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Opposite grain size dependence of strain rate sensitivity of copper at low vs high strain rates
Author’s Accepted Manuscript Opposite grain size dependence of strain rate sensitivity of copper at low vs high strain rates Z.N. Mao, X.H. An, X.Z. Liao, J.T. Wang
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S0921-5093(18)31214-0 https://doi.org/10.1016/j.msea.2018.09.018 MSA36899
To appear in: Materials Science & Engineering A Received date: 23 May 2018 Revised date: 23 August 2018 Accepted date: 7 September 2018 Cite this article as: Z.N. Mao, X.H. An, X.Z. Liao and J.T. Wang, Opposite grain size dependence of strain rate sensitivity of copper at low vs high strain r a t e s , Materials Science & Engineering A, https://doi.org/10.1016/j.msea.2018.09.018 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Opposite grain size dependence of strain rate sensitivity of copper at low vs high strain rates Z.N. Maoa, X.H. Anb, X.Z. Liaob*, and J.T. Wanga* a
School of Materials Science and Engineering, Nanjing University of Science and
Technology, Nanjing 210094, China. b
School of Aerospace, Mechanical & Mechatronic Engineering, The University of
Sydney, Sydney, NSW 2006, Australia.
* Corres. authors. Tel: +86 25 84315493; [email protected](J.T. Wang). Tel: +61 2 93512348; [email protected] (X.Z. Liao).
Abstract The grain size dependence of the strain rate sensitivity (SRS) of copper were systematically investigated via tensile deformation at strain rates of ~10-4 s-1 and ~103 s-1. In contrast to the general perception that SRS increases with decreasing grain size at low strain rates in FCC metals, the SRS increases monotonously with grain size under deformation at high strain-rates of ~103 s-1. Analytical formulation based on the Nemat-Nasser-Li (NNL) and Modified-Rusinek-Klepaczko (MRK) models was established to reveal the essential dependence of SRS on grain size, at the change of strain rate: that the opposite dependence of SRS on grain size at low vs high strain rates can be attributed to the transformation of the dominant rate-controlling 1
deformation mechanism from thermal activation at low strain rates to viscous drag at high strain rates. It is demonstrated that the thermal activation component of SRS,
m* , which is nearly strain rate independent, increases with reducing grain size; while the viscous drag component of SRS, mvs , which is enhanced significantly at high strain rates, decreases with reducing grain size. Microstructural observation based on local misorientation characterization and statistics confirms that the viscous-drag dominates at high strain-rate deformation, and becomes progressively influential as grain size increases. This unveils the essence of viscous drag in the opposite grain size dependence of SRS at low vs high strain rates.
Keywords: Opposite grain size dependence, Strain rate sensitivity, Dynamic tension, Viscous drag, Thermal activation. 1. Introduction It is well acknowledged that the plastic flow characteristics of materials are sensitive to the rate of loading, which can be quantitatively reflected by the strain rate sensitivity (SRS). The SRS, commonly defined as 𝑚 = 𝜕 𝑙𝑛𝜎/𝜕 𝑙𝑛𝜀̇ (where 𝜎 is the flow stress and 𝜀̇ is the strain rate), is widely recognized as the essential fingerprint for the thermodynamics and kinetics of deformation processes and is used to characterize the strain rate hardening phenomenon of materials [1, 2]. Therefore, the SRS has important implications in comprehending the mechanical behavior and deformation mechanisms of metals and alloys. Although face-centered cubic (FCC) metals generally exhibit a lower SRS than those with body-centered cubic (BCC)
2
structure, extensive investigations under a wide range of strain rates from quasi-static to ~102 s-1 revealed that the SRS is enhanced by reducing the grain size of FCC metallic materials [3-7]. Especially, ultrafine-grained and nanocrystalline materials presents an increase of SRS by up to an order of magnitude relative to their coarse-grained counterparts [7]. The introduction of nanoscale twins into ultrafine grains also reach the same trait of increased rate sensitivity [3, 8-10]. Regarding the significance of length scale on the SRS, several models based on the the thermally activated dislocation processes have been proposed to uncover the physical origins of the grain size effect on SRS for FCC metallic materials [7, 8].
Previous investigations have substantiated that the strain-rate dependence of flow stress for many FCC metals and alloys originates from thermal activation [11], which governs the rate-controlling mechanism at low strain rates [12]. With increasing the strain rate, there is less time available to overcome the thermally activated barrier [13]. In fact, dislocation theory manifests that viscous drag is another essential rate-controlling deformation mechanism that gradually becomes dominant at high strain rates [14]. Because the viscous drag stress does not have any effect on the flow stress until the strain rate reaches ~103 s-1 [15, 16], viscous or quasi-viscous drag processes are triggered only at high strain rates of above ~103 s-1[17]. Rather than thermal activation assistance for dislocations overcoming the resistance from short-range barriers (forest dislocations, Peierls stress, point defects etc.) [18], the dislocation motion becomes more continuous, controlled by drag mechanisms based on its interaction with phonons and electrons of the crystals [19]. With the influence 3
of viscous drag, a rapid upturn of flow stress with strain rate is observed at strain rate above ~103 s-1 [20], indicating a fast rise in SRS at high strain rate [21]. This signals that SRS is not constant but varies with strain rates. However, a basic gap in our knowledge for many years has been the lack of a quantitative understanding based on the deformation mechanism to account for the strain rate effect on SRS. In addition, the effect of grain size on the SRS derived from the experiments at high strain rates is still lacking. Thus, it is necessary to systematically investigate the evolution of SRS with strain rate and the grain size effect when different mechanisms dominate the plastic deformation. This fundamental exploration will significantly enrich and enliven our understanding of essential deformation dynamics and kinetics of materials at low and high strain rates, and will facilitate building a physical based model to strengthen our ability of predicting the plasticity of materials.
In this work, pure copper samples with five average grain sizes were tensile tested over a wide range of strain rates (from ~10-4 s-1 to ~103 s-1) to examine the effect of grain size on the SRS at low and high strain rates. A model describing strain rate dependence of SRS was established based on the Nemat-Nasser-Li (NNL) [22] and the Modified-Rusinek-Klepaczko (MRK) [15] constitutive relationships. The influence of the two rate-controlling deformation mechanisms (thermal activation and viscous drag) to SRS was analyzed separately. Opposite grain size dependences of SRS at low and high strain rates were revealed and their physical interpretation was provided. Microstructural observation based on local misorientation analysis conducted using the electron back scatter diffraction (EBSD) technique further 4
confirmed the explanation to this abnormal phenomenon.
2. Experimental procedures As-received oxygen free high conductivity copper (99.98% Cu, 0.0095% O) bars with dimensions of 32 × 32 × 160 mm3 were annealed at 600 ºC for 1 h, forming a homogeneous microstructure consisting of equiaxed grains with an average grain size of ~ 90 μm. To refine the coarse grains, equal-channel angular pressing (ECAP), one of the most popular severe plastic deformation (SPD) techniques, was conducted for 8 passes via route Bc, i.e., the billet was rotated 90° clockwise between two consecutive passes [23], using a die having an internal channel angle of 90° and an outer arc curvature of 20° at the ambient temperature with a pressing speed of 2 mm/s. After ECAP, ultrafine-grained samples were successfully obtained and they were kept in liquid nitrogen to suppress any potential recovery of the microstructure. In order to achieve homogeneous microstructures with different grain sizes, annealing treatments were conducted at 280ºC for 50s , 400ºC for 1min, 280ºC for 1h and 600ºC for 2h to achieve microstructures with grain sizes of about 3μm, 6μm, 12μm and 90μm respectively.
Tensile samples with a dog-bone shape were cut from the middle of the ECAP bars by wire electro-discharge machining. The gauge section geometry of the specimens was 10 × 2 × 3 mm3 and 12 × 2 × 4.8 mm3, for quasi-static and high-strain-rate tensile loading, respectively. The strain rate applied in this work varies from ~10-4/s to ~103/s. Low-strain-rate (~10-3/s) tensile tests were performed in
5
an Auto Graph AGS-X testing machine (Shimadzu Corporation). The split Hopkinson pressure bar (SHPB) was employed for high-strain-rate (~103/s) tensile experiments. According to the one dimensional elastic stress wave theory, the stress, strain and strain rate of tested specimen can be calculated as [24]: A As
s E
T ,
s
2C0 ls
t
0
R
d ,
s
2C0 R ls
where εT and εR are the transmitted and reflected strain pulses which can be measured by the strain gages stuck on the input and output bars, respectively; C0, E, and A denote the longitudinal elastic wave velocity, Young’s modulus and cross-sectional area of the loading bars; ls and As are the length and cross-sectional area of the specimen, respectively. The parameters used for SHPB tensile tests are presented in Table 1. At least three tensile tests were performed on each data point to substantiate the reproducibility of the stress-strain curves.
Samples used for microstructural characterization were mechanically ground on 4000-grit SiC papers and then polished using 1μm diamond paste. Subsequent electropolishing was conducted at 20 ºC in an electrolyte of 25% orthophosphoric acid, 25% ethanol and 50% distilled water at a voltage of 7 V and duration of 1 min. The EBSD investigations were undertaken in an FEI Quanta 250 field emission scanning electron microscope (SEM) equipped with an Oxford Instruments Aztec 2.0 EBSD system with operating voltage of 12 kV and a step size of 80 nm. The microstructural features were recorded using the Channel 5 software (HKL
6
Technology). Misorientation angles smaller than 2° were not considered due to the noise effect. High-angle grain boundaries (HAGBs) and low-angle grain boundaries (LAGBs) in this study are defined as grain boundaries with misorientation angles >15° and in the range of 2–15°, respectively.
3. Experimental results Typical microstructures of copper samples with 5 different grain sizes are exhibited in the EBSD maps in Fig. 1. The black and white lines in the EBSD maps denote HAGBs and LAGBs, respectively. After ECAP processing, although there was still a high density of LAGBs in the sample, the fingerprint of microstructures was uniformly distributed equiaxed ultrafine grains with the average grain size of ~0.5 μm in most areas, as detected in Fig. 1 (a). In contrast, few LAGBs were found in the four annealed samples that presented the average grain sizes of ~ 3 μm, ~ 6 μm, ~ 12 μm, and ~ 90 μm, as shown in Fig. 1 (b-e), respectively. Additionally, the orientation maps of both the as ECAP and recrystallized samples revealed that the microstructures have no strong texture, and therefore an insignificant effect of texture on the mechanical behavior is expected, which is in accordance with the results from [25, 26].
Figure 2 shows the tensile true stress–strain curves of samples with different grain sizes conducted at the strain rates of (a) 8.33 × 10-4 s-1 and (b) ~103 s-1. Only the first stress wave response during high strain-rate deformation is presented. As expected from the Hall-Petch effect [27, 28], remarkable increase in yield strength was achieved with reducing grain size under both low and high strain-rate
7
deformation. For low strain-rate deformation, large strain hardening capability was observed in all samples except for the sample with the average grain size of 0.5 μm due to its saturated work hardening with a high dislocation density caused by ECAP processing [29]. The ductility of the 4 recrystallized samples was slightly reduced with decreasing the grain size. Irrespective of the grain sizes, deformation at high strain rates led to much higher flow stresses than those at low strain rates due to the suppression of relevant thermally activated mechanisms at high strain rates [30, 31].
To evaluate the significance of strain rate and grain sizes during deformation, the flow stress of each sample at 2% true strain was plotted as a function of strain rate, as indicated by the scattered datum points in Fig. 3. Based on experimental data, MRK constitutive model [15] was applied for data fitting separately for samples with different grain size, and the results were also shown in Fig. 3 using solid lines. The distinct effects of strain rate on flow stress are clearly demonstrated for deformation at low and high strain rates. At the low strain rate range, the flow stress exhibits relatively weak sensitivity to the strain rate and then a slight enhancement in flow stress with increasing strain rate was identified. Similar to previous investigations [3, 4, 32], the dependence of the flow stress on strain rate, i.e. SRS increases with decreasing grain size. In contrast, a remarkable rapid upturn of flow stress with strain rate was observed at strain rates > ~103 s-1, which is attributed to the activation of the viscous drag mechanism. The fast increase of flow stress with strain rate not only implies the strong rate-dependence of SRS, but also shows apparent grain size effect. Different from those at the low strain rate range, the slope of the data in the 8
ultrafine-grained sample is lower than those for the coarse-grained samples, which surprisingly signals the inverse grain size effects on the SRS at the high strain rate regime.
The strain rate dependence of SRS calculated from the fitting curves in Fig. 3 for each grain size is illustrated in Fig. 4. Although SRS nearly remains invariant over a wide range of low strain rates, significant increase of SRS appears at high strain rate. This convincingly substantiates that SRS is not constant but varies with strain rate, especially at the high strain rate regime. The sudden change of the evolution trend of SRS with strain rate is highly related to the transformation of dominant rate-controlling mechanism from thermal activation to viscous drag. In addition, the grain size effect on the SRS is unexpectedly overturned, revealing the opposite trend at low and high strain rate regimes, which needs to be discussed in detail. The opposite grain size dependence of SRS in FCC metallic materials were not reported before. These results manifest two extraordinary messages: (1) the SRS value is not constant but varies with strain rate, and (2) grain size effects on SRS are different at low and high strain rates. To comprehensively understand these startling phenomena, it is necessary to establish a physical-based constitutive model to unveil the mechanism.
4. Model for strain rate sensitivity 4.1 Constitutive equation of flow stress Plastic deformation of metallic materials is a result of dislocations moving
9
through the crystal lattice under the rate-controlling mechanism. In the thermal activation analysis, dislocation motion is impeded by two types of obstacles: long-range and short-range barriers [33-35]. Long-range barriers are attributed to the structure of the material and cannot be surmounted by thermal activation [36, 37]. However, short-range barriers can be readily overcome by the assistance of thermal energy, which is highly sensitive to the temperature and strain rate and can reduce the applied stress needed for dislocations passing obstacles [38]. Therefore, the flow stress at a certain strain p , defined by the resistance to dislocation motion, can be decomposed into an athermal component and a thermal component according to the NNL constitutive relationship [22]:
p ,T
p
u p p ,T
p
(1)
Here u represents the athermal component of the flow stress introduced by the long-range barriers. This component of stress is a constant at a certain strain since it is insensitive to strain rate and temperature and varies only with plastic strain. Differently, is the thermal component of the flow stress, reflecting the strain rate dependent interactions between dislocations and short-range obstacles. As it denotes the rate-controlling deformation mechanism of thermal activation, the can be expressed as [18, 22]: 1/ p
MG p ,T 2 0 b
kT p 1/ q m 1 ln p G0 0
(2)
where k is the Boltzmann constant, M is the Taylor factor ( M ), G0 is the
10
reference free energy of thermal activation at 0 K. p and q are the exponent parameters with the variation of 0
p ,T
p
E T E0
u p
p ,T
p
vs
p
p
(3)
Herein, the E/E0 defines the evolution of the Young’s modulus with temperature and is strain rate independent. vs is the viscous drag component of flow stress, reflecting the dislocation drag effect [39]:
vs p 1 exp a p ,
a
M 2B mb2 u
(4)
where χ is a material constant, α represents an effective damping coefficient affecting the dislocation motion (B is the drag coefficient) [40, 41]. The Arrhenius-type equation that vs adopts reveals its insignificant effect on the flow stress at strain rate lower than ~103 s-1, while it increases sharply to become the predominant factor
11
of the flow stress when strain rate exceeds ~103 s-1 [18]. With the introduction of the viscous drag component, the physical mechanism behind fast upturn of flow stress at high strain rates is decoded essentially.
4.2 Constitutive model of SRS Among the three components of flow stress, only u is strain-rate independent, thus the parameter of SRS, m, can be denoted as follows: E T 1 E0
vs
ln 1 p ln ln p ln p E T 1 vs m* mvs p E0 ln ln p m
u
(5)
where m* and mvs are the contributions from and vs , respectively. By using Eq. (2) ~ (5), the expression of m* and mvs can be obtained as followings:
m*
E T E T ln E T 1 ln E0 ln p E0 ln p E0 1 u ln p
E T p E0 1 u
MG ln 2 0 b
1/ p
kT p 1/ q m 1 ln p G0 0 p
E T 1 kT kT p ln E0 1 u pqG0 G0 0 p
1/ q 1
1/ q kT p 1 ln p G0 0
(6) 1
a p exp a p M 2 B p M 2B p 1 vs mvs exp 2 ln p mb2 u mb u
(7)
These two equations recognize the strain rate dependence of SRS, and analyze
12
the specific contribution of thermal activation and viscous drag to SRS. In addition, previous investigation indicated that the dislocation density was proportional to the reciprocal of the grain size (d) at a given strain according to work-hardening theory [42]. Since the mobile dislocation density is a part of the total dislocation density and the scale factor is a constant for most metals [43], it is thus reasonable to postulate the following relation:
m d 1
(8)
where ξ is a material parameter. Therefore, according to Eqs. (5)–(8), the mechanistic model of grain size effect on SRS can be quantitatively established and is extended to the high strain rate range, which is significant to analytically decode the effects of strain rate and grain size on the SRS. 4.3 Model analysis and prediction Most of the parameters in the constitutive model of SRS represented by equations (5)-(8) can be obtained from literatures as shown in table 2 [15] except for ξ the material parameter links grain size to mobile dislocation density. Using these typical parameter values, the SRS model represented by equations (5)-(8) gives analytical results at low vs high strain rates for different values of ξ, as shown in Fig. 5. All the curves describing the dependence of m upon grain size d indicate that SRS increases monotonously with grain size at the high strain rate of 103 s-1, and decreases monotonously with grain size at the low strain rate of 10-3 s-1. And it is clear also from Fig.5 that this opposite dependence of SRS on grain size is not sensitive to the value
13
of ξ. This clearly shown that the SRS at high strain rates is opposite to the generally acknowledged trend that SRS increases with the reduction of grain size at low strain rate [4, 6, 44].
Extracted from the data in Fig. 4, An experimental relation of SRS with grain size at strain rates of ~10-3 s-1 and ~103 s-1 can thus be obtained, and is shown in Fig. 6 (a). The curve for ξ≈2×107 m-1 from Fig. 5—which is an analytical prediction of the SRS model—is also plotted in Fig. 6 (a). It can be seen that the data points from experiments fit fairly well with the analytical prediction for ξ≈2×107 m-1. As ξ is the material parameter links grain size to mobile dislocation density, the value it takes should be physical make sense on this respect. To estimate ξ from experimental data, note that for copper sample with grain size d=1.46μm, the total dislocation density ρ=1.2×1014m-2 [45], also it’s expected that ρm≈(0.1~0.5)ρ [46]. This yields a value of ξ=(1.75~8.76)×107 m-1, and the value ξ takes, 2×107 m-1, to make the model prediction fit the experimental results, falls just among this range. At low strain rate of 10-3 s-1, SRS increases from 0.008 to 0.019 when the average grain size decreases from 90 μm to 0.5 μm. Differently, at a high strain rate of 103 s-1, SRS presents a prominent reduction from 0.469 to 0.042 with the average grain size decreasing from 90 μm to 0.5 μm. For comparison, SRS data calculated from experimental data in the literatures [47, 48] at high strain rate for copper was also included in Fig. 6 (a), which illustrates the alike trend that SRS increases with grain size at high strain rate range. Therefore, the monotone increase of SRS with
14
grain size at this regime is an intrinsic phenomenon. Based on the Eqs. (5)-(8) of the constitutive model and using the typical values of the parameters (shown in Table 2) for copper, the grain size effect on SRS at different strain rate was predicted and displayed in Fig. 6 (b). According to the imposed strain rate, two kinds of dependence of SRS on the grain sizes were categorized. When the strain rate is below 100 s-1, the SRS increases slightly with the reduction of grain size, which is the well-recognized grain size effect on SRS [6]. When the strain rate is above 101 s-1, the value of SRS increases monotonically with grain size and this inverse grain size effect becomes progressively prominent with enhancing the strain rate.
The disparate influence of strain rate on the SRS at the low and high strain rate regimes illustrated in Fig. 4 meaningfully reminds us that it is crucial to elaborate the specific contribution of the thermal activation component ( m* ) and the viscous drag component ( mvs ) to the evolution of SRS with strain rate. By recourse to the typical parameter values in Table 2 and Eqs. (6)-(8), Fig. 7 exhibits the variation of m* and
mvs with strain rate at the five-selected grain sizes. For the thermal activation component of SRS shown in Fig. 7 (a), irrespective of grain sizes, m* exhibits very slow increase with strain rate. However, regarding the contribution of viscous drag mechanism, at low strain rates, mvs stabilized at zero, having no effect on SRS, due to the inactivation of this component, while its domination mainly functions at high strain rates, as illustrated in Fig. 7 (b). In the high rate regime, a dramatic increase in
mvs is achieved and it is strongly strain-rate dependent as well. In comparison with those in Fig. 4 that are based on experimental results, the combination of analytical 15
formulation of m* and mvs can ideally fit with the experimental curves at the low and high strain rate sections. More importantly, the analytical formulation enables the precise prediction of the dependence of SRS on strain rate for given grain sizes, while the opposite grain size effects on m* and mvs are essentially revealed. Therefore, the current constitutive model not only decodes the physical origination of the strain rate effect on the SRS, but also significantly extends our capacity of making reliable prediction about the evolution of SRS with strain rate.
It is widely recognized that the rate-controlling deformation mechanism will be transformed gradually from thermal activation to viscous drag with the increase of strain rate from quasi-static ones to very fast ones [15-17]. Therefore, the opposite grain size effects on the SRS at low and high strain rates are mainly related to the transition of the dominant mechanisms. Eqs. (4) and (6)–(8) predict that the thermal activation component and the viscous drag component of SRS are negatively and positively dependent to the grain size, respectively. Fig. 8 illustrates analytically the constitutive relationship between SRS and grain size at the strain rates of 10-3 s-1 and 103 s-1,. At the low strain rate of 10-3 s-1, since viscous drag mechanism is not activated, thermal activation plays the plenipotentiary role in plasticity, leading to a slight increase in SRS with reducing grain size. This grain size effect on the SRS is mainly attributed to the thermally activated dislocation motion [46], since reducing grain size leads to the decrease in the spacing of dislocation forests, which contribute to the enhancement of SRS with decreasing grain size. However, at high strain rate of 103 s-1, the viscous drag mechanism matters since the viscous drag component of SRS 16
generally exhibits a prominently increased value by up to an order of magnitude relative to that obtained from thermal activation component. The overall relationship between the rate-controlling mechanisms and SRS is thus governed by the viscous drag part of SRS, rendering the monotonous increase of SRS with grain size. Therefore, the mechanistic model shed light on the physical meaning about the effects of thermal activation and viscous drag mechanism on the grain size dependence of SRS at different strain rates.
5. Discussion Strain rate plays essential roles in the dislocation interactions and plasticity in materials. Based on the strain-rate-dependent triggering mechanisms, the above analytical model can excellently explain and predict the rapid increase of SRS at high strain rates. Especially, due to a crossover of dominant rate-controlling deformation mechanisms from thermal activation to viscous drag, SRS surprisingly increases with grain size at high strain rates, which overturns the well-recognized perception concerning the grain size effect on the SRS at low strain rates. Since the kinetics of dislocation–defect interactions are different at low and high strain rates, the characterisation of post-deformation microstructures is beneficial to cognize the distinct effects of thermal activation and viscous-drag.
During deformation, dislocations continuously encounter obstacles when they move through the lattice [13]. In a thermal activation process, dislocations can surmount the obstacles and continue to sweep across the glide planes, with the
17
activation from thermal agitation and applied stress field [16, 49]. For FCC pure copper, grain boundaries are the main long-range barriers to dislocation motion that cannot be overcome by thermal activation. Therefore, dislocations will be impeded and piled up at grain boundaries, resulting in a much higher dislocation density near grain boundaries than grain interior. In contrast, viscous drag, another rate-controlling mechanism that is not thermally activated and only arises at high strain rates [16], has been attributed to the dislocation damping effect from phonons and electrons that is related to the crystal structure [50]. In the viscous drag process, in lieu of long-range barriers, dislocations can be stored both at grain boundaries and in grain interior. Resultantly, homogeneous distribution of dislocation density will be fairly achieved both in grain interior and near the grain boundaries.
The distribution of dislocations or local density of dislocations can be qualitatively described by means of the local misorientation analysis, as shown in Fig. 9. Samples with average grain sizes of 3 μm, 12 μm and 90 μm were examined for the local misorientation distribution in uniformly deformed areas after deformation to failure at low and high strain rates. Only HAGBs are outlined with black line for brevity. For samples deformed at the low strain rate in which the thermal activation controlled the deformation, the misorientation is much significant at grain boundary areas than interior due to the extensive pile up of dislocations. However, for samples deformed at the high strain rate in which viscous-drag mechanism was evoked, the distribution of dislocation density was comparatively more homogeneous, evidenced by the weakened difference of misorientation intensity between grain boundary areas 18
and the grain interior. It should be admitted that the uniform elongation of the sample deformed at the high strain rate was slightly larger than that deformed at the low strain rate, as exhibited in Fig. 10(a). However, large change in crystallographic orientation adjacent to GBs and near triple junctions due to abundant dislocation pile ups, was observed as well in the samples subjected to quasi-static SPD with the strain much higher than that reached at the high strain rate shown in Fig. 10(a) [51, 52]. Therefore, the variance of the misorientation intensity distribution at the low and high strain rates should be mainly ascribed to the dominant activation of rate-controlling deformation mechanisms.
In the term of grain size effects, uniform elongation does not show much difference for samples with different grain size as displayed in Fig 10(a), much higher densities of dislocations were accumulated in the coarse-grained samples than those in fine-grained samples at both high and low strain rates. This is reflected by the increased fraction of LAGBs with grain size in Fig 10(b). Therefore, the invariable value of D - Q (see Fig 10 (a)) implies that the increase of FD-FQ with grain size is essentially relevant to the strain rate. Namely, high strain-rate deformation promotes dislocation activities and this promotion is progressively stronger with increasing grain size. This essentially signals that viscous-drag becomes increasingly influential as grain size increases, which can also be further validated by the more obvious difference of dislocation distribution between low and high strain-rate deformation in coarse-grained samples than that in fine-grained samples. Therefore, from the microstructural evolution perspective, the increase of SRS with grain size at high 19
strain rates is also reasonably comprehended.
In light of thermally activated dislocation processes, the common grain size effect on SRS at low strain rates has been extensively explored and fundamentally uncovered. Although the mechanistic model and microstructural observation provide feasible prediction and excellent explanation about the inverse grain size effect on SRS at high strain rates, which is attributed to the activation of viscous drag mechanism, the physical origination of this phenomenon is still mysterious. Since the drag mechanism is rooted in the interactions of dislocations with phonons and electrons [13], the increased scattering of phonons and electrons should prominently restrict this mechanism. It is well known that crystalline defects including dislocations and grain boundaries serve as scattering centre for phonons and electrons [53]. The increased density of grain boundaries via reducing grain size might remarkably supress the activation of the drag mechanism, as reflected in Fig. 3 and Fig. 4. Therefore, this inverse grain size effect on the SRS is physically intrinsic in FCC materials. The unexpected experimental phenomena and the mechanistic model that enables excellent prediction and wide applications fundamentally extend and enrich our perceptiveness of the dynamics and kinetics of plastic deformation.
6. Conclusions In summary, the grain sizes dependence of SRS was investigated for copper samples at low vs high strain rates, and the following conclusions can be drawn:
1. The SRS of copper increases monotonously with grain size under 20
deformation at high strain-rates of ~103 s-1, which is opposite to the general perception that SRS increases with decreasing grain size at low strain rates observed in FCC metals.
2. Analytical formulation of the constitutive model of SRS based on the NNL and MRK models was established to describe the essential dependence of SRS on grain size at different strain rate level, which reveals that the opposite dependence of SRS on grain size at low vs high strain rates is attributed to the transformation of the dominant rate-controlling deformation mechanism from thermal activation at low strain rates to viscous drag at high strain rates.
3. Comprehensive analysis shows that the thermal activation component of SRS,
m* , which is nearly strain rate independent, increases with reducing grain size; while the viscous drag component of SRS, mvs , which is enhanced significantly at high strain rates, decreases with reducing grain size. This indicates the monotonous increases of SRS with grain size at the high strain rate range is caused by viscous drag processes, which contributes to the rapid increase of flow stress.
4. Microstructural observation based on local misorientation characterization and statistics confirms that the viscous-drag dominates at high strain-rate deformation, and becomes progressively influential as grain size increases.
Acknowledgments This work was supported by National Key R&D Program of China (Grant No. 2017YFA0204403), the Natural Science Foundation of China (Grant No. 21
51520105001), and the Australian Research Council (Grant No. DE170100053 and DP150101121).
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[18] C. Gao, L. Zhang, Constitutive modelling of plasticity of fcc metals under extremely high strain rates, International Journal of Plasticity, 32 (2012) 121-133. [19] G. Regazzoni, U.F. Kocks, P.S. Follansbee, Dislocation kinetics at high strain rates, Acta Metallurgica, 35 (1987) 2865-2875. [20] A.R. Rosenfield, G.T. Hahn, Numerical description of the ambient low-temperature and high-strain rate flow and fracture behavior of plain carbon steel, ASM Trans Quart, 59 (1966) 962-980. [21] T. Suo, Y. Chen, Y. Li, C. Wang, X. Fan, Strain rate sensitivity and deformation kinetics of ECAPed aluminium over a wide range of strain rates, Materials Science and Engineering: A, 560 (2013) 545-551. [22] S. Nemat-Nasser, Y. Li, Flow stress of fcc polycrystals with application to OFHC Cu, Acta Materialia, 46 (1998) 565-577. [23] T.G. Langdon, Twenty-five years of ultrafine-grained materials: achieving exceptional properties through grain refinement, Acta Materialia, 61 (2013) 7035-7059. [24] H. Kolsky, An investigation of the mechanical properties of materials at very high rates of loading, Proceedings of the Physical Society. Section B, 62 (1949) 676. [25] G. Wang, S. Wu, L. Zuo, C. Esling, Z. Wang, G. Li, Microstructure, texture, grain boundaries in recrystallization regions in pure Cu ECAE samples, Materials Science and Engineering: A, 346 (2003) 83-90. [26] X. Molodova, G. Gottstein, M. Winning, R.J. Hellmig, Thermal stability of ECAP processed pure copper, Materials Science and Engineering: A, 460–461 (2007) 204-213. [27] E. Hall, The deformation and ageing of mild steel: III discussion of results, Proceedings of the Physical Society. Section B, 64 (1951) 747. [28] N. Petch, The cleavage strength of polycrystals, J. Iron Steel Inst., 174 (1953) 25-28. [29] Y. Wang, E. Ma, Three strategies to achieve uniform tensile deformation in a nanostructured metal, Acta Materialia, 52 (2004) 1699-1709. [30] A.S. Krausz, K. Krausz, Unified constitutive laws of plastic deformation, Elsevier, 1996. [31] T. Suo, Y. Li, F. Zhao, X. Fan, W. Guo, Compressive behavior and rate-controlling mechanisms of ultrafine grained copper over wide temperature and strain rate ranges, Mechanics of Materials, 61 (2013) 1-10. [32] R. Schwaiger, B. Moser, M. Dao, N. Chollacoop, S. Suresh, Some critical experiments on the strain-rate sensitivity of nanocrystalline nickel, Acta Materialia, 51 (2003) 5159-5172. [33] G.Z. Voyiadjis, F.H. Abed, Microstructural based models for bcc and fcc metals with temperature and strain rate dependency, Mechanics of Materials, 37 (2005) 355-378. [34] U. Kocks, H. Mecking, Physics and phenomenology of strain hardening: the FCC case, Progress in materials science, 48 (2003) 171-273. [35] N. Bonora, P.P. Milella, Constitutive modeling for ductile metals behavior incorporating strain rate, temperature and damage mechanics, International Journal of Impact Engineering, 26 (2001) 53-64. [36] F.J. Zerilli, R.W. Armstrong, Dislocation-mechanics-based constitutive relations for material dynamics calculations, Journal of Applied Physics, 61 (1987) 1816-1825. [37] S. Nemat-Nasser, J.B. Isaacs, Direct measurement of isothermal flow stress of metals at elevated temperatures and high strain rates with application to Ta and Ta W alloys, Acta Materialia, 45 (1997) 907-919.
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[38] A. Lennon, K. Ramesh, The influence of crystal structure on the dynamic behavior of materials at high temperatures, International journal of Plasticity, 20 (2004) 269-290. [39] R. Kapoor, S. Nemat-Nasser, Comparison between high and low strain-rate deformation of tantalum, Metallurgical & Materials Transactions A, 31 (2000) 815-823. [40] S. Nemat-Nasser, W.G. Guo, D.P. Kihl, Thermomechanical response of AL-6XN stainless steel over a wide range of strain rates and temperatures, Journal of the Mechanics and Physics of Solids, 49 (2001) 1823-1846. [41] W.G. Guo, S. Nemat-Nasser, Flow stress of Nitronic-50 stainless steel over a wide range of strain rates and temperatures, Mechanics of Materials, 38 (2006) 1090-1103. [42] H. Conrad, S. Feuerstein, L. Rice, Effects of grain size on the dislocation density and flow stress of niobium, Materials Science & Engineering, 2 (1967) 157-168. [43] G.Z. Voyiadjis, F.H. Abed, Effect of dislocation density evolution on the thermomechanical response of metals with different crystal structures at low and high strain rates and temperatures, Archives of Mechanics, 57 (2005) 299-343. [44] Y. Wang, E. Ma, Temperature and strain rate effects on the strength and ductility of nanostructured copper, Applied physics letters, 83 (2003) 3165-3167. [45] M.R. Staker, D.L. Holt, The dislocation cell size and dislocation density in copper deformed at temperatures between 25 and 700°C, Acta Metallurgica, 20 (1972) 569-579. [46] H. Conrad, Grain size dependence of the plastic deformation kinetics in Cu, Materials Science and Engineering: A, 341 (2003) 216-228. [47] D. Jia, K. Ramesh, E. Ma, L. Lu, K. Lu, Compressive behavior of an electrodeposited nanostructured copper at quasistatic and high strain rates, Scripta materialia, 45 (2001) 613-620. [48] W. Tong, R.J. Clifton, S. Huang, Pressure-shear impact investigation of strain rate history effects in oxygen-free high-conductivity copper, Journal of the Mechanics and Physics of Solids, 40 (1992) 1251-1294. [49] A. Kumar, F. Hauser, J. Dorn, Viscous drag on dislocations in aluminum at high strain rates, Acta Metallurgica, 16 (1968) 1189-1197. [50] W.G. Ferguson, Dislocation Damping in Aluminum at High Strain Rates, Journal of Applied Physics, 38 (1967) 1863-1869. [51] X.H. An, Q.Y. Lin, G. Sha, M.X. Huang, S.P. Ringer, Y.T. Zhu, X.Z. Liao, Microstructural evolution and phase transformation in twinning-induced plasticity steel induced by high-pressure torsion, Acta Materialia, 109 (2016) 300-313. [52] X.H. An, S.D. Wu, Z.F. Zhang, R.B. Figueiredo, N. Gao, T.G. Langdon, Evolution of microstructural homogeneity in copper processed by high-pressure torsion, Scripta Materialia, 63 (2010) 560-563. [53] L. Lu, Y. Shen, X. Chen, L. Qian, K. Lu, Ultrahigh strength and high electrical conductivity in copper, Science, 304 (2004) 422-426.
24
Fig. 1 Typical microstructures of copper samples with average grain sizes of (a) 0.5 μm, (b) 3 μm, (c) 6 μm, (d) 12 μm, and (e) 90 μm.
25
a
400
True stress (Mpa)
d=0.5μm
6μm
3μm
12μm
300
90μm 200
.
100
= 8.33×10-4 s-1 OFHC copper
0 0.0
0.1
0.2
0.3
0.4
True strain
600
True stress (Mpa)
500
b
OFHC copper
.
d=0.5μm
≈ 103 s-1
400
6μm
3μm
300
12μm
200
90μm
100
0 0.00
0.05
0.10
0.15
0.20
True strain
Fig. 2 True tensile stress–strain curves of copper samples with different grain sizes under (a) low strain-rate and (b) high strain-rate loading conditions.
26
Points: Experimental data Curves: Fitting with MRK model
Flow stress (Mpa)
500
400
d=0.5μm
300
3μm
200
6μm 100
12μm
90μm -4
-2
10
0
10
10
2
10
Strain rate (s ) -1
Fig. 3 Flow stresses at 2% true strain vs. strain rate for copper samples with different grain sizes
Strain rate sensitivity m
0.5
d=90μm 12μm
0.4
0.5μm
0.02
6μm
3μm
0.3
0.01
6μm 12μm
0.2
90μm 10
0.1
-4
10
-3
10
0.00
-2
10
-1
3μm 0.5μm
0.0 -3
10
-1
1
10
10
3
10
Strain rate (s ) -1
Fig. 4 The strain rate dependence of SRS for copper samples with different grain size obtained from the fitting curves in Fig. 3.
27
. . Dash curves: =10
3 -1
Solid curves: =10 s
-3 -1
s
-1
Strain rate sensitivity m
7
10
0.1
=
5 0.
(×
.5 (
-1
10
)
×10 7
0
10
5
2
1
=0
0.01
m
1
2
m -1)
1
10
Grain size d (μm)
5
2
10
Fig. 5 Dependence of SRS upon grian size at low vs high strain rates with different values of ξ.
28
Strain rate sensitivity m
a
Model Exp. 3 -1
10 s
-3 -1
10 s
0.1
= 2 ×107 m-1
= 2 ×107 m-1
0.01 5 -1
~10 s [48] 3
4 -1
~4×10 -~2×10 s [47] -2
2x10
-2
3x10
0
1
10
2
10
10
Grain size d (μm)
Strain rate sensitivity m
b
=104 s-1
3
-1
10 s
2
-1
10 s
0.1
1
-1
10 s
0.01
0
-1
10 s
= 2 ×107 m-1 -1
10
-3
-1
10 s 0
10
1
10
Grain size d (μm)
2
10
Fig. 6 Grain size dependence of SRS (a) at the low vs high strain rates from experiments and modelling (b) at strain rates from low to high from modelling.
29
Strain rate sensitivity m
0.04
a
m*
0.03
d=0.5μm 0.02
3μm 6μm
0.01
12μm 90μm 0.00 -3
-1
10
1
10
3
10
10
Strain rate (s )
b
0.5
Strain rate sensitivity m
-1
d=90μm
mvs
0.4
12μm 0.3
6μm
0.2
0.1
3μm
0.0
0.5μm -4
10
-3
10
-1
10
0
1
10
2
10
10
3
10
Strain rate (s ) -1
Fig. 7 The strain rate dependence of SRS with the contribution from (a) thermal activation and (b) viscous drag for copper samples with different grain sizes.
30
0.5
High strain rate: 103 s-1
Strain rate sensitivity m
0.4 0.3 0.2
Viscous drag
Overall
0.1
Thermal activation
0.0
Low strain rate: 10-3 s-1
0.02
Overall (Thermal activation) 0.01
Viscous drag
0.00 10
0
10
1
Grain size d (μm)
10
2
Fig. 8 The grain size dependence of SRS under the dominated mechanism of thermal activation and viscous drag at low and high strain rates.
31
Fig. 9 Local misorientation maps of copper sample with average grain sizes of (a) and (b) 3 μm, (c) and (d) 12 μm, (e) and (f) 90 μm after tensile deformation to failure at the strain rates of 8.33×10-4 s-1 (left column) and 0.8~1.4×103 s-1 (right column).
32
a
Uniform elongation
0.6
D
0.4
- D
0.2
0
Q
Q
1
10
2
10
10
Grain size d (μm)
Fraction of LAGBs F
0.8
b
FD FQ
0.6
0.4
0.10
FD-FQ 0.05
0.00 0 10
1
10
2
10
Grain size d (μm)
Fig. 10 (a) The variation of uniform elongation with grain size at the strain rates of 8.33×10-4 s-1 (εQ) and 0.8~1.4×103 s-1 (εD), and the difference between them; (b) Fraction of LAGBs as a function of grain size after tensile deformations at the strain rates of 8.33×10-4 s-1 (FQ) and 0.8~1.4×103 s-1 (FD), and the difference between them.
33
Table 1 The parameters used for SHPB tensile test parameter value
εT ~6.1×10-5
εR ~1.3×10-3
C0 (m/s) 5190
E (Gpa) 210
A (mm2) 314
ls (mm) 10
Table 2 Typical values of the constitutive parameters of the SRS model Constitutive parameters k T G0 p q E(T)/E0 0 p
u
χ b M B
values 1.3806×10-23 300 2.8176×10-19 [22] 1 [22] 2 [22] 0.99 [15] 4.2×106 [22] 30 [15] 249 [15] 2×10-10 [22] 3 9.06×10-11 [15]
34
Unit J/K K J / / / s-1 MPa MPa m / MPa•s
As (mm2) 7.5
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PII: DOI: Reference:
S0921-5093(18)31214-0 https://doi.org/10.1016/j.msea.2018.09.018 MSA36899
To appear in: Materials Science & Engineering A Received date: 23 May 2018 Revised date: 23 August 2018 Accepted date: 7 September 2018 Cite this article as: Z.N. Mao, X.H. An, X.Z. Liao and J.T. Wang, Opposite grain size dependence of strain rate sensitivity of copper at low vs high strain r a t e s , Materials Science & Engineering A, https://doi.org/10.1016/j.msea.2018.09.018 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Opposite grain size dependence of strain rate sensitivity of copper at low vs high strain rates Z.N. Maoa, X.H. Anb, X.Z. Liaob*, and J.T. Wanga* a
School of Materials Science and Engineering, Nanjing University of Science and
Technology, Nanjing 210094, China. b
School of Aerospace, Mechanical & Mechatronic Engineering, The University of
Sydney, Sydney, NSW 2006, Australia.
* Corres. authors. Tel: +86 25 84315493; [email protected](J.T. Wang). Tel: +61 2 93512348; [email protected] (X.Z. Liao).
Abstract The grain size dependence of the strain rate sensitivity (SRS) of copper were systematically investigated via tensile deformation at strain rates of ~10-4 s-1 and ~103 s-1. In contrast to the general perception that SRS increases with decreasing grain size at low strain rates in FCC metals, the SRS increases monotonously with grain size under deformation at high strain-rates of ~103 s-1. Analytical formulation based on the Nemat-Nasser-Li (NNL) and Modified-Rusinek-Klepaczko (MRK) models was established to reveal the essential dependence of SRS on grain size, at the change of strain rate: that the opposite dependence of SRS on grain size at low vs high strain rates can be attributed to the transformation of the dominant rate-controlling 1
deformation mechanism from thermal activation at low strain rates to viscous drag at high strain rates. It is demonstrated that the thermal activation component of SRS,
m* , which is nearly strain rate independent, increases with reducing grain size; while the viscous drag component of SRS, mvs , which is enhanced significantly at high strain rates, decreases with reducing grain size. Microstructural observation based on local misorientation characterization and statistics confirms that the viscous-drag dominates at high strain-rate deformation, and becomes progressively influential as grain size increases. This unveils the essence of viscous drag in the opposite grain size dependence of SRS at low vs high strain rates.
Keywords: Opposite grain size dependence, Strain rate sensitivity, Dynamic tension, Viscous drag, Thermal activation. 1. Introduction It is well acknowledged that the plastic flow characteristics of materials are sensitive to the rate of loading, which can be quantitatively reflected by the strain rate sensitivity (SRS). The SRS, commonly defined as 𝑚 = 𝜕 𝑙𝑛𝜎/𝜕 𝑙𝑛𝜀̇ (where 𝜎 is the flow stress and 𝜀̇ is the strain rate), is widely recognized as the essential fingerprint for the thermodynamics and kinetics of deformation processes and is used to characterize the strain rate hardening phenomenon of materials [1, 2]. Therefore, the SRS has important implications in comprehending the mechanical behavior and deformation mechanisms of metals and alloys. Although face-centered cubic (FCC) metals generally exhibit a lower SRS than those with body-centered cubic (BCC)
2
structure, extensive investigations under a wide range of strain rates from quasi-static to ~102 s-1 revealed that the SRS is enhanced by reducing the grain size of FCC metallic materials [3-7]. Especially, ultrafine-grained and nanocrystalline materials presents an increase of SRS by up to an order of magnitude relative to their coarse-grained counterparts [7]. The introduction of nanoscale twins into ultrafine grains also reach the same trait of increased rate sensitivity [3, 8-10]. Regarding the significance of length scale on the SRS, several models based on the the thermally activated dislocation processes have been proposed to uncover the physical origins of the grain size effect on SRS for FCC metallic materials [7, 8].
Previous investigations have substantiated that the strain-rate dependence of flow stress for many FCC metals and alloys originates from thermal activation [11], which governs the rate-controlling mechanism at low strain rates [12]. With increasing the strain rate, there is less time available to overcome the thermally activated barrier [13]. In fact, dislocation theory manifests that viscous drag is another essential rate-controlling deformation mechanism that gradually becomes dominant at high strain rates [14]. Because the viscous drag stress does not have any effect on the flow stress until the strain rate reaches ~103 s-1 [15, 16], viscous or quasi-viscous drag processes are triggered only at high strain rates of above ~103 s-1[17]. Rather than thermal activation assistance for dislocations overcoming the resistance from short-range barriers (forest dislocations, Peierls stress, point defects etc.) [18], the dislocation motion becomes more continuous, controlled by drag mechanisms based on its interaction with phonons and electrons of the crystals [19]. With the influence 3
of viscous drag, a rapid upturn of flow stress with strain rate is observed at strain rate above ~103 s-1 [20], indicating a fast rise in SRS at high strain rate [21]. This signals that SRS is not constant but varies with strain rates. However, a basic gap in our knowledge for many years has been the lack of a quantitative understanding based on the deformation mechanism to account for the strain rate effect on SRS. In addition, the effect of grain size on the SRS derived from the experiments at high strain rates is still lacking. Thus, it is necessary to systematically investigate the evolution of SRS with strain rate and the grain size effect when different mechanisms dominate the plastic deformation. This fundamental exploration will significantly enrich and enliven our understanding of essential deformation dynamics and kinetics of materials at low and high strain rates, and will facilitate building a physical based model to strengthen our ability of predicting the plasticity of materials.
In this work, pure copper samples with five average grain sizes were tensile tested over a wide range of strain rates (from ~10-4 s-1 to ~103 s-1) to examine the effect of grain size on the SRS at low and high strain rates. A model describing strain rate dependence of SRS was established based on the Nemat-Nasser-Li (NNL) [22] and the Modified-Rusinek-Klepaczko (MRK) [15] constitutive relationships. The influence of the two rate-controlling deformation mechanisms (thermal activation and viscous drag) to SRS was analyzed separately. Opposite grain size dependences of SRS at low and high strain rates were revealed and their physical interpretation was provided. Microstructural observation based on local misorientation analysis conducted using the electron back scatter diffraction (EBSD) technique further 4
confirmed the explanation to this abnormal phenomenon.
2. Experimental procedures As-received oxygen free high conductivity copper (99.98% Cu, 0.0095% O) bars with dimensions of 32 × 32 × 160 mm3 were annealed at 600 ºC for 1 h, forming a homogeneous microstructure consisting of equiaxed grains with an average grain size of ~ 90 μm. To refine the coarse grains, equal-channel angular pressing (ECAP), one of the most popular severe plastic deformation (SPD) techniques, was conducted for 8 passes via route Bc, i.e., the billet was rotated 90° clockwise between two consecutive passes [23], using a die having an internal channel angle of 90° and an outer arc curvature of 20° at the ambient temperature with a pressing speed of 2 mm/s. After ECAP, ultrafine-grained samples were successfully obtained and they were kept in liquid nitrogen to suppress any potential recovery of the microstructure. In order to achieve homogeneous microstructures with different grain sizes, annealing treatments were conducted at 280ºC for 50s , 400ºC for 1min, 280ºC for 1h and 600ºC for 2h to achieve microstructures with grain sizes of about 3μm, 6μm, 12μm and 90μm respectively.
Tensile samples with a dog-bone shape were cut from the middle of the ECAP bars by wire electro-discharge machining. The gauge section geometry of the specimens was 10 × 2 × 3 mm3 and 12 × 2 × 4.8 mm3, for quasi-static and high-strain-rate tensile loading, respectively. The strain rate applied in this work varies from ~10-4/s to ~103/s. Low-strain-rate (~10-3/s) tensile tests were performed in
5
an Auto Graph AGS-X testing machine (Shimadzu Corporation). The split Hopkinson pressure bar (SHPB) was employed for high-strain-rate (~103/s) tensile experiments. According to the one dimensional elastic stress wave theory, the stress, strain and strain rate of tested specimen can be calculated as [24]: A As
s E
T ,
s
2C0 ls
t
0
R
d ,
s
2C0 R ls
where εT and εR are the transmitted and reflected strain pulses which can be measured by the strain gages stuck on the input and output bars, respectively; C0, E, and A denote the longitudinal elastic wave velocity, Young’s modulus and cross-sectional area of the loading bars; ls and As are the length and cross-sectional area of the specimen, respectively. The parameters used for SHPB tensile tests are presented in Table 1. At least three tensile tests were performed on each data point to substantiate the reproducibility of the stress-strain curves.
Samples used for microstructural characterization were mechanically ground on 4000-grit SiC papers and then polished using 1μm diamond paste. Subsequent electropolishing was conducted at 20 ºC in an electrolyte of 25% orthophosphoric acid, 25% ethanol and 50% distilled water at a voltage of 7 V and duration of 1 min. The EBSD investigations were undertaken in an FEI Quanta 250 field emission scanning electron microscope (SEM) equipped with an Oxford Instruments Aztec 2.0 EBSD system with operating voltage of 12 kV and a step size of 80 nm. The microstructural features were recorded using the Channel 5 software (HKL
6
Technology). Misorientation angles smaller than 2° were not considered due to the noise effect. High-angle grain boundaries (HAGBs) and low-angle grain boundaries (LAGBs) in this study are defined as grain boundaries with misorientation angles >15° and in the range of 2–15°, respectively.
3. Experimental results Typical microstructures of copper samples with 5 different grain sizes are exhibited in the EBSD maps in Fig. 1. The black and white lines in the EBSD maps denote HAGBs and LAGBs, respectively. After ECAP processing, although there was still a high density of LAGBs in the sample, the fingerprint of microstructures was uniformly distributed equiaxed ultrafine grains with the average grain size of ~0.5 μm in most areas, as detected in Fig. 1 (a). In contrast, few LAGBs were found in the four annealed samples that presented the average grain sizes of ~ 3 μm, ~ 6 μm, ~ 12 μm, and ~ 90 μm, as shown in Fig. 1 (b-e), respectively. Additionally, the orientation maps of both the as ECAP and recrystallized samples revealed that the microstructures have no strong texture, and therefore an insignificant effect of texture on the mechanical behavior is expected, which is in accordance with the results from [25, 26].
Figure 2 shows the tensile true stress–strain curves of samples with different grain sizes conducted at the strain rates of (a) 8.33 × 10-4 s-1 and (b) ~103 s-1. Only the first stress wave response during high strain-rate deformation is presented. As expected from the Hall-Petch effect [27, 28], remarkable increase in yield strength was achieved with reducing grain size under both low and high strain-rate
7
deformation. For low strain-rate deformation, large strain hardening capability was observed in all samples except for the sample with the average grain size of 0.5 μm due to its saturated work hardening with a high dislocation density caused by ECAP processing [29]. The ductility of the 4 recrystallized samples was slightly reduced with decreasing the grain size. Irrespective of the grain sizes, deformation at high strain rates led to much higher flow stresses than those at low strain rates due to the suppression of relevant thermally activated mechanisms at high strain rates [30, 31].
To evaluate the significance of strain rate and grain sizes during deformation, the flow stress of each sample at 2% true strain was plotted as a function of strain rate, as indicated by the scattered datum points in Fig. 3. Based on experimental data, MRK constitutive model [15] was applied for data fitting separately for samples with different grain size, and the results were also shown in Fig. 3 using solid lines. The distinct effects of strain rate on flow stress are clearly demonstrated for deformation at low and high strain rates. At the low strain rate range, the flow stress exhibits relatively weak sensitivity to the strain rate and then a slight enhancement in flow stress with increasing strain rate was identified. Similar to previous investigations [3, 4, 32], the dependence of the flow stress on strain rate, i.e. SRS increases with decreasing grain size. In contrast, a remarkable rapid upturn of flow stress with strain rate was observed at strain rates > ~103 s-1, which is attributed to the activation of the viscous drag mechanism. The fast increase of flow stress with strain rate not only implies the strong rate-dependence of SRS, but also shows apparent grain size effect. Different from those at the low strain rate range, the slope of the data in the 8
ultrafine-grained sample is lower than those for the coarse-grained samples, which surprisingly signals the inverse grain size effects on the SRS at the high strain rate regime.
The strain rate dependence of SRS calculated from the fitting curves in Fig. 3 for each grain size is illustrated in Fig. 4. Although SRS nearly remains invariant over a wide range of low strain rates, significant increase of SRS appears at high strain rate. This convincingly substantiates that SRS is not constant but varies with strain rate, especially at the high strain rate regime. The sudden change of the evolution trend of SRS with strain rate is highly related to the transformation of dominant rate-controlling mechanism from thermal activation to viscous drag. In addition, the grain size effect on the SRS is unexpectedly overturned, revealing the opposite trend at low and high strain rate regimes, which needs to be discussed in detail. The opposite grain size dependence of SRS in FCC metallic materials were not reported before. These results manifest two extraordinary messages: (1) the SRS value is not constant but varies with strain rate, and (2) grain size effects on SRS are different at low and high strain rates. To comprehensively understand these startling phenomena, it is necessary to establish a physical-based constitutive model to unveil the mechanism.
4. Model for strain rate sensitivity 4.1 Constitutive equation of flow stress Plastic deformation of metallic materials is a result of dislocations moving
9
through the crystal lattice under the rate-controlling mechanism. In the thermal activation analysis, dislocation motion is impeded by two types of obstacles: long-range and short-range barriers [33-35]. Long-range barriers are attributed to the structure of the material and cannot be surmounted by thermal activation [36, 37]. However, short-range barriers can be readily overcome by the assistance of thermal energy, which is highly sensitive to the temperature and strain rate and can reduce the applied stress needed for dislocations passing obstacles [38]. Therefore, the flow stress at a certain strain p , defined by the resistance to dislocation motion, can be decomposed into an athermal component and a thermal component according to the NNL constitutive relationship [22]:
p ,T
p
u p p ,T
p
(1)
Here u represents the athermal component of the flow stress introduced by the long-range barriers. This component of stress is a constant at a certain strain since it is insensitive to strain rate and temperature and varies only with plastic strain. Differently, is the thermal component of the flow stress, reflecting the strain rate dependent interactions between dislocations and short-range obstacles. As it denotes the rate-controlling deformation mechanism of thermal activation, the can be expressed as [18, 22]: 1/ p
MG p ,T 2 0 b
kT p 1/ q m 1 ln p G0 0
(2)
where k is the Boltzmann constant, M is the Taylor factor ( M ), G0 is the
10
reference free energy of thermal activation at 0 K. p and q are the exponent parameters with the variation of 0
p ,T
p
E T E0
u p
p ,T
p
vs
p
p
(3)
Herein, the E
vs p 1 exp a p ,
a
M 2B mb2 u
(4)
where χ is a material constant, α represents an effective damping coefficient affecting the dislocation motion (B is the drag coefficient) [40, 41]. The Arrhenius-type equation that vs adopts reveals its insignificant effect on the flow stress at strain rate lower than ~103 s-1, while it increases sharply to become the predominant factor
11
of the flow stress when strain rate exceeds ~103 s-1 [18]. With the introduction of the viscous drag component, the physical mechanism behind fast upturn of flow stress at high strain rates is decoded essentially.
4.2 Constitutive model of SRS Among the three components of flow stress, only u is strain-rate independent, thus the parameter of SRS, m, can be denoted as follows: E T 1 E0
vs
ln 1 p ln ln p ln p E T 1 vs m* mvs p E0 ln ln p m
u
(5)
where m* and mvs are the contributions from and vs , respectively. By using Eq. (2) ~ (5), the expression of m* and mvs can be obtained as followings:
m*
E T E T ln E T 1 ln E0 ln p E0 ln p E0 1 u ln p
E T p E0 1 u
MG ln 2 0 b
1/ p
kT p 1/ q m 1 ln p G0 0 p
E T 1 kT kT p ln E0 1 u pqG0 G0 0 p
1/ q 1
1/ q kT p 1 ln p G0 0
(6) 1
a p exp a p M 2 B p M 2B p 1 vs mvs exp 2 ln p mb2 u mb u
(7)
These two equations recognize the strain rate dependence of SRS, and analyze
12
the specific contribution of thermal activation and viscous drag to SRS. In addition, previous investigation indicated that the dislocation density was proportional to the reciprocal of the grain size (d) at a given strain according to work-hardening theory [42]. Since the mobile dislocation density is a part of the total dislocation density and the scale factor is a constant for most metals [43], it is thus reasonable to postulate the following relation:
m d 1
(8)
where ξ is a material parameter. Therefore, according to Eqs. (5)–(8), the mechanistic model of grain size effect on SRS can be quantitatively established and is extended to the high strain rate range, which is significant to analytically decode the effects of strain rate and grain size on the SRS. 4.3 Model analysis and prediction Most of the parameters in the constitutive model of SRS represented by equations (5)-(8) can be obtained from literatures as shown in table 2 [15] except for ξ the material parameter links grain size to mobile dislocation density. Using these typical parameter values, the SRS model represented by equations (5)-(8) gives analytical results at low vs high strain rates for different values of ξ, as shown in Fig. 5. All the curves describing the dependence of m upon grain size d indicate that SRS increases monotonously with grain size at the high strain rate of 103 s-1, and decreases monotonously with grain size at the low strain rate of 10-3 s-1. And it is clear also from Fig.5 that this opposite dependence of SRS on grain size is not sensitive to the value
13
of ξ. This clearly shown that the SRS at high strain rates is opposite to the generally acknowledged trend that SRS increases with the reduction of grain size at low strain rate [4, 6, 44].
Extracted from the data in Fig. 4, An experimental relation of SRS with grain size at strain rates of ~10-3 s-1 and ~103 s-1 can thus be obtained, and is shown in Fig. 6 (a). The curve for ξ≈2×107 m-1 from Fig. 5—which is an analytical prediction of the SRS model—is also plotted in Fig. 6 (a). It can be seen that the data points from experiments fit fairly well with the analytical prediction for ξ≈2×107 m-1. As ξ is the material parameter links grain size to mobile dislocation density, the value it takes should be physical make sense on this respect. To estimate ξ from experimental data, note that for copper sample with grain size d=1.46μm, the total dislocation density ρ=1.2×1014m-2 [45], also it’s expected that ρm≈(0.1~0.5)ρ [46]. This yields a value of ξ=(1.75~8.76)×107 m-1, and the value ξ takes, 2×107 m-1, to make the model prediction fit the experimental results, falls just among this range. At low strain rate of 10-3 s-1, SRS increases from 0.008 to 0.019 when the average grain size decreases from 90 μm to 0.5 μm. Differently, at a high strain rate of 103 s-1, SRS presents a prominent reduction from 0.469 to 0.042 with the average grain size decreasing from 90 μm to 0.5 μm. For comparison, SRS data calculated from experimental data in the literatures [47, 48] at high strain rate for copper was also included in Fig. 6 (a), which illustrates the alike trend that SRS increases with grain size at high strain rate range. Therefore, the monotone increase of SRS with
14
grain size at this regime is an intrinsic phenomenon. Based on the Eqs. (5)-(8) of the constitutive model and using the typical values of the parameters (shown in Table 2) for copper, the grain size effect on SRS at different strain rate was predicted and displayed in Fig. 6 (b). According to the imposed strain rate, two kinds of dependence of SRS on the grain sizes were categorized. When the strain rate is below 100 s-1, the SRS increases slightly with the reduction of grain size, which is the well-recognized grain size effect on SRS [6]. When the strain rate is above 101 s-1, the value of SRS increases monotonically with grain size and this inverse grain size effect becomes progressively prominent with enhancing the strain rate.
The disparate influence of strain rate on the SRS at the low and high strain rate regimes illustrated in Fig. 4 meaningfully reminds us that it is crucial to elaborate the specific contribution of the thermal activation component ( m* ) and the viscous drag component ( mvs ) to the evolution of SRS with strain rate. By recourse to the typical parameter values in Table 2 and Eqs. (6)-(8), Fig. 7 exhibits the variation of m* and
mvs with strain rate at the five-selected grain sizes. For the thermal activation component of SRS shown in Fig. 7 (a), irrespective of grain sizes, m* exhibits very slow increase with strain rate. However, regarding the contribution of viscous drag mechanism, at low strain rates, mvs stabilized at zero, having no effect on SRS, due to the inactivation of this component, while its domination mainly functions at high strain rates, as illustrated in Fig. 7 (b). In the high rate regime, a dramatic increase in
mvs is achieved and it is strongly strain-rate dependent as well. In comparison with those in Fig. 4 that are based on experimental results, the combination of analytical 15
formulation of m* and mvs can ideally fit with the experimental curves at the low and high strain rate sections. More importantly, the analytical formulation enables the precise prediction of the dependence of SRS on strain rate for given grain sizes, while the opposite grain size effects on m* and mvs are essentially revealed. Therefore, the current constitutive model not only decodes the physical origination of the strain rate effect on the SRS, but also significantly extends our capacity of making reliable prediction about the evolution of SRS with strain rate.
It is widely recognized that the rate-controlling deformation mechanism will be transformed gradually from thermal activation to viscous drag with the increase of strain rate from quasi-static ones to very fast ones [15-17]. Therefore, the opposite grain size effects on the SRS at low and high strain rates are mainly related to the transition of the dominant mechanisms. Eqs. (4) and (6)–(8) predict that the thermal activation component and the viscous drag component of SRS are negatively and positively dependent to the grain size, respectively. Fig. 8 illustrates analytically the constitutive relationship between SRS and grain size at the strain rates of 10-3 s-1 and 103 s-1,. At the low strain rate of 10-3 s-1, since viscous drag mechanism is not activated, thermal activation plays the plenipotentiary role in plasticity, leading to a slight increase in SRS with reducing grain size. This grain size effect on the SRS is mainly attributed to the thermally activated dislocation motion [46], since reducing grain size leads to the decrease in the spacing of dislocation forests, which contribute to the enhancement of SRS with decreasing grain size. However, at high strain rate of 103 s-1, the viscous drag mechanism matters since the viscous drag component of SRS 16
generally exhibits a prominently increased value by up to an order of magnitude relative to that obtained from thermal activation component. The overall relationship between the rate-controlling mechanisms and SRS is thus governed by the viscous drag part of SRS, rendering the monotonous increase of SRS with grain size. Therefore, the mechanistic model shed light on the physical meaning about the effects of thermal activation and viscous drag mechanism on the grain size dependence of SRS at different strain rates.
5. Discussion Strain rate plays essential roles in the dislocation interactions and plasticity in materials. Based on the strain-rate-dependent triggering mechanisms, the above analytical model can excellently explain and predict the rapid increase of SRS at high strain rates. Especially, due to a crossover of dominant rate-controlling deformation mechanisms from thermal activation to viscous drag, SRS surprisingly increases with grain size at high strain rates, which overturns the well-recognized perception concerning the grain size effect on the SRS at low strain rates. Since the kinetics of dislocation–defect interactions are different at low and high strain rates, the characterisation of post-deformation microstructures is beneficial to cognize the distinct effects of thermal activation and viscous-drag.
During deformation, dislocations continuously encounter obstacles when they move through the lattice [13]. In a thermal activation process, dislocations can surmount the obstacles and continue to sweep across the glide planes, with the
17
activation from thermal agitation and applied stress field [16, 49]. For FCC pure copper, grain boundaries are the main long-range barriers to dislocation motion that cannot be overcome by thermal activation. Therefore, dislocations will be impeded and piled up at grain boundaries, resulting in a much higher dislocation density near grain boundaries than grain interior. In contrast, viscous drag, another rate-controlling mechanism that is not thermally activated and only arises at high strain rates [16], has been attributed to the dislocation damping effect from phonons and electrons that is related to the crystal structure [50]. In the viscous drag process, in lieu of long-range barriers, dislocations can be stored both at grain boundaries and in grain interior. Resultantly, homogeneous distribution of dislocation density will be fairly achieved both in grain interior and near the grain boundaries.
The distribution of dislocations or local density of dislocations can be qualitatively described by means of the local misorientation analysis, as shown in Fig. 9. Samples with average grain sizes of 3 μm, 12 μm and 90 μm were examined for the local misorientation distribution in uniformly deformed areas after deformation to failure at low and high strain rates. Only HAGBs are outlined with black line for brevity. For samples deformed at the low strain rate in which the thermal activation controlled the deformation, the misorientation is much significant at grain boundary areas than interior due to the extensive pile up of dislocations. However, for samples deformed at the high strain rate in which viscous-drag mechanism was evoked, the distribution of dislocation density was comparatively more homogeneous, evidenced by the weakened difference of misorientation intensity between grain boundary areas 18
and the grain interior. It should be admitted that the uniform elongation of the sample deformed at the high strain rate was slightly larger than that deformed at the low strain rate, as exhibited in Fig. 10(a). However, large change in crystallographic orientation adjacent to GBs and near triple junctions due to abundant dislocation pile ups, was observed as well in the samples subjected to quasi-static SPD with the strain much higher than that reached at the high strain rate shown in Fig. 10(a) [51, 52]. Therefore, the variance of the misorientation intensity distribution at the low and high strain rates should be mainly ascribed to the dominant activation of rate-controlling deformation mechanisms.
In the term of grain size effects, uniform elongation does not show much difference for samples with different grain size as displayed in Fig 10(a), much higher densities of dislocations were accumulated in the coarse-grained samples than those in fine-grained samples at both high and low strain rates. This is reflected by the increased fraction of LAGBs with grain size in Fig 10(b). Therefore, the invariable value of D - Q (see Fig 10 (a)) implies that the increase of FD-FQ with grain size is essentially relevant to the strain rate. Namely, high strain-rate deformation promotes dislocation activities and this promotion is progressively stronger with increasing grain size. This essentially signals that viscous-drag becomes increasingly influential as grain size increases, which can also be further validated by the more obvious difference of dislocation distribution between low and high strain-rate deformation in coarse-grained samples than that in fine-grained samples. Therefore, from the microstructural evolution perspective, the increase of SRS with grain size at high 19
strain rates is also reasonably comprehended.
In light of thermally activated dislocation processes, the common grain size effect on SRS at low strain rates has been extensively explored and fundamentally uncovered. Although the mechanistic model and microstructural observation provide feasible prediction and excellent explanation about the inverse grain size effect on SRS at high strain rates, which is attributed to the activation of viscous drag mechanism, the physical origination of this phenomenon is still mysterious. Since the drag mechanism is rooted in the interactions of dislocations with phonons and electrons [13], the increased scattering of phonons and electrons should prominently restrict this mechanism. It is well known that crystalline defects including dislocations and grain boundaries serve as scattering centre for phonons and electrons [53]. The increased density of grain boundaries via reducing grain size might remarkably supress the activation of the drag mechanism, as reflected in Fig. 3 and Fig. 4. Therefore, this inverse grain size effect on the SRS is physically intrinsic in FCC materials. The unexpected experimental phenomena and the mechanistic model that enables excellent prediction and wide applications fundamentally extend and enrich our perceptiveness of the dynamics and kinetics of plastic deformation.
6. Conclusions In summary, the grain sizes dependence of SRS was investigated for copper samples at low vs high strain rates, and the following conclusions can be drawn:
1. The SRS of copper increases monotonously with grain size under 20
deformation at high strain-rates of ~103 s-1, which is opposite to the general perception that SRS increases with decreasing grain size at low strain rates observed in FCC metals.
2. Analytical formulation of the constitutive model of SRS based on the NNL and MRK models was established to describe the essential dependence of SRS on grain size at different strain rate level, which reveals that the opposite dependence of SRS on grain size at low vs high strain rates is attributed to the transformation of the dominant rate-controlling deformation mechanism from thermal activation at low strain rates to viscous drag at high strain rates.
3. Comprehensive analysis shows that the thermal activation component of SRS,
m* , which is nearly strain rate independent, increases with reducing grain size; while the viscous drag component of SRS, mvs , which is enhanced significantly at high strain rates, decreases with reducing grain size. This indicates the monotonous increases of SRS with grain size at the high strain rate range is caused by viscous drag processes, which contributes to the rapid increase of flow stress.
4. Microstructural observation based on local misorientation characterization and statistics confirms that the viscous-drag dominates at high strain-rate deformation, and becomes progressively influential as grain size increases.
Acknowledgments This work was supported by National Key R&D Program of China (Grant No. 2017YFA0204403), the Natural Science Foundation of China (Grant No. 21
51520105001), and the Australian Research Council (Grant No. DE170100053 and DP150101121).
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24
Fig. 1 Typical microstructures of copper samples with average grain sizes of (a) 0.5 μm, (b) 3 μm, (c) 6 μm, (d) 12 μm, and (e) 90 μm.
25
a
400
True stress (Mpa)
d=0.5μm
6μm
3μm
12μm
300
90μm 200
.
100
= 8.33×10-4 s-1 OFHC copper
0 0.0
0.1
0.2
0.3
0.4
True strain
600
True stress (Mpa)
500
b
OFHC copper
.
d=0.5μm
≈ 103 s-1
400
6μm
3μm
300
12μm
200
90μm
100
0 0.00
0.05
0.10
0.15
0.20
True strain
Fig. 2 True tensile stress–strain curves of copper samples with different grain sizes under (a) low strain-rate and (b) high strain-rate loading conditions.
26
Points: Experimental data Curves: Fitting with MRK model
Flow stress (Mpa)
500
400
d=0.5μm
300
3μm
200
6μm 100
12μm
90μm -4
-2
10
0
10
10
2
10
Strain rate (s ) -1
Fig. 3 Flow stresses at 2% true strain vs. strain rate for copper samples with different grain sizes
Strain rate sensitivity m
0.5
d=90μm 12μm
0.4
0.5μm
0.02
6μm
3μm
0.3
0.01
6μm 12μm
0.2
90μm 10
0.1
-4
10
-3
10
0.00
-2
10
-1
3μm 0.5μm
0.0 -3
10
-1
1
10
10
3
10
Strain rate (s ) -1
Fig. 4 The strain rate dependence of SRS for copper samples with different grain size obtained from the fitting curves in Fig. 3.
27
. . Dash curves: =10
3 -1
Solid curves: =10 s
-3 -1
s
-1
Strain rate sensitivity m
7
10
0.1
=
5 0.
(×
.5 (
-1
10
)
×10 7
0
10
5
2
1
=0
0.01
m
1
2
m -1)
1
10
Grain size d (μm)
5
2
10
Fig. 5 Dependence of SRS upon grian size at low vs high strain rates with different values of ξ.
28
Strain rate sensitivity m
a
Model Exp. 3 -1
10 s
-3 -1
10 s
0.1
= 2 ×107 m-1
= 2 ×107 m-1
0.01 5 -1
~10 s [48] 3
4 -1
~4×10 -~2×10 s [47] -2
2x10
-2
3x10
0
1
10
2
10
10
Grain size d (μm)
Strain rate sensitivity m
b
=104 s-1
3
-1
10 s
2
-1
10 s
0.1
1
-1
10 s
0.01
0
-1
10 s
= 2 ×107 m-1 -1
10
-3
-1
10 s 0
10
1
10
Grain size d (μm)
2
10
Fig. 6 Grain size dependence of SRS (a) at the low vs high strain rates from experiments and modelling (b) at strain rates from low to high from modelling.
29
Strain rate sensitivity m
0.04
a
m*
0.03
d=0.5μm 0.02
3μm 6μm
0.01
12μm 90μm 0.00 -3
-1
10
1
10
3
10
10
Strain rate (s )
b
0.5
Strain rate sensitivity m
-1
d=90μm
mvs
0.4
12μm 0.3
6μm
0.2
0.1
3μm
0.0
0.5μm -4
10
-3
10
-1
10
0
1
10
2
10
10
3
10
Strain rate (s ) -1
Fig. 7 The strain rate dependence of SRS with the contribution from (a) thermal activation and (b) viscous drag for copper samples with different grain sizes.
30
0.5
High strain rate: 103 s-1
Strain rate sensitivity m
0.4 0.3 0.2
Viscous drag
Overall
0.1
Thermal activation
0.0
Low strain rate: 10-3 s-1
0.02
Overall (Thermal activation) 0.01
Viscous drag
0.00 10
0
10
1
Grain size d (μm)
10
2
Fig. 8 The grain size dependence of SRS under the dominated mechanism of thermal activation and viscous drag at low and high strain rates.
31
Fig. 9 Local misorientation maps of copper sample with average grain sizes of (a) and (b) 3 μm, (c) and (d) 12 μm, (e) and (f) 90 μm after tensile deformation to failure at the strain rates of 8.33×10-4 s-1 (left column) and 0.8~1.4×103 s-1 (right column).
32
a
Uniform elongation
0.6
D
0.4
- D
0.2
0
Q
Q
1
10
2
10
10
Grain size d (μm)
Fraction of LAGBs F
0.8
b
FD FQ
0.6
0.4
0.10
FD-FQ 0.05
0.00 0 10
1
10
2
10
Grain size d (μm)
Fig. 10 (a) The variation of uniform elongation with grain size at the strain rates of 8.33×10-4 s-1 (εQ) and 0.8~1.4×103 s-1 (εD), and the difference between them; (b) Fraction of LAGBs as a function of grain size after tensile deformations at the strain rates of 8.33×10-4 s-1 (FQ) and 0.8~1.4×103 s-1 (FD), and the difference between them.
33
Table 1 The parameters used for SHPB tensile test parameter value
εT ~6.1×10-5
εR ~1.3×10-3
C0 (m/s) 5190
E (Gpa) 210
A (mm2) 314
ls (mm) 10
Table 2 Typical values of the constitutive parameters of the SRS model Constitutive parameters k T G0 p q E(T)/E0 0 p
u
χ b M B
values 1.3806×10-23 300 2.8176×10-19 [22] 1 [22] 2 [22] 0.99 [15] 4.2×106 [22] 30 [15] 249 [15] 2×10-10 [22] 3 9.06×10-11 [15]
34
Unit J/K K J / / / s-1 MPa MPa m / MPa•s
As (mm2) 7.5