A unified dislocation-based model for ultrafine- and fine-grained face-centered cubic and body-centered cubic metals

Computational Materials Science 131 (2017) 1–10

Contents lists available at ScienceDirect

Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci

A unified dislocation-based model for ultrafine- and fine-grained facecentered cubic and body-centered cubic metals S.H. He a,b, K.Y. Zhu c, M.X. Huang a,b,⇑ a

Department of Mechanical Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong, China Shenzhen Institute of Research and Innovation, The University of Hong Kong, Shenzhen, China c ArcelorMittal Research, Voie Romaine-BP30320, 57283 Maizières-lès-Metz cédex, France b

a r t i c l e

i n f o

Article history: Received 27 August 2016 Received in revised form 8 January 2017 Accepted 10 January 2017 Available online 6 February 2017 Keywords: Ultrafine-grained Dislocation density Lüders strain Ductility Grain boundary dynamic recovery

a b s t r a c t A unified dislocation-based model is proposed in the present work to quantitatively predict the stressstrain behavior of various face-centered cubic (FCC) and body-centered cubic (BCC) metals with grain sizes ranging from submicron to microns. The effect of grain boundary dynamic recovery (GB-DRV) on the evolution of dislocation density was considered in the model. The presence of Lüders strain in metals with submicron grain size is well predicted by the model. Furthermore, the present model can explain well the poor tensile ductility of ultrafine-grained metals in terms of dislocation-based mechanisms. Ó 2017 Elsevier B.V. All rights reserved.

1. Introduction Severe plastic deformation (SPD) processes followed by different heat treatments have been widely used to produce finegrained (FG, 1–10 lm) and ultrafine-grained (UFG, 0.1–1 lm) metals [1]. Different to the coarse-grained (CG) counterparts, FG and UFG metals exhibit a strong dependence of grain size on their tensile stress-strain behavior. For instance, the uniform elongation drops significantly when the grain size is reduced from micron to submicron [2]. The mechanism responsible for such sudden decrease of uniform elongation is still under debate in literature [3,4]. More interestingly, some FG and UFG pure metals produced by SPD followed by heat treatment exhibit Lüders strains, which in general increase with the decrease of grain size [5,6]. The physical mechanisms to explain such Lüders strain are still not yet fully understood as it could not be explained by the classical mechanism related to the locking of dislocations induced by Cottrell atmosphere [7]. The stress-strain behavior of coarse-grained metals can be well captured by the Kocks-Mecking (K-M) model as the evolution of dislocation density during deformation can be predicted [8]. Nevertheless, the K-M model cannot explain the suppression of strain hardening in UFG metals and therefore fails to predict their tensile

stress-strain behavior. For nano-grained (NG, grain size < 100 nm) metals, Malygin proposed a dislocation-based model, which has successfully incorporated the enhanced dynamic recovery from grain boundary (GB) and can therefore predict the stress-strain curves of NG metals [9]. However, such annihilation mechanism at GB may not be applicable for UFG and FG metals, which limits its applicability to FG and UFG metals. Furthermore, the unique Lüders deformation phenomenon in FG and UFG metals produced by SPD method cannot be captured by the above two models [6]. The present work attempts to develop a unified dislocationbased model to predict the stress-strain behavior as well as the Lüders phenomenon of FG and UFG metals. The present model highlights the significance of grain boundary dynamic recovery (GBDRV) on the evolution of dislocation density. A possible mechanism based on residual defects including dislocations and vacancies is proposed to explain the presence of Lüders strain. Furthermore, the sharp decrease of uniform elongation from FG to UFG metals is well quantified by the present model. The model is applied to various pure face-centered cubic (FCC) and body-centered cubic (BCC) metals strained at different temperatures and strain rates, showing good agreements with experimental observations.

2. Model ⇑ Corresponding author at: Department of Mechanical Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong, China. E-mail address: [email protected] (M.X. Huang). http://dx.doi.org/10.1016/j.commatsci.2017.01.022 0927-0256/Ó 2017 Elsevier B.V. All rights reserved.

Extensive experiments have demonstrated a linear relationship between the increment of flow stress and the square root of

2

S.H. He et al. / Computational Materials Science 131 (2017) 1–10

dislocation density when the grain size is larger than 100 nm [8,10]. A proper estimation of the dislocation evolution is thus crucial for the modeling of stress-strain curves. In the original K-M model, the evolution of dislocation density q with plastic strain e is written as

pffiffiffiffi dq ¼ k1 q  k2 q de

ð1Þ

where k1 and k2 are proportionality factors to scale the capability of dislocation multiplication and annihilation [8]. This model has been verified by many CG materials and thus established a simple but effective connection between stress-strain behavior and dislocation density. The influence of GB on the evolution of dislocations was not incorporated in the original K-M model. A modified K-M model was later proposed to include the effect of grain size on the generation of dislocations [11]. Nevertheless, both the original and modified K-M models cannot address the low strain hardening capability in UFG metals as they did not incorporate the effect of grain size on the dislocation annihilation. Therefore, an extension of Eq. (1) from original K-M model is required to incorporate the effect of grain size on the annihilation of dislocations for UFG metals. In other words, the original K-M model will be extended in the present work to simulate the evolution of dislocation density for both FG and UFG metals. The suppression of work hardening in UFG metals is related to the lower capacity of dislocation storage [12]. This phenomenon may be realized by either a weaker dislocation multiplication or a stronger dislocation annihilation. Given that grain boundaries are sinks for dislocation annihilation [9], the suppression of strain hardening in UFG metals is more likely caused by the enhanced dislocation annihilation at GB, which is related to GB-DRV. This GB-DRV mechanism is incorporated in the evolution of dislocation density that is expressed as

dq dq dq dq ¼ þ þ de de de de þ

 in

 GB

ð2Þ

where qþ stands for the multiplication of dislocation density, while qin and qGB is the annihilation of dislocation density in grain interior and at GB, respectively. The annihilation at grain interior dq in =de can be reconstructed as

dqin dqin dt dqin 1 ¼ ¼ : de dt de dt e_

ð3Þ

Due to the rate control mechanism of the annihilation event, the increment of dislocation density with time is expected to have a linear relationship with the corresponding dislocation density at the current state [13]

dqin ¼ f in qin dt

ð4Þ

where f in and qin represents the corresponding thermal activation frequency and dislocation density related to this annihilation event, respectively. The minus sign on the right-hand side means the decrease of dislocation density due to annihilation. On the other hand, by the comparison with the K-M model shown in Eqs. (1) and (3) can be expressed as

dqin ¼ kin q de

ð5Þ

where kin counts for the capability of dislocation annihilation in grain interior. Combining Eqs. (3)–(5) leads to

1 kin q ¼ f in qin : e_

ð6Þ

Following the same analysis of grain interior, the annihilation at GB dq GB =de can be expressed as

dqGB 1 ¼ kGB q ¼ f GB qGB de e_

ð7Þ

where kGB , f GB and qGB are the capability of dislocation annihilation, the thermal activation frequency and dislocation density related to the annihilation at the boundary, respectively. The ratio between kin and kGB will then be translated into the proportion between their corresponding frequencies by using Eqs. (6) and (7):

kin =kGB ¼ f in qin =f GB qGB :

ð8Þ

The difference in annihilation rate between grain interior and grain boundary is due to their different effective area as illustrated in Fig. 1a. Only dislocations near GB are counted in qGB . In this way, we can define qin and qGB as

qin ¼

4ðd  2cÞ

2

2

d

q ¼ q; c  d

ð9Þ

and

qGB ¼

4ðd  cÞc 2

d

c d

q ¼ 4 q; c  d

ð10Þ

where c and d are the effective distance of GB-DRV and grain size illuminated in Fig. 1a, respectively. Eqs. (9) and (10) are the approximate relations based on the assumption that dislocation distribution is relatively homogeneous, and c is much smaller than d. Combining Eqs. (8)–(10), the relation between kGB and d is finally derived as

kGB ¼

4f GB c d0 kin ¼ f in d d

ð11Þ

where d0 ¼ 4f GB ckin =f in . Eq. (2) is thus rewritten as

pffiffiffiffi dq d0 ¼ k1 q  kin q  q: de d

ð12Þ

The purpose of this approximation procedure is to circumvent the estimation of the absolute value of the recovery mechanisms and corresponding parameters, given that parameters such as f in , f GB and c are different from case to case. These parameters could be measured in future experiments though the relative values between kin and kGB are sufficient to show the significant of GBDRV. In summary, Eq. (12) represents a modified K-M model to include the effect of GB-DRV on the evolution of dislocation density. With a given dislocation density, the flow stress predicted from dislocation strengthening (rdis ) is expressed as

pffiffiffiffi

rdis ¼ r0 þ Kd1=2 þ K b d1 þ aMlb q

ð13Þ

where r0 , K, K b , a, M, l, b are the lattice friction stress, Hall–Petch slope, back stress slope, Taylor constant, Taylor factor, shear modulus and Burgers vector, respectively. The four terms on the right side of Eq. (13) represent the contributions to the flow stress from the lattice friction, GB, back stress and dislocation strengthening, respectively. Here we have two terms related to d to estimate the 1=2

effect of grain size on the flow stress. The first one (Kd ) is commonly used in modeling of CG materials and is well known as the Hall-Petch relationship. According to the review by Bhadeshia [14], when the size of the cross-slip plane is below one micron, the effect of grain size on the flow stress can be expressed by the 1

second one (K b d ). In addition, some studies incorporating the contribution of back stress on the flow stress also obtain an identical relation between the flow stress and the linear inverse dependence 1

term (K b d ) for UFG metals [15,16]. Considering the grain size in the present study covers both UFG and FG, the present model therefore combines both terms to compromise the deviation among various grain sizes. During a tensile test, the first three terms on the

S.H. He et al. / Computational Materials Science 131 (2017) 1–10

3

Fig. 1. (a) Schematic illustration of effective area of GB-DRV (marked with gray color) in the vicinity of GB and (b) schematic illustration of residual dislocations in the grain interior.

right-hand side of Eq. (13) remain almost constant, while the evolution of dislocation density deals with the strain hardening during plastic deformation. Apart from the boundary effects, a high initial dislocation density may also lead to the suppression of work hardening [8]. However, the same UFG metals with low initial dislocation density prepared by spark plasma sintering methods also show that a finer grain leads to a lower strain hardening [17,18]. Therefore, the low strain hardening in UFG metals is mainly due to the boundary effect which results in a slow increase of dislocation density and therefore a low strain hardening. Moreover, transmission electron microscope (TEM) measurement has found that the initial dislocation density in most of the annealed UFG metals are low [6]. Therefore, a uniform initial dislocation density q0 ¼ 1  1013 =m2 for all cases was adopted for simplification in this study. This value is comparable with the synchrotron XRD measurement results of UFG 1100 Al which range from 2  1013 =m2 to 5  1013 =m2 [19], while the method used in their study to calculate the dislocation density will lead to an overestimated initial value [20,21]. Previous studies have proved that the Lüders strains widely observed in UFG pure metals should not stem from interstitial atoms or phase transformation [6]. Some studies indicated that Lüders strains in UFG metals are due to the lack of mobile dislocations after heat treatment [22,23]. This explanation is consistent with the experimental results that no yield point is observed after prior mobile dislocations are induced, and only the annealed samples have shown Lüders strains [6,23,24]. However, shortage of mobile dislocations alone cannot explain the weaker Lüders stress after a more thorough recovery with higher annealing temperature or longer annealing time. Here we propose an alternative mechanism that considers the residual defects after heat treatment being the obstacles for dislocation glide. The Lüders stress is defined as the critical stress for a mobile dislocation needed to bow out from the pinning of the neighboring residual defects. For the UFG metals produced by severe plastic deformation without post-annealing treatment, it is reasonable to assume that a high dislocation density is present in the sample. If such UFG metals are subjected to high-temperature annealing, most dislocations are annealed while some of them may rearrange themselves to form dislocation walls. For instance, if we use paralleled edge dislocations with the same Burgers vector to represent the dislocation structure after annealing treatment, these dislocations should form a dislocation wall to lower the internal energy. As presented schematically in Fig. 1b, the repulsive forces between dislocations will drive the leading dislocations into the grain boundary via dis-

location climb, resulting in dislocations annealed at GB and therefore less dislocations in the wall. Nevertheless, some grains with particular orientations can provide a repulsive force from GB against the leading dislocations and thus suppress the annealing of dislocations at GB [25]. Therefore, after high-temperature annealing treatment, some grains which can provide higher repulsive force from GB to the leading dislocation will have more residual dislocations in the grain interior. In addition, such residual dislocations in the grain interior could become Frank-Read (F-R) sources as well as the obstacles to other mobile dislocations. On the contrary, other grains may have lower repulsive forces to the leading dislocations, resulting in more dislocation annihilation at GB and consequently much less residual dislocations in the grain interior. In other words, after high-temperature annealing treatment, some grains will have more residual dislocations in the grain interior while others may have very little residual dislocations. Furthermore, the multiplication of dislocations in the subsequent tensile test could stem from the residual dislocations in the grain interior as they are F-R sources. Therefore, grains with residual dislocations may be the first ones to provide plastic strain and thus determine the critical flow stress to trigger Lüders deformation. Such concept of residual dislocations is consistent with the TEM observation in UFG metals and the tendency that weaker Lüders stress related to a more thorough recovery [1,22–24,26]. The distance of neighboring residual dislocations can be determined as follows. We assume that only the leading dislocation would be subjected to the repulsive stress sobs from grain boundary due to the short-range character of their interactions. At the end of annealing treatment, the dislocation configuration achieves its equilibrium state, and the net stress on each dislocation should be equal to zero. Then the position of each dislocation yi can be determined as follows:

sxxi;j ¼ fi ¼

lb ; 2pð1  mÞðyj  yi Þ

n X

sxxi;j þ f GB;i ¼ 0;

ð14Þ

ð15Þ

j¼1 j–i and

f GB;i

8 i¼1 > < sobs ; ¼ 0; 1 : sobs ; i ¼ n

ð16Þ

4

S.H. He et al. / Computational Materials Science 131 (2017) 1–10

where f i is the net stress on dislocation i. m is the Poisson’s ratio. sxxi;j and f GB;i are the interaction stress on dislocation i from dislocation j and GB, respectively [27,28]. The positions of dislocations are calculated in a normalized form by employing Eqs. (14) and (15), and are shown in Fig. 2a. The results reveal a clustering of dislocations near GB and the appearance of maximum neighboring distance Lmax at the grain interior. In addition, a linear relationship between the normalized maximum distance and the normalized square root of grain size is obtained and is shown in Fig. 2b assuming that sobs is a constant. According to the Frank-Read source model, Lmax is proportional to the critical Lüders stress [28]. Consequently, we find approximately the relation:

rL ¼ r0L þ K L d1=2

ð17Þ

where rL is the critical stress to initiate Lüders deformation. The two terms on the right sides of Eq. (17) represent the contribution from solid solution and residual dislocations in the grain interior, respectively. This estimation of the size effect on Lüders stress in Eq. (17) exhibits satisfactory agreement with the experiment results [1,22]. Finally, the model combines the above equations to simulate the entire stress-strain curve. If the material is prepared by powder metallurgy methods, the initial dislocations are mobile dislocations and the flow stress agree with the prediction of r from Eq. (13). The deformation is provided by the multiplication of dislocations while the dislocation density determines how much flow stress is required for further deformation. Consequently, no Lüders strain is observed in such samples. However, if the UFG metals are produced by SPD followed with post-annealing treatment, some residual dislocations will remain in some grains as described above. If the applied stress is lower than rL in Eq. (17), Lüders deformation will not take place. In the meantime, if the applied stress is higher than rdis in Eq. (13), a continuous deformation takes place. This means that the flow stress r, which represents the minimum value of applied stress to continue plastic deformation, should exceed either rdis or rL . This is expressed mathematically as

r ¼ maxðrdis ; rL Þ:

ð18Þ

According to the concept of residual dislocations, the Lüders stress rL only depends on the initial state of microstructures. rdis , on the other hand, will keep increasing with dislocation density or the plastic deformation. For the fully recovered metals, rL could be small enough and the whole plastic process is controlled by the dislocation hardening mechanism so that no Lüders deformation can take place. For the partially recovered metals, rL are larger

than the initial value of rdis so that Lüders band initiates at the stress concentration location. The front of Lüders band keeps deforming until

rdis ¼ rL :

ð19Þ

Then the front of Lüders band will temporarily cease because further deformation requires higher applied stress to overcome rdis . Meanwhile, the front of Lüders band moves toward the undeformed area until the end of Lüders deformation. After that, the homogeneous hardening stage begins until necking takes place. The present model assumes that the dislocation evolution model described in Eq. (12) remains the same for both samples with and without Lüders deformation. In other words, the same model (Eq. (12)) still holds after the end of Lüders deformation. This assumption may be true as the total gauge area of the sample has deformed uniformly from the end of Lüders deformation to necking [7]. For the fitting procedure, the value of k1 , kin and d0 are determined through the evolution of dislocation density firstly from Eq. (12). The evolution of dislocation density is calculated through Eq. (13) from the stress-strain curve, and the three fitting parameters are carefully selected so that the strain hardening of fitting results from yielding to necking are consistent with that from the experiment results of most cases. Given that all the stress contributors other than dislocation strengthening in Eq. (13) keep constant with deformation, this equation can be rewritten as:

pffiffiffiffi

rcons ¼ rdis  aMlb q:

ð20Þ

Then the value of K b , K and r0 is confirmed through a quadratic

and rcons . Meanwhile, all the six parameters fitting between d must be a positive value otherwise the fitting procedure needs to be repeated through the adjustment of k1 and all the other parameters. Detailed information will be given from Figs. 7 and 8 in the next part. 1=2

3. Results and discussion Various experimental data for different metals and deformation conditions have been used to verify the present model. The predictions from the present model fit well with these experimental results as showed in Figs. 3 and 4. All stress-strain curves shown in Figs. 3 and 4 illustrate that a smaller grain size leads to a lower strain hardening rate. This phenomenon can be quantitatively described by the present model because of the incorporation of the GB-DRV mechanism in the present model which is represented by the term dd0 q in Eq. (12). The d0 values used in the present model

Fig. 2. (a) the configuration of residual dislocations with the variation of the number of dislocations (b) the relation between the normalized maximum distance of neighboring dislocations and the square root of normalized grain size. The red line is a linear fitting of the calculation results. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

S.H. He et al. / Computational Materials Science 131 (2017) 1–10

are listed in Tables 1 and 2. In addition, a smaller grain size leads to a larger Lüders strain, which can also be well described by the present model as showed in Figs. 3 and 4. In the modeling results shown in Figs. 3 and 4, we modified the grain size in a few cases to obtain a better fitting. It is noted that the experimental method for measuring grain size may be varied among various studies, and thus leading to the uncertainty of grain size measurement, especially for submicron grain sizes. This could provide one justification for the slight modification of grain size in our modeling results. The modeling parameters used in Eqs. (12) and (13) are summarized in Table 1. Comparison among the cases can be combined with the experiment and the current modeling results. For the aluminum deformed at a lower temperature, TEM observation has illuminate that the deformation at 77 K leads to more dislocation density after deformation compared with their counterpart deformed at room temperature [22]. Meanwhile, the much higher strain hardening at the early deformation stage at 77 K also suggest a higher increase of dislocations from Taylor equation. Basically, the higher increase could only be achieved through either by a stronger multiplication (a higher k1 ) or a weaker annihilation (a lower kin or d0 ). Through the cases of deformation at room temperature (RT-Al) and deformation at 77 K (77 K-Al) in Table 1, the current fitting has found that both two mechanisms contribute to the increase of dislocations. This tendency is different from the original K-M model employed for the CG metals, which considers k1 as a constant value from 300 K to 800 K [8], though some previous models used irreversible thermodynamics theory also have stated that k1 can change with temperature [29]. The increase of k1 may arise from the abnormal dislocation structures at 77 K and causes the uniform elongation increased from about 25% to above 40%. For a similar reason, the dislocation density measured in high purity aluminum is lower than that in commercial purity Al [6], which may partially due to the decrease of k1 by the comparison between the cases of commercially pure (CP) Al and ultra-high purity (HP) Al in Table 1. Analysis of dynamic recovery from kin and d0 would be more difficult since they change with k1 . However, GB-DRV seems more sensitive to the temperature given that d0 at 77 K is only 16% of the value at room temperature while kin remains 56% of the value. This conclusion that d0 deceases faster than kin at lower temperature may not be affected by the uncertainty of fitting parameters because only the cases deformed at 77 K in Table 4 showing that the value of kin is higher than d0 , while nearly all the other cases deformed at room temperature showing the opposite way. One of the possible explanation could be that the more defects favored to stay at GB may act as efficient dislocation annihilation sources with the help of the thermal activation at ambient temperature.

5

TEM observations also suggest the enhanced GB-DRV at room temperature with the trapping of dislocations at boundaries [22]. Concerning the effective distance of GB-DRV (c) shown in Fig. 1a, Eq. (11) can be rewritten as

c  f GB =f in ¼ d0 =4kin :

ð21Þ

Since we have presumed the similarity of the dynamic recovery mechanisms in grain interior and at GB during deformation, it seems that Eq. (21) could be used to estimate c when

f GB =f in ¼ 1:

ð22Þ

The effective distance of GB-DRV should be smaller than the grain size, and larger than the mean free path of dislocations to ensure the existence of dislocations that could be annihilated near the boundary. These requirements are well followed from the results of Table 3. In the present model, we propose the concept of residual dislocations to estimate the Lüders stress based on Eq. (17). From Eq. (14), a reference distance Dx is defined as

Dx ¼

lb 2pð1  mÞsobs

ð23Þ

where sobs is obtained from Eq. (16). Dx determines the minimum grain size containing two dislocations in a grain when the boundary resistance is sobs . If the boundary resistance keeps constant, adding more dislocations in the same grain needs to increase the grain size d. The normalized defects distribution at equilibrium state is summarized in Fig. 2a and the maximum distance between neighboring dislocations L also depends on the number of dislocations. Thus, we obtain

L ¼ N L Dx

ð24Þ

and

d ¼ Nd Dx

ð25Þ

where N L and N d are coefficients calculated from the force equilibrium. The linear relationship shown in Fig. 2b can be approximately written as 1=2

0:9ðL=DxÞ ¼ ðd=DxÞ

:

ð26Þ

According to the Frank-Read mechanism [28], the critical stress that a dislocation needed to overcome the residual dislocations is expressed as

s ¼ lb=L:

ð27Þ

Comparing Eqs. (26) and (27) and Eq. (17), the relation between Dx and K L is revealed as

K L ¼ 0:9M lb=Dx1=2 :

ð28Þ

Fig. 3. Comparison between modeling (solid line) and experiments results (dots) of stress-strain curves for pure aluminum at (a) room temperature [6] and (b) 77 K [22]. The grain sizes in the brackets are the values used in the modeling to achieve a better fitting with experiments. rdiff is defined in Eq. (31).

6

S.H. He et al. / Computational Materials Science 131 (2017) 1–10

Fig. 4. Comparison between modeling (solid line) and experiments results (dots) of stress-strain curves in pure BCC metals: (a) Iron [1] and (b) Vanadium [42]. The grain sizes in the brackets are the values used in the modeling to achieve a better fitting with experiments.

Table 1 Parameters used in the modeling for Al at room temperature (RT-Al), 77 K (77 K-Al) [22], ultra-high purity Al at room temperature (HP-Al) [6], and commercial purity Al (CP-Al) at room temperature [1]. Fitting parameters are marked with bold type while other parameters are from Refs. [8,43]. Modeling metal

M

a

b (nm)

G (GPa)

k1

kin

d0 (lm)

Kb (MPa lm)

K (MPa lm1/2)

r0 (MPa)

RT-Al 77 K-Al HP-Al CP-Al

3.06 3.06 3.06 3.06

0.38 0.43 0.38 0.38

0.28 0.28 0.28 0.28

26 28.9 26 26

0.009 0.014 0.004 0.011

2.5 1.4 0.9 1.5

3.2 0.5 0.8 4

0 7.1 0 6.4

51.8 83.4 74.7 83.4

9.6 13.4 0.2 6.1

Table 2 Parameter used in modeling of Iron [1] and Vanadium [42]. Fitting parameters are marked with bold type while other parameters are from Ref. [43]. Modeling metal

M

a

b (nm)

G (GPa)

k1

kin

d0 (lm)

Kb (MPa lm)

K (MPa lm1/2)

r0 (MPa)

Iron Vanadium

2.77 2.77

0.38 0.38

0.25 0.262

80 46.7

0.01 0.04

1.3 5

1.8 3

192.2 107.5

135.7 153.5

22.6 8.0

Table 3 pffiffiffiffiffiffiffiffiffi c  f GB =f in is effective distance of GB-DRV; drange is the range of grain size simulated in the modeling; 1/ qUTS is the mean free path of dislocations at the ultimate tensile stress; Dx is the reference distance; sobs =l is the normalized boundary resistance stress. Modeling metal

c  f GB =f in (lm)

drange (lm)

pffiffiffiffiffiffiffiffiffiffi 1/ qUTS (lm)

Dx (lm)

sobs =l

RT-Al 77 K-Al HP-Al CP-Al Iron Vanadium

0.26 0.09 0.31 0.67 0.35 0.15

1.72–12 0.44–45 3.3–51 1.2–10 1.6–13 0.57–3.1

0.12–0.17 0.04–0.08 0.11–0.15 0.07–0.17 0.06–0.10 0.05–0.08

0.027 0.028 0.011 0.029 0.010 0.010

0.0024 0.0022 0.0055 0.0022 0.0054 0.0062

K L is fitted from experiment data, while Dx and sobs could be calculated from Eq. (28) and Eq. (23) subsequently. In Table 3, the comparison between different temperature and purity in the case of Al seems reasonable from the following points pffiffiffiffiffiffiffiffiffi of view. The decreasing of c and 1/ qUTS in 77 K is in good agreement with experimental observation [22]. Furthermore, some TEM observations also show that most GB were blurred and the portion of low angle boundary was increased [30], indicating that the dislocations near GB are reserved rather than recovered after deformation at 77 K. The decrease of Dx in lower temperature may stem from the stabilization of residual defects that act as extra obstacles for dislocation motion. However, a proper estimation of Dx and sobs relies on an accuracy measurement of K L . The dependence of uniform elongation on grain size is represented in Fig. 5a. The correlation between modeling and experimental results has been verified in Fig. 3b. From the modeling results shown in Fig. 3b, we estimate the elongation of Lüders strain as the corresponding strain when Eq. (19) holds. This elongation represents the exact plastic strain at the end of Lüders defor-

mation. Two mechanisms related to the decreasing tendency of uniform elongation can be summarized in Fig. 5a. The first mechanism is the weakening of strain hardening due to the significant GB-DRV. A smaller grain size leads to a higher GB-DRV and therefore a lower dislocation density as illuminated in Fig. 6b. A lower dislocation density will lead to a lower strain   hardening rate ddre according to Eq. (13). The uniform elongation is defined through Considère criterion

dr ¼ r: de

ð29Þ

This indicates that a smaller grain size leads to a lower strain hardening rate. This mechanism has been shown as the red curve in Fig. 5a which reflects an approximately linear relationship between uniform elongation and the reciprocal of the square root of grain size. The second mechanism is the existence of localized plastic deformation (or Lüders deformation) shown as the green curve in

7

S.H. He et al. / Computational Materials Science 131 (2017) 1–10

Fig. 5. Size effect on ductility from model calculation and experiment data [22]. Red curve represents the predicted uniform elongation, and the green curve represents the predicted Lüders strain (b) uniform elongation vs. grain size for Al deformed at room temperature (RT) and 77 K [22], ultra-high purity (HP) Al deformed at room temperature [6] and commercial purity (CP) Al deformed at room temperature [1]. Dots are experimental data and lines represent the modeling results. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 6. Dislocation density evolution with plastic deformation for the case of pure aluminum (a) at RT and [6] (b) at 77 K [22] correspond to Fig. 3. Dots are experimental data.

Fig. 5a. A sample with a smaller grain size has a higher critical stress to initiate plastic deformation and therefore obtains a larger Lüders strain. This change may owe to the smaller distance of residual defects, and the more deformation that is required in the front area to reach conditions described by Eq. (19). This mechanism may also explain the unappealing ductility in the materials with submicron-sized grains. Once the applied stress has overcome Lüders stress, dislocation hardening can no longer prevent the continuous deformation at the front area. Thus, necking takes place right after the yielding in these materials. In general, localized deformation may cause the sudden transition of the uniform elongation to nearly zero when the grain size is close to a critical value. The transition tendency of aluminum in the different cases has been demonstrated in Fig. 5b. It is noted that localized deformation only exists in metals produced by SPD followed with annealing. The sudden dropping of ductility may not happen if the UFG pure metals are prepared by other methods [17,18,31]. From the results shown in Table 1 and Fig. 5b, some possible methods other than annihilation of initial defects to improve the ductility of UFG metals can be suggested. Firstly, the increasing of grain size will bring less GB-DRV and higher strain hardening. Secondly, deformation at ultra-low temperature will dramatically enhance the dislocation multiplication as well as weakening the dislocation annihilation, resulting in improved strain hardening and therefore uniform elongation. Thirdly, the incorporation of proper strengthening elements should also be a good way to increase dislocation multiplication and hence the uniform elongation. This tendency has been shown by the data comparison between the cases of high purity and commercial purity metals as shown in Fig. 5b.

As for the evolution of dislocation density, the current estimated value for the cases of the aluminum tensile test at room temperature given in Fig. 6a is comparable with the torsion results ranging from 7:8  1012 =m2 to 9:6  1013 =m2 [32], and the in-situ results at the magnitude of 1013 =m2 [19]. The main reason for the slow increase of dislocation density during the early stage of plastic deformation at different temperature in Fig. 6 may be justified by the Taylor strengthening equation that describes a linear relation between the flow stress and the square root of dislocation density. Based on the Taylor hardening equation, the increase of dislocation density can be reflected by the increase of the flow stress in Fig. 3 which have shown that the flow stress indeed increases slowly during the early stage of plastic deformation. The fulfillment of Taylor equation has been demonstrated through the in-situ synchrotron diffraction experiment in 7020-T6 aluminum alloy from the end of yielding to the middle between yielding and ultimate tensile strength [33]. This relation has also been testified in micro-alloyed lath martensite through the 1% offset flow strength [34]. Nevertheless, the assumption of uniform initial value with varying grain size in the current model may cause an underestimation of dislocation density for the cases with finer grains. The strengthening from increasing initial dislocation density among these partially recovered cases may be implicitly con1

sidered as the term of K b d in Eq. (13). The uncertainties of the fitting parameters have been checked from two aspects. Considering the evolution of dislocation density in Eq. (12) is only affected by k1 , kin and d0 , k1 is set to be constant to show the sensitivity of kin and d0 . Changing of kin will leads to an obvious discrepancy both with large grain size from Fig. 7a or small

8

S.H. He et al. / Computational Materials Science 131 (2017) 1–10

Fig. 7. Modeling of dislocation density evolution with plastic deformation with the varying of kin and d0 for the case of pure aluminum at 77 K [22] with (a) d = 12 mm, (b) d = 0.59 mm. The origin curve (black dot) uses k1 = 0.014, kin = 1.4 and d0 = 0.5 from Table 1. Other curves are obtained through the changing of k2 (dash line) or d0 (solid line).

grain size from Fig. 7b, while d0 becomes significant to the dislocation density evolution only with smaller grain size in Fig. 7b. This means with a given k1 , kin and d0 could be basically determined through the cases with largest grain size and smallest grain size subsequently. Besides, the effect of k1 on the fitting results is illuminated in Fig. 8. With varying k1 , kin and d0 are determined afterwards to achieve the best fitting and the relation between the true strain and the stress difference Dr is plotted in Fig. 8a and b. Here Dr is defined as:

pffiffiffiffi Dr ¼ r  aM lb q:

ð30Þ

Based on Eq. (14), Dr should not change with deformation, thus a good fitting requires the curve in Fig. 8a to be a flat line. The stress peaks showing on the curves with the smaller grain size stem from the Lüders deformation and for these curves Dr would

become stable only after that rdis equals to r (which means rdis is bigger than rL ). From Fig. 8b, an improper value of k1 leads to a deviation between the fitting results and the flow stress from experiments in the early stage of deformation especially with larger grain size. This deviation could be estimated as:

rdiff ¼ rcons  Dre¼0:01 ;

ð31Þ

where rcons is set to the value of Dr when e is about to 0.1–0.2 and Dr keeps constant with further deformation until necking. The value of rdiff which has been marked in Fig. 3a approximately equals to the stress difference between the modeling results and experimental data at the initial state of deformation. Finally, K, K b and r0 are determined through a quadratic fitting from the relation:

rcons ¼ r0 þ Kd1=2 þ K b d1 :

ð32Þ

Fig. 8. (a) stress difference between flow stress and dislocation strengthening stress with increase of plastic strain, (b) effects of k1 on the stress difference and (c) corresponding relation between rcons and grain size for the case of aluminum at 77 K [22]. Red curves represent the quadratic fitting from Eq. (32). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

9

S.H. He et al. / Computational Materials Science 131 (2017) 1–10

Table 4 Parameter dependence on k1 for all the Al cases. The negative, asterisk and positive sign in the bracket from the first column mean a smaller, same and larger value of k1 used in Table 1, respectively. rdiffmax is chosen from the sample with the max value of rdiff and does not have Lüders deformation. R2 is the square of correlation coefficient from the quadratic fitting of Eq. (32). N is the number of points taking into account. Modeling metal

k1

kin

d0 (lm)

Kb (MPa lm)

K (MPa lm1/2)

r0 (MPa)

rdiffmax (MPa)

R2

N

RT-Al () RT-Al (*) RT-Al (+) 77 K-Al () 77 K-Al (*) 77 K-Al (+) HP-Al () HP-Al (*) HP-Al (+) CP-Al () CP-Al (*) CP-Al (+)

0.006 0.009 0.014 0.009 0.014 0.02 0.003 0.004 0.006 0.009 0.011 0.014

1.2 2.5 4 0.5 1.4 2.15 0.32 0.9 1.6 1 1.5 2.3

2.8 3.2 5.5 0.25 0.5 0.6 0.8 0.8 2 3 4 4.3

0 0 0 2.7 7.1 10.2 0 0 0 9.3 6.4 6.4

50.3 51.8 60.9 81.5 83.4 84.2 75.9 74.7 82 71.1 83.4 86.9

14.8 9.6 1.5 41.8 13.4 19.7 1.6 0.2 7.5 14.9 6.1 2.4

22.7 18.5 11.2 37.1 11.7 18.4 22.6 20.9 15.3 15.1 10.4 4

0.967 0.97 0.981 0.998 0.999 0.999 0.988 0.992 0.991 1 1 1

4 4 4 6 6 6 4 4 4 3 3 3

The relation between rcons and d corresponds to the case in Fig. 8a and b is plotted in Fig. 8c. Each point in this figure is transformed from one stress-strain curve from Fig. 3b and all the other fitting parameters are listed in Table 4. It seems the varying of k1 slightly change the value of K and K b , though r0 decrease a lot with the increase of k1 . With the same procedure, Table 4 summarized all the fitting results of aluminum cases with the varying of k1 . To maintain the physics requirement, the value of K and K b are forced to be a non-negative one. Generally, an underestimation of k1 would amplify rdiff . Since the yield stress of FG aluminum is only about 50–100 MPa from Fig. 3, rdiff with a value of 37.1 MPa in the case of 77 K-Al (–) implies a very large discrepancy between the modeling stress and corresponding data from experiment with the strain ranging from 0 to 0.1, and thus the fitting parameters would not be acceptable. Meanwhile, an overestimation of k1 would cause a negative value of r0 and thus be rejected as well. Therefore, the cases between RT-Al and 77 K-Al (or between HP-Al and CP-Al) cannot share a same value of k1 either by increase the value in RT-Al (or HP-Al) or vice versa. In summary, k1 is the decisive fitting parameter and all the other parameters would be uniquely fitted once k1 is decided. However, this parameter may also not be very flexible due to the requirement of physics meaning. Although the current model could be used for different metals, it still has several limitations at its present form. First, it needs more accurate as well as consistent measurement of physical parameters such as initial dislocation density and grain size. If the material has high initial dislocation density due to the inadequate heat treatment, the presumption of constant initial dislocation density will be no longer reasonable. In addition, different materials and treatments can form diverse boundary characters and distributions [35–38]. Such difference in microstructure may affect the final properties [23,24,39–41]. Consistent criteria of grain size measurement also have substantial importance for the modeling work. Furthermore, other deformation mechanisms in nano-grained metals such as GB sliding have not been incorporated in the current model. Therefore, modeling of nano-grained metals is apparently beyond the capability of the present model. It is also worth to note that the Lüders deformation only exists in metals with specific sample fabrication methods so that it should not be an intrinsic property of UFG metals. 1=2

4. Conclusions An extended Kocks-Mecking model is developed in the present work to study the stress-strain behavior of various pure metals with the grain size ranging from submicron to micron. This study highlights the significance of GB-DRV on the evolution of disloca-

tion density. To understand the Lüders deformation observed in UFG materials produced by SPD followed by heat treatment, a residual dislocation concept is proposed. This concept explains the relation between Lüders stress and the grain size by linking them with the maximum distance of neighboring residual dislocations. Modeling results show that GB-DRV has a remarkable effect on UFG metals by weakening the dislocation storage capacity, resulting in low strain hardening and therefore poor uniform elongation. The loss of uniform elongation in UFG metals are due to two mechanisms: (1) the weakening of strain hardening caused by the enhanced GB-DRV and (2) the existence of localized plastic deformation due to the residual dislocations. Acknowledgments M.X. Huang is grateful for financial support from the Steel Joint Funds of the Natural Science Foundation of China (Grant No. U1560204), the Natural Science Foundation of China (Grants No. 51301148), and the Research Grants Council of Hong Kong (Grants No. 712713, 17203014). References [1] N. Tsuji, Y. Ito, Y. Saito, Y. Minamino, Scripta Mater. 47 (2002) 893–899. [2] R. Song, D. Ponge, D. Raabe, J.G. Speer, D.K. Matlock, Mater. Sci. Eng. A 441 (2006) 1–17. [3] B. Raeisinia, C.W. Sinclair, W.J. Poole, C.N. Tomé, Modell. Simul. Mater. Sci. Eng. 16 (2008). [4] J. Liu, G. Zhu, W. Mao, S.V. Subramanian, Mater. Sci. Eng. A 607 (2014) 302– 306. [5] Y. Tomota, A. Narui, N. Tsuchida, ISIJ Int. 48 (2008) 1107–1113. [6] N. Kamikawa, X. Huang, N. Tsuji, N. Hansen, Acta Mater. 57 (2009) 4198–4208. [7] H. Halim, D.S. Wilkinson, M. Niewczas, Acta Mater. 55 (2007) 4151–4160. [8] U.F. Kocks, H. Mecking, Prog. Mater. Sci. 48 (2003) 171–273. [9] G.A. Malygin, Phys. Usp. 54 (2011) 1091–1116. [10] J. Gubicza, N.Q. Chinh, G. Krállics, I. Schiller, T. Ungár, Curr. Appl. Phys. 6 (2006) 194–199. [11] O. Bouaziz, N. Guelton, Mater. Sci. Eng. A 319–321 (2001) 246–249. [12] M. Dao, L. Lu, R.J. Asaro, J.T.M. De Hosson, E. Ma, Acta Mater. 55 (2007) 4041– 4065. [13] O. Bouaziz, Y. Estrin, Y. Bréchet, J.D. Embury, Scripta Mater. 63 (2010) 477– 479. [14] H.K.D.H. Bhadeshia, J.W. Christian, MTA 21 (1990) 767–797. [15] J. Li, A.K. Soh, Int. J. Plast. 39 (2012) 88–102. [16] C.W. Sinclair, W.J. Poole, Y. Bréchet, Scripta Mater. 55 (2006) 739–742. [17] G.M. Le, A. Godfrey, N. Hansen, Mater. Des. 49 (2013) 360–367. [18] P. Xue, B.L. Xiao, Z.Y. Ma, Mater. Sci. Eng. A 532 (2012) 106–110. [19] H. Adachi, Y. Miyajima, M. Sato, N. Tsuji, Mater. Trans. 56 (2015) 671–678. [20] D. Akama, T. Tsuchiyama, S. Takaki, ISIJ Int. 56 (2016) 1675–1680. [21] S. Takaki, K.L. Ngo-Huynh, N. Nakada, T. Tsuchiyama, ISIJ Int. 52 (2012) 710– 716. [22] C.Y. Yu, P.W. Kao, C.P. Chang, Acta Mater. 53 (2005) 4019–4028. [23] P.C. Hung, P.L. Sun, C.Y. Yu, P.W. Kao, C.P. Chang, Scripta Mater. 53 (2005) 647– 652. [24] P.L. Sun, C.Y. Yu, P.W. Kao, C.P. Chang, Scripta Mater. 52 (2005) 265–269. [25] Y. Shen, P.M. Anderson, J. Mech. Phys. Solids 55 (2007) 956–979. [26] Y. Okitsu, N. Takata, N. Tsuji, Scripta Mater. 64 (2011) 896–899.

10

S.H. He et al. / Computational Materials Science 131 (2017) 1–10

[27] A.K. Head, Phil. Mag. 44 (1953). [28] J.P. Hirth, J. Lothe, Theory of Dislocations, 1982. [29] M. Huang, P.E.J. Rivera-Díaz-Del-Castillo, O. Bouaziz, S. Van Der Zwaag, Mater. Sci. Technol. 24 (2008) 495–500. [30] F. Huang, N.R. Tao, J. Mater. Sci. Technol. 27 (2011) 1–7. [31] X.J. Zhen, P.Q. La, C.L. Li, S.L. Hu, J. Iron Steel Res. Int. 22 (2015) 324–329. [32] S. Khamsuk, N. Park, S. Gao, D. Terada, H. Adachi, N. Tsuji, Mater. Trans. 55 (2014) 106–113. [33] Z.Y. Zhong, H.G. Brokmeier, W.M. Gan, E. Maawad, B. Schwebke, N. Schell, Mater. Charact. 108 (2015) 124–131. [34] S.C. Kennett, G. Krauss, K.O. Findley, Scripta Mater. 107 (2015) 123–126. [35] B.L. Li, N. Tsuji, N. Kamikawa, Mater. Sci. Eng. A 423 (2006) 331–342.

[36] S.S. Hazra, E.V. Pereloma, A.A. Gazder, Acta Mater. 59 (2011) 4015–4029. [37] O. Saray, G. Purcek, I. Karaman, T. Neindorf, H.J. Maier, Mater. Sci. Eng. A 528 (2011) 6573–6583. [38] O. Saray, G. Purcek, I. Karaman, H.J. Maier, Mater. Sci. Eng. A 619 (2014) 119– 128. [39] W. Blum, X.H. Zeng, Acta Mater. 57 (2009) 1966–1974. [40] K. Zhu, O. Bouaziz, C. Oberbillig, M. Huang, Mater. Sci. Eng. A 527 (2010) 6614– 6619. [41] A.A. Gazder, S.S. Hazra, E.V. Pereloma, Mater. Sci. Eng. A 530 (2011) 492–503. [42] Y.B. Chun, S.H. Ahn, D.H. Shin, S.K. Hwang, Mater. Sci. Eng. A 508 (2009) 253– 258. [43] K. Edalati, Z. Horita, Acta Mater. 59 (2011) 6831–6836.