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Materials Science and Engineering A 520 (2009) 121–133

Contents lists available at ScienceDirect

Materials Science and Engineering A

journa l homepage: www.e lsev ier .com/ locate /msea

rain size, grain boundary sliding, and grain boundary interaction effects onanocrystalline behavior

. Shi, M.A. Zikry ∗

epartment of Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC 27695-7910, USA

r t i c l e i n f o

rticle history:eceived 13 March 2009eceived in revised form 6 May 2009ccepted 7 May 2009

eywords:

a b s t r a c t

A dislocation–density grain boundary (GB) interaction scheme, a GB misorientation dependentdislocation–density relation, and a grain boundary sliding (GBS) model are presented to account for thebehavior of nanocrystalline aggregates with grain sizes ranging from 25 nm to 200 nm. These schemes arecoupled to a dislocation–density multiple slip crystalline plasticity formulation and specialized finite ele-ment algorithms to predict the response of nanocrystalline aggregates. These schemes are based on slip

rain boundariesislocation–density grain boundary

nteractionsinite elementsrystal plasticityall–Petch relation

system compatibility, local resolved shear stresses, and immobile and mobile dislocation–density evo-lution. A conservation law for dislocation–densities is used to balance dislocation–density absorption,transmission and emission from the GB. The relation between yield stresses and grain sizes is consis-tent with the Hall–Petch relation. The results also indicate that GB sliding and grain-size effects affectcrack behavior by local dislocation–density and slip evolution at critical GBs. Furthermore, the predictionsindicate that GBS increases with decreasing grain sizes, and results in lower normal stresses in critical

y offs

rain boundary sliding locations. Hence, GBS ma

. Introduction

It is well known that mechanical properties, such as yieldtrength, of crystalline materials are closely related to the grain size.maller grain sizes can correspond to higher yield and flow stressesn crystalline materials, as indicated by the Hall–Petch relationship1,2]

y = �o + kd−1/2, (1)

here �o is the lattice friction stress to move individual dislo-ations [3], k is a constant commonly interpreted to representhe stress needed to extend dislocation activity into grains adja-ent to regions that have already yielded [4], and �y is the yieldtress of the current material. The Hall–Petch relationship is gen-rally applicable for grain sizes larger than the order of tens ofanometers. There are numerous experimental results that supporthe Hall–Petch relationship. For face centered crystal (f.c.c.) cop-er, experimental results by [3–7] have confirmed the Hall–Petchelation when the grain size is larger than 10 nm. When the

rain size is less than 10 nm, the Hall–Petch relationship seems toreak down, and an inverse Hall–Petch appears to be operative.t this regime, it is suggested that the yield stress of nanocrys-

als decreases as the grain size decreases, and it attains a lower

∗ Corresponding author. Tel.: +1 919 515 5237; fax: +1 919 515 7968.E-mail address: zikry@ncsu.edu (M.A. Zikry).

921-5093/$ – see front matter © 2009 Elsevier B.V. All rights reserved.oi:10.1016/j.msea.2009.05.012

et strength increases associated with decreases in grain size.© 2009 Elsevier B.V. All rights reserved.

bound limit corresponding to the yield stress of amorphous mate-rials [8,9].

Grain boundaries (GBs) play a critical role in the yield stressof materials in that normally they can act as barriers or initiatorsof dislocation activities, and can increase the threshold energy fordislocation movement [10]. The dislocation structures in the GB arevery different from those in the grain interior due to various types ofpartial dislocations [11]. The initial immobile dislocation–density(before deformation) of the GB region is closely related to themisorientation, and for low angle GBs, there is a simple relationthat gives the initial GB dislocation–density by accounting forGB misorientation angles [12]. Based on this relation, the initialGB dislocation–densities of polycrystalline materials with variousgrain sizes can be obtained. Since high strength GBs can repelincoming dislocations and emit dislocations, it is critical to modelthese GB effects. This is especially significant for aggregates withgrain sizes of several nanometers, where GB effects are more sig-nificant in comparison with coarse grained aggregates.

There are four major deformation modes that could affect thebehavior of crystalline materials: (i) grain boundary sliding (GBS)due to the atomic shuffling of the GB interface, (ii) collective GBmigration, (iii) stacking faults, and (iv) dislocation activities in the

interface and grain interiors [13]. The first two modes correspondto GB-mediated deformation, and the last two modes correspondto dislocation-mediated deformation. These deformation modeswork together in characterizing the overall inelastic behavior ofcrystalline materials.

1 and E

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22 J. Shi, M.A. Zikry / Materials Science

For coarse grain materials, the plastic deformation is mainly dueo the dislocation-mediated deformation such as full and partialislocation evolutions and annihilations, in which GBs act as barrierf dislocations movement, sink and source of dislocations. This maye understood by considering the sequence of events involved inhe initiation of plastic flow from a point source (within one grain)n the polycrystalline aggregates [14]. The strengthening providedy GBs depends on GB structure, misorientations and interactionsetween dislocations and GBs.

For crystalline materials with grain sizes of several nanometers,he plastic deformation is mainly attributed to the GB-mediatedeformation, such as GBS and GB migration. There are generallywo different types of GBS. The first is Rachinger sliding [15],hich refers to the relative displacement of adjacent grains where

he grains retain their original shape but displace with respect toach other. Since polycrystalline aggregates generally have irreg-lar grain shapes, Rachinger sliding has to be accommodated byome intragranular movement of dislocations within the adjacentrains. The other type of sliding is Lifshitz sliding [16], which referso the boundary offsets that develop as a direct consequence of thetress-directed diffusion of vacancies. The above types of sliding isue to thermal activation process, such as diffusion and atom shuf-ing, although some MD simulations [13,17] indicate that GB slidingay also happen at 0 K, which indicates that the GBS also contains

n athermal component.One major issue that has been extensively investigated is the

nderstanding on how dislocations interact with GBs. SEM andEM experimental observations have clearly indicated that disloca-ions can interact with GBs in different ways through direct transfer,bsorption, or transmission across the GB with residual GB dislo-ations and pile-ups that can result in dislocation emission or theucleation of intergranular cracks. From these observations, sev-ral criteria on dislocation transmission have been proposed forhese complex processes [18–22]. For example, Lee et al. have pro-osed three criteria for dislocation transmission: (i) that the angleetween the lines of intersection between the GB and each sliplane must be a minimum, (ii) that the magnitude of the Burg-rs vectors of the dislocations that remains in the GB must be ainimum; (iii) that the resolved shear stress on the outgoing slip

ystem must be a maximum. These three criteria have been usedo understand single dislocation transmissions and to rationalize

ultiple dislocation activities [23–25]. By understanding these GBffects, there are also experiments directly studying the yield stressf crystalline materials. For example the yield stresses of polycrys-alline copper with different grain sizes have been studied, and theall–Petch relation has been confirmed [4–6].

Computational simulations have also provided insights of GBctivities on different physical scales. Molecular dynamics (MD)imulations have shown that GBs can act as sinks and sources ofislocations [26,27]. Dislocations can be absorbed in the GB caus-

ng pile-ups [26,28] and cross-slip [28]. These MD methods providensights that may not be possible with TEM in-situ observations onhe nano-scale. But, severe limitations on time and length scales

ay render these simulations ineffective on the microstructuralhysical scale that pertains to the inelastic behavior and yield stressf crystalline aggregates with different grain sizes.

Finite element methods (FEM) and different crystalline plastic-ty formulations have provided further insights on GB behavior.shmawi and Zikry [29] have introduced interfacial GB regions to

rack slip and dislocation–density transmissions and intersectionsor a formulation based on dislocation–density based crystalline

lasticity. Other investigators [30–32] have coupled FEM with MDormulations for a multi-scale approach. There have been a lot offforts on modeling the grain size effects using FEM. Fu et al. [33]resent a crystal plasticity model, in which plastic flow is a func-ion of grain size incorporating different dislocation accumulation

ngineering A 520 (2009) 121–133

rates in GB regions and grain interiors. The material is modeledas a monocrystalline core surrounded by a mantle (GB region)with a high work hardening rate response defined by Voce equa-tion. Cheong et al. [34] proposed a non-local dislocation-mechanicsbased crystallographic theory to describe the evolution of disloca-tion mean spacings within each grain, and grain size effect is studiedby FEM incorporated with grain interaction effects. A three-phasemicromechanical scheme accounting for stress and strain rate jumpat the grain core/grain boundary interface has been developed byBenkassem et al. [35] to describe the size effect in the viscoplasticbehavior of pure f.c.c. polycrystalline materials. The above methodsconsider the dislocation-mediated deformation and use differentGB properties in the prediction on yield stress of polycrystals. How-ever, none of the above methods account for important physicalmechanism, such as the interactions between dislocations and GB,the initial GB dislocation–densities, and GB-mediated deformationmodes, which can render them inaccurate in the modeling of yieldstress of nanocrystalline materials.

As a part of this framework, a recently proposeddislocation–density grain boundary interaction (DDGBI) schemebetween GBs and adjacent grain interiors has been used. Thisscheme is based on accounting for three interrelated processes:dislocation–density absorption, emission and transmission. Aconservation law for dislocation–density is used to balance dis-location activities within GB regions. A step-by-step illustrationof the DDGBI scheme is given by [36,37]. By using this scheme,together with other physical properties of GBs in nanocrystallinematerials, the grain size effect of crystalline material can bemodeled and predicted. In this investigation, we will use thisdislocation–density GB interaction scheme, a GB misorientationdependent dislocation–density relation, and a GBS model topredict and understand the behavior of nanocrystalline aggregateswith different grain sizes. We will also investigate how these inter-related GB effects affect aggregates with a preexisting crack. Thisproposed framework is a bridge to link nanocrystalline phenomenato a microstructural relevant scale that would be associated withnanocrystalline aggregates.

This paper is organized as follows: the crystalline plasticity con-stitutive formulation is presented in Section 2, which include thecomputational approach, the proposed dislocation–density grainboundary interaction scheme, the misorientation dependence oninitial GB dislocation–density and GBS mechanism; computationalresults and discussions are presented in Section 3, which includethe process to illustrate the DDGBI scheme, the prediction on grainsize effects and a discussion on how GBS affect crack behavior innanocrystalline materials; a summary of the significant results isgiven in Section 4.

2. Constitutive formulation

2.1. Multiple-slip dislocation–density based crystallineformulation

In this section, a constitutive formulation for the finite deforma-tion of rate dependent multiple-slip crystal plasticity is outlined.The detailed presentation of this constitutive formulation is givenby Zikry and Kao [38].

It is assumed that the deformation gradient can be decomposedinto elastic and plastic components, starting from the decomposi-tion of the velocity gradient Vi,j, into symmetric and anti-symmetricparts as

Vij = Dij + Wij, (2)

where Wij is the anti-symmetric part, representing the spin tensor,and Dij is symmetric part, representing the deformation rate tensor.

and E

ID

D

W

petc

D

a

W

w

ωv

P

ω

ts

�

w

raTsd

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wmdh

t

�

sp

m

�

(ti

J. Shi, M.A. Zikry / Materials Science

t is assumed that the spin tensor Wij and deformation rate tensorij can be further decomposed into elastic and plastic parts as

ij = Dpij

+ D∗ij, (3a)

ij = Wpij

+ W∗ij . (3b)

The superscript * means that the elastic part and the superscriptsdenotes the plastic part. W∗

ijrepresents the rigid body spin. The

lastic components of the gradient D∗ij

correspond to the elastic lat-ice distortion and the inelastic parts are defined in terms of therystallographic slip rates as

pij

= P(˛)ij

� (˛), (4a)

ndpij

= ω(˛)ij

� (˛), (4b)

here ˛ is summed over all slip systems and the tensors P(˛)ij

and(˛)ij

are defined in terms of the unit normals and the unit slipectors as

(˛)ij

= 12

(s(˛)i

n(˛)j

+ s(˛)j

n(˛)i

), (5a)

(˛)ij

= 12

(s(˛)i

n(˛)j

− s(˛)j

n(˛)i

). (5b)

For a rate dependent formulation, the slip-rates are functions ofhe resolved shear stress and the reference shear stress on each slipystems in a power law form of

˙ (˛) = � (˛)ref

[�(˛)

�(˛)ref

][�(˛)

�(˛)ref

](1/m)−1

(no sum on ˛), (6)

here � (˛)ref

is the reference shear strain rate which corresponds to a

eference shear stress �(˛)ref

, and m is the rate sensitivity parameter,nd for most of the metals, m is close to 100 at room temperature.he reference shear stress is a modification of widely accepted clas-ical forms that relate the reference shear stress to a square-rootependence on the immobilized dislocation–density as

(˛)ref

= �(˛)y + aGb

12∑�=1

√�(�)

im, (7)

here b is the magnitude of burgers vector, G is the shearodulus, �(˛)

y is the static yield stress, �im is the immobileislocation–density, and a is interaction coefficients, and generallyave a magnitude of unity.

The resolved shear stress �(˛) on the ˛th slip system is given inerms of the Cauchy stress, �ij as

(˛) = P(˛)ij

�ij. (8)

For a deformed material, it can be assumed that the dislocationtructure of total dislocation–density �(˛) can be additively decom-osed into two components: immobile dislocation–density �(˛)

imand

obile dislocation–density �(˛)m as

(˛) = �(˛)m + �(˛)

im. (9)

During an increment of strain, the dislocation–density generatesdenoted by �(˛)+) and annihilates (denoted by �(˛)−). It is assumedhat there exist balance laws to account for dislocation generation,

nteraction, trapping and recovery as:

d�(˛)m

dt=� (˛)

(gsour

b2

(�(˛)

im

�(˛)m

)−gminter

b2exp(

− H

kT

)−gimmob

b

√�(˛)

im

),

(10a)

ngineering A 520 (2009) 121–133 123

d�(˛)im

dt= � (˛)

(gminter

b2exp(

− H

kT

)

+ gimmob

b

√�(˛)

im− grecov exp

(− H

kT

)�(˛)

im

), (10b)

where gsour is a coefficient pertaining to an increase in the mobiledislocation–density due to dislocation sources; gminter is a coeffi-cient related to the trapping of mobile dislocation due to forestinteractions; grecov is a coefficient related to the rearrangementand annihilation of immobile dislocations; gimmob is a coefficientpertaining to the immobilization of mobile dislocations; H is theactivation enthalpy; and k is the Boltzmann’s constant. As theseevolutionary equations indicate, the dislocation activities relatedto recovery and trapping are coupled to thermal activation.

2.2. The dislocation–density grain boundary interaction (DDGBI)scheme

In this section, a DDGBI scheme is presented to accountfor the GB dislocation–density activities. It is assumed that thedislocation–density interactions occur between slip systems oneach side of the GB interface. A brief outline of the method willbe given here, for detailed presentation, see [36].

2.2.1. Criteria 1A transmission factor is proposed based on two components.

The first one is the angle ˛, which is the angle between the inter-section lines of the slip planes and GB planes; the second part is theangle ˇ between slip directions of these two slip systems. Hence,the transmission factor M˛ˇ can be denoted as

M˛ˇ = cos ˛ · cos ˇ, (11)

where

˛ = arccos(�l1 · �l2) in which (�l1 = �n1 · �nGB and�l2 = �n2 · �nGB), (12a)

and

ˇ = arccos(�s1 · �s2). (12b)

2.2.2. Criteria 2The ratio of resolved shear stress to the reference shear stress

of the outgoing slip system (stress ratio) should be greater than acritical value ccr, which is approximately one.

If these transmission criteria are satisfied, a dislocation–densitycan emit from a higher dislocation–density region to alower dislocation–density region. It is assumed that thedislocation–densities will be redistributed according to thesetransmission factors as shown in Fig. 1. Based on this, a balanceof dislocation–densities due to these different dislocation–densityinteractions, ��(ˇ)

out , can be defined as

��(ˇ)out =

M˛ˇ(�(ˇ)out/�(ˇ)

ref)∑m

i=1(M˛i�(i)out/�(i)

ref)��(˛)

inc, (13)

where ˛ is an incoming slip system, ˇ is an outgoing slip system, �(ˇ)out

is the corresponding resolved shear stress, and �(ˇ)ref

is the referenceshear stress of the outgoing slip system ˇ. m is the number of allpossible outgoing slip systems, and M˛ˇ is the transmission factorbetween the incoming and outgoing slip systems.

When dislocation–densities emit from one element to the other,

a balance for the dislocation–densities can be obtained by consid-ering dislocation–density conservation [39].

∂�(˛)

∂t= c(˛) − div�j(˛), (14a)

124 J. Shi, M.A. Zikry / Materials Science and E

Fig. 1. Dislocation–densities redistribute according to GB transmission factors andrddi

w

d

wtidigd

c

c

S

aro

esolved shear stresses. When dislocation–density absorptions occurs, the mobileislocation–density is immobilized in the GB; emission occurs when immobileislocation–density is mobilized into the grain interior; transmission occurs when

mmobile dislocation–densities transmit through GB.

here

iv�j(˛) =∂(∫

(�(˛)/t)�ndA)

∂x+

∂(∫

(�(˛)/t)�ndA)

∂y, (14b)

here �j(˛) represents the flux of dislocation–densities in slip sys-em ˛, A is area of integration, �n is the GB surface normal, �(˛)

s the summation of the dislocation–densities changes in theomain indicated by Eq. (13), and c(˛) represents the generation,

mmobilization or annihilation of dislocations densities. c(˛) isiven by the statistical distribution of the immobile and mobileislocation–densities from Eqs. (15a) and (15b).

ˆ(˛)im

= � (˛)

(gminter

b2exp(

− H

kT

)

+ gimmob

b

√�(˛)

im− grecov exp

(− H

kT

)�(˛)

im

), (15a)

ˆ(˛)m =� (˛)

(gsour

b2

(�(˛)

im

�(˛)m

)−gminter

b2exp(

− H

kT

)−gimmob

b

√�(˛)

im

).

(15b)

o that the evolution of dislocation–densities is updated as

d�(˛)im

dt= � (˛)

(gminter

b2exp(

− H

kT

)

+gimmob

b

√�(˛)

im−grecov exp

(− H

kT

)�(˛)

im

)−div�j(˛), (16a)

d�(˛)m

dt= � (˛)

(gsour

b2

(�(˛)

im

�(˛)m

)

− gminter

b2exp(

− H

kT

)− gimmob

b

√�(˛)

im

)− div�j(˛). (16b)

Dislocation–density activities can occur in the GB interfacend GB-interior interfaces (Fig. 1). In the proposed GB rep-esentation, we assume that dislocation–density transmissionccurs when immobile dislocation–densities transmit through

ngineering A 520 (2009) 121–133

the GB interfaces to compatible slip systems in the neighbor-ing grain. Dislocation–density absorption occurs when mobiledislocation–densities from the grain interior transmit into theGB, but do not transmit out of it. In this case, all of themobile dislocation–densities are then immobilized to immobiledislocation–densities, and are assumed to be absorbed in the GB.Some of these immobile dislocation–densities in the GB regionsmay pile-up, but as the deformation evolves can activate neighbor-ing slip systems. This can result in dislocation–density emission, inwhich the immobile dislocation–densities in the GB become mobiledislocation–densities and emit into the neighboring grain. Hence,all three processes of transmission, absorption, and emission canoccur simultaneously on different slip systems within the GB regionand between neighboring grains.

2.3. Initial dislocation–densities dependence on GBmisorientations

GB properties such as dislocation–densities, dislocation net-works, and GB energy are closely related to the crystallographicproperties of GBs, such as GB misorientations. Read and Shockley[12] have noted that there is a relation between misorientation andinitial dislocation–densities (before deformation) in low angle GBs.That relation is

�x = �

acos �, (17a)

�y = �

asin �, (17b)

where �x and �y are the line density of initial dislocation at x andy directions; � is the tilt misorientation of the GB, a is the latticeparameter; � is the angle between lattice and GB, and 0 < � < /2.Assuming that the grain has a rectangular shape, and the grain sizeis d, the initial dislocation–density due to GB misorientation can begiven as

�im = 2d

(�x + �y) = 2�

ad(cos � + sin �). (18)

According to this equation, the initial dislocation–density isgrain size and misorientation dependent. This can be critical, sincefor nanocrystalline materials, the initial dislocation–densities in theGB region are usually higher than the grain interior [12,40,41].

2.4. GBS mechanism

For nanocrystalline materials, GBS can be an operative defor-mation mechanism [42–44]. There are several GBS models basedon thermal activation mechanisms. In this study, we incorporatethe model proposed by Raj and Ashby within our proposed frame-work [43]. The overall GB nodal velocity ut is a combination ofdislocation-mediated deformation and GB-mediated deformationas

ut = udis + ugbs, (19)

where udis is the dislocation-mediated velocity and ugbs repre-sents the GBS velocity. The GBS rate of nanocrystalline material isaccounted for by the Raj and Ashby equation as

ugbs = dugbs

dt= 2

�a˝

kT

�

h2DV

(1 + ı

�

DB

DV

)(20)

where �a is the shear stress along the direction of the GB, ˝ is theatom volume, � is the wavelength, and h is the amplitude of theGB. DV and DB are volume and boundary diffusion coefficients, ıis the GB width, k is Boltzmann constant, and T is temperature. At

J. Shi, M.A. Zikry / Materials Science and Engineering A 520 (2009) 121–133 125

F ment 1, (c) simple shear for GB element 2. The overall GBS rate is a linear combination ofs

tt

u

Imcae

Table 1Parameters for Raj and Ashby GBS rate.

Fd

ig. 2. The simple shear model for GBS: (a) GB normals; (b) simple shear for GB eleimple shear of elements 1 and 2.

emperature significantly below melting point, DB � DV, and hencehe Raj and Ashby equation can be given as [33,42]

˙ gbs = 2ıDB˝�a

kT

1h2

. (21)

The detailed values of the above parameters are given in Table 1.

n this study, the shear stress �a is obtained by coordinate transfor-

ation on the stress from the global coordinate system to the localoordinate system of the GB. The GBS is updated at each time steps illustrated in Fig. 2. The GB region is modeled as two layers oflements, one layer belongs to grain 1, and the other layer belongs

ig. 3. Finite element mesh for different grain sizes, loading and boundary conditions of= 100 nm (d) d = 200 nm.

Properties k (J K−1) DB ˝ (m3) ı (nm) h (d)

Value 1.3807 × 10−23 1 × 10−21 8.7 × 10−27 10 0.5

to grain 2. The shear stress �a along the tangential direction of GB is

calculated from the stress state in the global coordinate system. TheGBS rate (ugbs) is calculated from Eq. (21). The deformation modeof GBS is assumed to be simple shear along the tangential directionof the GB element. Assuming GBS only occurs in the GB elements,

nanocrystalline aggregates for different grain sizes: (a) d = 25 nm; (b) d = 50 nm; (c)

1 and Engineering A 520 (2009) 121–133

aob1Td

2

tTdgttifiwcnfs

3

cuaa(bi

Table 2Material properties of polycrystalline copper.

Properties E (GPa) m �ref (s−1) b (10−10 m) �y (MPa) � �critical (s−1) A

Fd

26 J. Shi, M.A. Zikry / Materials Science

nd the GBS at the interfacial nodes are given by simple shearingf the lower element (Fig. 2b), the GBS on the top nodes are giveny the summation of the GBS of the interfacial nodes in elementand shear displacement of the top nodes in element 2 (Fig. 2c).

he total nodal velocity can then be updated by the summation ofislocation-mediated displacement and GBS by Eq. (19).

.5. Computational methods

The total deformation rate tensor Dij, and the plastic deforma-ion rate tensor Dp

ijare needed to update the material stress state.

he method used here is one developed by Zikry [45] for rate depen-ent crystalline plasticity formulations, and a brief outline will beiven here. An implicit finite element method is used to obtain theotal deformation rate tensor Dij. To overcome numerical instabili-ies associated with stiffness, an adaptive explicit-implicit methods used to obtain the plastic deformation rate tensor Dij. An explicitfth-order Runge–Kutta method is used most of the times, andhen numerical stiff occurs, the computation scheme automati-

ally switches to the implicit backward Euler method. This adaptiveumerical scheme is also used to update the evolutionary equations

or the mobile and immobile dislocation–densities and the resolvedhear stress.

. Results

The multiple-slip crystal plasticity dislocation–density basedonstitutive formulation and the FE computational scheme aresed to investigate the behavior of f.c.c. copper polycrystalline

ggregates with different grain sizes. The material properties thatre used here (Table 2) are representative of crystalline coppersee, for example, [45]). In the grain interior, the initial immo-ile dislocation–density, �ims, was chosen as 1010 m−2, and the

nitial mobile dislocation–density �ms was chosen as 107 m−2.

ig. 4. Initial immobile dislocation–density of the most active slip system [0 1 1](1 1 1)islocation–densities are higher when the grain size is smaller. GBs effects are significant

Value 100 0.01 0.001 2.5 100 0.3 104 0.3

The values of the initial and the saturated dislocation–densitiesare representative of copper [46]. In the GB region, the initialimmobile dislocation–density is a function of misorientation andgrain size, which is given by Eqs. (17a) and (17b). Based on thescheme developed by Zikry and Kao [47], we have obtained thecoefficient values needed for the evolution of the immobile andmobile dislocation–densities and the specified material proper-ties. These values are gminter = 5.53, grecov = 6.67, gimmob = 0.0127,gsour = 2.7 × 10−5, and H/k = 3.289 × 103 K.

We used 18 hexagonal grains in the polycrystalline aggregate,and each grain is assumed to have a grain interior and a distinct GBregion. To account for grain size effects, aggregates with grain sizesof 25 nm, 50 nm, 100 nm and 200 nm were used (Fig. 3a–d). Thewidth of the GB region is assumed to be 10 nm in these aggregates.Based on a convergence analysis, 1152 elements were used. Sym-metric boundary conditions were applied, and the aggregates areunder tensile displacement with a loading rate of 1 × 10−4 s−1. Themechanical properties, such as static yield stress, Young’s modulus,Poisson ration and slip systems in GB elements are assumed to bethe same as in grain interior.

Two sets of GB misorientations were used. The first set is forrandom low angle (low angle case), in which the misorientationbetween each neighboring grains is less than 5◦. Since the Read andShockley relation [12] still holds for misorientation angles up to 20◦,

the second case is the higher range of low angle GBs (intermediateangle), in which the misorientations are random angles between10◦ and 20◦.

for different grain sizes: (a) 25 nm; (b) 50 nm; (c) 100 nm; (d) 200 nm. The initialas the grain size decreases.

J. Shi, M.A. Zikry / Materials Science and Engineering A 520 (2009) 121–133 127

Fig. 5. Immobile dislocation–density of the most active slip system [0 1 1](1 1 1) undeformed and after 20% nominal strain (grain size = 100 nm): (a) undeformed, withoutconsidering initial dislocation–density based on GB misorientations; (b) undeformed, considering initial dislocation–density based on GB misorientations; (c) at 20% nominalstrain, without the DDGBI scheme; (d) at 20% nominal strain, with the DDGBI scheme. For nanocrystalline materials, the initial dislocation–densities in the GB region havea magnitude of 1015 m−2, which is in line with the prediction based on misorientations shown in (b). At 20% nominal strain, the dislocation–densities in the GB region arehigher than the grain interior region when the DDGBI scheme is used.

Fig. 6. (a) normal stress without the DDGBI scheme; (b) normal stress with the DDGBI scheme; (c) plastic slip without the DDGBI scheme; (d) plastic slip with the DDGBIscheme. GB effects result in an increase in the normal stress in the GB region, and the GB region become a barrier of plastic deformation.

128 J. Shi, M.A. Zikry / Materials Science and Engineering A 520 (2009) 121–133

F nm; (S e GBS

3

dtFi1goirima

Fa

ig. 7. GBS for different grain sizes (unit: m): (a) grain size = 25 nm; (b) grain size = 50ignificant GBS occurs for 25 nm aggregate, but for grain sizes larger than 50 nm, th

.1. Initial dislocation–densities for different grain sizes

First a comparison is made for the initial immobileislocation–densities of an aggregate with and without consideringhe misorientation dependence on initial GB dislocation–densities.ig. 4a–d shows the immobile dislocation–densities (normal-zed by initial immobile dislocation–density of grain interior× 1010 m−2) of the most active slip system [0 1 1](1 1 1) of aggre-ates with different grain sizes when misorientation dependencef initial GB dislocation–densities is considered. As we can see

n Fig. 4a–d, the initial immobile dislocation–densities in the GBegion of nanocrystalline materials are much higher than the grainnterior. The dislocation–densities in the GB region are in the

agnitude of 1014–1015 m−2, and the highest dislocation–densitiesre in the GB with the highest misorientation of 3.26◦. We

ig. 8. Total GBS for different misorientations (unit: m): (a) low angle GB (misorientatiggregates with higher misorientations. The maximum GBS rate for random intermediate

c) grain size = 100 nm; (d) grain size = 200 nm. GBS increases as grain sizes decreases.is negligible.

can also see that smaller grain size correspond to higher GBdislocation–densities. The aggregates with 25 nm grain size has amaximum immobile dislocation–density of 1.3 × 1015 m−2, whilethe aggregate with a 200 nm grain size has a maximum immobiledislocation–density of 1.6 × 1014 m−2. The dislocation–densitiespredicted by our model is in line with TEM observations [7], inwhich the dislocation–densities are also a function of grain sizeand the densities for an aggregate with 200 nm grain size are ofthe magnitude of 1014 m−2.

3.2. Material behavior during deformation

The immobile dislocation–densities of the most active slip sys-tem [0 1 1](1 1 1) for a 100 nm aggregate, with a distribution ofrandom low angle GBs, is shown in Fig. 5a with no GB effects

ons < 5◦); (b) intermediate angle GB (10◦ < misorientations < 15◦). GBS is higher inangle GBs is about 50% higher than aggregates with random low GBs.

J. Shi, M.A. Zikry / Materials Science and Engineering A 520 (2009) 121–133 129

Fig. 9. Nominal Stress–strain curve for aggregates with different GB misorienta-tions: (a) low angle GBs (within 5◦); (b) intermediate low angle GBs (10–20◦). Theyyls

(riia3dngecinctatdaaatTmt1t

Fig. 10. Comparison between Hall–Petch relation and predictions: (a) comparisonbetween random intermediate angle GBs and random low angle GBs aggregate; (b)comparison between experimental results and low angle GB aggregates predictions.

ield stress of nanocrystalline aggregates increases as the grain size decreases. Theield stress for random intermediate angle GBs is approximately 40% higher thanow angle GBs. If the DDGBI scheme is not used, the yield stresses are exactly theame for different grain sizes.

no DDGBI scheme, no initial dislocation–density misorientationelation and no GBS). We see that the initial dislocation–densitys 1010 m−2 when no GB effects are considered. Fig. 5b shows thenitial dislocation–densities when the GB effects are considered,nd the maximum initial dislocation–density is approximately.2 × 1014 m−2. Fig. 5c–d shows the corresponding immobileislocation–densities after 20% nominal strain. If GB effects areot considered, the dislocation–densities mostly evolve in therain interior and the maximum value is 7.5 × 1013 m−2. If the GBffects are considered, the dislocation–densities in the GB regionontinue to increase as the deformation evolves, and the max-mum immobile dislocation–densities reaches 7 × 1014 m−2 at aominal strain of 20%. This is approximately 10 times higher inomparison for the case where GB effects are not considered. Fur-hermore, the normal stresses, plastic deformation, and pressurere also significantly affected by the dislocation–density activi-ies at the GB regions. At 20% nominal strain, there are significantifferences in normal stress and shear slip distribution betweenn aggregate where GB effects are not considered (Fig. 6a and c)nd with an aggregate where GB effects are considered (Fig. 6bnd d). The normal stresses and shear slip are homogenously dis-ributed in the grain interior when GB effects are not considered.he maximum normal stress is approximately 380 MPa and the

aximum plastic slip is 0.49. When GB effects are considered,

he maximum normal stress in the GB region is approximately20% higher than stresses in the grain interior. This is due tohe higher dislocation–density concentration in the GB region. Fig. 11. Crack model with d = 25 nm and d = 5 nm.

130 J. Shi, M.A. Zikry / Materials Science and Engineering A 520 (2009) 121–133

F vior aa cation d; (e)

Trrra

3

atih

mai1as

ig. 12. The effect of GBS for an aggregate with grain sizes of 25 nm on crack behactive slip system [0 1 1](1 1 1) when GBS is not considered; (b) the immobile disloormal stress when GBS is not considered; (d) normal stress when GBS is considere

he plastic slip (the slip on all the active slip systems) in the GBegion is lower in comparison with plastic slip in the grain inte-ior, because the high dislocation–densities and stress in the GBegion impede the further accumulation of dislocation–densitiesnd plastic strains.

.3. GBS

Fig. 7a–d shows the total GBS displacement for random lowngle aggregates with various grain sizes. At 20% nominal strain,he maximum GBS of the 25 nm grain-sized aggregate is approx-mately 2 nm, which is comparable with the grain size. Also theigher GBS occurs at the GBs with higher normal stresses (Fig. 7a).

For different grain sizes, the total GBS differs significantly. Theaximum total GBS, at 20% nominal strain for a 25 nm grain-sized

ggregate is approximately 2 nm (Fig. 7a). In contrast, the total GBSs 0.19 nm for 50 nm grain-sized aggregate (Fig. 7b), 0.015 nm for00 nm sized aggregate (Fig. 7c) and 1.6 × 10−3 nm for 200 nm sizedggregate (Fig. 7d). It is obvious that the GBS decreases as the grainize increases, and when the grain size is larger than 50 nm, the GBS

t a nominal strain of 1.6% strain: (a) the immobile dislocation–density of the mostn–density of the most active slip system [0 1 1](1 1 1) when GBS is considered; (c)shear slip when GBS is not considered; (f) shear slip when GBS is considered.

is less than 0.4% of the grain size and is negligible compared to thetotal plastic deformation.

GB misorientations also play an important role on GBS sinceat low angle regime, higher misorientations correspond to higherGB energy [12,48] and higher GBS rate [10,49,50]. At high angleregime, there is no direct relation between misorientation and GBenergy, as well as GBS rate [51,52]. Fig. 8a shows the GBS for a 25 nmgrain-sized aggregate for random low angle GBs. The highest totalGBS is approximately 2 nm at the GB with a misorientation angle of3.26◦. The maximum GBS for a random intermediate angle modelis approximately 3 nm and occurs at a GB with misorientation of19.0◦ (Fig. 8b), and this is approximately 50% higher than the lowangle GB aggregate.

3.4. Yield stress of polycrystalline aggregates

The yield stress in this investigation is determined by the0.2% offset yield strength of the material as determined bythe stress–strain curve for aggregates with different grain sizes.Fig. 9a–b shows the normalized nominal stress–strain curve of dif-

J. Shi, M.A. Zikry / Materials Science and Engineering A 520 (2009) 121–133 131

F ior ats sity ofw hen G

fiagasa3aiawrlc

3

t

ig. 13. The effect of GBS for an aggregate with grain sizes of 5 nm on crack behavystem [0 1 1](1 1 1) when GBS is not considered; (b) the immobile dislocation–denhen GBS is not considered; (d) shear slip when GBS is considered; (e) shear slip w

erent grain-sized aggregates with random low angle and randomntermediate angle GBs. As we can see from the figures, if GB effectsre not considered, the yield stresses for different grain-sized aggre-ates are exactly the same at a value of 260 MPa. When GB effects areccounted for, there are significant differences for different grainized aggregates. For instance, the yield stress for the random lowngle GB aggregate is 530 MPa for the 25 nm grain-sized aggregate,80 MPa for the 50 nm grain-sized aggregate, 310 MPa for 100 nmggregate and 280 for 200 nm aggregate (Fig. 9a). The yield stresss also affected by GB misorientations: the yield stress for a 25 nmggregate is 670 MPa for the intermediate GB aggregate (Fig. 9b),hich is approximately 22% higher than the aggregate with the

andom low angle GB aggregate. It should also be noted that at theowest grain size, the stresses soften beyond the yield point, thisan be due to the increased values of GBS.

.5. Comparison with the Hall–Petch relation

The yield stresses of these aggregates have been compared tohe Hall–Petch relation. Fig. 10 shows the relation between the

1.6% nominal strain: (a) the immobile dislocation–density of the most active slipthe most active slip system [0 1 1](1 1 1) when GBS is considered; (c) normal stressBS is not considered; (f) shear slip when GBS is considered.

inverse of square root of grain sizes and the yield stresses fromour modeling and the experimental results. The parameter �o andk can be obtained by a first-order curve fitting of the predictedresults. For example, four different yield stresses with respect to theinverse of square root of four grain-sized aggregates with randomlow angle GB are shown in Fig. 10a. By fitting these four points bya straight line, �o and k can be obtained. For the random low angleGB distribution, k is 1999 MPa/nm−1/2 and for the intermediate GBmisorientations k is 3103 MPa/nm−1/2 (Fig. 10a). Fig. 10b shows acomparison between the experimental results and our predictions.The experimental data are from several sources from hardness andtensile experiments, which have been compiled by Meyers et al.[10] and Conrad [53]. The parameter k in the experiments is equal to2106 MPa/nm−1/2, which is approximately 5% higher than the pre-dicted value for the random low angle GB aggregate. The prediction

of friction stress �o is 123 MPa for the random low angle GB aggre-gate, and 20 MPa for the intermediate low angle GBs. The frictionstress �o extracted from experimental results is 170 MPa, which isapproximately 40% higher than our simulation. These differencesbetween the predicted and experimental values can be due to vari-

1 and E

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32 J. Shi, M.A. Zikry / Materials Science

tions in grain sizes and GB orientations. However, the predictionsre consistent with behavior based on Hall–Petch relations and thexperimental values of k.

.6. The influence of GBS on nanocracks

A crack is introduced in the 25 nm and 5 nm grain-sized aggre-ates to investigate how GBS can affect crack behavior with differentrain sizes. The crack was initially oriented normal to the loadingurface, and it was generated across a GB region and within onerain by having free nodes associated with the crack opening dis-lacement (Fig. 11). The crack has a normalized length of a/w = 0.25,here a is the crack length and w is the plate width.

Fig. 12 shows various results for a 25 nm grain-sized aggre-ate at a nominal strain of 1.6%. The total GBS displacement ispproximately 0.15 nm at this strain. Fig. 12a shows the immobileislocation–densities of the most active slip system [0 1 1](1 1 1)hen GBS is not updated, and Fig. 12b shows the immobileislocation–densities of the same slip system when GBS is updated.s can be seen in these two figures, the GBS does not significantlyffect the immobile dislocation–densities. This is probably due tohe fact that the GBS is not a dislocation-mediated deformation,ence will not directly change the dislocation–densities in the GBegion. Fig. 12c–d shows the normal stresses without and with GBS.

e can see that there is almost no difference between these twoases: at the GB region, the difference between normal stresses forhese two cases is within 1%. The shear slip is shown in Fig. 12e–f. Inhe GB region ahead of the crack tip, when GBS is updated, the shearlip is approximately 7% higher than the case with no GBS. GBS alsoas some minor effects on lattice rotation and temperature.

Fig. 13 shows the results for a 5 nm grain-sized aggregate at 1.6%ominal strain. Due to the small grain sizes, the GBS is significant.he total GBS displacement at 1.6% nominal strain is approximatelynm, which is 20% of the grain size. Fig. 13a shows the immobileislocation–densities of the most active slip system [0 1 1](1 1 1)hen GBS is not considered, and Fig. 13b shows the immobileislocation–densities of the same slip system when GBS is con-idered. As can be seen in these two figures, the GBS slightlyncreases the immobile dislocation–densities by approximately 5%.he average normal stress in the GB region is 5% lower when GBSs accounted for (Fig. 13c–d), but for some specific regions, such ast triple junctions, the normal stress is higher. This uneven distri-ution of normal stresses may result in intergranular crack growth

n the direction of the high stress regions. We can also see that inhe GB region ahead of the crack tip, when GBS is considered, thehear slip is about 30% higher in comparison for the case when GBSs not considered (Fig. 13e–f). This increase in shear slip is due toBS. This sliding also has significant effects on lattice rotation and

emperature. It results in increases in the lattice rotation by approx-mately 10% and temperature by approximately 5% in comparison

ith the case when GBS is not considered. The temperatures herere updated by considering only changes due to adiabatic heating.

. Conclusion

The proposed DDGBI scheme has been used with differentrain-sized aggregates. This scheme is coupled to a multiple slipislocation–density based crystalline formulation and a specializednite element scheme. Dislocation–density transmission, emission

nd absorption are based on slip system compatibility, immo-ile and mobile dislocation accumulation and conservation. Initialislocation–densities are also calculated with respect to GB misori-ntation angle and grain size. GBS is accounted assuming thermallyctivated process related to the Raj–Ashby model.

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ngineering A 520 (2009) 121–133

Based on these interrelated mechanisms, the yield stress ofpolycrystalline aggregates with different grain sizes have beenobtained and comparisons with experimental results indicateexcellent correlations with Hall–Petch effects for different experi-mental results. If the GB effects are not accounted for, the predictedyield stresses are independent of the grain size. If GB interactionsare represented, different GB-mediated and dislocation-mediateddeformations can be accounted for as a function of grain size andinitial dislocation–densities. More significantly, the yield stress isshown to be a function of grain size. When the grain size decreases,the yield stress increases. Furthermore, higher GB misorientationsincrease the initial dislocation–density in the GB region, whichresults in increases in the yield stress. The predicted results areconsistent with the Hall–Petch relation, and the parameter k and �o

have been obtained for random low angle and intermediate angleGBs.

It has also been shown that GBS increases when grain sizedecreases. Furthermore, GBS has an effect on crack behavior forgrain sizes of 5 nm in that there are increases in lattice rotation andplastic slip, and distortion in normal stresses ahead of the crack-tip. This significant change in stresses due to GBS can change thedirection of crack growth or even cause intergranular crack, sincethe high stress in the GB region increase the possibility of voidsnucleation and low stress in the GB region decrease this possibility.The crack will probably grow along the high stress GB region. Foraggregates with larger grain size of 25 nm, the effect of GBS on thecrack behavior is negligible. Furthermore, the predictions indicatethat GBS increases with decreasing grain sizes, and results in lowernormal stresses in critical locations. Hence, GBS may offset strengthincreases associated with decreases in grain size.

Acknowledgements

Support from the Office of Naval Research through GrantN000140510097, is gratefully acknowledged. The computationswere performed in the High Performance Computation (HPC) Cen-ter of North Carolina State University. The support and assistancefrom employees in HPC center are highly appreciated.

References

[1] N.J. Petch, J. Iron. Steel Inst. 174 (1953) 25–28.[2] E.O. Hall, Proc. Phys. Soc. Lond. 64 (1951) 747–753.[3] L. Thilly, F. Lecouturier, J. von Stebut, Acta Mater. 50 (2002) 5049–5065.[4] N. Hansen, B. Ralph, Acta Metall. 30 (1982) 411–417.[5] P.G. Sanders, J.A. Eastman, J.R. Weertman, Acta Mater. 45 (1997) 4019–4025.[6] G.W. Nieman, J.R. Weertman, R.W. Siegel, J. Mater. Res. 6 (1991) 1012–1027.[7] M. Hommel, O. Kraft, Acta Mater. 49 (2001) 3935–3947.[8] T.G. Nieh, J. Wadsworth, Scripta Metall. Mater. 25 (1991) 955–958.[9] C.A. Schuh, T.G. Nieh, T. Yamasaki, Scripta Mater. 46 (2002) 735–740.

[10] M.A. Meyers, A. Mishra, D.J. Benson, Prog. Mater. Sci. 51 (2006) 427–556.[11] A. Gemperle, J. Gemperlova, N. Zarubova, Interface Sci. 10 (2002) 59–65.12] W.T. Read, W. Shockley, Phys. Rev. 78 (1950) 275–289.

[13] F. Sansoz, J.F. Molinari, Acta Mater. 53 (2005) 1931–1944.[14] W. Soboyejo, Mechanical Properties of Engineered Materials, Marcel Dekker,

Inc., Princeton, NJ, 2003.[15] W.A. Rachinger, J. Inst. Metals 81 (1952) 33.[16] I.M. Lifshitz, Sov. Phys. Jetp-USSR 17 (1963) 909–920.[17] H. Van Swygenhoven, A. Caro, Phys. Rev. B 58 (1998) 11246–11251.[18] W.A.T. Clark, R.H. Wagoner, Z.Y. Shen, T.C. Lee, I.M. Robertson, H.K. Birnbaum,

Scripta Metall. Mater. 26 (1992) 203–206.[19] T.C. Lee, I.M. Robertson, H.K. Birnbaum, Metall. Trans. A 21 (1990) 2437–2447.20] J.D. Livingston, B. Chalmers, Acta Metall. 5 (1957) 322–327.21] Z. Shen, R.H. Wagoner, W.A.T. Clark, Scripta Metall. 20 (1986) 921–926.22] B.W. Lagow, M. Jouiad, D.H. Lassila, T.C. Lee, H.K. Birnbaum, Mater. Sci. Eng. A

309 (2001) 445–450.

23] T.C. Lee, I.M. Robertson, H.K. Birnbaum, Philos. Mag. A 62 (1990) 131–153.24] J. Gemperlova, A. Jacques, A. Gemperle, T. Vystavel, N. Zarubova, M. Janecek,

Mater. Sci. Eng. A (2002).25] J. Gemperlova, M. Polcarova, A. Gemperle, N. Zarubova, J. Alloys Compud. 378

(2004) 97–101.26] J. Schiotz, Scripta Mater. 51 (2004) 837–841.

and E

[

[[[[

[

[[[[[[[[

[[[[[[[[

[[50] R.Z. Valiev, T.G. Langdon, Acta Metall. Mater. 41 (1993) 949–954.

J. Shi, M.A. Zikry / Materials Science

27] H. Van Swygenhoven, P.M. Derlet, A.G. Froseth, Acta Mater. 54 (2006)1975–1983.

28] V. Yamakov, D. Wolf, S.R. Phillpot, H. Gleiter, Acta Mater. 51 (2003) 4135–4147.29] W.M. Ashmawi, M.A. Zikry, Mech. Mater. 35 (2003) 537–552.30] M.P. Dewald, W.A. Curtin, Modell. Simul. Mater. Sci. Eng. 15 (2007) S193–S215.31] M. de Koning, R. Miller, V.V. Bulatov, F.F. Abraham, Philos. Mag. A 82 (2002)

2511–2527.32] M. de Koning, R.J. Kurtz, V.V. Bulatov, C.H. Henager, R.G. Hoagland, W. Cai, M.

Nomura, J. Nucl. Mater. 323 (2003) 281–289.33] H.H. Fu, D.J. Benson, M.A. Meyers, Acta Mater. 49 (2001) 2567–2582.34] K.S. Cheong, E.P. Busso, A. Arsenlis, Int. J. Plast. 21 (2005) 1797–1814.

35] S. Benkassem, L. Capolungo, M. Cherkaoui, Acta Mater. 55 (2007) 3563–3572.36] J. Shi, M.A. Zikry, Comp. Mater. Sci. (submitted).37] J. Shi, M.A. Zikry, J. Solid. Mech. (submitted).38] M.A. Zikry, M. Kao, Comput. Syst. Eng. 6 (1995) 225–240.39] E.C. Aifantis, Int. J. Plast. 3 (1987) 211–247.40] H. Van Swygenhoven, D. Farkas, A. Caro, Phys. Rev. B 62 (2000) 831–838.

[

[

[

ngineering A 520 (2009) 121–133 133

41] H. Gleiter, Scripta Metall. 11 (1977) 305–309.42] B.N. Kim, K. Hiraga, K. Morita, Acta Mater. 53 (2005) 1791–1798.43] R. Raj, M.F. Ashby, Metall. Trans. 2 (1971) 1113.44] R.L. Coble, J. Appl. Phys. 34 (1963) 1679.45] M.A. Zikry, Comput. Struct. 50 (1994) 337–350.46] H. Mughrabi, Mater. Sci. Eng. 85 (1987) 15–31.47] M.A. Zikry, M. Kao, Scripta Mater. 34 (1996) 1115–1121.48] T. Watanabe, H. Yoshida, T. Saito, T. Yamamoto, Y. Ikuhara, T. Sakuma, Mater. Sci.

Forum 304 (1999) 601–606.49] K.S. Kumar, H. Van Swygenhoven, S. Suresh, Acta Mater. 51 (2003) 5743–5774.

51] R. Monzen, Y. Sumi, K. Kitagawa, T. Mori, Acta Metall. Mater. 38 (1990)2553–2560.

52] T. Watanabe, M. Yamada, S. Shima, S. Karashima, Philos. Mag. A 40 (1979)667–683.

53] H. Conrad, Metall. Mater. Trans. A 35A (2004) 2681–2695.