Mechanical, microstructural behaviour and modelling of dual phase steels under complex deformation paths

International Journal of Plasticity xxx (2016) 1e22

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Mechanical, microstructural behaviour and modelling of dual phase steels under complex deformation paths
de
ric Barlat a, c, Juan Liao a, *, Jose A. Sousa b, Augusto B. Lopes b, Xin Xue a, Fre a
nio B. Pereira Anto a

Centre for Mechanical Technology and Automation, Dep. Mechanical Engineering, University of Aveiro, 3810-193, Aveiro, Portugal Department of Materials and Ceramic Engineering, CICECO e Aveiro Institute of Materials, University of Aveiro, 3810-193, Aveiro, Portugal c Materials Mechanics Laboratory, Graduate Institute of Ferrous Technology, Pohang University of Science and Technology, 77 Cheongamro, Nam-gu, Pohang, Gyeongbuk, 790-784, Republic of Korea b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 30 November 2015 Received in revised form 22 February 2016 Available online xxx

This paper aims to identify the mechanisms associated to the transient hardening behaviour of dual phase steels under strain path changes, and to capture the observed material behaviours with appropriate constitutive models. First, three DP steel sheets with different amounts of martensite were tested under monotonic and various strain path changes. Second, microstructural analysis of the materials before and after strain path change were performed by means of SEM, TEM, and EBSD. The contribution of texture evolution on the mechanical behaviour was also assessed using the visco-plastic selfconsistent (VPSC) polycrystal plasticity model. Transient hardening behaviour and permanent softening were observed in the tensionetension tests for all the studied DP steels. These behaviours were explained by the development of strain gradients during the first load resulting from strain accommodation incompatibilities between the ferrite and martensite phases. For the purpose of describing the macroscopic material behaviours, the enhanced homogeneous anisotropic hardening (HAH) model (Barlat et al., 2014) integrated with the Yld2000-2d anisotropic yield function were adopted for constitutive modelling. The simulation results were discussed in view of the microstructure evolution. © 2016 Elsevier Ltd. All rights reserved.

Keywords: B. Constitutive behaviour B. Anisotropic material C. Mechanical testing A. Microstructures C. Characteristics

1. Introduction Dual phase (DP) steels are of high commercial importance as engineering material due to their unique properties of strength and ductility (Khan et al., 2012; Matsuno et al., 2015). They have been extensively used in the production of automotive components with reduced weight but improved crash performance (Bouaziz et al., 2013). Their mechanical behaviours can be interpreted from their microstructure, which is predominantly composed of soft ferritic matrix with hard martensitic particles. The hard martensite provides substantial strength while soft ferrite phase is associated with good ductility.

* Corresponding author. E-mail addresses: [email protected], [email protected] (J. Liao). http://dx.doi.org/10.1016/j.ijplas.2016.03.010 0749-6419/© 2016 Elsevier Ltd. All rights reserved.

Please cite this article in press as: Liao, J., et al., Mechanical, microstructural behaviour and modelling of dual phase steels under complex deformation paths, International Journal of Plasticity (2016), http://dx.doi.org/10.1016/j.ijplas.2016.03.010

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In many cases, metallic components can be economically produced by sheet metal forming operations. These operations always involve large strains and complex loading paths that can significantly change the sheet metal formability and promote plastic instabilities with negative consequences on the quality of the final products. The optimization of these operations can be attained numerically with finite element (FE) simulations. However, the success of this approach relies on the accuracy of the constitutive equations that describe the mechanical behaviours of the metals under the processing conditions. Thus, they should be based as much as possible on the physical mechanisms of plastic deformation under monotonic and complex loading conditions. Flow stress, strain hardening and plastic anisotropy are crucial parameters that have to be considered in the constitutive equation formulations. In the un-deformed state, plastic anisotropy is usually controlled by the crystallographic orientation of the grains while strength and strain hardening are strongly affected by other microstructural features such as solutes and second phases. After a certain amount of plastic deformation, anisotropy and strain hardening are also affected by the dislocation structure (Akbarpour and Ekrami, 2008; Gracio et al., 2004; Lopes et al., 2003; Rauch et al., 2007). These microscopic features tend to interact with each other and their evolution leads to a complex anisotropic mechanical behaviour. The dislocation microstructure developed in the plastic deformation depends on the distribution of slip on the active systems, which change with the loading conditions, and material characteristics such as crystal structure, grain orientations, stacking fault energy, second phases and solutes, etc. (Rauch, 2000). In polycrystalline materials, the dislocation microstructure also depends on the local stresses associated with the compatibility of plastic deformation between neighbour grains or second phases (Abid et al., 2015). Due to prior thermo-mechanical processing, the grains of polycrystalline materials usually exhibit preferred crystallographic orientations that evolve during plastic deformation. The effect of crystallographic texture on the flow stress can be estimated using the average value of the macroscopic stress to the critical resolved shear stress ratios (). Lower or higher values of correspond to, respectively, more or less favourable grain orientations for slip due to the applied stress. Premature plastic flow localization can also occur due to a change of strain path, which affects the dislocation substructure evolution. In turn, the stressestrain curve depends on this evolution and, in particular, on the amplitude of the strain path change, which can be characterized by the parameter a proposed by Schmitt et al. (1994).

ε $ε εp kεk

a ¼

p



(1)

In the above relationship, εp and ε correspond to the strain tensors of the pre-strain and subsequent loading, respectively. The highest and lowest values of a, i.e., 1 and 1, represent monotonic and reverse loading tests, for which the slip system is reactivated in the same and opposite directions, respectively. A value of a equal to zero corresponds to a cross-loading condition for which the prior activated slip systems become latent during reloading (Rauch et al., 2011). In the past decades, most of the studies involving complex strain paths were performed on single phase metals (Haddadi et al., 2006; Haddag et al., 2007; Holmedal et al., 2008; Khadyko et al., 2016; Kitayama et al., 2013; Manik et al., 2015; Rauch et al., 2007; Resende et al., 2013; Rousselier et al., 2010; Wen et al., 2015). For these materials, the dislocation structures formed in the pre-strain become unstable during reloading and, by mechanisms involving multiplication and mutual annihilation of dislocations, a new structure, characteristic of the new strain path, develops (Nesterova et al., 2001a,b; Rauch et al., 2002). Depending on the pre-strain amount and amplitude of strain path change (a), this evolution may lead to transient strain hardening at the earlier reloading stage (Gardey et al., 2005; Vincze et al., 2005) and to premature strain localization (Da Rocha et al., 2009). In recent years, many work have been dedicated to the mechanical behaviours of DP steels (Franz et al., 2009; Gardey et al., 2005, 2006; Ha et al., 2013; Larsson et al., 2011; Marcadet and Mohr, 2015; Resende et al., 2013; Sun and Wagoner, 2013; Tarigopula et al., 2008, 2009; Weiss et al., 2015; Yoshida et al., 2011; Yu and Shen, 2014). Their results revealed that the mechanical behaviours of DP steels present some distinctions compared with single phase steel, due to the existence of dispersed hard martensitic particles in the softer matrix. However, the influences of the local stain incompatibilities between the two phases on the macroscopic behaviour of these important advanced high strength steels (AHSS) during strain path change are not completely understood and need to be further investigated. To capture the anisotropic work-hardening behaviours of materials under complex strain paths with the final aim to be used in the metal forming simulations, various types of constitutive models have been developed. For instance, the widely used kinematic hardening-based models (Chun et al., 2002a,b; Chung et al., 2005; Geng and Wagoner, 2002; Taherizadeh et al., 2015; Yoshida and Uemori, 2002; Yoshida et al., 2015) and multi-surface representations (Lee et al., 2007) were introduced to reproduce the complex hardening behaviours of materials observed under reverse loading conditions, such as transient hardening at high rate, flow stress stagnation and permanent softening. A detailed review of these models was addressed by Chaboche (2008) and further by Yoshida et al. (2015). Recently, considerable attention has been focused on the constitutive description of the anisotropic hardening behaviours of DP steels. The developed models can be divided into two categories: physical approaches and phenomenological approaches. Concerning the physical approaches, some micromechanical models (Franz et al., 2009; Kim et al., 2012; Lai et al., 2015; Resende et al., 2013; Wei et al., 2015; Yoshida et al., 2011) have been proposed by incorporating a number of microstructural parameters, which were used to describe the mutual interactions between phases, dislocationegrain-boundary interactions and other microstructural features. The Please cite this article in press as: Liao, J., et al., Mechanical, microstructural behaviour and modelling of dual phase steels under complex deformation paths, International Journal of Plasticity (2016), http://dx.doi.org/10.1016/j.ijplas.2016.03.010

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predictions provided by these models are in agreement with the macroscopic experiments and microscopic observations. However, the calculation of these models require extensive computation time and resources. Therefore, time-efficient phenomenological models based on continuum concepts (Chongthairungruang et al., 2012; Kuwabara and Nakajima, 2011; Sun and Wagoner, 2013; Tarigopula et al., 2008, 2009) were preferred for the large-scale forming applications. Nevertheless, it is still difficult to reproduce satisfactorily all the features occurring in load reversals and other strain path changes simultaneously with the same model. Lately, Barlat et al. (2011) introduced a distortional hardening model, the so-called homogeneous anisotropic hardening (HAH) model. Its initial version, which was proposed as an alternative to kinematic hardening, mainly accounts for reverse loading. Later, it was extended (Barlat et al., 2013) and further enhanced (Barlat et al., 2014) to include latent hardening (i.e., hardening due to the activation of previously latent slip systems) effects and transient hardening behaviour in cross-loading, simultaneously. The prediction results for both single and dual phase steels, i.e. EDDQ and DP780, agreed well with the experiments (Barlat et al., 2014). However, its performance should be further evaluated with more materials and complex deformation paths. The structure of this paper consists of three main sections. In the first one, three DP steels with different amounts of martensite are subjected to monotonic and complex strain path tests, namely, uniaxial tension, bulge and two-step sequential tension tests conducted in different directions. The second section deals with the microstructural analysis of the materials under the above strain paths. The texture/microstructure evolutions of DP steels during plastic deformation are presented and its correlation with the mechanical behaviour is revealed. In the last section, the enhanced HAH model is evaluated for constitutive description. The results are discussed in consideration of the microstructural analysis. 2. Material characterization 2.1. Materials Three grades of DP steels with the same thickness (0.8 mm) were considered: DP500, DP600 and DP780. The microstructures of these steels observed by SEM are shown in Fig. 1. DP500 is characterized by the presence of martensite islands and a coarse-grained, 22 mm in average, ferrite phase. A more refined microstructure is observed for DP600 and DP780 in which the average ferrite grain size is 10 and 6 mm, respectively. The volume fraction of martensite in the three steels was measured to about 14%, 21% and 37%, respectively. The chemical composition for each material is listed in Table 1. It is of interest to note the addition of a small amount of Nb in DP600 and DP780. This micro-alloying element has a positive effect on production of a more homogeneous and finer grained DP microstructure, resulting in a higher strength and better ductility (Lee et al., 2012). 2.2. Mechanical tests 2.2.1. Uniaxial tension tests Standard uniaxial tensile tests were carried out at an angle (q1) equal to 0 , 45 and 90 from the rolling direction (RD), for all the three grades of steels. These tests were conducted at room temperature in a tensile testing machine (SHIMADZU 100kN) with a strain rate of 103 s1. Five specimens were repeated for each condition in order to assess the scatter in the results but only one was used as representative for the modelling. The strains of the specimens were determined continuously with a digital video extensometer. 2.2.2. Bulge tests The balanced biaxial bulge test allows the determination of the flow curve for a higher strain range compared with the uniaxial tensile test. Therefore, for a better mechanical characterization, this test was conducted in a hydraulic bulge test device for each of the three DP steels. The opening diameter of the circular die was 150 mm. Specimen with a diameter of 250 mm was bulged via the die opening, following the experimental procedures illustrated in Martins et al. (2013).

Fig. 1. SEM images of the microstructure of a) DP500, b) DP600 and c) DP780 steels.

Please cite this article in press as: Liao, J., et al., Mechanical, microstructural behaviour and modelling of dual phase steels under complex deformation paths, International Journal of Plasticity (2016), http://dx.doi.org/10.1016/j.ijplas.2016.03.010

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Table 1 Chemical composition of DP steels in weight percent.

DP500 DP600 DP780

C

Mn

Si

P

S

N

Cr

Ni

Cu

Al

Nb

V

B

0.079 0.089 0.138

0.65 0.85 1.52

0.31 0.2 0.2

0.009 0.014 0.011

0.003 0.004 0.002

0.003 0.004 0.003

0.03 0.03 0.03

0.03 0.03 0.04

0.01 0.01 0.01

0.038 0.049 0.038

e 0.016 0.014

0.01 0.01 0.02

0.0003 0.0003 0.0002

2.2.3. Two-step tension tests Two-step tension tests were conducted on the three steels to compare the influence of various strain path change on their plastic behaviour. Large and sub-size specimens used for the first and the subsequent steps, respectively, are depicted in Fig. 2. First, the large tensile specimens were pre-strained in the RD to the strain of 4% or 7%. Thereafter, sub-size tensile specimens were machined at angle q2 ¼ 0 , 45 and 90 to the pre-strain axis in the uniform strain region of the large pre-strained specimens. Subsequent tension tests were then carried out on the sub-size specimens to evaluate the mechanical behaviour after an abrupt change of the tensile direction. For an isotropic material, the reloading in uniaxial tension at 0 , 45 and 90 from the pre-strain tensile axis corresponds to a values equal to 1 (monotonic loading), 0.2 (pseudo cross-loading) and 0.5 (orthogonal tension), respectively. The pre-strain and the subsequent tension tests were conducted on SHIMADZU tensile machines with load capacities of 1000 KN and 100 KN, respectively. The strains were determined continuously using a digital video extensometer. 2.3. Microstructural examinations The microstructural analysis of the initial materials, after pre-strain and after reloading was conducted using the EBSD and TEM. A Bruker CrystAlign QC 400 EBSD system interfaced to a Hitachi SU-70 SEM was employed to map the crystallographic orientations of the grains. Because of the small crystallite size and lattice distortion of the martensite phase, only the orientations of the ferrite grains were measured during these tests. EBSD data were obtained to characterize the crystallographic texture of the samples. This data was also used as input for crystal plasticity calculations using the Visco-plastic self
(1993), to evaluate the effect of the crystallographic texture of the consistent (VPSC) model proposed by Lebensohn and Tome ferrite grains on the mechanical behaviour of the DP steels. In all the performed VPSC calculations were used stress-imposed boundary conditions. The dislocation structure evolution during plastic deformation was detected using a transmission electron microscope (TEM) Hitachi H-9000. All the specimens were extracted from the mid-plane of the sheet and prepared by mechanical thinning followed by electropolishing at room temperature. Features such as the size, shape and homogeneity of the dislocation patterns were measured in order to understand how plastic deformation is accommodated at the microscopic scale. 3. Result and discussion 3.1. Mechanical behaviour 3.1.1. Monotonic loading The uniaxial true stressestrain curves for all the three steels are depicted in Fig. 3. The corresponding standard mechanical properties are summarized in Table 2. The strength of the steel increases with the grade of the DP steel, but the uniform elongation tends to decrease. The higher strength is ascribed to the larger volume fraction of hard martensite phase (Zhang et al., 2015). A simple way to explain the decrease of elongation for DP780 is, the plastic deformation is predominantly € m et al., localized in the ferrite, while the martensite remains, in a first approximation plastically un-deformed (Bergstro 2010). Hence, the local strains in the ferrite are much larger than that recorded globally for the whole material in a conventional tension test. Fig. 3 also indicates that the flow stress anisotropy of DP780 is more prominent than that of DP500 or

Fig. 2. Dimensions of large specimens and sub-size specimens used in the tensionetension tests.

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1000 DP780 True stress (MPa)

800

DP600

600

DP500

400 θ1=0o

200

θ1=45o θ1=90o

0 0.00

0.04

0.08

0.12

0.16

True strain Fig. 3. Standard tension tests for DP500, DP600 and DP780 at different directions from RD.

DP600. The strain anisotropy of the tensile specimens expressed by the R-value, defined as the ratio of the width-to-thickness strains for the different loading direction, is presented in Fig. 4. In the as-received state, the anisotropy of a material is usually due to its initial crystallographic texture, which will be addressed specifically in the next section. 3.1.2. Complex loading The results of tensionetension tests are presented in Fig. 5. The true stress-true strain curves of the second tensile test, measured at 0 , 45 and 90 (q2 ) from the first loading direction, for each DP steels pre-strained up to 4% and 7%, is plotted together with the corresponding monotonic curves. The results indicate that the re-yield stress and the strain hardening rate evolution are dependent on the reloading angle. These differences are discussed in more details below. a) Monotonic interrupted loading For the reloading angle at q2 ¼ 0 , a is equal to 1 and the changes of slip system activity and grain orientations during both, interrupted and uninterrupted tests should be similar. Therefore, no differences are expected in the mechanical behaviour of the DP steels during both loadings. Indeed, for all DP steels and pre-strain values, the reloading stressestrain curves at 0 superimpose the uninterrupted monotonic curve in the reloading direction (monotonic reference curve). b) Pseudo cross-loading For the reloading angle at q2 ¼ 45 , the change of strain path is close to the cross-loading condition (a z 0.2). This corresponds to a second loading step in which new slip systems are activated. For single phase metals, e.g. the EDDQ steel (Ha et al., 2013), this change leads to latent hardening, i.e. the reloading yield stress overshots the monotonic flow stress for the same accumulated strain value. The amplitude of this transient phenomenon depends on the pre-strain value and may result in a premature plastic flow localization and decrease of the uniform strain (Da Rocha et al., 2009; Manik et al., 2015). For the investigated materials and pre-strain values, a completely different mechanical behaviour during reloading is observed (Fig. 5). All the reloading curves represent a transient hardening behaviour in the initial stage of reloading. Furthermore, it is interesting to observe that the reloading flow stress, after the transient stage, is always smaller than the monotonic reference curve while, independently of the pre-strain amount, the reloading curve for q1 ¼ 0 become similar the monotonic curve. This lower flow stress characterizes a permanent softening, since a persistent offset is observed between the reloading and the monotonic curves after the transient period. Tarigopula et al. (2008) and Yoshida et al. (2011) also detected similar results for DP steels in their two-step tension tests. c) Orthogonal tension tests The condition of orthogonal tension tests (q2 ¼ 90 ) is between cross and reverse loading (a z 0.5). For single phase steels, the reloading curve is similar to that observed in cross-loading, but with a smaller overvalue of the re-yield stress (Ha et al., 2013). For all the DP steels investigated in this work, the observed anisotropic hardening behaviour, in these loading conditions, are qualitatively similar to those of the pseudo cross-loading condition. However, the features are more pronounced Please cite this article in press as: Liao, J., et al., Mechanical, microstructural behaviour and modelling of dual phase steels under complex deformation paths, International Journal of Plasticity (2016), http://dx.doi.org/10.1016/j.ijplas.2016.03.010

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Table 2 Parameters extracted from the monotonic tests. Material

Loading direction, q1

Yield stress, s0.2 (MPa)

Tensile strength, sT (MPa)

Uniform elongation, e (%)

DP500

0 45 90 0 45 90 0 45 90

390 409 398 424 435 433 535 555 572

660 674 669 742 744 750 932 967 990

15.2 14.3 14.8 14.9 15.4 14.3 12.0 11.9 11.8

DP600

DP780

than those of cross-loading with, namely, a lower reloading yield stress, a longer transient deformation stage and an increased permanent softening. For the purpose of comparing the anisotropic hardening features of all the materials for the second tension step, a flow stress ratio r, proposed by Ha et al. (2013), was used. Its definition is:



rq ¼

sq   sref 

(2) WP

where sq and sref denote the flow stress at reloading in the direction q and that of the monotonic reference test, respectively. W p is a given specific plastic work. The comparison for the three DP steels and the two investigated pre-strain levels are shown in Fig. 6. The results calculated at small amounts of plastic work (0.5 MPa, 1 MPa) indicate that, for q2 angles of 45 and 90 , all the DP steels present a sq value smaller than sref , which is consistent with the transient behaviour that characterize the beginning of the reloading. This effect is more pronounced for the reloading angle of 90 , larger pre-strain level (7%) and DP steels with a higher ratio of martensite (DP780). For larger amount of plastic work, the reloading flow stress is still less than the monotonic for all the three DP steels, which indicates the existence of a permanent softening. This effect also increases with the amount of martensite and pre-strain level. It should be noted that both transient behaviour and permanent softening, that normally occur for single phase materials only in reverse loading, is observed for dual phase steels during both orthogonal tension and pseudo cross-loading tests. This behaviour, which was not observed for single phase materials, such as the low carbon steel (Ha et al., 2013), is believed to be a consequence of strain incompatibilities due to the existence of the hard martensite particles in the softer matrix. Tarigopula et al. (2008), who observed similar results, attributed this phenomenon to a deformation-induced stress state building-up during the predeformation of the DP steel. However, no further evidence and explanation for this effect was provided.

3.2. Microstructural behaviour 3.2.1. Monotonic loading In Fig. 7, the {110} and {111} pole figures of the ferrite phase, measured by EBSD in the initial materials, give a description of the crystallographic textures. The distributions of crystallographic orientations are typical of rolled steels, which are usually identified as a g-fiber ({111}) texture with a reinforcement of the {111}<110> component (Hong and Lee, 2002; Gardey et al., 2005). To assess the contribution of the ferrite texture on plastic anisotropy, as presented for the initial materials in Figs. 3 and 4, the EBSD data were input in the VPSC model to calculate the average value of the macroscopic stress to the critical resolved shear stress ratios () and R values at 0.2% plastic strain, for each DP steel and loading direction (Figs. 8 and 9, respectively). In these numerical calculations, it was assumed that all the plastic deformation was accommodated by the ferrite phase and the influence of the martensite particles was neglected. For all the DP steels, the evolution of the calculated and R values with the loading directions agree qualitatively with the evolution experimentally observed for the yield stress and plastic strain ratio. More specifically, the model correctly predicted an increase of the yield stress (increase of ) with the loading angle for all the materials, and a decrease for DP500 loaded at 90 . It also predicted an increase of the R value for the loading at 45 of DP600 and DP780 and a decrease for DP500. This suggests that the selected polycrystal model can be used with the EDSD data to qualitatively assess the influence of crystallographic texture on the behaviours of DP steels and that the initial anisotropy observed during the monotonic tests has a crystallographic texture origin. In Fig. 10, the EBSD misorientation maps of the initial DP780 steel and those after 7% of strain are presented, in which the colour of each pixel represents the local misorientation angle with regard to the average crystallographic orientation of the considered ferrite grain. These misorientations are caused by strain gradients (Dillien et al., 2010; Calcagnotto et al., 2015; Abid et al., 2015), therefore, blue and red colours (in the web version) in these maps represent a small and a large strain gradient inside the ferrite grains, respectively. Fig. 10 (a) shows that the misorientation values for the initial material are very small, as expected for an undeformed metal. However, for the deformed sample (Fig. 10 (b)) different results are obtained, Please cite this article in press as: Liao, J., et al., Mechanical, microstructural behaviour and modelling of dual phase steels under complex deformation paths, International Journal of Plasticity (2016), http://dx.doi.org/10.1016/j.ijplas.2016.03.010

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DP500 DP600 DP780

1.1 1.0

R value

7

0.9 0.8 0.7 0.6

0

45

90

o

θ1 ( ) Fig. 4. Plastic strain ratio (R) of DP500, DP600 and DP780 for tensile loading at different directions from RD.

namely, large misorientation angles were measured often near the ferrite-martensite phase interfaces and smaller misorientation angles more frequently in the middle of a ferrite grains. It was found that these misorientation gradients, which originate due to the local strain incompatibilities between the ferrite grains and the martensite particles (Yoshida et al., 2011; Tasan et al., 2014; Abid et al., 2015) increase at a higher rate during the initial stages of plastic deformation and for DP steels with a higher volume fractions of martensite. However, no specific differences were observed in the misorientation maps presented by same material after a given strain in different directions. The microscopic plastic strain distribution in DP steel can also be examined by TEM analysis (Fig. 11). In the as-received steel, the dislocation structure is characterized by a low density in the middle of ferrite grains and higher density near the interfaces of ferrite and martensite (Fig. 11 (a)). The origin of this initial heterogeneity is usually attributed to the residual stresses developed as a result of the austenite-to-martensite phase transformation during the DP steel production (Ramazani et al., 2013; Goto et al., 2015). With the accumulation of the plastic strain, the average density of dislocation increases, giving rise to structures with different dislocation density and degrees of organization along the same ferrite grain. It is worth noting that evidence of plastic deformation of the martensite particles was never observed. Moreover, the dislocation cells developed in some areas of the ferrite, exhibited shape, size and orientation with different characteristics within a single grain (Fig. 11 (b)), confirming the strong inhomogeneous strain distribution in the ferritic grains as detected by the EBSD analysis. It should be stressed that local hardness measurements have also confirmed the development of strain hardening gradients in DP steels (Kadkhodapour et al., 2011; Zhang et al., 2015), which is believed to have a strong effect on the plastic €m et al., 2010; Abid et al., 2015). However, only monotonic loading was analyzed in deformation of these materials (Bergstro these studies. 3.2.2. Complex loading Depending on the loading conditions, material and initial orientation of the grains, the crystallographic texture of a metal can evolve during the pre-strain and strongly influences its mechanical response during the reloading. For example, the evolution during plastic strain of an initial strong cube {100}<001> texture is at the origin of large changes in the flow stress and formability in aluminium sheets (Lopes et al., 2003). In contrast, the good deep drawability of low carbon steels sheets is usually attributed to a development of stable g-fiber-type recrystallization texture (Hong and Lee, 2002). For the DP steels, the stability of the crystallographic orientations is evidenced by the weak evolution of the experimental pole figures (Fig. 12 compared to Fig. 7 (c)) and the small differences between the values of the as-received materials and after 7% of strain (Table 3). It is also interesting to observe that the evolutions of and yield stress with the reloading angle are opposites. Thus, except for DP500 reload at 90 , increases with the reloading angle, indicating a texture effect for the re-yield stress. However, the experimental flow stress values of the second strain path decrease with the loading angle. This shows that the mechanical behaviour differences observed during the strain path change should have a microstructural origin other than crystallographic texture. The TEM and EBSD analysis performed on the reloaded samples reveal that the microstructures of the DP steels is also characterized by a strong heterogeneity of the strain distribution along the ferrite grains. Excluding changes in grain shape and orientation due to plastic deformation, the main microstructural evolution observed after reloading was the change of misorientation values in the ferrite grains (Fig. 13). It was found that the strain gradient changes were more pronounced for non-proportional loading and for DP steels with higher content of martensite. However, for high strain values after the strain path change, no obvious differences were detected in the microstructure developed in the samples for any reloading direction. These microstructural analysis suggest that the flow stress at the beginning of the reloading is controlled by the plastic deformation of weakly strain hardened and favourably oriented regions in ferrite grains with respect to the new local stress Please cite this article in press as: Liao, J., et al., Mechanical, microstructural behaviour and modelling of dual phase steels under complex deformation paths, International Journal of Plasticity (2016), http://dx.doi.org/10.1016/j.ijplas.2016.03.010

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700

DP500

True stress (MPa)

600 500

θ1=0o, θ2=0o

400

4%

θ1=0o, θ2=45o

7%

θ1=0o, θ2=90o

Monotonic θ1=0o

300

Monotonic θ1=45o Monotonic θ1=90o

200 0.00

0.04

0.08 True strain

0.12

0.16

a) 800

DP600

True stress (MPa)

700 600 θ1=0o, θ2=0o

500 4%

400

θ1=0o, θ2=45o

7%

θ1=0o, θ2=90o

Monotonic θ1=0o

300

Monotonic θ1=45o Monotonic θ1=90o

200 0.00

0.04

0.08 True strain

0.12

0.16

b)

True stress (MPa)

1000

DP780

800 θ1=0o, θ2=0o

600

400

θ1=0o, θ2=45o

4%

θ1=0o, θ2=90o

7%

Monotonic θ1=0o Monotonic θ1=45o

200

0.00

Monotonic θ1=90o 0.04

0.08

True strain

0.12

0.16

c) Fig. 5. Tensionetension tests of a) DP500, b) DP600, c) DP780 at q2 ¼ 0 , 45 and 90 , after 4% and 7% pre-strain along RD (q1 ¼ 0 ).

conditions imposed by the loading angle change. In this situation, it is reasonable to assume that the softening effect promoted by these preferentially activated volumes increases with the amplitude of the local stress changes and the strain gradient developed in the ferrite grains during the pre-strain. This explanation is in line with the decrease of the yield stress with higher reloading angle, pre-strain value and martensite volume fraction, experimentally observed for all DP steels in this work and previous experimental results (Tarigopula et al., 2008; Sun and Wagoner, 2013; Ha et al., 2013). Please cite this article in press as: Liao, J., et al., Mechanical, microstructural behaviour and modelling of dual phase steels under complex deformation paths, International Journal of Plasticity (2016), http://dx.doi.org/10.1016/j.ijplas.2016.03.010

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1.1

0.8

DP500 DP600 DP780

0.7

0

45

W p=0.5 MPa

1.0 Flow stress ratio

Flow stress ratio

Pre-strain 7% (RD, θ1=0o)

W p=0.5 MPa

0.9

0.6

1.1

Pre-strain 4% (RD, θ1=0o)

1.0

0.9 0.8

0.6

90

DP500 DP600 DP780

0.7

0

45

1.1

Flow stress ratio

Flow stress ratio

0.8

DP500 DP600 DP780

0.7

0

45

W p=1 MPa

1.0

W p=1 MPa

0.9

0.6

Pre-strain 7% (RD, θ1=0o)

Pre-strain 4% (RD, θ1=0o)

1.0

0.9 0.8

0.6

90

DP500 DP600 DP780

0.7

0

1.1

1.0

1.0

Pre-strain 4% (RD, θ1=0o) W p=35 MPa DP500 DP600 DP780

0.7 0.6

0

45 o

θ2 ( )

90

Flow stress ratio

Flow stress ratio

1.1

0.8

45

90

θ2 (o)

θ2 ( o)

0.9

90

θ2 (o)

θ2 (o) 1.1

9

0.9

Pre-strain 7% (RD, θ1=0o) W p=35 MPa DP500 DP600 DP780

0.8 0.7 0.6

0

45

90

o

θ2 ( )

Fig. 6. Flow stress ratio of DP steels for 4% and 7% pre-strain value along RD (q1 ¼ 0 ) and at plastic work of 0.5 MPa, 1 MPa and 35 MPa in the subsequent tension test.

After the activation of the first regions, the deformation also proceeds in other portions of the ferrite grains, initially more strain hardened or less favourable oriented. Therefore, the macroscopic flow stress tends to increase quickly, defining a transient deformation stage that ends when the strain hardening rate reaches a similar value for all the reloading directions. During this process, the reloading strain is accommodated by the local activation of different sets of slip systems that leads, through mechanisms of multiplication and mutual annihilation of dislocations, to changes in the strain gradients and dislocation structures of ferrite grains. These heterogeneous plastic strain accommodation processes during reloading are also supported by the histogram showing the evolution of the misorientations measured for DP780 after pre-strain and reload (Fig. 14). Indeed, this figure shows that, for 2.5% reloading curve, an increase of fraction (frequency) of ferrite grains with lower misorientation and a decrease of that with higher misorientation. Therefore, the results indicate a decrease of the mean misorientation value during the transient deformation stage (curve for 2.5% reloading strain), as expected if the plastic deformation occurs through the preferential activation of less strain hardened volumes of the ferrite grains. For higher reloading strains (curve for 5.5% reloading strain), an increase of the mean misorientation value is detected, which is consistent with the end of the transient stage and the increase of the strain heterogeneity as observed during monotonic loading. Please cite this article in press as: Liao, J., et al., Mechanical, microstructural behaviour and modelling of dual phase steels under complex deformation paths, International Journal of Plasticity (2016), http://dx.doi.org/10.1016/j.ijplas.2016.03.010

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Fig. 7. {110} and {111} experimental pole figures for the initial materials a) DP500, b) DP600, c) DP780.

Fig. 8. values for monotonic loading of the initial materials DP500, DP600 and DP780 along 0 , 45 and 90 from RD (For comparison, the yield stress values of Table 2 are also represented).

1.35

VPSC calculations

1.30

DP500 DP600 DP780

1.25 R value

1.20 1.15 1.10 1.05 1.00 0.95

0

45

90

Angle Fig. 9. Predicted plastic strain ratio values for monotonic loading of the initial materials along 0 , 45 and 90 from RD.

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11

Fig. 10. EBSD misorientation maps of DP780 steel a) in initial state and b) after strain of 7% along the RD.

It is particularly important to emphasize that, for all the DP steels, reloading directions and pre-strain values, the strain hardening rate after the transient stage is similar to that of monotonic loading along the pre-strain direction. This trend clearly shows that the mechanical behaviours of the DP steels during the reloading are controlled by the microstructure developed during the pre-strain. It seems that the effects of strain gradients developed during the pre-strain are so strong (at least for the pre-strain values investigated in this work) that it controls the mechanical behaviour of DP steels during the reloading, overcoming the effects of the initial anisotropy. This strong influence of the pre-strain microstructure supports the above hypothesis stating that the flow stress at the beginning of the reloading is controlled by the preferential deformation of weakly strain hardened and more favourably oriented regions of the ferrite grains originating from the strain incompatibilities between the ferrite and martensite phases. From Fig. 5, it is also interesting to verify that, except for DP500 reloaded at 45 , a superposition of the 7% pre-strain curves after the transient stage and the monotonic curves in the pre-strain direction is obtained by shifting the reloading curve to the left, along the strain axis, by an amount equal to the extension of the transient deformation stage. This coincidence of flow stress values is clear evidence that the permanent softening observed after a change of loading direction is a direct consequence of the strong effect of the pre-strain microstructure on the mechanical behaviour during reloading and the occurrence of a transient stage after the change of strain path. The mechanical behaviour of DP500 steel during reloading at 45 is worth discussing further. Considering the monotonic curve in the pre-strain direction as the reference curve, strong similarities can be seen between the mechanical behaviour of DP500 steel and single phase materials during the reloading at 45 , namely, the overshot of the reloading flow stress by an amount that increases with the pre-strain, and the decreases of the total uniform strain for higher pre-strain values (Da Rocha et al., 2009; Franz et al., 2013). This suggests that the mechanical behaviour of the DP500 steel during reloading is strongly influenced by the latent hardening phenomena, which occurs in single phase metals during similar changes of loading conditions. However, the decrease of the yield stress and the transient deformation stage for both pre-strain values show that the effect of the strain gradients also influences the mechanical behaviour of the DP500 steel, at least during the beginning of the reloading. The higher impact of the latent hardening effects observed during the reloading at 45 of DP500 is explained by the lower fraction of martensite, that decreases the strain gradients in ferrite grains, and the amplitude of strain path change near cross-loading, which intensify the latent hardening effects. For higher reloading angle (90 ), the lower latent hardening effects due to the decrease of a and the stronger effect of strain gradients developed during the pre-strain, are consistent with the closer similarities of the reloading behaviour between DP500 and the other investigated DP steels. 4. Constitutive modelling The complex, distinct mechanical behaviour of DP steels under strain path changes can affect the formability and springback of parts after forming. To build robust numerical modelling for the large scale forming process, accurate and efficient constitutive models that can reproduce all these behaviours are pursued. In this work, a recent proposed hardening model, i.e., HAH in its enhanced version, was evaluated for all the three DP steels, according to the performance of this model. With regards to the yield function, the anisotropic yield function Yld2000-2d (Barlat et al., 2003), was adopted to combine with the HAH model. 4.1. The enhanced HAH model The initial version of HAH model is expressed as follows (Barlat et al., 2011):

FðsÞ ¼

h

q q 1       q b b q b q b : s  þ f 2  h : s  : s  h : s þ h fq ðsÞ þ f1  h ¼ sðεÞ

(3)

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Fig. 11. Dislocation structures of DP600 at the a) as-received steel and b) 10% strain along RD. Note that in a) M identifies a martensite particle.

Fig. 12. {110} and {111} experimental pole figures of DP780 after 7% pre-strain along RD.

Table 3 values for the initial material and after 7% strain along RD. Material

Loading direction

0 45 90 0 45 90 0 45 90

DP500

DP600

DP780

Initial

After 7% strain

2.35 2.44 2.38 2.35 2.37 2.42 2.37 2.39 2.43

2.36 2.43 2.38 2.37 2.38 2.42 2.37 2.38 2.43

where q is a constant exponent. sðεÞ is a monotonic reference curve characterizing isotropic hardening. f is any positively homogeneous yield criterion. s is the stress deviator. fk are state variables enabling the distortion of the stable yield surface f b is a microstructure deviator representing the previous deformation history. The initial value of it for the Bauschinger effect. h is set the same as the normalized stress deviator, which is defined as 0 0 b h ij ¼ bs hl ¼ shl

,rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 s :s : 3 hl hl

(4)

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Fig. 13. EBSD misorientation maps of a ferrite grain of DP780 steel a) after a pre-strain of 7% along RD and b) reloading at 45 up to an additional 4% strain. The misorientation maps are superimposed to the respective EBSD quality map.

In the enhanced version (Barlat et al., 2014), the stable yield functionfðsÞ in Eq. (3) was replaced by

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi     1 4ð1  gs Þ zðsÞ ¼ f2 sc þ so þ f2 so gL gL

(5)

where gs and gL are two state variables related to latent hardening and cross contraction, respectively. sc and so , which are b decomposed from the stress deviator s , represents two components coaxial or orthogonal to the microstructure deviator h, respectively. This modification was proposed in order to achieve a yield surface extension or contraction in the direction b orthogonal to h. In the enhanced version, the formulae account for the two different features in cross-loading, i.e., latent hardening and contraction, can be activated simultaneously or separately. For only cross-loading contraction, the enhanced HAH model b assumes that the stable surface contracts in the appropriate direction, but more intensive in any direction orthogonal to h. The state variable gs is proposed to attain this. Its evolution is given by

h i dgs ¼ ks 1 þ ðS  1Þcos2 c  gs dε

(6)

b i.e. the angle between the two tensors. S is a contraction coefficient less or equal to 1. It should be where,cos c ¼ 83 ðb s : hÞ, noted that gs already evolves gradually to the value S during the previous proportional loading. When cos c ¼ 0, i.e., crossloading occurs, gs asymptotically recovers back to 1.0 with a rate given by ks. In the enhanced HAH model, the evolution of the microstructure deviator, which can be viewed as a ‘rotation’ in the pplane, is refined as

 

  b 3 cos c1=z dh b  þ gR bs  cos c h ¼ ksgnðcos cÞ   dε 8

(7)

b g is a variable where the coefficient k control the evolution of h. R

h   i dgR ¼ kR k0R 1  cos2 c  gR dε

(8)

In Eq. (7), the recommended value for z, kR , k0R are 5, 15 and 0.2, respectively. However, this parameters may change if deemed necessary. 4.2. Parameters identification In this work, the Hockett-Sherby model was used for the reference isotropic hardening description for the three DP steels. The expression of this model is:

s ¼ A  ðA  BÞeC$ε

n

(9)

Please cite this article in press as: Liao, J., et al., Mechanical, microstructural behaviour and modelling of dual phase steels under complex deformation paths, International Journal of Plasticity (2016), http://dx.doi.org/10.1016/j.ijplas.2016.03.010

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Fig. 14. Histograms of the misorientation angles for DP780 after pre-strain of 7% and reloading at 90 .

Due to the decoupling feature of the HAH model, the parameter identification procedure was carried out step by step. First, the eight coefficients of the Yld2000-2d yield criterion were identified in a general way. They were calculated using four yield stresses sq (q¼0 , 45 , 90 ),sb , and four r-values, i.e., rq (q¼0 , 45 , 90 ), rb in the corresponding direction. In this work, sq and rq were obtained from the uniaxial tension tests. sb was obtained from the biaxial bulge test and rb was calculated with the other yield function Yld96 (Barlat et al., 2003), since the in-plane balanced biaxial test was not available. The exponent m for the function was put to 6 as suggested for BCC metals. Second, the coefficients for the Hockett-Sherby model were identified by best approximations of the bulge stressestrain curve. Finally, the coefficients associated with the enhanced HAH model were determined. The exponent q was set to 2, as generally recommended by Barlat et al. (2014). The coefficients ks , S controlling the cross-loading contraction and the coefficient k controlling the microstructure deviator evolution were optimized as best approximations of the stressestrain curve from the sequential tension test, i.e., 7% pre-strain at q1 ¼ 0 followed by subsequent tension at q2 ¼ 45 . Then, the five coefficients k1 , k2 , k3 , k4 and k5 accounting for reverse loading (Barlat et al., 2011), were determined by another stressestrain curve, hence 7% pre-strain at q1 ¼ 0 followed by subsequent tension at q2 ¼ 90 . The coefficients z, kR , k0R for microstructure deviator evolution in Eq. (7) were set as recommended values. All resulting coefficients for three DP steels are listed in Tables 4e6. 4.3. Comparison between enhanced HAH predictions and experiments 4.3.1. Proportional loading The monotonic tension and bulge experimental data are represented with the enhanced HAH model calculations in Figs. 15 and 16, respectively, indicating that the predictions for the DP steels agree well with the experiments. For proportional loading, the enhanced HAH model predicts the same results as the standard yield criterion combined with the reference isotropic hardening law as expected. In other words, the results demonstrate that the initial anisotropy and hardening behaviour of the three DP steels, under proportional loading, were well predicted by the Yld2000-2d yield function and the Hockett-Sherby model. 4.3.2. Non-proportional loadings Figs. 17e19 illustrate the predicted true stress-true strain curves for DP500, DP600 and DP780 in the two-step tension tests, respectively. The enhanced HAH model captures all the anisotropic hardening responses of DP500 very well, as depicted in Fig. 17. For DP600 and DP780, the model represents the transient hardening behaviour pretty well for the subsequent tension at both 45 and 90 after pre-strain of 4% and 7%. It also captures the permanent softening for tension reloading at 90 . However, there is a deviation between simulated curve and the reloading experimental curve at 45 at large strain for both

Table 4 Coefficients for the reference isotropic hardening model. Hockett-Sherby

A (MPa)

B

C

n

DP500 DP600 DP780

974 961 1080

358 391 501

1.79 2.89 5.05

0.506 0.567 0.598

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15

Table 5 Coefficients for the Yld2000-2d yield function. Yld2000-2d

a

a1

a2

a3

a4

a5

a6

a7

a8

DP500 DP600 DP780

6 6 6

0.988 0.935 0.973

0.983 0.968 0.879

0.889 0.890 0.864

0.962 0.981 0.948

0.987 1.017 0.999

0.910 0.939 0.860

0.946 0.987 0.949

1.050 1.072 1.030

Table 6 Coefficients for the enhanced HAH model. HAH

q

k

k1

k2

k3

k4

k5

ks

S

DP500 DP600 DP780

2 2 2

13.9 43.8 45

106.3 200.7 90

33.4 16.1 26.2

0.48 0.31 0.30

0.92 0.85 0.80

5 3 5

179 148 110

0.833 0.825 0.7

True plastic stress (MPa)

DP780

DP600

DP500

o

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

True plastic strain Fig. 15. Experimental and predicted curves from the standard tensile tests at q1 ¼ 0 , 45 and 90 from RD, for DP500, DP600 and DP780.

DP600 and DP780, and this discrepancy increases for the higher grade of DP, i.e., DP780. The curves predicted by enhanced HAH model approach gradually to the corresponding monotonic curves, while the experimental curve is still lower than the monotonic curve, even at large strain values. This means that the enhanced HAH model is not able to capture the long-term softening that occurs in the pseudo cross-loading condition for these two materials. The predicted results for DP500 is good since the permanent softening is not pronounced for this lower grade of steel. 4.3.3. Discussion In order to illustrate the discrepancy occurring in reloading at 45 for DP600 and DP780, Figs. 20 and 21 present the evolution of the state variable gs and the corresponding yield surface evolution, in the same loading history, using the input data of the DP780 steel listed in Tables 4e6, i.e., the material was first pre-strained in uniaxial tension along the RD to a strain of ε ¼ 7% after which, the load changed abruptly to a state near plane strain tension along the TD. The two corresponding stress deviators are of the form

2

3 2 1 0 0 0 0 5 and ½s2  ¼ z2 4 0 ½s1  ¼ z1 4 0 0:5 0 0 0:5 0

3 0 0 1 0 5 0 1

(10)

b : s equals to zero just after the change, this denotes a cross-loading condition. Since the quantity h The contraction of the yield locus is controlled by the state variable gs , whose evolution law relies on cos c, hence, on the b which reflects a loading change. The evolution of cos c and corresponding gs in the aforementioned angle between b s and h, condition is plotted in Fig. 20. When gs is less than one, it means that the yield surface starts contraction. The figure illustrates that the evolution of gs is not suddenly altered after the loading change, but is already active during the first loading step and asymptotically saturates to a specific value, as shown in Fig. 20 (b). This was adopted in order to account for a lower reloading yield stress just after cross-loading. After the loading change, the evolution of gs increases at high rate towards 1.0, but soon decreases again to accommodate any further cross-loading change. Please cite this article in press as: Liao, J., et al., Mechanical, microstructural behaviour and modelling of dual phase steels under complex deformation paths, International Journal of Plasticity (2016), http://dx.doi.org/10.1016/j.ijplas.2016.03.010

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1200

True plastic stress (MPa)

DP780 1000 DP600

800

DP500 600

Exp. DP500 bulge Exp. DP600 bulge Exp. DP780 bulge Predicted results

400

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 True plastic strain Fig. 16. Experimental and predicted curves from the bulge tests for DP500, DP600 and DP780.

The corresponding yield surface evolution of DP780 is shown in Fig. 21. In the end of the RD tension, the yield surface, which is represented as “7%”, not only flattens at the opposite side to the corresponding load, but also contracts in a direction b After 2% strain near plane strain tension, the contraction of the yield surface tends to disappear, which is orthogonal to h. controlled by the fast recovery of the state variable gs . The main transformation of the yield surface is a rotation resulting from

700

700

600

500

True stress (MPa)

True stress (MPa)

DP500

θ1=0o o

θ1=0 (7%), θ2=0

o

θ1=0o (4%), θ2=0o

400

DP500

600

θ1=0o

500

θ1=45o θ1=0o(7%),θ2=45o θ1=0o(4%),θ2=45o

400

HAH 0.00

0.04 0.08 0.12 True plastic strain

HAH

0.16

0.00

a)

0.08

True plastic strain

0.12

0.16

b)

700

True stress (MPa)

0.04

DP500

600 θ1= 0o

500

θ1= 90o θ1=0o(7%), θ2=90o θ1=0o(4%), θ2=90o

400

HAH 0.00

0.04 0.08 0.12 True plastic strain

0.16

c) Fig. 17. Tensionetension tests for DP500 in q2 ¼ a) 0 , b) 45 , c) 90 after 4% and 7% pre-strain in RD (q1 ¼ 0 ).

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800

800

DP600

DP600 700

700

600

θ1=0

500

o

θ1=0o(7%),θ2=0o

True stress (MPa)

True stress (MPa)

17

θ1=0o(4%),θ2=0o

400

0.04 0.08 0.12 True plastic strain

θ1=0o θ1=0o(7%),θ2=45o

0.16

0.00

a)

θ1=0o(4%),θ2=45o HAH 0.04

0.08 0.12 True plastic strain

0.16

b)

800

True stress (MPa)

θ1=45o

500 400

HAH

0.00

600

DP600

700 600

θ1=0o

500

θ1=90o

400

θ1=0o(4%),θ2=90o

0.00

θ1=0o(7%),θ2=90o HAH 0.04 0.08 0.12 True plastic strain

0.16

c) Fig. 18. Tensionetension tests for DP600 in q2 ¼ a) 0 , b) 45 , c) 90 after 4% and 7% pre-strain in RD (q1 ¼ 0 ).

the microstructure deviator, which evolves gradually toward the second stress state. After an additional 10% strain in plane strain tension, the final yield locus and the corresponding microstructure deviator are marked at a total strain of 17%. At this strain level, the microstructure deviator is almost coincident with the second loading state. The distortion of the yield locus at the direction orthogonal to the first loading has already recovered back to the stable yield locus, which indicates that the flow stress becomes coincident with the monotonic loading corresponding to the second tension step. However, this does not agree with the experiment results showing that the flow stress should be still lower than the monotonic curve, for all the DP steels in pseudo cross-loading. In the enhanced HAH model, the initial anisotropic yield surface expanding isotropically, i.e., fðsÞ ¼ sðεÞ, is selected as a stable reference yield surface (dotted line). During cross-loading, a combination of distortion and rotation of the yield locus contributes to the transient reloading regime of the material. After that, the yield surface recovers back to the stable yield surface in the vicinity of the current stress state. However, from the previous microstructural analysis, it is clear that the influence of the strain gradients developed during the previous loading is so strong that it dominates the reloading hardening response of DP steels, overcoming the effects of the initial anisotropy. Thus the reloading flow stress of DP steels (especially those with higher amount of martensite) after strain path change couldn't resume the monotonic one determined by the initial anisotropy even at large strain. The assumption of the enhanced HAH need to be refined in the future for further improvement of this feature. To illustrate the influence of critical parameters in the HAH model on the material behaviour after cross-loading, a sensitivity analysis of the three key parameters S, ks, k was conducted for the DP780. Fig. 22 shows the effect of the parameter S on the evolution of state variable gs and the flow stress of reloading when the material subjects a tension axis change of 45 after the pre-strain on the RD. Since the variable gs saturates towards 1 þ ðS  1Þcos2 c, the parameter S controls the extent to which the yield surface contracts. As the contraction of the yield surface at current stress state is related to the stress ratio s0 =sðεÞ, the constant S determines the onset of re-yielding. It can be seen from Fig. 22 (b) that an earlier re-yielding happens when S is smaller. From the previous comparison, it was clear that the DP steel with higher volume fraction of martensite

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1000

DP780

800

600

True stress (MPa)

True stress (MPa)

1000

θ1=0o o

θ1=0 (7%),θ2=0

o

θ1=0o(4%),θ2=0o

400

DP780

800

θ1=0o

600

θ1=45o θ1=0o(7%),θ2=45o θ1=0o(4%),θ2=45o

400

HAH 0.00

0.05 0.10 True plastic strain

HAH 0.15

0.00

0.05 0.10 True plastic strain

a)

b)

1000

True stress (MPa)

0.15

DP780

800 θ1=0o

600

θ1=90O θ1=0o(7%),θ2=90o θ1=0o(7%),θ2=90o

400

HAH 0.00

0.05 0.10 True plastic strain

0.15

c) Fig. 19. Tensionetension tests for DP780 in q2 ¼ a) 0 , b) 45 , c) 90 after 4% and 7% pre-strain in RD (q1 ¼ 0 ).

exhibited smaller re-yielding stress, which can be reflected by an obvious decrease of the value of S for the three steels, i.e. DP500, DP600 and DP780 in Table 6. The effect of parameter ks is demonstrated in Fig. 23. This parameter controls the evolution rate of the variable gs, in both contraction and recovery processes. Fig. 23 (a) shows that a higher value of the ks is not only related to a faster contraction of the yield surface before the loading change, but also means a faster recovery just after the loading change. It should be noted that this parameter mainly controls the transient hardening rate, it has limited influence on the magnitude of the re-yield onset. Fig. 23 (b) indicates that the hardening rate become higher if the value of ks is larger. This parameter reflects the length of transient region of the material to some extent, since a larger ks means a shorter transient region. For the DP steel with higher volume fraction of martensite, the strain gradient was more pronounced and its influence on the flow stress was more significant, leading to a longer transient region, as shown previously in Fig. 5. A decrease of the ks for the three steels can be seen from Table 6, which is in line with this observation. The parameter k is related to the evolution rate of the microstructure deviator, e.g. the “rotation” of yield surface viewed in b are collinear during proportional loading. When the loading path changes, the tensor the p-plane. Initially, the tensors s and h b evolves in a manner to become collinear with the current s or s, at a rate controlled by the parameter k. The rotation rate of h the microstructure deviator also affects the evolution of the variable gs, as illustrated in Fig. 24 (a). The second contraction of the yield surface occurs earlier when k is doubled. Fig. 24 (b) shows that this parameter has a dominate influence on the flow stress after the transient stage. A larger k contributes to a faster recovery of flow stress. This is because after the transient stage, the recovery of the flow stress is mainly controlled by the rotation of the yield surface. A smaller rotation rate somewhat delays the recovery process but it cannot capture the permanent softening. Model adjustment of the evolution rate of the microstructure deviator and the evolution rate of the contraction might be a potential way to improve the predictions. The parameter k can be viewed as a reflection of the reorganization of the stress interactions between phases when loading changes. The rotation of the microstructure deviator is faster for DP steel with higher volume fraction of martensite than the

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DP780

DP780

1.0

1.0

19

0.5

gs

cosχ

0.9

0.8

Loading change 0.7

0.0 0.00

0.05

0.10

0.15

Loading change

0.20

0.0

0.1

0.2

Equivalent plastic strain

Equivalent plastic strain

a)

b)

Fig. 20. Evolution of a) cos c and b) state variable gs for DP780.

s2

π-plane

s ^ h 17% ^h ^ 9% h

17%

s1

7%

7%

s3

9% Isotropic hardening

Fig. 21. HAH predictions for DP780 showing cross-direction contraction of normalized stable surface after uniaxial pre-strain of ε ¼ 7%, and after additional crossloading in near plane strain for an additional 2% and 10%.

lower one, as listed in Table 6. This is presumably because, the effect of the martensite is to induce high internal stresses in the material. When the stress state changes, the internal stresses reorient according to the new loading. This reorientation is much faster than the microstructure evolution. However, the microstructure deviator in HAH reflects more the influence of the internal stress than the microstructure itself. Thus, the effect of martensite is likely to increase the rate of evolution of the 1000

1.0

S=0.7 S=0.5 True stress (MPa)

0.9

gs

0.8 0.7 0.6 0.5 0.00

0.05

0.10

Equivalent plastic strain

a)

0.15

DP780

800

600

S=0.7 S=0.5

400 0.00

0.05

0.10

0.15

True plastic strain

b)

Fig. 22. The influence of the parameter S on the a) state variable gs, b) the flow stress of reloading.

Please cite this article in press as: Liao, J., et al., Mechanical, microstructural behaviour and modelling of dual phase steels under complex deformation paths, International Journal of Plasticity (2016), http://dx.doi.org/10.1016/j.ijplas.2016.03.010

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J. Liao et al. / International Journal of Plasticity xxx (2016) 1e22

1000

1.0

ks=60

gs

0.9

0.8

0.7 0.00

0.05

0.10

0.15

Equivalent plastic strain

True stress (MPa)

ks=110

DP780

800

600

ks=110 400 0.00

ks=60 0.05

0.10

0.15

True plastic strain

a)

b)

Fig. 23. The influence of the parameter ks on the a) state variable gs, b) the flow stress of reloading.

1000

1.0

True stress (MPa)

k=45 k=90

gs

0.9

0.8

0.7 0.00

0.05

0.10

Equivalent plastic strain

a)

0.15

DP780

800

600

k=45 k=90 400 0.00

0.05

0.10

0.15

True plastic strain

b)

Fig. 24. The influence of the parameter k on the a) state variable gs, b) the flow stress of reloading.

microstructure deviator (the parameter k in the model). However, due to the limitation of this model to the permanent softening in this case, it should be further consolidated from the refinement of the model.

5. Conclusions The strain hardening response and microstructural behaviour of three dual phase steels, i.e. DP500, DP600 and DP780, deformed in monotonic and non-proportional strain paths, were analyzed. The two-step tension test sequences at various angles were conducted to cover a broad range of strain path change, e.g. proportional loading, pseudo cross-loading and orthogonal tension. Microstructural analysis, namely, SEM, EBSD, TEM and simulation with the VPSC polycrystal plasticity model were conducted in order to identify the mechanisms associated to the observed mechanical behaviour. For the constitutive modeling, the so-called enhanced HAH anisotropic hardening model integrated with the Yld2000-2d yield criterion were adopted to reproduce the hardening responses in these tests. The following conclusions can be extracted from this work: (1). The three DP steels show similar characteristic in both pseudo cross-loading and orthogonal tension. That is (a) yielding at a lower stress compared to that before unloading, followed by a rapid, transient strain hardening; (b) A long range (ε  12%) flow stress, lower than that of corresponding monotonic deformation; (c) A strain hardening rate approaching to the monotonic curve after all strain path change. These features are more pronounced for material with higher amount of martensite and higher pre-strain level. (2). The initial anisotropy of the three DP steels during the monotonic tests has a crystallographic texture origin. However, the evolution of the crystallographic texture during the pre-strain is very weak and cannot explain the mechanical behaviour during a change of strain path. Please cite this article in press as: Liao, J., et al., Mechanical, microstructural behaviour and modelling of dual phase steels under complex deformation paths, International Journal of Plasticity (2016), http://dx.doi.org/10.1016/j.ijplas.2016.03.010

J. Liao et al. / International Journal of Plasticity xxx (2016) 1e22

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(3). The plastic deformation process in DP steels is very inhomogeneous. This results from the deformation incompatibilities between the soft ferritic matrix and the hard martensitic particles. This heterogeneity of the strain distribution along the ferrite grains increases with the amount of strain and the martensite volume fraction. (4). Results from microstructural investigation along with the analysis of the stressestrain curves suggest that the mechanical behaviour during the beginning of the reloading is controlled by the plastic deformation of weakly strain hardened regions in the ferrite grains, developed during the pre-strain as a result of strain incompatibilities between the two phases. Both, the transient stage and the evolution of the reloading yield stress with the pre-strain value, reloading angle and martensite volume fraction can be ascribed to this inhomogeneous plastic strain of ferrite grains. (5). The effect of strain gradients developed during the previous loading is so strong that it determines the reloading hardening response of DP steels, overcoming the effects of the initial anisotropy due to the crystallographic texture. (6). The plastic behaviour of all the DP steels investigated in this work during the monotonic tests, as well as the transient hardening behavior observed in sequential tension tests, were well reproduced by the enhanced HAH model. However, this model failed to represent permanent softening in pseudo cross-loading conditions, especially for DP600 and DP780. Further refinement should be put on the current HAH formulation to improve this situation. In addition, special attention will also be paid to incorporate microscopic scale parameters, such as martensite content or ferrite grain size, to keep stronger physical relevance in describing the transient behaviours for multi-phase steels under strain path changes.

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Please cite this article in press as: Liao, J., et al., Mechanical, microstructural behaviour and modelling of dual phase steels under complex deformation paths, International Journal of Plasticity (2016), http://dx.doi.org/10.1016/j.ijplas.2016.03.010