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Materials Science & Engineering A journal homepage: www.elsevier.com/locate/msea

Modeling of critical grain size for shifting plasticity enhancement to decrease by reﬁning grain size J. Liu a, G. Zhu a,b,n, W. Mao a, S.V. Subramanian c a

University of Science and Technology Beijing, Beijing 100083, China Anhui University of Technology, Maanshan, Anhui 243002, China c McMaster University, Hamilton, Ontario, Canada L8S 4M1 b

art ic l e i nf o

a b s t r a c t

Article history: Received 19 January 2014 Received in revised form 31 March 2014 Accepted 2 April 2014 Available online 13 April 2014

The effect of grain size on plasticity was investigated based on the theories of dislocation pile-up and Frank–Read (FR) source activation in ultraﬁne grained materials. The possible reason of the experimental phenomena of decrease in plasticity as grain size reﬁned in ultraﬁne grained materials was theoretically analyzed and the critical grain size for plasticity decrease was proposed. The results predicted that there was a critical grain size of about 4–5 μm with the present parameters of X80 microalloyed steel where the plasticity would shift from increase to decrease as grain reﬁned, which well agreed with the experimental data described in the references in ultraﬁne grained materials. The mechanism of decreasing in plasticity as grain size reﬁned was dominantly due to the decrease of probability of activation of FR source. The modeling for predicting the critical grain size has been built up which would be very helpful to optimize microstructure for obtaining excellent combination of strength and plasticity. & 2014 Elsevier B.V. All rights reserved.

Keywords: Plasticity Grain reﬁnement Dislocations Frank–Read source

1. Introduction Over the past several decades, reﬁnement of grain size was considered as the only strengthening method without plasticity decrease. Based on the traditional physical metallurgy theory, it is concluded that reﬁnement of grain size from hundreds of microns to dozens of microns could improve the plasticity [1]. The mechanism could be attributed to more uniform deformation and more difﬁcult for propagation of crack in reﬁned grain size materials. However, yield/ tensile strength ratio increase and work hardening rate decrease were recently observed in the materials in which the grain size was reﬁned to several microns and sub-microns. Furthermore, the experimental data exhibited that the plasticity would decrease as grain size further decreased when the grain size was in the range of submicro/ nanoscales [2–8]. These experimental phenomena have attracted much interest of researchers because its importance in design of microstructures of ultraﬁne grained materials. Some researchers tried to explain the experimental results, for example, competition between recovery softening and work hardening result in decrease of work hardening rate [9], as well as localization of deformation result in instability of plasticity [10]. However, these interpretations did not specify the mechanism of deformation behaviors in ultraﬁne grained

n Correspondence to: School of Materials Science and Engineering, University of Science and Technology Beijing, Beijing 100083, China. Tel.: þ 86 1340950197. E-mail address: [email protected] (G. Zhu).

http://dx.doi.org/10.1016/j.msea.2014.04.012 0921-5093/& 2014 Elsevier B.V. All rights reserved.

materials, i.e. what mechanism and what condition for recovery softening surpasses work hardening, and/or when and why deformation localized especially in ultraﬁne grained materials. In recent years, discrete dislocation dynamics and molecular dynamics have been widely used for numerical simulation in this ﬁeld [11,12]. This was only for numerical simulation. However, the physical mechanism of the experimental results was still vague. The range of critical grain size where plasticity shifted from increase to decrease as the grain size reﬁned was not physically analyzed and predicted. The strain at a given stress was taken as plasticity consideration in the present work. The dislocation slipping, their interaction with grain boundaries, the relationship between dislocation FR source and grain size were applied to discuss the possibility of plasticity decrease with grain size decrease. In this model, the critical grain size where the plasticity shifted from increase to decrease as grain size further reﬁned was physically predicted.

2. Modeling of displacement induced by dislocation slipping in single grain 2.1. Effect of grain size on displacement induced by dislocation slipping Plastic strain was determined by the free path length for dislocation movement. The free path length for dislocation

J. Liu et al. / Materials Science & Engineering A 607 (2014) 302–306

movement could be deﬁned as about half of the grain diameter in the case of interaction between dislocation and grain boundary without consideration of the interaction among dislocations. In fact the interaction among dislocations was too complicated and difﬁcult to calculate. And the problem here discussed was the effect of grain size on dislocation movements. Therefore, the assumption of above was reasonable. Also, here it was assumed that there was no texture and precipitation in single phase with spherical grains. When the applied shear stress is larger than the critical resolved shear stress of given slip system, the dislocations emitted from the FR sources would slip forward until meeting the grain boundaries. Dislocations would be piled up on grain boundaries because the slip system was not activated in adjacent grain because of different orientations. The number of dislocations in pile-up group was determined by the applied shear stress and the length of the pile-up group. As shown in Fig. 1, two dislocation pile-up groups were obtained in the simpliﬁed case. Here, only the edge dislocation was taken into account in the model. The slip plane contained x-axis is perpendicular to the paper, and the shear stress applies on the x positive direction. The displacement caused by each dislocation in the pile-up group should be Di ¼

ðL xi Þb 2L

ð1Þ

where L is the grain radius, b is the value of Burgers vector, and xi is the x-axis of the ith dislocation in the pile-up group counted from the grain boundary. The location of dislocation in the pile-up group depended on the resultant force of each dislocation including applied shear stress and stresses applied by other dislocations, and the length of one dislocation pile-up group was equal to the grain radius L approximately. The location of each dislocation in the group could be calculated as follows [13]: xi ¼

ðGbÞ2 16ð1 νÞ2 τ2 L

ði 1Þ2

ð2Þ

where τ is the shear stress applied on slip system, ν is the Poisson's ratio, and G is the shear modulus. Based on the dislocation theory, the

number of dislocations in one pile-up group could be calculated as N¼

ð1 νÞπτL Gb

ð3Þ

When the dislocation slipped along the slip system, it would induce displacement as plastic strain. The displacement induced by a dislocation pile-up group could be calculated as N

D0 ¼ ∑ ðL xi Þb=2L i¼1

ð4Þ

As discussed above, there are two pile-up groups in a single grain. Therefore the displacement in each grain induced by one FR source (DFR) could be twice the D0, as expressed in Eq. (5). Taking X80 pipeline steel in which the tensile stress was around 760 MPa for example [14], the relationship between DFR and grain diameter is shown in Fig. 2, where the parameters G, b, ν are of 77.5 GPa, 0.248 nm and 0.273 respectively [15]. N

DFR ¼ 2D0 ¼ ∑ ðL xi Þb=L i¼1

ð5Þ

2.2. Effect of grain size on the number of applicable activated dislocation FR sources It should be noted that only one dislocation FR source was taken into account here, which should be corrected. Actually the number of dislocation FR source was dependent on the grain size, where the dislocation FR source was deﬁned as applicable to be activated for emitting dislocations. Some literatures [16] have reported that whether the FR source could be activated is dependent on the ratio of FR source length (lFR) and grain diameter (d), i.e. lFR/d. When the ratio of lFR/d is larger than a critical value of about 1/3, the FR source with the corresponding length could not be activated. In order to investigate the applicable number of activated FR sources in a given grain diameter, it is necessary to calculate the length distribution of FR sources in polycrystalline materials. Unfortunately, the length of all dislocation sources in a sample of polycrystalline material cannot be characterized in practice [17]. Shishvan et al. [17] have drawn upon the analogy with the distribution of size of many other objects, such as grains [18,19] and particles, to assume that the distribution of source lengths follows a log-normal function. The length distribution of FR source could be expressed with a lognormal distribution (Eq. (6)) in polycrystalline materials [17]: f ðlFR Þ ¼

Fig. 1. Schematic diagram of dislocation pile-up groups inside individual grain.

303

2 1 2 pﬃﬃﬃﬃﬃﬃe ðln lFR mÞ =2s lFR s 2π

Fig. 2. Displacement in each grain versus grain diameter.

ð6Þ

304

J. Liu et al. / Materials Science & Engineering A 607 (2014) 302–306

Fig. 3. Log-normal distribution of FR source lengths (lFR) and the dash area is the range of the lengths of activated FR sources.

Fig. 5. Actual displacement of each grain at a given stress of 760 MPa versus grain diameter.

of m and s were given, the length distribution of FR source could be derived explicitly. Based on Ohashi's work [16], FR source could be very difﬁcult to be activated when the ratio of lFR/d is larger than 1/3. That means applicable number of activated FR sources would be decreased as the grain diameter reﬁned which could be integrated as Eq. (10): Z lmax nuc FðLÞ ¼ f ðxÞ dx ð10Þ lmin nuc

max

Fig. 4. The probability of activation of FR source at a given stress of 760 MPa versus grain diameter.

where F(L) is the probability of activation of FR source, lnuc is the max max length of activated FR source, lnuc is the minimum length of activated FR source, dx is the inﬁnitesimal of the length of FR source. Taking X80 pipeline steel as an example, in which the tensile stress was around 760 MPa [14], the probability of activation of FR source as a function of grain size could be drafted as shown in Fig. 4. It could be seen that applicable probability of activated FR source would be decreased as grain size decreases. 2.3. Displacement induced by dislocation slipping in a single grain

where lFR is the length of FR source, m and s are the average value and standard deviation of the length of FR sources respectively. Fig. 3 shows a schematic density function of FR source lengths (lFR). When grain diameter is d1, the probability of activation of FR sources is the min

dash area between the two vertical lines of d1/3 and lnuc . When grain diameter is d2, the probability of activation of FR source is the dash min

area between the two vertical lines of d2/3 and lnuc . Using the isotropic line tension model, the length of FR source decided the nucleation strength of the FR source, as follows [20]: τnuc ¼ Gb=lFR

ð7Þ

where τnuc is the nucleation strength of the FR source, when the stress applied on FR source is larger than τnuc, the FR source is activated and began to emit dislocations. The conﬁdence interval of the normal density function of ln τnuc is 99.9%. So the parameters of m and s could be calculated as follows [17]: min m ¼ 12 ðln τmax nuc þ ln τ nuc Þ

ð8Þ

min s ¼ 12 ðln τmax nuc ln τnuc Þ

ð9Þ

where the maximum value τmax nuc was given as applied shear stress (τ), and the minimum value of t min nuc was deﬁned by the maximum max length of FR source lFR through Eq. (7) [20]. Once the parameters

As discussed above, the displacement induced by one dislocation FR source in a single grain could be expressed as Eq. (5). If the number of dislocation FR source was involved into the model, the displacement induced by dislocation slipping in single grain should be Z lmax N nuc DSG ¼ DFR FðLÞ ¼ ∑ ðL xi Þb=L f ðxÞ dx ð11Þ i¼1

lmin nuc

According to Eq. (11), the actual displacement of each grain at a given stress of 760 MPa [14] could be calculated as shown in Fig. 5, in which the displacement would be decreased as grain size decreases.

3. Strain induced by dislocation slipping in polycrystalline materials As discussed above, the displacement (DSG) induced by dislocation slipping in a single grain should be decreased with reﬁning grain size due to number of dislocations in pile-up group and applicable number of activated FR source. However, in a unit volume of materials, the number of grains (n) would be increased as the reﬁning grain size. That means it is possible that the total strain induced by dislocation slipping in polycrystalline materials

J. Liu et al. / Materials Science & Engineering A 607 (2014) 302–306

would be increased due to increase of grain number, which would be investigated as following. At ﬁrst, the strain could be calculated approximately through adding displacement induced by dislocation slipping in a single grain. In a unit volume, assuming the displacement in a single grain was DiSG , the total elongation (DT) of the sample should be n

DT ¼ ∑ DiSG i¼1

ð12Þ

DiSG

is the displacement in the ith grain and n is the number of where grains in unit volume which could be calculated approximately by n ¼ 1=L3

ð13Þ

However, it should be noted that the displacement induced by dislocation slipping was not in the same direction because the orientation of grains. Thus, the DT should be corrected by multiplying orientation factor. The orientation factor for a given grain could be calculated by equation as following [21]: μ ¼ cos ϕ cos λ

936

i¼1

case of random orientation distribution assumed in the present work which is about 0.45. Here, the total strain of the unit volume of the materials could be obtained by ! Z lmax N ðL x Þb nuc 1 n i μ n j ε ¼ μ ∑ DSG ¼ ∑ f ðxÞ dx ð16Þ ∑ min a i¼1 ai¼1 j¼1 L lnuc where a is the dimension of the sample, equal to 1 for unit volume, μ is average orientation factor, equal to 0.45 for random orientation distribution, n is the number of grains in unit volume samples, N is the number of dislocation in dislocation pile-up group and L is the grain radius. Again taking X80 as an example [16], the total strain induced by dislocation slipping in polycrystalline materials as a function of grain size could be drafted in Fig. 6. The total strain is calculated assuming Shear modulus G ¼77.5 GPa, Poisson's ratio ν ¼0.273, and the value of Burgers vector b ¼0.248 nm [15]. It could be seen in Fig. 6, a critical grain diameter of about 4 μm where the plasticity shifted from increase to decrease as grain size reﬁned.

ð14Þ

where ϕ is the angle between direction of applied stress and slip direction and λ is the angle between direction of applied stress and normal direction of slip plane. For polycrystalline materials, the calculation of orientation factor, called as average orientation factor, should consider the strength of texture component and orientation symmetry. The average orientation factor could be calculated by Sachs model as follows [22]: μ ¼ ∑ μðg i Þf ðg i Þ

305

ð15Þ

where the total orientation space for all grain orientations could be divided into 936 special orientation units (gi), in which each orientation unit could have one component strength (f(gi)). μ(gi) is the orientation factor for a given orientation unit. In the calculation, μ(gi) was determined by the maximum orientation factor of 24 slip systems of {110}〈111〉 and 24 slip systems of {211}〈111〉 where slip systems of {321}〈111〉 was not considered because its slipping could be obtained by combination of {110}〈111〉 and {211} 〈111〉. It should be noted that the critical resolved shear stress on {110}〈111〉 is different from that on {211}〈111〉. In the present work, it is approximate that critical resolved shear stress on {110}〈111〉 is equal to 0.95 that on {211}〈111〉 [23]. According to the discussion above, the average orientation factor could be calculated in the

Fig. 6. Critical grain size and total strain predicted versus grain diameter.

4. Discussion and validation In the present work, it could be seen that for a given ﬂow stress, the strain should be increased as grain size reﬁned. However, when the grain size was reﬁned to a critical size, the work hardening and plasticity would be conversely decreased as the grain size further reﬁned. This critical grain size is very important for the microstructure design in order to optimize the mechanical properties of the materials. The critical grain size for shifting from plastic enhanced to decrease as grain size reﬁned could be calculated by dislocation pipe-up and FR source model, here it was calculated of about 4 μm of experimental X80 microalloyed steel. Of course, this number was obtained in some assumptions for simpliﬁed, for example the interaction of dislocation sources. Further work should be done in future work. A modeling has been proposed to exhibit that there is a critical grain size for plasticity shift from increasing in traditional materials to decreasing in ultraﬁne grained materials as grain size reﬁned, and the physical principle was partially explored in the present work. These are well agreement with numerous experimental data as shown in Fig. 7 [24–26]. These indicated that the grain size should be reﬁned to the critical grain size where the comprehensive mechanical properties including strength and plasticity could be achieved.

Fig. 7. Experimental data of plasticity versus grain diameter.

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J. Liu et al. / Materials Science & Engineering A 607 (2014) 302–306

5. Summary A modeling of critical grain size for shifting plasticity from enhancing to decreasing as grain size reﬁned was developed. The model was built up based on displacement induced by dislocation slipping, effect of grain size on applicable activated dislocation source and polycrystalline deformation. The calculation resulted from the modeling showed that there was a critical grain size where the strain at a given ﬂow stress would shift from increasing to decreasing as the grain size reﬁned which agreed with experimental data reported in the references. The critical grain size would be about 4 μm in case of X80 line pipe steels. This critical value was obtained with some of the assumptions. Further work should be done in the future work. However, physic base and modeling has been proposed in the present paper. Acknowledgments The authors gratefully acknowledge ﬁnancial support from the National Natural Science Foundation of China (Grant no. 51071026) and Ministry of Education of the People's Republic of China. References [1] T. Gladman, B. Holmes, F.B. Pickering, J. Iron Steel Inst. 208 (1970) 172–183. [2] Y. Wang, M. Chen, F. Zhou, E. Ma, Nature 419 (2002) 912–915. [3] J. Gil Sevillano, J. Aldazabal, Scr. Mater. 51 (2004) 795–800.

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