JOURNAL OF APPLIED PHYSICS 99, 076103 共2006兲

Strain rate sensitivity of face-centered-cubic nanocrystalline materials based on dislocation deformation Jianshe Lian,a兲 Changdong Gu, Qing Jiang, and Zhonghao Jiang Key Laboratory of Automobile Materials, Ministry of Education, College of Materials Science and Engineering, Jilin University, Nanling Campus, Changchun 130025, China

共Received 22 December 2005; accepted 15 February 2006; published online 5 April 2006兲 The relationship between strain rate sensitivity and activation volume for face-centered-cubic metals is proposed based on the bow-out model of single dislocation from its source, which gives reasonable prediction of the enhanced strain rate sensitivity that occurs in nanostructured and ultrafine grained Ni and Cu. © 2006 American Institute of Physics. 关DOI: 10.1063/1.2186981兴 Coarse grained 共CG兲 fcc metals deformed by dislocation motion usually have very low strain rate sensitivity of stress 共defined as m = ⳵ ln ␴ f / ⳵ ln ␧˙ , where ␴ f and ␧˙ are the flow stress and strain rate, respectively兲. High strain rate sensitivity value of ⬃0.5– 1.0 is observed for superplasticity materials at elevated temperature and Coble creep of nanostructured Ni at room temperature 共RT兲,1–3 where grain boundary 共GB兲 sliding or GB diffusion is the dominant deformation mechanism. Recently, investigations of the mechanical behavior of nanocrystalline 共nc兲 and ultrafine grained 共UFG兲 fcc metals at RT have concentrated on the strain rate sensitivity and the work hardening of these materials in order to compare their behaviors with those of their CG counterparts.4–17 For example, an evident increase of the strain rate sensitivity to about 0.02 and 0.04 is usually observed in nc or UFG Ni and Cu,4–17 compared with their CG counterparts, respectively. However, the mechanisms underlying the unusual rate sensitivity of deformation of nc metals are not fully understood at this time. It is still not clear whether it relates to the participation of GB deformation due to the increase of GB fraction as grain size reduced to nanoscales, just like what is assumed by a simple finite element analysis based on the grain boundary affected zone model by Schwaiger et al.8 However, it seems to be more or less accepted that for the grain sizes typically larger than about 20– 30 nm for Ni and Cu, deformation is dominated by dislocation process supported by numerous experimental results9–19 and molecular dynamics 共MD兲 computer simulations.20–22 That is, the dislocations are emitted from GB sources,15,16,18–21 and traverse the tiny grain under the applied stress to be reincorporated into the opposing GB. And the GB-defect-assisted dislocation generation can well be a thermally activated process.10 A relationship between strain rate sensitivity and activation volume based on the bow-out model of a single dislocation will be proposed to explain the increase of strain rate sensitivity of nc and UFG fcc metals. The activation volume 共␷兲 is widely used to determine the possible deformation mechanisms of nc and UFG materials.10–17 The thermal activated component of plastic Author to whom correspondence should be addressed; FAX: ⫹86-4315095876; electronic mail: [email protected]

deformation in a material is generally expressed in terms of an activation volume:11

␷eff = −

共1兲

where ␷eff is the effective activation volume, ⌬G is the minimum Gibbs free energy that has to be supplied to overcome obstacles at a given temperature, and ␶e is the effective shear stress. In addition, the experimental activation volume can be measured by16

␷ = 冑3kT

⳵ ln ␧˙ , ⳵␴

共2兲

where k is the Boltzmann constant, T is the absolute temperature, and ␴ and ␧˙ are the flow stress and strain rate, respectively. In fact, the reciprocal of activation volume may represent physical strain rate sensitivity. The experimental strain rate sensitivity m of stress is generally expressed as m=

⳵ ln ␴ . ⳵ ln ␧˙

共3兲

Combining Eqs. 共2兲 and 共3兲, we have m=

冑3kT ␴␷

共4兲

.

For materials based on the dislocation mechanism deformation, the flow stress of plastic deformation is generally determined by the dislocation line length. As to CG and UFG materials, the average dislocation length is determined by the mobile dislocation density. However, for the nc materials with grain size larger than 20– 30 nm, it may be determined by the length of existing GB dislocations or source in grains. In either case, an activation event could be expressed by the bow out of an existing dislocation from its source. The bowout model of single dislocation in a grain had been proposed to account for the mechanical behavior of nc materials.23,24 The critical stress for the bow out of an edge dislocation source is generally expressed as24

␶=

a兲

0021-8979/2006/99共7兲/076103/3/$23.00

⳵⌬G , ⳵␶e

冋冉冊 册

0.12Gb L ln − 1.65 L b

or

99, 076103-1

© 2006 American Institute of Physics

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076103-2

J. Appl. Phys. 99, 076103 共2006兲

Lian et al.

␴=␣

冋冉冊 册

Gb L ln − 1.65 , L b

共5兲

where ␶ is the shear stress, ␴ is the uniaxial flow stress, L is the dislocation line or source length, G is the shear modulus, b is the Burgers vector, and ␣ is a constant. For the dislocation-based deformation in an activation event of a dislocation source, the activation volume is related to the length L of the dislocation segment involved in the thermal activation by13

␷ = Lb2 or L =

␷ . b2

共6兲

The single dislocation can be understood as a dislocation segment in a grain or GB dislocation source. Taking Eqs. 共5兲 and 共6兲 into Eq. 共4兲, it gives m=

冑3kT ␣Gb

3

冋冉 冊 册 ln

␷ − 1.65 b3

−1

.

共7兲

This is the relationship between strain rate sensitivity and activation volume based on the deformation model of the bow out of single dislocation. Bulk nc Ni with a thickness of about 350 ␮m is electrodeposited from a modified Watts-type electrolyte, containing low content of saccharin 共1 g / l兲.25 Two nc Ni–Co alloys of Ni– 1.7 wt % Co 共Ref. 26兲 and Ni– 8.6 wt % Co 共Ref. 27 are obtained by adding different concentrations of cobalt sulfate in the bath. The addition of Co results in a refinement of grain size. The average grain sizes of Ni, Ni– 1.7 wt % Co and Ni– 8.6 wt % Co alloys are 40, 25, and 13 nm, respectively, determined by transmission electron microscope observations. The dog-bone shaped specimens with gauge size of 8.0⫻ 2 ⫻ ⬃ 0.25 mm3 are cut from the deposited sheets. The tensile tests are carried out on the MTS-810 system at strain rates from 1.04⫻ 10−6 to 1.04 s−1 and RT. The detailed mechanical behaviors and microstructure observations of these Ni and Ni–Co alloys are reported in Refs. 25–27. The strain rate sensitivity and activation volume of these materials are presented and analyzed here. It should be noted that 13 nm Ni– 8.6 wt % Co alloy showed three stages of strain rate sensitivity at strain rates of 1 ⫻ 10−6 to 1 s−1 and different strain rate sensitivity values relate to different deformation mechanisms.27 In this study, only the strain rate sensitivity value of 0.012 for the Ni– 8.6 wt % Co alloy arising from the dislocation movements is considered. Figure 1 shows flow stress ␴ versus strain rate ␧˙ curve in logarithm at 1.0% plastic strain of these three materials. The values of strain rate sensitivity 共m兲 measured from the slopes of these curves are shown, which is in the range reported for nc Ni 共0.01–0.02兲.7,10 The inset is the plot of ln ␧˙ vs ␴ transformed from Fig. 1 wherein the experimental activation volumes 共␷兲 calculated from the slopes are indicated. Figure 2 shows the experimental data between strain rate sensitivity and activation volume of Ni from our work 共Fig. 1兲 and literatures.7,10,14,15 The relationship between m and ␷ predicted by Eq. 共7兲 is plotted by the dashed line. Figure 3 shows the same experimental data of Cu from literatures12–14 and the relationship predicted by Eq. 共7兲. Good agreements

FIG. 1. Flow stress ␴ f vs strain rate ␧˙ curves in logarithm at 1.0% plastic strain for nc Ni and two Ni–Co alloys. Strain rate sensitivity is calculated from the slopes. The inset is the ln ␧˙ vs ␴ curves where the activation volumes are calculated.

between experimental data and the prediction by Eq. 共7兲 are achieved for both Ni 共and Ni–Co alloys兲 and Cu, the two typical fcc nc metals. The constant ␣ is chosen to be 0.36 for Ni and Cu in this study. Equation 共5兲 with the value of 0.36 for ␣ was used successfully to simulate the relationship between hardness and grain size for many nc materials.24 Therefore, the enhanced strain rate sensitivity exhibited by nc/UFG fcc materials can be predicted by the dislocationbased model. That is, the refinement of grain size to nanometer range leads to evident reduction of dislocation source 共or line兲 length and thereby small activation volume, which consequentially results in an increase of strain rate sensitivity of flow stress predicted by Eq. 共7兲. The enhanced strain rate sensitivities of 0.02–0.04 for nc Cu and 0.01–0.02 for nc Ni are only results of the reduction of dislocation length to about 10– 100b. It has no direct relations to the GB deformation. It is anticipated that a nc material with further

FIG. 2. Summary of strain rate sensitivity m vs activation volume ␷ for Ni and Ni–Co alloys. The dashed line plotted by the theoretical model by Eq. 共7兲 is in well agreement with the experiment results from literature plotted by the scattered dots.

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076103-3

J. Appl. Phys. 99, 076103 共2006兲

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FIG. 3. Summary of strain rate sensitivity m vs activation volume ␷ for nc, UFG and CG Cu. The dashed line plotted by the theoretical model by Eq. 共7兲 is in well agreement with the experiment results from literature plotted by the scattered dots.

smaller grain size 共for example, smaller than 15– 20 nm兲 or with dominant high-angle GB, the participation of GB-based deformation would produce an even higher strain rate sensitivity and much smaller activation volume.10,27 In summary, the enhanced strain rate sensitivities of 0.02–0.04 for nc Cu and 0.01–0.02 for nc Ni are suggested as only results of the reduction of dislocation length to about 10– 100b. It has no direct relation to the GB deformation mechanism. The relationship between strain rate sensitivity and activation volume for fcc metals is proposed based on the bow-out model of a single dislocation from its source, which gives well prediction of the strain rate sensitivity of fcc metals, such as Cu and Ni. This work was supported by the Foundation of National Key Basic Research and Development Program 共No. 2004CB619301兲.

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