Prediction of flow stress and surface roughness of stainless steel sheets considering an inhomogeneous microstructure

Author’s Accepted Manuscript Prediction of flow stress and surface roughness of stainless steel sheets considering an inhomogeneous microstructure Cong Hanh Pham, Sandrine Thuillier, Pierre-Yves Manach www.elsevier.com/locate/msea

PII: DOI: Reference:

S0921-5093(16)31184-4 http://dx.doi.org/10.1016/j.msea.2016.09.101 MSA34186

To appear in: Materials Science & Engineering A Received date: 11 September 2016 Revised date: 25 September 2016 Accepted date: 26 September 2016 Cite this article as: Cong Hanh Pham, Sandrine Thuillier and Pierre-Yves Manach, Prediction of flow stress and surface roughness of stainless steel sheets considering an inhomogeneous microstructure, Materials Science & Engineering A, http://dx.doi.org/10.1016/j.msea.2016.09.101 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Prediction of flow stress and surface roughness of stainless steel sheets considering an inhomogeneous microstructure Cong Hanh Phama , Sandrine Thuilliera,∗, Pierre-Yves Manacha a

Univ. Bretagne Sud, FRE CNRS 3744, IRDL, F-56100 Lorient, France

Abstract The aim of this study is to model the mechanical behavior of ultra-thin sheets of stainless steel taking into account the heterogeneous distribution of the mechanical properties of the grains. EBSD analysis and micro-hardness measurements are performed to characterize the grain size and shape as well as the local hardness distribution. A finite element model of a representative volume, based on the grain size distribution and Vorono¨ı polyhedrons, is developed using the dedicated free software Neper and the material inhomogeneity is introduced via a discrete distribution of the flow stress of the grains. Four different flow stress distributions are considered, by changing either the flow stress range or the corresponding volume fractions. A uniaxial tensile test of the representative volume is simulated and compared to experimental data. Then, the influence of the grain heterogeneity on the hysteresis of loading-unloading sequences in tension and on the Bauschinger effect in simple shear is investigated numerically and compared with experiments. In particular, it is shown that the type of the flow stress distribution influences significantly the magnitude of the Bauschinger effect. Finally, the surface roughness is characterized numerically both in tension and simple shear and is shown to reproduce faithfully the experimental values. Keywords: ultra-thin metallic sheet, surface roughness, grain size, material inhomogeneity, mechanical behavior 1. Introduction Industrial metallic alloys are polycrystalline materials, that are composed of millions of grains of different sizes, shapes and orientations. Size and orientation are well-known to influence the mechanical properties, e.g. the distribution of crystallographic orientations, ∗

Corresponding author Email address: [email protected] (Sandrine Thuillier)

Preprint submitted to Materials Science & Engineering A

September 27, 2016

5

or texture, leads to anisotropy of the flow stress and plastic anisotropic coefficients whereas the lower the grain size, the larger the flow stress is, as described by the Hall-Petch relation. Though they have the same crystallographic structure in single phase polycrystalline materials, the grains may have different mechanical properties depending on the initial dislocation density, on the inclusion or precipitate or phase volume fractions. Further-

10

more, surface grains may have a different microscopic behavior compared to the grains in the bulk material (Song et al., 2004). All these effects lead to a rather large inhomogeneity of the mechanical properties of polycrystalline metallic alloys. Additionally, these effects are more pronounced in the case of ultra-thin sheets, with thickness of the order of 0.10 mm, where the number of grains through the thickness influences also significantly

15

the mechanical properties. Numerous experimental and numerical investigations have been carried out to highlight the relationship between the inhomogeneity of the material and the Bauschinger effect (D´epr´es et al., 2011), the mutiphase microstructure (Tasan et al., 2014), the surface roughness (Furushima et al., 2009a; Lu et al., 2013; Rossiter et al., 2013; Yoshida, 2014) or failure mechanisms (Chan and Fu, 2012; Kim and Yoon,

20

2015; Meng and Fu, 2015; Zhang and Dong, 2016). Though frequently neglected within phenomenological modeling approaches, taking into account such an inhomogeneity may benefit the representation of the mechanical behavior. As an example, the modeling of texture and its variation upon straining can lead to a very good representation of the material anisotropy.

25

Within a grain, the inhomogeneity of the plastic deformation is caused by the mechanism itself, i.e. production and movement of dislocations. They introduce imperfections into the crystallographic structure, thus leading to plastic straining and their multiplication and interactions are responsible for strain hardening. The difference in the dislocation density between grains, in the intragranular structures of dislocations (pile-ups, entan-

30

glements, walls) is also responsible of the grain inhomogeneity (Meyers and Ashworth, 1982; Chan et al., 2011; Fu and Chan, 2011). Since the flow stress is closely related to the hardness of the material (Meyers et al., 2006; Glaeser et al., 2010), the scatter of nanoor micro-hardness has been proposed to illustrate the heterogeneity of the mechanical properties of grains.

35

In order to investigate experimentally the grain interactions during the deformation, Sachtleber et al. (2002) measure the deformation at the grain scale of a coarse-grained recrystallized polycrystalline aluminum in uniaxial tension. They show that the defor2

mation at the grain scale is non-uniform at an early stage of plastic straining and the heterogeneity becomes even more obvious after an increase of the macroscopic deforma40

tion. For example, after 8% reduction of the sample thickness, some grains support as little as 1% von Mises equivalent plastic strain while others show a maximum deformation above 16%, particularly in areas located in the vicinity of grain boundaries. Song et al. (2004) have also investigated the inhomogeneity of each grain and sub-grain during the deformation, showing that the deformation inhomogeneity increases with the applied

45

strain. They also confirm that the surface roughness is mainly determined by the inhomogeneity of the deformation of a surface layer, with a layer thickness around 3-4 grains; the relationship between surface topography and the distribution of accumulated microstrains is also evidenced in (Becker, 2003). Such an inhomogeneity is also evidenced by near-field high-energy X-ray diffraction microscopy of polycrystalline copper (Pokharel et al., 2015)

50

and spherical nano-indentation of a polycrystalline aluminium (Vachhani et al., 2016). It is therefore interesting to develop modeling approaches able to represent, at least partially, these inhomogeneities, in order to investigate their influence on the macroscopic mechanical behavior. Several inhomogeneous models are proposed in literature. Lai et al. (2008) develop a mixed model based on surface and inner grains; the stress level of the

55

resulting material is a combination of a size independent term, which corresponds to a conventional polycrystalline model, and a size dependent term related to the ratio of the grain size over the thickness. It is shown that the flow stress decreases when the ratio of the thickness over the gran size is below 10, leading to a size dependent effect. This model has been used widely to investigate the size effects on material behavior of ultra-

60

thin sheet metals in tension (Xu et al., 2013; Liu et al., 2011) and hydraulic bulging (Mahabunphachai and Ko¸c, 2008). Chan et al. (2010) proposed a heterogeneous model to estimate grain properties and grain size effect in micro-compression tests of pure copper. The decrease of the flow stress, the increase of the scatter of data and inhomogeneous deformation that occur with the decrease of the ratio of specimen size to grain size are

65

predicted. Straffelini et al. (2013) propose an approach that accounts for microstructural inhomogeneities by considering the material as a composite made of a matrix (main component) and regions of slightly higher and lower strength. This approach is found to be suitable to model the deformation behavior in tension, and it succeeds to correctly capture the onset of shear localisation of AA5182 aluminium sheets. Additionally, crystal

70

plasticity finite element models (CPFEM), based on texture representation and crystal 3

plasticity constitutive equations, is suggested for predicting inhomogeneous deformation both at microscopic and macroscopic scales (Raabe et al., 2002a). A more or less realistic representation of the grain shapes is chosen, e.g. using octagons grains Kim and Yoon (2015) or Vorono¨ı polyhedrons, e.g. (Barbe and Quey, 2011; Zhang et al., 2016) and 75

eventually a direct mapping from EBSD information to a 3D mesh (e.g. Zhao et al. (2008); Choi et al. (2013)).Macroscopic phenomena such as surface roughening (Becker, 1998; Yue, 2005; Zhao et al., 2008) and orange peel (Raabe et al., 2002b), grain interactions, grain size effects, strain gradient effects (Diard et al., 2005; Barbe et al., 2001a,b), forming, deep drawing, spring-back (Nakamachi et al., 2007) can be predicted. The advantages

80

of CPFEM is that the numerical predictions can be compared to experiments in a very detailed way, probing a large variety of data, for examples grain shape change, forces, strains, texture evolution and size effects one-to-one at different scales (Roters et al., 2010). Furthermore, virtual database of the mechanical properties can be generated, in order to identify material parameters of phenomenological models (Zhang et al., 2016).

85

However, high performance computers and long calculation times are needed in case of CPFEM for three dimensional and applications in forming processes (Beaudoin et al., 1993; Zhang et al., 2016). Though promising, CPFEMs usually make use of a phenomenological description of kinematic hardening, at the grain scale, to represent the Bauschinger effect, e.g. (Barbe

90

and Quey, 2011); it should be emphasized that this effect is related to internal stresses arising from strain incompatibilities between the deforming grains. However, taking account only the texture as a source of material inhomogeneity is not rich enough to reproduce the Bauschinger effect. Recently, Adzima et al. (2016) show the influence on material inhomogeneity by introducing a distribution of the critical resolved shear stress; the authors

95

obtain a good representation of the Bauschinger effect for copper alloy sheets, at least in a limited range of strains. As a simpler alternative, Furushima et al. (2009a, 2011, 2013) suggest a finite element (FE) model considering mesoscopic inhomogeneities, in order to investigate the effect of free surface roughening on the ductile fracture of copper sheet and ultra-thin sheet in

100

uniaxial and bi-axial tension. In their model, the grains, assumed to be of a cubic form and meshed with many finite elements, are divided into seven groups and a flow stress distribution is chosen, based on micro-indentation measurements, to assign a given flow stress level to each group of grains. This model is also used to predict the occurrence of 4

necking phenomenon during a uniaxial tensile test and the localized deformation caused by 105

surface roughening before occurrence of diffused necking can be confirmed. However, with the proposed distribution, this model can not predict the contribution of the kinematic hardening of material and applicability of this model to micro-forming for metal foils has not been verified yet. Similar models have also been developed by Utsunomiya et al. (2004a), Utsunomiya et al. (2004b) in the case of rolling of aluminum sheets and by Lu

110

et al. (2013) in the micro-compression test and micro-cross wedge rolling of pure copper. As these models are based on the real microstructure, they are developed and used mainly for ultra-thin sheets, that are characterized by a small number of grains in the thickness, and are more prone to a thickness-dependent behavior (Pham et al., 2015). In parallel, D´epr´es et al. (2011) and Tabourot et al. (2012) propose a compartmen-

115

talized model, based on the distribution of dislocations, for the mechanical behavior of titanium and high carbon steel. The macroscopic mechanical behavior of the material is predicted by taking into account the distribution of dislocation density. Each grain, of cubic form, is assigned a bi-linear hardening law, with the initial yield stress distributed according to a Rayleigh distribution. Three parameters should be adjusted to obtain a

120

correct reproduction of the overall behavior of the material in tension. The advantage of this model is that common models based on finite element codes can be used without additional complexity and it becomes possible to reproduce the Bauschinger effect and localization of deformation (D´epr´es et al., 2011). The aim of this paper is to introduce a phenomenological FE model based on a flow

125

stress distribution at the grain scale, in order to predict the mechanical behavior in tension and simple shear, during loading and also unloading and reverse loading. The originality of this study is to introduce a realistic shape of the grains, i.e. irregular polyhedrons instead of cubes, and to investigate the influence of the flow stress distribution on the macroscopic mechanical behavior. Moreover, both strain paths in tension and simple shear

130

are considered. Firstly, the material microstructure is introduced as well as its mechanical characterization performed at room temperature. Micro-indentation measurements are also carried out, in order to highlight the flow stress distribution. In a second step, the 3D polycrystalline aggregate is modeled with a Vorono¨ı tessellation, using the dedicated free software Neper, and experimental and numerical grain shapes and sizes are compared.

135

Then, a Gaussian distribution of the flow stress of the grains is introduced to represent the material inhomogeneity. A representative 3D sample is submitted numerically to tension 5

and simple shear tests. The results focus on the comparison of predicted stress-strain curves with experimental data, during monotonic loading and loading-unloading sequences in tension, monotonic simple shear and also reverse loading. The influence of the flow 140

stress distribution is also investigated. Finally, the surface roughness is measured both experimentally and numerically in tension and simple shear and the values are compared. 2. Material and experimental procedure 2.1. Microstructure characterization The material is an austenitic stainless steel provided by ArcelorMittal of AISI 304 type

145

(X4CrNi18-9). The material is supplied as cold rolled sheets of thickness t = 0.15 mm, in coils of 28 mm width in a shining annealed final state. The weight chemical composition of the material is given in Table 1. Electron Backscatter diffraction (EBSD) analysis has been performed both on the sheet plane and on the edge of the sheet. Surfaces are polished first mechanically and then electro-chemically. Fig. 1 displays an overview of

150

the grain distribution, both in the sheet plane and on the sheet edge. An average grain size d of approximately 17.50 µm and average grain surface of 335 µm are determined. It corresponds to about 8 grains through the thickness of the sheet. Table 1: Chemical composition in weight percent of material AISI 304

C

Mn

Si

S

P

Ni

Cr

0.040

1.310

0.440

0.003

0.026

8.580

18.240

2.2. Macroscopic mechanical behavior The mechanical behavior has been investigated experimentally at room temperature 155

and along the rolling direction, at a strain rate of the order of 1 × 10=3 s=1 . Tensile and simple shear tests are performed with a local strain measure, using the non-contacting Digital Image Correlation (DIC) system Aramis (GOM GmbH). Sample dimensions for tension are fixed according to ISO6892-1, with a width of 12.50 mm and a gauge area of constant section of 57 mm (Pham et al., 2015). And simple shear specimens are rect-

160

angular, with a gaude area of 28 × 1.40 mm2 . Monotonic tests are performed, as well as tests composed of loading-unloading-reloading sequence in the same direction in tension, in order to investigate the unloading non-linear behavior. Finally, tests composed of loading-unloading-reloading sequence in the reverse direction in simple shear are carried 6

S1

S

S

S

S

200μm

(a)

(b) Figure 1: EBSD analysis of the microstructure of stainless steel AISI 304: observations (a) in the sheet plane and (b) on the edge of the sheet. Sections Si , i = 1 − 5 are defined on the surface to analyze the roughness

out to characterize the Bauschinger effect. Fig. 2 shows the stress-strain curves obtained 165

in uniaxial tension and simple shear. The material displays a very high hardening rate in tension, going from stress around 300 up to 1000 MPa over a strain span of 0.5 and a slight Bauschinger effect. The chord modulus EU is defined as the slope of the straight line connecting the intersection point of the unloading curve with the following reloading curve and the end point of the unloading stress–strain curve; its variations with the equivalent plastic strain is investigated in tension, with repeated loading-unloading at several strains. Such a variation can be represented by Eq. 1 (Yoshida et al., 2002): EU = E0 − (E0 − Esat ) [1 − exp(−ξεp )] 7

(1)

with εp the equivalent plastic strain, E0 = 206.20 GPa, Esat = 147 GPa and ξ =45. A plastic model associated to von Mises isotropic yield criterion is selected to represent 170

the mechanical behavior. A previous study has shown that such a yield criterion leads to a good description of the flow stress both in tension and simple shear (Pham et al., 2015). Furthermore, isotropic hardening is assumed based on a Voce type law, with the addition of a linear term to prevent excessive strain localization from the stress saturation (Eq. 2); the yield stress σY depends on four material parameters, its initial value σ0 , the

175

saturation value Q and rate of saturation b and finally the slope of the linear term H. Material parameters are identified in order to minimize the gap between experimental and numerical data, both in tension and simple shear and are given in Table 2. The numerical prediction of the stress levels in tension and simple shear is also plotted in Fig. 2. A fairly good description is obtained, though the model fails to capture the stress

180

level in simple shear at the early stage of straining. Moreover, the material displays a Bauschinger effect, not taken into account with this isotropic hardening model, and the difference between experiments and the numerical prediction during the reverse loading in simple shear highlights the magnitude of this effect. σY = σ0 + Q [1 − exp (−bεp )] + Hεp

(2)

Table 2: Material properties and material parameters for the isotropic hardening model of Eq. 2

Material

t (mm)

d (μm)

N=t/d

σ0 (MPa)

Q (MPa)

b

H (MPa)

AISI 304

0.15

17.5

8

287.5

551.7

4.36

727.2

In addition, some tensile and simple shear tests are stopped at different strains to 185

measure the variation of surface roughness and each type of test is performed at least three times to check the reproducibility of the results. The surface roughness is measured by contact measurement (TR100 surface roughness tester) along a length of 2.00 mm, perpendicular to the loading direction in tension and in the loading direction in simple shear test. The initial surface roughness Rz = 1.40 µm and its variation with plastic strain

190

are then evaluated (Pham et al., 2015). 2.3. Micro-hardness measurements The material heterogeneity is analyzed via micro-hardness tests over the sheet surface, using the Shimadzu HMV-2 Micro-hardness tester. Indeed, the grain hardness, noted HV , 8

1200 1000 800

σ (MPa)

600 400 200 0 -200 Tensile test Shear test Isotropic hardening

-400 -600 -0.3

-0.2

-0.1

0

0.1 0.2 ε, γ

0.3

0.4

0.5

0.6

Figure 2: Stress-strain curves obtained in uniaxial tension, monotonic and Bauschinger simple shear.

is an effective parameter to estimate the deformability of individual grains. Specimens are directly cut in the sheet. The specimen surface is polished down to diamond paste to remove any deformation layer at the surface then etched during 5 min using Glyceregia etching solution (Small et al., 2008). A load of 0.10 N and a holding time of the indenter of 5 s are chosen to get a print size about 7 µm. Four hundred measures have been performed inside the grains, visible after etching, and the grain hardness distribution is derived and shown in Fig. 3. This distribution displays two peaks, a high one around HV = 200 and a smaller one close to HV = 300. The first peak is related to austenite and the smaller could arise from ferrite. Indeed, depending on the steel chemical composition and the heat treatment, up to 10% of delta ferrite could be observed at room temperature in 304 stainless steel, either as a continuous network or as isolated cores (Tseng et al., 1994; Padilha et al., 2013). A bimodal normal distribution (Eq. 3) can be used to fit the experimental results. This bimodal distribution is the combination of two Gaussian distributions, associated with the average hardness of the austenite (H a ) and a mixture of austenite and ferrite phases (H f ). This result is in good agreement with the results of nano-indentation of multiphase materials performed by Nohava et al. (2010) and Pavlina et al. (2015) and the values of hardness for austenite are close to the ones presented in Horvath et al. (1998); additionally, hardness of ferrite is higher than hardness of austenite.

9



1 f = a exp − 2



HV − H a b

2 



1 + c exp − 2



HV − H f d

2  (3)

with f the frequency of having a grain of a given HV , a, b, c, d material constants. Two levels of heterogeneity are then highlighted: austenite/ferrite-rich zone and the distribution within the austenitic phase, with a hardness ranging from 160 up to 240. For 195

simplicity’s sake, only the Gaussian normal distribution corresponding to the austenite phase is considered in the remaining part of this study. 20 18 16 14

f [%]

12 $XVWHQLWH

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4 2 0

50 !3 0 32 0 31 0 30 0 29 0 28 0 27 0 26 0 25 0 24 0 23 0 22 0 21 0 20 0 19 0 18  0 17 60

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Figure 3: Statistical distribution of grain hardness measured over 400 areas. The image in the top right corner shows the indentation prints. Each measure corresponds roughly to one grain.

3. Description of the FE model 3.1. General principles A numerical model, based on the representation of a flow stress distribution at the grain 200

scale as proposed by Furushima et al. (2009b), is developed with the FE code Abaqus. In this model, grain size and its distribution, as characterized by EBSD, are considered to build the FE mesh. A representative volume of dimensions 645 × 2000 × 75 µm3 is selected. It corresponds to half the thickness and is similar to the area investigated for surface roughness measurements in the sheet plane. This representative volume contains

205

around four grains in the thickness and is divided into 15840 Vorono¨ı polyhedrons with

10

an arbitrary shape generated by the free software Neper (Quey et al., 2011). These grains are meshed using four-node tetrahedral elements with linear interpolation and full integration scheme (C3D4 in Abaqus) with an average element size of 5 µm, leading to about 300 elements per grain. Input data for Neper is the number of grains, the mesh 210

size and type of finite element. Fig. 4(a) displays the numerical grain distribution in the sheet plane, for the representative volume, as well as the mesh of some of the grains. Sections, as illustrated in Figs. 1(a) and 4(a), have been defined in the sheet plane, in order to measure an average value of the intersecting segments. The average numerical grain size is 18 µm, to be compared to the experimental value of 17.50 µm. Different

215

sections are defined in order to measure the size of each grain in the prediction and in experiments. Fig. 4(b) shows a good correlation of the numerical grain size distribution with experimental values. In both cases, the size of grains can be represented in the form of a normal distribution, with most part of grains having a size between 17 and 22 µm. The next step is to fix the flow stress distribution. First of all, the flow stress at a

220

grain level of a single phase material results from the grain crystallographic orientation, the inclusion/precipitates volume fraction and the dislocation density. Since the hardness of the material is closely related to the flow stress, the flow stress distribution could follow a similar trend as in Fig. 3. However, Furushima et al. (2009a) has shown that the FE analysis using such a distribution obtained directly from micro-hardness measurements is

225

not in full agreement with experimental results for the prediction of surface roughness in tension. But a Gaussian distribution could be used to describe the volume fraction of the flow stress of the grains. It should be emphasized that other distribution functions are proposed in literature, e.g. Utsunomiya et al. (2004a) make use of a bi-linear or triangular distribution function dependent on three characteristics values and D´epr´es et al. (2011)

230

propose a Rayleigh distribution function dependent on a unique parameter. Though the influence of the type of distribution has not been investigated yet, it is out of the scope of this study. The grains of the representative volume are then divided into different groups and a total of seven groups is considered in this study, as proposed in (Furushima et al., 2013).

235

The grains of a same group have all the same flow stress, whereas it is different for each group; the flow stress of a given group i, called σi , is calculated by introducing a material parameter αi in Eq. 2, i.e. σi = αi ∗ σY . It should be emphasized that the overall flow stress variation is modified for each group, and not only the initial yield stress as proposed 11

2000 μm

645 μm

S1 S2 S3

Y Z

S4 S5

X n° 2693

n° 10528

n° 13213

n° 8638

20 μm (a)

Exp. Sim.

14 12

f

10 8 6 4 2 0

 5 7

  1 15 17 19 21 23 25 27 29 31 33 9 11            !35 G(—m)

(b) Figure 4: Finite element model: (a) 15840 Vorono¨ı polyhedral in the model and example of the grains meshed by C3D4 type (b) comparison of experimental and numerical grain sizes

in (Utsunomiya et al., 2004a; D´epr´es et al., 2011), as illustrated in Fig. 5. In a first step, 240

the volume fraction of each group is imposed from a normal distribution within a range from 0.46 to 1.54; the corresponding values are given in Table 3. Such a range is the one proposed by Furushima et al. (2013) and serves as a first trial. Other studies make use of lower ranges, e.g. Utsunomiya et al. (2004a) propose a variation of around 44% for an

12

aluminium alloy whereas Furushima et al. (2011) try several ranges, up to 25%. It should 245

be emphasized that the stress level distribution is plotted in Fig. 5 for the same strain range; however, grains with the higher values of αi are expected to support a very limited amount of plastic strain, thus limiting the flow stress level. Moreover, Fig. 6 illustrates the normal distribution: a variation of ±50% around the average value of one is chosen and the normal distribution parameter (and thus the volume

250

fraction of each group of grains) is adjusted by a trial-and-error method in order to fit the stress level in tension. The volume fraction V4 , corresponding to α4 = 1, is maximum and represents the dominant characteristic of the grains. The number of grains of each group is then quantified from the total number of grains and the respective volume fractions and grains are randomly distributed in the mesh by

255

using a program developed in the Scilab environment (Fig. 7(a)). Finally, the variation of the chord modulus with the plastic strain (cf. Eq. 1) is also introduced in the FE model via an external subroutine (USDFLD for Abaqus) as a variation of the elastic modulus with the equivalent plastic strain. 3000

α7 = 1.54 α6 = 1.36

2500

α5 = 1.18

σ (MPa)

2000

α4 = 1.00 α3 = 0.82

1500

α2 = 0.64

1000

α1 = 0.46

500

/PSNBM
EJTUSJCVUJPO
G A

0 0

0.5

1.5

1

2

2.5

εp

Figure 5: A schematic illustration of different flow stress of grains by considering material inhomogeneous parameter αi used in the FE model.

Calculations are run on a Linux computer (16 Intel cores and 32 Go of ram) with six 260

microprocessors in parallel. It takes 69 hours for the simulation of the tensile test up to a strain of 0.58. The higher the macroscopic strain, the more difficult is the convergence, due to the strain localization.

13

7



 7



7

7

G
D

  7

7

7



 7



7

7

 

















D

Figure 6: Normal distribution of the parameters αi for model 1. s = 0.23, μ = 1. Table 3: Material inhomogeneous parameters and the corresponding volume fractions of each group of grains. 4 sets of parameters were considered in this study.

Model 1

Model 2

Model 3

Model 4

i

1

2

3

4

5

6

7

αi

0.46

0.64

0.82

1.00

1.18

1.36

1.54

Vi (%)

4

9

22

30

22

9

4

αi

0.10

0.40

0.70

1.00

1.30

1.60

1.90

Vi (%)

4

9

22

30

22

9

4

αi

0.10

0.40

0.70

1.00

1.30

1.60

1.90

Vi (%)

3

7.5

20

30

24

10.5

5

αi

0.10

0.40

0.70

1.00

1.50

2.00

2.50

Vi (%)

4

9

22

30

22

9

4

3.2. Influence of the parameters of the flow stress distribution In order to investigate the influence of the distribution parameters and to investigate 265

the capability of the model to reproduce the non-linearity observed during unloading as well as the Bauschinger effect, three extra sets of parameters were chosen and the corresponding values are given in Table 3. Model 1 refers to the normal distribution illustrated in Fig. 6. As it is not able to reproduce none of these effects, the idea is to enhance the level of heterogeneity by extending the range of variation of the inhomogeneous parameters αi ,

270

from 0.1 and up to 1.9; this new set of parameters is referred to as model 2. However, the stress level in tension is under-estimated with model 2, and an alternative solution is to increase the volume fractions corresponding to αi > 1 and to decrease the ones related to

14

αi < 1, leading to model 3. Finally, in order to still enhance heterogeneities, and for the same volume fractions as models 1 and 2, the range of variation for αi is increased, but 275

only for the higher contributions to the flow stress, i.e. up to to 2.5 and the values of αi for i = 5, 6, 7 are adjusted to reproduce the macroscopic stress-strain curve in tension.  The numerical mean initial yield stress σ0 7i=1 αi Vi is 312.60 MPa, compared to σ0 = 287.50 MPa, to find the best prediction of the stress-strain curve in tension. This last set corresponds to model 4.

280

A tensile test is simulated numerically using the representative volume, with boundary conditions displayed in Fig. 7(a). Equal magnitude but opposite displacements are imposed in the tensile direction and symmetry conditions are used in the width and in Δ the thickness. The numerical macroscopic deformation (ε) is calculated as ε = ln(1 + ) l0 with l0 = 645 µm, the initial length in y direction (see frame definition in Fig. 4(a)). The

285

Cauchy stress is calculated from the reaction load at one extremity. Fig. 7(b) shows the numerical Cauchy stress variation with the macroscopic strain as well as a comparison with experimental results and with the phenomenological approach based on isotropic hardening. It can be seen that the macroscopic stress level in monotonic tension is nicely reproduced for models 1,3 and 4 whereas it is under-estimated with model

290

2. The aim of the following section is to investigate the predictions for several complex strain paths as well as to illustrate the heterogeneities at the representative volume scale as well as at the grain scale. 4. Results 4.1. Heterogeneity

295

The inhomogeneous strain distribution over the representative volume, arising from the flow stress distribution of the grains, is investigated in uniaxial tension. Fig. 8(a) shows the range of variation for the local equivalent plastic strain. The macroscopic flow stress of the heterogeneous material is displayed, as well as for several macroscopic strains, the minimum and maximum equivalent plastic strains and von Mises equivalent stresses

300

extracted over all grains. The deformation is already non uniform at the very early stage of the plastic straining. This result is caused by the difference of initial yield stresses of each group of grains. For example, for a macroscopic stress of 150 MPa, the grains in the group corresponding to α2 has just exceeded the yield stress whereas the grains in the

15

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6$ A

A

A

A

A

A

A

(a) 1400 1200

σ (MPa)

1000 800 600 Exp Isotropic hardening Model-1 Model-2 Model-3 Model-4

400 200 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

ε

(b) Figure 7: (a) FE model illustrating the grain distribution and boundary conditions in tension. The different colors correspond to the αi , i = 1, .., 7 values given in Table 3 (b) Comparison of the stress–strain curve predicted by the FE model with the experiment in tension.

group related to α1 were deformed plastically but the grains in groups αi , i = 3, 4, 5, 6, 7 305

are still in the elastic range. This tendency to accommodate the imposed strain in a heterogeneous way becomes even more obvious by increasing the macroscopic strain. At a value of 0.2, while some grains carry as little as an equivalent plastic strain of 0.02, others support a maximum deformation above 0.45, particularly in the vicinity of grain boundaries.

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Fig. 8(b) shows the local equivalent plastic strain, calculated at the central node of each element, of 100 elements of a given grain as a function of the macroscopic strain. This grain belongs to the group corresponding to α5 , which explains that the equivalent 16

3000

0odel ε=0. ε=0.2 ε=0.3 ε=0.4

2500

σ (MPa)

2000 1500 1000 500 0 0

0.2

0.4

0.6

0.8

1

ε (a)

0.3

B

εS

0.2

0.1

0 0

0.1

0.2

0.3

0.4

0.5

ε (b) Figure 8: Illustration of the model heterogeneity in tension: (a) increase of the range of the von Mises equivalent stress and equivalent plastic strain for all elements in the model and (b) variation of the equivalent plastic strain with the macroscopic strain for 100 elements selected randomly in a given grain (α5 , 271 elements). Isovalues of the equivalent plastic strain correspond to a macroscopic strain ε = 0.41. Model 4 was used for the calculation of the macroscopic stress.

17

plastic strain is smaller than the applied one. The distribution of strain increases as the applied strain increases and the upper and lower values correspond to areas close to the 315

grain boundary, especially for grains with an angular shape and softer neighboring grains. Some of the grains reveal a relatively homogeneous strain field within their borders. A rotation of some grains is also recorded during the tension. 4.2. Sequential loading–unloading tests Sequential loadings–unloadings in tension, in the plastic strain range, are simulated

320

with the inhomogeneous model and compared with experimental results, in order to verify the capability of the model to predict the non-linear and non-reversible behavior recorded during unloading. Indeed, the experiments show that unloading after pre-strains of 0.1 and 0.2 and the subsequent reloadings are non linear, forming hysteresis loops that are more pronounced for higher flow stresses prior to unloading. The shape of the unloading

325

loop is close to linear initially, with the slope approximately equal to the Young’s modulus. The slope reduces progressively with the strain; the reloading curve has similar features (Fig. 9). This shape is also observed for other metallic alloys, for example 6022-T4 and a high strength steel (Cleveland and Ghosh, 2002), deep drawing quality steel material (Yang et al., 2004) and DP780 and DP980 dual phase steels (Sun and Wagoner, 2011).

330

This phenomenon can not be predicted with classical elastic-plastic model, even with a decreasing unloading slope but is a key issue in the numerical prediction of springback Wagoner et al. (2013). Fig. 9 displays the predictions obtained with model 4. It can be seen that the hysteresis loops are nicely reproduced without any dedicated parameter to gauge this phenomenon.

335

Similar unloading features are obtained with either the phenomenological elastic-plastic model or the inhomogeneous one, down to half the stress reached at the end of the loading stage, corresponding to a linear unloading, with all the grains in the elastic range. Then, below this value, the unloading slope reduces progressively due to a local plastic deformation in some areas in the grains, and as a consequence, the recoverable strain

340

after unloading increases. Fig. 10 shows the distribution of the stress component σ22 for some selected grains, during the unloading, after reaching a macroscopic strain of 0.2. At the first stages of unloading, almost all finite elements display a tensile state, with σ22 > 0. However, at the end of unloading, the stress component evolves from negative to positive values, highlighting the fact that some areas are submitted to a compression 18

600 500

σ (MPa)

400 300 200 Exp Isotropic, Econst 0odel, Econst 0odel, EPRG

100 0 0.101

0.102

0.103

0.104

0.105

0.106

ε

(a) 800 700 600 σ (MPa)

500 400 300 200

Exp Isotropic, Econst 0odel, Econst 0odel EPRG

100

0 0.201 0.202 0.203 0.204 0.205 0.206 0.207 0.208 0.209 ε

(b) Figure 9: Verification and validation of the FE model in sequential loading–unloading tension with prestrain (a) ε = 0.1 and (b) ε = 0.2.

345

state. This range of variation of the internal stress seems related to the hysteresis loop during unloading. Indeed, with models 1 and 2, such hysteresis is not recorded and it is necessary to have a non symmetric range either for the αi parameters or for the volume fractions (cf. Table 3) in order to predict this phenomenon. Moreover, the variation of the slope with the equivalent plastic strain was necessary in order to fit the hysteresis

350

loop.

19

            

B

C

D

Figure 10: Variation of the Cauchy stress component σ22 at (a) the beginning (b) half of the pre-stress and (c) end of unloading, with a pre-strain ε = 0.2.

4.3. Monotonic and Bauschinger simple shear A simple shear test is also simulated with the inhomogeneous model. The corresponding boundary conditions are imposed as shown in Fig. 11: one of the long edge of the sample is clamped while the other one is submitted to an imposed displacement. Stress 355

and strain states are not homogeneous, in particular near the free edges, but this can be neglected for shear strains below 0.5 and therefore, a reduced area is defined in order to calculate an average macroscopic shear strain over this area. Fig. 12 shows the influence of the flow stress distribution on the prediction of the Bauschinger effect. It can be seen that model 1 prediction is very close to the one obtained with the isotropic

360

hardening, enhancing the fact that the corresponding flow stress distribution is not able to reproduce the lower yield stress or the rounded yield point at reloading in the reverse direction. However, both models 3 and 4 are able to catch these phenomena, though with an over-estimation of the flow stress after the elastic-plastic transition. In particular, the predicted yield point is lower than the one reached before unloading, evidencing the

365

capability of the flow stress distribution to represent this effect. Moreover, model 4 gives a slightly better prediction than model 3, pointing out the large influence of the choice of the distribution. Fig. 13 compares also the prediction of model 4 with experiments and the low stress calculated with a phenomenological elastic-plastic model, using either isotropic hardening

370

or mixed hardening (Pham et al., 2015). Though the mixed hardening model still gives the closest prediction of the Baushinger effect, it can be seen that model 4 is quite close. Further investigation should be performed in order to fix more precisely the flow stress distribution. Fig. 14 shows the von Mises equivalent stress-equivalent plastic strain curves of several 20

$

Figure 11: Boundary conditions and definition of the gauge area to avoid the edge effects in simple shear. Distribution of the shear strain ε12 for an imposed shear strain γ = 0.2

600

σ (MPa)

Exp. Isotropic hardening 400 Model-1 Model-2 Model-3 200 Model-4 0 -200 -400 -600 -0.3

-0.2

-0.1

0 γ

0.1

0.2

0.3

Figure 12: Influence of the model on the prediction of the Bauschinger effect in simple shear for a pre-strain γ = 0.1. 375

finite elements selected randomly in the model and the dots correspond to the local deformation of each element at the same macroscopic strain. This strain distribution is also observed within a same grain. Several authors observed that the introduction of a kinematic hardening contribution to the flow stress of the grains describe more accurately the material response during strain reversal in case of a polycrystalline model (Li et al.,

380

2012). Further investigation is needed to have a better understanding of the relationship between the flow stress distribution and the Bauschinger effect for FE models considering the microstructure inhomogeneity.

21

1200 Exp. Isotropic Mixed 0odel

1000 800

5FOTJPO

σ (MPa)

600 400

4IFBS

200 0 -200 -400 -600 -0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

ε, γ

Figure 13: Comparison of the stress-strain curves predicted by the inhomogeneous model with the experiments for uniaxial tensile test, monotonic and Bauschinger simple shear tests. Predictions with an isotropic hardening model and mixed hardening model are also added, out of comparison’s sake. 1400 Model Element-1 Element-2 Element-3 Element-4 Element-5

1200

σeq (MPa)

1000 800 600 400 200 0 0

0.05

0.1

0.15 εeq

0.2

0.25

0.3

Figure 14: Heterogneous deformation of several elements selected randomly in the model. The dots correspond to the same macroscopic strain. The macroscopic stress-strain curve is transformed into the von Mises equivalent stress versus the equivalent strain.

4.4. Variation of surface roughness in tension and simple shear One of the main application of models representing the microstructure inhomogeneity 385

and based on a finite element modeling of the grains, of cubic shape, is the prediction of surface roughness. Indeed, the knowledge of the surface roughness is of great importance in the forming process of ultra-thin metallic sheets because it governs the success, or not, of the drawing of any industrial piece. Roughness is directly related to the grains and

22

evolves with the plastic strain. In this study, the numerical prediction of surface roughness 390

is performed both in tension and simple shear and then compared with experimental values (Pham et al., 2015). Only model 4 is used in this section, as the prediction of the macroscopic flow stress is closer to experiments than for the other models. The calculation of the surface roughness is carried out using 5 sections, perpendicular to the rolling direction, on the surface of the specimen, defined in Fig. 4(a). The maximum

395

height difference of each profile is calculated and the average of these values is taken as the roughness Rz . In order to study the variation of the roughness, an equivalent strain √ according to the von Mises criterion, defined as (γ/ 3), is used in simple shear. The variation of Rz predicted with the FE model, both in tension and simple shear, is shown in Fig. 15. It can be seen that the predicted results are close to experimental values,

400

especially in simple shear. 10.0 9.0 8.0

Exp. tension Exp. shear Sim. tension Sim. shear

Rz (μm)

7.0 6.0 5.0 4.0 3.0 2.0 1.0 0.0 0.0

0.1

0.2

0.3 εeq

0.4

0.5

0.6

Figure 15: Variation of surface roughness as a function of the macroscopic deformation in tension and simple shear. Dotted lines represent the linear interpolation.

The numerical simulation represents well the linear variation of the roughness as a function of the macroscopic deformation for both stress states. The roughness of the free surface of the specimen in tension is around twice higher than that of the specimen in simple shear test. For a macroscopic deformation εeq lower than 0.5, the roughness evolves 405

quite linearly with the macroscopic deformation and this variation can be represented by Rz = 10.2εeq for tension and Rz = 5.6εeq for simple shear test. Beyond this value, the 23

roughness increases suddenly and the trend is no longer linear. This result is coherent with the experimental results and with other studies, e.g. (Wilson et al., 1981; Utsunomiya et al., 2004b; Wouters et al., 2005; Becker, 1998). 410

Figs. 16(a) and 16(b) show the numerical height variation along the profile S3 on the surface, for different value of deformation in tension and simple shear respectively. These figures show clearly the change of the thickness of specimen during the tests. In tension test, the thickness clearly decreases when the deformation increases and is around 15 µm (20% of thickness) at εeq = 0.4. In simple shear test, at this same value of deformation, the

415

thickness only decreases by around 1 µm and can be neglected in the calculation. For both cases, and as expected, the grains with a lower flow stress present a larger deformation. This difference becomes more and more important as the strain increases and causes a significant variation of the roughness (Furushima et al., 2011). The surface roughness should be evaluated relatively to the actual thickness and the

420

ratio of the roughness over the actual thickness Rz /t is used. Moreover, the thickness decrease during a tensile test leads to an increase of the relative influence of surface roughness. Fig. 17 shows the relationship between the ratio of surface roughness to the sample thickness and the equivalent strain εeq in tension and simple shear. In simple shear test, the roughness remains small compared to the specimen thickness, that does

425

not evolve during the test (cf. Fig. 16(b)). This relationship can be expressed by a linear increase as a function of the equivalent deformation. This value is less than 2% during the test and can be neglected in the measurement of the cross section. However, in tension, the specimen thickness decreases and causes an important increase of the ratio Rz /t. This increase can be expressed by a parabolic function. This result shows that at εeq = 0.5,

430

this relationship can reach up to 5%, meaning that the roughness represents 5% of the thickness of the material and its influences may not be neglected during the test. Fig. 18 shows the displacement distribution in the thickness direction in tension at εeq =0.58. The blue zone is the concave part formed by soft grains with a lower flow stress and becomes more and more concave with the increasing deformation. For example, the

435

maximum displacement calculated is 28.30 µm compared to an average of around 20 µm and causes an important localized deformation before the occurrence of diffused necking. This strain localization becomes a germination site for necking in the thickness and may cause the premature fracture without diffused necking in the width of specimen compared to a thicker material, as observed experimentally in (Klein et al., 2001) for pure copper, 24

80.0 εyy = 0.0

75.0

εyy = 0.1 Z (mm)

70.0

εyy = 0.2

65.0

εyy = 0.3

60.0

εyy = 0.4 εyy = 0.5

55.0 50.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 X (mm)

(a)

76.0

Z (μm)

75.0

74.0

73.0 εeq = 0.1 εeq = 0.2 εeq = 0.3

72.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

X (mm)

(b) Figure 16: Position along the thickness direction of elements on the free surface, at the origin of surface roughness in tension and simple shear. Profile S3 is used to output the data for different values of deformation in tension (a) and simple shear test (b). The decrease of the profile length expresses the deformation in the width of the specimen in tension test and the displacement of profile in simple shear test expresses a slight displacement of the sample during the test. It should be emphasized that the scale on the y-axis is not the same for the two graphs.

440

(Diehl et al., 2008) for SE-Cu 58 and (Xu et al., 2013) for stainless steel. 5. Conclusion A finite element model of a representative volume of an austenitic stainless steel, with an average grain size of 18 µm, is developed. The model takes into account the grain 25

10.0 9.0 8.0

Exp. tension Exp. shear Sim. tension Sim. shear

Rz/t [%]

7.0 6.0 5.0 4.0 3.0 2.0 1.0 0.0 0.0

0.1

0.2

0.3 εeq

0.4

0.5

0.6

Figure 17: Relationship between ratio of roughness to thickness Rz /t and equivalent strain εeq in tension and simple shear.

Figure 18: Displacement in the thickness direction of specimen in tension at εeq = 0.58.

size distribution, characterized experimentally by EBSD analysis, as well as a realistic 445

polyhedric shape of the grains. Ultra-thin sheets are considered in this study, leading to acceptable calculation times with 8 grains in the thickness. The heterogeneity is introduced via a flow stress distribution quantified into seven groups; both the initial yield point and the hardening law are modified. Four different distributions are considered in this study, in order to investigate its influence on the macroscopic stress-strain curves.

450

The mechanical behavior at room temperature of the material is numerically investigated

26

in tension and simple shear and compared with experimental results. The FE model gives a very good representation of the mechanical behavior in tension, both monotonic and during loading-reloading sequences; it is also able to reproduce the hysteresis loops classically recorded during unloading for metallic materials. The Bauschinger effect is 455

partially taken into account, in particular the rounded and lower yield point upon reloading in the reverse direction though the stress level is still over-estimated in absolute value. Finally, the surface roughness is investigated numerically in tension and simple shear. The inhomogeneous model leads to a very good representation of the roughness for both strain paths. Such result opens interesting perspective as the study of surface roughness

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