Damage initiation and fracture loci for advanced high strength steel sheets taking into account anisotropic behaviour

Journal of Materials Processing Tech. 248 (2017) 218–235

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Research Paper

Damage initiation and fracture loci for advanced high strength steel sheets taking into account anisotropic behaviour K. Charoensuka, S. Panichb, V. Uthaisangsuka,

MARK

⁎

a

Department of Mechanical Engineering, Faculty of Engineering, King Mongkut’s University of Technology Thonburi, 126 Pracha Uthit Road, Bang Mod, Thung Khru, Bangkok 10140, Thailand b Department of Production Engineering, King Mongkut’s University of Technology North Bangkok, 1518 Pracharat 1 Road, Wongsawang, Bangsue, Bangkok 10800, Thailand

A R T I C L E I N F O

A B S T R A C T

Keywords: Advanced high strength steels Anisotropy Fracture loci Damage initiation Forming limit curve

In this work, ductile failure loci for advanced high strength (AHS) steel sheet grade 780 and 1000 were determined by using a combined approach between experiments and FE simulations. Effects of anisotropic behaviour of the steels were considered. Tensile tests of sheet specimens with various geometries taken from the rolling, transverse and diagonal direction were performed. During the tests, the direct current potential drop (DCPD) method and digital image correlation (DIC) technique were applied for identifying damage initiation on the micro–scale and fracture occurrence of the steels, respectively, under different states of stress. Subsequently, FE simulations of the tensile tests were carried out and stress triaxialities, equivalent plastic strains and Lode angles were evaluated for the corresponding critical areas. Hereby, the von Mises, Hill’s 48 and Yld2000–2d yield criteria coupled with the Swift hardening law were applied. The threshold values within the triaxiality range of 0–0.577 obtained from different samples were plotted as the 3D failure loci of the examined steels. The predicted damage initiation states were also verified by SEM analysis. Then, influences of different material orientations and yield functions on the shape alteration of the determined failure locus were studied. To investigate formability of the steel sheets the damage initiation and fracture loci were transformed to strain based forming limit curves (FLCs). Additional limiting strains for the shear stress region were here incorporated. Finally, a non–symmetric rectangular cup test was conducted for the investigated steels until fracture. Then, plastic strain distributions, strain paths of the critical areas and achieved drawing depths were evaluated by the FLCs. It was found that the proposed FLCs could fairly predict fracture incidence of the formed parts under shear deformation. Moreover, damage initiations were predicted at about 70% of the final drawing depth.

1. Introduction Nowadays, the automotive industries have been rapidly grown up and investments are greatly increased due to much higher technological competitions. By the manufacturing of automotive parts and components, sheet metal forming technology belongs to one of the most important sections. To achieve lighter vehicles with reduced fuel consumption but improved safety performance advanced high strength (AHS) sheet steels have been progressively applied. Such steel grades exhibit superior strength and fair elongation when comparing with other low carbon steels with the same strength. However, these steels have still shown difficulties during their forming processes because of their complex microstructural characteristics and consequently unexpected failure behaviour. On the one hand, design of forming procedures, dies as well as part shapes and geometries needed to take into

⁎

Corresponding author. E-mail address: [email protected] (V. Uthaisangsuk).

http://dx.doi.org/10.1016/j.jmatprotec.2017.05.035 Received 27 November 2016; Received in revised form 22 May 2017; Accepted 27 May 2017 Available online 29 May 2017 0924-0136/ © 2017 Elsevier B.V. All rights reserved.

account this concern. On the other hand, small micro–cracks could occur in formed components and subsequently cause premature failure in operation or lowered durability. Therefore, methods or approaches for predicting damage occurrences of AHS steel parts during their forming processes with higher accuracies are necessary. Until now, various fracture criteria for ductile materials have been studied and developed. One of the mostly applied ductile fracture models was introduced by Gurson (1975). The model was developed on the basis of void nucleation and growth, which was supposed to be the main factor for causing failure in porous ductile materials, as reported in Gurson (1977). This model was further enhanced by Tvergaard (1981) and Tvergaard and Needleman (1984), in which secondary voids, which took place after a certain deformation, accelerated void coalescences and three adjusted parameters were included. It was shown that more precise prediction could be hereby achieved. The model proposed by

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used shear tests for sheet metal characterization was provided. Otherwise, various failure criteria and approaches have been also developed for predicting damage and fracture of sheet materials. However, in the case of sheet metal parts with complex geometries or made of higher steel grades, more accurate tool or criterion is yet needed for the manufacturing process. Bao and Wierzbicki (2004) emphasized that stress triaxiality was the most important factor, which governed ductile fracture initiation, beside the strain magnitude. A series of tests including upsetting tests, shear tests and tensile tests of an aluminum alloy was carried out. Based on experimental and numerical results it was observed that fracture mode of the material could be divided into three distinct regions. For negative stress triaxialities, fracture was governed by shear deformation. For large stress triaxialities, void evolution was the dominant failure mode. At low stress triaxialities between the two regimes, a combination of shear fracture and void growth was found. Bai and Wierzbicki (2010) applied the Mohr–Coulomb (M–C) fracture criterion for describing ductile fracture of isotropic crack–free solids. It was shown that the hydrostatic pressure and Lode angle parameter significantly controlled fracture appearance of ductile metals. In this work, the M–C criterion was transformed to the spherical coordinate system containing the axes of equivalent strain to fracture, stress triaxiality and normalized Lode angle parameter. An aluminum alloy and high strength steel grade TRIP690 were used to calibrate and validate the proposed fracture model. It was found that the fracture locus could precisely predict material ductility in dependence on the stress triaxiality. For conventional deep drawing steel sheets, necking has been the dominant failure mode. Nevertheless, AHS steel sheets showed a typical shear fracture, which could not be simply predicted by the forming limit curve (FLC). Li et al. (2010) applied the fracture locus, which was represented on the plane of the equivalent strain to fracture and the stress triaxiality, to predict crack initiation and propagation of the AHS steel grade HCT690T during a series of deep drawing–punch test. This fracture locus was based on the modified Mohr–Coulomb fracture criterion (MMC). Furthermore, the 2D fracture locus was transformed to the space of principal strains with two new branches, which exhibited the formation of shear–induced fracture. Gruben et al. (2011) studied the fracture behaviour of AHS steel sheet under quasi–static loading conditions. The fracture occurrences were characterized by using the digital image correlation (DIC) technique in combination with FE simulations of various mechanical tests. Hereby, a method for determining the stress triaxiality and Lode angle parameter based on the DIC measurements was shown. On the other hand, Lian et al. (2012) reported that the onset of damage and subsequent damage evolution were the key factors in the application of AHS steel sheets. Therefore, a microscopic description for describing the damage onset was necessary by the modeling of ductile damage. In this study, a non–quadratic yield function with consideration of the Lode angle effect was applied. It was found that this model could predict the plastic behaviour of the examined steel more precisely than the conventional J2 plasticity model. Note that only isotropic material behaviour was yet assumed for the calculations. Additionally, a phenomenological criterion, which incorporated influences of the stress triaxiality and Lode angle parameter, was introduced for describing the damage initiation of the steel. The stress state significantly affected the ductile crack initiation locus. Later, investigation of damage initiation on the micro–scale of DP steel under various stress states was done by Lian et al. (2014) by means of a numerical method based on representative volume element (RVE) FE simulations on the microstructural level. Hereby, the plastic strain localization was used as the criterion for damage initiation in the RVE modeling without any other damage models or imperfections. It was stated that the local damage initiation also showed the dependency on both stress triaxiality and Lode angle. Sirinakorn et al. (2015) investigated influences of microstructure characteristics on forming limits behaviour of DP steel sheets. Hereby, micromechanics models were applied to predict failure occurrence in the microstructures by considering plastic instability due

Gurson (1977) and Tvergaard and Needleman (1984) has been widely applied to predict failure of ductile materials in many previous works. Besson et al. (2001) developed a damage model based on the Gurson–Tvergaard–Needleman (GTN) model for presenting the crack growth in round bars and plane strain specimens, which showed cup and cone fracture at the end. However, procedures for determining model parameters were not described. Also, the predictions provided by the model were not verified with any experimental results. One crucial concern of the GTN model has been the large number of material parameters and extensive identification procedures. This led to a limitation in industrial applications. Faleskog et al. (1998) and Gao et al. (1998) used cell model as a predictive tool for nonlinear fracture analysis of the examined material. A key feature of this computational model was the description of material in front of the crack represented by a layer of similarly–sized cubic cells. Each cell contained a spherical void of initial volume fraction of f0 and subsequently resulted in void growth and coalescence as defined by the GTN model. The micromechanics model was firstly calibrated taking into account both strain hardening and strength of the material. Then, the model was successfully applied to predict load, displacement and crack growth histories in specimens considering two crack geometries with different crack tip constraints and crack resistance behavior. Lemaitre (1985) presented an integrated model of ductile plastic damage, which was developed on a thermodynamic and effective stress concept. It was shown that the damage was linear with equivalent strain and was largely influenced by the triaxiality. Its validity range was limited by the assumption of isotropic plasticity, isotropic damage and constant triaxiality ratio during loading. Dhar et al. (2000) applied Lemaitre's model in large deformation elastic–plastic FE simulations for studying mode I ductile fracture in AISI1095 steel. It was stated that the proposed criterion is acceptable by predicting the critical load for crack growth initiation in the material. The ductile fracture process was directly influenced by both the plastic strain and triaxiality. Chaboche (1993) proposed another phenomenological approach of continuum damage mechanics for describing fracture processes of elastic solids under consideration of anisotropic and unilateral damage. Hereby, anisotropic damage was basically assumed and tensorial damage variables were introduced. A relationship between small crack occurrences and strain direction was examined. Hammi et al. (2003) described anisotropic ductile damage behaviour of Al–Si–Mg alloys. The developed damage–plasticity coupling model was based on the effective stress concept, in which effects of damage tensors on the deviatoric and hydrostatic part were taken into account. Therefore, the induced damage anisotropy was mainly driven by the void nucleation as a function of the plastic strain rate tensor. For sheet metal forming, Behrens et al. (2012) reported an approach using experimental and numerical analyses for characterizing flow and fracture behaviour of cold rolled dual phase (DP) steels under a wide range of plane stress states. A modified Miyauchi shear test was here carried out, in which plastic strain localization at the sample edges was noticeably reduced. Therefore, shear fracture behaviour, which was critical for deformed steel sheets, could be more precisely examined. Björklund and Nilsson (2014) clearly reported that failure in ductile sheet metals was principally induced by tensile fractures, shear fractures or localized instability. The state of stress occurred during plastic deformation of DP steel sheets strongly affected their failure mechanisms, which was verified by scanning electron microscope (SEM). Furthermore, FE simulations were performed to determine effective plastic strains at failure as a function of the average stress triaxiality and average Lode parameter. Generally, shear test has been frequently used for sheet metals in order to achieve large deformations without plastic instability. Yin et al. (2014) developed different shear tests for extensively investigating material behavior under shear conditions. In this work, results of the shear test proposed by Miyauchi, using sample according to the ASTM standard and an in–plane torsion test were compared. Here, a unique overview of the most commonly 219

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to incompatible deformation of each existing phase and governing stress state. With regard to the effects of material anisotropy of AHS steels, Mohr et al. (2010) evaluated the predicative capability of different yield functions for multi–axial loading conditions of AHS steel sheets under large deformation. In this study, the isotropic von Mises, the orthotropic Hill’s 48 and a non–associated yield model were considered. Although, the tensile stress–strain curves of these steels appeared to be direction–independent, the r–values showed a pronounced dependency on the in–plane direction. It was observed that by using the non–associated flow rule the resulted force–displacement curves and the position of strain localization band were more accurately predicted. Björklund and Nilsson (2014) investigated two AHS steel grades, in which a constitutive model with consideration of plastic anisotropy and mixed isotropic–kinematic hardening was used. To describe ductile and shear fracture the models presented by Cockroft–Latham and Bressan–Williams were applied. The results were presented in the form of FLC and prediction of force–displacement response of the Nakajima test. Luo et al. (2012) studied the anisotropic ductile fracture of extruded 6260–T6 aluminum alloy by means of a hybrid experimental–numerical approach. Tensile tests of various samples covering a wide range of stress states and different material orientations were conducted. A strong dependency of the strain to fracture on the material orientation with respect to the loading direction was observed. Additionally, it was shown that the proposed uncoupled non–associated anisotropic fracture model could provide accurate predictions of the onset of fracture for the tested specimens. Therefore, it was clearly illustrated that fracture and damage initiation of AHS steels play an important role for their successful manufacturing and applications. Moreover, plastic anisotropy of such steels could influence their failure behaviour so that it should be taken into account by the formability prediction. In this work, ductile damage initiation and fracture of two AHS steel grade were investigated and described by a relationship between plastic strain, stress triaxiality and Lode angle parameter. Tensile tests of sheet samples with various geometries for a wide range of stress triaxiality values, in particular shear deformation, were carried out. The direct current potential drop (DCPD) method was carried out in order to experimentally characterize the instant and location of microcrack initiation during tensile tests of sheet samples with various geometries. In parallel, the crack appearances on the microstructural level were validated by SEM analyses. The strain and stress distribution in the critical zones of samples determined by DIC technique and FE simulations, respectively, were compared and discussed. The influences of different material orientations and yield criteria on the shape alteration of the determined failure locus were examined. Finally, the failure loci were transformed to strain based forming limit curves and used to evaluate the damage initiation and fracture of examined steels during a non–symmetric rectangular cup test.

Table 1 Chemical composition of the investigated DP780 and DP1000 steel (in wt.%). Steel grade

C

Si

Mn

P

S

Al

DP780 DP1000

0.130 0.134

0.602 0.840

2.126 2.190

0.016 0.018

0.002 0.004

0.019 0.026

similar carbon and manganese contents. The microstructures of the steels taken by light optical microscope are presented in Fig. 1. Generally, the microstructures exhibited fine martensitic islands (gray color) dispersed in ferritic matrix (white color). Nevertheless, the DP1000 steel showed somewhat higher martensitic phase fraction. From the phase analysis, it was found that the DP780 steel sheet consisted of about 42% martensitic phase fraction, whereas the DP1000 steel sheet contained about 50% martensitic and 8% bainitic phase fractions. These microstructure characteristics would significantly lead to different mechanical properties of the steels. Uniaxial tensile tests of the investigated steels were performed on a universal testing machine using the ASTM E8 standard specimen. To study anisotropic behaviour, the sheet samples were prepared in three directions, namely, 0°, 45°, and 90° with regard to the rolling direction (RD). The constant strain rate of 0.001 s−1 was kept for all the tests. During the tests, longitudinal elongation and width reduction of the gauge length region were measured by means of both extensometer and DIC technique. Then, stress–strain curves and all characteristic tensile properties including r–values of the steels were determined for different sample orientations. Additionally, hydraulic bulge tests were carried out on an Erichsen bulge/FLC tester model 161 for determining the flow behavior of the steels under biaxial stress state. The velocity of the punch of 12 mm/min was used. The oil pressure was recorded by a pressure sensor, whereas in–plane elongations were measured by means of a combination of extensometer and a linear variable differential transformer (LVDT) attached on specimens. The membrane stresses and thickness strains were calculated for obtaining the flow stress curves in balanced biaxial tension. The detailed experimental procedure and tooling geometries of the test can be found in Panich et al. (2013). Besides, disk compression tests were carried out for examining plastic anisotropy of the steels under a balanced biaxial stress state. The procedure for the through thickness disk compression test was described in details in Panich et al. (2013) as well. Afterwards, the r–values and the balanced biaxial r–value were obtained by a linear approximation of the true width and thickness strains measured at about 14% of the total elongation. Moreover, the r–values were also determined by means of the DIC technique in order to validate the values gathered from both methods. From these results, the required material parameters of applied yield criteria were subsequently calculated for the investigated steels. 2.2. Identification of crack initiation and fracture state

2. Experimental The plastic deformation of metals is directly connected with the movement of dislocations. It was shown in Stroh (1954) that a crack was formed when a number of dislocations piled up under the stress magnitude took place in a cold–worked metal. These developed stresses were found to be with respect to the initiation of crack. The crack length likely depended on the amount of plastic flow occurring around the crack tip. On the other hand, void evolution in metals during plastic deformation is considered as the main driving factors for ductile fracture. The development of ductile failure proceeds in three successive steps, namely, void nucleation, void growth, and void coalescence. Hereby, the last step could be referred to the state of damage or crack initiation in the material. Moreover, voids formation in steels could also be caused by the interface cracking between matrix and precipitates or second phases after a certain plastic strain. Such kind of micro–cracks, which are potentially induced in a component during forming process,

In this work, it was aimed to determine threshold locus for damage initiation and fracture of two AHS steels by using a hybrid approach between experiments and FE simulations. Hereby, the anisotropic behaviour of steel sheets was taken into account. Tensile tests of samples with various geometries taken from three materials orientations were investigated. The DCPD method and DIC technique were used to identify the moment of microcrack initiation in the specimens during plastic deformation. 2.1. Material characterization The steel sheets examined in this work were the AHS steels grade DP780 and DP1000 with the initial thickness of 1.4 mm. The chemical compositions of the steels are given in Table 1. Both steels have nearly 220

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Fig. 1. Observed microstructure of (a) the DP780 steel and (b) the DP1000 steel.

Fig. 2. (a) Setup of the DCPD method on a universal testing machine and (b) obtained force–time and voltage–time curves of a tested specimen.

Fig. 3. Geometries of tested sheet specimens for tensile tests.

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Fig. 4. Determined true stress–strain curves of (a) the DP780 steel and (b) the DP1000 steel from each direction. Table 2 Tensile properties in different directions of the investigated DP780 steel.

Table 8 Anisotropic coefficients of the Yld2000–2d model for the DP1000 steel.

Test direction

YS

UTS

Elongation (%)

(degree)

(MPa)

(MPa)

Uniform

Total

0° 45° 90°

466.2 460.8 468.7

833.6 810.8 836.3

15.2 14.2 14.5

24.3 23.6 22.3

YS

UTS

Elongation (%)

(degree)

(MPa)

(MPa)

Uniform

Total

0° 45° 90°

630.5 660.0 691.4

990.5 984.7 1034.7

9.70 7.20 7.76

18.8 16.2 14.5

0°

45°

90°

Balanced biaxial

Normalized flow stress r–value

1.000 0.99

0.988 1.08

1.005 1.16

1.069 0.95

Table 5 Normalized flow stresses and r–values of the investigated DP1000 steel. DP1000

0°

45°

90°

Balanced biaxial

Normalized flow stress r–value

1.000 1.08

1.046 0.95

1.096 0.98

0.989 1.22

Table 6 Anisotropic coefficients of the Hill’s 48 model for the DP780 and TRIP780 steels. Steel grade

F

G

H

N

DP780 DP1000

0.4282 0.5232

0.5028 0.4808

0.4972 0.5192

1.4157 1.4670

Table 7 Anisotropic coefficients of the Yld2000–2d model for the DP780 steel. α1

α2

α3

α4

α5

α6

α7

α8

0.971

0.991

0.876

0.950

0.967

0.811

0.994

1.087

α3

α4

α5

α6

α7

α8

1.136

0.813

1.210

0.978

0.960

0.918

0.960

0.862

Steel grade

K

ε0

n

DP780 DP1000

1326 1414

0.002 0.004

0.136 0.110

can cause an earlier failure or reduced product life time. The DCPD method, as shown in Fig. 2a, which has been commonly used to identify the beginning of stable crack in various fracture mechanics tests, was applied in this work because of its simplicity and ability to determine material degradation on the micro–scale (Lian et al., 2012; Panich et al., 2016). The DCPD approach is based on the Ohm's law, which exhibits the relationship between electric voltage and resistance. A DC power supply with controller was used to provide a constant electrical current to the tested samples, while a multimeter was used to measure potential drop between the gauge length or notch areas, as depicted in Fig. 2. The current was chosen with regard to the dimension of used specimens in order to prevent a critical heating on the specimens. Since the magnitude of applied current was small so that insulators between samples and fixtures could be omitted. Note that it was found that experiments using plastic insulators showed no noticeably varying results. Generally, the determined potential slightly increased at the beginning because of the reduced cross section area of specimens. After a certain plastic deformation of tested sheet samples, void coalescence or microcrack formation occurred that caused severe defects and a sudden increase of electrical resistance and potential. Therefore, the abrupt slope change of potential curve measured for the critical location could be used to indicate the state of damage onset of a concerned material. Hereby, the points of slope change were approximately obtained from a second order derivative. It was supposed that coalescences of voids and resulting formation of micro–cracks took place at the first discontinuity of the voltage–time curve, as illustrated in Fig. 2b. The distinct increase of electric potential at the nearly final stage was due to the final macroscopic fracture. Note that the similar DCPD method was also utilized by Lian et al. (2012) and Lian et al. (2015) to detect microcrack initiation of high strength low alloy steels. Furthermore, interrupted tensile tests of different samples were carried out until the corresponding critical forces determined from the potential curves. Subsequently, metallographic analyses were done for the deformed specimens within gauge area to verify the microcrack initiation. The obtained results can be found in the following section. To describe damage initiation and fracture of the examined steels under various states of stress tensile tests of different sample geometries

Table 4 Normalized flow stresses and r–values of the investigated DP780 steel. DP780

α2

Table 9 Materials constants of the Swift hardening model for the investigated steels.

Table 3 Tensile properties in different directions of the investigated DP1000 steel. Test direction

α1

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Fig. 5. Determined force–voltage–time curves from tensile tests and DCPD method of (a) shear butterfly, (b) combined load, (c) R5 notch and (d) U notch sample in different directions to RD for the investigated steel DP780.

Fig. 6. Determined force–voltage–time curves from tensile tests and DCPD method of (a) shear ASTM, (b) combined load, (c) R30 notch and (d) V notch sample in different directions to RD for the investigated steel DP1000.

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Fig. 7. Comparisons between experimentally and numercially determined force–displacment curves from tensile tests of (a) butterfly shear, (b) combined load and (c) R30 notch sample from 0° to the RD for the steel DP780 in case of using the Yld2000–2d model.

Fig. 8. Comparisons between experiemntally and numerically determined force–displacment curves from tensile tests of (a) ASTM shear, (b) combined load and (c) R30 notch sample from 0° to the RD for the investigated steel DP1000 in case of using the Yld2000–2d model.

Fig. 9. SEM micrographs of the (a) shear sample, (b) combined load sample, c) R30 notch sample (zone a) and (d) R30 notch sample (zone b) at the state of damage initiation of the investigated steel DP780.

were gathered. Total seven different types of sheet specimens were taken into account, namely, shear (ASTM), shear (butterfly), combined load, R30 notched, R5 notched, V notched and U notched samples, as depicted in Fig. 3. These varying sample geometries would induce

were performed in combination with the DCPD method and DIC technique. From the tensile tests with the DCPD method, electric potential curves between the gauge section, which was the most critical area of each specimen and where micro–cracks were supposed to occur, 224

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Fig. 10. SEM micrographs of the (a) shear sample, (b) combined load sample, c) R30 notch sample (zone a) and (d) R30 notch sample (zone b) at the state of damage initiation of the investigated steel DP1000.

the balanced biaxial r–value from the disk compression tests are shown in Tables 4 and 5 for the DP780 and DP100 steel sheets, respectively. It was found that the r–values from the linear approximation and DIC method deviated from each other less than 10 percent. Thus, these r–values could accurately describe the anisotropic behaviour of the steels. Obviously, the r–values of the DP780 and DP1000 steels at different directions were close to 1. Nevertheless, these small differences of r–values could affect material formability of the steels, as seen in Tables 2 and 3. Then, the anisotropic coefficients of the Hill’s 48 yield criterion could be directly calculated from the r–values at 0°, 45°, and 90° to the RD and the yield stress in the RD, as illustrated in Table 6 for both steel sheets. From the tensile tests and hydraulic bulge test, the normalized flow stresses of examined steels were also calculated and given in Tables 4 and 5. In case of the Yld2000–2d model, the eight experimentally obtained properties, namely, flow stresses and r–values at 0°, 45°, 90° to the RD and balanced biaxial state were used in conjunction with a system of non–linear algebraic equations to determine the necessary anisotropic coefficients. Since the DP780 and DP1000 steels mostly consisted of the ferritic matrix, the m exponent of the Yld2000–2d function was set to be 6, which was recommended for metals with BCC lattice. Finally, the anisotropic coefficients of the Yld2000–2d yield criterion were obtained and summarized for the DP780 and DP1000 steels in Tables 7 and 8, respectively. Afterwards, the Swift hardening law was used to describe the isotropic strain hardening behaviors of the steels. The material constants K, n and εo were determined for the samples taken from the 0° to the RD and are given in Table 9 for the DP780 and DP1000 steel. It could be observed that the DP780 steel exhibited somewhat larger work hardening rate than the DP1000 steel.

different stress triaxialities and Lode angle in the material during plastic deformation. The stress triaxiality values between 0 and 0.5774 were possible by this specimen shape series. All specimen types were prepared along the 0°, 45° and 90° to the RD from as–delivered steel plates. During the tensile tests, the moments of damage or microcrack initiation were evaluated for various stress states by means of the DCPD measurement. Furthermore, the DIC technique was used to precisely identify the critical location and state shortly before fracture of each deformed sample. Hereby, local strains developed on the tested specimens were determined. Afterwards, FE simulations of the tensile tests were performed using various yield functions for all used specimen types. Force and displacement curves from the experiments and simulations were compared and the moments of damage initiation in each specimen were then determined in the simulations. To distinguish the states of fracture in the simulations, local strain paths within the critical area of each sample were gathered and compared with those obtained from the DIC technique.

3. Resulted tensile properties and material parameters The true stress–strain curves of the DP780 and DP1000 steels, which were determined under different loading directions, are depicted in Fig. 4. It is seen that the DP1000 steel exhibited considerably higher strength and work hardening rate than the DP780 steel. Additionally, yield strength, ultimate tensile strength, uniform elongation, total elongation and r–values, which were gathered for each testing direction, are summarized for the DP780 and DP1000 steels in Tables 2 and 3, respectively. For both steels, specimens from the 90° to the RD exhibited the highest yield and tensile strengths, but the lowest total elongations in comparison to those from other directions. In contrast, specimens from the 0° to the RD showed significantly larger uniform and total elongations than others. Hereby, anisotropic yield and damage behaviour of the steels played an important role. The r–values determined at about 14% of the total elongation, and

4. Determination of damage initiation and fracture loci To generate failure curve for representing damage initiation of the examined steels, force–time and voltage–time curves obtained from the tensile tests and DCPD method were firstly correlated to each other and 225

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Fig. 11. Comparisons of plastic strain distributions on each test specimen at a state shortly before fracture determined by the DIC method and calculated by FE simulations coupled with the Yld2000–2d model for the investigated DP1000 steel.

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Fig. 12. (a) Damage initiation and fracture loci calculated by different yield functions and (b) fracture loci calculated by the Yld2000–2D yield model for the DP780 steel under various loading directions.

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Fig. 13. (a) Damage initiation and fracture loci calculated by different yield functions and (b) fracture loci calculated by the Yld2000–2D yield model for the DP1000 steel under various loading directions.

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hardening model was employed to describe isotropic work hardening characteristic of the steels. The elastic–plastic materials data, as discussed in the above section, were given. The 4 node shell element was used and the element size within the critical area was defined to be about 0.2 × 0.2 mm2. Geometries of the specimens and applied boundary conditions were defined with regard to those in the experiments. To verify the simulation results, force–displacements curves for different loading directions and sample shapes obtained from the experiments and simulations were compared, as shown in Figs. 7 and 8, for the steel DP780 and DP1000 in the RD, respectively, as examples in case of applying the Yld2000–2d yield criterion. It is obvious that the experimental curves acceptably agreed with the calculated curves until the maximum points. Note that after these maximum loads all experimental results were overestimated by the predictions, since no damage criterion was specified in the simulations. For the samples from other directions, similar tendencies were obtained. Furthermore, the r–values predicted by FE simulations coupled with different yield functions were compared with the corresponding experimental results of both steels. It was observed that the numerical results fairly agreed with the experimental values. Therefore, the simulations could be further used to evaluate local stress and strain responses of each deformed specimen. Subsequently, the moments of damage initiation in the simulations were identified according to the results from the DCPD method. At these states, stress triaxiality, equivalent plastic strain and Lode angle of the critical elements with highest plastic strain from each specimen in different testing directions were then gathered. These values were further used to construct damage initiation loci of the steels. In addition, SEM analyses were performed for different sheet samples of both steels, which were deformed by tensile test until the moment of damage initiation provided by the DCPD method. For this examination, the shear, combined load and R30 notch specimens, which represent low and high stress triaxiality conditions, were taken into account. To observe damage occurrence in the steels, sheet samples after interrupted tensile tests were cut in the middle along the loading direction. Then, SEM observation was conducted on the thickness plane within the critical area of each specimen. It was supposed that these locations were directly governed by the characteristic stress triaxiality value of each sample shape. The SEM images taken from the different samples of the DP780 and DP1000 steel are presented in Figs. 9 and 10, respectively. Obviously, hints of damage appearance in the microstructure of both steels were found in form of voids in different sizes and small cracks resulted from void growth and void coalescence. Observed voids or micro–cracks mostly emerged in the vicinity of phase boundaries. However, they were characterized as reported by Lian et al. (2014). It was shown that the pattern of damage initiation in such DP steels could be classified into two types. First, micro–scale damage took place at the interfaces between ferrite and martensite, which are revealed by circular marks in Figs. 9 and 10. Second, damage could occur due to the breaking of martensite islands, which are marked by rectangular symbols in Figs. 9 and 10. Note that the zone a and zone b in Figs. 9 and 10 located close to each other within the gauge or notch region. It was aimed to present that micro–cracks could be observed anywhere in the critical region of the R30 notch specimens at the instant identified by the DCPD method. The damage initiations were found in the samples loaded under both low and high stress triaxiality regions, similar to the results in Lian et al. (2014) and Ahmad et al. (2000), in which damage mechanism of

Table 10 Determined material constants of fracture model according to Bai and Wierzbicki (2008) for the fracture loci of DP780 steel in the 0° direction. Yield model

D1

D2

D3

D4

D5

D6

Von Mises Hill 48 Yld2000

1.67 1 0.83

−0.11 −3.03 −3.34

1.29 0.79 0.75

1.65 −0.29 −0.39

7.21 13.06 14.05

0.14 −0.92 −0.89

Table 11 Determined material constants of the fracture model according to Bai and Wierzbicki (2008) for the fracture loci of DP1000 steel in the 0° direction. Yield model

D1

D2

D3

D4

D5

D6

Von Mises Hill 48 Yld2000

1.72 1.67 0.54

4 −0.27 −3.59

0.87 1.25 0.52

1.45 2.43 −0.83

17.11 5.68 6.19

0.5 −0.45 −1.74

moments of damage occurrence were evaluated. The points, when the voltage–time curves apparently changed their slopes, were determiend by the second order derivative. At the same points, the current forces and displacements were gathered and compared with those from the FE simulations to specify the damage initiation states. For example, Fig. 5a–d show the obtained force–voltage–time curves from tensile tests and DCPD method of the DP780 steel for butterfly shear, combined load, R5 notch and U notch samples in different directions to RD, respectively. It is seen that the force–time curves of sheet samples from varying directions were slightly deviated. For the DP780 steel, the results by the DCPD method exhibited that damage initiations took place in the 90° samples earlier than other samples at all states of stress. By the same manner, damage initiation states in the DP1000 steel sheets during the tensile tests were also evaluated. For instance, the force–voltage–time curves of the steel are illustrated in Fig. 6a–d for ASTM shear, combined load, R30 notch and V notch samples in different directions to the RD, respectively. It was observed that the 90° samples showed damage initiation at earlier states than samples from other directions under all loading conditions, which was similar to the DP780 steel. These determined damage initiation states at varying directions were well correlated with the elongations from the tensile tests of both investigated steels. All the damage initiation points took place between the yield points and maximum loads. Note that in case of the ASTM shear samples the states of damage occurrences were difficult to detect, since the samples were broken very early, when voltage values were quite small. FE simulations of tensile tests in different directions were carried out for the samples with various shapes, in which the yield criteria, namely, von Mises, Hill’s 48 (Hill, 1948) and Yld2000–2d (Barlat et al., 2003), were applied for considering plastic anisotropic behavior of the investigated steels grade 780 and 1000. It is obvious that material anisotropy could have significant influences on local material properties and thus resulting stress and strain responses of steel sheet during forming. For the von Mises yield criterion, the flow stress curve from each direction was defined in the corresponding simulation. In case of FE simulations with the Hill’s 48 model, the anisotropic parameters F, G, H and N and yield stress in the RD of the steels were used, while FE simulations with Yld2000–2d criterion required the anisotropic parameters α1–α8 and yield stress in the RD. The rate–independent Swift

Table 12 Determined critical strains at damage initiation of the investigated DP780 steel at varying loading directions and states of stress. Direction

Butterfly

ASTM

Combined loading

R30 notch

R5 notch

U notch

V notch

0° 45° 90°

23.4 25.1 22.1

23.5 23.0 23.7

26.2 23.5 25.0

37.5 34.0 33.7

24.7 23.2 22.1

16.7 15.7 14.5

24.5 22.3 20.4

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Table 13 Determined critical strains at damage initiation of the investigated DP1000 steel at varying loading directions and states of stress. Direction

Butterfly

ASTM

Combined loading

R30 notch

R5 notch

U notch

V notch

0° 45° 90°

22.4 22.1 20.1

23.8 23.0 23.1

23.0 20.1 20.5

32.5 30.1 28.0

23.4 21.5 20.3

14.5 13.6 11.5

21.5 20.3 18.5

Fig. 14. (a) relationship between stress triaxiality and plastic fracture strain of the investigated DP1000 steel and (b) relationship between stress triaxiality and Lode angle parameter of the investigated DP780 steel in comparison with other AHS steel and aluminium alloy.

provided higher plastic strains. The R30 notch sample had the largest critical plastic strain. Subsequently, local stresses and strains from the most critical element of each sample at the identified fracture states were determined. They were used to calculate the corresponding Lode angle values for each sample under different loading directions and fracture loci of the steels could be finally generated. Note that effects of different yield criteria on both damage initiation and fracture loci were also examined. By means of the tensile tests of different samples in combination with the DCPD, DIC method and FE simulation coupled with von Mises, Hill’s 48 and Yld2000–2D yield criteria, the damage initiation loci and fracture loci as relationships between critical plastic strain and stress triaxility were determined for the DP780 and DP1000 steels in varying loading directions, as depicted in Figs. 12 a and 13 a , respectively. For the simulations of each loading direction using the von Mises yield criterion, corresponding flow stresss curves from each direction were applied. Obviously, in case of low stress triaxiality, fracture of the steels occurred directly after reaching the maximum load, whereas under high stress triaxiality the steels showed some further deformation after the maximum point. In this work, the obtained fracture loci rather represented the maximum points on the force–displacement curves of each sample. In sheet metal forming, significant strain localization, which basically took place after the maximum load, already caused an improper defect and should be referred as the final failure criterion. It is seen that different yield fucntions led to noticeably varied fracture and damage initiation loci, especially in case of the DP1000 steel sheets. For both steels under all loading directions, the Yld2000–2D yield model provided lower fracture strains in the low triaxiality domain, but higher critical strains in the high triaxiality domain. In the middle triaxiality range of about 0.33, effect of the yield model on fracture strain was less significant. Generally, the von Mises model led to both damage initiation and fracture loci with larger values. It is noticed that the DP1000 steel showed much larger deviations of the critical strains for fracture and damage initiation calculated by various yield functions than the DP780 steel. The influences of the yield functions on failure

DP steel was investigated using shear and notched specimens. Otherwise, inclusions in the steel could be another reason for void initiation, though they were present in a very limited amount, as shown in Depover et al. (2016). The high strength DP steel certainly composed of ferritic matrix with fine dispersed martensite islands. However, the DP1000 steel exhibited significantly larger phase fraction of the hard martensite. Therefore, it is seen that damage occurrences in the DP780 steel were governed only by the interface failure. On the other hand, in the DP1000 steel with higher amount of martensite damage initiations were affected by both interface and martensite cracking. AvramovicCingara et al. (2009) reported that the fracturing of martensite could be observed in tensile specimens of DP steel between the yield point and elevated stress before the uniform elongation. To generate failure curve based on the final fracture of the examined steels, the DIC technique was applied during tensile tests of different samples. In parallel, elastic–plastic FE simulations of the tensile tests were carried out. Firstly, local equivalent plastic strains of the critical area in each specimen at the states close to the final fracture, which were determined by the DIC and FE calculations, were compared. At large deformation, pronounced strain localization occurred on formed sheet samples so that it was necessary to more precisely determine the fracture state. The moments, when plastic strain distributions within the critical region from experiments were in accordance with those calculated by FE simulations, were used for evaluating local stress and strain responses. The distributions of equivalent plastic strains on the samples with all used geometries obtained by the DIC and simulations are illustrated in Fig. 11 for the DP1000 steel at the states shortly before fracture. Note that DP780 sheet samples showed similar tendency of plastic strain distributions, but significantly higher magnitudes in all stress range. It was found that the FE results obtained by using the Yld2000–2d yield function acceptably agreed with the experimental ones, even at large deformation states, as seen in Fig. 11. The U notch, V notch and R5 notch samples with high stress triaxiality values exhibited considerably lower failure plastic strains. The ASTM shear, butterfly shear and combined loading samples with low stress triaxiality values

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Fig. 15. Forming limit curves determined from fracture loci of the investigated (a) DP780 steel and (b) DP1000 steel using different yield criteria.

fracture loci for varying test directions were considerably different. The DP1000 steel exhibited stronger effect of the anisotropic behaviour on the failure curves. It was found that the determined damage inititaion and fracture loci for different loading directions were in accordance with the corresponding elongations from the tensile tests, in which samples from 90° to the RD commonly showed the lowest failure limits and those from 0° to the RD had the highest ones. Nevertheless, in case of the DP1000 steel the sequences of fracture limit strains for varying loading directions at low and high Lode angle parameter regions were changed. Hence, both anisotropic yield and failure charactersitics of the investigated steels significantly affected the prediction of damage initiation and fracture and should be taken into account. Note that

loci of each loading direction were strengthened by increasing plastic deformation. Afterwards, 3D fracture loci including the Lode angle parameter were determined by the Yld2000–2D yield criterion and are then illustrated in Figs. 12 b and 13 b for the DP780 and DP1000 steels in all test directions, respectively. The Lode angle, which could be calculated according to the relationships shown in Bai and Wierzbicki (2008), represented shear stress state of each deformed specimen. For example, the Lode angle parameter of the uniaxial tensile and shear specimen was 1.0 and 0.03, respectively. Bai and Wierzbicki (2010) and Kim et al. (2011) showed noticeable effects of the Lode angle on shear fracture appearance in aluminum alloy series 2000 and DP steel. The 231

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et al., 2013) and aluminum grade 2024–T351 (Luo et al., 2012) taken from literatures in Fig. 14a. It is seen that fracture of the DP1000 steel with higher strength took place much earlier than that of the DP600 steel with lower strength. The aluminum alloy exhibited the lower failure curve than both AHS steels. Nevertheless, the fracture limits of the AHS steels and aluminium alloy showed the similar tendencies for various stress triaxiality ranges. Fig. 14b depicts the relationships between stress triaxiality and Lode angle of the examined steel DP780, the Docol600 and the aluminium alloy. The similar tendencies could be also observed. Note that there are two methods for applying the obtained damage initiation and fracture loci to predict forming limit of the steels, as shown in Li et al. (2010) and Mohr and Oswald (2008). By the first method or so–called “coupled approach”, the failure loci were incorporated in a fracture model, by which damage accumulation and resulting load carrying capacity loss could be presented. The constitutive fracture model needed to be implemented in the FE code. By the second method or so–called “uncoupled approach”, damage initiation and fracture of the steels were predicted when local stress and strain of sheet sample reached the failure loci. Hereby, yield potential of the steel was not affected. In this study, the damage initiation and fracture loci were further verified by means of the uncoupled approach.

Fig. 16. A formed downsized industrial part at failure.

the fracture model according to Bai and Wierzbicki (2008) were employed to construct the fracture loci in Figs. 12 b and 13 c. The material constants of the model for the DP780 and DP1000 steel in 0° to the RD with consideration of different yield criteria are given in Tables 10 and 11 respectively, for example. Otherwise, the critical equivalent plastic strains at damage initiation of the investigated DP780 and DP1000, which were calculated by the Yld2000–2D yield model, were given in Tables 12 and 13, respectively, for all used samples. It is obvious that different states of stress and loading directions also caused varying failure strains for premature damage initiations. In general, the critical strains for damage initiation of the DP780 steel were larger than those of the DP1000 steel. However, within the low triaxiality region or shear stress state the failure strains of both examined steels were rather close to each other. This result was similar to those reported in Malcher et al. (2012) and Björklund and Nilsson (2014). The obtained damage initiation loci for the studied stress triaxiality range exhibited the similar tendency as that of the fracture loci. It is obvious that the critical strains at damage initiation were sensitive to the loading directions. Therefore, it is recommnded to describe the fracture behaviour of AHS steels by anisotropic fracture criteria, such as presented in Luo et al. (2012) and Lou and Yoon (2017). By this manner, the prediction accuracy of anisotropic ductile fracture characteristics may be improved. Moreover, for each specimen type, the fracture locus of the DP1000 steel was compared with that of the DP steel grade Docol600 (Björklund

5. Transformation of failure locus to forming limit curve In the automotive industry, the forming limit curve or FLC has been widely used for predicting material formability of various sheet metals during a forming process analysis. The FLC described forming threshold of a sheet metal on the basis of the in–plane principal strains. Generally, experimentally obtained FLC was restricted to such forming process within the stress triaxiality range from uniaxial to biaxial deformation and work hardening of material was not taken into account. Therefore, in this work FLCs of the investigated AHS steels under varying directions were subsequently determined from the failure loci for fracture and damage initiation. Hereby, the corresponding Lode angle values were used to calculate the forming limit strains, as shown in Li et al. (2010) and Lou and Huh (2013). The FLC could be then obtained by transforming failure locus represented in the (εpl, θ) space into the principal strain space (εmajor, εminor). The relationships between equivalent plastic strain and maximum, middle and minimum principal strains, as reported in Lou and Huh (2013) and Lou et al. (2014), which are shown in Eqs. (1)–(3), respectively, were taken. The principal

Fig. 17. Experimentally determined strain distributions on the formed part at failure for the steel DP780 and DP1000.

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Fig. 18. Determined plastic strain distribution on defromed part at failure and strain path of critical area calculated by FE simulations until fracture on the FLCs based on fracture loci for the investigated steels (a) DP780 and (b) DP1000.

strains for the shear region of FLC were calculated according to Eqs. (1) and (3), while those for the region between uniaxial and balanced biaxial tension of FLC were computed by Eqs. (1) and (2). The transformation was carried out on the basis of a proportional deformation. Note that the strains in the biaxial state of the determined FLCs were extrapolated with regard to the experimental results, since no data of failure loci from this stress state was available.

εI =

εII =

3−θ 2 θ2 + 3

2θ 2 θ2 + 3

εpl

εpl

εIII =

−3 − θ 2 θ2 + 3

εpl

(3)

The Lode angles in the range of 0 and 1, which covered the states of stress between shear deformation and plane strain state due to the used sample shapes, were taken. The FLCs transformed from the fracture loci of the DP780 steel and DP1000 steel, which were previously gathered by using the von Mises, Hill’s 48 and Yld2000–2d yield functions, are then illustrated in Fig. 15a and b, respectively. The FLCs determined by the conventional Nakazima test of both steel sheets were also compared. Note that FLCs for each testing direction were obtained from the corresponding fracture loci. In Fig. 15a and b, a presented single FLC described the FLCs from all three different loading directions, in which

(1)

(2) 233

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Fig. 19. Comparison between drawing depths at fracture and damage initiation determined by experiments and predicted by FLCs using different yield criteria for the investigated steels.

point until fracture were plotted on the FLCs, as illustrated in Fig. 18a and b for the DP780 and DP1000 steel, respectively. Note that only the strain paths from the Yld2000–2d yield function were presented here. Other yield models also provided the similar tendencies. It is seen that both experimental and numerical strains located on the left hand side of the FLC in the shear stress region. The plastic strains occurred on the experimentally failed part were all below the FLCs and some larger strains from the area of the part radius located very close to the FLCs. On the other hand, the strain paths until fracture from the simulations terminated slightly above the FLCs. However, if the conventional FLCs were used instead, then no failure would be predicted at this moment. Moreover, the drawing depths of deformed parts at failure were measured in the experiments and then compared with those predicted by the FLCs based on different yield functions, as shown in Fig. 19 for the examined steels. In addition, calculated drawing depths at the states of damage initiation were evaluated and provided. It is seen that the used yield models significantly affected the predicted drawing depths at fracture, whereas the computed drawing depths at damage initiation were slightly influenced. The Yld2000–2d yield criterion exhibited the lowest drawing depths at failure for both steels, which were most closely to the experimental results in comparison to the other yield models. The von Mises and Hill’s 48 criteria considerably overestimated the fracture limits of formed part, especially when the DP780 steel sheets were used. The effects of yield functions became much stronger at large deformation. From the predictions, it can be also noticed that damage initiation took place in both steels when the part was deformed until about 70% of the final drawing depth. This should be taken into account when the part will be further employed under cyclic loading. Note that in case of this demonstrated part shear stress was dominant for the failure occurrence. When other types of deformation were applied, different prediction tendencies could be expected. The FLCs obtained by the failure loci for fracture and damage initiation could precisely describe the forming limits of the investigated steels.

the upper and lower bound of the individual FLCs were taken into account. By this manner, the determined FLCs could be used to predict forming threshold of the examined steel sheet in a single analysis step, in which their anisotropic characteristics were incorporated. It is seen that forming limit strains in the shear stress region between ε1 = −ε2 and ε1 = −2ε2 much differed from those of the conventional FLC, whereas those between the uniaxial and biaxial region fairly agreed with the experimental curves. The results were in accordance with that reported by Beese et al. (2010) for an aluminum alloy grade 6061–T6. The experimental FLCs of both steels predicted fracture states slightly earlier than the proposed FLCs in all stress ranges. For both steels, the forming limit strains under shear deformation were stronger influenced by the applied yield functions than those under other conditions. The Yld2000–2d yield model provided the lowest threshold strains in the shear stress region, whereas the Hill’s 48 model showed the lowest strains in the common range of conventional FLC. The von Mises model generally led to the highest critical strains in all states of stress. Obviously, the critical strains of shear stress areas could strongly affect prediction accuracy of the FLCs. For such AHS steels, it was shown that shear fracture frequently occurred at part radii between blank and punch or blank and die. Thus, it is improper to predict this failure type of steel sheets by using the conventional FLC. In contrast to others state of stress, failure strains of steel sheets under shear deformation would be largely overestimated by the experimental FLC. 6. Verification In order to verify the resulted FLCs based on the failure loci for fracture and damage initiation, experimental forming tests of a downsized industrial part were performed for the investigated steels until fracture, as depicted in Fig. 16. The examined part was a non–symmetric rectangular cup. It is seen that the occurred fracture zone was commonly observed at the part radius. The plastic strain distributions on the deformed parts, especially at the critical areas, were afterwards determined by the DIC technique, as shown in Fig. 17. It is obvious that local developed plastic strains of failed parts from the DP780 steel sheet were basically higher than those of the DP1000 steel sheet. However, fracture of both steels took place on the similar areas, but at different moments. Additionally, FE simulations of the forming tests were carried out under consideration of the von Mises, Hill’48 and Yld2000–2d yield functions. All boundary conditions were defined as those in the experiment. Then, the principle strain paths were evaluated from the critical area of simulated part until the fracture state as observed in the experiments. The experimentally determined plastic strains on the entire formed sample at failure and calculated strain path of the critical

7. Conclusions The failure loci for damage initiation and fracture of the AHS steel grade DP780 and DP1000, which were presented in the space of stress triaxiality and effective plastic strain, were determined. The anisotropic behaviour of the steels were considered, in which three different loading directions, namely, 0°, 45° and 90° to the RD were examined. Tensile tests of sheet specimens with various geometries were performed in combination with the DCPD and DIC technique for identifying the states of damage initiation and fracture. Afterwards, corresponding FE simulations were carried out for evaluating stress triaxialities, Lode angle and critical plastic strains of failed regions at the 234

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given states. The yield criteria according to the von Mises, Hill’s 48 and Yld2000–2d model coupled with the Swift hardening law were applied. The results of this work can be concluded as following. – The determined failure loci for damage initiation and fracture provided a wide range of the stress triaxiality between about 0 and 0.6 and Lode angle between 0 and 1. – Damage initiation in the investigated steels characterized by the DCPD method were evidenced by SEM, in which micro–scale damage occurred either at the interfaces between ferrite and martensite or due to the fracturing of martensite islands. – For both steels under all testing directions, the Yld2000–2D yield criterion provided lower critical strains in the low triaxiality range, but higher fracture strains in the high triaxiality region. In the middle triaxiality range of about 0.33, effect of the yield model was less significant. The von Mises model led to both damage initiation and fracture loci with higher magnitudes. The influences of the yield functions on damage initiation and fracture strains were more pronounced for the DP1000 steel than the DP780 steel and became much larger by increasing plastic deformation. – Obviously, the failure loci of the DP1000 steel were stronger dependent on the anisotropic behaviour than the DP780 steel. The determined damage inititaion and fracture loci for different loading directions were in accordance with the corresponding elongations from the tensile tests. The sheet samples from 90° to the RD commonly showed the lowest failure limits and those from 0° to the RD had the highest thresholds in all stress states. Nevertheless, the sequence of fracture limit strains for varying loading directions at low Lode angle values were different from that at high Lode angle values in case of the DP1000 steel. – The failure loci of the steels were transformed into FLCs including the shear stress domain. The obtained FLCs were verified by the industrial part with a non–symmetric rectangular cup shape, in which strain path in the shear stress state was induced. It was seen that the proposed FLCs could fairly described the states of shear fracture occurrence at part radius, whereas no failure would be predicted at this moment by the conventional FLCs. – The Yld2000–2d yield criterion exhibited the lowest drawing depths at failure for both steels, which were most closely to the experimental results in comparison to the other yield models. The von Mises and Hill’s 48 criteria considerably overestimated the fracture limits of formed part, especially when the DP780 steel sheets were used. Additionally, it was predicted that damage initiation would emerge in both steels when the part was deformed until about 70% of the final drawing depth. Acknowledgements The authors would like to acknowledge Office of the Higher Education Commission, Thailand Research Fund (TRF) and King Mongkut’s University of Technology Thonburi (KMUTT) for financial supports (TRG5880258) and the “KMUTT 55th Anniversary Commemorative Fund”. References Ahmad, E., Manzoor, T., Ali, K.L., Akhter, J.I., 2000. Effect of microvoid formation on the tensile properties of dual–phase steel. J. Mater. Eng. Perform. 9, 306–310. Avramovic-Cingara, G., Ososkov, Y., Jain, M.K., Wilkinson, D.S., 2009. Effect of martensite distribution on damage behaviour in DP600 dual phase steels. Mater. Sci. Eng. A 516, 7–16. Bai, Y.L., Wierzbicki, T., 2008. A new model of metal plasticity and fracture with pressure and Lode dependence. Int. J. Plast. 24, 1071–1096. Bai, Y.L., Wierzbicki, T., 2010. Application of extended Mohr–Coulomb criterion to ductile fracture. Int. J. Fract. 107, 1–20. Bao, Y.B., Wierzbicki, T., 2004. On fracture locus in the equivalent strain and stress triaxiality space. Int. J. Mech. Sci. 46, 81–98. Barlat, F., Brem, J.C., Yoon, J.W., Chung, K., Dick, R.E., Lege, D.J., Pourboghrat, F., Choi,

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