International Journal of Solids and Structures 48 (2011) 1610–1616

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International Journal of Solids and Structures

journal homepage: www.elsevier .com/locate / i jsolst r

Constitutive modeling of strain rate effects in nanocrystallineand ultrafine grained polycrystals

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

a r t i c l e i n f o a b s t r a c t

Article history:Received 9 August 2010Received in revised form 9 February 2011Available online 16 February 2011

Keywords:Constitutive modelingRate dependenceNanocrystalsCrystal plasticityGrain size

0020-7683/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.ijsolstr.2011.02.013

⇑ Corresponding author.E-mail addresses: ercan.gurses@kaust.edu.sa (E. G

t.edu.sa (T. El Sayed).

We present a variational two-phase constitutive model capable of capturing the enhanced rate sensitivityin nanocrystalline (nc) and ultrafine-grained (ufg) fcc metals. The nc/ufg-material consists of a grain inte-rior phase and a grain boundary affected zone (GBAZ). The behavior of the GBAZ is described by a rate-dependent isotropic porous plasticity model, whereas a rate-independent crystal-plasticity model whichaccounts for the transition from partial dislocation to full dislocation mediated plasticity is employedfor the grain interior. The scale bridging from a single grain to a polycrystal is done by a Taylor-typehomogenization. It is shown that the enhanced rate sensitivity caused by the grain size refinement is suc-cessfully captured by the proposed model.

� 2011 Elsevier Ltd. All rights reserved.

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1. Introduction

The inelastic deformation behavior of polycrystalline materials,in particular the influence of grain size (d) on material propertiesand active deformation mechanisms, has long been the subject ofintensive research. After the synthesis of polycrystalline materialswith grain sizes lower than �100 nm, which are often denoted asnanocrystalline (nc) materials, the research efforts in this area havefurther grown and numerous review articles have been published;comprehensive lists of literature may be found in recent reviews(Dao et al., 2007; Gleiter, 2000; Koch, 2007; Kumar et al., 2003;Meyers et al., 2006; Saada and Dirras, 2009; Weertman, 2007; Wolfet al., 2005) and monographs (Cherkaoui and Capolungo, 2009;Ramesh, 2009). Nc-materials are known to posses several distinctfeatures when compared to coarse grained polycrystals. These in-clude high strength and fatigue resistance, low ductility, pro-nounced rate and temperature dependence, tension–compressionasymmetry and susceptibility to plastic instability. In this paperwe will focus on the pronounced rate dependence of nc-fcc metals.

The strain rate sensitivity of metals is often described as thevariation of flow stress with strain rate at constant temperaturefor a given strain level. The non-dimensional strain rate sensitivityexponent is also defined as (Asaro and Suresh, 2005; Wei et al.,2004)

ll rights reserved.

ürses), tamer.elsayed@kaus-

m ¼ 3kBTryV

; ð1Þ

where kB is the Boltzmann constant, T is the absolute temperature,ry is the flow stress and V is the activation volume, i.e., the deriva-tive of the activation enthalpy with respect to the effective shearstress (Asaro and Suresh, 2005; Kocks et al., 1975; Wei et al.,2004). The rate sensitivity exponent m and the activation volumeV can be measured from quasistatic tensile experiments as (Caillardand Martin, 2003; Wei, 2007)

m ¼ @ ln ry

@ ln _eand V ¼

ffiffiffi3p

kBT@ ln _e@ry

; ð2Þ

where _e is the applied tensile strain rate. Conventional fcc metalshave a large activation volume, e.g. V � 100 � 1000b3 where b isthe Burgers vector, which leads to lower values for the rate sensitiv-ity exponent m through the relation (1). The strain rate sensitivity mis approximately 0.002–0.006 for most coarse grained fcc metals(Dalla Torre et al., 2005; Wang et al., 2006; Zehetbauer and Seumer,1993; Zhang et al., 2009). These large values for the activation vol-ume are associated with running dislocations cutting through exist-ing forest dislocations in grain interiors. On the other hand, theactivation volumes for grain boundary (GB) diffusion and slidingprocesses are much lower, e.g. V � 1 � 10b3, representing a lowerbound for the activation volume in polycrystalline materials. There-fore, GB-diffusion mediated diffusional creep and the GB-slidingmechanisms entail rate sensitivity exponents of m = 1.0 andm = 0.5, respectively (Wang et al., 2006). Note that although thereis a marked reduction in activation volume with grain size refine-ment as reported in Chen et al. (2006), Guduru et al. (2007), Wang

E. Gürses, T. El Sayed / International Journal of Solids and Structures 48 (2011) 1610–1616 1611

et al. (2006), Wei et al. (2004) and Zhang et al. (2009), a recent studyby Trelewicz and Schuh (2007, 2008) showed increase in activationvolume for Ni–W alloys through nano-indentation tests when thegrain size was further reduced down to (almost amorphous limit)values less than 10 nm.

In one of the earliest studies on rate dependence of nc-metals,Wang et al. (1997) investigated room temperature creep behaviorof electrodeposited nc-Ni with grain size in the range of 6–40 nm.They found that nc-Ni is highly rate sensitive at room temperature.Lu et al. (2001) have conducted tensile tests on electrodepositednc-Cu (d � 28 nm) for strain rates 6 � 10�5–1.8 � 103 s�1. Thestrain-rate sensitivity (m) was reported to be around 0.036. In thiswork, contrary to the most of the experimental results in the liter-ature, the nc-material showed anomalously low flow stress valuesand very ductile behavior, which is possibly due to the existence ofhigh number of low angle GBs in the sample. Uniaxial compressivebehavior of electrodeposited nc-Cu (d � 28 nm) has been studiedby Jia et al. (2001) both at quasistatic (4 � 10�4 s�1) and high strainrates (3 � 102–2 � 104 s�1). The nc-Cu in this work, in a similarway to Lu et al. (2001), possesses high number of low angle GBsand shows low flow stress values, high ductility and relativelyweak strain rate dependence. On the other hand, Dalla Torreet al. (2002), who have compared two commercially available elec-trodeposited nc-Ni (d � 20 nm) with respect to the effect of micro-structure and impurities on the mechanical response, reported adecrease in ductility as the strain rate was increased from5.5 � 10�5 to 5.5 � 10�2 s�1. The strain-rate sensitivity was mea-sured to be around 0.01–0.03 for these loading rates. The ratedependency of electrodeposited nc-Ni (d � 40 nm) and ufg-Ni(d � 320 nm) has been investigated by Schwaiger et al. (2003)through tensile and indentation tests. Tensile tests with strainrates 3 � 10�4–3 � 10�1 s�1 have been conducted and an evidentincrease in the flow stress with strain rate has been observed fornc-Ni, while almost no rate dependence was found for ufg-Ni.Cheng et al. (2005) have performed tensile tests on in situ consol-idated nc-Cu (d � 54–62 nm) for loading rates 10�4–10�2 s�1. Therate sensitivity parameter was quoted as 0.027, and a high strengthand decent ductility combination has been achieved. Strain ratesensitivity of nc-Cu (d � 24–26 nm) synthesized by electric brushplating technique has been studied by Jiang et al. (2006) throughcreep and tensile tests. They considered strain rates in the range1.04 � 10�6–4.17 � 10�1 s�1 and measured m to be 0.104. Jianget al. (2008) have performed compressive tests on pulse electricbrush-plated nc-Cu (d � 20–25 nm) for strain rates from1 � 10�5 s�1 to 3 s�1. They observed a change in the rate sensitivityexponent with loading rate such that m = 0.086 for strain ratessmaller than 10�2 s�1 and m = 0.041 otherwise. A similar depen-dence of m on the loading rate was reported by Wang and Ma(2003) in tensile tests of ufg-Cu (d � 190–300 nm, but havingmicrostructural features such as subgrains and domain structuressmaller than 100 nm) produced by ECAP. The rate sensitivity expo-nent m was 0.025 when the strain rate _e ¼ 6� 10�7 s�1, anddropped to m = 0.010 when _e ¼ 1� 10�4 s�1. Chen et al. (2006)have studied the rate dependent behavior of Cu via nano-indenta-tion tests for a wide variety of grain sizes synthesized with threedifferent techniques, i.e., ECAP for ufg-Cu (d � 190 nm), a surfacemechanical attrition treatment (SMAT) for nc-Cu (d � 40 nm) andmagnetron sputtering (MS) for nc-Cu (d varies from 10 to 30 nmdepending on the substrate temperature). The following valueswere quoted for the rate sensitivity exponent: m = 0.02 for ufg-Cu (ECAP), m = 0.032 for nc-Cu (SMAT) and m = 0.038–0.06 fornc-Cu (MS). Electrodeposited nc-Cu (d � 75 nm) and ball-millednc-Cu (d � 25 nm) have been studied by Guduru et al. (2007)through tensile and shear punch tests. According to the tensiletests results, m was found to be around 0.059–0.064 for electrode-posited nc-Cu for strain rates in the range 2.5 � 10�4–

2.5 � 10�2 s�1. Zhang et al. (2009) have reported m = 0.029 forpulse electrodeposited nc-Cu (d � 33 nm) tested under tensionfor strain rates 1.04 � 10�5–1.04 s�1.

Although some inconsistencies exist in the experimental find-ings by the different research groups (mostly arising from differentsample synthesis techniques and testing methods), most of thedata indicates a substantial increase in m for fcc metals when dis reduced down to the nc regime (see e.g. Wei et al. (2004) andWei (2007) for nickel, Chen et al. (2006), Dao et al. (2007) andWei (2007) for copper and Wei (2007) for aluminum). It can beconcluded from these figures that m values exhibit a slight increasewith a decrease in d from the coarse grain level to the sub-micronscale, whereas an obvious enhancement in the rate dependence oc-curs when d is reduced further below 100 nm. The enhanced strainrate sensitivity in nc fcc metals as compared to coarse grainedcounterparts is often explained as a result of the presence of grainboundary deformation mechanisms (particularly d below 20 nm)and/or the increase in interactions of dislocations with GBs, e.g.dislocation nucleation and dislocation de-pinning at the GBs. Threepossible scenarios of dislocation-GB interaction have been dis-cussed in Wang et al. (2006).

In this paper, we extend the rate-independent two-phase consti-tutive model proposed in Gürses and El Sayed (2011) for nc-fcc-materials to the rate-dependent case. To this end, the isotropic por-ous plasticity model employed in the previous work is modified totake into account the enhanced rate-sensitivity caused by GBdeformation mechanisms. The paper is organized as follows: therate dependent constitutive model is presented in Section 2. In Sec-tion 3, the predictive capability of the proposed model throughcomparisons against experimental data from the literature is illus-trated. Finally, we present our conclusions and final remarks inSection 4.

2. Constitutive model

In the sequel, an extension of the rate-independent model pro-posed in Gürses and El Sayed (2011) to rate-dependent case is pre-sented. In a similar manner to Gürses and El Sayed (2011) a singlegrain is assumed to consist of a crystalline core region and a grainboundary affected zone (GBAZ). However, different from the for-mer work the behavior of the GBAZ is assumed to be rate-depen-dent in order to model enhanced rate-sensitivity in nc-fccmaterials.

Following the classical multiplicative decomposition frame-work (Lee, 1969), the total deformation gradient F = FeFp is as-sumed to decompose into an elastic part Fe and a plastic part Fp

with corresponding Jacobians with J, Je and Jp. The free energy ofa grain consisting of an interior phase and a grain boundary phase(GBAZ (Schwaiger et al., 2003)) can be written in a simple volumeaverage form W = nWgi + (1 � n)Wgb where n is the volume fractionof the grain core region, Wgi and Wgb denote the free energies of thegrain interior and boundary phases, respectively. We further as-sume that the multiplicative decomposition holds for both phasesseparately, i.e., F ¼ Fe

giFpgi and F ¼ Fe

gbFpgb. The average first Piola–

Kirchhoff stress P = nPgi + (1 � n)Pgb of a particular grain is also gi-ven by a volume average. Provided that the grains have cubical(Carsley et al., 1995) shapes and the thickness of grain boundaryzone dgb is constant, the volume fraction of grain cores reads

n ¼ ðd� dgbÞ3=d3: ð3Þ

Following porous plasticity model (Weinberg et al., 2006; El Sayedet al., 2008) the plastic deformation rate is assumed to have bothvolumetric and deviatoric contributions, i.e.,

_FpgbFp�1

gb ¼ _�pMþ _hpN; ð4Þ

Table 1Material constants for nanocrystalline Cu.

Grain interior Grain boundary

Elastic model parameters c11 = 160 GPa E = 118.8 GPac12 = 115.3 GPa m = 0.34 [–]c44 = 71.6 GPa

Plastic model parameters G = 33.6 GPa �p0 ¼ 0:05 [–]

b = 0.255 nm n = 1.5 [–]C = 0.004 [–] _�p

0 ¼ 0:00032 [–]h0 = 80 GPa m = 1.0 [–]d0 = 180 nm r0 = 0.06 GPan = 12 [–] N = 100 mm�3

q = 0 [–] a0 = 0.1 mmdgb = 2.0 nm

1612 E. Gürses, T. El Sayed / International Journal of Solids and Structures 48 (2011) 1610–1616

where _�p and _hp are the plastic multipliers with irreversibility con-straints _�p P 0 and _hp P 0. The tensors M and N are the directionsof the deviatoric and volumetric plastic deformation rates, respec-tively. The free energy of the GBAZ is assumed to have an additivestructure

WgbðF;Fpgb; �

p; hpÞ ¼WegbðF

egbÞ þWp

gbð�p; hpÞ; ð5Þ

where WegbðF

egbÞ and Wp

gbð�p; hpÞ are the elastic and plastic stored en-ergy densities, respectively. The plastic density is assumed to addi-tively decompose into deviatoric and volumetric parts

Wpgbð�

p; hpÞ ¼Wp;devgb ð�pÞ þWp;vol

gb ðhpÞ ð6Þ

which are solely functions of deviatoric and volumetric plastic mul-tipliers, respectively. Specific forms of the deviatoric and volumetricparts of the plastic energy and the elastic energy are given in Gürsesand El Sayed (2011) and therefore, they are not repeated here.

The enhanced rate-sensitivity in nc/ufg fcc-metals is oftenattributed to a change in the rate controlling mechanism. In con-ventional coarse grained metals the rate controlling process isthe forest dislocations cutting mechanism (Wei, 2007). This leadsto activation volumes V of several hundred to couple of thousandtimes b3 where b is the Burgers vector (Meyers et al., 2006). Onthe other hand, low levels of activation volumes, i.e., V � 10–100b3 (Asaro and Suresh, 2005; Dalla Torre et al., 2005; Daoet al., 2007; Wang et al., 2006) in nc/ufg-fcc metals cannot be ex-plained by forest dislocation mechanisms. It is often attributed todeformation mechanisms related with grain boundaries, e.g. inter-action of dislocations with grain boundaries (Dalla Torre et al.,2005; Li et al., 2007; Wang et al., 2006). Therefore, we assume thatthe rate-dependence of the proposed two-phase model is solelydue to the grain boundary phase. Following Ortiz and Stainier(1999) the rate-dependent response is incorporated into the grainboundary model through a rate potential. The rate potential is as-sumed to decompose additively into volumetric and deviatoricparts, i.e.,

w�gbð _�p; _hp; JpÞ ¼ w�;volgb ð _h

p; JpÞ þ w�;devgb ð _�pÞ: ð7Þ

The volumetric and deviatoric contributions are originally given inWeinberg et al. (2006), i.e.,

w�;volgb ð _h

p; JpÞ ¼ m2r0 _�p0

mþ 1N

4pa3

3ð1� f 1=mÞ 2 _a

_�p0a

��������

mþ1m

;

w�;devgb ð _�pÞ ¼ mr0 _�p

0

mþ 1_�p

_�p0

� �mþ1m

;

ð8Þ

where m is the rate sensitivity exponent, _�p0 is a reference plastic

strain rate, N is the void density, f is the volume fraction of voids,a is the mean void radius and r0 is the yield stress.

The rate-independent multisurface plasticity model of Gürsesand El Sayed (2011) which accounts for the transition from partialdislocation to full dislocation mediated plasticity is employed todescribe the response of the grain interior region. Having obtainedthe single grain behavior through volume averaging, a Taylor-typeaveraging scheme is used for the transition from single grain scaleto polycrystalline scale. Nanocrystalline sample is assumed to pos-ses a lognormal grain size distribution. We refer to Gürses and ElSayed (2011) for further details of the grain interior model andthe averaging scheme.

3. Numerical examples and validation

It is well known that the Taylor-type averaging schemes requirelarge number of grains to be sampled in order to be predictive. Inthe previous work (Gürses and El Sayed, 2011) we conducted anumerical study to identify the effect of number of grains on the

average stress response and observed that the average stress re-sponse does not change for ensembles having more than 200grains. Hence, in all simulations the Taylor averaging scheme is uti-lized with 200 grains.

3.1. Grain size dependence of rate sensitivity

To illustrate the rate dependent behavior of the proposed twophase model, we set the material parameters to be representativefor nc-Cu, see Table 1. Experimental observations (Gleiter, 1989;Nieman et al., 1991; Sanders et al., 1997; Shen et al., 1995) andmolecular dynamics simulations (Schiøtz et al., 1998, 1999) haveshown that the elastic constants of nc-metals are generally smallerthan the coarse grained polycrystals. Therefore, we reduce the elas-tic properties of the grain interior phase and the GBAZ by 5% com-pared to the coarse grained polycrystalline Cu. We first investigatethe tensile response of the model for various grain sizes (d = 10, 25,50, 100, 250 nm) and strain rates ð _e ¼ 1� 10�5;1� 10�4;1�10�3;1� 10�2 s�1). In this investigation, a uniform grain size distri-bution is assumed, i.e., the variance ~r is assumed to be zero.

The tensile stress curves are given for four different grain sizes inFig. 1(a)–(d) where for each d the stress–strain curves are plotted forfour different strain rates. It is clear from the figures that the re-sponse is highly rate dependent for d = 10 nm and becomes lessand less rate sensitive as the grain size increases. Fig. 2(a) and (b)shows the effect of grain size for two strain rates, i.e.,_e ¼ 1� 10�5 s�1 and _e ¼ 1� 10�2 s�1. It is evident from this plot thatthe proposed model exhibits a strong grain-size-dependent stressresponse. The mechanical behavior of the sample with d = 10 nm issofter than the one with d = 25 nm for the slow loading rate_e ¼ 1� 10�5 s�1 resulting in an inverse Hall–Petch behavior, seeFig. 2(a). However, the model does not exhibit the inverse Hall–Petch behavior when the strain rate is increased to_e ¼ 1� 10�2 s�1, see Fig. 2(b). A similar change in the inverse Hall–Petch response with strain rate has also been predicted by the mod-els proposed in Kim and Estrin (2005) and Wei and Gao (2008).

The variation of flow stress at e = 0.01 with respect to strain rateis depicted in a double logarithmic plot in Fig. 3(a) for five differentgrain sizes. In the figure discrete points are the simulation resultsand the solid lines are the best linear fits. Except for the cased = 10 nm, the model predicts the linear relation (which is com-monly observed in almost all metals) between the logarithm offlow stress and the logarithm of strain rate very well. Note thatthe slopes of the lines give the rate sensitivity exponent m, whichreads m = 0.0714, 0.0316, 0.0180, 0.0104 and 0.0046 for d = 10, 25,50, 100 and 250 nm, respectively. The rate sensitivity exponent mversus grain size plot is provided in Fig. 3(b) along with a compileddata from the literature. The black solid line is taken from Dao et al.(2007) as an eye-guide while red plus signs are the predictions ofthe model. The general trend of m is captured reasonably well withthe proposed model.

0

500

1000

1500

2000

0 0.02 0.04 0.06 0.08 0.1

Ten

sile

Str

ess

[MPa

]

Tensile Strain [-]

1e-2/s

1e-3/s

1e-5/s

0

200

400

600

800

1000

1200

1400

1600

0 0.02 0.04 0.06 0.08 0.1

Ten

sile

Str

ess

[MPa

]

Tensile Strain [-]

1e-2/s

1e-3/s

1e-5/s

(a) (b)

0

200

400

600

800

1000

1200

0 0.02 0.04 0.06 0.08 0.1

Ten

sile

Str

ess

[MPa

]

Tensile Strain [-]

1e-2/s

1e-3/s

1e-5/s

0

200

400

600

800

1000

0 0.02 0.04 0.06 0.08 0.1

Ten

sile

Str

ess

[MPa

]

Tensile Strain [-]

1e-2/s

1e-3/s

1e-5/s

(c) (d)

Fig. 1. Tensile stress–strain curves for various strain rates ð _e ¼ 10�5;10�4;10�3;10�2;10�1 s�1Þ and grain sizes: (a) d = 10 nm, (b) d = 25 nm, (c) d = 50 nm and (d) d = 100 nm.The response is highly rate dependent for d = 10 nm and becomes less and less rate sensitive as the grain size increases.

0

200

400

600

800

1000

1200

0 0.02 0.04 0.06 0.08 0.1

Ten

sile

Str

ess

[MPa

]

Tensile Strain [-]

d=10 nm

d=25 nm

d=50 nm

d=100 nm

d=250 nm

0

200

400

600

800

1000

1200

1400

1600

1800

0 0.02 0.04 0.06 0.08 0.1

Ten

sile

Str

ess

[MPa

]

Tensile Strain [-]

d=10 nm

d=25 nm

d=50nm

d=100nm

d=250nm

(a) (b)

Fig. 2. Tensile stress–strain curves for various grain sizes (d = 10, 25, 50, 100, 250 nm) and strain rates: (a) _e ¼ 1� 10�5 s�1, (b) _e ¼ 1� 10�2 s�1. The mechanical behavior ofthe sample with d = 10 nm is softer than the one with d = 25 nm for the slow loading rate _e ¼ 1� 10�5 s�1 resulting in an inverse Hall–Petch behavior, while the model doesnot exhibit the inverse Hall–Petch behavior when the strain rate is increased to _e ¼ 1� 10�2 s�1.

E. Gürses, T. El Sayed / International Journal of Solids and Structures 48 (2011) 1610–1616 1613

3.2. Application to nanocrystalline copper

In this section we will use the proposed model to simulate therate-dependent tensile behavior of nanocrystalline copper andshow that the model is able capture the experimental data fromGuduru et al. (2007). We consider an average grain size of 75 nm

as experimentally investigated in Guduru et al. (2007), and modelthe polycrystalline aggregate with 200 randomly oriented grains.The variance of the grain size distribution ~r is chosen as 75 nm2.The set of material parameters used in simulations is summarizedin Table 2. Four different strain rates, namely _e ¼ 2:5� 10�4;2:5�10�3;4� 10�3;2:5� 10�2 s�1 are studied. The stress–strain curves

1000

750

500 1e-06 1e-05 0.0001 0.001 0.01

Flow

Str

ess

[MPa

]

Strain Rate [1/s]

d=10 nm

d=25 nm

d=50nm

d=100nm

d=250nm

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

10000 1000 100 10

Rat

e Se

nsiti

vity

Exp

onen

t m [

-]

Grain size [nm]

(a) (b)

Fig. 3. (a) Double logarithmic plot of the flow stress vs. strain rate to determine the rate sensitivity exponent m for various grain sizes (d = 10, 25, 50, 100, 250 nm). Discretepoints are the simulation results and the solid lines are the best linear fits. (b) The rate sensitivity exponent m vs. the grain size d (red plus) together with the summary of datafrom the literature (green cross (Chen et al., 2006), blue asterisk (Lu et al., 2001), empty magenta square (Wei et al., 2004), black filled circle (Carreker and Hibbard, 1953),empty orange upward triangle (Zehetbauer and Seumer, 1993), empty red downward triangle (Lu et al., 2005) (nano-twinned Cu), filled cyan square (Elmustafa et al., 2002),filled blue downward triangle (Gray et al., 1997), filled gray upward triangle (Lu et al., 2005)). (For interpretation of the references to colour in this figure legend, the reader isreferred to the web version of this article.)

Table 2Material constants used for simulation of nc-Cu in Guduru et al. (2007).

Grain interior Grain boundary

Elastic model parameters c11 = 160 GPa E = 118.8 GPac12 = 115.3 GPa m = 0.34 [–]c44 = 71.6 GPa

Plastic model parameters G = 33.6 GPa �p0 ¼ 0:005 [–]

b = 0.255 nm n = 1.5 [–]C = 0.004 [–] _�p

0 ¼ 0:00032 [–]h0 = 80 GPa m = 1.0 [–]d0 = 180 nm r0 = 60 GPan = 12 [–] N = 100 mm�3

q = 0 [–] a0 = 0.1 mmdgb = 2.0 nm

0

200

400

600

800

1000

1200

0 0.02 0.04 0.06 0.08 0.1

Ten

sile

Str

ess

[MPa

]

Tensile Strain [-]

Fig. 4. Comparison of model predictions with the tensile stress–strain data fromGuduru et al. (2007) for nanocrystalline Cu. The solid lines are simulation results(red for _e ¼ 2:5� 10�4 s�1, green for _e ¼ 2:5� 10�3 s�1, blue for _e ¼ 4� 10�3 s�1,magenta for _e ¼ 2:5� 10�1 s�1) and the points are experimental data (red squarefor _e ¼ 2:5� 10�4 s�1, green cross for _e ¼ 2:5� 10�3 s�1, blue circle for_e ¼ 4� 10�3 s�1, magenta triangle for _e ¼ 2:5� 10�1 s�1). (For interpretation ofthe references to colour in this figure legend, the reader is referred to the webversion of this article.)

Table 3Material constants used for simulation of nc-Ni in Schwaiger et al. (2003).

Grain interior Grain boundary

Elastic model parameters c11 = 179.6 GPa E = 143 GPac12 = 106.9 GPa m = 0.31 [–]c44 = 90.9

Plastic model parameters G = 45 GPa �p0 ¼ 0:011 [–]

b = 0.248 nm n = 35 [–]C = 0.004 [–] _�p

0 ¼ 0:0045 [–]h0 = 240 GPa m = 1.1 [–]d0 = 150 nm r0 = 0.04 GPan = 12.5 [–] N = 100 mm�3

q = 0 [–] a0 = 0.1 mmdgb = 1.7 nm

0

200

400

600

800

1000

1200

1400

1600

1800

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

Ten

sile

Str

ess

[MPa

]

Tensile Strain [-]

Fig. 5. Comparison of model predictions with the tensile stress–strain data fromSchwaiger et al. (2003) for nanocrystalline Ni. The solid lines are simulation results(red for _e ¼ 3� 10�4 s�1, blue for _e ¼ 1:8� 10�2 s�1, magenta for _e ¼ 3� 10�1 s�1)and the points are experimental data (square for _e ¼ 3� 10�4 s�1, cross for_e ¼ 1:8� 10�2 s�1, circle for _e ¼ 3� 10�1 s�1). (For interpretation of the referencesto colour in this figure legend, the reader is referred to the web version of thisarticle.)

1614 E. Gürses, T. El Sayed / International Journal of Solids and Structures 48 (2011) 1610–1616

E. Gürses, T. El Sayed / International Journal of Solids and Structures 48 (2011) 1610–1616 1615

computed by the model are compared with the experimental datain Fig. 4. A substantial increase in the flow stress (almost 50%) ispredicted when the strain rate increased from _e ¼ 2:5� 10�4 s�1

to _e ¼ 2:5� 10�2 s�1. The stress–strain behavior of nc-Cu for differ-ent strain rates is very well-predicted by the proposed model.

3.3. Application to nanocrystalline nickel

We further validate the proposed model by comparing againstthe experimental data on nanocrystalline nickel given in Schwaigeret al. (2003). The set of material parameters utilized in the simula-tion is given in Table 3. A set of strain rates, _e ¼ 3�10�4;1:8� 10�2;3� 10�1 s�1, is studied. The average grain size isset to d = 40 nm as reported in Schwaiger et al. (2003). The vari-ance of the polycrystal is assumed to be ~r ¼ 25 nm2 since the sam-ple was characterized as having a narrow grain size distribution(Schwaiger et al., 2003). The simulation results provided in Fig. 5show that the model successfully captures the rate-dependent re-sponse of nc-Ni as well.

4. Conclusion

The rate-independent multiscale constitutive model proposed byGürses and El Sayed (2011) for nanocrystalline fcc metals is ex-tended to a rate-dependent case. Similar to the original model,the proposed extension has a variational structure and modelsthe behavior of nanocrystalline fcc metals in the finite deformationregime. The nc-material is considered as a two phase material con-sisting of a rate-independent grain interior phase and a rate-depen-dent grain boundary affected zone (GBAZ). A rate-dependentisotropic porous plasticity model which accounts for void growthis employed to describe the GBAZ, whereas a rate-independentcrystal-plasticity model which accounts for the transition frompartial dislocation to full dislocation mediated plasticity is usedfor the plasticity of the grain core regions. The single grain behav-ior is given by a volume average of the two phases and the poly-crystalline response is obtained by a Taylor-type homogenizationmethod. The grain size dependent rate-sensitivity of the proposedmodel has been demonstrated by a plot of the strain rate versus theflow stress and a plot of the grain size d versus the rate sensitivityexponent m. Finally, it has been shown that the model is able toreproduce successfully the experimental results of Guduru et al.(2007) on nc-copper and Schwaiger et al. (2003) on nc-nickel.

Acknowledgment

This work was fully funded by the KAUST baseline fund.

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