described above, the required coefficients were found using two distinct procedures. One uses the experimental data to fit the parameters introduced above. The other method takes into account, and enforces, the H/V Model accuracy at large strain using a fitting procedure to be described. The two procedures use the same data: 3 uniaxial loading tests at each of 4 test temperatures, plus 1 additional tensile test to determine friction coefficient.3 The 3 tests used for mechanical measurement at each temperature consist of 1 monotonic

3 Note that this friction coefficient applies only to the CT test conditions and is determined from CT testing itself, and has a small effect on the final stressstrain results. It does not apply to the DBF testing, which involves markedly different process conditions. The determination of DBF friction coefficient, which is important for the results in this paper, is accomplished differently, under DBF-like conditions.

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tensile test, and 2 compression-tension (CT) reverse tests with the strain rate reversals at strains of 0.04 and 0.08. In all cases, data was used up to the maximum engineering stress to ensure accurate conversion to true stress-true strain. To be clear, the determination of Chaboche parameters and related thermal variations of them is done by minimizing the standard error of fit to three sets of data simultaneously: a) monotonic tensile test, b) CT test subsequent to the path reversal at 0.04 strain,4 and c) CT test subsequent to the path reversal at 0.08 strain.3 The procedures are illustrated using detailed example results for one material, DP 780. Final summary results for the other alloys will be presented in the Results section. 4.1. Materials Six AHSS representing a broad class of alloys of current interest for automotive applications were acquired from Posco Steel (Posco). The materials and their standard mechanical properties are shown in Table 1. “DP” refers to a dual-phase (ferrite-martensite) steel, “TRIP” refers to a transformation-induced plasticity steel, “CP” refers to a complex phase steel, and “TWIP” refers to a twinning-induced plasticity steel. The TWIP steel was part of a special experimental lot provided for this research. In all cases the numbers in the labels refer to the nominal ultimate tensile strength (UTS) of that grade. These AHSS have nominal ultimate tensile strengths ranging from 590 to 1180 MPa, a factor of 2, and tensile elongations (engineering strains to fracture) ranging from 0.05 to 0.51, a factor of 10. Accurate 1-D (tensile) constitutive equations incorporating strain, strain rate, and temperature have been developed for these alloys based on proportional, uniaxial loading (Smith et al., 2014). These forms were confirmed at the critical high strain range by balanced-biaxial bulge testing (Smith et al., 2014, Lee et al., 2013). For all materials except TWIP, H/V-type temperature-dependent strain hardening forms were deemed to be optimal. For TWIP, a temperature-dependent Swift law was constructed. 4.2. Tensile and compression-tension tests In order to obtain the data for determining the required Chaboche parameters and friction coefficients, 2 tensile tests (with and without side plate support) and 2 CT tests were performed for DP 590, DP 780, DP 980, and TWIP 980, 0.04 and 0.08 pre-strain were used. Smaller pre-strains of 0.01 and 0.02 were used for CP 1180 and DP 980 because they buckle at lower compressive strains.5 In DBF tests of DP and TRIP steels (Sung et al., 2012), the maximum temperature measured at the onset of necking was 120 C under severe testing conditions (strain rates up to 10/s, minimum relative bending radius R/t of 2.2, unlubricated). Therefore, an approximately commensurate temperature range (room temperature up to 100 C) was chosen for testing. Testing was done using an MTS 810 tensile frame at room temperature, 50 C, 75 C, and 100 C. The tensile tests utilized 2% tapered, ASTM E8-08 (ASTM-E8/E8M-08, 2008) specimens oriented in the rolling direction of the material. The CT tests utilized a special parallel specimen with wide shoulders optimized to extend the compression range attainable before buckling (Boger et al., 2005; Piao et al., 2012a, 2012b). The CT testing procedures, including compensation for biaxiality and friction have been described in the literature (Piao et al., 2012a). Aluminum heating plates with Teflon sheets bonded to them were electronically-controlled for temperature and were used to obtain quasi-isothermal conditions6 in all tests as well as to constrain buckling in compression. Friction coefficients for such testing were established by comparing the room temperature tensile tests with and without side plate support (a side force of 4.4 kN was applied in all testing with side plates) and then choosing a friction coefficient (plus or minus 0.01) that minimized the difference in stress for the strain range 0.02 to εu (uniform elongation). The friction coefficients obtained in this manner are shown in Table 1.7 The test-to-test stress scatter of the duplicate CT tests (48 pairs of CT tests comprising 6 materials, 4 temperatures, and 2 prestrains) was 26 MPa. This scatter is computed as the average standard deviation between the two tests in each pair based on the tensile portion from the strain at the stress reversal (i.e. zero stress) to the maximum engineering stress (uniform strain, εu). If instead the first 0.01 of strain following the reversal is excluded, the average test-to-test scatter for the strain range 0.01to εu was reduced to 15 MPa.

4

For the tested alloys with limited tensile ductility, DP980 and CP1180, strain reversals were alternately set at 0.01 and 0.02. In fact, two duplicate tests of each kind were performed but in all cases the results were insignificantly different, so one was selected randomly for further analysis for each set of conditions. 6 The presence of the heat-conducting aluminum side plates reduces deformation-induced heating during tensile testing. While we refer to “isothermal tensile tests” in the current work, they are of course not perfectly isothermal at finite strain rate. Simulations of the tensile tests were performed both isothermally and thermo-mechanically with FE models and thermal constants presented in Piao et al., 2012. For one of the test materials, TRIP 780, which is highly temperature-sensitive, the simulated stress-strain curves for the two conditions differed by more than the experimental uncertainty. For the others, the differences from isothermality were insignificant. 7 Note that the friction coefficients appearing in Table 1 are for CT testing conditions only. Alternate values corresponding to draw-bend conditions are presented later. 5


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4.3. Model 2: Chaboche model fit directly Model 2 is the modified Chaboche model using one nonlinear and one linear evolution equation for the backstress (Sun, 2011) fit directly from the tension and CT data. An isothermal version of this model was shown effective for the simulation of draw-bend springback (DBS) tests.8 The back stress is thus composed of two components:

dX1 ¼ c1 dεp g1 X1 dp

(11)

dX2 ¼ c2 dεp

(12)

where the coefficients to be determined are c1, c2, and g1 plus the isotropic hardening coefficients b, Ro, and Rs from Equations (7) and (8). Note that in this model, the isotropic hardening component saturates to a stress Rs while the reverse hardening component increases without bound via c2. The nonlinear least-squares fitting method lsqnonlin in Matlab 7.0 (MathWorks, 2010) was used for data-fitting of Model 2 and Model 3. The fitting procedure is as follows: Step 1: Using Equations (7), (8), (11) and (12), fit all of the Chaboche coefficients except the initial yield stress, Ro, individually for temperatures 25 C, 50 C, 75 C and 100 C. The initial yield stresses Ro at each temperature were constrained to be the experimental 0.2% yield offset values as shown in Table 3. Step 2: Determine the form of a satisfactory relationship between the Chaboche coefficients and temperature by plotting and inspecting. Step 3: Adopting the form of temperature-dependence established in Step 2, optimize a single set of coefficients for all data, including temperature-dependent terms, starting from trial values averaged from the individual temperatures as determined in Step 1. Avoiding local minima is an important consideration in fitting a complex (multi-parameter) nonlinear function to extensive data. Since there is no theoretical test for identifying or proving the existence of a global optimum, a practical method was used to test the uniqueness of the fitting results by using varied initial values as described by Smith et al., 2014. The base initial Chaboche coefficients for the fitting were from previous work (Sun and Wagoner, 2013). Then the fitting procedure was repeated by using 12 additional initial values constructed by multiplying each Chaboche coefficient by 1/2 and 2 and by multiplying 1/2 and 2 for one coefficient while keeping the other initial coefficients unchanged. For Step 1, lsqnonlin produced uniform optimal parameters (within the precision required) for all of the 13 initial value sets. In Step 1, a set of the Chaboche coefficients (C1, g1, C2, Rs and b) was determined for each of 4 temperatures, as shown in Table 3. The constitutive models fit the data at each temperature with standard errors of less than or equal to approximately 2% of average stress values over all strains. Fig. 1 shows the difference between the experimental data and the Chaboche model fit to the data, as well as the associated standard deviations. The trial of additional initial value sets, the standard errors within the scatter of experimental data itself, and the conceptually correct shape of curves in Fig. 1 verified the ‘‘uniqueness’’ and the “correctness” of the optimal coefficients in a practical way. The Chaboche back stress is known to evolve with strain following an abrupt path change, most notably within a limited subsequent strain range where the deviation from isotropic hardening is most apparent. As noted above, CT reversal data at reversal strains of 0.04 and 0.08 strain (0.01 and 0.02 for DP980 and CP1180) was used to find the best-fit parameters. In order to test whether the strain range was large enough to establish accurate, stable Chaboche parameters, an alternate analytical technique was pursued. In this case, only the data from the reversal at 0.08 was used to find the parameters. The resulting alternate constitute model was then compared with the original one. The results for this exercise for example alloy DP780 appear in Fig. 2. Some minor differences of the fits can be seen, i.e. they fit the 0.08 reversal data better but the 0.04 data, which they ignore, worse. Nonetheless, the standard error of fit of the alternative model is insignificantly changed by focusing on this reversal of this larger strain; the average for the original model for 4 temperatures was 16 MPa in both cases. Even when compared only for the 0.04 prestrain data (which in one case was used to fit the coefficients and the other is a prediction vs. measurement), the standard error of fit increases insubstantially, from 17 MPa to 25 MPa. These standard errors of fit are less than the average test-to-test scatter of the CT tests themselves, 27 MPa (23 MPa for DP780 alone). Therefore, in view of the numerical results, the experimental scatter, and graphical information shown in Figs. 1 and 2, the fit at these prestrains is sufficient to ignore further evolution of Chaboche parameters at higher prestrains. When plotted versus temperature, the fit Chaboche coefficients obtained from Step 1 and Ro exhibit a linear relationship with the temperatures, as shown in Fig. 3. Therefore, a simple linear function was adopted for each coefficient as follows:

8 DBS tests use identical tooling and draw speeds as DBF tests, the principal difference being that the back force is maintained constant in DBS tests while the displacement rate of the back grip is maintained constant in the DBF test. Thus, except for a rising or constant back force throughout the tests, the mechanics is identical. Under these conditions, it was anticipated as a first approximation that an accurate and effective constitutive model verified for DBS test simulations would be equally applicable to DBF test simulations.

128


Fig. 1. Comparisons of experimental data and Model 2 for tensile and CT tests at each tested temperature for DP 780: (a) 25 C; (b) 50 C; (c) 75 C; (d) 100 C.

Fig. 2. Comparisons of experimental data and Model 2 with one pre-strain for tensile and CT tests at each tested temperature for DP 780 (a) 25 C (b) 50 C (c) 75 C (d) 100 C.


xi ¼ kxi þ mxi ðT TR Þ i ¼ 1; 2; …; 6

129

(13)

where xi(i ¼ 1,2,…,6) represents for the coefficients C1, g1, C2, Ro, Rs and b, and TR is the room temperature. The temperature coefficients kxi and mxi for each Chaboche coefficient obtained by linear fitting are listed in Table 4. These coefficients were used as initial values for the overall optimization in Step 3. In Step 3, a set of the temperature-dependent Chaboche coefficients xT ¼ {kc1,mc1,kg1,mg1,kc2,mc2, kR0, mR0,kRs,mRs,kb,mb} was optimized by fitting to all of the data from 12 tests (room temperature, 50, 75, 100 C, each temperature has 3 tests). No constraint of the initial yield stress was imposed. In order to test the results for uniqueness, 26 additional initial sets of coefficients were constructed by using the same way introduced in Step 1 except that each coefficients were assigned a boundary from 0.2*initial value to 5*initial value to speed up the fitting procedure. All of these optimizations had very close standard error of fit, within a derivation less than 0.1%. Therefore, the proposed method is robust.

Fig. 3. Linear relationship between the coefficients within the trial Chaboche model 2 and temperature for DP 780: (a) C1 (T); (b)g1 (T); (c) C2 (T); (d)R0 (T); (e) Rs (T); (f) b (T).

130


Table 4 Temperature-dependent Chaboche coefficients fit to original DP 780 test data and H/V extrapolated data, Models 2 and 3. Parameter

C1

g1 C2

g2 Ro Rs b Std. Error a

Unit

kc1 mc1 kg1 mg1 kc2 mc2 kg2 mg2 kR0 mR0 k Rs m Rs kb mb

MPa MPa/K /K MPa MPa/K /K MPa MPa/K MPa MPa/K /K MPa

Fit to original test data (Model 2)

Extrapolated H/V(Model 3)

Trial

Opti-mized

Trial

Opti-mized

26685 190.24 108 0.60 645 3.5

22569 29.80 128 0.21 461.41 1.72

14144 195.94 103.47 0.91 667.7 2.81 2.6827 0.011 490.82 1.60 168.27 0.28 13.32 0.125 16.14a

13405 106.7 102.91 0.62 749 1.77 3.0514 0.009 491.19 1.05 163.7 0.17 15.014 0.073 22.38

488 1.60 179.50 0.34 8.2 0.128 15.92a

445 0.76 186.04 0.46 10.84 0.095 15.56

The average standard errors of separately fit results for four temperatures.

The coefficients from the overall optimization are also listed in Table 4. The standard error of the overall optimization was 15.6 MPa, while the average value of the fitting errors for four temperatures in step 1 was 15.9 MPa. The negligibly small difference between the fitting errors further verified that the Chaboche coefficients are linearly dependent on temperature. Model 2 is compared with the experimental data used to generate it in Fig. 4. When Model 2 is extrapolated to true strains of ~0.5 (the strain range of interest to shear fracture), it is clear that the large-strain strain hardening is larger than for isotropic hardening (Model 1). However, the isotropic hardening curves of Model 1 have been verified as accurate, even under

Fig. 4. Comparisons of Models 1, 2 , and 3 with experimental tensile and CT data (symbols) and the large-strain hardening behavior reflected by Model 1 (a) 25 C (b) 50 C (c) 75 C (d) 100 C.


131

Table 5 Best-fit H/V coefficients for tensile and CT tests for DP 780. Temp ( C)

Test type

a

H1 (MPa)

H2

V1 (MPa)

V2

V3

25

Tension CT 4% CT 8% Tension CT 4% CT 8% Tension CT 4% CT 8% Tension CT 4% CT 8%

0.719 0.719 0.719 0.706 0.706 0.706 0.694 0.694 0.694 0.681 0.681 0.681

1200 1022 856 1200 1172 848 1200 1131 904 1200 1529 920

0.197 0.410 0.8 0.197 0.301 0.8 0.197 0.353 0.8 0.197 0.124 0.8

1376 2130 2804 1377 1596 2663 1377 1690 2511 1377 1382 2404

0.283 1.24 1.017 0.283 2.475 1.16 0.283 1.747 1.277 0.283 0.963 1.37

13.7 147 106 13.7 182 122 13.7 160 131 13.7 0.64 133

50

75

100

reverse-strain conditions similar to the CT strain paths (Smith et al., 2014).9 Therefore, the enhanced large-strain strain hardening of the directly-fit Chaboche model was of concern. As noted by Geng et al. (2002), multiple choices of Chaboche coefficients are often capable of accurately fitting a single set of test data. These alternate sets may extrapolate quite differently. Therefore, in order to introduce the desired nonproportional hardening effects while preserving the accuracy high-strain extrapolation behavior of the H/V model, a hybrid procedure for determining the Chaboche coefficients was developed: Model 3. 4.4. Model 3: Chaboche model with H/V large-strain behavior The H/V model is based on the common observation that steels exhibit Hollomon-like behavior (Hollomon, 1945) at low temperature and more Voce-like behavior (Voce, 1948) at higher temperature (Sung et al., 2010). A linear combination of the Hollomon and Voce hardening laws was used to represent the hardening behavior at various temperatures, which can be expressed as follows (Sung et al., 2010; Smith et al., 2014):

s ¼ asH þ ð1 aÞsV ¼ aH1 pH2 þ ð1 aÞV1 ð1 V2 expðV3 pÞÞ

(14)

where s is equivalent stress, H1 and H2 are material constants for Hollomon type hardening and V1, V2, and V3 are material constants for Voce type hardening. The coefficient a determines the proportion of Hollomon character and Voce character of the overall hardening law. The parameter a therefore varies only with temperature. Since the H/V model has a high accuracy in test data extrapolation for AHHS (Smith et al., 2014), it was adopted in the current work to extrapolate not only the monotonic tension curve but also the tension curves following a path transition in the CT tests. Thea values and corresponding H1, H2, V1, V2, and V3 coefficients calibrated by Smith et al. (2014) were used for extrapolating tensile test data in the current work. The same a values were also used for fitting tension curves subsequent to the reversal in CT tests based on the assumption that the proportion of Hollomon and Voce type hardening will remain unchanged in one cycle, because the temperature is fixed. The remaining H/V coefficients were fit by least-squares to the test data. The complete procedure for extrapolating test data and fitting the Chaboche coefficients for Model 3 is as follows: Step 1: Find the H/V model coefficients a, H1, H2, V1, V2, and V3 from the monotonic tension curves at various temperatures following the procedures proposed by Smith et al. (2014). Step 2: Extrapolate the tensile test curve using the H/V model with coefficients obtained in the previous step. Step 3: Use the a values obtained in step 1 and fit the remaining H/V coefficients H1, H2, V1, V2, and V3 for each tension curve subsequent to the reversal in CT tests. Step 4: Extrapolate each tension curve of the CT tests using the H/V model and the coefficients obtained in step 3. Step 5: Fit the Chaboche coefficients except the initial yield stress to the extrapolated test data at each investigated temperatures. Step 6: Identify the functional relationship between the Chaboche coefficients and temperature. Step 7: Use the temperature-dependent Chaboche coefficients from step 6 as initial values and perform an overall optimization. Steps 1e4 use the H/V model to extrapolate the CT test data. The best-fit H/V coefficients are shown in Table 5. Steps 5e7 are similar to the Model 2 except that the extrapolated data are used instead of the original test data for fitting. Another

9 It would be pertinent and illuminating to obtain strain hardening data at large strains following a strain reversal to reproduce more closely the conditions encountered in the DBF test. In fact, such experiments have been attempted based on pre-rolling and then balanced-biaxial tension. So far, they have proven unsuccessful because of cracking for the high-strength AHSS materials of interest.

132


Table 6 Temperature-dependent Chaboche coefficients fit to original test data and H/V extrapolated data, Models 2 and 3, for 6 materials. Materials

DP 780

DP 590

DP 980

TRIP 780

TWIP

CP 1180

Parameter

Unit

M2

M3

M2

M3

M2

M3

M2

M3

M2

M3

M2

M3

kc1 mc1 g1 kg1 mg1 C2 kc2 mc2 g2 kg2 mg2 Ro kR0 mR0 Rs k Rs m Rs b kb mb Std. Error

MPa MPa/K

22569 29.8 128.0 0.2 461 1.7

13405 106.7 102.9 0.6 749 1.8 3.1 0.0 491.2 1.1 163.7 0.2 15.0 0.1 22.4

6517 32.3 92.9 0.3 302 0.5

6312 28.2 89.9 0.3 313 0.8 0.5 0.0 424.0 0.7 137.0 0.1 12.1 0.0 30.6

12615 78.7 127.8 0.0 1555 1.0

773 13.3 0.0 0.0 13073 66.7 90.0 0.0 880.0 0.0 187890.0 2380.0 0.0 0.0 17.1

2540 21.5 26.6 0.3 380 1.8

873 0.7 2.0 0.0 21889 932.0 877.0 2.0 456.0 1.0 304.0 2.0 18.0 0.0 35.7

30182 75.8 560.9 0.3 783 0.0

929 0.0 1.0 0.0 23148 0.0 458.0 1.0 523.0 1.0 1443.0 6.0 1.0 0.0 52.3

44073 13.3 221.4 0.0 1820 13.3

1669 14.4 10.4 0.0 33580 48.1 172.4 0.2 935.8 1.0 240.4 1.1 4.0 0.1 25.5

C1

/K MPa MPa/K /K MPa MPa/K MPa MPa/K /K MPa

445.0 0.8 186.0 0.5 10.8 0.1 15.6

424.3 0.8 149.7 0.2 9.5 0.0 29.5

853.7 0.4 130.0 10.0 2.3 0.0 10.7

503.5 0.2 321.4 2.2 7.8 0.0 29.6

514.0 0.7 847.4 1.6 2.1 0.0 33.6

892.0 4.2 0.1 4.2 6.8 0.0 70.4

difference is that the linear backstress component of Model 2 is replaced by a nonlinear component, i.e. the backstress is now composed of two nonlinear components.

dXi ¼ ci dεp gi Xi dp i ¼ 1; 2

(15)

where the new coefficientg2 varies in the usual manner:

g2 ¼ kg2 þ mg2 ðT TR Þ:

(16)

The best-fit Chaboche coefficients for Model 3 are listed in Table 4 together with those obtained from Model 2. Fig. 4 compares Model 3 with Models 0/1 and 2. Note that the procedure used to generate Model 3 accomplishes its goal, that is, to represent the transient behavior after a path change captured by the CT tests while maintaining the large-strain behavior of the H/V model, as confirmed by large-strain balanced biaxial bulge tests. Table 6 shows temperature-dependent Chaboche coefficients10 fit to original test data and H/V extrapolated data, Models 2 and 3, for 6 materials. 5. DBF tests and simulations As reviewed earlier, shear fracture can be reliably reproduced in the laboratory by the DBF test. In order to assess the performance of the nearly-devised Chaboche models with a temperature dimension, Models 2 and 3, as well as the proposed fitting procedures, DBF tests and corresponding simulations were performed. 5.1. DBF tests Fig. 5 illustrates the DBF test schematically. A 25.4 mm wide strip aligned in the rolling direction is wrapped around a fixed roller and clamped by front and back grips at the front (1) and back (2) sides. Linear actuators are used to independently control the front and back speeds of grips, V1 and V2, respectively. For all DBF tests in the current work, the back grip was stationary (V2 ¼ 0) and the speed of front grip was set to displace at 51 mm/s corresponding to a maximum strain rate of about 5/s at the bending region for R/t ¼ 3.2 mm. This is approximately half of the maximum industrial stretching strain rate (Sung et al., 2012). Two fixed rollers with radii of 3.2 mm and 6.4 mm were used: the radius for each material was selected as shown in Table 7 in order to insure shear fracture. Thus, the materials more susceptible to shear fracture make use of larger R/t values. The principal output data from DBF tests is the formability, as defined by the draw-distance-to-fracture, Uf. All of the tests were run under a standard lubricated condition for which the friction coefficients (m) were measured, with results presented in Table 7. The values in Table 7 are different from those in Table 1 because they reflect different conditions,

10 There are two sources of difference in the values of the parameters presented in Table 6 from the corresponding ones presented in Lee et al. (in press) for the alloy used in common, DP780. The values of C in Table represent the form of the Chaboche equations and parameters that presented here (e.g. Eq. (4) and Eq. (7)) while the corresponding Lee et al. (in press) ones have 2/3 of that magnitude because of a difference in the equation representation. These differences represent identical material behavior. There is also a slight difference of the values for all of the parameters because in Table 6 they were fit and optimized using compression-tension data for 4% and 8% prestrains while in Lee et al. (in press) only the 4% prestrain data was used. Examination of simulated stress-strain curves for compression-tension tests shows that the differences in the final constitutive model are insignificant.


133

Fig. 5. Schematic of the DBF test.

i.e. draw-bend conditions with high contact pressure over a curved contact region versus flat tensile testing at low contact pressure. Since friction is related to such differences (Schey, 1983; Kim et al., 2014), it is critical to measure friction separately. The friction determination method under DBF conditions is based on the draw-bend springback (DBS) test (Wagoner et al., 1997; Carden et al., 2002) and parallel FE simulation. The DBS test is the same as the DBF test as shown in Fig. 5 except that the back grip is not fixed. Instead, a constant restraining force (Fb) is applied. For friction determination under the employed method, two back forces, 0.8 ss and 0.9 ss (where ss is the yield stress), were used in two otherwise-identical DBS tests. The differential forces from front to back of the strip are compared for the two tests using a computed value, F ¼ (Ff2 Fb2) (Ff1Fb1), where Fb2 ¼ 0.9 ss, Fb1 ¼ 0.8 ss, and Ff2 and Ff1 are the correspondingly measured front force. In parallel, a series of thermo-mechanical finite element simulations was conducted using a range of assumed friction coefficients. The friction coefficient is the value that allows the measured F is equal to the simulated F. The analysis was repeated with variations of constitutive models (including the most important choice of yield function type) to check the sensitivity to the simulation accuracy. Such choices have little effect on the measured friction value, no more than 0.01. A 3D thermo-mechanical (TM) finite element model of the DBF tests was developed previously (Kim et al., 2011) for shear fracture prediction, utilizing ABAQUS Standard Version 6.10. The newly proposed temperature-dependent Chaboche model was implemented in a UMAT subroutine. An associated flow rule with Hill's 1948 anisotropic yield criterion (Hill, 1950) was used. The nonlinear unloading behavior of AHHS steels has a significant effect on springback in DBS tests, but has little influence on DBF tests. A constant Young's modulus (E ¼ 211 GPa) was used for each model to simplify the comparisons for strain hardening. 5.2. FE implementation The return-mapping algorithm and implicit backward Euler integration scheme were adopted to implement the proposed Chaboche model (Simo and Hughes, 1998; Sun and Wagoner, 2011). Considering the temperature effect and rate sensitivity, the general yield function can be defined as follows:

f ¼ se ðs XÞ R$gð_ε; TÞ 0

(17)

where se is the equivalent stress (defined by the particular yield function), ε_ is plastic strain rate and the function g represents strain rate hardening. In the current work, a temperature-dependent power-law strain rate hardening model is used (Sung et al., 2010):

ðm1 þm2 ðTT0 ÞÞþm3 ln ε_ ε_ 0 ε_ gð_ε; TÞ ¼ ε_ 0

(18)

134


Table 7 The experimental and simulation results of DBF tests. Material

Thick- ness (mm)

Roller radius (mm)

Friction coefficient

Exp. UF (mm)

M0

DUF

M1 UF%

DUF

M2 UF%

(mm)

M3 /UF/% %

DUF (mm)

UF%

(mm)

DP590

1.4

3.2

0.16

28.9

16.9

58%

DP780

1.2

DP980

1.2

3.2 6.4 6.4

0.06 0.05 0.04

23.6 40.1 11

13.1 20.8 6

56% 52% 55%

CP1180

1.2

6.4

0.05

7

8.1

116%

TRIP780

1.4

3.2

0.06

35.9

2.5

7%

TWIP 980

1.4

3.2

0.1

71.9

46.5

65%

Average/UF/for all

31.2

16.3

52%

Average Error of DP and CP steel/UF/% Average/UF/for non-TF

22.12

13.0

59%

7.8

26%

13.1

56%

Average Error of non-TF DP and CP steel/UF/%

D UF

e

e

0.6 4.2

3% 10%

e

e

e

e

e

e

e

e

e

e

e

e

e

e

e

e

9.6

33%

0.1

0%

2.3 16.1 5.21

10% 40% 47%

0.1 2.3 0.2

0% 6% 2%

43

614%

0.1

1%

5.05

14%

10.1

28%

28.3

39%

21.3

30%

15.7

50%

4.9

16%

8.3

38%

0.6

3%

11.1

31%

4.9

16%

8.3

32%

0.6

3%

Key:

DjUF j ¼ UFExp: UFSim: . P jDUF j DjUF j% ¼ P *100%. UF

where ε_ 0 ¼ 0:001 is the reference strain rate and m1, m2, m3 are material constants. The evolution of the back stress X and the size of yield surface R is defined in Equations (9) and (10). The flow vector (norm to the yield surface) is:

n¼

vf vse vse vðs XÞ vse ¼ ¼ ¼ : vs vs vðs XÞ vðsÞ vðs XÞ

(19)

The schematic of backward Euler return algorithm is shown in Fig. 6. The point A represents the stress sA at the beginning of time step t. Assuming plastic loading during the time span Dt, the point C represents the updated stress sC at time t þ Dt. In the general return-mapping algorithm, a trial stress outside the yield surface is chosen by elastic predictor. This trial stress is then updated with a plastic correction to bring it back onto the yield surface. After an elastic predictor, the trial stress and back stress for the point B is defined as:

Fig. 6. Schematic of backward Euler return algorithm.


135

Fig. 7. Comparisons of experimental and simulation results using Models 1e3 for DBF tests. (a) DP 780, 3.2 mm; (b) DP780, 6.4 mm; (c) DP 590, 3.2 mm; (d) DP 980, 6.4 mm; (e) CP 1180, 6.4 mm; (f) TRIP 780, 3.2 mm; (g) TWIP 980, 3.2 mm.

136


sB ¼ sA þ CDε

(20)

XB ¼ XA ;

(21)

where C is the constant elastic stiffness matrix, XA is the back stress at point A and Dε is the given strain increment of the step. In the backward Euler integration scheme, the flow vector was evaluated at stress point C. Therefore, the Backward Euler stress is defined as:

sC ¼ sB DlnC ;

(22)

where Dl is the plastic multiplier. A first guess of the plastic multiplier Dl can be derived from the Taylor expansion of yield function at the backward Euler stress point with respect to the trial stress sB. Usually, the new stress after this correction is not exactly on the updated yield surface. Therefore, iterative schemes are necessary to restore the plastic consistency condition. The Newton Raphson method is used to find suitable plastic multiplier Dl. The residual stress and back stress are defined as the difference between the current values and the Backward Euler estimations:

r ¼ s ðs Dln Þ s B X C : rX ¼ X XB þ dXi

(23)

Let R ¼ [rs, rX], if fC < TOL1 and R < TOL2, where TOL1 and TOL2 are pre-specified numerical tolerance, the iteration will be terminated and all state variables will be updated. 5.3. DBF simulation The FEM model reflects the physical DBF geometry. Eight-noded temperature-dependent solid elements (C3D8RT) were used for temperature-displacement coupled simulation. The thermal constants are from Sung et al. (2010): the thermal expansion coefficient varies linearly from 1.54 106/K at 25 C to 2.59 106 J/gK at 200 C and the heat capacity from 0.45 J/gK at 25 C to 0.52 J/gK at 200 C. The data for thermal conductivity are presented by a piecewise linear variation: 36.7 W/m K at 25 C, 36.9 W/m K at 70 C, 36.8 W/m K at100 C and 36 (W/m K) at 200 C. Elastic properties are temperatureindependent and are taken from handbooks; this approach has no significant effect on the simulations or conclusions. The mesh used for the simulations has elements sized 0.3 mm by 1.1 mm (length by width) with 11 equal-thickness elements through the thickness. A coarse mesh was used for verification: 0.6 mm by 2.2 mm (length by width), with 11 elements through the thickness for all regions of the strip that come into contact with the fixed roller. Except for the mesh size, all other settings are the same. The change of mesh produced insignificant differences, as follows: 6% difference for Uf and 4% difference for the peak sf. Fig. 7a and b compare the DBF simulations using Models 0e3 with experimental results for DP 780 with R ¼ 3.2 mm and 6.4 mm, respectively (Note: Unless marked by “TF” (tensile fracture), all curves represent shear fracture, that is, the fracture in the bending/unbending region of the strip in contact with the tooling.). The use of the von Mises yield function in Model

Fig. 8. H/V fit vs. tensile tests at elevated temperatures for TRIP 780.


137

Fig. 9. Linear relationship between the coefficients within the trial Chaboche model 3 and temperature for TRIP 780 (a) C1 (T) (b) g1 (T) (c) C2 (T) (d) g2 (e) R0 (T) (f) Rs (T) (g) b (T).

138


0 leads to significant over-prediction of formability as compared with measurements. Model 1 based on Barlat Yield'91 corrects that error, leading to very accurate predictions of formability. More germane to the current development is the comparison of Models 2 and 3, which include Chaboche transient effects and differ primarily in their extrapolation to large strains. For R ¼ 3.2 mm (Fig. 7a), Model 2 overestimates the formability slightly, but well within acceptable scatter. However, for R ¼ 6.4 mm, Model 2 is overestimates formability by a significant margin that is greatly reduced (by approximately 75%) when Model 3 is used. Clearly, the dominant feature of all such models for accuracy is the large-strain strain hardening, not the details of the transient response. In fact, inclusion of the Chaboche treatment of the transient response is not essential to the accurate prediction of shear fracture, while the large-strain strain hardening is essential for accuracy. Fig. 7ceg are similar to Fig. 7a or b but are for the remainder of the materials. Except for TRIP 780, Models 0 and 2 overestimate the actual formability and Model 3 predicts it accurately. The only exception is TRIP 780, Fig. 7f, which is the only alloy that undergoes phase transformation during straining. The thermo-plastic behavior of TRIP 780 is more complex and is not captured perfectly by the existing constitutive format. As shown in Fig. 8, the H/V fit underestimates the strain hardening at the higher strains. Also, TRIP 780 exhibits a nonlinear relationship between Chaboche coefficients and temperatures as shown in Fig. 9. Neither of these differences was accommodated in the current work by refining the constitutive model further. Note: strain rate sensitivity m value was set to be zero for Model 3 simulations for CP 1180 and TWIP steel in Fig. 7, due to convergence problem. This is acceptable for two reasons: first, it has been known that relatively high strength steels usually have smaller m values, such as m z 0.00054 for CP 1180 and m z 0.001 for TWIP at room temperature; second, the strain rate range for DBF test (R ¼ 3.2 mm, most sever case compared with other larger rollers) is about 2.5/s (Kim et al., 2011), the strain rate effect is reasonable to be negligible. The comparison of the formabilities from DBF simulations and experiments is summarized in Table 7. Absolute values are used for the last four rows calculations. The predictions of Models 0 and 2 overestimate the shear fracture formability, by, on average, 26% and 31%, respectively. These errors are reduced by a factor of 2 by Model 3, to 16%. Perhaps more significantly, when the two alloys with complex hardening mechanisms are eliminated from the comparison, Models 0 and 2 overestimate shear fracture formability from 56% and 32%, respectively, to 3%, approximately 1 order of magnitude! Clearly, the large-strain extrapolation of strain-hardening must be handled properly in order to predict shear fracture. Equally clearly, when such care is taken, shear fracture is closely predicted by accurate constitutive equations for plasticity, thus confirming the essential nature of shear fracture. 6. Summary and conclusions A temperature-dependent Chaboche model has been constructed and implemented in Abaqus Version 6.10 using two alternate procedures for fitting: a typical method using original test data and a novel method using the data obtained from H/ V model extrapolation. The results of each method are compared with DBF test results. The follow conclusions were reached: 1. The dominant role in the plastic constitutive model for the prediction of shear fracture is the extrapolation of large-strain strain hardening. The details of the transient behavior following path reversals are not critical to subsequent shear fracture at much larger strains. 2. A modified temperature-dependent Chaboche model, fit to extrapolated data from the H/V hardening form, reproduces both the transient effects and the large strain behavior properly and thus predicts DBF fracture accurately. 3. A method was introduced to make use of the H/V parameter a (characterizing the nature of the strain-hardening curve at a single temperature) in order to extrapolate strain hardening to large strain. Using this method greatly improves the prediction of shear fracture. The success suggests that a has fundamental material significance in characterizing strain hardening, i.e. a represents a material constant that varies with temperature but little with details of the strain path. This also suggests that the nature of strain hardening following path changes is essentially a transient phenomenon. 4. A linear relationship between the Chaboche coefficients and the temperature for DP 590, DP 780, DP 980, TWIP 980, CP 1180 is accurate allows for simple implementation. For TRIP 780, a more complex relationship and additional internal variables in Chaboche model to describe the effect of temperature history (Chaboche, 1993) are needed to improve the accuracy, presumably required by phase transformation during deformation.

Acknowledgments The authors are grateful for the many kinds of support offered by Posco Steel, the GIFT program at the Pohang University of Science and Technology (Postech), and the National Natural Science Foundation of China (Grant No. 50805050) and China Scholarship Council (Grant No. 2011615504). MGL appreciates the support by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIP) (NRF-2014R1A2A1A11052889). Thanks are offered in particular to Jaebok Nam, Dong-Jin Kim, Hong Woo Lee, and Jae Wook Lee, all of Posco.Ken Kushner and Ross Baldwin are gratefully acknowledged for assistance with mechanical testing in the Department of Materials Science and Engineering at The Ohio State University (OSU).


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