Contents lists available at ScienceDirect

Journal of the Mechanics and Physics of Solids

Journal of the Mechanics and Physics of Solids 59 (2011) 732–749

0022-50

doi:10.1

� Cor

E-m

journal homepage: www.elsevier.com/locate/jmps

A variational multiscale constitutive model fornanocrystalline materials

Ercan Gurses, 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

Article history:

Received 23 March 2010

Received in revised form

14 October 2010

Accepted 30 October 2010Available online 4 November 2010

Keywords:

Variational updates

Nanocrystals

Voids

Crystal plasticity

Grain size

96/$ - see front matter & 2010 Elsevier Ltd. A

016/j.jmps.2010.10.010

responding author.

ail addresses: ercan.gurses@kaust.edu.sa (E. G

a b s t r a c t

This paper presents a variational multi-scale constitutive model in the finite deformation

regime capable of capturing the mechanical behavior of nanocrystalline (nc) fcc metals.

The nc-material is modeled as a two-phase material consisting of a grain interior phase

and a grain boundary effected zone (GBAZ). A rate-independent isotropic porous plasticity

model is employed to describe the GBAZ, whereas a crystal-plasticity model which

accounts for the transition from partial dislocation to full dislocation mediated plasticity

is employed for the grain interior. The constitutive models of both phases are formulated

in a small strain framework and extended to finite deformation by use of logarithmic and

exponential mappings. Assuming the rule of mixtures, the overall behavior of a given

grain is obtained via volume averaging. The scale transition from a single grain to a

polycrystal is achieved by Taylor-type homogenization where a log-normal grain size

distribution is assumed. It is shown that the proposed model is able to capture the inverse

Hall–Petch effect, i.e., loss of strength with grain size refinement. Finally, the predictive

capability of the model is validated against experimental results on nanocrystalline

copper and nickel.

& 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Nanocrystalline (nc) materials are known to posses several distinct features when compared to coarse grainedpolycrystals. These include high strength and fatigue resistance, low ductility, pronounced rate dependence, tension–compression asymmetry and susceptibility to plastic instability. After the publication of the seminal work of Gleiter (1989),nanocrystalline materials have become the subject of intense research over the past two decades. Numerous review articleson both the experimental and modeling aspects of nc-materials have been published; recent reviews and comprehensive listsof literature may be found in Dao et al. (2007), Gleiter (2000), Kumar et al. (2003b), Meyers et al. (2006), Saada and Dirras(2009), Weertman (2007) and Wolf et al. (2005). Despite the extensive research activities on nc-materials, there are still somepoints of dispute about their properties and operative deformation mechanisms due to the fact that alternative synthesistechniques may produce substantially different microstructures with unique processing-induced defects. However, thefollowing points seem to be well agreed upon: (i) There is a strong competition between dislocation mediated crystalplasticity in the grain interiors and inelastic deformation mechanisms, i.e., atomic shuffling, grain boundary diffusionand sliding, in the GBAZ. (ii) Dislocations are mainly emitted from grain boundaries which act also as sinks for dislocations.(iii) Nc face centered cubic-materials exhibit stronger strain rate dependence as the grain size decreases.

ll rights reserved.

urses), tamer.elsayed@kaust.edu.sa (T. El Sayed).

E. Gurses, T. El Sayed / J. Mech. Phys. Solids 59 (2011) 732–749 733

The flow stress and hardness of coarse grained polycrystals are known to increase with grain size refinement. The relationof flow stress sy to grain size is given by the well-celebrated empirical Hall–Petch (H–P) relation (Hall, 1951; Petch, 1953):

sy ¼ s0þKHPffiffiffi

dp , ð1Þ

where s0, KHP and d are the friction stress, the Hall–Petch slope and the grain size, respectively. The physical mechanisms behindthe H–P relation have been explained by several dislocation-interaction-based approaches, such as dislocation pile-ups (Cottrell,1958; Eshelby et al., 1951), dislocations emitted by grain boundary ledges (Li, 1963) and geometrically necessary dislocations(Ashby, 1970). Contrary to coarse grained polycrystals, experimental evidence indicates that with further grain refinement belowa grain size of 10–50 nm, the strength of nc-materials decreases (Chokshi et al., 1989; Jeong et al., 2003; Lu et al., 1990; Masumuraet al., 1998; Sanders et al., 1997a; Schuh et al., 2002; Youngdahl et al., 1997) or the slope of H–P plot becomes smaller (Gertsmanet al., 1994; Hofler and Averback, 1990; Jang and Koch, 1990; Nieman et al., 1989, 1991) or the flow stress in H–P plot saturates byexhibiting a plateau type behavior (El-Sherik et al., 1992; Volpp et al., 1997). A comprehensive summary of experimental data fromthe literature on the effect of grain size on the flow stress of Cu can be found in Conrad (2003, 2004) and Figs. 15 and 19 of Meyerset al. (2006). The decrease of strength with grain refinement in the nanoscale is often denoted as the inverse Hall–Petch

relationship. In this regime, Coble creep (Coble, 1963) and grain boundary deformation mechanisms, such as atomic shuffling,grain boundary sliding and void initiation and growth, apparently constitute the dominant deformation modes. Van Vliet et al.(2003) presented a two-dimensional bubble raft model and observed directly the softening with grain refinement that isaccompanied by a transition in the primary deformation mechanism.

Most of our current theoretical understanding on operative deformation mechanisms in nc-materials has been obtainedby large-scale molecular dynamics (MD) simulations, see e.g. Schiøtz et al. (1998), Schiøtz and Jacobsen (2003), Schiøtz et al.(1999), Van Swygenhoven and Caro (1997), Van Swygenhoven et al. (2006), Van Swygenhoven et al. (1999b), Wolf et al.(2005) and Yamakov et al. (2002). MD simulations of nc-Cu (Schiøtz et al., 1999), nc-Ni (Van Swygenhoven et al., 1999b; VanSwygenhoven and Caro, 1997) and nc-Pd (Yamakov et al., 2002) have shown that deformation in nc-materials is mainlyconcentrated at the grain boundary region and the close neighboring atoms. Moreover, MD simulations successfullypredicted the dislocation nucleation from, and annihilation at, the grain boundaries (Schiøtz et al., 1998; Van Swygenhovenet al., 2002) and the crossover from dislocation-based to grain boundary-based deformation mechanisms with decreasinggrain size (Schiøtz and Jacobsen, 2003; Van Swygenhoven et al., 1999a; Yamakov et al., 2002). Although valuable informationabout atomic level details of deformation mechanisms has been gained by MD simulations, these models are still restricted tosmall specimen sizes and extremely high loading rates which do not represent current experimental conditions.

Continuum models have also been developed to describe the behavior of nc-materials. The first continuum based model fornanocrystalline metals was proposed by Carsley et al. (1995), where the strength and hardness dependence on grain size isstudied. The model is based on the assumption that the polycrystal is composed of two phases, i.e., a soft grain boundary phase anda hard intragranular region with the rule of mixtures. Assuming the hardness of the intragranular region is given by the classicalH–P relation and the hardness of the soft grain boundary phase is related to an amorphous metal, the inverse H–P relation wasestimated for grain sizes below 4 nm. Later, an improvement of Carsley et al. (1995) was provided by Konstantinidis and Aifantis(1998) where both phases are assumed to obey the classical H–P relation (1) but with a corrected slope KHP for the grain boundaryphase. In Kim et al. (2000), an alternative phase mixture model that considers diffusional flow of matter for the grain boundaryphase and isotropic viscoplastic with diffusion creep constitutive model for the crystalline phase was developed. The model wasable to predict inverse H–P behavior for samples having grains smaller than 16–100 nm depending on the loading rate. Kim andEstrin (2005) extended this model by introducing an additional parameter, i.e., a critical grain size, below which the dislocationcontribution to plastic strain in crystalline phase vanishes. Another two phase model was proposed by Fu et al. (2001), where thethickness of the grain boundary zone is assumed to depend on the grain size. The model predicted that the slope of the H–P plotmonotonously decreases and asymptotically approaches a plateau when the grain size is reduced. Wei and Gao (2008) developedan elastic–viscoplastic model as an extension to the work of Zhu et al. (2005). The model was not based on a phase mixtureassumption and did not introduce an additional parameter to quantify the volume fraction of grain boundaries, however, themodel incorporated strongly coupled deformation mechanisms of grain boundary sliding, grain boundary diffusion, grain interiordiffusion and grain interior plasticity, and provided a more rigorous computation of rate of dislocation emission from grainboundaries. The model predicted the inverse H–P behavior for grain sizes lower than 15 nm. It is important to note that none of theaforementioned models took into account the plastic anisotropy of the intragranular region, which is dictated by the crystallinestructure and slip systems.

Recently, as an alternative to phase mixture models, but still assuming nanocrystals as two phase materials,micromechanics-based homogenization schemes have been developed (Capolungo et al., 2005a, 2005b; Jiang and Weng,2004; Lebensohn et al., 2007; Mercier et al., 2007). Capolungo et al. (2005a) used a self-consistent scheme where grain coreswere treated as an inclusion into a matrix representing both grain boundaries and triple junctions. The dislocation basedviscoplastic constitutive model of Estrin (1996) was used for the inclusion phase. On the other hand, a thermally activatedmechanism, which takes into account the penetration of dislocations into grain boundaries and emission of dislocations fromgrain boundary ledges, was employed for the matrix phase. Capolungo et al. (2005b) in a self-consistent formulation similarto Capolungo et al. (2005a) extended the behavior of the inclusion phase by considering both the dislocation based model(Estrin, 1996) and the Coble (1963) creep mechanism, while the boundary phase was modeled as an amorphous materialhaving perfect elastic-plastic behavior. The critical grain size at which inverse H–P behavior is observed was found to be

E. Gurses, T. El Sayed / J. Mech. Phys. Solids 59 (2011) 732–749734

35 nm in Capolungo et al. (2005a) and in the range 25–100 nm in Capolungo et al. (2005b), depending on the applied strainrate. As an extension to Capolungo et al. (2005a), Benkassem et al. (2007) proposed a three-phase self-consistent model basedon the original work of Christensen and Lo (1979). In this work, the nanocrystalline material was assumed to be composed ofcoated inclusions embedded into an equivalent homogeneous medium. The coating represents both the grain boundaries andthe triple junctions, whereas the inclusion represents the grain cores. Although a change in the H–P slope was successfullypredicted, the model did not yield a softening with progressive grain refinement. Jiang and Weng (2004) developed ageneralized elastoplastic self-consistent formulation where grain interiors and grain boundaries were modeled by a rate-independent crystallographic multi-surface plasticity and an isotropic pressure-dependent Drucker’s-type plasticity,respectively. Lebensohn et al. (2007) presented a micromechanical model based on fast Fourier transforms to study theeffect of grain size, strain rate and pressure on the local and overall behavior of nc-materials under quasi-static and shockloading conditions. Mercier et al. (2007) proposed an extension of the work of Kim et al. (2000) by incorporating the effects ofelastic deformations and grain size distribution. In Mercier et al. (2007), a lognormal grain size distribution was assumed anda Taylor-type homogenization scheme was employed for the overall behavior of the nc-material.

Another approach, different from the previously discussed models, was followed in Schwaiger et al. (2003), Fu et al. (2001,2004) and Wei et al. (2006) where nanocrystalline polycrystals were still assumed to consist of two phases. However, thegrain structure was considered explicitly by the finite element discretization and different constitutive laws were assigned tothe elements at the grain boundary and the interior. Schwaiger et al. (2003) assumed a simple isotropic power-law type ratedependent constitutive response with a strain-based failure criterion for the grain boundary region. The grain boundary zonewas assumed to have a lower initial yield stress than the interior which was described by isotropic, linear hardening behavior.They also studied the rate-dependence of nanocrystalline and ultra-fine crystalline nickel for tensile and indentation testsand were able to capture pronounced rate sensitivity at the nanocrystalline range. Fu et al. (2001) took elastic anisotropy intoaccount in a very simple manner by assigning different elastic moduli to isotropic grains, which correspond to soft, mediumand hard orientations with respect to loading (Diehl, 1956), without considering plastic anisotropy due to the existence of slipsystems. The grain boundary region was described by a Voce-hardening law and its thickness was assumed to depend on thegrain size through a phenomenological relation. The model was implemented in a Eulerian finite element framework and thedecrease in the slope of the H–P plot due to grain size refinement was captured. Later Fu et al. (2004) extended this model bytaking both elastic and plastic anisotropy into account explicitly. The proposed model assumed crystal plasticity constitutivelaws with different material parameters for both the grain boundaries and the interiors. Furthermore, a hardening functionthat depends on the spatial position in the polycrystal, i.e., the distance from the closest grain boundary, was proposed in Fuet al. (2004). Wei et al. (2006) developed a rate-dependent amorphous plasticity model, which accounts for cavitation andrelated failure phenomena, to represent the grain boundary response. The grain interiors were simulated by using acontinuum crystal plasticity model in which the slip system resistances were assumed to be inversely proportional to thegrain size. The employed crystal plasticity model was based on Asaro et al. (2003) and Zhu et al. (2005), which accounted forthe transition from partial to complete dislocations depending on a critical grain size. The model was applied for grain sizes of10–50 nm and it was observed that the yield strength does not increase monotonously as the grain size decreases. It wasshown that a marked rate dependency and a tension-compression asymmetry were predicted as well.

Wei and Anand (2004) and Warner et al. (2006) coupled classical continuous finite elements with discontinuous interfaceelements in a two-dimensional finite element framework. A single crystal plasticity model was used for continuous finiteelements describing the intragranular region while a cohesive interface model was employed for the grain boundary zone.Wei and Anand (2004) studied grain sizes in the range of 14–61 nm and observed an inverse H–P relation, whereas Warneret al. (2006) found a change in the slope of H–P curve, but not an inverse relation for grain sizes of 5–50 nm. Later, Jerusalemet al. (2007) proposed a three-dimensional interface model for grain boundaries while assuming grain interiors to be elasticand isotropic. They investigated nanocrystals under both static and dynamic loading conditions and observed only an inverseH–P effect for grains smaller than 33 nm. Jerusalem and Radovitzky (2009) studied the response of nc-materials under shockloading by extending the former model (Jerusalem et al., 2007) to include isotropic rate-dependent plasticity for thedeviatoric response and an equation of state for the volumetric response of grain interiors. They investigated the hindering ofgrain boundary sliding due to friction and high pressure that inevitably occur under shock loading conditions.

In this paper, we introduce a rate-independent isotropic porous plasticity model accounting for the existence of voids inGBAZ and an anisotropic crystal plasticity model accounting for the transition from partial to full dislocation mediatedplasticity in the grain interior. The paper is organized as follows: The constitutive models for the single grain behavior, i.e., theporous plasticity model for GBAZ and the crystal plasticity model for grain interiors, and the Taylor-type averaging schemeare presented in Section 2. In Section 3, we summarize the numerical implementation of the constitutive models. In Section 4,the predictive capability of the proposed model through comparisons against experimental data from the literature isillustrated. Finally, we present our conclusions and final remarks in Section 5.

2. Model description

2.1. Single grain model

We first present a two-phase model for single grain behavior. A single grain is assumed to consist of a crystalline coreregion where deviatoric dislocation mediated plasticity is the dominating deformation mechanism and a grain boundary

E. Gurses, T. El Sayed / J. Mech. Phys. Solids 59 (2011) 732–749 735

affected zone (GBAZ) where mechanisms other than dislocations, such as atomic shuffling, grain boundary sliding and voidinitiation and growth are operative. The overall response of a single grain is then computed as the volume average of the twophases by using the mixture theory.

2.1.1. Single grain mixture model

Let u : X/x be the nonlinear deformation map at time t of a nanocrystalline solid body. u maps points X of the referenceconfiguration onto points x¼uðX,tÞ of the current configuration. F¼ruðX,tÞwith J¼ detF40 are the deformation gradientand the Jacobian, respectively. In what follows we consider the classical multiplicative decomposition of the deformationgradient F¼ FeFp into an elastic part Fe with Je ¼ detFe40 and a plastic part Fp with Jp ¼ detFp40 as first suggested by Lee(1969). Regarding a particular grain as a composite material consisting of a grain interior phase and a grain boundary phase,which we denote as GBAZ (Schwaiger et al., 2003), the free energy function can be written in a simple volume average form

W ¼ xWgiðF,Fpgi,Q giÞþð1�xÞWgbðF,Fp

gb,Q gbÞ, ð2Þ

where x is the volume fraction of the grain core region, Fpgi and Fp

gb denote the plastic part of the deformation gradient in thegrain interior and boundary phases, respectively. Qgi and Qgb in (2) denote some suitable collections of internal variables. Notethat the deformation gradient is the same for both phases, i.e., F¼ Fgb ¼ Fgi, whereas the plastic part of the deformationgradient is not necessarily the same, i.e., Fp

gbaFpgi. Similar to the free energy function (2), the average first Piola–Kirchhoff

stress P in a grain is given by

P¼ xPgiþð1�xÞPgb: ð3Þ

The volume average stress P, the grain interior stress Pgi and the grain boundary stress Pgb are computed from Coleman’srelations

P¼@W

@F, Pgi ¼

@Wgi

@F, Pgb ¼

@Wgb

@F: ð4Þ

As seen in (3) the volume fraction x determines how strongly the grain boundary mechanisms influence the overall stressresponse of a single grain. Therefore, the determination of x is of great importance for a predictive constitutive model.Assuming grains have cubical (Carsley et al., 1995) or tetrakaidecahedral (Palumbo et al., 1990) shape and the thickness ofgrain boundary zone dgb is constant, the volume fraction of grain cores can be evaluated as

x¼ðd�dgbÞ

3

d3: ð5Þ

In coarse grained polycrystals, x� 1 and the material is almost completely occupied by the grain interiors. Thus, the materialbehavior is governed by the grain interior phase. Note that the nature of the intercrystalline grain boundary zones dependsstrongly on the synthesis procedure of the nanocrystalline material. Schwaiger et al. (2003) observed for electrodepositednanocrystalline nickel that the grain boundary affected zone spans over 7–10 lattice parameters from the atomically sharpgrain boundary. On the other hand, Ranganathan et al. (2001) reported for vapor deposited nanocrystalline palladium andtitanium that the GBAZ is about 2–3 lattice plane spacings. Nevertheless, atomic simulations Van Swygenhoven et al. (2000)have indicated that the structure of grain boundaries in nc-materials are similar to that found in coarse grained polycrystals,and does not change with the grain size. Therefore, in what follows it will be assumed that the thickness of the grain boundaryzone dgb does not depend on the grain size.

2.1.2. Grain boundary model

We employ the isotropic porous plasticity model developed by Weinberg et al. (2006) to describe the plastic response ofthe grain boundary affected region while the elastic response is governed by isotropic elasticity. We briefly summarize themodel here and refer to El Sayed et al. (2008) and Weinberg et al. (2006) for further details. The response of the GBAZ isassumed to be independent of the grain size.

The plastic deformation rate is assumed to obey the following flow rule:

_FpgbFp�1

gb ¼_epMþ _ypN, ð6Þ

where multipliers _ep and _yp

are subject to irreversibility constraints, i.e., _epZ0, _y

pZ0. The tensors M and N are the directions

of the deviatoric and volumetric plastic deformation rates, respectively. The plastic flow directions are assumed to satisfy

trM¼ 0, M : M¼ 32 , N¼71

3I ð7Þ

with the plus sign in N corresponding to void expansion, and the minus sign to void collapse. Note that the tensors M and N areunknown and need to be determined as part of the solution through the use of, for example, Lagrange multipliers.

The free energy is assumed to have an additive structure

WgbðF,Fpgb,Ep,yp

Þ ¼WegbðF

egbÞþWp

gbðEp,ypÞ, ð8Þ

where Wegb(Fe

gb) and WpgbðE

p,ypÞ are the elastic and plastic stored energy densities, respectively. Material frame indifference

requires that Wegb has to depend on Fe

gb ¼ FFp�1gb through the elastic right Cauchy–Green tensor Ce

gb ¼ FeTgbFe

gb. In order to

E. Gurses, T. El Sayed / J. Mech. Phys. Solids 59 (2011) 732–749736

incorporate a logarithmic strain formulation, which allows the development of large strain constitutive models in small strainframework with purely geometric pre- and post-processing operations (Cuitino and Ortiz, 1992; Miehe et al., 2002), wealternatively express the elastic energy density

Wegb ¼We

gbðeegbÞ ð9Þ

in terms of the logarithmic elastic strain

eegb ¼

12logðCe

gbÞ: ð10Þ

The details of implementation of logarithmic elastic strain formulation are given in Section 3.3. In what follows, the particularchoices of the elastic and plastic free energy density functions are specified.

Elastic response: Due to the perturbed atomic arrangements in the GBAZ, the elastic behavior in the grain boundary phase isassumed to be isotropic. To this end, we consider an isochoric-volumetric split of the elastic strain tensor

eegb ¼

13trðee

gbÞ1þdevðeegbÞ, ð11Þ

and describe the elastic response of the grain boundaries by the classical quadratic energy storage function

Wegbðe

egbÞ ¼

12ktr2ðee

gbÞþmdevðeegbÞ : devðee

gbÞ, ð12Þ

where k and m are the bulk modulus and the shear modulus, respectively.Plastic response: We briefly summarize the porous plasticity model developed by Weinberg et al. (2006). The rate

sensitivity, microinertia effects and the temperature dependence of the original model are not considered here. The initiallocal volume fraction of voids is

f0 ¼N4pa3

0

3, ð13Þ

where N is the void density, i.e., the number of spherical voids per unit volume, and a0 is the initial mean void radius. Wedenote the mean void radius in the plastically deformed configuration by a. We consider a spherical shell in a plasticallyincompressible matrix, and neglect the elastic volume change of the voids. These assumptions lead to

Jp ¼ 1�f0þN4pa3

3, f ¼

f0þ Jp�1

Jp, ð14Þ

where Jp is the determinant of plastic part of the deformation gradient and f is the volume fraction of voids in the deformedconfiguration. A simple additive decomposition of the stored plastic energy into deviatoric and volumetric parts is assumedto be

WpgbðE

p,ypÞ ¼Wp,dev

gb ðEpÞþWp,volgb ðy

pÞ: ð15Þ

The deviatoric part is modeled by a conventional power-law of hardening

Wp,devgb ðEpÞ ¼

ns0Ep0

nþ11þ

Ep

Ep0

!ðnþ1Þ=n

, ð16Þ

where n is the hardening exponent, s0 is the yield stress, and Ep0 is a reference deviatoric plastic strain. The volumetric part of

the plastic energy is attributed to void growth. The stored energy for a spherical void in a power-law hardening material wasgiven in Ortiz and Molinari (1992). Void initiation and growth in nc-materials have been attributed to different sources.Kumar et al. (2003a) studied deformation and failure behavior of electrodeposited nanocrystalline nickel and reported thatvoids exist at grain boundaries and triple points, and they nucleate by dislocations emitted from the grain boundaries or theformation of wedges at triple junctions and cavities on ledges due to grain boundary sliding. Our model considers only theexisting voids and their growth under loading. To this end, mechanisms such as void nucleation due to dislocation emissionand grain boundary sliding, and void coalescence are not considered explicitly. The final form of the volumetric plastic energyfor a spherical shell model is (Weinberg et al., 2006)

Wp,volgb ðy

pÞ ¼

ns0Ep0

nþ1N

4pa3

3gðyp,nÞ, ð17Þ

where

gðyp,nÞ ¼

Z 1=f

11þ

2

3Ep0

logx

x�1þf0

f0þexpyp�1

0BB@

1CCAðnþ1Þ=n

dx: ð18Þ

Note that the use of a combined deviatoric–volumetric plasticity formulation with void growth for the GB phase naturallyprovides a criterion for failure initiated at grain boundaries. This is a common failure mechanism as observed experimentallyin Kumar et al. (2003a). To this end, a simple failure criterion is proposed in Section 4.3 that is based on two competinginelastic deformation modes (deviatoric and volumetric) in the grain boundary zone.

E. Gurses, T. El Sayed / J. Mech. Phys. Solids 59 (2011) 732–749 737

2.1.3. Grain interior model

The response of the grain interior region is described by a rate-independent crystallographic multi-surface plasticitymodel. The plastic deformation rate is assumed to obey the classical flow rule (Rice, 1971):

_FpgiF

p�1gi ¼

XN

a ¼ 1

_gasa �ma, ð19Þ

where _ga is the rate of crystallographic slip on slip system a; sa and ma are orthogonal unit vectors defining the slip directionand the slip plane normal corresponding to slip system a.

The free energy is assumed to have an additive structure

WgiðF,Fpgi,g

aÞ ¼WegiðF

egiÞþWp

giðgaÞ, ð20Þ

where Wegi(F

egi) and Wp

giðgaÞ are the elastic energy density and stored plastic energy, respectively. The elastic energy We

gi issolely a function of elastic deformation Fe

gi ¼ FFp�1gi and determines the elastic response of grain interiors, while the plastic

part Wpgi depends on the plastic slips ga in the slip systems and describes the hardening behavior. Similar to the constitutive

model of GBAZ, we can alternatively formulate the elastic energy density in terms of the logarithmic elastic strainee

gi ¼12 logðCe

giÞ, i.e.,

Wegi ¼We

giðeegiÞ: ð21Þ

In the sequel, the particular choices of the elastic free energy density and the plastic stored energy functions for grain corephase are specified.

Elastic response: The elastic response of the grain core phase is described by the following quadratic energy storagefunction:

Wegiðe

egiÞ ¼

12e

egi : C : ee

gi, ð22Þ

where the moduliChas three independent material parameters, c11, c12 and c44, characterizing the elastic anisotropy of cubiccrystals. The matrix representation of C with respect to a coordinate system parallel to the axes of a cubic crystal reads

C¼

c11 c12 c12 0 0 0

c12 c11 c12 0 0 0

c12 c12 c11 0 0 0

0 0 0 c44 0 0

0 0 0 0 c44 0

0 0 0 0 0 c44

26666666664

37777777775: ð23Þ

Plastic response: Transmission electron microscope observations on nanocrystalline metals have shown that only very fewdislocations are stored in grain interiors and, for very small grain sizes, nanoscale grains are essentially free of dislocations(Ke et al., 1995; Kumar et al., 2003a; Legros et al., 2000; Youngdahl et al., 2001). Similar results have been obtained bymolecular dynamics simulations (Schiøtz et al., 1998; Schiøtz and Jacobsen, 2003; Van Swygenhoven et al., 1999a; Yamakovet al., 2003). Dislocations exist in nanocrystalline metals, but mainly located at the grain boundaries, in addition to thosepresent in the boundaries accounting for the boundary misorientations. MD simulations (Derlet et al., 2003; Wolf et al., 2005)suggest that, after coming out of one side of the grain and traversing the grain, the dislocations usually disappear into the GBson the opposing side, such that no debris is left. In other words, the dislocation activity, e.g. dislocation nucleation andde-pinning, is highly localized at the grain boundaries.

In the following, we present a grain size dependent constitutive model for the grain interior phase that takes into accountthe transition in deformation mechanism from full dislocations to partial dislocations. The transition mechanism isincorporated into the model by using the dislocation model of Asaro et al. (2003), Asaro and Suresh (2005), Zhu et al. (2005) inwhich both full and partial dislocations are emitted from and absorbed into grain boundaries. The resolved shear stress valuerequired for emission of a dislocation from a grain boundary is given by (Asaro et al., 2003; Asaro and Suresh, 2005; Zhu et al.,2005)

tacr ¼ tcr ¼

Gb

dfor d4dcr ,

Gb

3dþ

d�deq

dGG for dodcr ,

8>><>>: ð24Þ

where G is the shear modulus, b is the Burgers vector and deq is the equilibrium spacing of partial dislocations which can beapproximated as (Asaro and Suresh, 2005; Zhu et al., 2005)

deq �1

12pb

G: ð25Þ

G¼ gsf =Gb is the dimensionless reduced stacking fault energy where gsf is the intrinsic stacking fault energy. dcr is the criticalgrain size below which emission of partial dislocations from grain boundaries start instead of full dislocations. The critical

E. Gurses, T. El Sayed / J. Mech. Phys. Solids 59 (2011) 732–749738

grain size is computed by Asaro and Suresh (2005) and Zhu et al. (2005)

dcr ¼2

3

d

d�deq

Gb2

gsf

: ð26Þ

Assuming d=ðd�deqÞ � 1, the critical grain size is estimated from (26) as dcr � 40 nm for Cu having relatively low stacking faultenergy and dcr � 16 nm for Ni having relatively high stacking fault energy. Therefore, stacking faults are more likely to beobserved in Cu as a result of partial dislocation emission from grain boundaries.

The slip systems of Cu when full dislocations are operative, i.e., d4dcr , are the standard twelve f1 1 1g/1 1 0S systems.The slip systems when partial dislocations become active, i.e., dodcr , are identical to the twelve f1 1 1g/1 1 2S twin systemsfor fcc materials, see e.g. Staroselsky and Anand (2004). Note that the twinning systems are polar in nature and therefore theresistance to slip in positive and negative directions is different. In what follows, it is assumed that only the easy direction ofslipping can occur for partial dislocations. The slip systems of nanocrystalline copper for full and partial dislocations aresummarized in Table 1.

The change in the critical resolved shear stress is expressed in the following rate form:

_taðgÞ ¼Xb

habðgÞ _gb, ð27Þ

where hab is the hardening matrix and taðg¼ 0Þ ¼ tcr given in (24). In order to obtain a variational framework of the hardeningrelations we follow Ortiz and Stainier (1999) and treat the hardening matrix explicitly at the beginning of the time step. Thisleads to the following quadratic approximation of the plastic energy density for symmetric hardening matrices:

Wpgiðg

anþ1Þ ¼Wp

gi,nþXa

tanDgaþ

1

2Dga

Xb

habn Dgb0@

1A, ð28Þ

where tan is the critical resolved shear stress for slip system a at time tn, and habn is the hardening matrix at time tn. We adoptthe classical hardening matrix from Peirce et al. (1982)

habðgÞ ¼ ½qþð1�qÞdab�hðgÞ, ð29Þ

where g ¼P

aga is the accumulated plastic slip on slip systems, dab is the Kronecker delta and q is the latent hardeningcoefficient. The hardening function hðgÞ is assumed to have the following form (Peirce et al., 1982):

hðgÞ ¼ h0 sechh0g

ts�tcr

� �: ð30Þ

The hardening function has three material parameters, namely the initial hardening modulus h0, the initial critical resolvedshear stress tcr and the saturation value of the critical resolved shear stress ts. We slightly modify (30) in order to incorporatea grain size dependence in the hardening behavior. First, the initial critical resolved shear stress tcr is a grain size dependentconstant given in (24). Furthermore, we assume that the initial hardening rate after plastic yielding is also grain sizedependent in such a way that the rate of hardening decreases with grain refinement. Grain size dependent hardening ismotivated by experimental findings that nc-materials often exhibit less hardening with grain refinement and they becomealmost ideally plastic after about 2% of plastic strain, see e.g. Champion et al. (2003), Cheng et al. (2005), Iyer et al. (1999) andWang and Ma (2004). A similar hardening behavior, i.e., a change in rate of hardening with grain refinement, has also beenobserved by Kim and Estrin (2005) in the simulation of low strain rate loading of nc-copper. As a result of the observed

Table 1The slip systems of fcc crystals for full and partial dislocations.

Full dislocations for d4dcr Partial dislocations for dodcr

Slip direction Plane normal Slip direction Plane normal

7 ½0 1 1� (1 1 1) ½2 1 1� (1 1 1)

7 ½1 0 1� (1 1 1) ½1 2 1� (1 1 1)

7 ½1 1 0� (1 1 1) ½1 1 2� (1 1 1)

7 ½1 0 1� ð1 1 1Þ [2 1 1] ð1 1 1Þ

7 ½0 1 1� ð1 1 1Þ ½1 2 1� ð1 1 1Þ

7 ½1 1 0� ð1 1 1Þ ½1 1 2� ð1 1 1Þ

7 ½0 1 1� ð1 1 1Þ ½2 1 1� ð1 1 1Þ

7 ½1 0 1� ð1 1 1Þ ½1 2 1� ð1 1 1Þ

7 ½1 1 0� ð1 1 1Þ ½1 1 2� ð1 1 1Þ

7 ½1 0 1� ð1 1 1Þ ½2 1 1� ð1 1 1Þ

7 ½0 1 1� ð1 1 1Þ [1 2 1] ð1 1 1Þ

7 ½1 1 0� ð1 1 1Þ ½1 1 2� ð1 1 1Þ

E. Gurses, T. El Sayed / J. Mech. Phys. Solids 59 (2011) 732–749 739

hardening behavior of nc-materials, we assume the initial hardening rate to be grain size dependent as well and employ thefollowing hardening function in (29):

hðg,dÞ ¼ h0d

d0sech

h0gtcrðdÞðx�1Þ

� �, ð31Þ

where d0 is a reference grain size and x¼ ts=tcrðdÞ is the ratio of the saturation critical resolved shear stress ts to initial one tcrðdÞ.Thus, the hardening behavior of grain interior phase is controlled by parameters h0, d0 andx in addition to the grain size d and totalplastic slip g. Note that the proposed grain size dependent hardening model is phenomenological in nature. Nevertheless, themodel is motivated by several experimental observations. It is well known that, in coarse grained polycrystals, the hardening ismainly governed by different types of dislocation multiplications yielding various types of interactions, e.g. formation of strongLomer–Cottrell sessile dislocations, weak sessile jogs as a result of orthogonal slip systems and others (Havner, 2005; Hull andBacon, 2001). Owing to low dislocation densities in nc-metals, a decrease in hardening with grain size refinement is an expectedoutcome in the absence of hardening mechanisms other than dislocation interactions.

2.2. Transition from single grain to polycrystal description

Taylor-type averaging is used to bridge the single grain and polycrystalline scales. In this framework, it is assumed that allthe grains are subject to the same deformation, i.e., the compatibility among grains is satisfied a priori. Taylor-type models donot take the spatial distribution of grains into account explicitly and the equilibrium condition among grains is notnecessarily fulfilled.

Furthermore, we assume that the grain size is not constant and has a particular distribution through the polycrystallinesample. The grain size distribution in polycrystalline materials can usually be well represented by a lognormal distributionthat is often used in various numerical models, see e.g. Mercier et al. (2007), Raeisinia et al. (2008) and Zhu et al. (2005, 2006).The probability density function P(d) of a lognormal grain size distribution reads as follows:

PðdÞ ¼1ffiffiffiffiffiffi

2pp

sdexp �

1

2

lnðd=d0Þ

s

� �2 !

, ð32Þ

where d0 (standard deviation of lnðdÞ) and s are parameters describing the shape of the distribution function andZ 10

PðdÞ dd¼ 1: ð33Þ

The mean grain size d and the variance ~s are the quantities that can be measured in experiments, and are related to d0 and sthrough the following identities:

d ¼ d0exps2

2

� �, ~s ¼ d2

0expðs2Þðexpðs2Þ�1Þ: ð34Þ

Two lognormal grain size distributions having different mean grain size and variance are depicted in Fig. 1. In the figure thered solid line corresponds to d ¼ 50 nm and ~s ¼ 50 nm2, while the green solid line is obtained for d ¼ 100 nm and~s ¼ 250 nm2. The blue and magenta steps correspond to the discrete realization of continuous distribution functions with200 grains.

0

0.01

0.02

0.03

0.04

0.05

0.06

0 50 100 150 200 250

Prob

abili

ty [

-]

Grain Size [nm]

Fig. 1. Lognormal grain size distribution. Two grain size distributions and their discrete realization are depicted. d ¼ 50 nm and ~s ¼ 50 nm2 for the red solid

line, while d ¼ 100 nm and ~s ¼ 250 nm2 for the green solid line. Blue and magenta steps correspond to the discrete realization of continuous distribution

functions with 200 grains. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

E. Gurses, T. El Sayed / J. Mech. Phys. Solids 59 (2011) 732–749740

Assuming a nanocrystalline sample consists of N randomly oriented grains with volumes Vi, i¼ 1 . . .N, the followingrelations hold among macroscopic and grain level quantities:

F¼ F1 ¼ � � � ¼ FN , P¼1

VXN

i ¼ 1

ViPi, ð35Þ

where V ¼PN

i ¼ 1 Vi is the total volume. In other words, all the grains deform with the same macroscopic deformation and theoverall homogenized response of the nc-material is given by a volume average over all grains. Note that as a result of the averagingscheme (35), finer grains influence the homogenized overall response less than coarser grains in a polycrystalline sample.

3. Constitutive updates

In this section, an incremental solution procedure is presented for the time integration of the constitutive equations forboth the GBAZ and the grain interior phase.

3.1. Constitutive updates for grain boundaries

We consider a typical time interval [tn, tn + 1], and assume that Fpgb,n,Ep

n and ypn are known at time tn, and the deformation

gradient Fn +1 at time tn +1 is given. The set of internal variables, i.e., Fpgb,nþ1,Ep

nþ1 and ypnþ1, the plastic flow directions M and N

and the first Piola–Kirchhoff stress Pgb,n + 1 at time tn + 1 need to be computed by the constitutive update algorithm.Discretization of the flow rule (6) using the exponential mapping (Cuitino and Ortiz, 1992) gives

Fpgb,nþ1 ¼ expðDEpMþDypNÞFp

gb,n, ð36Þ

where DEp ¼ Epnþ1�E

pn and Dyp

¼ ypnþ1�y

pn. The update of the remaining internal variables Ep

nþ1 and ypnþ1 and the direction of

plastic flow M, N is performed by a variational minimization problem. The constitutive update can be written in the followingvariational form:

CgbðFnþ1Þ ¼ minEp

nþ 1,yp

nþ 1M,N

fgbðFnþ1,Epnþ1,yp

nþ1,M,NÞ, ð37Þ

where the incremental objective function

fgbðFnþ1,Epnþ1,yp

nþ1,M,NÞ ¼Wegbðe

egb,nþ1ÞþWp

gbðEpnþ1,yp

nþ1Þ ð38Þ

is defined in terms of the elastic and plastic energies given in (12) and (15), respectively. Note that in (38) the logarithmicelastic strain ee

gb,nþ1 depends on Fn +1, M and N through (10) and (6). The minimization problem (37) is subject to thekinematic constraints (7), which can be solved by Lagrange multipliers and the plastic irreversibility constraints

DEpZ0, Dyp

Z0: ð39Þ

The plastic flow direction M can be solved analytically from the constraint conditions (7), and therefore, does not need to besolved by a numerical minimization, see e.g. Fancello et al. (2008), Ortiz and Stainier (1999) and Weinberg et al. (2006). It hasbeen shown in Ortiz and Stainier (1999) and Weinberg et al. (2006) that the effective work of deformation Cgb acts as apotential, and determines the first Piola–Kirchhoff stress tensor Pgb,n +1 at time tn + 1 similar to a hyperelastic material response

Pgb,nþ1 ¼@Cgb

@Fnþ1: ð40Þ

3.2. Constitutive updates for grain interiors

Similar to the previous section, we consider a typical time interval [tn, tn+ 1], and assume that Fgi,np and gan are known at time

tn, and the deformation gradient Fn +1 at time tn +1 is given. The set of internal variables, i.e., Fgi,n+ 1p and ganþ1 the first Piola–

Kirchhoff stress Pgi,n + 1 at time tn + 1 are to be computed by the constitutive update algorithm.Discretization of the flow rule (19) using the exponential mapping in Ortiz and Stainier (1999) leads to

Fpgi,nþ1 ¼ exp

XN

a ¼ 1

Dgasa �ma

!Fp

gi,n, ð41Þ

where Dg¼ gnþ1�gn. The update of the plastic slips ganþ1 is performed by a variational minimization problem.The constitutive update for the grain interiors is given by the minimization problem

CgiðFnþ1Þ ¼minga

nþ 1

fgiðFnþ1,ganþ1Þ, ð42Þ

where the incremental objective function

fgiðFnþ1,ganþ1Þ ¼Wegiðe

egi,nþ1ÞþWp

giðganþ1Þ ð43Þ

E. Gurses, T. El Sayed / J. Mech. Phys. Solids 59 (2011) 732–749 741

is defined in terms of the elastic and plastic energies given in (22) and (28), respectively. In this formulation, the plus andminus directions of a slip system are treated as different slip systems, and therefore, the minimization problem (42) is subjectto the irreversibility constraint

DgaZ0: ð44Þ

Note that the active set, i.e., the set of slip systems on which plastic slip occurs, is not known a priori, and needs to bedetermined as a part of the minimization problem. Thus, an active set search algorithm (Miehe and Schroder, 2001; Ortiz andStainier, 1999) is employed. Analogous to (40), the effective work of deformation Cgi acts as a potential, and determines thefirst Piola–Kirchhoff stress tensor Pgi,n + 1 at time tn + 1, i.e.,

Pgi,nþ1 ¼@Cgi

@Fnþ1: ð45Þ

3.3. Logarithmic elastic strain formulation

The implementation of the constitutive updates is done following the logarithmic elastic strain formulation presented inCuitino and Ortiz (1992) and Ortiz and Stainier (1999). This formulation reduces finite deformation constitutive updates tosmall strain updates with material independent, pure kinematic, pre- and post-processing operations. Elastic-predictorCauchy–Green tensors are defined for GBAZ and grain interiors

Ce,pregb,nþ1 ¼ Fp�T

gb,nCnþ1Fp�1gb,n,

Ce,pregi,nþ1 ¼ Fp�T

gi,n Cnþ1Fp�1gi,n : ð46Þ

Assuming that the commutation relation holds between the elastic-predictor Cauchy–Green tensors and the deviatoric flowdirections, it can be shown that

eegb,nþ1 ¼ ee,pre

gb,nþ1�DEpM�DypN,

eegi,nþ1 ¼ ee,pre

gi,nþ1�XN

a ¼ 1

Dgasa �ma, ð47Þ

where

ee,pregb,nþ1 ¼

12logðCe,pre

gb,nþ1Þ,

ee,pregi,nþ1 ¼

12logðCe,pre

gi,nþ1Þ ð48Þ

are the trial logarithmic elastic strains in the GBAZ and grain core phase, respectively. As a result of the relations (46)–(48), theconstitutive updates obtain an additive structure of elastic and plastic deformations analogous to small strain plasticityformulations.

The first Piola–Kirchhoff stresses (40) and (45) are obtained by an application of the chain rule

Pgb,nþ1 ¼@Cgb

@eegb,nþ1

:@ee

gb,nþ1

@Fnþ1,

Pgi,nþ1 ¼@Cgi

@eegi,nþ1

:@ee

gi,nþ1

@Fnþ1: ð49Þ

In (49) the first derivatives on the right hand side are computed by a constitutive model formulated in the small strainframework, while the last terms on the right hand side are purely kinematic tensorial derivatives which are independent ofthe material model. The details of the chain rule operations can be found in Weinberg et al. (2006).

4. Numerical examples and validation

It is well known that the Taylor-type averaging schemes require large number of grains to be sampled in order to bepredictive. In the following, we conduct a numerical study to identify the effect of number of grains on the average stressresponse. To this end, we ran a set of tensile simulations with the same material parameters but having different number ofgrains. In Fig. 2, the tensile stress–strain curves are plotted for ensembles of 100, 200, 500 and 1000 grains. As observed in thefigure, the average stress response does not change for ensembles having more than 200 grains. Therefore, in the followingsimulations the Taylor averaging scheme is utilized always with 200 grains.

4.1. Application to nanocrystalline copper

In this section we will use the proposed model to simulate the tensile behavior of nanocrystalline copper and present thepredictive capability of the model through comparisons with the experimental data from Sanders et al. (1997b). Five differentaverage grain sizes are considered (16, 22, 26, 49 and 110 nm) as experimentally investigated in Sanders et al. (1997b).

0

100

200

300

400

500

600

0 0.002 0.004 0.006 0.008 0.01 0.012

Eng

inee

ring

Str

ess

[MPa

]

Engineering Strain [-]

100 grains

200, 500 and 1000 grains

Fig. 2. Dependence of the overall stress response on the number of grains. Four different grain ensembles consisting of 100, 200, 500 and 1000 grains are

considered. The average stress response does not change for ensembles having more than 200 grains.

E. Gurses, T. El Sayed / J. Mech. Phys. Solids 59 (2011) 732–749742

The variance of the grain size distribution ~s is chosen as 20 nm2 for all grain sizes. The set of material parameters used in allsimulations is summarized in Table 2. Sanders et al. (1997b) measured the elastic constants, i.e., Young’s modulus and theshear modulus, of nanocrystalline Cu through mechanical and ultrasonic tests and showed that they can be as low as 80% ofthe coarse grained polycrystalline Cu. Schiøtz et al. (1998, 1999) conducted molecular dynamic simulations of nc-Cu andfound a decrease of 15–25% in the elastic modulus compared to coarse grained Cu. Furthermore, Jiang and Weng (2003, 2004)have simulated the same data (Sanders et al., 1997b) with a self-consistent homogenization method where Young’s moduluswas reduced up to a 20% for grain interiors and 30% for GBAZ. Therefore, the elastic constants c11, c12 and c44 of Cu singlecrystal in Table 3 are reduced accordingly from the values c11 ¼ 168:4 GPa, c12 ¼ 121:4 GPa and c44 ¼ 75:4 GPa given inCuitino and Ortiz (1992). A similar decrease is applied to Young’s modulus E for the isotropic GBAZ as well. The Burgers vectorb and the reduced stacking fault energy G are taken from Asaro and Suresh (2005). The critical grain size dcr in (26) and theequilibrium spacing of partial dislocations deq in (25) for nc-Cu are computed as dcr ¼ 42:5 nm (assuming d=ðd�deqÞ � 1) anddeq ¼ 1:69 nm by using the values given in Table 2. In the averaging procedure, the grains that are smaller than dcr exhibitcrystallographic slip on the slip systems for partial dislocations (see the second column of Table 1), whereas for the grains thatare bigger than dcr dislocations move on the classical fcc slip systems (see the first column of Table 1). Since the nc-metals donot show intra-grain dislocation structures, classical hardening mechanisms due to the interaction of dislocations are notexpected to play a dominant role in their stress response. Hence, only self-hardening is considered and latent-hardening iscompletely disregarded by setting the latent hardening coefficient q¼ 0. Concerning the thickness of GBAZ, there aredifferences in the experimental literature. Schwaiger et al. (2003) observed that the grain-boundary affected zone spans over7–10 lattice parameters while Ranganathan et al. (2001) reported that the GBAZ is about 2–3 lattice parameters. Therefore,staying in the range of experimental observations (0.5–3.5 nm), we set the thickness of the GBAZ dgb ¼ 2 nm. The rest of theparameters are fitted to capture the experimental data.

In Fig. 3, stress strain curves of nc-Cu are plotted together with the experimental data (Sanders et al., 1997b) for a uniaxialtension test where three different grain sizes are considered. Both the simulation results and the experimental data show thatthe response for the sample with d¼ 49 nm is substantially stronger than the one with d¼ 110 nm. On the other hand, a slightincrease in the strength is observed by reducing the grain size further from 49 nm to 26 nm. The comparison in Fig. 3 showsthat the stress–strain behavior for different grain sizes is well-predicted with the model proposed in this work.

Table 2Material constants for nanocrystalline Cu.

Grain interior Grain boundary

Elastic model parameters c11 ¼ 124:74 GPa E=95 GPa

c12 ¼ 89:93 GPa n=0.34 (–)

c44 ¼ 55:85 GPa

Plastic model parameters G¼ 25 GPa Ep0 = 0.013 (–)

b¼ 0:255 nm n=35 (–)

G=0.004 (–) s0 =0.05 GPa

h0=67.5 GPa N=100 mm�3

d0=225 nm a0=0.1 mm

x=11 (–) dgb=2 nm

q=0 (–)

0

100

200

300

400

500

600

0 0.005 0.01 0.015 0.02 0.025

Eng

inee

ring

Str

ess

[MPa

]

Engineering Strain [-]

Fig. 3. Comparison of the model response with the tensile stress–strain data from Sanders et al. (1997b) for nanocrystalline Cu. The solid lines are simulation

results (red for d=110 nm, blue for d=49 nm, green for d=26 nm) and the points are experimental data (square for d=110 nm, cross for d=49 nm, triangle for

d=26 nm). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

E. Gurses, T. El Sayed / J. Mech. Phys. Solids 59 (2011) 732–749 743

The grain size dependence of the model is further demonstrated through a Hall–Petch plot shown in Fig. 4. In the figure, thered solid line corresponds to the response of the proposed model, whereas blue crosses with error-bars are from Sanders et al.(1997b). Note that the Hall–Petch plot of coarse grained Cu, the solid green line, is also included in the plot for completeness.The inverse Hall–Petch effect that can be seen clearly in the experimental data is reproduced successfully by our model. Thestrongest size, i.e., the grain size at maximum strength is also well-captured. The model is shown to have the ability toreproduce grain size effects, e.g. a maximum strength for a grain size in the range 20–40 nm.

Fig. 5 shows how the relative contributions from the grain boundary model and the grain interior model vary with grainsize. In Fig. 5a the evolution of the total plastic strain, i.e., the deviatoric plastic strain Ep plus the volumetric plastic strain yp inGBAZ and plus the sum of plastic slips ga in grain interiors, in a tensile test is plotted for various grain sizes. The total amountof plasticity monotonously decreases with grain refinement. Figs. 5b and c demonstrate the ratio of the deviatoric plasticstrain to the total plastic strain and the volumetric plastic strain to the total plastic strain, respectively. As expected, thecontribution of the deviatoric and volumetric plastic strains, which are solely due to the inelastic deformations in GBAZ,increases as the grain size decreases, and reaches nearly 40% at the final deformed state for the 16 nm grain size. Thecontribution of the grain interiors to total plastic deformation is depicted in Fig. 5d. The contribution of dislocation mediatedplastic deformations monotonously decreases with progressive grain refinements in agreement with the results presented inFig. 3a in Wei and Gao (2008).

250

300

350

400

450

500

550

600

0.1 0.15 0.2 0.25

16nm222649110nm

Flow

str

ess

[MPa

]

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

Grain size d [nm]

Extrapolation ofHall-Petch for coarse

grained Cu

Fig. 4. Hall–Petch plot for nanocrystalline copper. Comparison of simulation results (red solid line) with experimental data (blue crosses) from Sanders et al.

(1997b). Experimental data is plotted with the error-bars based on the uncertainty (75 GPa) provided in Sanders et al. (1997b). The maximum strength

experimentally observed for a grain size in the range of 20–40 nm is captured by the numerical simulations. The green line is the extrapolation of the coarse-

grained Hall–Petch relation for Cu (Meyers and Chawla, 1984) to the nanocrystalline range. (For interpretation of the references to color in this figure legend,

the reader is referred to the web version of this article.)

E. Gurses, T. El Sayed / J. Mech. Phys. Solids 59 (2011) 732–749744

4.2. Application to nanocrystalline nickel

We further validate the proposed model by comparing against the experimental data on nanocrystalline nickel given inDalla Torre et al. (2005). The set of material parameters utilized in the simulation is given in Table 3. The elastic parametersare determined as follows: Dalla Torre et al. (2005) estimated the effective elastic modulus of their setup (the specimenmachine assembly) from the slope of the elastic part of their tensile stress–strain curves. They found that the elastic modulusof the assembly is � 1=3 of the elastic modulus of coarse grained Ni. Therefore, the elastic constants c11, c12 and c44 of the graininterior phase are adopted accordingly from the values c11 ¼ 247 GPa, c12 ¼ 147 GPa and c44 ¼ 125 GPa given in Wei et al.(2006), see Table 3. A similar decrease is applied to the Young’s modulus E for the isotropic GBAZ as well. The parameters forthe determination of initial critical resolved shear stress are determined as follows: The Burgers vector b and the reducedstacking fault energyG are taken from Asaro and Suresh (2005) and the shear modulus G is from Dalla Torre et al. (2005). Thecritical grain size dcr in (26) and the equilibrium spacing of partial dislocations deq in (25) for nc-Ni are computed asdcr ¼ 16:5 nm (assuming d=ðd�deqÞ � 1) and deq ¼ 0:66 nm by using the values given in Table 3. The thickness of the GBAZ, dgb,and the latent hardening coefficient, q, are assumed to be the same as in the previous nc-Cu example. The rest of theparameters are fitted to capture the experimental data.

In the simulations, two different grain sizes are considered as experimentally studied by Dalla Torre et al. (2005). Theaverage grain size for the two samples studied in Dalla Torre et al. (2005) are reported based on two different measurementtechniques, X-ray diffraction (XRD) method and transmission electron microscopy (TEM). It was reported that both methodsmeasured an average grain size of 21 nm for the finer sample. However, there was a substantial difference in the results of the

0

0.005

0.01

0.015

0.02

0.025

0.03

0 0.002 0.004 0.006 0.008 0.01 0.012

Tot

al P

last

ic S

trai

n [-

]

Engineering Strain [-]

d = 110 nm

d = 49 nm

d = 26 nm

d = 22 nm

d = 16 nm

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.002 0.004 0.006 0.008 0.01 0.012

Rat

io o

f D

evia

tori

c Pl

astic

Str

ain

[-]

Engineering Strain [-]

d = 110 nm

d = 49 nm

d = 26 nm

d = 22 nm

d = 16 nm

0.25

0.2

0.15

0.1

0.05

00 0.002 0.004 0.006 0.008 0.01 0.012

Rat

io o

f V

olum

etri

c Pl

astic

Str

ain

[-]

Engineering Strain [-]

d = 110 nm

d = 49 nm

d = 26 nm

d = 22 nm

d = 16 nm

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.002 0.004 0.006 0.008 0.01 0.012

Rat

io o

f C

ryst

allin

e Pl

astic

Slip

[-]

Engineering Strain [-]

d = 110 nm

d = 49 nm

d = 26 nm

d = 22 nm

d = 16 nm

Fig. 5. Grain size dependence of plastic contributions from GBAZ and grain interiors. (a) Change in total plastic strain, i.e., the deviatoric plastic strain Ep , the

volumetric plastic strain yp in GBAZ and the sum of plastic slips ga in grain interiors during tensile loading for various grain sizes. (b) Ratio of the deviatoric

plastic strains to total plastic strains. (c) Ratio of the volumetric plastic strains to total plastic strains. (d) Ratio of the sum of crystalline plastic slips to total

plastic strains.

E. Gurses, T. El Sayed / J. Mech. Phys. Solids 59 (2011) 732–749 745

two measurement techniques for the coarser specimen, i.e., XRD measured the average grain size as 37 nm while TEMmeasurement yielded 105 nm. In the simulations we set the mean grain size for the finer sample to 21 nm, whereas theaverage grain size is taken as 60 nm for the coarser sample as an intermediate value of the results of the two measurementtechniques. It was also reported in Dalla Torre et al. (2005) that the coarser specimen has a significantly broader grain sizedistribution than the finer one. Therefore, different variance values are assigned for two grain ensembles. We assume~s ¼ 50 nm2 and ~s ¼ 240 nm2 for the finer and coarser samples, respectively.

Fig. 6 shows the tensile stress–strain curves for nc-Ni for two different grain sizes. Both the simulation results and theexperimental data show that the response for the sample with d=21 nm is significantly stronger than the one with d=60 nm.As shown in Fig. 6, the stress–strain behavior for different grain sizes is successfully reproduced by the proposed model. Notethat the proposed model is also able to capture the difference in elastic behavior, i.e., the finer grained specimen has an elasticmodulus lower than the coarser one.

Dalla Torre et al. (2005) have investigated only two grain sizes which is not sufficient to generate a H–P plot. Therefore, wepresent a comparison of H–P plots for nc-Ni with the work of El-Sherik et al. (1992). Except for the elastic constants, the samematerial parameter set is used as given in Table 3. The elastic parameters c11, c12 and c44 are taken from Wei et al. (2006). Thenormalized Hall–Petch plots in Fig. 7 present the simulation results plotted against the experimental data from El-Sherik et al.(1992). The H–P data in El-Sherik et al. (1992) was provided in terms of hardness values, and therefore, they are presented inFig. 7 by normalizing to the hardness value measured for d¼ 11 nm. Similarly, simulation results are normalized to the flowstress computed for d¼ 11 nm. Experimentally observed progressively decreasing Hall–Petch slope is captured by thenumerical simulations. Note that, experimental data is available down to 11 nm, and the inverse Hall–Petch behavior has notbeen observed down to this grain size. In order to demonstrate that the proposed model exhibits the inverse H–P behavior,two further simulations with mean grain sizes d¼ 9,8 nm are performed. As can be seen from Fig. 7 a further refinement of thegrain size indeed results in a decrease in the flow stress leading to the inverse H–P behavior.

0

200

400

600

800

1000

1200

1400

1600

1800

2000

0 0.01 0.02 0.03 0.04 0.05 0.06

Eng

inee

ring

Str

ess

[MPa

]

Engineering Strain [-]

Fig. 6. Comparison of the model response with the tensile stress–strain data from Dalla Torre et al. (2005) for nanocrystalline Ni. The solid lines are

simulation results and the points are experimental data. Both the simulation results and the experimental data show that the response for the sample with

d=21 nm (blue line and blue crosses) is significantly stronger than the one with d=60 nm (red line and red squares). (For interpretation of the references to

color in this figure legend, the reader is referred to the web version of this article.)

Table 3Material constants for nanocrystalline Ni.

Grain interior Grain boundary

Elastic model parameters c11=89.82 GPa E=72.73 GPa

c12=53.45 GPa n=0.31 (–)

c44=45.45 GPa

Plastic model parameters G=75 GPa Ep0 = 0.013 (–)

b=0.248 nm n=35 (–)

G= 0.01 (–) s0 =0.05 GPa

h0=200 GPa N=100 mm�3

d0=180 nm a0=0.1 mm

x=5 (–) dgb=2 nm

q=0 (–)

0

0.2

0.4

0.6

0.8

1

1.2

0.1 0.15 0.2 0.25 0.3 0.35

8nm9nm11nm1521100nm

Nor

mal

ized

Flo

w s

tres

s / H

ardn

ess

[-]

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

Grain size d [nm]

Fig. 7. Normalized Hall–Petch plot for nanocrystalline nickel. Comparison of simulation results (red solid line) up to grain size of d¼ 8 nm with experimental

data (blue crosses) up to d¼ 11 nm from El-Sherik et al. (1992). (For interpretation of the references to color in this figure legend, the reader is referred to the

web version of this article.)

E. Gurses, T. El Sayed / J. Mech. Phys. Solids 59 (2011) 732–749746

4.3. Failure criterion

Finally, we demonstrate that the proposed model naturally offers a failure criterion that is initiated at grain boundaries asexperimentally observed in Kumar et al. (2003a). We propose a failure criterion that is based on two competing inelasticdeformation modes (deviatoric and volumetric) in the grain boundary zone. In other words, we specify a critical deviatoricplastic deformation Ep

c and a critical volumetric plastic deformation ypc such that the material is considered to fail if one of the

following conditions hold:

Ep4Epc , yp4yp

c : ð50Þ

If the first condition holds, then the material fails due to more dominant deviatoric inelastic deformations in the GB, e.g. GBsliding, whereas fulfillment of the second condition corresponds to volumetric inelastic deformation being the moredominant cause in failure in the GB due to significant void growth. To this end, a set of simple tension tests of nc-Cu aresimulated for various grain sizes. Furthermore, variation of failure strain with flow strength is investigated and the results arecompared qualitatively with experiments. Fig. 8a which is adapted from Ma (2003) summarizes the results of severalexperiments on nc-Cu as discrete points together with a solid eye guide line. Most experimental results agree with the generaltrend of reduction in ductility with increase in yield strength. In Fig. 8b the ductility-flow stress relation obtained by the

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 100 200 300 400 500 600 700 800

Tota

l Elo

ngat

ion

to F

ailu

re [-

]

Yield Strength [MPa]

0

0.2

0.4

0.6

0.8

1

0.7 0.75 0.8 0.85 0.9 0.95 1

Nor

mal

ized

Tot

al E

long

atio

n to

Fai

lure

[-]

Normalized Flow Stress [-]

Fig. 8. (a) Variation of tensile failure strain with yield strength for various nanocrystalline- and ultrafine-grained Cu polycrystals. The plot is adapted from

Ma (2003) where discrete points correspond to various experimental results and the solid blue line is an eye guide. (b) Normalized plot of flow stress vs.

ductility obtained by the proposed model. Four different grain sizes, d¼ 25,50,75,100 and 150 nm, are considered and both flow stress and failure strain

values are normalized to corresponding maximum values. (For interpretation of the references to color in this figure legend, the reader is referred to the web

version of this article.)

E. Gurses, T. El Sayed / J. Mech. Phys. Solids 59 (2011) 732–749 747

proposed model is presented in a normalized plot. Four different grain sizes, d¼ 25,50,75,100 and 150 nm, are considered inthe numerical tests and both the flow stress and the failure strain values are normalized to corresponding maximum values.Contrary to the plot in Fig. 8a in which the results for a wide range of grain sizes are plotted, i.e., from coarse grained Cu tonc-Cu, numerical results in Fig. 8b cover only nanocrystalline range. In other words, our numerical results approximatelycorrespond to the range of flow stress values � 3002600 MPa. The proposed model exhibits a loss of ductility with increasingflow stress in a qualitative agreement with the experimental results.

5. Summary and concluding remarks

A variational multiscale constitutive model in the finite deformation regime capable of capturing the mechanical behaviornanocrystalline fcc metals has been presented. The nc-material is considered as a two phase material consisting of a graininterior phase and a grain boundary effected zone (GBAZ). A rate-independent isotropic porous plasticity model whichaccounts for void growth is employed to describe the GBAZ, whereas a crystal-plasticity model which accounts for thetransition from partial dislocation to full dislocation mediated plasticity is used for the plasticity of the grain core regions.Assuming the rule of mixtures, the overall grain behavior is obtained by a volume average of the responses of two phases. Thescale transition from single grain to polycrystal is performed by a Taylor-type homogenization method where a log-normalgrain size distribution is assumed. It has been shown that the proposed model is able to capture the inverse Hall–Petch

behavior. The predictive capability of the model has been tested through comparisons against the experimental results ofSanders et al. (1997b) on nc-copper and Dalla Torre et al. (2005) and El-Sherik et al. (1992) on nc-nickel. Finally, we haveshown that the inclusion of porous plasticity enables the introduction of a physical failure criterion based on two competitiveinelastic deformation modes, i.e., volumetric and deviatoric plastic strain. Thus, the proposed model together with the failurecriterion can be used in the modeling of GB-failure and fracture in nanocrystalline fcc metals.

The strain rate dependence of nanocrystalline metals is known to be substantially stronger than their coarse grainedcounterparts, see e.g. Schwaiger et al. (2003) and Wei et al. (2004). The low activation volume in nc-metals, which is due tohighly localized dislocation activity at the grain boundaries, has been considered as a possible reason for the enhanced ratesensitivity (Asaro and Suresh, 2005; Wang and Ma, 2004). Future work will include the study of rate-dependence effects andthe contribution of Coble (1963) creep to the deformation in GBAZ.

Acknowledgments

The authors are grateful to the very relevant suggestions made by the reviewers. The authors would like to thank theKAUST Research Computing team for their technical support. This work was fully funded by the KAUST baseline fund.

References

Asaro, R.J., Krysl, P., Kad, B., 2003. Deformation mechanism transitions in nanoscale fcc metals. Philosophical Magazine Letters 83, 733–743.Asaro, R.J., Suresh, S., 2005. Mechanistic models for the activation volume and rate sensitivity in metals with nanocrystalline grains and nano-scale twins.

Acta Materialia 53, 3369–3382.Ashby, M.F., 1970. Deformation of plastically non-homogeneous materials. Philosophical Magazine 21, 399.Benkassem, S., Capolungo, L., Cherkaoui, M., 2007. Mechanical properties and multi-scale modeling of nanocrystalline materials. Acta Materialia 55,

3563–3572.Capolungo, L., Cherkaoui, M., Qu, J., 2005a. A self-consistent model for the inelastic deformation of nanocrystalline materials. Journal of Engineering

Materials and Technology—Transactions of the ASME 127, 400–407.Capolungo, L., Jochum, C., Cherkaoui, M., Qu, J., 2005b. Homogenization method for strength and inelastic behavior of nanocrystalline materials.

International Journal of Plasticity 21, 67–82.Carsley, J.E., Ning, J., Milligan, W.W., Hackney, S.A., Aifantis, E.C., 1995. A simple, mixture-based model for the grain-size dependence of strength in

nanophase metals. Nanostructured Materials 5, 441–448.Champion, Y., Langlois, C., Guerin-Mailly, S., Langlois, P., Bonnentien, J.L., Hytch, M.J., 2003. Near-perfect elastoplasticity in pure nanocrystalline copper.

Science 300, 310–311.Cheng, S., Ma, E., Wang, Y.M., Kecskes, L.J., Youssef, K.M., Koch, C.C., Trociewitz, U.P., Han, K., 2005. Tensile properties of in situ consolidated nanocrystalline

Cu. Acta Materialia 53, 1521–1533.Chokshi, A.H., Rosen, A., Karch, J., Gleiter, H., 1989. On the validity of the Hall–Petch relationship in nanocrystalline materials. Scripta Metallurgica 23,

1679–1684.Christensen, R.M., Lo, K.H., 1979. Solutions for effective shear properties in 3 phase sphere and cylinder models. Journal of the Mechanics and Physics of Solids

27, 315–330.Coble, R.L., 1963. A model for boundary diffusion controlled creep in polycrystalline materials. Journal of Applied Physics 34, 1679–1682.Conrad, H., 2003. Grain size dependence of the plastic deformation kinetics in Cu. Materials Science and Engineering A—Structural Materials Properties

Microstructure and Processing 341, 216–228.Conrad, H., 2004. Grain size dependence of the flow stress of Cu from millimeters to nanometers. Metallurgical and Materials Transactions A 35, 2681–2695.Cottrell, A.H., 1958. The 1958 institute of metals division lecture—theory of brittle fracture in steel and similar metals. Transactions of the American Institute

of Mining and Metallurgical Engineers 212, 192–203.Cuitino, A.M., Ortiz, M., 1992. A material-independent method for extending stress update algorithms from small-strain plasticity to finite plasticity with

multiplicative kinematics. Engineering Computations 9, 437–451.Dalla Torre, F., Spatig, P., Schaublin, R., Victoria, M., 2005. Deformation behaviour and microstructure of nanocrystalline electrodeposited and high pressure

torsioned nickel. Acta Materialia 53, 2337–2349.Dao, M., Lu, L., Asaro, R.J., De Hosson, J.T.M., Ma, E., 2007. Toward a quantitative understanding of mechanical behavior of nanocrystalline metals.

Acta Materialia 55, 4041–4065.

E. Gurses, T. El Sayed / J. Mech. Phys. Solids 59 (2011) 732–749748

Derlet, P.M., Hasnaoui, A., Van Swygenhoven, H., 2003. Atomistic simulations as guidance to experiments. Scripta Materialia 49, 629–635.Diehl, J., 1956. Zugverformung von Kupfer–Einkristallen. 1. Verfestigungskurven und Oberflachenerscheinungen. Zeitschrift fur Metallkunde 47,

331–343.El Sayed, T., Mota, A., Fraternali, F., Ortiz, M., 2008. A variational constitutive model for soft biological tissues. Journal of Biomechanics 41, 1458–1466.El-Sherik, A.M., Erb, U., Palumbo, G., Aust, K.T., 1992. Deviations from Hall–Petch behaviour in as-prepared nanocrystalline nickel. Scripta Metallurgica

et Materialia 27, 1185–1188.Eshelby, J.D., Frank, F.C., Nabarro, F.R.N., 1951. The equilibrium of linear arrays of dislocations. Philosophical Magazine 42, 351–364.Estrin, Y., 1996. Dislocation-density-related constitutive modeling. In: Krausz, A.S., Krausz, K. (Eds.), Unified Constitutive Laws of Plastic Deformation.

Academic Press, New York, pp. 69–106 (Chapter 2).Fancello, E., Vassoler, J.M., Stainier, L., 2008. A variational constitutive update algorithm for a set of isotropic hyperelastic–viscoplastic material models.

Computer Methods in Applied Mechanics and Engineering 197, 4132–4148.Fu, H.H., Benson, D.J., Meyers, M.A., 2001. Analytical and computational description of effect of grain size on yield stress of metals. Acta Materialia 49,

2567–2582.Fu, H.H., Benson, D.J., Meyers, M.A., 2004. Computational description of nanocrystalline deformation based on crystal plasticity. Acta Materialia 52,

4413–4425.Gertsman, V.Y., Hoffmann, M., Gleiter, H., Birringer, R., 1994. The study of grain size dependence of yield stress of copper for a wide grain size range.

Acta Metallurgica et Materialia 42, 3539–3544.Gleiter, H., 1989. Nanocrystalline materials. Progress in Materials Science 33, 223–315.Gleiter, H., 2000. Nanostructured materials: basic concepts and microstructure. Acta Materialia 48, 1–29.Hall, E.O., 1951. The deformation and ageing of mild steel. Part III: discussion and results. Proceedings of the Physical Society of London B 64, 747–753.Havner, K.S., 2005. On lattice and material-frame rotations and crystal hardening in high-symmetry axial loading. Philosophical Magazine 85, 2861–2894.Hofler, H.J., Averback, R.S., 1990. Grain growth in nanocrystalline TiO2 and its relation to Vickers hardness and fracture toughness. Scripta Metallurgica et

Materialia 24, 2401–2406.Hull, D., Bacon, D.J., 2001. Introduction to Dislocations, fourth ed. Butterworth-Heinemann.Iyer, R.S., Frey, C.A., Sastry, S.M.L., Waller, B.E., Buhro, W.E., 1999. Plastic deformation of nanocrystalline Cu and Cu–0.2 wt% B. Materials Science and

Engineering A—Structural Materials Properties Microstructure and Processing 264, 210–214.Jang, J.S.C., Koch, C.C., 1990. The Hall–Petch relationship in nanocrystalline iron produced by ball milling. Scripta Metallurgica et Materialia 24, 1599–1604.Jeong, D.H., Erb, U., Aust, K., Palumbo, G., 2003. The relationship between hardness and abrasive wear resistance of electrodeposited nanocrystalline

Ni–P coatings. Scripta Materialia 48, 1067–1072.Jerusalem, A., Radovitzky, R., 2009. A continuum model of nanocrystalline metals under shock loading. Modelling and Simulation in Materials Science and

Engineering 17, 025001.Jerusalem, A., Stainier, L., Radovitzky, R., 2007. A continuum model describing the reverse grain-size dependence of the strength of nanocrystalline materials.

Philosophical Magazine 87, 2541–2559.Jiang, B., Weng, G.J., 2003. A composite model for the grain size dependence of yield stress of nanograined materials. Metallurgical and Materials Transactions

A 34, 765–772.Jiang, B., Weng, G.J., 2004. A generalized self-consistent polycrystal model for the yield strength of nanocrystalline materials. Journal of the Mechanics and

Physics of Solids 52, 1125–1149.Ke, M., Hackney, S.A., Milligan, W.W., Aifantis, E.C., 1995. Observation and measurement of grain rotation and plastic strain in nanostructured metal thin

films. Nanostructured Materials 5, 689–697.Kim, H.S., Estrin, Y., 2005. Phase mixture modeling of the strain rate dependent mechanical behavior nanostructured materials. Acta Materialia 53, 765–772.Kim, H.S., Estrin, Y., Bush, M.B., 2000. Plastic deformation behavior of fine-grained materials. Acta Materialia 48, 493–504.Konstantinidis, D.A., Aifantis, E.C., 1998. On the anomalous hardness of nanocrystalline materials. Nanostructured Materials 10, 1111–1118.Kumar, K.S., Suresh, S., Chisholm, M.F., Horton, J.A., Wang, P., 2003a. Deformation of electrodeposited nanocrystalline nickel. Acta Materialia 51, 387–405.Kumar, K.S., Van Swygenhoven, H., Suresh, S., 2003b. Mechanical behavior of nanocrystalline metals and alloys. Acta Materialia 51, 5743–5774.Lebensohn, R.A., Bringa, E.M., Caro, A., 2007. A viscoplastic micromechanical model for the yield strength of nanocrystalline materials. Acta Materialia 55,

261–271.Lee, E.H., 1969. Elastic–plastic deformation at finite strains. Journal of Applied Mechanics 36, 1–6.Legros, M., Elliott, B.R., Rittner, M.N., Weertman, J.R., Hemker, K.J., 2000. Microsample tensile testing of nanocrystalline metals. Philosophical Magazine A—Physics

of Condensed Matter Structure Defects and Mechanical Properties 80, 1017–1026.Li, J.C.M., 1963. Petch relation and grain boundary sources. Transactions of the Metallurgical Society of AIME 227, 239.Lu, K., Wei, W.D., Wang, J.T., 1990. Microhardness and fracture properties of nanocrystalline Ni–P alloy. Scripta Metallurgica et Materialia 24, 2319–2323.Ma, E., 2003. Instabilities and ductility of nanocrystalline metals. Scripta Materialia 49, 663–668.Masumura, R.A., Hazzledine, P.M., Pande, C.S., 1998. Yield stress of fine grained materials. Acta Materialia 46, 4527–4534.Mercier, S., Molinari, A., Estrin, Y., 2007. Grain size dependence of strength of nanocrystalline materials as exemplified by copper: an elastic–viscoplastic

modelling approach. Journal of Materials Science 42, 1455–1465.Meyers, M.A., Chawla, K.K., 1984. Mechanical Metallurgy. Prentice-Hall, Englewood Cliffs, NJ.Meyers, M.E., Mishra, A., Benson, D.J., 2006. Mechanical properties of nanocrystalline materials. Progress in Materials Science 51, 427–556.Miehe, C., Apel, N., Lambrecht, M., 2002. Anisotropic additive plasticity in the logarithmic strain space: modular kinematic formulation and implementation

based on incremental minimization principles for standard materials. Computer Methods in Applied Mechanics and Engineering 191, 5383–5425.Miehe, C., Schroder, J., 2001. A comparative study of stress update algorithms for rate-independent and rate-dependent crystal plasticity. International

Journal for Numerical Methods in Engineering 50, 273–298.Nieman, G.W., Weertman, J.R., Siegel, R.W., 1989. Microhardness of nanocrystalline palladium and copper produced by inert-gas condensation. Scripta

Metallurgica 23, 2319–2323.Nieman, G.W., Weertman, J.R., Siegel, R.W., 1991. Mechanical behavior of nanocrystalline Cu and Pd. Journal of Materials Science 6, 1012–1027.Ortiz, M., Molinari, A., 1992. Effect of strain hardening and rate sensitivity on the dynamic growth of a void in a plastic material. Journal of Applied

Mechanics—Transactions of the ASME 59, 48–53.Ortiz, M., Stainier, L., 1999. The variational formulation of viscoplastic constitutive updates. Computer Methods in Applied Mechanics and Engineering 171,

419–444.Palumbo, G., Thorpe, S.J., Aust, K.T., 1990. On the contribution of triple junctions to the structure and properties of nanocrystalline materials. Scripta

Metallurgica et Materialia 24, 1347–1350.Peirce, D., Asaro, R.J., Needleman, A., 1982. An analysis of nonuniform and localized deformation in ductile single-crystals. Acta Metallurgica 30, 1087–1119.Petch, N.J., 1953. The cleavage strength of polycrystals. The Journal of the Iron and Steel Institute 174, 25–28.Raeisinia, B., Sinclair, C.W., Poole, W.J., Tome, C.N., 2008. On the impact of grain size distribution on the plastic behaviour of polycrystalline metals. Modelling

and Simulation in Materials Science and Engineering 16, 025001.Ranganathan, S., Divakar, R., Raghunathan, V.S., 2001. Interface structures in nanocrystalline materials. Scripta Materialia 44, 1169–1174.Rice, J.R., 1971. Inelastic constitutive relations for solids—an internal-variable theory and its application to metal plasticity. Journal of the Mechanics and

Physics of Solids 19, 433–455.Saada, G., Dirras, G., 2009. Mechanical properties of nanograined metallic polycrystals. In: Hirth, J.P., Kubin, L. (Eds.), Dislocations in Solids. Elsevier,

pp. 200–248.

E. Gurses, T. El Sayed / J. Mech. Phys. Solids 59 (2011) 732–749 749

Sanders, P.G., A, Y.C.J., Weertman, J.R., 1997a. The strength of nanocrystalline metals with and without flaws. Materials Science and Engineering A—StructuralMaterials Properties Microstructure and Processing 234, 77–82.

Sanders, P.G., Eastman, J.A., Weertman, J.R., 1997b. Elastic and tensile behavior of nanocrystalline copper and palladium. Acta Materialia 45, 4019–4025.Schiøtz, J., Di Tolla, F.D., Jacobsen, K.W., 1998. Softening of nanocrystalline metals at very small grain sizes. Nature 391, 561–563.Schiøtz, J., Jacobsen, K.W., 2003. A maximum in the strength of nanocrystalline copper. Science 301, 1357–1359.Schiøtz, J., Vegge, T., Di Tolla, F.D., Jacobsen, K.W., 1999. Atomic-scale simulations of the mechanical deformation of nanocrystalline metals. Physical Review B

60, 11971–11983.Schuh, C.A., Nieh, T.G., Yamasaki, T., 2002. Hall–Petch breakdown manifested in abrasive wear resistance of nanocrystalline nickel. Scripta Materialia 46,

735–740.Schwaiger, R., Moser, B., Dao, M., Chollacoop, N., Suresh, S., 2003. Some critical experiments on the strain-rate sensitivity of nickel. Acta Materialia 51,

5159–5172.Staroselsky, A., Anand, L., 2004. Inelastic deformation of polycrystalline face centered cubic materials by slip and twinning. Journal of the Mechanics and

Physics of Solids 46, 671–696.Van Swygenhoven, H., Caro, A., 1997. Plastic behavior of nanophase Ni: a molecular dynamics computer simulation. Applied Physics Letters 71, 1652–1654.Van Swygenhoven, H., Derlet, P.M., Frøseth, A.G., 2006. Nucleation and propagation of dislocations in nanocrystalline fcc metals. Acta Materialia 54,

1975–1983.Van Swygenhoven, H., Derlet, P.M., Hasnaoui, A., 2002. Atomic mechanism for dislocation emission from nanosized grain boundaries. Physical Review B 66,

024101.Van Swygenhoven, H., Farkas, D., Caro, A., 2000. Grain-boundary structures in polycrystalline metals at the nanoscale. Physical Review B 62, 831–838.Van Swygenhoven, H., Spaczer, M., Caro, A., 1999a. Microscopic description of plasticity in computer generated metallic nanophase samples: a comparison

between Cu and Ni. Acta Materialia 47, 3117–3126.Van Swygenhoven, H., Spaczer, M., Caro, A., Farkas, D., 1999b. Competing plastic deformation mechanisms in nanophase metals. Physical Review B 60, 22–25.Van Vliet, K.J., Tsikata, S., Suresh, S., 2003. Model experiments for direct visualization of grain boundary deformation in nanocrystalline materials. Applied

Physics Letters 83, 1441–1443.Volpp, T., Goring, E., Kuschke, W., Arzt, E., 1997. Grain size determination and limits to Hall–Petch behavior in nanocrystalline NiAl powders. Nanostructured

Materials 8, 855–865.Wang, Y.M., Ma, E., 2004. Strain hardening, strain rate sensitivity, and ductility of nanostructured metals. Materials Science and Engineering A—Structural

Materials Properties Microstructure and Processing 375, 46–52.Warner, D.H., Sansoz, F., Molinari, J.F., 2006. Atomistic based continuum investigation of plastic deformation in nanocrystalline copper. International Journal

of Plasticity 22, 754–774.Weertman, J.R., 2007. Mechanical behavior of nanocrystalline metals. In: Koch, C.C. (Ed.), Nanostructured Materials: Processing, Properties, and Applications.

William Andrew Publishing, pp. 537–564.Wei, Q., Cheng, S., Ramesh, K.T., Ma, E., 2004. Effect of nanocrystalline and ultrafine grain sizes on the strain rate sensitivity and activation volume: fcc versus

bcc metals. Materials Science and Engineering A—Structural Materials Properties Microstructure and Processing 381, 71–79.Wei, Y., Gao, H., 2008. An elastic–viscoplastic model of deformation in nanocrystalline metals based on coupled mechanisms in grain boundaries and grain

interiors. Materials Science and Engineering A—Structural Materials Properties Microstructure and Processing 478, 16–25.Wei, Y.J., Anand, L., 2004. Grain-boundary sliding and separation in polycrystalline metals: application to nanocrystalline fcc metals. Journal of the

Mechanics and Physics of Solids 52, 2587–2616.Wei, Y.J., Su, C., Anand, L., 2006. A computational study of the mechanical behavior of nanocrystalline fcc metals. Acta Materialia 54, 3177–3190.Weinberg, K., Mota, A., Ortiz, M., 2006. A variational constitutive model for porous metal plasticity. Computational Mechanics 37, 142–152.Wolf, D., Yamakov, V., Phillpot, S.R., Mukherjee, A., Gleiter, H., 2005. Deformation of nanocrystalline materials by molecular-dynamics simulation:

relationship to experiments? Acta Materialia 53, 1–40.Yamakov, V., Wolf, D., Phillpot, S.R., Gleiter, H., 2002. Grain-boundary diffusion creep in nanocrystalline palladium by molecular-dynamics simulation.

Acta Materialia 50, 61–73.Yamakov, V., Wolf, D., Phillpot, S.R., Mukherjee, A.K., Gleiter, H., 2003. Deformation mechanism crossover and mechanical behaviour in nanocrystalline

materials. Philosophical Magazine Letters 83, 385–393.Youngdahl, C., Weertman, J., Hugo, R., Kung, H., 2001. Deformation behavior in nanocrystalline copper. Scripta Materialia 44, 1475–1478.Youngdahl, C.J.A., Sanders, P.G., Eastman, J.A., Weertman, J.R., 1997. Compressive yield strengths of nanocrystalline Cu and Pd. Scripta Materialia 37,

809–813.Zhu, B., Asaro, R.J., Krysl, P., Bailey, R., 2005. Transition of deformation mechanisms and its connection to grain size distribution in nanocrystalline metals.

Acta Materialia 53, 4825–4838.Zhu, B., Asaro, R.J., Krysl, P., Zhang, K., Weertman, J.R., 2006. Effect of grain size distribution on the mechanical response of nanocrystalline metals: part II.

Acta Materialia 54, 3307–3320.