Materials Letters 65 (2011) 3391–3395

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Materials Letters

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A constitutive model of nanocrystalline metals based on competing grain boundaryand grain interior deformation mechanisms

Ercan Gürses, Tamer El Sayed ⁎Computational Solid Mechanics Laboratory (CSML), Division of Physical Sciences and Engineering, King Abdullah University of Science and Technology (KAUST), Saudi Arabia

⁎ Corresponding author. Tel.: +966 28082985.E-mail address: tamer.elsayed@kaust.edu.sa (T. El Sa

0167-577X/$ – see front matter © 2011 Elsevier B.V. Adoi:10.1016/j.matlet.2011.07.039

a b s t r a c t

a r t i c l e i n f o

Article history:Received 8 February 2011Accepted 14 July 2011Available online 23 July 2011

Keywords:Nanocrystalline materialsDiffusionGrain boundariesSimulation and modeling

In this work, a viscoplastic constitutive model for nanocrystalline metals is presented. The model is based oncompeting grain boundary and grain interior deformation mechanisms. In particular, inelastic deformationscaused by grain boundary diffusion, grain boundary sliding and dislocation activities are considered. Effects ofpressure on the grain boundary diffusion and sliding mechanisms are taken into account. Furthermore, theinfluence of grain size distribution on macroscopic response is studied. The model is shown to capture thefundamental mechanical characteristics of nanocrystalline metals. These include grain size dependence of thestrength, i.e., both the traditional and the inverse Hall–Petch effects, the tension–compression asymmetry andthe enhanced rate sensitivity.

yed).

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© 2011 Elsevier B.V. All rights reserved.

1. Introduction

Nanocrystalline (nc) materials, because of their distinct features, e.g.,high strength and fatigue resistance, low ductility, pronounced ratedependence, tension–compression (T/C) asymmetry and susceptibility toplastic instability, have become the subject of intense research over thepast twodecades [1,2]. Owing to a largevolume fractionof grainboundary(GB) atoms, the deformationmechanisms in nc-metals are different fromtraditional coarse grained polycrystals. Indeed, there are several inelasticdeformationmechanisms, e.g.,GB-diffusion,GB-sliding, dislocations, grainrotation, grain growth and GB-migration, that can be simultaneouslyoperative in nc-metals. In thiswork, a viscoplastic constitutivemodel thatis based on competing GB and grain interior deformation mechanisms ispresented. The model extends the work byWei and Gao [3] with respectto four aspects. These are considerations of (i) the pressure dependenceof the GB-diffusivity, (ii) the normal stress dependence of the GB-sliding,(iii) both the partial and full dislocation emissions from GBs and finally(iv) amoregeneral lognormal grain sizedistributions insteadof aRayleighdistribution. As a result of first two points, the model proposed in thiswork is able to demonstrate the T/C-asymmetry in an agreement withexperimental observations [4–9] and molecular dynamics (MD) simula-tions [10–13]. The distinction of the partial or full dislocation emission isachieved by adapting the works [14,15]. The Rayleigh distributionemployed in the original model [3] is described by a single parameterwhich is themean grain size. The spread of the grain size distribution ina Rayleigh distribution automatically increases as the mean grain sizeincreases. The final aspect, i.e., the use of a lognormal grain sizedistribution, allows to study the effects of the mean grain size and thevariance separately.

2. Model description

Three different types of deformation mechanisms are assumed to beoperative for the grain sizes considered. These are theGB-diffusion (Coblecreep [16]), GB-sliding [17], and the grain interior dislocation mediatedplasticity. Thefirst twomicro-mechanisms areGB-based,whereas the lastone is related with the grain interior. It has been shown in [3] that theNabarro–Herring creep [18] through lattice diffusion does not playsignificant role in comparison to other mechanisms. Therefore, in whatfollows the effect of lattice diffusion is neglected. The inelastic strain ratedue to GB-diffusion has been given by the Coble creep [3,16].

ε̇gbd d;σ ; θð Þ = 47σΩa

kbθδDgb θ; pð Þ

d3; ð1Þ

where Ωa is the atomic volume, δ is the GB-thickness, Dgb(θ,p) is thetemperature and pressure-dependent GB-diffusivity, d is the grain size,σ is the stress, θ is the temperature, p is the pressure, kb and R are the gasconstant and the Boltzmann constant, respectively. The GB-diffusivity istemperature dependent through a traditional Arrhenius type relation.Furthermore, experimental studies [19–22] andMD simulations [23,24]have shown that the GB-diffusivity is sensitive to the pressure.

To this end, the GB-diffusivity is given by

Dgb θ;pð Þ = D0;gb exp −Qgb−pΔV

Rθ

� �; ð2Þ

where Qgb is the activation energy for the GB-diffusion, D0, gb is a pre-exponential multiplier and ΔV is the activation volume. Note that p isdefined as negative for compression and positive for tension, i.e.,p=σ. Consequently, the GB-diffusivity (Eq. (2)) and the deformationcaused by GB-diffusion (Eq. (1)) decreases as the pressure increases.

Table 1Physical constants and material parameters of the model for nc-Cu.

kb=1.38×10−23J K−1 R=8.31J K−1mol−1 Ωa=1.18×10−29m3 δD0, gb=5×10−15m3s−1

Qgb=104×103J mol−1 b=0.256×10−9m νd=6.25×1012s−1 ΔF=104×103J mol−1

λ=0.2 [−] ΔV=0.6Ωa β0=1 [−] ΔG*=115×103J mol−1

G=42×109Pa Γ=0.004 E=135×109Pa

3392 E. Gürses, T. El Sayed / Materials Letters 65 (2011) 3391–3395

The inelastic strain rate caused by the collective response of GB-sliding was given by Conrad and Narayan [17] as

ε̇gbs d;σ ; θð Þ = 6bνd

dsinh

Ωaτe σð Þkbθ

� �exp

−ΔFkbθ

� �; ð3Þ

where νd is the Debye frequency of lattice vibration, b is the Burgersvector, ΔF is the Helmholtz free energy of activation and τe is theeffective shear stress. Recent MD simulations [25,26,11] havedemonstrated that the GB-plasticity is affected by normal stressesacting on GBs. Furthermore, Jerusalem and Radovitzky [27] havefound through finite element simulations that GB-sliding is preventedwhen nc-metals are subjected to high levels of pressure. Therefore,different from [17], we assume that the effective shear stress dependson the normal traction acting on the GB through

τe = σ + λσnð Þ=ffiffiffi3

p; ð4Þ

where σn=sign(σ)σ is the normal traction and λ is a materialparameter. As a result of Eq. (4), the GB-sliding is greater in tensionwhen compared to compression.

The plastic strain due to collective behavior of dislocations havebeen derived in [3] for the one-dimensional case

ε̇gip d;σ ; θð Þ = β0νd sign σð Þ exp −ΔG⁎kbθ

� �exp

τα σð Þτ f

� �: ð5Þ

Here, ΔG⁎ is the activation energy for dislocation emission,τα ≈ jσ j =

ffiffiffi3

pis the slip resistance and τf is the stress required to

activate dislocation emission from GBs. Different from the modelproposed in [3], both the emission of partial and full dislocations fromthe GBs are considered here. Following [14,15,28], the stress requiredfor the emission of a dislocation from the GB reads

τ f =Gb= d for d > dcr

Gb= 3dð Þ + d−δeq� �

ΓG= d for d < dcr;

(ð6Þ

where G is the shearmodulus and δeq is the equilibrium spacing of partial

dislocationswhich canbe approximated as δeq ≈1

12πbΓ[15,28]. Γ=γsf/Gb

0.08

0.06

0.04

0.02

0100500

Prob

abili

ty [

1/nm

]

Grain Size [nm]

200 nm2

100 nm2

50 nm2

25 nm2

a

Fig. 1. (a) Various lognormal grain size distributions having the same mean d = 20 nm and(magenta). (b) Tensile stress strain curves corresponding to the grain size distributions sho

is the dimensionless reduced stacking fault energy where γsf is theintrinsic stacking fault energy. Below the critical grain size [15,28]

dcr =23

dd−δeq

Gb2

γsfð7Þ

the emission of partial dislocations from GBs start instead of fulldislocations.

Note that as pointed out in [3], the rate of plastic deformationgiven in Eq. (5) is valid for grain sizes up to∼100 nm. Themacroscopicplastic strain rate is given as the sum of deformation mechanismsmentioned above

ε̇p σ ; θð Þ = ε̇gbd σ ; θð Þ + ε̇gbs σ ; θð Þ + ε̇gip σ ; θð Þ: ð8Þ

The bar in Eq. (8) describes an averaging of the local fields over thepolycrystalline sample by using grain size distribution functions. Inother words, the macroscopic inelastic strain rates are obtained by

ε̇α σ ; θð Þ =∫∞0ε̇α d;σ ; θð ÞP dð Þkd3dd

∫∞0P dð Þkd3dd

where α = gbd; gbs; gip: ð9Þ

P(d) denotes the grain size distribution function and it isassumed that the volume of a grain with diameter d is kd3. One ofthe simplest possibilities for P(d) is a Rayleigh distribution, i.e.,Pray dð Þ = d=d

� �2exp −0:5d2 = d

2� �. Wei and Gao [3] have chosen the

Rayleigh distribution to be able to perform integrations analyticallyin their model. The main problem with the Rayleigh distributionis that it is described by a single parameter, so called the mean value,and therefore, it is not possible to control the spread of the distribution.In other words, as the mean value increases the distribution becomeswider. Moreover, there have been very few studies [28–31] on theeffects of the grain size distribution, particularly the variance, on themechanical behavior of nc-metals. Therefore, in what follows we em-ploy a more general lognormal grain size distribution

P log dð Þ = 1ffiffiffiffiffiffi2π

pΣd

exp −12

ln d=d0ð ÞΣ

� �2� �; ð10Þ

0

100

200

300

400

500

600

700

800

0 0.02 0.04 0.06 0.08 0.1

Stre

ss [

MPa

]

Strain [-]

25 nm2

50 nm2

100 nm2

200 nm2

b

different variances Σ̃ = 25 nm2 (red), 50 nm2 (blue), 100 nm2 (green), and 200 nm2

wn in (a) for a loading rate of ε̇ = 1 × 10−6s−1.

0

200

400

600

800

1000

-0.1 -0.05 0 0.05 0.1

Stre

ss [

MPa

]

Strain [-]

15 nm

20 nm

40 nm

60 nm

80 nm100 nm

15 nm

20 nm

40 nm

60 nm

80 nm

100 nm

0.4

0.5

0.6

0.7

0.8

0.9

1

20 40 60 80 100 120

Tens

ile s

tres

s/C

ompr

essi

vest

ress

[-]

Grain Size [nm]

ba

Fig. 2. T/C-asymmetry of the model for nc-Cu. (a) Tensile and compressive stress-strain curves for various grain sizes, i.e., d = 15 nm (red), d = 20 nm (blue), d = 40 nm (green),d = 60 nm (magenta), d = 80 nm (cyan), d = 100 nm (brown), The variance Σ̃ = 25 nm2 for all grain sizes. (b) The change in the ratio of tensile stress to compressive stress as afunction of grain size. Red solid line with squares are the results of the proposedmodel, whereas blue triangles, green circles andmagenta diamonds are the predictions of the numericalmodels of [34], [35] and [36] for nc-Cu, respectively.

3393E. Gürses, T. El Sayed / Materials Letters 65 (2011) 3391–3395

where d0 (standard deviation of ln(d)) and Σ are parametersdescribing the shape of the distribution function. The mean grainsize d and the variance Σ̃ can be measured in experiments, and arerelated to d0 and Σ through

d = d0 exp Σ2= 2

� �; Σ̃ = d20 exp Σ2

� �exp Σ2

� �−1

� �: ð11Þ

Note that the grain size distribution in polycrystalline materialsare usually well represented by a lognormal distribution that is oftenused in numerical models, see e.g., [29,28,31].

The one-dimensional macroscopic stress–strain relation is governedby the classical elastoplasticity σ̇ = E ε̇− ε̇

pσ ; θð Þ

� �where E is the

Young's modulus. In a process where the strain rate is prescribed, thestress–strain relation can be integrated to obtain a nonlinear equation

r σn+1� �

= σn+1−σn−E εn+1−εn−εpn+1 σn+1; θ� �

+ εpn� �

= 0 ð12Þ

which can to be solved for the current value of the stress σn+1.

3. Results and discussion

In this section, the capabilities of the model in capturing the mainmechanical characteristics of nc-metals, i.e., the inverseHall–Petcheffect,the T/C-asymmetry and the enhanced rate sensitivity, are demonstrated.Set of material parameters employed in numerical simulations areadapted from [3,32] except for the dimensionless stacking fault energy Γ,

300

400

500

600

700

800

900

1000

1100

0.1 0.15 0.2 0.25

2025304060100

Flow

str

ess

[MPa

]

d-1/2 [nm-1/2]

Grain size d [nm]

25 nm2

50 nm2

100 nm2

200 nm2

a

Fig. 3. Grain size dependence of (a) tensile and (b) compressive flow stress for different gr(green circles), 200 nm2 (magenta diamonds). Inverse H-P behavior is clearly seen in bothcritical grain size in compression is smaller than in tension.

the activation volume of diffusion ΔV and λ, see Table 1. Γ is taken from[14] and ΔV is known to be in the range of 0.6–1.0 Ωa [20]. MDsimulations [11] have shown that the parameter λ is in the range 0.14–0.25. The effect of variance on themacroscopic stress response is studiedfirst. In Fig. 1a five different grain size distributions with the same meangrain size (d=50 nm) but having different variances (Σ̃=25, 50, 100,200 nm2) are depicted. Simulations of the tensile behavior are conductedfor a loading rate of ε̇=1×10−6s−1. The significant effect of the varianceon the tensile stress–strain behavior is demonstrated in Fig. 1b. As can beseen from thefigure, thewidening of the grain size distribution results ina softer response in agreement with themodel predictions of [30,31,33].This result clearly shows the that the variance, in addition to the meangrain size, is an important microstructural parameter effecting themacroscopic behavior significantly.

Next, the T/C-asymmetry of the proposed model is demonstrated.Tensile and compressive tests are conducted for a loading rate ofε̇=1×10−6s−1 and Σ̃=25 nm2. The corresponding stress–straincurves are given in Fig. 2a for grain sizes from 100 nm down to 15 nm.As can be seen from the figure, the T/C-asymmetry becomes evident forgrain sizes smaller than 60 nm. The grain size dependence of the T/C-asymmetry is further demonstrated in Fig. 2b where the absolute valueof the ratio of the tensile and compressive stresses is plotted against thegrain size. For the grain sizes considered (100 nm≥d≥15 nm) the ratiodrops from100% to55%. In thefigure, T/C-asymmetry innc-Cupredictedby three other numerical models [34–36] are provided for comparison.

d-1/2 [nm-1/2]

300

400

500

600

700

800

900

1000

1100

0.1 0.15 0.2 0.25

2025304060100

Flow

str

ess

[MPa

]

Grain size d [nm]

25 nm2

50 nm2 100 nm2

200 nm2

b

ain size distributions, i.e., Σ̃=25 nm2 (red squares), 50 nm2 (blue triangles), 100 nm2

tensile and compressive stress–strain curves. Note that in agreement with [35,34] the

6000 5000

4000

3000

2000

1500

1000

750 1e-06 1e-05 0.0001 0.001 0.01 0.1

Flow

Str

ess

[MPa

]

Strain Rate [1/s]

80 nm

60 nm

40nm

20nm

10nm

0

0.05

0.1

0.15

0.2

0.25

10000100010010

Rat

e Se

nsiti

vity

Exp

onen

t m [

-]

Grain size [nm]

ba

Fig. 4. (a) Double logarithmic plot of the flow stress vs. strain rate to determine the rate sensitivity exponent m for various grain sizes for Σ̃=25 nm2. Discrete points are thesimulation results for nc-Cu and the solid lines are the best linear fits. (b) The rate sensitivity exponent m predicted by the model for Σ̃=25 nm2 (red pluses) together with thesummary of experimental data for nc-Cu from the literature (magenta squares). The literature is adapted from Fig. 5 of [3] and Fig. 6 of [1].

3394 E. Gürses, T. El Sayed / Materials Letters 65 (2011) 3391–3395

Note that the T/C-asymmetry of the proposed model is solely due toGB-deformation mechanisms, i.e., GB-sliding and GB-diffusion. There-fore, the T/C-asymmetry vanishes very rapidly for grain sizes greaterthan 60 nm. The model can be improved in this aspect by taking intoaccount the pressure dependence of the dislocation emission from GBsas proposed by Cheng et al. [37].

The grain size dependence of the model is demonstrated throughtraditional Hall–Petch plots in Fig. 3. The change of the tensile and com-pressive ultimate stress values with the mean grain size are depictedin Fig. 3a and b for different Σ̃ values, respectively. The model predictsthe inverse Hall–Petch effect, i.e., the loss of strength with grain sizerefinement, both in tension and compression. However, in line with[35,34] the critical grain size belowwhich the softening starts is found tobe smaller in compression than in tension.

Finally, the strain-rate dependence of themodel is demonstrated. InFig. 4a the variation of the flow stress with strain rate is depicted forvarious grain sizes with Σ̃=25 nm2. In the strain-rate sensitivity cal-culations the flow stress is measured at a total strain of 10% where thestress seems to have reached a plateau. Discrete points in the figure arethe simulation results and the solid lines are the best linear fits. Thestrain-rate sensitivity exponent m = ∂ lnσ = ∂ ln ε̇ is found as 0.0995,0.0991, 0.105, 0.123, 0.156 for grain sizes d=80, 60, 40 20, 10 nm,respectively. In Fig. 4b the change ofmwith grain size is presented (redpluses) togetherwith the summary of data from the literature (magentasquares) [1,3]. Themodel predicts successfully a strong enhancement ofrate sensitivity with grain size refinement in accordance with theexperimental observations.

4. Conclusion

Wehavepresentedaviscoplastic constitutivemodel fornc-metals thatis based on competing GB and grain interior deformation mechanisms.Different frommost constitutive models where nc-metals are consideredas a multi-phase composites with corresponding volume fractions, theproposed model does not assume a parameter for a GB-thickness orGB affected zone. The proposed model extends [3] with respect to con-siderations of (i) the pressure dependence of the GB-diffusivity, (ii) thenormal stress dependence of the GB-sliding, (iii) both the partial and fulldislocation emissions from GBs and (iv) a more general lognormal grainsize distribution instead of a Rayleigh distribution. It has been shownthat themodel captures the traditional and the inverse Hall–Petch effects,the enhanced rate sensitivity and the T/C-asymmetry. Furthermore, theeffects of the grain size distribution have been studied and it is found thatnot only the mean grain size but also the spread of grain size distributionare important microstructural parameters.

Acknowledgments

This work was fully funded by the KAUST baseline fund.

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