Accepted Manuscript Title: An elastic-viscoplastic model of deformation in nanocrystalline metals based on coupled mechanisms in grain boundaries and grain interiors Authors: Yujie Wei, Huajian Gao PII: DOI: Reference:

S0921-5093(07)01067-2 doi:10.1016/j.msea.2007.05.054 MSA 22971

To appear in:

Materials Science and Engineering A

Received date: Revised date: Accepted date:

20-3-2007 14-5-2007 15-5-2007

Please cite this article as: Y. Wei, H. Gao, An elastic-viscoplastic model of deformation in nanocrystalline metals based on coupled mechanisms in grain boundaries and grain interiors, Materials Science & Engineering A (2007), doi:10.1016/j.msea.2007.05.054 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 proof before it is published in its final 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.

3. Manuscript

Yujie Wei and Huajian Gao

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An elastic-viscoplastic model of deformation in nanocrystalline metals based on coupled mechanisms in grain boundaries and grain interiors

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Division of Engineering, Brown University, Box D, Providence RI 02912, USA

Abstract

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An elastic-viscoplastic model based on coupled deformation mechanisms of grainboundary sliding, grain-boundary diffusion, grain-interior diffusion and grain-interior plasticity is developed to investigate the mechanical behavior of nanocrystalline metals. Application of the model to nanocrystalline copper shows that overall plastic deformation from the coupled deformation mechanisms depends on both grain size and strain rate. A sharp strain rate sensitivity transition occurs from about 0.01 to 1, as strain rates change to below 10−6 /s in nanocrystalline Copper. Grain-boundary sliding and diffusion dominate inelastic deformation when the strain-rate sensitivity approaches 1. For grain sizes larger than a critical value about 15nm, increasing slip resistance in grain interiors causes the strength to increase as the grain size is reduced. However, further reduction in grain sizes to below 15nm results in softening in strength due to enhanced grain-boundary sliding and diffusion.

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Key words: Nanocrystalline metals; Strain-rate sensitivity; Grain-boundary diffusion; Grain-boundary sliding; dislocation emission-absorbtion

Introduction

Nanocrystalline (nc) materials are known to exhibit a number of distinct mechanical properties: high strength, enhanced strain-rate sensitivity and softening in strength for grain sizes less than about 15nm. These properties are ∗ Corresponding author. Email addresses: yujie [email protected] (Yujie Wei), huajian [email protected] (Huajian Gao).

Preprint submitted to Elsevier Science

14 May 2007 Page 1 of 23

presumably associated with competing deformation mechanisms operating in both grain boundaries and grain interiors. For nc materials, the following deformation mechanisms are relatively well accepted [1–12]:

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• There is a strong interplay between dislocation-based deformation mechanisms in the crystalline grain interiors and inelastic deformation mechanisms in the grain-boundary regions. • Grain boundaries act as both sources and sinks for dislocations. Graininterior plastic deformation is accommodated by dislocations emitted from grain boundaries.

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To develop predictive methods for analysis of structures made of nc materials, we need physically sound models capable of capturing different deformation mechanisms active in both grain interiors and grain boundaries. Much of our current theoretical understanding of deformation mechanisms in nc materials has been based on large-scale molecular dynamics (MD) studies published in the past few years, as documented by Kumar et al. [4], Wolf et al. [9], and Van Swygenhoven et al. [13]. Although the MD studies have provided valuable insights into the atomic level details of deformation mechanisms in nc materials, these methods are at present not suitable for simulating deformation under conditions comparable to those found in experiments.

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Recently, some investigators have developed mesoscopic models based on continuum finite element method (FEM) simulations of inelastic deformation and failure response of nc materials in which grain boundaries and grain interiors are treated as regions of different constitutive behaviors [12, 14–18]. These mesoscopic models have provided additional insights which can not be easily obtained from atomistic simulations or experiments, although only a limited number of grains can be taken into account. Constitutive models suitable for understanding deformation behavior of nc materials at a macroscopic level are currently lacking. Zhu et al. [17] have made a first effort to develop a macroscopic model for nc materials accounting for grain-interior plasticity and grainboundary sliding. Mercier et al. [19] have extended the ’Mantle-Core’ type of model used by Fu et al. [16] and developed an elastic-viscoplastic model for nc materials. Only grain-interior plasticity and grain-boundary diffusion is considered in their formulation. The model introduced here is an extension of that by Zhu et al. [17]. To be specific, (1) we develop a more rigorous dislocationemission rate in grain boundaries in nc materials; (2) we take into account the deformation by GB diffusion in nc materials; (3) a simple elastic-viscoplastic model is formulated, which allow us to study the competing deformation mechanisms in nc materials, i.e., grain-boundary sliding, grain-boundary diffusion and grain-interior plasticity. 2 Page 2 of 23

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Deformation mechanisms in grain boundaries and grain interiors

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In nc materials, there are four major inelastic deformation mechanisms, two of which, occurring in grain boundaries, are grain-boundary diffusion and grainboundary sliding; the other two, facilitated by grain boundaries, are dislocations emitted in grain boundaries and lattice diffusion. Macroscopic deformation in nc materials should reflect a combinations of all four mechanisms. Differing from approaches in [16, 19], where a volume fraction parameter is needed to quantify the dependence of grain-boundary ‘phase’ on grain sizes d, our formulation will take into account the dependence of competing deformation mechanisms on grain size directly. There is no need to introduce an extra parameter to quantify volume fraction of different “phases”. For materials with typical grain size d subjected to uniaxial stress σ, with the assumption that each grain is subjected to the same stress, the strain rates associated with each deformation mechanism can be evaluated as follows.

˙gbd =

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(i) As suggested by Gleiter [1], nc materials should deform via Coble creep even at relatively low temperatures. Also, MD simulations by Keblinski et al. [20] and Yamakov et al. [21] showed grain-boundary diffusion controlled creep in nc materials. The macroscopic strain rate due to collective grain-boundary diffusion has been given by Coble [22] as 47σΩa δDgb exp(−Qgb /RT ) kB T d3

(1)

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where Ωa is the atomic volume, D is the GB diffusivity and δ is the thickness of a layer in which interface diffusion is supposed to take place, Qgb is the activation energy for grain-boundary diffusion in unit of J/mol, R and kB are the gas constants and Boltzman constants respectively, and T is temperature. (ii) Grain-boundary sliding may be divided into thermally activated sliding and athermal sliding. The athermal grain-boundary sliding is a process involving relative sliding of two grain boundaries when the resolved shear stress overcomes a threshold resistance. Such direct sliding differs from the thermally activated sliding in that it is essentially rate-independent. In our analysis, only the thermally assisted grain-boundary sliding is taken into consideration. one of which is a thermally activated, rate-dependent process for which Conrad and Narayan [23] have developed a grain-boundary shearing model that gives a macroscopic shearing rate


˙gbs =






Ωa τe 6bνd ∆F exp − sinh d kB T kB T



(2)

for a grain √ with size d. Here, ∆F is the Helmholtz free energy of activation, τe ≈ σ/ 3 is the effective shear stress, νd is the Debye frequency of lattice 3 Page 3 of 23

vibration, and b is the Burgers vector. There can also be a threshold stress τth below which no sliding is active. In this case, the macroscopic shearing rate could be rewritten as


with H(τe − τth ) =

 0 1



if τe < τth , if τe ≥ τth .

H(τe − τth )

(3)

t

˙gbs



cri p



Ωa (τe − τth ) 6bνd ∆F sinh = exp − d kB T kB T

(iii) Inelastic deformation due to lattice diffusion in grain interiors have been given by Nabarro [24] and Herring [25] as 1 10σΩa DL exp(−QL /RT ) kB T d2

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˙gid =

(4)

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where DL denotes lattice diffusivity and QgL is the activation energy for grain-boundary diffusion in units of J/mol. (vi) Plastic deformation in grain interiors of nc materials is not controlled by gliding of preexisted dislocations. For nc materials, plastic deformation in grain interiors is achieved by dislocation nucleation-absorbtion facilitated by grain boundaries. Complete or partial dislocations may be emitted in a boundary of a nano-sized grain, sweep across the whole grain, and be absorbed in an opposite boundary. Such a dislocation nucleation-absorbtion process in grain interiors could be formulated in a macroscopic fashion as follows. For one emission on a slip system α, the strain increment of ∆γ (α) and the strain rate of γ˙ (α) can be written as Zhu et al. [17]

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1 ∆γ (α) = b(α) ⊗ m(α) , d ∆γ (α) γ˙ (α) = , ∆τ

(5) (6)

where b(α) is the Burgers vector, m(α) is the normal of the slip plane, and ∆τ is a characteristic time for such an emission, which consists of an incubation time ∆tincubation for a nucleation event and a travelling time ∆ttravel for an emitted dislocation to sweep the whole grain [26], i.e. ∆τ = ∆ttravel + ∆tincubation

(7)

Assuming dislocations move at a velocity close to the shear wave speed cs , 1

Though we have included the influence of lattice diffusion in our model, it turns out that this term is negligible for deformation in nc material at room temperature. Indeed, molecular dynamics simulation by Yamakov et al. [21] showed that lattice diffusion becomes important only at an averaged grain size less than 6nm.

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the travelling time for a dislocation to sweep across the grain is d . cs

∆ttravel ≈

(8)

1 νem

.

(9)

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∆tincubation =

t

The incubation time ∆tincubation depends on the frequency of thermal vibration. Assume dislocation emission frequency at a typical site in grain boundary is νem ,

Usually, ∆tincubation  ∆ttravel and hence ∆τ ≈ ∆tincubation . We can then write the total plastic strain rate as a summation of all potential sites in a grain boundary along all possible slip systems, N 

1 νem × n(d) b(α) ⊗ m(α) d α=1

d2 b , Ω

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γ˙ p =

and n(d) = β ∗

(10)



an

where n(d) is the potential sites for dislocation emission in a grain boundary, Ω ≈ b3 and β ∗ is a geometrical factor on the order of 1. The activation rate νem of the inelastic process, the emission of dislocations, can be given approximately as,

dM

G+ =νl exp − kB T

νem



,

(11)

where G+ is the activation energy for dislocation emission and νl a cluster frequency which can be related to Debye frequency νd crudely as [27]

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b νl = νd , l

(12)

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l being the space between obstacles. Let G∗ be the activation energy when there is no applied stress; since the applied stress will bias thermal activities, the activation energy can be written as G+ = G∗ − τ (α) vs where vs denotes the shear activation volume [28]. Only forward dislocation emissions are considered. Inserting the formula for G+ into Equ. (11) yields


νem = νl exp −

G∗ kB T



exp


α  τ v s

kB T

(13)

We can assume that grain size d ∼ l in nc materials. Substituting Equ. (12) and Equ. (13) into Equ. (10), we have γ˙ p =

N 

1 νR × n(d) b(α) ⊗ m(α) d α=1


G∗ = β νd exp − kB T ∗

 N

α=1

exp

(14)
α  τ v s

kB T

b(α) ⊗ m(α) . b

(15)

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Let τ f = kBvsT define a thermal resistance to dislocation movement, corresponding to the stress required to activate dislocation emission in grain boundaries. For emissions of complete dislocations in a grain boundary, τ f could be approximated as [29], τf ≈

Gb . d

(16)



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G = G0 1 − 0.54

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We are aware that such critical shear resistance may be impractical for grains with size of d > 100 nm, where dislocation nucleation in grain inte√ riors may be active. In that circumstance, a dependence of τ = τ0 + k/ d should be used, with τ0 being a size independent slip resistance and k is a material constant. The shear modulus is taken to be temperature-dependent as [30] T − 300 TM

(17)

an

where TM is the melting temperature and G0 is the shear modulus at T=300 K. Macroscopically, for the one-dimensional case, the collective strain rate along the loading axis contributed by slip rates from all slip systems can be simplified to be ˙gip



dM








∆G∗ σd = β0 νd exp − exp √ , kB T 3Gb

(18)

3

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where a simple relation √ between macroscopic tensile strength σ and slip α α resistance τ of τ ≈ σ/ 3 is taken; β0 is a coefficient on the order of 1. Note that information for actual crystallography in crystals is discarded. Such a treatment is reasonable for bulk nanostructured materials without strong initial texture. Otherwise, β0 should be adjusted correspondingly to account for the fact that plastic deformation may be dominated by several specific slip systems.

Elastic-viscoplastic Model

For stress-strain relation in an isotropic elastic-viscoplastic model, the stress increment with time is typically expressed as σ˙ = E(ε˙ − ε˙p )

(19)

for an applied strain rate ε˙ and a material with Young’s modulus E. Using Equ. (1), Equ. (3), Equ. (4) and Equ. (18), the macroscopic plastic strain rate ε˙p can be written as a collective effect of the microscopic deformation 6 Page 6 of 23

mechanisms, ε˙p = ε˙ gbd + ε˙ gbs + ε˙ gid + ε˙ gip , where



∞ ˙∗ p(x)kx3 dx ε˙ ∗ = 0 ∞ = 3 0 p(x)kx dx



(20)

˙∗ p(x)x3 dx , < x3 >

(21)



us



x2 x p(x) = 2 exp − 2 d0 2d0

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with * denoting any mechanism. The integral represents a volume average of one type of plastic strain rate by assuming the volume of a grain with diameter x has a volume of kx3 , k being a constant. For simplicity in integration, the grain size distribution p(x) is assumed to follow a Rayleigh distribution in Equ. (22) instead of the well-known log-normal distribution, (22)

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With Equ. (21) and Equ. (22), we can write the averaged plastic strains as 47σΩa δDgb exp(−Qgb /RT ) kB T < x3 > 
 12bd20 Ωa (τe − τth ) ∆F ˙εgbs = νd exp − H(τe − τth ) sinh < x3 > kB T kB T 10σΩa < x > DL exp(−QL /RT ) ε˙ gid = kB T < x3 > ∗

β0 νd exp − ∆G 1 5α α3 kB T + ε˙ gip = { < x3 > d20 2β 4β 2

   

3 α4 3α2 π + + 2 1 + erf(α β) exp βα2 } + 2 2β 8β β

(23a) (23b) (23c) (23d)

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ε˙ gbd =

Ac ce

where α, β, < x >, < x3 > and erf(x) are σd2 1 α = √ 0 , β = 2 , < x >= 2d0 3Gb 2 erf(x) = √ π

 x 0



π d0 ; , 2

3



< x >= 3

π 3 d , and 2 0

2

e−t dt, hence erf(∞) = 1.

Using Equ. (19), Equ. (20) and Equ. (23), we can calculate, for any given applied strain rate ε, ˙ the stress-strain relationship of nc materials with a known grain-size distribution, i.e. a known d0 in Equ. (22) for the case of Rayleigh distribution. In the next section we will apply the model to investigate the deformation of nc Cu for different grain sizes and different strain rates. 7 Page 7 of 23

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Application to nanocrystalline Cu

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To illustrate the behavior of the coupled deformation model described in Equ. (19) and Equ. (23), we specify the model parameters for Cu, including physical constants {kB , R, vd }, material constants {E, G0 , b, Ωa }, material parameters {δDgb , Qgb , DL , QL , ∆F, ∆G∗ , τth }, as well as parameter d0 for grainsize distribution. The temperature is fixed at T = 300K. Material parameters {δd Dgb , Qgb , DL , QL } are obtained from [30]. For grain-boundary sliding, we assume τth = 0 and ∆F , the activation energy, is taken to be the same as that for grain boundary diffusion, i.e. ∆F ≈ Qgb . The activation energy ∆G∗ for dislocation emission is assumed to be higher than Qgb and lower than QL . Considering that emission of dislocations will inevitably induce vacancies in a grain-boundary region, we let ∆G∗ ≈ Evacancy . The vacancy-formation energy in Cu is taken to be 1.19 ev [31]. Note that material parameters associated with diffusion are taken from those known for coarse-grained materials, assuming their changes are small when grain sizes change from hundreds of microns to tens of nanometers. Such an assumption can be improved if more accurate data become available. Detailed information about these parameters are listed in Table 1.

kB = 1.38×10−23 J/K

R = 8.31 J/K mol

νd = 6.25×1012 /s

E = 135 GPa

G0 = 42 GPa

b = 0.256nm

Ωa = 1.18×10−29 m3

δDgb =5.0×10−15 m3 /s

Qgb = 104 kJ/mol

∆F ≈ Qgb

∆G∗ = 115 kJ/mol

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QL = 197 kJ/mol TM =1356 K DL =2.0×10−5 m2 /s Table 1 Physical constants and material parameters used for nanocrystalline Cu.

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We first investigated relative fractions of plastic-strain contributed by different mechanisms for a given grain-size distribution. Plastic strain contributed by grain-boundary diffusion, for an average grain size of 30nm, shows a monotonic increase with increasing strain rates, as shown in Fig. 1(a). Similar trends have been observed for plastic strain due to grain-boundary sliding, Fig. 1(b). In contrast to contributions from grain-boundary mechanisms, the strain due to grain -interior plasticity increases rapidly with increasing total strain at high strain rates. At low strain rates, however, the strain due to grain-interior plasticity changes less rapidly with total strain and, as a result, at high strain rates and large strains (> 2%), its relative contribution drops significantly, Fig. 1(c). These results suggest that plastic strains contributed by grain-boundary sliding and diffusion tend to dominate at low strain rates while dislocation processes dominate at high strain rates. Fig. 2 shows the stress-strain relationships for different grain sizes and strain rates. Rate-sensitivity is observed to enhance 8 Page 8 of 23

when strain rate or grain size decreases. If the applied strain rate is fixed at 3 × 10−5 /s, the plastic strain contributed by grain interior is found to decrease with decreasing grain size. For average grain sizes of 10nm or below, there is a sharp drop for strain contributed by grain-interior plasticity, indicating that grain-boundary sliding and diffusion become the dominant mechanisms at the current strain rate, Fig. 3.

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Fig. 4 depicts the changing of strain rate sensitivity m, defined as m = ∂lnσ/∂lnε, ˙ as a function of the applied strain rate at different grain sizes. In our calculation, the strain-rate sensitivity is measured at a total strain of 10% where flow stress seems to have reached a plateau. This analysis shows that strain rate sensitivity in nanocrystalline materials depends not only on grain size (the microstructure of materials) but also on the applied strain rate. The rate sensitivity rises rapidly as the applied strain rate is reduced. At relatively high strain rates, the rate sensitivity increases slowly with decreasing grain size. As we reduce the applied strain rate, the rate sensitivity rises quickly with decreasing grain size. For strain rates around 10−6 /s or below, the rate sensitivity approaches m = 1. Such trends match qualitatively with experimental observations, as seen in Fig. 5 and Fig. 6. We note that a similar trend is presented by Mercier et al. [19] without considering deformation by grain-boundary sliding. Since both GB diffusion and viscous GB sliding will result in a strain-rate sensitivity of m = 1, the similar trends for m will be obtained when deformation shifts from grain-interior plasticity to grain-boundary plasticity even sliding is not taken into account. However, we believe that one have to allow GB sliding and diffusion simultaneously for a compatible deformation field in nc materials. Deformation by grain-boundary sliding in nc materials can not be neglected.

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At the strain rate of 3 × 10−5 /s, the stress strain relationship for different average grain sizes are shown in Fig. 7. The yield or ultimate strengths increase with decreasing grain sizes from 100nm to about 15nm, correlating with the dependence of slip resistance on grain size shown in Equ. (16). As the grain size is further reduced from 15nm to 10nm and 7nm, there is a softening of strength due to the influence of grain-boundary sliding and grain-boundary diffusion.

Discussion

We have shown that the influence of grain sizes and strain rates on stress strain behaviors of nanocrystalline materials can be understood from the competition among different deformation mechanisms in grain boundaries and grain interiors. Brief discussions for a few other aspects of deformation like the origin of strain-rate sensitivity, the influence of rate-sensitivity on ductility and 9 Page 9 of 23

the influence of pressure on diffusion are given below.

5.1 Origin of strain-rate sensitivity

Ωa

α0 Ωa d/b

for vacancy-assisted diffusion, for dislocation emission-absorbtion,

dM

v∗ =

 ∼

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Strain-rate sensitivity in materials is broadly associated with the activation volume v ∗ , defined as the rate of decrease of activation enthalpy with respect to flow stress at fixed temperature, as suggested by Conrad [32] to quantify the rate-controlling mechanisms in the plastic deformation of engineering metals and alloys, √ 3kB T m= . (24) σv ∗ The smallest activation volume, in the case of vacancy-assisted diffusion in grain boundaries, is approximately equal to atomistic volume Ωa , since only one or a few atoms might be involved. Activation volume associated with dislocation emissions in boundaries may depend on the grain size. The least number of activated atoms during a dislocation emission process, only counting atoms along the dislocation line, should be proportional to the grain size d, i.e., the activation volume in these two case may be given as, (25)

Ac ce

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where α0 is a geometrical factor. As the grain size decreases, the volume of grain-boundary regions increases, which will enhance macroscopically observed rate-sensitivity if vacancy-assisted diffusion is dominant in boundaries. According to Equ. (25), the reduced grain size also reduces the activation volume during dislocation emission. Asaro and Suresh [10] and Wang et al. [33] investigated the correlation between the activation volumes and grain sizes for nc Cu and Ni, and found that the activation volumes are in the range of 3b3 ∼ 100b3 for truly nc metals. Indeed, the activation volume decreases monotonically with decreasing grain sizes, and rate-sensitivity enhances while the grain size decreases, as seen in Fig. 5. Bcc nc materials differ from fcc and hcp nc materials in that most bcc nc materials show a significant decrease in m while the grain size becomes smaller, as seen in the collective experimental data in Fig. 8. Meyers et al. [11] argued that the activation volume is already low in the conventional polycrystalline bcc metals. However, such an argument could not explain why rate-sensitivity drops as the grain size decreases. Jia et al. [34] have developed a constitutive model for bcc iron to explain the reduction in strain-rate sensitivity in bcc nano-grained metals. Their model did not explain the influence of grainboundary deformation on the observed reduction in m as the grain size is reduced. A decreasing rate sensitivity with decreasing grain size also seems 10 Page 10 of 23

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to contradict the trend predicted by Coble creep. In our understanding, the grain-boundary mobility in bcc metals is expected to be much lower compared to fcc metals. The energy barrier for diffusion is usually higher in bcc metals than that in fcc and hcp metals. Due to the high energy barrier for diffusion in bcc nc metals, contribution of rate-controlled grain-boundary deformation to the overall plastic strain can be very small at room temperature. On the other hand, the rate-controlled plastic deformation in grain interiors by dislocation glide becomes more difficult as the grain size is reduced. In this case, the inelastic deformation could be dominated by direct grain-boundary sliding, which is essentially rate-independent, hence the observed decreasing rate sensitivity with decreasing grain size.

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5.2 Rate sensitivity and ductility

1 ∂σ + m − 1. σ ∂

dM

f (x) =

an

Rate sensitivity not only has significant influence on stress, but also affects ductility in nc materials. High rate-sensitivity is believed to help retard necking and sustain homogeneous deformation in nc materials, as suggested by the Hart’s instability criterion [35]: (26)

pte

If f > 0, no necking occurs; otherwise, deformation will localize. More precisely, due to the fact that nanocrystalline material may also show significant pressure dependence in yielding and failure [e.g. 36–41], the instability criterion could be reformulated as f=

1 ∂σ ∂p + +m−1 σ ∂ ∂σ

(27)

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where p is hydrostatic pressure, being positive during compression and negative versus vice. The term ∂p/∂σ is usually greater than zero, suggesting that hydrostatic tensile stress tends to degrade the materials and may result in premature strain localization. The pressure dependence can be sensitive to the fabrication process. Nc materials produced by powder-consolidation, for example, seem to be more vulnerable to hydrostatic tensile stress since grain boundaries in those materials may be far from equilibrium. Following Equ. (27), we have at least two strategies to increase ductility in nc materials. One method is to tailor the structure of grain boundaries, a process known as grain-boundary engineering. Alternatively, we could design special microstructures to enhance strain-hardening. Well controlled nc materials with bi-modal (or multi-modal) grain-size distribution seem to sustain more uniform elongation [46, 47] without much loss in strength. Some progresses have indeed been made in synthesizing nc materials with desired microstructures and properties. For example, nc Cu with dominantly high angle grain boundary, therefore high 11 Page 11 of 23

diffusivity and rate sensitivity, has been reported by Jiang et al. [43]; polycrystalline Cu with nanoscale twins [44, 45] has been synthesized and shown to exhibit high strength, enhanced rate sensitivity and significantly improved ductility.

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5.3 Pressure and diffusivity

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Rate sensitivity is not only associated with microstructures and applied strain rates, but also dependent on the applied pressure when diffusion becomes the primary rate-controlled deformation mechanism in a material. During diffusion-controlled plastic deformation, the pressure can influence the rate of diffusion through increasing energy barriers for an atom to jump from one site to another. Significant annihilation of vacancies can also be induced by the pressure which could increases the vacancy-formation energy without changing the configuration entropy. For a given pressure p, an extra vacancy-formation energy of pVf is required [30], δGf = ∆Gf 0 + pVf ,

(28)

dM

where ∆Gf 0 is the vacancy-formation energy at zero pressure and ∆Gf the energy at pressure p. The diffusivity in metals under pressure p is then modified through the change in vacancy concentration nv since




(29)

pte

∆Gf 0 + pVf . nv = exp − kB T

Following the kinetic theory of self-diffusion by vacancy [48], the diffusivity can be written as (30) D = α2 a2 nv Γf ,

Ac ce

where α2 is a geometric constant associated with crystal structure, a is the lattice parameter and Γf is a trying frequency. The equation for diffusivity in Equ. (30) should be applicable to both grain-boundary diffusion and graininterior diffusion, although parameters may change significantly in these two cases. Following Equ. (30), we can draw a conclusion that rate effects should be more significant in tensile tests than in compressive tests in nc materials, if the primary plastic deformation is controlled by vacancy assisted diffusion.

6

Conclusion

We have formulated an elastic-viscoplastic model of deformation in nc metals, taking into account competition among different deformation mechanisms in 12 Page 12 of 23

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grain boundaries and grain interiors, including grain-boundary sliding, grainboundary diffusion, grain-interior diffusion and grain-interior plasticity. The model incorporates the known deformation mechanisms and provides a qualitative understanding on the influenced of grain size and applied strain rate on the deformation behavior of nc materials. Application of the model to nc Cu shows that the relative contribution of plastic deformation by different mechanisms strongly depends on both grain size and the applied strain rate. At a strain rate around 10−6 /s or below, grain boundary deformation dominates and strain-rate sensitivity approaches 1. Above a grain size around 15nm, the strength increases due to increasing slip resistance in grain interiors. Further reduction in grain size may result in softening due to enhanced grain-boundary sliding and diffusion.

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The model formulated here represents only a first effort towards a constitutive theory for this important class of materials. Much work remains to be done to improve our understanding of the deformation mechanisms and their coupling in nc materials. For instance, in fcc nc materials, there exists a critical grain size dc ≈ (2/3)(Gb2 /Γ), where Γ is the stacking fault energy, G the nominal shear modulus, and b the Burgers vector. For grain sizes above dc , plastic deformation in the grain interior occurs by the emission of complete dislocations from grain boundaries; below dc , deformation occurs by partial dislocations [17, 29, 51]. In our model, only complete dislocations are considered for grain-interior plasticity, which may explain the overestimated yield strength for materials with grain sizes close to or below dc .

Ac ce

We have also assumed that all grain boundaries are involved in diffusion with the same diffusivity. The actual topology of a nanostructured material is not taken into account. In reality, diffusivity and sliding resistance in grain boundaries may heavily depend on boundary structures such as lattice misorientations and boundary thickness. Diffusivity in some boundaries could be several order of magnitude faster than those in others. Analogous to percolation, atoms could diffuse easily along paths of high diffusivity but may be blocked in slow diffusion boundaries. The differences in grain-boundary diffusivity may lead to formation of crack-like diffusion wedges [49, 50] and consequently induce stress concentrations that could result in cavitation and decohesion in the material. In order to develop truly predictive models of deformation and failure in nc materials, much work remains to be done in areas such as heterogeneous grain-boundary diffusivity, grain-boundary decohesion and dislocation emission-absorbtion in grain boundaries. 13 Page 13 of 23

Acknowledgement

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The research support at Brown University of the MRSEC Program of the National Science Foundation, under Award DMR-0520651, is gratefully acknowledged.

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References

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[1] Gleiter H, Prog. Mater. Sci. 33(1989)223-315. [2] Schiotz J, Tolla FDD, Jacobsen KW, Nature 391(1998)561. [3] Weertman JR, 2002. In Nanostructured materials, Koch C, editor. NY: Noyes; 2002. p. 393 [4] Kumar KS, Van Swygenhoven H, Suresh S, Acta Mater. 51(2003)5743. [5] Yip S, Nature Materials 3(2004)11. [6] Van Vliet KJ, Tsikata S, Suresh S. Appl Phys Lett 83(2003)1441. [7] Yamakov V, Wolf D, Phillpot SR, Mukherjee AK, Gleiter H, Nature Materials 3(2004)43. [8] Sansoz F, Molinari JF, Scripta Mater. 50(2004)1283. [9] Wolf D, Yamakov V, hillpot SR, Mukherjee A, Gleiter H, Acta Mater. 53(2005)1. [10] Asaro RJ, Suresh S, Acta Mater. 53(2005)3369. [11] Meyers MA, Mishra A, Benson DJ, Progress Mat. Sci. 51(2006)427. [12] Wei YJ, Su C, Anand L, Acta Mater. 54(2006)3177. [13] Van Swygenhoven H, Derlet PM, Froseth AG, Acta Mater. 54(2006)1975. [14] Schwiager R, Moser B, Dao M, Chollacoop N, Suresh S, Acta Mater. 51(2003)5159. [15] Wei YJ, Anand L, J of the Mechanics and Physics of Solids 52(2004)2587. [16] Fu HH, Benson DJ, Meyers MA, Acta Mater. 52(2004)4413. [17] Zhu B, Asaro RJ, Krysl P, Bailey R, Acta Mater. 53(2005)4825. [18] Warner DH, Sansoz F, Molinari JF, Int J of Plasticity 22(2006)754. [19] Mercier S, Molinari A, Estrin Y, J. Mater. Sci. 42(2007)1455. [20] Keblinski P, Wolf D, Gleiter H, Interface Sciences 6(1998)205. [21] Yamakov V, Wolf D, Phillpot SR, Gleiter H, Acta Mater. 50(2002)61. [22] Coble RL, J. of Applied Physics 34(1963)1679. [23] Conrad H, Narayan J, Scripta Mater. 42(2000)1025. [24] Nabarro FRN. 1948. Report of a conference on the strength of solids, pp. 75, Physical society, London. [25] Herring C, J. of Applied Physics 21(1950)437. [26] Meyers MF, Dynamic Behavior of Materials, 1994, pp. 323-381, JohnWiley & Sons, New York. [27] Kocks UF, Argon AS, Ashby MF, Prog. Mater. Sci. 19(1975)1. [28] Argon AS, in: Physical Metallurgy, 4th, eds. Cahn RW and Haasen P. 1996, p1891. [29] Asaro RJ, Krysl P, Kad B, Philos. Mag. Lett. 83(2003)733. [30] Frost H, Ashby MF, Deformation Mechanism Maps: the plasticity and creep of metals and ceramic 1982, Pergamon Press, Oxford. [31] Hehenkamp Th, Berger W, Kluin J-E, Ludecke Ch, Wolff J, Phys. Rev. B 45(1992)1998. [32] Conrad H. In: Zackey VF, editor. High strength materials; 1965. [33] Wang YM, Hamza AV, Ma E, Appl. Phys. Lett. 86(2005)241917. [34] Jia D, Ramesh KT, Ma E, Acta Mater. 51(2003)3495. 15

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Sci. Eng. A 358(2003)266. [67] Wei Q, Jiao T, Ramesh KT, Ma E, Scripta Mater. 50(2004)359.

17 Page 17 of 23

0

1e−6/s

−5

10

1e−4/s

−10

t

1e−2/s 1/s

10

1e+2/s 10

−5

−4

−3

10

10 Total Strain

−2

10

0

10

an

1e−4/s 1e−2/s −5

1/s 1e+2/s

−10

10

0.02

0.04 0.06 Total Strain

0.08

0.1

pte

0

dM

Strain Fraction of GB Sliding

1e−6/s

10

−1

10

us

(a)

cri p

Strain Fraction of GB Diffusion

10

(b)

0

Ac ce

Strain Fraction of GI Plasticity

10

−5

10

1e−7/s 1e−6/s 1e−5/s 1e−4/s

1e−2/s 1/s

−10

10

1e+2/s

−15

10

10

−5

−4

10

−3

10 Total Strain

−2

10

−1

10

(c) Fig. 1. Fractions of plastic strain due to different mechanisms versus the total strain for samples with an average grain size of 30nm at different strain rates. (a) Plastic strain due to grain-boundary (GB) diffusion; (b) Plastic strain due to grain-boundary sliding; (c) plastic strain by grain-interior (GI) dislocations.

18 Page 18 of 23

1/s

6000

6000 1e−4/s

4000

0.05 Strain

an

1e−6/s

0 0

0.1

4000

0.1

(b)

2000

1/s

1/s

1500

1e−2/s 2000

pte

1e−4/s

1e−6/s

1000

Ac ce 0.05 Strain

0.1

(c)

Stress (MPa)

3000 Stress (MPa)

1e−6/s

0.05 Strain

dM

(a)

0 0

1e−4/s

2000

2000 0 0

1e−2/s

us

1e−2/s

Stress (MPa)

Stress (MPa)

8000

4000

t

8000 1/s

cri p

10000

1e−2/s 1e−4/s

1000 1e−6/s 500

0 0

0.05 Strain

0.1

(d)

Fig. 2. Stress-strain curves for different strain rates and grain sizes: (a) 10nm. (b) 20nm. (c) 50nm. and (d) 100nm

19 Page 19 of 23

0

100 70 50 30 20

−1

10

t

10nm

−2

10

0

0.02

0.04 0.06 Total Strain

0.08

0

30 50 70

−4

10

−6

100nm

10

−8

10

−10

10

0.02

0.04 0.06 Total Strain

0.08

0.1

0.08

0.1

pte

0

an

10 20

−2

dM

Strain Fraction of GB Diffusion

10 10

0.1

us

(a)

cri p

Strain Fraction of GI Plasticity

10

(b)

0

Ac ce

Strain Fraction of GB Sliding

10

−2

10

10 20 30 50 70

100nm

−4

10

−6

10

0

0.02

0.04 0.06 Total Strain

(c) Fig. 3. Fractions of plastic strain due to different mechanisms versus the total strain for different average grain size at a strain rate of 3 × 10−5 /s.(a) plastic strain by grain-interior dislocations; (b) Plastic strain due to grain-boundary diffusion; (c) Plastic strain due to grain-boundary sliding.

20 Page 20 of 23

1

20

t

0.6

cri p

Rate sensitivity m

0.8

10nm 0.4

100

70 50

0

−5

10

30 0

10 Strain rate (1/s)

us

0.2

dM

an

Fig. 4. The rate sensitivity index m versus the applied strain rate for different grain sizes.

0.15

pte

0.2

Ac ce

Strain rate sensitivity: m

0.25

0.1

0.05

0 1 10

2

10

3

10 Grain size (nm)

4

10

10

5

Fig. 5. Compiled data of strain rate sensitivity m versus grain size for fcc Copper with grain size ranging from hundreds of microns to a few nanometers. ( [52]; + [53];
[54]; o[55]; x [56]; [45]; [42]; ✸[57]; [58]; [59]; *[60];[43]).

21 Page 21 of 23

0.8

cri p

0.6

t

Cheng et al. 2005 Gray et al. 1997 Lu et al. 2001 Valiev et al. 2002 Jiang et al. 2006 Wei et al. 2004

0.4 0.2 0

−5

us

Strain−rate sensitivity: m

1

0

10 Strain rate (1/s)

an

10

10

5

2000

Ac ce

Stress (MPa)

2500

pte

3000

dM

Fig. 6. Compiled data of strain rate sensitivity m versus applied strain rates for fcc Copper.

15 10 20

7

30

1500

50

1000

70

500

0 0

0.01

100nm 0.02 0.03 Strain

0.04

0.05

Fig. 7. Stress-strain curves for different grain sizes at a fixed strain rate of 3×10−5 /s.

22 Page 22 of 23

t cri p us an

0.05 0.04 0.03

dM

Strain rate sensitivity: m

0.06

0.02

0 1 10

pte

0.01 2

10

3

10 Grain size (nm)

4

10

10

5

Ac ce

Fig. 8. Compiled data of strain rate sensitivity m versus grain size in bcc Iron with average grain size ranging from hundreds of microns to nanosized grains. (♦ [61];  [34];  [62]; o [63];
[64];  [60]; +[66]; x[67] )

23 Page 23 of 23