Deformation-induced anisotropy of uniaxially prestrained steel sheets

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International Journal of Solids and Structures 0 0 0 (2017) 1–10

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Deformation-induced anisotropy of uniaxially prestrained steel sheets Shakil Bin Zaman∗, Frédéric Barlat∗∗, Jin-Hwan Kim Graduate Institute of Ferrous Technology (GIFT), Pohang University of Science & Technology (POSTECH), 77 Cheongam-ro, Nam-gu, Pohang, Gyeongbuk 37673, Republic of Korea

a r t i c l e

i n f o

Article history: Received 12 April 2017 Revised 27 August 2017 Available online xxx Keywords: Sheet metal forming Mechanical testing Dual phase Distortional hardening Strain path change HAH model Yield surface distortion R-Value Bauschinger effect

a b s t r a c t This article investigates the macroscopic behavior of uniaxially pre-strained large-scale DP780 and CHSP45R sheet specimens. A novel grip system was designed for the pre-straining of the large tensile specimens. After this first loading, the uniformly strained gauge-section of the large-scale specimen was machined to smaller standard specimens. After pre-strain, further uniaxial tension tests at 45° and 90° from the pre-tensile direction, compression test and in-plane biaxial tension tests at 1:1, 2:1 and 1:2 force ratios were conducted. The flow curves and the instantaneous r-values of the pre-strained steel in the aforementioned uniaxial loading directions were compared with their monotonic response. In addition, the pre-strained compression test was incorporated to the π -plane at several incremental offset strains. The yield function- and isotropic hardening-based homogeneous anisotropic hardening (HAH) model (Barlat et al., 2014) was selected to predict the material response after non-linear strain path change. The Swift law characterized the isotropic hardening, while the Yld20 0 0-2d anisotropic yield function (Barlat et al., 2003a) represented the yield locus. All the coefficients of the distortional plasticity model were manually determined and validated with an optimization algorithm. The HAH-predictions of the pre-strained flow curves and yield loci at several offset strains were in reasonable agreement with the experimental data, while the instantaneous r-value evolution was qualitatively well captured. © 2017 Elsevier Ltd. All rights reserved.

1. Introduction 1.1. Sheet metal forming The manufacturing of automotive parts via forming process involves permanent shape change coupled with fields of non-linear deformation paths in the sheet sample. The material response of metals strongly depends on the loading path (Gérard et al., 2013; Barlat et al., 2014; Tutyshkin et al., 2014), which changes abruptly during sheet metal forming (De et al., 2014; Hashemi and Abrinia, 2014). Considering the entire loading history, sheet materials after a particular loading stage behaves in a unique manner, with distinct yield locus, hardening mode and mechanical response compared to that of the as-received material. The mechanical behavior of sheet materials during forming process can be well perceived and mimicked if the metal sheet is investigated under conditions of pre-strain and combined stress. Such investigation could lead to a better understanding of the behavior of sheet materials and

∗ Corresponding author at University of Twente, Nonlinear Solid Mechanics. Faculty of Engineering Technology, P.O. Box 217, 7500 AE Enschede, The Netherlands. ∗∗ Corresponding author. E-mail addresses: [email protected], [email protected] (S.B. Zaman), [email protected] (F. Barlat).

establish a basis for a more satisfactory strain-hardening model. It was a common practice to assume isotropic hardening for the simulations of sheet metal forming process. However, it has been shown in a number of articles that the yield surface does not expand in an isotropic fashion (Tozawa, 1978). 1.2. Experimental corroboration In case of sheet materials, a linear or non-linear strain path change always leads to an anisotropic work hardening response (Fernandes and Schmitt, 1983; Wagoner and Laukonis, 1983; Rauch and Schmitt 1989; Wilson and Bate, 1994; Barlat et al., 2003a; Bouvier et al., 2005), which occurs whenever there is a substantial variation in the loading axis (Tarigopula et al., 2008). For instance, sheet materials subjected to tension followed by compression in continuous strain reversal test (Boger et al., 2005) exhibit Bauschinger effect. Also, low carbon steels deformed in forwardreverse simple shear may, in addition, present transient regions of strain hardening stagnation (Rauch et al., 2011). Note that, in this article, “strain path change” is used interchangeably with “nonproportional loading” as it indicates any deviation from the monotonic stress strain behavior. A combination of two linear strain paths, uniaxial tension followed by simple shear tests on aluminum alloy sheet samples af-

https://doi.org/10.1016/j.ijsolstr.2017.10.029 0020-7683/© 2017 Elsevier Ltd. All rights reserved.

Please cite this article as: S.B. Zaman et al., Deformation-induced anisotropy of uniaxially prestrained steel sheets, International Journal of Solids and Structures (2017), https://doi.org/10.1016/j.ijsolstr.2017.10.029

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ter 450 , 900 , and 135° strain path changes elicited transient effects in cross- and reverse-loading tests (Barlat et al., 2003b). Furthermore, after a sequence of two uniaxial tension tests, in which the load axes are different, for instance at 45° and 90° from each other, various hardening behavior is observed depending on the material (Lloyd and Sang, 1979). For example, with low carbon steels, the flow stress after 45° strain path change overshoots the corresponding monotonic hardening (Barlat et al., 2013), whereas with dual phase (DP) 780 steels, early re-yielding occurs at reloading near cross-loading conditions, followed by the Bauschinger effect (Ha et al., 2013; Barlat et al., 2014), which is usually observed in the load reversal case. In a prior study by Sugimoto et al. (1985), cross-loading contraction behavior was also observed when DP steels were first pre-strained via the first-step plane strain or uniaxial tension followed by second-step tension test near cross-loading. Regarding the plastic strain ratio after load shift, Manik et al. (2015) observed sudden fluctuation of the r-value (width-to-thickness strain increment ratio) in aluminum alloys for the same type of two-step tension tests. Experiments involving manifold loading history were also operated to characterize the anisotropic response. For instance, Kim and Yin (1997) conducted three-step tension tests, by means of an exaggerated large-scale specimen for the first and second loadings. Similarly, Vincze et al. (2013) conducted double strain path change experiments using low carbon steels, with the 1st and 3rd loadings superimposed along the rolling (RD), while the 2nd loading is varied at different angles from the RD. Although the load or strain path changes in an actual forming operation may be quite complex, it is useful to investigate the material behavior after one single strain path change where the amplitude of the change can be characterized using the parameter ϕ (Schmitt, 1985).

ϕ=

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dε1 : dε2 ||dε1 ||||dε2 ||

(1)

The strain increment tensors dε1 and dε2 denote the first and second strain paths, respectively. For monotonic loading, the default value of ϕ is always fixed at unity. According to this relationship, for an isotropic material, the monotonic uniaxial RD tension in the first step followed by equibiaxial, TD plane strain, orthogonal tension, or RD compression in the second step sequence are quantitatively represented by ϕ as 0.4, 0, −0.5, and −1, respectively. In any monotonic tensile test, the r-value is usually constant due to the unchanging loading axis or strain path. Therefore, the p p instantaneous r-value (dεww /dεtt ) is identical to the average rp p value (εww /εtt ) throughout the first strain path (the superscript ‘p’ refers to the plastic portion of the total strain, whereas the subscript ‘ww’ and ‘tt’ denote the width and thickness directions, respectively). However, after a strain path change, depending on its severity, the average r-value fails to represent the instantaneous rvalue at every progression of strain. Therefore, the evolution of instantaneous r-value in the second loading must be calculated from p p the gradient of εww and εtt for each increment of strain. A detailed description of the r-value calculation at second step loading is given in Appendix A. In this work, the sheet materials are assumed to follow associated flow rule (AFR). In particular, for steel and other metallic sheets, Hecker (1976) and Kuwabara (2007) did not find any inconsistency between the plastic potential and yield surface normal; i.e., the plastic strain increment is perpendicular to the yield surface slope, n, which can be evaluated from the instantaneous rvalues (n = (r + 1)/r). Therefore, any deviation to the yield surface normal after strain path change can be indirectly observed from the r-value fluctuations. The yield surface is usually evaluated through monotonic uniaxial and biaxial tests at different loading angles and force ra-

tios, respectively (Kuwabara et al., 1998). The mechanical behavior of the first loading path is represented by an isotropic hardening law since the material deforms with a linear strain path. However, the anisotropic nature of yielding predominates when the material is deformed in one loading path and reloaded in another. Teodosiu and Hu (1998) specified that the accumulation of strain in sheet metals disrupts the shape of the material’s stable yield surface. In this paper, to determine the extent of surface distortion after plastic deformation, the uniaxial and biaxial tests were conducted after pre-straining the sheet in one direction. 1.3. Constitutive modeling In plasticity, the usage of a multi-scale modeling is useful because such a model dictates the macroscopic mechanical behavior based on the evolution of microscopic dislocations at different scales (McDowell, 2010). However, since sheet metal industries require practical and time-efficient material models, comprehensive multi-scale models are incompatible in satisfying the manufacturing demand. Meanwhile, plasticity models at a continuum level are relatively fast and can roughly predict the macroscopic behavior of sheet materials. The emergence of plasticity models spurred with the development of yield surface, which defines, in monotonic loading condition, the plastic anisotropy of sheet materials (Hill, 1948; Hershey, 1954). Having said that, plastic anisotropy can also be described via the strain rate potential concept (Hill, 1987; Barlat and Chung, 1993; Chung et al., 1998). Anyway, we know that the plastic deformation of metal sheets is mainly caused by dislocation glide on favorable slip systems. Consequently, the effect of the crystallographic slip was roughly captured using continuum non-quadratic yield functions (Hosford, 1985; Barlat et al., 2003a), which, by the way, are only limited to capturing the monotonic loading behavior of sheet materials (Yoon et al., 2004). Therefore, these aforementioned yield functions are unsuitable for characterizing sheet samples at non-proportional loading conditions. In order to incorporate the material response after multiple or continuous non-linear strain path changes, a translation-based strain-hardening model was first proposed, also known as the kinematic hardening model. Briefly speaking, the kinematic concept allows yield surface translation via the evolution of the socalled back-stress. This linear yield surface translation along the direction of hardening captured the Bauschinger effect at load reversal, (Prager, 1949; Ziegler, 1959), while a non-linear evolution (Chaboche et al., 1979) predicted, in addition, other transient strain-hardening features (Chaboche, 1986). During tensioncompression test, the reloading material behavior was modeled by the Yoshida-Uemori model, which utilizes the non-linear backstress terms to characterize permanent softening (Yoshida and Uemori, 2002; Yoshida et al., 2002; Yoshida and Uemori, 2003). Recently, kinematic hardening approaches successfully captured work-hardening stagnation and springback (Geng, 20 0 0; Chung, 2005; Lee et al., 2007), along with the accurate material characterization under cyclic loading conditions (Chaboche, 2008; Dafalias and Feigenbaum, 2011). The kinematic framework has solely been used to characterize the Bauschinger effect until recently when a distortion-based hardening model, prominently known as the homogeneous anisotropic hardening (HAH) model, was proposed by Barlat et al. (2011). This concept was originally developed at a continuum level based on the overall dislocation behavior of singlephase sheet materials; later, the model also incorporated the characteristics of dual-phase steels (Barlat et al., 2013). Unlike translation in kinematic theories, the HAH model obeys isotropic expansion followed by distortion and rotation of the yield surface to capture the anisotropic flow curves, r-values and yield loci after multiple strain path changes. This concept captures the

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HSS is manufactured by adding substitutional alloying elements (P and Mn) amongst relatively small ferrite, resulting in high internal friction stress and lattice distortion. The microstructure of CHSP45R allows greater elongation but lower strength than that of DP780. The material response in different monotonic loading conditions were evaluated via the standard uniaxial tensile test. The yield point and r-value at different angle from the rolling are listed in Table 1, which is used to calibrate the initial anisotropy of both materials. The yield stresses were measured using the 0.2% offset strain method, whereas the instantaneous r-values were estimated from the linear regression line between the plastic strains measured in two orthogonal directions. The terms σ bb and rbb refer to the yield point and r-value, respectively in the balanced-biaxial (bb) stress state.

2.2. Standard measurements Fig. 1. Schematic illustration of the selection of standard specimens from the uniform region of the large-scale specimen.

Bauschinger effect and other transient effects via a loading axisbased fluctuating component, which responds to the change in loading direction and subsequently prescribes an evolved yield locus. So far, this model is upgraded to accurately characterize the strain hardening stagnation, yield surface flattening and crossloading effects in both single- and dual-phase steels after multiple strain path changes (Barlat et al., 2013; Barlat et al., 2014). Furthermore, the accurate prediction of springback and Bauschinger effect, which are essential for press-forming applications are also well captured by this model (Lee et al., 2012). 1.4. Objective of this study The main purpose of this work is to analyze the feasibility of the current HAH model (Barlat et al., 2014) in predicting the flow stresses, r-values and evolved yield surface in uniaxial and biaxial loading conditions after the first loading. This investigation required an experimental setup in scale large enough to perform a two-step tensile operation in uniaxial and biaxial loading conditions, for which an additional grip fixture was devised. Afterwards, the standard uniaxial and cruciform specimens nested within the pre-strained large-scale specimen were extracted to perform nonmonotonic tests in several loading conditions (Fig. 1). The corresponding pre-strained flow curves and r-values after 45° and 90° loading axes change were compared to that of the undeformed sheet material by conducting two-step uniaxial tension tests. Consequently, these tests are predicted via the HAH model. In addition to the tension tests, the evolution of yield surface after pre-strain is investigated by conducting pre-strained biaxial tension tests at several force ratios and tension-compression tests. Section 2 describes experimental details with special emphasis to the largescale tensile system. Section 3 explains the nature and behavior of the distortion-based HAH model, including the significance of material coefficients. Finally, Section 4 presents and discusses the experimental and model results. 2. Experimental strategy 2.1. Sheet materials used For the investigation of non-linear strain path change, two different grades of steel, namely DP780 AHSS and single-phase CHSP45R re-phosphorized high strength steel (HSS) were used. The 1 mm thick DP780 steels consist of hard martensitic islands dispersed in a soft ferrite sea, producing high levels of strength along with high total elongation, whereas the 1.2 mm thick single-phase

The yield and r-value data that were used to define the initial anisotropy (Table 1) of as-received sheet materials were extracted from standard uniaxial and biaxial tensile tests. The aforementioned standard tests were also performed on the pre-strained steel in order to investigate the evolution of anisotropy after nonlinear strain path change. All the tests were machined using laserwire electrode cutting technique in order to minimize the stress concentration around the corners, holes and slits. Each tension sequence was repeated at least twice at room temperature and only the duplicate data are acquired. The uniaxial tests were performed via the Instron 3382 Universal Testing Systems with 100 kN load-cell capacity, using standard uniaxial specimens abiding by the ASTM E8 specification (ASTM, 2010). The crosshead velocity of all the uniaxial tension tests was fixed at 2 mm/min until fracture. The longitudinal strain accumulated in the gauge section of the uniaxial specimen was measured using static axial clip-on extensometers. The same test was repeated in the MTS machine using the DIC methodology, which measures strain via the deformation of the speckle pattern printed on the specimen gauge. Different tensile testing method was adopted to validate the yield point acquired from Instron machine and, simultaneously, measure the r-value in each tested material orientation (RD, 450 and TD). Regarding the biaxial tester, the 260 × 260 mm2 cruciform specimens were adapted with respect to the specifications of Kuwabara et al. (1998) (also standards ISO 16,842:2014) and tested via Kokusai in-plane biaxial testing machine (KBAT-100). The strain gauges were glued to the gauge section of the cruciform specimen to measure the average strain accumulated in each loading axis until fracture. Several additional tests were conducted for further validation of the yield locus. The large-strain balanced biaxial bulge test is performed on a 300-mm square sheet using the Erichsen bulge/FLC tester (Model 161) for each as-received sheet metals. In every test, the blank holding force was fixed at 10 0 0 kN with 8 mm/min drawing speed and the thickness strain was measured via the DIC system. The tension-compression test was implemented by means of a Sheet Metal Tension/Compression Tester (Model No. RB319). This test was executed at 20 kN anti-buckling force and 1 mm/min displacement rate. The strain imposed on the standard dog-bone type uniaxial specimen during the process was measured using laser extensometer, which encompasses 8–127 mm of measuring range. Simple shear test (Rauch and Schmitt, 1989; Rauch, 1998) was also conducted on the universal tensile machine by means of two rigid grips, one of which moves away from the other along the sample longitudinal direction (Choi et al., 2015). The 60 × 15 mm2 rectangular samples were clamped between the shear grips and translated back and forth at a crosshead velocity of 2 mm/min to obtain forward-reverse stress-strain response of each tested materials.

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S.B. Zaman et al. / International Journal of Solids and Structures 000 (2017) 1–10 Table 1 Material characterization and plastic anisotropy in the uniaxial (0°, 45° and 90°) and balanced-biaxial (bb) stress states. E (GPa) DP780 (1t) 185 CHSP45R (1.2t) 165

σ 0 (MPa)

σ 45 (MPa)

σ 90 (MPa)

σ bb (MPa)

r0

r45

r90

rbb

548

506

534

552

0.70

0.97

0.82

0.90

334

331

345

344

1.07

1.36

0.95

0.86

Fig. 3. Grip system for the large-scale specimen. Fig. 2. Large-scale uniaxial tensile specimen.

2.3. Experimental conditions for non-linear deformation path A linear strain path execution via the uniaxial tensile operation is the simplest method to build-up strain in sheet samples. After pre-straining a material, any subsequent deviation of the corresponding flow curves and yield locus can only be analyzed if the pre-deformed sheet sample is large enough in dimension and uniform enough in strain distribution to perform further tensile tests in non-coaxial directions. The direction of pre-strain should be restricted to either the rolling or the transverse direction (TD) because any other pre-loading angle might hamper the uniformity of strain due to the pertaining anisotropic characteristics in sheet materials (see Appendix B). In this study, the preliminary large-scale test specimens for first-step tension test were aligned to the TD, which shows reasonably uniform and consistent strain distribution. 2.4. Large-scale uniaxial tensile system As discussed earlier concerning the dimensional requirement, the gauge-section of the first-step specimen after straining should be large enough for the subsequent tensile or biaxial sequence at any orientation. Therefore, a large-scale tensile specimen (1040 × 400 mm2 ) was designed for this purpose, with the aspect ratio (gauge length : gauge width) fixed at 2:1 (Fig. 2). The length of the exaggerated specimen was limited by the maximum longitudinal clearance of the tensile machine. After tension, the largescale tensile data for tested materials showed negligible difference with that of the standard uniaxial tensile specimen. Therefore, any observation of the subsequent non-proportional loadings via the large-scale specimen is the result of strain path change and not due to specimen design or machining flaws. A novel gripping fixture (Fig. 3) was also designed, connecting the large-scale specimen to the existing 500 kN MTS 810 machine. The overall design of the fixture, including the number, diameter and location of the bolt-holes were acquired after several iterations of trial and error based on experimental tests and finite element simulation. The orientation of bolts throughout the grip fixture was distributed such that steady load is distributed within the system. The grip system consists of three separate sections- two

Fig. 4. Strain maps in the AOI of DP780 (left) and CHSP45R (right).

arms and a hinge, the latter being installed to the servo-hydraulic MTS machine. During assembly, all the aforementioned grip segments and the specimen were bolted together along a fixed line of symmetry, co-linear with the loading plane. One arm was deliberately made thinner than the other for ease in unmounting the arm and inserting the specimen. The hinge was made of a material with properties higher in strength and hardness than the arms because firstly, the hinge stayed fixed once it was installed to the tensile machine, and secondly, it was prone to the enormous gripping pressure (∼50 0 0 MPa) from the MTS wedge grips throughout the tensile operation. 2.5. Strain distribution analysis In order to assemble the large-scale tensile system, the grip fixture was first installed and aligned to the machine followed by alignment of the large-scale specimen to the latter. The crosshead velocity of the machine was fixed at 6 mm/min for all the largescale tensile tests and interrupted at 45 mm and 42 mm longitudinal displacements for DP780 and CHSP45R steels, respectively. The strain distribution in the central rectangular region (420 × 300 mm2 ) were obtained using the digital image correlation (DIC) technique. The longitudinal strain (ε yy ) maps in Fig. 4 show reasonably uniform and recurrent strain distribution for both materials. Furthermore, since the ε yy -range in the figure was restricted to within 1%-strain, the strain distribution in the central region is found stable and therefore, assumed uniform.

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3. Modeling strategy 3.1. HAH formulation In the original formulation (Barlat et al., 2011), the plastic potential  of the HAH modular framework is expressed as a function of the deviatoric stress s-

 1 (s ) = ω (s )q + φ (s )qh q        1  ˆ  ˆ q ˆ  ˆ q q = ω (s )q + f1q h : s − h : s + f2q h : s − h : s

(2)

The stable component ω corresponds to an isotropic or anisotropic yield function homogeneous of first degree with respect to the stress. The fluctuating yield function φ (s)h has a distortional effect on the yield surface corresponding to ω(s) and the term q is a constant exponent. The distortion aligns with the miˆ , a normalized tensor that mimics the gradcrostructure deviator h ual evolution of the dislocation structure when the strain path changes. The state variables f1 and f2 control the degree of yield ˆ. surface distortion that is coaxial to h In an enhanced formulation (Barlat et al., 2014), ω(s) provides cross-hardening contraction or latent hardening, with maximum ˆ . In absence of the crosseffects for stress states orthogonal to h hardening features, the aforementioned expression reduces to the usual yield function ω(s) (See Barlat et al. (2014) and Appendix C). The current HAH framework is established on the rotation of the microstructure deviator, which relies on the amplitude of strain ˆ : s ≥ 0, path change. For instance, if the stress state is such that h then the reloaded flow stress eventually follows the monotonic curve in absence of the cross-hardening effects. However, when ˆ : s < 0, the HAH approach allows lower flow stress at reloading h than at unloading. When a yield surface is under plastic deformation, the isotropic hardening model suggests that the surface enlarges proportionately in every direction with increasing strain, thus preserving the global shape of the yield surface. Kinematic hardening proposes an adjustment to the back-stress (α), which allows the yield surface to translate along the direction of loading in terms of the linear (Ziegler, 1959) or non-linear (Chaboche, 1986) evolution of α. Unlike the isotropic or kinematic hardening concepts, the distortional HAH model predicts a yield surface flattening opposite to the active loading direction. In case of proportional loading, the HAH distortion-based hardening model adopts the yield surface normal and flow stress dictated by the isotropic hardening (Manopulo et al., 2015). The HAH-response deviates from the monotonic behavior whenever there is a strain path change imposed to the sheet material. However, with sufficient straining in the final loading stage, the model gradually recovers the flow curve and strain increment direction (r-value) towards that of the isotropic hardening, except for the flow curves which are permanently softened by the state variables corresponding to prior loading history. Therefore, in the π -plane, the surface flattening due to the first step loading allows a larger stress range during second step loading in the orthogonal direction than during the initial (Fig. 5). The distortion also results in deviation of the plastic strain increment direction, as shown by the (blue) arrows on the HAH-distorted yield surface along the s2 - and plane strain-axis. In contrast to an isotropic condition after the first loading, the isotropic hardening, presumably, shows identical stress range and yield surface normal in any second loading case.

Fig. 5. Yield loci predictions after isotropic (IH) and distortional (HAH) hardening after pre-strain along s1 followed by second step loadings in s1, near plane strain and s2 axis. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) Table 2 Anisotropic parameters of the Yld20 0 0-2d. a

α1

α2

DP780 (1t) 6 0.955 0.960 CHSP45R (1.2t) 6 1.094 0.894

α3

α4

α5

α6

α7

α8

1.006

0.991

1.014

0.983

1.006

1.035

0.806

1.976

0.984

0.908

1.042

1.085

Fig. 6. Experimental and HAH-predicted DP780 flow curves before and after ∼6% uniaxial TD pre-strain in uniaxial tension at (a) 90° and (b) 45° from TD.

3.2. HAH model calibration rule The application of HAH model requires a sequential characterization of the yield function and the isotropic hardening flow rule. In this paper, the anisotropic Yld20 0 0-2d yield function (Barlat et al., 2003b) and the Swift law were used to represent and identify, in a classical manner, the yield locus and monotonic strain hardening curves, respectively. The anisotropic yield function coefficients (α 1 − α 8 ) are listed in Table 2. Afterwards, the HAHcoefficients for each material were determined from the best approximation of experimental flow stresses, r-values and yield locus after strain path change. For this work, the so-called reverse loading coefficients (k1 − k5 ) in the HAH model were manually determined from the flow curve after 90° tensile axis change. Other coefficients, such as the microstructure deviator rotation rate (k) and the cross-loading coefficients ( kL ,L, kS , kS and S) were obtained from the best approximation of the cross-tension flow curve and yield locus after pre-strain. The isotropic and distortional hardening coefficients are presented in Table 3. 4. Results and discussion 4.1. Flow stress evolution A transient response is observed in the reloading flow curves whenever there is a strain path change. Figs. 6 and 7 show the experimental (symbols) and HAH-predicted (lines) flow curves of the

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S.B. Zaman et al. / International Journal of Solids and Structures 000 (2017) 1–10 Table 3 Isotropic (Swift) and distortional (HAH) hardening coefficients. K

ε0

DP780 (1t) 1315 5.0 × 10−5 CHSP45R (1.2t) 733 6.5 × 10−3

n

k

k1

k2

k3

k4

k5

kL

L

S

ks & ks

0.141

140

130

20

0.47

0.94

8.0

•

•

0.46

8.0

0.167

35

250

6.0

0.45

0.95

15

70

5.6

0.50

7.0

• No latent hardening for DP780.

Fig. 7. Experimental and HAH-predicted CHSP45R flow curves before and after 6% uniaxial TD pre-strain in uniaxial tension at (a) 90° and (b) 45° from TD.

two-step uniaxial tension tests conducted on DP780 and CHSP45R sheet samples, respectively. The dotted vertical lines in every diagram refer to the uniform elongation point, after which the material starts necking. As mentioned earlier, the Swift law characterizes the monotonic stress-strain curves, while the HAH model predicts the flow curve transients after tensile axes change. Note that all the modeled graphs presented in this article before and after pre-strain are reproducible using the hardening coefficients listed in Table 3. Figs. 6a and 7a represent the reloaded stressstrain curves after 90° tensile axis change (ϕ ∼ = − 0.5). Both materials exhibit early re-yielding and pseudo-Bauschinger effect followed by permanent softening, all of which are precisely anticipated by the HAH-curves until the uniform elongation point. Figs. 6b and 7b show the tensile stress-strain curves after 45° tensile axis change (ϕ ∼ = 0). Early re-yielding was observed with DP780 whereas slight stress overshooting was noticed with CHSP45R, after which both materials re-aligned with their respective monotonic behavior. After early re-yielding and/or stress overshooting, the current version of the HAH model also suggests reloading flow stress identical to monotonic; thus, showing good agreement to both the experimental outcome.

Fig. 8. Experimental and HAH-predicted DP780 instantaneous r-values before and after 6% uniaxial TD pre-strain in uniaxial tension at (a) 90° and (b) 45° from TD.

Fig. 9. Experimental and HAH-predicted CHSP45R instantaneous r-values before and after 6% TD pre-strain in uniaxial tension at (a) 90° and (b) 45° from TD.

4.2. Evolution of the r-value After non-linear strain path change, the fluctuation of the instantaneous r-value represents a shift in the yield surface normal from its monotonic direction. The initial upsurge of the r-value is dependent on the magnitude of a single strain path change (ϕ in Eq. (1)), after which the instantaneous r-value gradually becomes identical to the monotonic r-value. Figs. 8 and 9 show the instantaneous r-value on DP780 and CHSP45R sheet samples after two-step uniaxial tension tests, respectively. Figs. 8a and 9a represent the instantaneous r-value evolution after 90° tensile axis change (ϕ ∼ = − 0.5). The HAH-curves predict an initial fluctuation followed by a relatively more rapid convergence of the r-value than what experimentally measured. Eventually, both the model and experimental data qualitatively comply with the convergence of the r-value. After 45° tensile axis change near cross-loading conditions (ϕ ∼ =0), due to smaller strain path change, Fig. 8b shows rather insignificant r-value fluctuation, which is well represented for DP780. Interestingly, the single-phase CHSP45R shows a permanent reduction of the r-value, which cannot be captured via the current HAH version.

Fig. 10. Experimental and HAH-predicted DP780 yield locus before and after 6% uniaxial TD pre-strain in (a) 2D- and (b) π -plane.

4.3. Yield surface evolution Figs.10 and 11 represent the normalized yield loci of DP780 and CHSP45R, respectively as a function of plastic work before and after 6% TD pre-strain. All the measured yield loci are plotted using uniaxial tension tests at orthogonal orientation, tensioncompression tests, and biaxial tension tests at 1:1, 1:2 and 2:1 load ratios at several plastic work points. Note that only these tests were used for the identification of yield surface distortion. The measured undeformed yield surface, which stabilizes at large strain values, was normalized and selected based on the plastic work approximately corresponding to 1.5% strains for DP780 and 3% for CHSP45R whereas the modelled yield surface, via the Yld20 0 02d anisotropic yield function, was established by incorporating the normalized flow stresses and r-values from the monotonic uniaxial and balanced-biaxial bulge tests at 100 MPa plastic work (ε¯ ∼ = 12%) for both steels. After pre-strain, Fig. 10a shows two distorted yield

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strain. In addition, the pre-strain causes greater spatial plastic flow range in the biaxial region than in the direction of pre-strain. Such a behavior cannot be predicted by any isotropic hardening concepts. 4.5. Discussion

Fig. 11. Experimental and HAH-predicted CHSP45R yield locus before and after 6% uniaxial TD pre-strain in (a) 2D- and (b) π -plane.

Fig. 12. Second loading plastic flow distribution in the DP780 after first step tension in the TD.

∼0.2%) and 3 MPa (ε¯r ∼ loci for DP780 at ∼1 Mpa (ε¯r = = 0.5%) plastic works (the subscript ‘r’ in ε¯r refers to reloading). According to Fig. 10b, two additional reloaded yield surfaces are plotted at ∼6 MPa (ε¯r ∼ =1%) and 13 MPa (ε¯r ∼ = 2%) plastic works. Similarly, Fig. 11a portrays two distorted yield loci at ∼1 MPa (ε¯r ∼ = 0.2%) and 2 MPa (ε¯r ∼ = 0.3%) plastic works followed by another yield locus at 4 MPa (ε¯r ∼ =0.7%) in the π -plane (Fig. 11b). It is important to note that all the distorted yield loci were equivalent to small plastic work because the surface distortion is only noticeable at the onset of reload. At moderately high plastic work, the distorted yield locus recovers and erases any forms of yield surface distortion near the active stress state. Furthermore, Figs. 10b and 11b include tensioncompression yield points opposite to pre-strain (s2 ) axis. To represent the overall evolution of yield surface, the 6% TD tension followed by compression test was conducted and the results are plotted in the π -plane in terms of the same plastic work for each material as mentioned above. The compression points show excellent agreement with the HAH model for DP780, but not for CHSP45R. Apart from the compression points in CHSP45R, the other prestrained uniaxial and biaxial loading progressions were successfully captured via the HAH approach for both the single- and dual-phase steels at several incremental offset strains. 4.4. Distortional hardening validation As discussed earlier in Section 3.1 (Fig. 5), the distortional HAH framework, after pre-strain, propounds a disproportionate distribution of the yield points in each principal stress axis in the π -plane. A similar distribution was experimentally observed with both the tested materials. Fig. 12 shows the plastic work distribution of DP780 steel in second-step loading condition after TD pre-strain. Due to the yield surface distortion, the DP780 sheet sample reyields at a lower stress in the RD than in the direction of pre-

The non-standard large-scale specimen, with 1:2 aspect ratio and ∼70° fillet angle, produced, after uniaxial tension, reasonably homogeneous strain in its gauge-section. Large-scale specimens with other combinations of aspect ratio and fillet angle were not taken into account in this work, although further investigation is needed to design a standard large-scale specimen and characterize the nature of the strain distribution. A quasi large-scale tensile setup for pre-strain was utilized by Yu and Shen (2014) and based on FE simulation, a large portion of the gauge was assumed to have uniform strain distribution after pre-strain. After straining large specimens, some sort of grid capturing method was adopted by Sun and Wagoner (2013) to locate sections of homogeneous strain, which was then used to cut sub-size specimens from the center of the gauge and also near the shoulder. From our study of strain distribution via the DIC method, it is clear that for a largescale specimen with 1:2-gauge aspect ratio, approximately 30–40% of the total gauge length constitutes of uniform strain. The rest of the gauge consists of other strain levels. In addition, recall that the large-scale specimens, in this work, were aligned to the TD (also possible with RD); any other alignment might hamper the homogeneity of strain. This assumption can primarily be investigated by analysing the strain distribution at fixed plastic deformation of standard uniaxial tensile specimens aligned at various angles from the rolling (see Appendix B). The reloaded flow stress after 90° tensile axis change is, somewhat, analogous to that of the reverse loading, with Bauschingerlike effect followed by permanent softening. Similar elastic-plastic reloading behavior of DP steels after non-linear strain path change were observed in Tarigopula et al. (2008), Sun and Wagoner (2013) and Ha et al. (2013). This is not surprising for such a loading sequence because ϕ as defined in Eq. (1) is ∼ -0.5; i.e., half way between cross-loading (ϕ = 0) and reverse loading (ϕ = −1). Besides, in terms of the amplitude of strain path change, the results show that the reloading flow stress is not recovered when ϕ < 0, although further tests involving small intervals in tensile axis change around cross-loading region is required for proper experimental validation. In this paper, the r-value is assumed to be constant for monotonic conditions. The plastic strain ratio is calculated at a strain range stretching from the yield point to the uniform elongation (UE) point or to 20% longitudinal strain if UE > 20% longitudinal strain. Note that r-value is not considered a material constant since it is dependent on the selected strain range in which it is measured and the subsequent crystallographic texture (Savoie et al., 1995). Hu (1975) postulated that the change of r-value with respect to strain depends on its deviation from unity. However, the evolution of plastic strain ratio does not depend on its magnitude when it is measured in segments of 2–3% longitudinal strains (An et al., 2013). This method of measuring r-value was defined as the incremental r-value method. For steel and aluminum alloys, the incremental r-value at large strains always decreases, which is not taken into account in this study, since the decrease is quite insignificant during monotonic loading. However, after strain path change, the fluctuations in r-value is rather drastic which is why the instantaneous r-value method is selected in preference to the incremental r-value because the former portrays plastic anisotropy of the pre-deformed material immediately after the strain path change. In addition, the change in the loading axis contributes largely to the changes in plastic anisotropy and the amplitude of that change

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is reflected precisely via the instantaneous strain ratio. In terms of the yield surface normal, the change in the tensile axis fluctuates the direction of plastic strain increment, the fluctuation being dependent on the value of ϕ (Eq. (1)). Larger change in ϕ from its monotonic value (ϕ = 1) results in higher fluctuation of the yield surface normal, which converges to its original monotonic orientation when the loading is unchanged for substantial strains; however, the latter behavior is not observed for CHSP45R near crossloading condition. From this study, it could be concluded that the instantaneous r-value fluctuation is inversely correlated to ϕ , with severe fluctuations transpiring whenever ϕ < 0. As mentioned above, the evolution of r-value during multiple loading sequence is presumably a consequence of the changing crystallographic texture, which also influences the yield surface. A polycrystal and texture-based plasticity (CTFP) model (An et al., 2011) used the textures of the initial and uniaxially deformed interstitial-free (IF) steels and observed early re-yielding in the biaxial direction (An et al., 2013). In addition, Sugimoto et al. (1985) reported that the cross-softening effect in DP steels is directly correlated to the amount of uniaxial pre-strain. The workhardening behavior of steel after non-linear strain path change was observed and modeled by Larsson et al. (2011) to conclude that a stand-alone kinematic hardening concept is insufficient to describe the material response during non-proportional loading conditions. Sun and Wagoner (2013) incorporated the Modified Chaboche model to capture the elastic-plastic behavior of DP steels after 45° and 90° strain path changes; however, the model reproduced the reloaded stress-strain curves only qualitatively, by overestimating the transient softening behavior. From these observations, we can conclude that for these non-linear strain path sequences, the current HAH model is likely to provide better prediction of the reloaded stress-strain curves than the modified Chaboche model in Sun and Wagoner (2013). The HAH coefficients for both materials are selected based on trial and error and the corresponding results showed that the socalled distortion-based model can be manually operated with reasonable accuracy. The coefficients were also simultaneously identified with an optimization algorithm, utilizing the cyclic simple shear test for optimizing the reverse loading parameters (k1 -k5 ). The optimization produced similar coefficients with negligible variation to the output results. In this article, two different steel grades were selected, namely DP780 and CHSP45R. In case of DP780, the dual-phase AHSS shows cross-loading contraction behavior upon strain path change. In Table 3, the coefficients controlling the latent hardening were inactive for DP780. In contrast, the CHSP45R demonstrated a coupling of latent extension and cross-loading contraction upon strain path change. The latent extension was observed after 45° strain path change while cross-loading contraction was noticed during pre-strained biaxial second-step loading sequence. From this study, it is clear that the current version of HAH model shows excellent results when it is utilized for sheet materials experiencing one of these two effects (Barlat et al., 2014) but fails to accurately capture the material response when both the effects are active in the same material. This work analyses the accuracy of the current HAH model to simultaneously capture the flow stresses, instantaneous r-values and yield surface after pre-strain using a single set of manually selected coefficients. Although the coefficients can be accurately predicted via the optimization algorithm, the current HAH model is also feasible and practical for manual parameter identification. Moreover, this investigation provides a clear understanding of what needs to be improved or modified for the next HAH formulation.

5. Conclusion The present article provides an exhaustive analysis of the behavior of sheet metals subjected to several non-linear strain paths in terms of the stress-strain, instantaneous r-value and yield surface evolution. The experimental and modeling strategies is cumulated to construct the following conclusions: • A novel design of a large-scale specimen and grip fixture for the application of pre-strains was successfully validated. • The pre-strained stress-strain curves showed early re-yielding, stress overshooting, Bauschinger-like effect and permanent softening. • After the first loading, the subsequent instantaneous r-value and yield surface showed considerable yield locus distortion. • The HAH approach successfully predicted the pre-strained flow curves and the distorted yield surface at several offset strains. • The distortional HAH model partially captured the instantaneous r-value evolution after the strain path change. • After the first-step loading, the disproportionate plastic flow range in the principal loading axes is successfully represented via the distortion-based HAH concept. Acknowledgements The authors are very grateful to POSCO for supporting this research. They would also like to express their appreciation and gratitude to Ms. Manon Wennagel for her co-operation in the largescale tensile experiments, Mr. Sylvain Fenot for optimizing the HAH coefficients via the simple shear tests and Dong-Do Engineering Co. for technical support. Appendix A. Instantaneous r-value formulation The instantaneous r-value and the average r-value is identical for monotonic loading. However, after any form of strain path change, the instantaneous r-value in the second loading sequence fluctuates, which, in this article, is evaluated using a 10th-order fitted polynomial, as shown in Fig. A1. The gradient of the curve gives the instantaneous r-value at every strain increment.

Fig. A1. Instantaneous r-value and average r-value calculation in the first and second loadings.

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state variable that controls cross loading contraction. Note that, in the absence of cross-loading contraction and latent extension, i.e.,  when gL = gS = 1,

(ω (s )2 + ω (sp )2 reduces to ω(s), Lastly, ac-

cording to the principles and operations of convex functions, since φ (s) is convex, the same is true for (s) (Rockafellar, 1972). Evoluˆ , g1 , g2 , gL , gS and two additional variables that tion equations for h characterize permanent softening complete the entire description of the model and can be found in Barlat et al. (2014). References

Fig. B1. Strain distribution inside the gauge section of a standard uniaxial DP780 specimen after tensile strain in 0°, 45° and 90° from the rolling direction.

Appendix B. Strain maps at different tensile loading angle Since one of our primary objectives was to procure homogeneous strain distribution after pre-strain, the first step loading operation was strictly performed along the TD (also possible with RD). Any other tensile loading angle would not produce uniform strain in the gauge-section because the strain distribution in rolled sheet metals are orientation-dependent. In order to verify this statement, tensile tests were performed with standard uniaxial DP780 specimens at different angles and the strain distribution was relatively more homogeneous along the RD and TD than at 45° from the rolling (Fig .B1). In this article, the large-scale specimens were loaded along the TD because the as-received sheet was not long enough to accommodate the large specimen along the rolling. Appendix C. HAH model chronicle The HAH model is defined by the homogeneous yield function

and the evolution equations. In this paper, an extension of the original model of Barlat et al. (2011), which is described by the following yield condition

 q (s ) ψ (s )2 + ψ (sp )2  

 

q 

 

 

q 1q 

+ f q 1 h : s−h : s + f q 2 h : s+h : s

= σ (ε¯ ),

(3)

is utilized (Barlat et al., 2014). In this enhanced HAH model, the two state variables f1 and f2 , associated to reverse loading, can be expressed with a more convenient set of variables g1 and g2 as



fk =

3 8

 1q

1 −1 gqk

(4)

σ (ε¯ ) is a monotonic reference curve characterizing isotropic hardening. In order to account for cross-loading, a yield surface extenˆ is introduced sion or contraction in  a direction orthogonal to h though the expression

ω (s )2 + ω (sp )2 , which replaces ω(s) in

Eq. (2). The function ω(s  ) is defined using the decomposition of the stress deviator in two terms, collinear and orthogonal to ˆ , namely, s = sC + sO . The state variable that capture latent hardh ening, gL = 1/η, is used in the linear transformation s  = sC + ηsO . ˆ )h ˆ , with H a constant, and It can be shown that sC = H (s : h (1−g ) sO = s − sC is used in the relationship sP = 4 g S sO . gS is the L

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