IOP PUBLISHING

JOURNAL OF PHYSICS D: APPLIED PHYSICS

J. Phys. D: Appl. Phys. 40 (2007) 7440–7446

doi:10.1088/0022-3727/40/23/027

Experimental and modelling investigations on strain rate sensitivity of an electrodeposited 20 nm grain sized Ni C D Gu1,2,3 , J S Lian1 , Q Jiang1 and W T Zheng1 1

Key Lab of Automobile Materials, Ministry of Education, Department of Materials Science and Engineering, Jilin University, Nanling Campus, Changchun, 130025, People’s Republic of China 2 Department of Mechanical Engineering, Hong Kong University of Science and Technology, Clearwater Bay, Kowloon, Hong Kong SAR, People’s Republic of China E-mail: changdong [email protected]

Received 23 July 2007, in final form 16 October 2007 Published 16 November 2007 Online at stacks.iop.org/JPhysD/40/7440 Abstract Two experimental techniques of nanoindentation and tensile testing were used at room temperature to investigate the strain rate sensitivity of an electrodeposited Ni with a mean grain size (d) of 20 nm, respectively. It was found that the nanocrystalline (nc) Ni possessed a higher strain rate sensitivity exponent (m) during nanoindentation than during tensile testing. Furthermore, a higher m was accompanied by a smaller activation volume (V ). It is believed that the higher stress concentration could activate a shorter dislocation line length (L), which should be responsible for the higher m value during the nanoindentation. Based on a model of dislocation nucleation or bowing-out mechanism, the relationship between m and d for Ni and its alloys was investigated. In the end, a simple and straightforward equation relating m to d was proposed in aid of a simple assumption associating L with d, which implied that the enhanced m in nc Ni and its alloys with d >∼ 6 nm should be due to the reduction of the dislocation line length. (Some figures in this article are in colour only in the electronic version)

1. Introduction The functionality and reliability of next-generation microelectromechanical systems (MEMS), nanoelectromechanical systems (NEMS), integrated circuits and micro- and nanoscale devices are closely tied to the deformation mechanism of the nanocrystalline (nc) metals/alloys of which they are constructed. Much interest in the plastic deformation mechanism of nc and ultra-fine grained (UFG) materials at room temperature (RT) is currently focused on the strain rate sensitivity and work hardening of these materials [1–15]. Significantly, the strain rate sensitivity exponent (m, which is defined as m = ∂ ln σf /∂ ln ε˙ , where σf and ε˙ are the flow stress and the strain rate, respectively) and activation volume (V ) are usually 3

Author to whom any correspondence should be addressed.

0022-3727/07/237440+07$30.00

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used to determine the possible deformation mechanisms of nc and UFG materials because these two parameters could provide quantitative measures of the sensitivity of flow stress to the loading rate and also give insights into the deformation mechanisms [7, 8, 12, 16–18]. The enhanced strain rate sensitivity is widely observed in nc metals compared with their coarse grained (CG) counterparts. For example, m of nc Ni is commonly in the range 0.01–0.03 at RT, which is about ten times as high as that of CG Ni (0.001–0.004) [5,10,11]. However, the mechanisms underlying the unusual rate sensitivity of deformation for nc metals are still under discussion at present [3, 19]. The strain rate sensitivity can also be expressed as (e.g. see [7, 8]) √ √ 3 3kT 3kT m= = , (1) V · σf V ·H

Printed in the UK

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Strain rate sensitivity of an electrodeposited 20 nm grain sized Ni

where k is the Boltzmann constant, T is the absolute temperature, H is the hardness, which is usually assumed to be three times the value of flow stress, σf , and V is the activation volume for plastic deformation which is directly related to the physical mechanism of deformation. It is generally accepted that a higher m is indicative of a smaller V [8,16]. In addition, the activation volume can be measured by [7, 8, 12, 16–18] V = or

√

√

3kT

V = 3 3kT




∂ ln ε˙ ∂σf ∂ ln ε˙ i ∂H

(2a)  ,

(2b)

where ε˙ i is the indentation strain rate for the indentation testing. Among the available experimental techniques to measure m or V values of nc metals, e.g. uniaxial tensile testing [5, 6, 11] and nanoindentation [3, 8, 20, 21] at different strain rates, respectively, strain rate jump test [12, 13, 22, 23], stress relaxation [10, 22–24], the first two methods are favourable for being used. However, tensile testing gives the panoramic mechanical behaviours of specimens, while nanoindentation presents the locally mechanical response of specimens [25]. There are differences in loading mode, stress state, specimen geometry, and so on, between these two experimental techniques. Therefore, it is very necessary to perform systematic experiments on the same kind of fully dense nc metal by the aforementioned two testing techniques to compare the results of m or V . In this contribution, m or V values of a fully dense electrodeposited 20 nm grain sized Ni were systematically measured by nanoindentation and tensile testing, respectively, and compared with each other. It is worth noting that in most of the publications [3, 5, 10, 11, 17, 18, 22, 26–29], research used only one testing technique or was lacking a comparison work between different testing techniques when studying the strain rate sensitivity of nc metals. Based on a proposed dislocation nucleation or bow-out model, the mechanisms underlying the strain rate sensitivity of nc Ni and its alloys were also investigated. Furthermore, this work is complementary to the limited data available to date on the strain rate sensitivity of nc Ni and its alloys [3, 5, 10, 11, 17, 18, 22, 26–29].

2. Experimental details A sheet of fully dense (99.6 ± 0.4% of theory density) nc Ni with a thickness of about 400 µm was fabricated by a so-called surfactant-assisted direct-current electrodeposition technique [30], which had been applied to synthesize various nanostructured Ni and Ni-based alloys [11, 17, 18]. The crystallographic structure and microstructure of the as-deposited nc Ni were analysed by an x-ray diffractometer (XRD, D/max 2500PC) and a transmission electron microscope (TEM, H-800), respectively. The chemical analysis by the inductively coupled plasma atomic emission spectrometry (ICP-AES, Plasma/1000) and carbon/sulfur determinators showed that the nc Ni had the main impurities of about 110 ppm S, 640 ppm C, 510 ppm Pb, 120 ppm Co and 132 ppm B. The dog-bone shaped specimens for tensile testing had a gauge cross-section of 2.0 mm × 0.3 mm and a gauge length

Figure 1. XRD and SAD (inset) patterns of as-deposited nc Ni.

of 8.0 mm and the specimens for nanoindentation testing were rectangular with about 10 mm side length. These specimens were all cut from the as-deposited nc Ni sheet by using an electro-discharging-machine (EDM) and then polished to a mirror-like finish surface. Tensile tests were carried out on the MTS-810 system at strain rates of 2.08×10−5 –2.08×10−1 s−1 and RT. For each strain rate, two or three specimens for tensile testing were tested. Nanoindentation experiments were performed on a Nanoindenter XPT (MTS Inc., TN) with a diamond Berkovich tip over a wide indentation strain rate range of 2.5 × 10−3 – 2.0 × 10−1 s−1 . At first, the tip was brought into contact with the specimen. Subsequently, the specimen was indented at a constant strain rate to a depth of 2 µm. Then a dwell period of 10 s was imposed at the maximum depth. Finally, the specimens were unloaded to 10% of the maximum load and held constant for 20 s to calibrate the thermal drift effect in the measurements. Continuous stiffness measurement (CSM) was used to acquire the hardness and elastic modulus variations with increasing penetration depth [31]. For each sample and indentation strain rate, at least four indents were performed.

3. Results and discussions 3.1. Microstructure characterization The as-deposited Ni is characteristic of a mean d of 20 nm and a preferential orientation along the (2 0 0) planes revealed by XRD as shown in figure 1. Electrodeposits are known for giving numerous, well-defined preferred orientations depending on the electrodeposition conditions, i.e. electrolyte composition, temperature, pH, current density, stirring and organic additions, etc [5,11,32]. The selected area diffractions (SAD) on TEM are well formed ring patterns (see figure 1 (inset)) which confirm the small grain sized structure of the electrodeposited Ni. In addition, no additional peaks can be measured in the inset SAD patterns, indicating that there is no second phase in the nc Ni. The details on the grain structure of the nc Ni can be found in the following TEM observations and analysis. Figure 2(a) gives the TEM observation of the nc Ni from a lower magnification view. It is found that the grains are roughly equiaxed and have a very uniform distribution. Moreover, the higher magnification bright field image in figure 2(b) indicates 7441

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(a) 2.0x10-1 s-1 1.0x10 -1 s-1 5.0x10 -2 s-1 1.0x10 -2 s-1 5.0x10 -3 s-1 2.5x10-3 s-1

Load (mN)

400

300

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0

0

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(b)

250

8

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150 100

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2

2.0x10-1 s-1 5.0x10-2 s-1 5.0x10-3 s-1 0

400

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1.0x10-1 s-1 1.0x10-2 s-1 2.5x10-3 s-1 1600

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Elastic Modulus (GPa)

10

50 0

Displacement Into Surface (nm)

Figure 3. Load–depth curves (a) and hardness and elastic modulus variations (b) for the 20 nm grain sized Ni from nanoindentation experiments performed at different strain rates.

(c) 80

Number

60

40

20

0

6

8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

Grain size (nm)

Figure 2. TEM observations: lower magnification (a), higher magnification (b) and the statistical distributions for grain size (c) of the electrodeposited nc Ni.

that the grain clusters possessing small misorientation angles between grain are formed in some regions. Grains can be distinguished by their change in contrast at grain boundaries (GBs). A statistical analysis of ∼600 grains indicates that the electrodeposited Ni has an average d of about 20 nm and a narrow grain size distribution, as shown in figure 2(c). The microstructure of the present Ni is largely different from our previous electrodeposited nc Ni [11], which was characteristic of a bimodal or wide grain size distribution. 3.2. Nanoindentation results During the nanoindentation of the 20 nm grain sized Ni, six different indentation strain rates were applied and the corresponding results are shown in figures 3(a) and (b). The 7442

load–displacement curves given in figure 3(a) for the nc Ni show a distinct and experimentally detectable effect of indentation strain rates. With increasing indentation strain rate, a higher indentation load is required in order to impose the same displacement. A similar strain rate effect in nanoindentation was also shown in nc Ni with different grain sizes by Schwaiger et al [3] and nano-sized twin Cu by Lu et al [8], respectively. In figure 3(b), the hardness and elastic modulus are plotted versus the depth of indentation at various indentation strain rates as determined continuously during indentation for the nc Ni, respectively. It is found that the hardness values of the nc Ni increase with increasing indentation strain rate and the values are in the range 5.1–6.0 GPa at a maximum indentation depth. The elastic modulus of the nc Ni from the nanoindentation is independent of the indentation strain rate and the value is about 230 ± 15 GPa, which is slightly higher than the value of 210 GPa for bulk CG Ni. The elastic modulus of the nc Ni possesses a bulk value of elastic modulus, which should be attributed to the fully dense structures of the electrodeposited Ni. The logarithmic plots of the hardness value (H ) from nanoindentation as a function of the indentation strain rate ε˙ i for the nc Ni are given in figure 4(a). The strain rate sensitivity exponent , m, was estimated to be about 0.033 from the slope of the double logarithmic line of H versus ε˙ i by the following definition [20, 33]: ∂ log H . (3) m= ∂ log ε˙ i The activation volume, V , is estimated to be about 7b3 from the slope of the linear fit in the plot of ln ε˙ i versus H , as

Strain rate sensitivity of an electrodeposited 20 nm grain sized Ni

(a)

2000

True stress (MPa)

ln (hardness, GPa)

7

6

m = 0.033 5

1600

2.08X10-5 s-1 1.04X10-4 s-1 1.04X10-3 s-1 1.04X10-2 s-1 1.04X10-1 s-1 2.08X10-1 s-1

1200

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4 1E-3

0.01

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4

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7

8

9

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(a)

2000

V = 7 b3

-4

Flow stress (MPa)

ln(strain rate, s-1)

2

Figure 5. True stress–strain curves for nc Ni from tensile testing at different strain rates and RT.

(b)

-2

-6

-8

1

True strain (%)

ln(strain rate, s-1) 0

0

0.1

4800

5000

5200

5400

5600

5800

6000

m = 0.016

6200

Hardness (MPa)

1000 1E-5

Figure 4. (a) Logarithmic plots of hardness value (H ) from nanoindentation as a function of indentation strain rate ε˙ i for the nc Ni. m was estimated from the slope of the linear fit. (b) Plots of ln ε˙ i versus H transformed from (a). The V was estimated from the slope of the linear fit using equation (2b).

3.3. Tensile testing results The mechanical properties of 20 nm grain sized Ni were evaluated by uniaxial tensile testing performed at strain rates of 2.08 × 10−5 –2.08 × 10−1 s−1 . It is found that the nc Ni exhibits very high ultimate tensile strength (UTS) of about 1540–1920 MPa and limited plastic strain of about 1.2–3.1%. The yield strength σ0.2 is in the range 1030–1280 MPa. The corresponding true stress–strain curves for the 20 nm grain sized Ni at different strain rates are shown in figure 5. The flow stress and plastic strain increase with increasing strain rates from 2.08 × 10−5 to 2.08 × 10−1 s−1 . The specimen deformed at the highest strain rate (2.08×10−1 s−1 ) gives some evidence of necking and maximum flow stress (about 1990 MPa). The present nc Ni is characterized by a fully dense structure and a fine and uniform grain size distribution, which is suggested to be responsible for its enhanced UTS in comparison with the other nc Ni specimens [3, 11, 34]. Figure 6(a) gives the logarithmic plots of flow stress at 1% plastic strain (σ1% ) as a function of strain rate ε˙ for the nc Ni performed by tensile testing. The m value is estimated

1E-3

0.01

0.1

1

Strain rate (s-1) 4

(b)

0

ln(strain rate (s-1)

shown in figure 4(b) using equation (2b), where b is the Burgers vector for Ni. A similar average activation volume of 7b3 was recently observed in the tensile testing studies of electrodeposited 13 nm grain sized Ni–8.6wt% Co alloy [21].

1E-4

-4

V = 14 b3 -8

-12

-16

-20 1500

1550

1600

1650

1700

1750

1800

1850

1900

Flow stress (MPa)

Figure 6. (a) Logarithmic plots of fl ow stress at 1% plastic strain (σ1% ) as a function of strain rate ε˙ for nc Ni. The m value was estimated from the slope of the linear fit. (b) Plots of ln ε˙ versus σ1% transformed from (a). The V value was estimated from the slope of linear fit using equation (2a).

to be about 0.016 from the slope of the linear fit, which is similar to that (0.02) of 21 nm Ni measured by strain rate jump tests [22] and that (0.02) of 29 nm Ni by stress relaxation experiment [10]. The activation volume, V , is estimated to be about 14b3 from the slope of linear fit in the plots of ln ε˙ versus σ1% (as shown in figure 6(b)) using equation (2a). Table 1 summarizes the m and V values obtained by nanoindentation and tensile testing, respectively, for the present nc Ni. It is found that the m value of 0.016 measured by tensile testing is about half of the one 7443

C D Gu et al

Table 1. The summary of m and V obtained by the nanoindentation and tensile testing, respectively, for the nc Ni in the study. m

V

Strain rate range

Nanoindentation Tensile testing

0.033 0.016

7b3 14b3

2.5 × 10−3 –2.0 × 10−1 s−1 2.08 × 10−5 –2.08 × 10−1 s−1

Activation volume, V / b3

1000

predicted by model [5] Ni [18] Ni-Co alloy [11] Ni [35] Ni [29] Ni this study-nanoindentation this study-tensile testing [27] Ni-Fe alloy [17] Ni-Co alloy [23] Ni [22] Ni [10] Ni

0.03

0.02

0.01

0.00

10

100

1000

Activation volume (V / b3)

100 [21] Ni [28] Ni [33] Ni [26] Ni-Fe alloy [26] Ni this study-tensile testing this study-nanoindentation [17] Ni-Co alloy [18] Ni-Co alloy [11] Ni [5] Ni [26] Ni [22] Ni

10

100

1000

Figure 8. Summary of strain rate sensitivity m versus activation volume, V /b3 , for Ni and Ni-based alloys from our work and the literature [5, 10, 11, 17, 18, 22, 23, 27, 29, 35]. The dashed line plotted by the theoretical model by equation (5) is in well agreement with the experiment results from the literatures plotted by the scattered dots.

10000

Grain size d (nm)

Figure 7. The variation of activation volume, V /b3 , as a function of the grain size, d, for Ni and its alloys [5, 11, 17, 18, 22, 23, 27, 29, 35].

measured by nanoindentation (0.033, as shown in figure 4(a)). Correspondingly, V of 14b3 obtained by the tensile testing is two times higher than the one obtained by nanoindentation. Therefore, it can be seen that the nanoindentation would give a much higher m or a smaller V than the tensile testing for certain nc metals. 3.4. Deformation mechanism of the 20 nm grain sized Ni A summary of available data [5, 11, 17, 18, 22, 23, 27, 29, 35] on the effect of d on V for Ni and its alloys is given in figure 7. Despite some inconsistency in the absolute values obtained using different synthesis techniques or different testing methods, a consistent trend can be concluded: with decreasing d, V falls monotonically from a thousand b3 in the CG region to about tens of b3 in the nc region, which indicates that the refinement of d could effectively restrain the dislocation line length in an activation event and hence the activation volume. It is more or less accepted that for Ni and Cu with d > 20–30 nm, deformation is dominated by the dislocation process supported by numerous experimental results [7, 8, 10, 12, 15, 21, 22, 29, 36, 37] and molecular dynamics computer simulations [38–40]. That is, the dislocations are emitted from GB sources and traverse the tiny grain under the applied stress to be reincorporated into the opposing GB. In a previous study, a relationship between m and V based on the bow-out model of a single dislocation was proposed to explain the increase in strain rate sensitivity of nc and UFG metals [19]. Firstly, the critical stress σ for the bow-out of an edge dislocation source is generally expressed [41]: 

 L 0.36Gb − 1.65 , (4) ln σ = b L where L is the dislocation line length or the activation length and G is the shear modulus. Then, for the dislocation-based 7444

Strain rate sensitivity (m)

Experimental technique

0.04

deformation in an activation event of a dislocation source, V is assumed to be V = Lb2 . The single dislocation can be understood as a dislocation segment or a GB dislocation source in nc grains. Finally, the equation between m and V is given as below: √ 
 −1 V 3kT ln 3 − 1.65 . (5) m= 0.36Gb3 b Figure 8 summarizes strain rate sensitivity m versus activation volume V for Ni and Ni-based alloys from the literature [5, 10, 11, 17, 18, 22, 23, 27, 29, 35] and this work. It is found that the dashed line plotted by the theoretical model by eqaution (5) gives an impressive agreement with the experiment results from the literature plotted by the scattered dots. It is deduced that the enhanced strain rate sensitivity of 0.01–0.03 for nc Ni is suggested as only a result of the reduction of the dislocation line length to about tens of b. Significantly, as to the present Ni, although nanoindentation and tensile testing give the different couples of m and V values, respectively, as shown in table 1, the correlations of m and V obtained from the two methods are also well simulated by the theoretical model of equation (5). It is indicated that the dislocation-motion or the dislocation nucleation mediated plastic deformation occurs in the 20 nm grain sized Ni during both nanoindentation and tensile testing. Furthermore, the present Ni exhibits very high stress levels of 5.1–6.0 GPa at a maximum indentation depth in the nanoindentation process, which is much higher than that of about 1.9 GPa in tensile testing. The higher the stress applied on the nc Ni, the shorter the dislocation line length, L, that could be activated according to equation (4). The emission of partial or perfect dislocations from stress concentrations at GBs would lead to estimates of a smaller V in the range of 3b3 –10b3 [16]. In addition, the nucleation of a small loop at a GB facet crack or a GB triple point would also be possible by the high stress concentration [15,16,42]. Therefore, the higher stress concentration during the nanoindentation process could activate a shorter dislocation line length, which might be the reason why the m value obtained by nanoindentation is higher than the one by tensile testing.

Strain rate sensitivity of an electrodeposited 20 nm grain sized Ni [17] Ni-Co alloy [18] Ni-Co alloy [11] Ni [21] Ni [5] Ni [5] Ni [10] Ni [22] Ni [33] Ni [3] Ni [22] Ni [42] Ni [42] Ni [26] Ni-Fe alloy this study-nanoindentation this study-tensile testing predicted by Eq. (7)

Strain rate sensitivity, m

0.03

0.02

0.01

0.00

10

100

1000

10000

100000

Grain size, d (nm)

Figure 9. The variation of strain rate sensitivity, m, as a function of grain size, d, for Ni and its alloys from the literature [5, 10, 11, 17, 18, 22, 23, 27, 35, 43] and this study. The solid line in figure is predicted by equation (7).

3.5. Modelling the correlation between strain rate sensitivity and grain size It is rational that the length of dislocation line, L, should depend on the grain size, d. Conceptually, for very small d, L/d approaches unity, and for very large d, L/d asymptotically tends to very small values [41]. Therefore, it is physically reasonable to assume that L/d ∼ d −n

with n < 1.

Comparing equation (4) with the Hall–Petch relationship for large grain sized materials, n = 1/2 is used. Thus, L = c · d 1/2 ,

(6)

where c is a constant with a unit of nm1/2 . It is believed that the emission of a small segment of dislocation, escaping from the GB edge and the closely spaced trapping obstacles at and surrounding a GB, would entail a rather small experimental activation volume, ∼10b3 [10, 16]. Therefore, it is reasonably assumed that the smallest activation length of the nucleation or emission of partial/perfect dislocations from GBs would be ∼10b (∼2.49 nm) because this scale is precisely a fraction of the edge length for an octagon grain with d = 6.5 nm. Taking L = 2.49 nm and d = 6.5 nm into equation (6), it gives c = 0.98 nm1/2 . Substituting equation (6) and V = Lb2 into equation (5), the relationship between m and d can be given as below: √ 
1/2  −1 cd 3kT m= . (7) ln − 1.65 0.36Gb3 b Figure 9 gives the variation of strain rate sensitivity, m, as a function of grain size, d, for Ni and its alloys from the literature [5, 10, 11, 17, 18, 22, 23, 27, 35, 43] and this study. The solid line in the figure is predicted by the theoretical model of equation (7). A clear trend can be seen that the m values hold at a level of 0.003–0.006 when decreasing d from the micrometre to the submicronmetre scale, while an obvious jack-up appears when d is reduced below 100 nm or so, as shown in figure 9. The m value for Ni or Ni alloys with d of 20–30 nm is increased to 0.02–0.03. A similar trend

was also found in the studies of Cu and Cu alloys [20, 21]. Significantly, the theoretical model of equation (7) is well in agreement with the experimental data plotted by the scattered dots in figure 9. Therefore, it can be seen that decreasing d in nanometre regions would effectively restrain the nucleation or the length of mobile dislocations which are the main carrier of plastic deformation in CG metals. This may be the reason why nc metals with strictly narrow grain size distributions usually exhibit the very limited plasticity at RT, just as in our case and others [2, 3, 5, 17, 44, 45]. Also, it could be deduced that the enhanced strain rate sensitivity of Ni and Ni alloys with d > ∼6 nm should be a result of the reduction of the dislocation line length in the plastic deformation, as indicated by the proposed dislocation nucleation or bowing-out model of equation (5) and (7).

4. Conclusions 1. Nanoindentation and tensile testing at different strain rates were, respectively, used to systematically investigate the strain rate sensitivity of the same kind of nc metal, i.e. the electrodeposited Ni with a mean d of 20 nm. With the comparison between the two experimental results, it is found that the m value of 20 nm Ni measured by nanoindentation is 0.033, which is much higher than the one by tensile testing, 0.016. According to a theoretical model based on the bow-out of single dislocation from its source, the higher stress concentration during nanoindentation could activate a shorter dislocation line length, which should be responsible for the higher m value in nanoindentation deformation. 2. A simple and straightforward equation relating m to d for Ni and its alloys is first given in this paper and it works well when compared with the experimental data, which implies that the decreasing d in nanometre regions would effectively restrain the nucleation or the length of mobile dislocations and hence lead to the enhanced strain rate sensitivity of Ni and its alloys with d > ∼6 nm. 3. Considering the present nc Ni possessing a strong (2 0 0) texture structure, some novel nanostructured materials with texture-free and strictly grain size distribution are being expected to be synthesized to confirm the experimental and theoretical results presented in this paper.

Acknowledgments The authors thank Dr T An of the Department of Materials Science and Engineering, Jilin University, for kind help in nanoindentation testing. The work was supported by the Foundation of National Key Basic Research and Development Program (No 2004CB619301) and the Project 985-automotive engineering of Jilin University.

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