Journal for Multiscale Computational Engineering8(5), 509–522 (2010)

CALIBRATION OF NANOCRYSTAL GRAINBOUNDARY MODEL BASED ON POLYCRYSTALPLASTICITY USING MOLECULAR DYNAMICSSIMULATIONS

Sangmin Lee & Veera Sundararaghavan∗

Department of Aerospace Engineering, University of Michigan, Ann Arbor, MI 48109, USA

∗Address all correspondence to Veera Sundararaghavan E-mail: vs85@cornell.edu

Decohesion parameters are computed for the tilt grain boundaries through molecular simulations and the parametersare employed in a elastoplastic deformation model of a face-centered-cubic nanocrystal. The calibrated continuum grainboundary model accounts for reversible elastic and irreversible inelastic separation sliding deformations. The intra-granular plasticity was modeled using a rate-independent single-crystal plasticity model. Atomistic calculations arepresented for a planar, copper grain boundary interface with a tilt lattice misorientation for cases of loading and un-loading. The interface models are deformed to full separation and then relaxed to study inelastic behavior. Plots of stressversus displacement show a distinctly different deformation response between normal and tangential interface loadingconditions. Two-dimensional microstructures uniaxially loaded using the calibrated cohesive model indicate that themacroscopically observed nonlinearity in the mechanical response is mainly due to the inelastic response of the grainboundaries. Plastic deformation in the interior of the grains prior to the initiation of grain boundary cracks was notobserved. Although key features of the molecular simulation results have been introduced in the cohesive model, a fewdiscrepancies between the behavior of cohesive model when compared to molecular simulations are noted.

KEY WORDS: nanocrystals, fracture, crystal plasticity, molecular dynamics, cohesive elements

1. INTRODUCTION

Metallic nanocrystals are interesting engineering materi-als due to their high yield strength and high strain hard-ening rates, leading to very high tensile strengths. Recentexperiments [transmission electron microscopy study inKumar et al. (2003)] as well as molecular simulations(Swygenhoven et al., 1999; Schiøtz et al., 1999; Swygen-hoven, 2004; Yamakov et al., 2006, 2001) have indicatedthat inelastic deformation in nanocrystals primarily orig-inates from grain boundary accommodation effects, suchas grain boundary sliding and separation. Grain boundarymodels have been developed in the past to describe thebrittle failure behavior of metallic materials. Grain bound-aries have been incorporated as volume elements withsimple isotropic-type hardening rules (Fu et al., 2004).These approaches fail to explain the sliding and fracturebehavior observed in grain boundaries.

In recent years, cohesive interface models haveemerged as attractive methods to numerically simu-late fracture initiation and growth by the finite elementmethod (Needleman, 1990, 1992; Tvergaard et al., 1993,1996; William, 1989; Camocho et al., 1997; Xu andNeedleman, 1994; Ortiz and Pandolfi, 1999; Zavattieriand Espinosa, 2001). Typically, a cohesive interface isintroduced in a finite element discretization of the prob-lem by the use of special interface elements, which obeya nonlinear interface traction separation constitutive re-lation. These relations provide a phenomenological de-scription for the complex atomistic processes that leadto the formation of new traction-free crack faces. Thegrain boundary parameters in these models are eitherphenomenologically assumed or calibrated from experi-ments. Thus, these models consider the macroscopic ef-fects without resolving the atomistic scale details. Al-though potentials for interplanar glide at the contin-

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510 Lee & Sundararaghavan

uum scale can be defined based on atomic level details[e.g., dislocation nucleation (Rice, 1992)], they do nothave a strong physical basis for coupling shear and nor-mal tractions and do not account for irreversibilities inthe tractionseparation process. These irreversibilities in-clude path history effects associated with nonuniform dis-tribution of bond ruptures and dislocation emission atnanoscales.

Recently, Wei and Anand (2004) have proposed anelasticplastic interface model that accounts for irre-versible inelastic separation and sliding deformations atthe interface prior to failure. In addition to a traction-separation relation for normal separation across an in-terface, appropriate relations for combined opening andsliding are also developed in an approach similar tosingle-crystal dislocation plasticity formulations. In theapproach, parameters were obtained by calibration withexperimental stress-strain curves, which makes it diffi-cult to unambiguously define the GB properties such asthe strengths in the tensile and shear modes. Furthermore,continuum models cannot be directly verified withoutatomistic calculations because their validity is contingenton the accuracy of parameters used in the phenomenolog-ical model. The nanoscopic length scales involved in themodel is directly accessible from molecular simulationsand will allow one to calibrate the model directly fromatomistic simulations, although at high deformation rates.With the advent of parallel processing, large-scale atom-istic calculations have been performed to address sucheffects (Nakano et al., 1994; Holian and Ravelo, 1995;Zhou et al., 1998; Cleri et al., 1997, 1998). Recently, qua-sicontinuum simulation of the interface (Warner et al.,2006) has been used to identify the normal and shearstrengths of the GBs for use in a continuum model. Meth-ods for calibrating such cohesive elements using resultsfrom molecular dynamic MD simulations has also beenrecently studied (Spearot et al., 2004; Gall et al., 2000;Sundararaghavan and Zabaras, 2006). In this approach,tensile and shear tests are performed on a bicrystal in-terface using molecular dynamics to obtain the cohesiveresponse of the interface. We further investigate the useof this method (Spearot et al., 2004) to calibrate cohe-sive models and perform continuum finite element simu-lations of nanocrystals. This continuum model uses rate-independent crystal plasticity theory for the grain interiorto model nanoscale intragranular behavior. Interface con-stitutive model employed in this work allows for inelastic-ity at the interface prior to failure. To capture the cohesivelaw obtained from MD simulations, we focus on calibrat-ing the parameters in the equations governing evolution

of scalar state variables of the interface. The mechanicalresponse of a nanocrystal with initially random orienta-tion distribution is investigated using this approach, andthe physical origin of macroscopically observed nonlin-earity in the stress-strain response is investigated. In ad-dition, these simulations are used to obtain informationon the effect of stress concentrations at the tips of grainboundary cracks and at triple junctions on overall plasticdeformation. The applicability of the cohesive model isstudied and discrepancies occurring between the contin-uum model and the molecular simulations are identified,especially, in the unloading and reloading behavior for thenormal separation mode.

The paper is organized as follows. In Section 2, thecontinuum grain boundary model is described. An elasticpotential is introduced for interface separation, which ad-mits a dependence of a cohesive interface potential on aset of state variables (for the isothermal case) that charac-terize the interface structure and morphology. To accountfor separation associated with irreversible processes, de-fined as plastic separation, the rate of change of the con-stituents that characterize the internal structure within theinterface region are tracked by a set of state variables overthe course of the deformation. In Section 3, MD simula-tion results are presented for normal and tangential sepa-ration of a copper bicrystal with a planar interface struc-ture. The continuum formulation that uses the molecularsimulation results is based on calibration of state variableevolution equations for the interface. In Section 4, simula-tion results for crack initiation processes in a nanocrystalare presented.

2. COHESIVE ELEMENT TECHNIQUE FORSIMULATING MECHANICAL RESPONSE INNANOCRYSTALLINE MATERIALS

Let x : Bn → B represent the nonlinear deformation mapof the nanocrystal at timet, andF = ∇nx the associatedtangent map.F maps pointsX ∈ Bn onto pointsx(X, t)of the current configurationBn+1.

The kinematic problem for deformation employs theupdated Lagrangian framework. Here, the total deforma-tion gradientF n+1 at time t = tn+1 of configurationBn+1 with respect to the initial undeformed configurationB0 (with pointsX0) at timet = 0 is assumed to be de-composed as

F n+1 = ∇0x(X0, tn+1)= ∇nx(Xn, t)∇0x(X0, tn)= F rF n = F eF p

(1)

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Calibration of Nanocrystal Grain Boundary Model Using MD Simulations 511

whereF e andF p are the elastic and plastic deformationgradients respectively at timen + 1, F r is the relativedeformation gradient of the current configuration with re-spect to the configuration at timen, andF n refers to thetotal deformation gradient in the reference configuration(Bn) with respect to the initial undeformed configuration.

The equilibrium equation is expressed in the referenceconfigurationBn as

∇n · P r = 0 (2)

where the PK-I stressP r(Xn, t) is expressed in terms ofthe Cauchy stress (T ) asP r(Xn, t) = (detF r)TF−T

r .A crack at time stepn in a cohesive formulation is rep-

resented as a surfaceδBintn (refer Fig. 1). Tractionstn at

the interface perform work on the displacement jump[|u|]across the cohesive interface. The solution of a genericloading increment involves the solution to the principle ofvirtual work (PVW) given as follows: Calculatex(Xn, t)such that

∫

Bn

P r · ∇nudVn +∫

δBintn

tn · [|u|]dA = 0 (3)

for every admissible test functionu expressed over thereference configurationBn. The weak form is solved in anincremental iterative manner as a result of material non-linearities. The finite element (FE) technique is used forthe solution of the weak form. Bilinear quadrilateral el-ements are used for modeling grain interiors, and four-noded 2D linear elements with initial zero thickness areused to model the cohesive interface. FE procedure hasbeen implemented in an object-oriented and parallel envi-ronment in C++ and PetSc parallel toolbox building fromour earlier work (Sundararaghavan and Zabaras, 2006b).

2.1 Constitutive Equations for the GrainBoundaries

The model presented here pertains to a two-dimensionalanalysis. We consider two grains separated by an inter-face with the local coordinate defined using a normalnand tangentτ to the interface defining thex- andy-axis,respectively. A rotation matrixR transforms the samplecoordinate system to the interface local coordinates. Letδ denotes the displacement jump across the cohesive sur-face at current time with respect to the undeformed con-figuration (as shown in Fig. 1). We assume that the dis-placement jump may be additively decomposed as

δ = δe + δp (4)

FIG. 1: Cohesive element between two crystals (B+ andB−) is shown. The middle surface is taken as the cohesivesurface (δBn). The coordinate system for the cohesive el-ement is also shown.

whereδe andδp respectively, denote the elastic and plas-tic parts of the displacement jump. A simple quadraticfree energy of the interface is assumed asφ =1/2δe.Kδe with K, the interface elastic stiffness tensor,taken to be positive definite. The tractiont is then givenby

t =∂φ

∂δe = Kδe (5)

The interface model is taken to be isotropic in its tan-gential response with interface stiffness tensor taken asK = KNn⊗ n + KT τ⊗ τ with KN > 0 andKT > 0the normal and tangential elastic stiffness moduli, respec-tively. The interface tractiont is decomposed into normaland tangential parts,tN andtT , respectively, over the in-terface. Scalar state variabless(1) and s(2) are used torepresent the deformation resistance for the normal sep-aration and the shear sliding mechanism.

The flow rule for the interface is written as

δp = γ(1)n + γ(2)τ (6)

whereγ is the inelastic deformation rate with the relativeplastic displacements defined asγ(i) =

∫ t

0γ(i)dt. Trac-

tion t attains a value ofs(i) on the systems where plasticdisplacements occur. Thus, forγ > 0 the following rela-tionship holds:

|tN | − s(1) = 0 (7)

|tT | − s(2) = 0 (8)

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512 Lee & Sundararaghavan

Interface hardening is defined using an evolution equa-tion for the scalar state variables (from initial values(i)(0) = s

(i)0 ) as

s(i) =2∑

j=1

h(ij)γ(i) (9)

To capture the cohesive law obtained from MD simu-lations, we focus on obtaining the right parameters in theabove-hardening law. In this work, the evolution of scalarstate variables is defined using an equivalent relative plas-tic displacement defined as

γ =√

γ(1)2 + αγ(2)2 (10)

with α as the coupling constant, as follows:

s(i) = h(i)0 1− s(i)

s∗(i)a(i) ˙γ for γ ≤ γc (11)

s(i) = −h(i)soft

˙γ for γc < γ ≤ γfail (12)

s(i) = 0 for γ > γfail (13)

Here,h0 andhsoft are hardening and softening coef-ficients, respectively; ands∗ is the threshold for the in-terfacial resistance. Here,γc andγfail denote the criticaleffective displacement and the effective displacement forinterface failure. The state variable evolution is calibratedso that the cohesive law as obtained from molecular sim-ulations can be realized.

3. MOLECULAR DYNAMICS FORGENERATING TRACTION SEPARATIONPROFILES FOR GRAIN BOUNDARIES

The MD simulations were designed to mimic simpleshear and simple tension for different crystal orientationsat different temperatures. The computational setup is il-lustrated in Fig. 2. The atoms are packed in the domainsuch that a grain boundary occurs in the middle of thedomain. The misorientation between the atoms in the toppart of the bicrystal and the bottom part was varied from15 to 45 deg. There is no rotation about thex- or y-axes,meaning the grain boundary does not have twist charac-teristics. The width and height of the model after ther-mal equilibration was 10.84 and 8.31 nm, respectively.This corresponds to about 16 planes of base lattice and 24planes of the rotated lattice (top) in they-direction, andthe resulting grain boundary was semicoherent. In previ-ous work (Horstemeyer et al., 2001), it was shown that in

FIG. 2: Initial configuration after equilibration: To de-form the grain boundary model with 28000 atoms, uni-form displacement rate boundary conditions are appliedto the atoms at the top and bottom of the cell.

fixed end simulations, the global continuum stress satu-rates if thez-direction is four or more unit cells in thick-ness. The thickness in our simulations is 3.615 nm, corre-sponding to 20 atomic planes. After creating the sampleswith a desired crystal orientation, atoms within a thick-ness along they-direction of 3.615 A at the bottom and5.425 A at the top were frozen by assigning zero forceson these atoms.

The embedded atom method (Daw et al., 1993) is usedto model the interaction potential between copper atomsin the lattice. Velocities of the interior (or mobile atoms)are initialized using the Maxwell-Boltzmann distributionat room temperature (300 K). These atoms are allowedto relax to minimum energy to accommodate any surfacerelaxation at the two remaining free surfaces (yz-planes)as well as the grain boundary through equilibration pro-cedure. If just the top row of atoms initially experiencedthe prescribed velocity without the active internal atomsexperiencing the same velocity field, then a shock wouldbe induced into the block of material because of the highstrain rates. In our calculations, we introduced an initial

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Calibration of Nanocrystal Grain Boundary Model Using MD Simulations 513

velocity field that mitigated the shock wave and then ap-plied the boundary velocity fields. To accomplish this, theinterior atoms in the model were also given an initialx-velocity (superposed on their thermal velocities) that var-ied linearly from zero to the prescribed velocity at thetop atomic plane, depending on theiry-coordinates in thesimulation box.

To apply tension to the the grain boundary model, uni-form displacement rate boundary condition is applied tothe top and bottom layer atoms, along the positive andnegativey-direction, respectively, as shown in Fig. 2. Forboth tension and simple shear, the top and bottom layeratoms in Fig. 2 are constrained so that the net force oneach atom is set to zero at each time step during the sim-ulation, creating a planar surface. Under application ofa prescribed velocity condition, each rigid boundary willmove as a single planar and parallel unit during the entiredeformation process. For tension simulation, the systemis placed in a periodic box along thex andz directions.In the case of simple shear, the computational block ofmaterial has free surface in thex direction and was pe-riodic in thez direction. For simple shear, a deformationrate was applied to the block of atoms by setting thex ve-locity of the topxz plane to a constant value. The bottomatomic plane had a prescribedx velocity of zero for theduration of the simulation. The displacement of the inter-face region is defined as the sum of the displacements ofthe loading planes (far-field displacement).

Following initialization, a constant number of atoms,constant volume, and constant temperature (NVT) simu-lation was performed with a 0.001 ps time step until theblock of atoms had undergone complete fracture. Becausestraining via moving the frozen planes adds considerableenergy to the active atoms, a Nose-Hoover thermostat[implemented in LAMMPS software (Plimpton, 1995)]was used during the MD simulation to keep the activeatoms at constant temperature. The thermostat applies adamping (or acceleration) factor to the active atoms basedon the difference between their current temperature andthe desired temperature. To compute the instantaneoustemperature of the ensemble of active atoms, one needsto subtract from each atom the nonthermalx velocity thatwas initially prescribed, since the active atoms essentiallymaintain the same component ofx velocity for the simu-lation. To qualitatively use the results of the MD simula-tion to calibrate the nanocrystal continuum model, the av-erage atomic stress (for free atoms) is calculated againstboundary displacement in both normal (tensile loading)and tangential (shear loading) directions and the resultsare discussed in Section 3.1. At each atom, the dipole

force tensor,β (known as the local stress tensor), is givenby

βikm =

1Ωi

N∑

j(6=i)

f ijk rij

m (14)

wherei refers to the atom in question andj refers to theneighbor atom,fk is the force vector between atoms,rm

is a displacement vector between atomsi andj, N is thenumber of neighbor atoms, andΩi is the atomic volume.If stress could be defined at an atom, thenβ would be thestress tensor at that point. Since stress is defined at a con-tinuum point, we determine the stress tensor (termed asthe global continuum stress hereafter) as a volume aver-age over the block of material,

σkm =1

N∗

N∗∑

i

βimk (15)

in which the stress tensor is defined in terms of the totalnumber of active atoms,N∗, in the block of material.

3.1 MD Results

The grain boundary interface model is loaded to fullseparation in order to study the quantities that are im-portant for development of a continuum cohesive modelfor the nanocrystal interface. The quantities include thepeak stress, the critical displacement (at peak stress), andthe failure displacement (when stress goes to zero). Fig-ure 3 shows the normal stress-displacement response ofthe grain boundary interface model along with the snap-shots of the grain boundary at different times (for a 45 degmisorientation case). Peak stress for tensile failure modewas noted to be about 14 GPa, which is the same as thatreported recently (Spearot et al., 2004). Failure displace-ments cannot be directly compared as this work employsa comparatively larger system size to overcome domainsize effect on failure displacements. Although the loca-tion of crack depends on the misorientation of the grainboundary, it is generally seen that cracks form at the grainboundaries for misorientations of> 15 deg and the cracksbegins with void formation.

We have investigated the stress displacement responseof the bicrystal for various values of the misorientations ata temperature of 300 K at an imposed deformation rate of1 A/ps and 3 fs time steps. We focus our attention on thethree parameters that are used for calibration of the cohe-sive zone model, namely, the peak stress,σp, the criticaldisplacement,δcritical, and the area under the traction sep-aration law (fracture energy). Figure 4 plots the normal

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514 Lee & Sundararaghavan

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0

2

4

6

8

10

12

14

No

rma

l str

ess (

GP

a)

Far-field normal displacement (A)

Stress (GPa)

FIG. 3: Normal stress displacement response of the interface model: During tensile separation, the normal stressdisplacement response shows a dominant peak with associated peak stress. Peak stress is around 14 GPa as reportedin (Spearot et al., 2004).

0

2

4

6

8

10

12

14

16

1 6 11 16 21 26 31

Far Field Normal Displacement (Angstroms)

Norm

al s

tress (

GP

a)

25

45

15

35

FIG. 4: Normal stress displacement response for various misorientation angles (in degrees) for the tilt GB

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Calibration of Nanocrystal Grain Boundary Model Using MD Simulations 515

stress displacement response curves for a set of bicrystalswith varying degrees of misorientations 15, 25, 35, and45 deg loaded at a set temperature of 350 K. The displace-ment associated with peak stress is the same for all thesecases. It is seen that peak stress reaches a minimum at amisorientation angle of 45 deg. Note that for a fcc bicrys-tal, a 45 deg misorientation represents the maximum tiltangle between the two crystals. The slope of the elasticpart of the deformation is nearly uniform over the rangeof misorientations.

Figure 5 plots the tangential stress displacement re-sponse curves for 45 deg bicrystal loaded at a temperatureof 300 K. Although the critical displacement for tensilemode and the shear mode occurs between 1 and 2 nm,the peak stresses for the shear mode are much smallercompared to the tensile mode. In addition, the soften-ing part of the traction separation law occurs for a muchlonger duration and zero stresses are not reached untilabout 31.7 nm (compared to 2.125 nm for tensile loadingcase).

It is to be noted that shear simulations contain freeboundaries along theyz planes. We note that the appli-cation of free boundaries may lead to a softer response(compared to that experienced by an equivalent cohesiveelement) owing to absorption of dislocations through re-arrangement of atoms along the free surfaces. In addition,

we note that the MD simulation is performed at the rateof 0.1 nm/ps whereas the continuum model is assumedto be rate independent. We performed molecular simu-lations at strain rates of 0.1 and 0.001 nm/ps to identifydeformation rate effects. Over two orders of magnitudedecrease in deformation rate, the decrease in peak stresswas modest (∼ 4.6%). The slope of the elastic part of thetraction separation law was unchanged over these strainrates. The critical displacement (at peak stress) increasedby 10.3% while fracture energy decreased by 11.6% overthe two orders of magnitude decrease in deformation rate.Although a rate-dependent constitutive models based ongrain boundary shearing rates [or based on Coble-typecreep mechanisms (Yamakov et al., 2002)] are being de-veloped (Wei et al., 2006), in this paper we restrict our-selves to a simpler rate-independent model. Indeed, thephysical experiments at quasistatic strain rates (Torre etal., 2002) also reveal only a very slight strain rate sensi-tive response (in the case of nanocrystalline nickel). Suchsmall strain rates can be accessed through molecular stat-ics simulations (Warner et al., 2006) or through develop-ment of scaling laws [e.g., based on Zener-Holloman scal-ing (Zener et al., 1944)]. Nevertheless, continuum modelscannot be directly verified without atomistic calculationsbecause their validity is contingent on the accuracy of pa-rameters used in the model.

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Stress (GPa)

Far-field tangential separation (A)

Tangential str

ess (

GP

a)

FIG. 5: Tangential stress versus displacement plot of the interface model in shear mode of separation.

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3.2 Construction of Continuum ModelParameters

We consider nanocrystalline fcc copper with slip occur-ing in the 12111〈110〉 slip systems. The anisotropicelasticity tensor for fcc copper can be specified in termsof the three stiffness parameters asc11 = 170 GPa,c12 = 124 GPa, andc44 = 75 GPa. Interface stiffnesses,KN andKT are defined based on the values of isotropicpolycrystalline Young’s modulus (E) and shear modulus(G) of copper and an assumed grain boundary thickness(g) of 1 nm asKN = E/g andKT = G/g which leads tovalues ofKN = 130 GPa/nm andKT = 48 GPa/nm.The slip system resistance for grain interiors is calcu-lated based on the notion that the grain boundaries arethe sources of dislocations and the grain interiors are freeof dislocations. The criterion for grain boundary emis-sion and absorption of dislocations in nanocrystals is pro-vided in (Wei et al., 2004) assα = G′b/D, whereG′ =

√(c11 − c12)c44/2, b is the magnitude of Burgers

vector, andD is the average grain size. This representsa criteria for grain boundary emission and absorption ofdislocations in nanocrystalline polycrystals (Asaro et al.,2003). For fcc copper, these values areG′ = 41.5 GPaand b = 0.26 nm. Additionally, initial elastic region(linear regime) in the cohesive law and was assumed tobe valid up to a nominal stress of 1 GPa in both ten-sile and shear mode beyond which hardening begins (i.e.,s(1)0 = s

(2)0 = 1 GPa).

To effectively address the dependence of separationon interface structure, the set of state variables shouldincorporate parameters that describe physical attributessuch as composition, dislocation density, and a measureof the number density of broken bonds (damage param-eters) within the interfacial separation boundary layer. Inthe adopted model (Wei et al., 2004), a simplified practi-cal description is provided using state variables that gov-ern evolution of resistance of interface to deformation[Eq. (11)]. The parameters for inelastic response of theinterface are obtained by systematically analyzing variousregimes (initial elastic followed by hardening and soften-ing) in a traction-separation curve predicted from molec-ular dynamics simulations. Key parameters of interest forthe fitting procedure are the peak stress (tN andtT ), criti-cal displacement (δc(N) andδc(T )) and the area under thecohesive law (fracture energy) for the interface under ten-sion and shear modes at a misorientation of 45 deg and ata temperature of 300 K.

The following state variables at the interface are con-structed from molecular calculations: The hardening ex-

ponentai, threshold resistances∗(i), hardening coeffi-cient h(i)

0 , softening coefficienth(i)soft, effective displace-

ment to failureγfail, critical displacementγc, and nor-mal/tangential coupling parameterα.

A value of the coupling parameterα = 0.3932 is ob-tained based on an assumedγc = 1.16 nm, which is thecritical displacement in tensile mode obtained from MDsimulations. This value of the coupling parameter leadsto a value ofγc = 1.85 nm when the interface startsto soften in shear (as observed from molecular simula-tions). The slope of the initial hardening region of the co-hesive law (region up to the peak stress) was varied untilit matched the molecular simulation result. The thresholdresistance [s∗(i)] was subsequently varied until the peakstress as predicted by MD simulation is achieved. A sim-ple linear evolution law is assumed for the evolution ofinterfacial resistancea(i) = 1. This reduces the optimiza-tion problem of determining the state variables to the de-termination of two parameters:[γfail, h

(1)soft] subject to the

constraints that the variables are positive andt = 0 forγ > γfail. The objective was to minimize the error inthe area under the cohesive law predicted by the cohe-sive model. The values obtained through this procedureare listed in Tables 1 and 2. The final traction separationresponse obtained is shown in Fig. 6.

In a related work (Spearot et al., 2004), the validityof the cohesive laws was tested during the unloading andreloading process in Cu nanobicrystal. The unloading be-havior is a long-time behavior and needs to be analyzedusing quasistatic or long-time molecular dynamics simu-lations. From results of MD simulations, it was postulatedthat although tangential separation produces a residualplastic displacement upon unloading, the normal mecha-nism would unload to zero displacement elastically whileretaining the original interface strength. In the shear sep-aration mode, the continuum cohesive model performs asexpected. However, in the normal separation mode, thecontinuum law leads to a permanent plastic displacement

TABLE 1: For bicrystal Cu, the following continuum pa-rameters were calibrated from MD simulations

Parameter ValuePeak stress in tension (GPa) 14.50Peak stress in shear (GPa) 4.23

Critical displacement in tension (nm) 1.16Critical displacement in shear (nm) 1.85

Failure displacement in tension (nm) 2.125Failure displacement in shear (nm) 31.7

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Calibration of Nanocrystal Grain Boundary Model Using MD Simulations 517

TABLE 2: For bicrystal Cu, the following parameters were identified from MD simulations

Parameter ValueHardening exponent a(1) = a(2) = 1Threshold resistance s∗(1) = 20 GPa ands∗(2) = 4.7 GPa

Hardening coefficient h(1)0 = 21 GPa/nm andh(2)

0 = 8.3 GPa/nm

Softening coefficient h(1)soft = 14 GPa/nm andh(2)

soft = 0.3 GPa/nmEffective displacement to failure γfail = 16.0 nm

Critical displacement γcritical = 1.16 nmNormal/tangential coupling parameter α = 0.3932

0 100 200 300 4000

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Tangential displacement (in angstroms)

Ta

ng

en

tia

l tr

actio

n (

GP

a)

MD CalculationCohesive law

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Normal displacement (in angstroms)

No

rma

l tr

actio

n (

GP

a)

MD calculationCohesive law

FIG. 6: Depiction of traction-displacement laws identified for simulation of grain boundary sliding and separation.

upon unloading. Upon reloading, the continuum cohesivelaw follows the original curve from the point at which itwas unloaded.

The unloading and reloading behavior in the normalseparation mode was tested using molecular simulationsand the results are shown in Fig. 7. The zero force con-straint and velocity boundary conditions on the top andbottom boundary planes were removed to simulate theunloading process. Upon release of the load, the normalstress relaxes almost immediately. This is followed by adynamic behavior where the inward momentum of atomsat the outward boundaries competes with the outward mo-mentum of atoms at the crack and the system elasticallyrebounds. Over a long time, these oscillations die downand the system slowly relaxes back such that the interfaceis regenerated. The configuration of atoms during the un-loading and the associated per atom stresses are shownin Fig. 7. A small amount of residual plastic deforma-tion is seen after the elastic oscillations die down (after 1ns simulation time). Upon reloading, the system followsa cohesive law similar in nature to the original cohesive

law, although with a relatively smaller peak stress. In areal system, dislocations that are emitted from the inter-face region have the ability to sweep further away fromtheir source. It is noted that MD model is sensitive to thesize of the domain used and the use of a restricted domainwould mask the larger length scales involved in plasticity(dislocation motion). The use of fixed boundaries would,in fact, drive the dislocations emitted back to the crack,which can affect the relaxation behavior. Clearly, there isfurther scope for interpretation of molecular simulationresults to understand unloading and reloading behavior inquasistatic regimes.

4. CONTINUUM RESPONSE PREDICTION

Standard crystal plasticity models for intragranular defor-mation is inadequate to represent limited plasticity [e.g.,one or two dislocation partials] in nanocrystalline materi-als. However, elastic anisotropy and crystallographic tex-ture are still important considerations since the disloca-tions are expected to move on the slip systems. Crystal-

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15

Far Field displacement (Angstroms)

No

rma

l str

ess (

GP

a)

Original cohesive law Unloading behavior (MD)Cohesive law upon reloading (MD)

unload

(continuum)

reload

(continuum)

-40

-30

-20

-10

0

10

20

30

4 3

2

1

4321

Stress (GPa)

FIG. 7: Depiction of normal traction-displacement laws in the case of unloading. The figure identifies discrepanciesin the model used in Wei and Anand (2004) and molecular dynamics model. The atom configuration at various pointsduring unloading are indicated

lographic slip and reorientation of crystals are modeledin this work using a rate-independent constitutive model(Anand and Kothari, 1996). It is assumed that deforma-tion takes place through dislocation glide and the evolu-tion of the plastic flow is given by

Lp = F p(F p)−1 =∑α

γαSα0 sign(τα) (16)

whereSα0 = mα⊗nα is the Schmid tensor andγα is the

plastic shearing rate on theαth slip system.mα andnα

are the slip direction and the slip plane normal, respec-tively. In the constitutive equations to be defined next, theGreen elastic strain measure (defined on the plasticallydeformed stress-free configuration) is utilized,

Ee

=12

(F eT F e − I

)(17)

A conjugate stress measure is then defined as

T = detF e(F e)−1T (F e)−T (18)

whereT is the Cauchy stress for the crystal in the samplereference frame. The constitutive relation for the stressmeasure is given by

T = Le[E

e]

(19)

whereLe is the fourth-order anisotropic elasticity tensorfor fcc Cu (Anand and Kothari, 1996), expressed in termsof the crystal stiffness parameters and the orientationr.The resolved shear stressτα = T · Sα

0 attains a criticalvalue sα on the systems where slip occurs with plasticshearing rate on theαth slip systemγα > 0. Further-more, the resolved shear stress does not exceedsα on theinactive systems withγα = 0. Thus, the following relationholds:

Journal for Multiscale Computational Engineering

Calibration of Nanocrystal Grain Boundary Model Using MD Simulations 519

|τα| − s(α) = 0 for γα > 0 (20)

Slip system resistances (Wei and Anand, 2004) are as-sumed to be constants, i.e.,sα(i) = 0.

A polycrystal (Fig. 8) with 64 grains was generated us-ing a standard Voronoi construction employed in the de-sign examples of Sundararaghavan and Zabaras (2006b).Based onsα = G′b/D as explained previously, and av-erage grain size ofD = 26 nm leads to a slip system re-sistance of 415 MPa. To overcome the dependency of thesolution on the mesh size, the polycrystal was meshed atthe highest mesh density using our inhouse meshing tool,which led to 19,525 quadrilateral elements, including thefour-noded zero-thickness cohesive zone elements at thegrain boundaries. Random orientations were assigned tothe grains such that all grain boundaries were of tilt char-acter. The polycrystal was subjected to tensile loadingalong they direction. In this work, the equivalent strainis computed based on the volume average of the deforma-tion rate (D = 〈D〉). The average effective plastic strainεeff is defined asεeff =

∫ t

0

√2/3D · Ddt. The equiva-

lent stress for the polycrystal is represented using the von

Mises normσ =√

(3/2)T′ · T ′

, T′= T − 1/3tr(T )I.

Figure 9 provides the stress contour and the equivalentstress strain curve when plastic strains are first observed.As the separation progresses, high stresses develop close

FIG. 8: Initial polycrystal for the crack problem with 64grains. Initial texture was randomly assigned.

to sharp crack corners, as seen in Fig. 9(a). The crackpath followed during this uniaxial loading process oc-curs approximately at the midline of the polycrystal. InFig. 9(b), we also show the equivalent stress strain curvewhen grain boundary deformation modes are deactivated(cohesive elements are removed). The plastic yielding be-gins at a much larger stress when grain boundary ac-commodation effects are not accounted for. The equiv-alent plastic strain plots in Fig. 10 clearly indicate therole of interfaces in nanocrystals. The initial plastic re-sponse observed at the nanoscale was localized at regionsof grain boundary cracks, especially at the sharp cracktips. As seen from Fig. 10, the interior of grains show nosign of plastic strains while crack tips are the zone forplastic deformation. This result is agreeable with results(Wei and Anand, 2004) that reveal that the macroscopi-cally imposed deformation is initially accommodated bygrain boundary accommodation effects. Note that the ini-tial yield strength of 26 nm Cu polycrystal is observedat 340 MPa (compared to 365 MPa value experimentallyidentified in several other papers (Sanders et al., 1997;Gertsman et al., 1994; Suryanararayan et al., 1996). Thegrain boundary sliding and separation effects at such lo-cations of high plastic strains are shown in Fig. 10. Thisreveals that when grain boundary deformation cannot beaccommodated due to geometric restrictions at locationsof crack tips or triple junction in Fig. 10, local stress con-centrations develop to cause plastic strains (dislocationemission) at these locations. Please note that a crystalplasticity-based description of the inelastic deformationof the grain interior for nanocrystalline materials doesnot account for the emission or absorption of dislocationsfrom grain boundaries. The simulations do not show sig-nificant plastic strain in the interior of grain, and the inte-rior of the grains can be considered elastic during the ini-tial part of the stress strain curve. In light of some of dis-crepancies mentioned previously in the continuum modeland the molecular simulation results (notably, unloadingbehavior, finite size, strain rate effects, and intragranu-lar plasticity laws), further study is still necessary beforecomparing stress-strain curves based on calibrated con-tinuum laws with experiments performed at quasistaticstrain rates.

5. CONCLUSIONS

This paper presents an initial study of the use of MDsimulations to calibrate an isothermal, rate-independentelasticplastic grain boundary model for pure nanocrys-talline copper. The continuum model introduces a variety

Volume 8, Number 5, 2010

520 Lee & Sundararaghavan

Equivalent Stress (MPa): 0 89 177 266 355 444 532 621 710

Equivalent plastic strainE

qu

iva

len

tstr

ess

(MP

a)

5E-07 1E-06 1.5E-06100

200

300

400

500

600

5E-07 1E-06 1.5E-06100

200

300

400

500

600

No GB effects

With GB sliding

and separation

FIG. 9: The stress contours for the polycrystal. The initial plastic strain is due to the effect of grain boundaries. Thecurve obtained when grain boundaries are deactivated is also shown.

0 0.0005 0.001 0.0015 0.002

FIG. 10: The plastic strain locations during initiation ofplasticity. The plastic strains are concentrated on grainboundary triple points or sharp corners

of new physics compared to existing models, including ir-reversible plastic deformation prior to failure and abilityto explain combined shear and tensile loading behavior.To guide the continuum formulation, atomistic calcula-tions are presented for a planar, copper grain boundaryinterface with a tilt lattice misorientation. The interface

models are deformed to full separation with both tensileand shear deformations. Plots of stress versus displace-ment show a distinctly different deformation response be-tween normal and tangential interface loading conditions.It is seen that loading that promotes tensile separation ex-hibits a rapid increase in stresses, while shear deforma-tion of the interface region produces a much slower evo-lution rate and smaller stresses by comparison. Further-more, MD simulations provide us with traction separationlaws that account for dependence on grain boundary mis-orientations and deformation rates.

The data were used to calibrate the state variables inthe traction separation law. Microstructures simulated us-ing cohesive model of the grain boundary indicates thatthe macroscopically observed nonlinearity in the stress-strain response is mainly due to the inelastic response ofthe grain boundaries. Plastic deformation in the interior ofthe grains prior to the initiation of grain boundary crackswas not observed. The stress concentrations at the tips ofthe distributed grain boundary cracks, and at grain bound-ary triple junctions, cause a limited amount of plastic de-formation in the high-strength grain interiors. However,some discrepancies were noted when the results of un-loading and reloading in the continuum model are com-pared to molecular simulations. The continuum modelmodels irreversible inelastic normal separation at the in-terface prior to failure. Results from long-time MD sim-

Journal for Multiscale Computational Engineering

Calibration of Nanocrystal Grain Boundary Model Using MD Simulations 521

ulations indicate that cracks formed due to normal loadswould unload to zero displacement elastically while re-taining the original interface strength. Clearly, there ismerit for future study in examining traction separationbehavior under long-time relaxation and reloading pro-cesses. However, the size of the domain and the strain-rateeffects need to be carefully accounted for to accuratelymodel the phenomena at continuum scales.

ACKNOWLEDGEMENTS

This work was sponsored by NASA Constellation Uni-versity Institutes Project under Grant No. NCC3-989 withClaudia Meyer as the project manager.

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