On the Roles of Grain Size Dispersion and Microscale Hall Petch Relation on the Plastic Behavior of Polycrystalline Metals

On the Roles of Grain Size Dispersion and Microscale Hall

­

Petch Relation

on the Plastic Behavior of Polycrystalline Metals

Stéphane Berbenni

, Francis Wagner

and Nathalie Allain-Bonasso

1_{Laboratoire d’Etudes des Microstructures et de Mecanique des Materiaux (LEM3), UMR CNRS 7239, University of Lorraine,}

Ile du Saulcy, F-57045 Metz Cedex, France

2_{Laboratory of Excellence on Design of Alloy Metals for low-mAss Structures (‘LabEx DAMAS’), University of Lorraine,}

F-57045 Metz Cedex, France

Micro-macro scale transition theories were essentially developed to model the inelastic behavior of polycrystals starting from the local behavior of the grains. The anisotropy of the plastic behavior of polycrystalline metals was mainly explained by taking into account the crystallographic textures. Issues like plastic heterogeneities due to grain size dispersion, involving the Hall­Petch relation at the microscale, were not often taken into account in the literature, because only the role of a mean grain size was investigated. Here, the coupled roles of grain orientation and size distributions on the plastic deformation state of grains are examined for a tensile deformed IF steel in the light of EBSD experimental data based on misorientation parameters and of a micromechanics-based modeling regarding grain and overall behaviors.

[doi:10.2320/matertrans.MA201307]

(Received July 23, 2013; Accepted September 25, 2013; Published December 25, 2013)

Keywords: grain size dispersion, Hall­Petch relation, micro-macro scale transition, plasticity, interstitial free (IF) steel

1. Introduction

Grain size effects on the tensile responses of polycrystal-line metals like low-carbon steels were essentially captured at the macroscopic scale using a Hall­Petch relationship1,2)

· ¼·0þ Kyffiffiffiffi D

p ð1Þ

whereDis the average grain size of the material,·0 andKy are materials constants usually referred as the friction stress and the Hall­Petch slope. Both parameters can be obtained from classic tensile tests at room temperature. For cold-rolled and annealed low-carbon steels®not ultra-fined grained steels®it was found3)that the Hall­Petch coefficientKy is about 500 MPa·µm1/2 for a 0.13C­0.67Mn mass% steel sheet. Generally, body centered cubic (b.c.c.) metals like low carbon steels have usually higher Ky by a factor of 2­3 compared to face centered cubic (f.c.c.) metals.4)

Plastic heterogeneity is often discussed as depending only on crystallographic texture and associated slip activity inside grains. However, the role of microstructural heterogeneities such as grain sizes and morphologies as well as their dispersions within the aggregate were not deeply studied. Polycrystalline models like Taylor,5)_{self-consistent}6)_orfinite

element7) _{models should be developed to study combined}

effects due to statistical orientation and size distributions on the local and overall behaviors.

A first self-consistent scheme8) focused on the effect of local grain size effect in polycrystalline copper. In this approach, the average grain size was incorporated through the critical shear stresses of slip systems by a microscale Hall­Petch type law. Later, a microscale Hall­Petch behavior coupled with grain shape effects and called“directional Hall­ Petch effect” was introduced in a Taylor model9) _{to study}

overall plastic anisotropy coefficients. More recently, new

studies tried to tackle the effects of grain size distribution using log-normal or bimodal distributions10­14)on the overall

flow stress of the material using either averaging rule of mixture10)_{or self-consistent models.}11­14)_{Assuming a}

micro-scale Hall­Petch relation within grains and isotropic plasti-city, these studies showed that an increase of the standard deviation within the grain size distribution enhances a decrease of the effective plasticflow stress and consequently a decrease of the overall Hall­Petch slope here referenced as Ky. This leads not only to depict a classic Hall­Petch effect based on an average grain size but a representative grain size containing grain size dispersion as well.

The effects of correlations between crystallographic orientations and grain sizes on the mechanical behavior of polycrystalline aggregates were recently shown using viscoplastic Taylor15) and elastoviscoplastic self-consistent (EVPSC)16)models. Both studies assumed a microscale Hall­ Petch relationship for the shearflow strength within each slip system

¸c¼¸0þ kyffiffiffiffiffiffi DI

p ð2Þ

where DI _{is an individual grain size and ky is a microscale}

Hall­Petch slope characterizing the grain boundary resistance to slip. The microscale Hall­Petch relationship was recently obtained by nanoindentation through pop-in loads.17­19) Interestingly, slip transmission mechanisms across grain boundaries, grain boundary geometry, prior cold work and carbon interstitial content can modify ky.19,20)Theoretically, this one was found to be proportional to the deformation energy required to eject dislocations from grain bounda-ries.21) However, nanoindentation measurements are gener-ally focused on local configurations with well identified grain boundaries where it is somewhat difficult to transpose results to real statistical polycrystalline metals.

Here, the coupled roles of crystallographic and grain size heterogeneities arefirst examined in the light of EBSD data +_{Corresponding author, E-mail: stephane.berbenni@univ-lorraine.fr}

Special Issue on Strength of Fine Grained Materials®60 Years of Hall­Petch®

to obtain some indicator of plastic deformed state, and, second using an EVPSC model with crystal plasticity to quantify the role of microscale Hall­Petch parameter on the plastic strainfluctuations between grains.

2. Experimental and Modeling

2.1 Material

The material used in this study was a ferritic single phase interstitial free steel (IF). Table 1 shows the chemical composition of the IF steel.

The several samples were in a recrystallized state with a mean grain size of³16.0 µm. The measured grain diameters (Fig. 1) follow approximately a statistical log-normal density probability function defined by a standard misorientation criterion. The orientation distribution function (ODF) shows the classical texture for such a steel with much orientations distributed along the£fiber (h111i//ND) and a little amount of ¡fiber (h110i//RD).

2.2 Mechanical tests and EBSD data

Tensile tests were carried out ex-situat room temperature up to four tensile strains 4.7, 6.7, 12 and 17%. The tensile direction was the same as the rolling direction. The samples consisted of small dogbone-shaped tensile test specimens with total length=61.5 mm, gauge length=25 mm, width=4 mm and thickness=0.7 mm. Tensile tests were performed on an Instron 8513 5 kN tensile machine. The crosshead speed in the tensile stage was set to 0.02 mm·s¹1 which corresponds to an initial strain rate of 8©10¹4s¹1. The EBSD (electron backscatter diffraction) technique was

used to investigate the microstructure and crystallographic orientation of the IF-steel.22) _{The EBSD data were obtained}

using a 6500F JEOL FEG-SEM equipped with an HKL Channel 5 system on samples tilted at 70°. The maps were carried out on a regular square grid with ¦x=¦y=0.5 µm steps on areas corresponding to 700 to 2200 grains. The EBSD maps were madeex-situon the same area before and after tensile deformation tests in order to study the micro-structure changes due to plastic deformation. The raw indexing rate was over 90% in each case and was over 99% after noise reduction. Grains were detected as sets of contiguous pixels with less than 5° of misorientation. Grains with less than 4 pixels (i.e., lower than 1 µm2_{) were ignored.}

The samples were prepared using standard mechanical polishing and electro-polishing. Finally, the samples were cleaned in an ultrasonic bath.

Here, an indirect method based on the values of the misorientation angles determined from EBSD was used to evaluate plastic deformation state of the different grains as a function of their grain size. The misorientation between a

fixed orientation denoted i and another one denoted k is

ªði; kÞ. For a given grain of average orientationi, the“Grain Orientation Spread”(GOS) is given by

GOSðiÞ ¼_np1 X

np

k¼1

ªði; kÞ ð3Þ

wherenpis the total number of pixels included in the grain of

orientation i. Then, the average GOS on a set of ng grains

becomes

GOSðngÞ ¼ 1

ng Xng

i¼1

GOSðiÞ: ð4Þ

Note that similar quantities were used by Kamaya et al.22)

to quantify the plastic deformation state in grains of 304 austenitic stainless steel, which motivated the present study on IF steel. The evolutions of GOSðngÞ are reported on Fig. 2 for two subpopulations referred as “smallest” and “largest” grains. The “smallest grains” correspond to a subpopulation of about 1000 grains with average grain size close to 10 µm and the “largest grains” correspond to a subpopulation of about 100 grains with average grain size close to 34 µm.

Figure 2 shows a first strong increase of GOS values with plastic deformation up to ³12% of tensile strain and a much slower increase after 12%. It is observed that thefirst increasing stage is due to the formation of “geometrically necessary dislocations” (GND) in the vicinity of grain boundaries and the stabilized stage which occurs earlier for the smallest grains is more due to intra-granular dislocation walls and cells under multiple slip. Furthermore, the highest GOS values were found for the largest grains subpopulation. Similar trends were found with misorientation parameters using an orientation imaging microscopy (OIM) analysis regarding polycrystalline Cu after 11% tensile strain.23) Furthermore, it was also shown22) that strong orientation gradients between grain boundaries and grain centers were also observed in the different grains at 17%of tensile strains. In the next modeling part and disregarding local situation with interactions between dislocations and grain boundaries, the analysis is based on grain boundary strengthening

Table 1 Chemical composition of the IF steel.

Element Ti C S N Mn P Si Cu Ni Cr Al

©10¹3

mass% 89 1.6 6 2.3 113 74 8 6 16 21 37

0 10 20 30 40 50

0 150 300 450 600 750 900 1050 1200 1350 1500

Grain diameter (µm)

described by the microscale Hall­Petch parameterkyusing an

elastoviscoplastic self-consistent model (EVPSC).

2.3 Modeling

For each grain denoted I, the constitutive behavior is supposed to be elastoviscoplastic. For the sake of simplicity, a linearized small strain kinematics is assumed and elastic properties are supposed homogeneous, so that the total strain rate splits into an elastic part and a viscoplastic part

_

¾I

ij ¼¾_eIijþ¾_pIij¼sijkl·_Iklþ

X

sðIÞ

RsðIÞ

ij £_sðIÞ ð5Þ

In eq. (5), the homogeneous elastic compliances sijkl are supposed isotropic and are defined by the elastic shear modulus µ and the elastic Poisson ratio¹. In eq. (5),Rijs(I)is the Schmid tensor of the slip system s in grainI. Here, for each grainI, and on the slip system s(I), the slip rate is given by

_

£sðIÞ _{¼£0_} j¸sðIÞj

¸scðIÞðμsðIÞ;£sðIÞ; DI;. . .Þ

n

signð¸sðIÞ_{Þ ð6Þ}

wherenis a parameter corresponding to the inverse of the slip rate sensitivity of the material, £_0 is a reference slip rate, ¸sðIÞ _¼RsðIÞ

ij ·Iijis the resolved shear stress on slip system s(I).

The reference shear stress ¸s_cðIÞ depends on the mean dislocation densities on the different active slip systemsμhðIÞ and on the individual grain diameterDI_{. Thus, the assumed}

microscale Hall­Petch type law on slip system s(I) for a given plastic deformation state reads

¸sðIÞ

c ¼¸s0ðIÞþ ksðIÞ

y ffiffiffiffiffiffi DI p þ¡®b

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X

hðIÞ

asðIÞhðIÞ_μhðIÞ

s

ð7Þ

where¸s0ðIÞ is an initial critical shear stress containing lattice friction and solid solution strengthening,b is the magnitude of the Burgers vector, ¡ is a material parameter (with 0<¡<1). Here, the microscale Hall­Petch coefficientksyðIÞ is introduced on each slip system s(I) to depict individual grain size effect on the initial critical shear stress. In eq. (7), asðIÞhðIÞ _{is the crystallographic part of the hardening matrix}

in any grain I. Then, eq. (7) accounts for both self and latent hardening during plastic deformation. Furthermore, the variation of dislocation density on each slip system is

supposed to depend on the dislocation density on the same system (due to annihilation of dislocation dipoles) but also on other slip systems (creation of new dislocations due to forest obstacles) as follows

_

μsðIÞ_¼1

b

1

DI þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX

uðIÞ6¼sðIÞ

μuðIÞ

r

K 2ycμsðIÞ 0

B B @

1 C C

Aj£_sðIÞj ð8Þ

In eq. (8), the mean free path of dislocations is constrained by both grain boundaries and forest dislocations. Here, for a considered recrystallized IF steel, it is initially on the order of the individual grain size DI _{which fixes parameter K.}

Furthermore, yc is a critical distance for dislocation dipole

annihilation.

An elastoviscoplastic self-consistent (EVPSC) procedure using the translatedfields technique was developed11,12,16)_to

determine the effective properties of the Representative Vol-ume Element (RVE) describing the polycrystal. The volVol-ume fractionfIis calculated asfI¼³D3I

6V whereDIis an equivalent

diameter deduced from EBSD data assuming spherical grains and Vis the total volume of the RVE. The interaction law between the average stress rate·_I_ijand plastic strain rate¾_pI_ijin each grainIand overall stress and plastic strain rates _ij,E_p_ij reads

_

·I

ij¼ _ijþCijklðIklmnSeshklmnÞðAImnpqE_ppq¾_pImnÞ ð9Þ

with __ij¼P_IfI·_I_ij; E_p_ij ¼P_IfI¾_pI_ij. In eq. (9), S_ijklesh is the elastic Eshelby tensor associated with the homogeneous elastic moduliC_ijkl,AI_ijklis the viscoplastic strain rate concentration tensor related to the effective viscoplastic moduli Be_ijkl for the polycrystal obtained through an iterative self-consistent procedure.11,12,16)

3. Results and Discussions

The crystal plasticity-based EVPSC model introduced in section 2.3 was applied to the studied IF steel. The RVE is constituted of a total of N grains including for each grain its size and mean orientation as determined from the large EBSD map of the initial microstructure. For the calculations, isotropic elastic properties are characterized by a shear modulus of µ=80 GPa and Poisson ratio ¯=0.3. The magnitude of the Burgers vector is b=2.5©10¹10m. The matrix asðIÞhðIÞ in eq. (7) is a 24©24 matrix with two major slip systems {110}h111i and {112}h111i considered in the case of this IF steel (b.c.c. metal). The latent hardening ratio present in asðIÞhðIÞ is denoted q. The microscale Hall­Petch

coefficient ks_yðIÞ introduced in eq. (7) is assumed to be constant on all the slip systems and identified on uniaxial tensile curves. This is found to be 250 MPa·µm1/2, which about half the one obtained macroscopically for low-carbon steels3) (i.e., 500 MPa·µm1/2). The materials parameters present in eqs. (6)­(8) are reported in Table 2. They were obtained through comparisons between the simulated and the experimental uniaxial tensile stress­strain responses.

Here, the correlations between internal plastic strains and microscale Hall­Petch parameter ky are examined and the

results are reported on Fig. 3 for a macroscopic tensile strain of 12%. For the sake of clarity, the statistics are based on 708

different grains with grain size ranging from 1.38 to 50.61 µm. These grains were also identified experimentally before and after tensile deformation using a homemade program to automatically correlate grains between two EBSD maps on the same area.25)

Two major results are highlighted with Fig. 3. Thefirst one is that an increase of the microscale Hall­Petch parameterky

from 70 to 500 MPa·µm1/2_{leads to an increase of¦¾}p

11for the

same grain size dispersion especially due to the lowest amount of plastic strain in the smallest grains, see Figs. 3(a) to 3(c). Second, larger normalized plastic strain (with respect to 1 which corresponds to the case of Taylor-based assumption) is observed for both largest grains in size and grains with high Schmid tensor component along the longitudinal tensile direction due to eq. (6). Considering the GOS value as an indicator of plastic deformation in grains, this may explain why the increase of GOS values with tensile strain for smallest grains is seen to be retarded with respect GOS evolution for largest grains. The present results are consistent with recent

finite element calculations performed with non uniform grain size distribution and microscale Hall­Petch relation.26) _In

contrast with Taylor’s uniform plastic strain based approach or uniform grain size approaches with self-consistent schemes (see Fig. 3(d)), the present model captures the existing correlations between grain sizes and orientations on the

development of plastic heterogeneities. However, the model is not able in its present form to consider shear banding classically observed infinite element simulations7,26)_{and in}

experiments.27) _{Such strain localization process may change}

plastic deformation states within grains crossed by shear bands. Furthermore, slip interaction mechanisms certainly needs multiple microscale Hall­Petch parametersky

depend-ing on slip system, grain boundary geometry and character as observed with nanoindentation.17­19)

4. Conclusion

Experimental misorientation parameters like GOS (“Grain Orientation Spread”) and an “augmented” mean field elastoviscoplastic self-consistent (EVPSC) model were developed to identify the coupled roles of grain size and orientation distributions on the plastic responses of poly-crystalline metals. After introducing a microscale Hall­Petch slope ky on each slip system, it is concluded that the high

value of this microstructural parameter in IF steel may play a key role on the plastic strain development in grains which could not be described by uniform plastic strain Taylor-type models or scale transition models with uniform grain size. Further works may focus on the variability of this parameter due to the material type and grain boundary-dislocation interactions to refine the present analysis.

Acknowledgments

This work is supported by the French State through the National Research Agency (ANR) under the program “Investment in the future” (LabEx DAMAS referenced as ANR-11-LABX-0008-01).

(a) (b)

(c) (d)

Fig. 3 Normalized plastic strains per grain along the longitudinal uniaxial tensile direction predicted by the EVPSC model with different

“microscale Hall­Petch slope”kyand non uniform experimental grain size distribution for 708 grains experimentally identified before

and after the tensile test: (a) 70 MPa·µm1/2_{, (b) 250 MPa·µm}1/2_{, (c) 500 MPa·µm}1/2_{. For comparisons, predictions given by the same}

EVPSC model with (d) uniform grain size (dotted line) and (a)­(d) Taylor based uniform plastic strain model (solid line).

Table 2 Identified materials parameter as input of the present elastovisco-plastic self-consistent model based on crystal elastovisco-plasticity.

¸0

(MPa)

ky

(MPa·µm1/2₎ _ £0

(s¹1₎ n K

2yc

(nm) q ¡

μ0

(m¹2₎

REFERENCES

1) E. O. Hall: Proc. Phys. Soc. London B64(1951) 747­753. 2) N. J. Petch: J. Iron Steel Inst.174(1953) 25­28.

3) R. Song, D. Ponge, D. Raabe, J. G. Speer and D. K. Matlock:Mater. Sci. Eng. A441(2006) 1­17.

4) M. A. Meyers and K. K. Chawla:Mechanical Behavior of Materials, 1st ed., (Prentice-Hall, Upper Saddle River, NJ, 1999).

5) G. I. Taylor: J. Inst. Metals61(1938) 307­324.

6) M. Berveiller and A. Zaoui:J. Mech. Phys. Solids26(1978) 325­344.

7) D. Raabe, M. Sachtleber, Z. Zhao, F. Roters and S. Zaefferer: Acta

Mater.49(2001) 3433­3441.

8) G. J. Weng:J. Mech. Phys. Solids31(1983) 193­203.

9) H. J. Bunge, F. Wagner, P. I. Welch and P. Van Houtte:J. Phys. Lett.46

(1985) 1109­1113.

10) K. J. Kurzydłowski:Scr. Metall. Mater.24(1990) 879­883.

11) S. Berbenni, V. Favier and M. Berveiller:Int. J. Plast.23(2007) 114­ 142.

12) S. Berbenni, V. Favier and M. Berveiller: Comput. Mater. Sci.3(2007) 114­142.

13) B. Raeisinia, C. W. Sinclair, W. J. Poole and C. N. Tomé: Model.

Simul. Mater. Sci. Eng.16(2008) 025001.

14) B. Raeisinia and C. W. Sinclair:Mater. Sci. Eng. A525(2009) 78­82.

15) B. S. Fromm, B. L. Adams, S. Ahmadi and M. Knezevic:Acta Mater.

57(2009) 2339­2348.

16) N. Nicaise, S. Berbenni, F. Wagner, M. Berveiller and X. Lemoine:Int. J. Plast.27(2011) 232­249.

17) W. A. Soer, K. E. Aifantis and J. T. M. De Hosson:Acta Mater.53

(2005) 4665­4676.

18) T. Ohmura, K. Tsuzaki and F. Yin:Mater. Trans.46(2005) 2026­2029.

19) T. A. Britton, D. Randman and A. J. Wilkinson: J. Mater. Sci.24(2009) 607­615.

20) R. W. Armstrong, I. Codd, R. M. Douthwaite and N. J. Petch:Philos. Mag.7(1962) 45­58.

21) V. Bata and E. Pereloma:Acta Mater.52(2004) 657­665.

22) N. Allain-Bonasso, F. Wagner, S. Berbenni and D. P. Field:Mater. Sci. Eng. A548(2012) 56­63.

23) M. Kamaya, A. J. Wilkinson and J. M. Titchmarsh:Nucl. Eng. Des.

235(2005) 713­725.

24) R. A. Lebensohn, R. Brenner, O. Castelnau and A. D. Rollett: Acta Mater.56(2008) 3914­3926.

25) F. Wagner, N. Allain-Bonasso and S. Berbenni: Mater. Charact.62

(2011) 681­683.

26) Y. Wang, J. Zhou, T. D. Shen, D. Hui and S. Zhang:Comput. Mater. Sci.58(2012) 175­182.