Investigating Damage Evolution at the Nanoscale: Molecular Dynamics Simulations of Nanovoid Growth...

Investigating Damage Evolution at the Nanoscale:Molecular Dynamics Simulations of Nanovoid Growthin Single-Crystal Aluminum

M.A. BHATIA, K.N. SOLANKI, A. MOITRA, and M.A. TSCHOPP

Nanovoid growth was investigated using molecular dynamics to reveal its dependence on voidsize, strain rate, crystallographic loading orientation, initial nanovoid volume fraction, andsimulation cell size. A spherical nanovoid was embedded into a periodic face-centered cubic (fcc)Al lattice, and a remote uniaxial load was applied to elucidate dislocation nucleation and shearloop formation from the void surface as well as the subsequent void growth mechanisms. Thenucleation stresses and void growth mechanisms were compared for four different strain rates(107 to 1010 seconds�1), five different simulation cell sizes (4-nm to 28-nm lengths), four differentinitial nanovoid volume fractions, and seven different tensile loading orientations representativeof the variability within the stereographic triangle. The simulation results show an effect of thesize scale, crystallographic loading orientation, initial void volume fraction, and strain rate onthe incipient yield stress for simulations without a void (single-crystal bulk material). Forinstance, the crystallographic orientation dependence on yield stress was less pronounced forsimulations containing a void. As expected, dislocations and shear loops nucleated on variousslip systems for the different loading orientations, which included orientations favored for bothsingle slip and multiple slip. The evolution of the nanovoid volume fraction with increasingstrain is relatively insensitive to loading orientations, which suggests that the nanoscale plasticanisotropy caused by the initial lattice orientation has only a minor role in influencing thenanovoid growth rate. In contrast, a significant influence of the initial nanovoid volume frac-tions was observed on the yield stress, i.e., a ~35 pct decrease in yield stress was caused byintroducing a 0.4 pct nanovoid volume fraction. Furthermore, a continuum-scale bridgingparameter m—which is a material rate sensitivity parameter in continuum damage mechan-ics—was calculated and found to be close to 1. Consequently, atomistic simulations of this typecan indeed inform continuum void growth models for application in multiscale models.

DOI: 10.1007/s11661-012-1082-z� The Minerals, Metals & Materials Society and ASM International 2012

I. INTRODUCTION

PREDICTING failure in ductile materials requiresunderstanding the complex process of damage evolu-tion, whereby voids nucleate, grow, and coalesce.[1–23]

This type of failure mechanism is intrinsically a multi-scale process that spans from the nanoscale to themacroscale. The void growth mechanism has beenstudied extensively at the microscale and the macro-scale.[1,2,11–13,24–31] In recent years, physically motivateddamage evolution laws have been used to try to revealthe nature of the ductile failure process in materials.These laws are supported by numerous experiments thathave measured void growth and coalescence as a

function of the applied stress.[1,2] However, these exper-iments are often difficult to conduct, and in many cases,the characterization techniques for measuring voidvolume fraction (vvf) are destructive. There differentcontinuum models are proposed to model void growthin two and three dimensions.[3–5,32] Continuum modelsare based on the physical understanding that in thecourse of plastic deformation, microvoids nucleate andgrow until a localized plastic necking or fracture of theintervoid matrix occurs, which causes the coalescence ofneighboring voids. One of these models was proposedby Gurson[3] and was extended phenomenologically byNeedleman and Tvergaard[4] (the so-called GTN mod-el). Other micromechanical damage models based oncontinuum damage mechanics and thermodynamicshave also been proposed, such as one by Rousselier.[33]

Previous on isolated void growth work conducted byRice and Tracey,[11] McClintock,[20] and Edelson andBaldwin[24] shows its dependence on both stress triax-iality and volume fraction. Moreover, Fleck et al.[12]

proved numerically that lower stress triaxiality causesthe Gurson model to overestimate softening caused byvoid nucleation. Additionally, Needleman[13] developeda continuum-based cohesive zone interface model for

M.A.BHATIA,GraduateResearchAssistant, andK.N. SOLANKI,Assistant Professor, are with the School for Engineering of Matter,Transport, andEnergy (SEMTE), Arizona StateUniversity, Tempe,AZ85287. Contact e-mail: [email protected] A. MOITRA, Postdoc-toral Fellow, is with the Department of Chemical Engineering, ThePennsylvania State University, University Park, PA 16802. M.A.TSCHOPP, Assistant Research Professor, is with the Center forAdvanced Vehicular Systems, Mississippi State University, MississippiState, MS 39762.

Manuscript submitted March 21, 2011.Article published online February 10, 2012

METALLURGICAL AND MATERIALS TRANSACTIONS A VOLUME 44A, FEBRUARY 2013—617

nucleation (debonding) and void growth for a weakermatrix material. Similarly, Pardoen and Hutchinson[16]

developed a void cell model for void growth andcoalescence. However, these continuum damage modelsoften cannot account for the nature of dislocationbehavior associated with void growth caused by theintrinsic length and time scales associated with disloca-tion nucleation, propagation, and its interactions.

Void nucleation, growth, and coalescence is a multi-scale deformation phenomena spanning across differentlength and time scales. Becker et al.[17] conductedexperiments and developed a model of microvoidnucleation and growth at grain boundaries. Theyconcluded that void nucleation and growth at grainboundaries has a strong dependence on the width ofgrain boundaries (a lower length scale feature) and wasaffected mildly by grain size. Steglich et al.[18] studiedcrack propagation in a metal matrix composite material(Al3Ti) and concluded that failure occurs by linkup ofmicrocracks (i.e., voids) in front of the crack tip.Atomistic simulations can provide insight into voidgrowth at the nanoscale where dislocations evolvenaturally. Several prominent studies have been con-ducted already in face-centered cubic (fcc)[13,34–38] andhexagonal close-packed (HCP) systems.[39] For example,Belak and coworkers[6–8] used molecular dynamics(MD) simulations to show that shear loop formationfrom growing voids is the primary mechanism of radialmaterial transfer required for void expansion. Potirnicheet al.[34] used molecular dynamics to investigate voidgrowth and coalescence using cylindrical and infinitevoids in a Ni single crystal and concluded that voidgrowth was more dominant than void coalescence at thenanoscale. In contrast, Groh et al.[39] used MD simula-tions of an HCP single crystal to show that nucleationand coalescence of microvoids were the two mostimportant damage mechanisms, independent of theloading direction. Solanki et al.[40] carried out large-scale MD simulations in copper and aluminum toinvestigate the collapse of preexisting voids by shock-induced loading. Validated by experiments when con-sidering their deformation shapes, their results show adecrease of the yield stress with an increase of the voidradius. Additionally, Traiviratana et al.[41] used MDsimulations on single-crystal and bicrystal copper toreveal that the emission and outward expansion of shearloops, nucleated from nanosized voids, were the primarymechanisms of void growth. Since then, several otherMD studies have observed both prismatic and shearloops emanating from voids.[6–10]

Molecular dynamics simulations have not been used tostudy fully the influence of crystallographic loadingorientation on void growth and the subsequent evolutionof the void volume fraction. When loading is applied in acertain crystallographic direction, the applied stress isresolved onto the available slip systems for deformation.Typically, only the resolved shear stress is neededbecause it is well known that dislocation motion in fccsingle crystals is governed by the critical resolved shearstress via Schmid’s law. However, recent MD studieshave also shown that the stress normal to the slip planemight play an important role in dislocation nucleation as

well.[35,36,42,43] Therefore, crystallographic orientationplays an important role in how the Schmid stress andstress normal to the slip plane contribute to bothhomogeneously[35,36,42,43] and heterogeneously[43–46]

nucleate dislocations in fcc Cu. Moreover, the impor-tance of the resolved stress normal to the slip plane leadsto an asymmetry in the nucleation stresses betweentensile and compressive stress states for single crys-tals[36,42,43] and grain boundaries,[43,45] partly because ofthe tensile normal stress in tension and the compressivenormal stress in compression. Because void growth at thenanoscale begins with dislocation nucleation from thevoid surface, considering the influence of crystallo-graphic orientation on void growth is important. Inaddition to the resolved stress components, some orien-tations have a resolved shear stress that is equal onseveral slip systems (multiple slip), whereas other orien-tations have one dominant slip system (single slip).Horstemeyer et al.[37] used MD simulations to show theeffect of crystal orientation on the stress–strain relation-ship under simple shear and to study the phenomena ofmultiple slip verses single slip in a bulk material (withoutany nanovoids). They found that dislocation–dislocationreactions, interactions, and trapping are important inthose orientations oriented for multiple-slip systems.[37]

At high strain rates, Horstemeyer et al.[38] proposed thatvoid growth is more dominant than void coalescence.Potirniche et al.[6] used crystal plasticity to investigatevoid growth and coalescence in fcc single crystals; theydetermined that plasticity anisotropy caused by theinitial lattice orientation has only a minor role in voidgrowth for microscale regions. Stress triaxiality is themain influencing factor controlling void growth andcoalescence. Accordingly, Liu et al.[27] investigated theinfluence of crystallographic orientation on void growthand coalescence in an fcc single crystal using finiteelement method. They concluded that crystallographicorientation influenced both the direction of growth andthe shape of the void.Hence, this work uses atomistic simulations to exam-

ine void growth by dislocation nucleation in single-crystal aluminum with the goal of casting the atomisticresults in terms of continuum parameters, specificallywith regard to the vvf. Aluminum was chosen as an idealfcc material. Aluminum and its alloys are used widely instructural component and aerospace industries becauseof its low density, moderate strength, and resistance tocorrosion. This study will examine uniaxial loading overa range of crystallographic orientations, initial nanovoidvolume fractions, strain rates, and simulation cell sizesto explore their influence on the incipient yield stress, thevvf, the void shape evolution, and the mechanisms fornanovoid growth.

II. SIMULATION METHODOLOGY

A parallel molecular dynamics code (LAMMPS[47])that incorporates spatial decomposition is used todeform the single-crystal atomistic models. The atompositions are generated first in the desired crystallo-graphic orientation with a void at the center of the

618—VOLUME 44A, FEBRUARY 2013 METALLURGICAL AND MATERIALS TRANSACTIONS A

simulation cell (Figure 1(a)). Then, the configuration isequilibrated using MD in isobaric-isothermal (NPT)ensemble at a pressure of 0 bar and a temperature of300 K (27 �C). The equilibration step allows the tem-perature to stabilize and the simulation cell bounds torelax to a zero-pressure condition by generating atomicpositions and velocities that sample the NPT ensemble.Thereafter, the simulation cell is deformed in uniaxialtension by applying a constant strain rate of 107, 108,109, or 1010 seconds�1 to the loading direction (xdirection) while prescribing a zero-strain condition onthe other two simulation cell boundaries (y and zdirections), as in References 2 and 3. This boundarycondition is implemented by decoupling the boundary inthe loading direction from the canonical (NVT) ensem-ble equations of motion (i.e., similar to Melchionnaet al.[48]). The four strain rates used in this study spanthree orders of magnitude and are combined with a widerange of simulation cell sizes (4-nm to 28-nm lengths, upto ~1 million atoms) and crystallographic orientations(seven orientations; Figure 1(b)). The system stress iscalculated by using virial definition without the kineticportion as

rab ¼1

V

XN�

i

XN

j 6¼ið Þ fijar

ijb ½1�

where fija is the force vector between atoms i and j in thedirection a, rb is the distance vector in direction b, N isthe number of neighbor atoms for atom i, N* is the totalnumber of atoms, and V is the simulation cell volume.The volume-averaged stress tensor is the same as theglobal stress used in previous MD simulations thatinvestigated length scale effects on yield.[4] Throughoutthis work, the stress required for dislocation nucleationfor void growth is the maximum von Mises stress. Thecentrosymmetry parameter[49] used for visualizationshows that dislocations nucleate from the void surface(causing the void to grow) at a stress close to maximumstress for all single-crystal configurations.

An embedded-atom method potential for alumi-num[50] is employed in this study. This embedded atommodel potential uses a basic set of cubic splines to fit the

ab initio and experiment database, similar to the methodused by Ercolessi and Adams.[51] The potential wasoptimized by dividing the database into two parts[52]:one part for fitting the potential and another part fortesting the potential. The Mishin et al.[50] potentialprovides a good description of both unstable and stablestacking fault energies and has been used previously tomodel dislocation nucleation at the nanoscale[43,44,46,53]

and void growth.[2,3,7,11,14,24–27] The unstable stackingfault energy determines the activation barrier for dislo-cation nucleation and plays an important role in theplasticity of metals. Because first principles calculationsof the stacking fault energies were used as fittingparameters, the Mishin et al. potential captures theessential deformation behavior and is sufficient tocharacterize void growth by dislocation nucleation insingle-crystal aluminum.Three-dimensional periodic boundary conditions are

used to investigate void growth by dislocation nucle-ation in single-crystal Al under tensile loading. Theinfluence of the simulation cell size (a cube with a sidelength of 4 to 28 nm) on void growth was examined firstto test for convergence of the dislocation nucleationstress from the voided surface. To investigate the strainrate (107 to 1010 seconds�1) and orientation effects, asimulation cell with 16-nm side lengths was used.Figure 1(b) shows the loading axis for seven crystal-

lographic orientations examined in this study for voidgrowth within the basic stereographic triangle with[100], [110], and [111] vertices. A similar methodologyhas been used to study homogeneous dislocation nucle-ation in single-crystal Cu.[11,12,50] Here, however, weexamine the effect of loading directionality on voidgrowth, whereby each single crystal is deformed intension at 300 K (27 �C) with an initial vvf of 0.4 pct.The evolution of vvf is calculated using a numerical

tool developed by the molecular biology commu-nity[54,55] to study the structure and function of mole-cules. The technique is based on the idea of rolling aprobe sphere over a molecule to generate a smooth outercontour, where the radius of the probe sphere is chosento represent a solvent molecule. In the current work,

Fig. 1—(a) The MD void growth unit cell with the initial vvf of 0.4 pct and periodic boundary condition in the x, y, and z directions (not toscale). (b) Stereographic triangle showing the seven crystallographic orientations investigated for void growth in this study.


instead of a solvent molecule, however, this tool is usedto calculate the void volume fraction and its evolution.This tool has been used previously for calculating thevoid growth in magnesium.[39]

III. SIMULATION RESULTS AND DISCUSSION

This section presents the stress–strain and nucleationstresses for single-crystal aluminum specimens with/without a single void as a function of the applieduniaxial strain. The influence of simulation cell size,initial vvf, strain rate, and crystallographic loadingorientation on the stress–strain response, vvf, and theshape evolution are analyzed subsequently.

A. Cell Size Dependence

The influence of the simulation cell size on voidgrowth was examined to test for convergence of thedislocation nucleation stress. Too small of a simulationcell size can produce artificially high stresses associatedwith interactions through the periodic boundaries,which might suppress certain deformation modes andmechanisms. For these simulations, a single crystal wasgenerated with 100h i crystallographic orientations forthe three directions and a vvf of 0.4 pct. The temper-ature was set to 300 K (27 �C). Multiple simulation cellinstantiations were generated with increasing cell sizes inall three directions with same initial vvf. After equili-bration, the configurations were then deformed inuniaxial tension at a constant strain rate of 108

seconds�1 with a strain-free condition for the twolateral boundaries. Figure 2 shows the size scale effectin the stress–strain response for a single-crystal alumi-num with 0.4 pct vvf, respectively. As the size scaleincreases, yield stress decreases as shown in Figure 2.Nucleation of dislocation loops on multiple slip planesat yield point for different cell sizes can also be observedin Figure 2. For an initial vvf of 0.4 pct, the peak stressfrom the stress–strain curve converges at 16 nm, asshown in Figure 3.The onset of dislocation emission/activity or the yield

stress is the stress at which the linear elastic behavior isno longer obeyed. Figure 3 shows the normalized yieldstress (with shear modulus G) values as a function ofnormalized void radius (with Burgers vector b). As thevoid radius increases for a fixed initial vvf, the yield stressrequired for dislocation nucleation from the surface ofthe void decreases. The atomistic results agree with theanalytical calculations by Lubarda et al.[56] (similar toTraiviratana et al.[41]), which are obtained from

ry

G¼ b

R

1ffiffiffi2p

pð1� vÞ

1þffiffi2p

R Rcore

� �4þ1

1þffiffi2p

R Rcore

� �4�1

½2�

where ry is the yield stress, G is the shear modulus, R isthe void radius, Rcore is the dislocation core size, v is thePoisson’s ratio, and b is the Burgers vector.Dislocation nucleation causes differences in the

stress–strain behavior and deviation from Lubarda’smodel. Interestingly, our results follow Lubarda’s modelwith Rcore = b for large void radii. For smaller radii

Fig. 2—Stress as a function of strain for different cell sizes with an initial vvf of 0.4 pct loaded along [100] direction. The simulation resultsreveal size scale effects on the incipient yield stress, which saturates at 16-nm cell size (also see Fig. 3).


with constant vvf, a deviation from Lubarda’s model iscaused by interactions through the periodic boundary;Lubarda’s model with Rcore = 2b is plotted to showthat the deviation is toward a larger Rcore value. Thisdeparture from Rcore = b might be expected, however.For instance, previous MD simulations on dislocation

nucleation have found that smaller dimensions actuallycan suppress the nucleation of dislocations on certainslip systems or increase the nucleation stress from thenucleating dislocation interacting with itself through theperiodic boundary.[43] Another reason for this departureat small length scales is that as cell size decreases with afixed initial vvf, the surface area/volume ratio will alsodecrease (as volume decreases faster than surface area),which leads to a higher stress concentration and lowerstresses required for dislocation nucleation. Hence atsmaller cell sizes, the physics governing the deformationbehavior might change. Figure 3 reveals also size-scaleeffects on the incipient yield stress, which saturates at1.6-nm void radius that corresponds to 16-nm cell size.Because we are interested in the void growth mechanism(and not void coalescence), all other orientations andstrain rates in this study used a minimum simulation celldimension of 16 nm in each direction with an initial vvfof 0.4 pct.

B. Strain Rate Dependence

The dislocation nucleation event and void growth is alsosensitive to the strain rate employed.Again, a single crystalwas generated with 100h i crystallographic orientations forthe three directions with a vvf of 0.4 pct. The temperaturewas set to 300 K (27 �C). The configuration was thendeformed in uniaxial at strain rates of 107, 108, 109, and1010 seconds�1 with a strain-free condition for other twolateral boundaries. High strain rates can produce adverseartifacts in the simulations, e.g., stress waves and largeatomic movements that can cause fracture. It is importantto select a strain rate that is low enough to cause sufficientconvergence of the nucleation stress.Figure 4 shows the stress–strain response for a

nanovoid strained in tension along with images of

0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

Normalized Void Radius, R/b

Nor

mal

ized

Yie

ld S

tres

s, σ

Y/G

Lubarda Model with Rcore

=b

Lubarda Model with Rcore

=2b

Atomistic Calculations

ν=0.333

G=35 GPa

Fig. 3—Normalized yield stress (with shear modulus, G) as a func-tion of normalized void radius (with Burgers vector, b) for a fixedinitial vvf when loaded along [100] direction and its comparison withLubarda model with different Rcore radii.[18] The dotted lines repre-sent increments of 0.2b in Rcore: 1.2b, 1.4b, 1.6b, and 1.8b. The simu-lation results reveal size-scale effects on the incipient yield stress,which saturates at 1.6-nm void radius that corresponds to 16-nm cellsize.

Fig. 4—Stress as a function of strain was plotted for different strain rates with a cell size of 16 nm (253,509 atoms), an initial vvf of 0.4 pct at300 K (27 �C) and loaded along [100] direction. The simulation results revealed strain rate effects on the incipient yield stress, which saturateswhen the applied strain rate is less than 5 9 108 s�1.


dislocation loop nucleation from the void at strain ratesranging from 107 to 1010 seconds�1. At high strain rates,a large number of dislocations is nucleated simulta-neously at the void surface because of the large affinedeformation with each time step. Moreover, the stress–strain response is influenced highly by the strain rate.However, at lower strain rates (107 to 108 seconds�1),the stress–strain response is similar as well as themechanism, i.e., several dislocations are nucleated atthe void surface. A strain rate of 108 seconds�1 wasselected for the remaining void growth simulations as acompromise of computational expense and the fidelityof results. Table I summarizes the yield stress as afunction of strain rate for bulk simulations (without anyvoid) and for simulation cells that contain a void (initialvvf = 0.4 pct).

Interestingly, in both cases, the strain rate effects onthe incipient yield stress saturates when the applied strainrate is below 5 9 108 seconds�1. Moreover, introducinga nanovoid with 0.4 pct volume fraction leads to a~35 pct decrease in yield strength. The ~35 pct decreasein yield stress is consistent with the macroscale degrada-tion of stress associated with introducing a holethat results in a stress concentration value of 3 (e.g.,Bammann and Solanki[30] and Solanki and Bammann[31]).

C. Loading Orientation Dependence

The effect of crystallographic orientation on voidgrowth was also examined by loading in multiple single-crystal orientations from around the stereographictriangle, including single slip ([321]), double slip ([210],[211], [221]), and multiple slip ([100], [110], [111])orientations. The simulation cells were generated inseven different crystallographic orientations using aninitial vvf of 0.4 pct at the center and a length of 16 nmfor each dimension. The temperature was set to 300 K(27 �C). The configuration was deformed in tension at aconstant strain rate of 108 seconds�1 with a zero-straincondition for other two lateral boundaries.

In fcc crystals, dislocation motion is observed com-monly to obey Schmid’s law along the {111} 110h i familyof slip systems. However, dislocation nucleation hasbeen shown to rely on non-Schmid terms in fcc Cu,[35,36]

whereby the nucleation event exhibits a strong depen-dence on resolved stress components in directions otherthan the slip direction. Because nanovoid growth occursas a result of dislocation motion—as dislocationstransport atoms from the void surface into the bulk

single-crystal regions—it is vital to understand howdislocations are first nucleated from the void surface.Previous experimental research demonstrated that

dislocation nucleation and void growth are extremelysensitive to material anisotropy under high strain rates(e.g., Stevens et al.[5]). However, the exact nature ofdislocation generation and evolution cannot be obtainedfrom these experimental observations. Therefore, toelucidate the effect of crystal orientation, a series of MDsimulations with different orientations was performed(as shown in Figure 1(b)). The stress–strain curves alongwith the nucleation of initial dislocation loop for theseven loading orientations are shown in Figure 5. Theeffect of loading orientation on the dislocation nucle-ation stress (onset of plasticity) is minimal for the single-crystal material with a void, except for the [100] loadingorientation, which has a higher nucleation stress. How-ever, different dislocation loop formation was observedas the crystallographic orientation is varied (Figure 5).As the loading orientation is varied, different slip planesare activated simultaneously and cooperatively as aresult of resolved stresses, and consequently, dislocationloops form on different planes. The initial dislocationloop nucleation and subsequent structure formation iscritical as it dictates how void will evolve (e.g., spherical,elliptical, or faceted void growth, i.e., Figure 7).The stress required to nucleate these dislocations in a

single crystal with a void and without a void can beimportant for predicting void growth. Table II showsthe dislocation nucleation stresses with and without avoid for the different loading directions.The range of stresses required for nucleation from the

void surface in various orientations can be attributed tothe influence of resolved stress components. Varyingdislocation patterns/shear loops occur according to thedifferent loading orientations, thereby confirming theeffect of crystallographic orientation, as illustrated inFigure 5. The calculated Schmid factors vary from0.4899 for the [210] loading orientation with two activeslip systems 11�1

� �½101� and �11�1

� ��1�10� �

to 0.2722 for the[111] loading orientation with three active slip systems1�1�1� �

½101�; �11�1� �

½011�, and 11�1� �

½011�. For a single-crystal lattice without a void, there is a much strongerdependence of the yield stress on the crystallographicloading orientation (range of ~2.5 GPa). However, for asingle-crystal lattice with a void, the yield stress isrelatively insensitive to loading orientations (range of~0.87 GPa with [100] and ~0.24 GPa without [100]); thisfinding suggests that the nanoscale plastic anisotropyresulting from the initial lattice orientation plays only aminor role in influencing the nanovoid growth rate. Incontrast, there is a significant influence of the initialnanovoid volume fractions on the yield stress, i.e., up toa 35 pct decrease in yield stress by introducing a 0.4 pctnanovoid volume fraction (Table II).The resolved stress components and the number of

slip systems can influence the nucleation of dislocationsfrom the void surface. Figure 6 shows the stress–strainbehavior with and without a void for the [210] loadingdirection for different initial vvf: 0.4 pct, 0.5 pct, and1.0 pct. The images show the evolution of the shear loopfrom the void at different strains for an initial vvf of

Table I. Yield Stress at Different Strain Rate

with and without Nanovoid

StrainRate, s�1

Yield Stresswithout Void

(GPa)

Yield Stresswith 0.4 pct vvf

(GPa)

PercentDifference in

Yield Stress (pct)

107 8.78 5.76 34.36108 8.89 5.85 34.195 9 108 8.97 5.93 33.86109 9.12 6.07 33.371010 10.25 7.48 27.04


0.4 pct. Initially, a single partial dislocation nucleates ona maximum Schmid factor slip plane and propagates inthe single-crystal lattice, creating a stacking fault as ittransverses the lattice (4.82 pct). Eventually, a secondpartial dislocation emerges from the void to create a fulldislocation with a stacking fault connecting the twoShockley partial dislocations (4.84 pct). At higherstrains, another partial dislocation is nucleated andemitted into the lattice on a second shear plane. Asshown in the plot, the presence of the void decreases thestress required to nucleate the shear loops when com-pared with the perfect single-crystal calculation.

The time evolution of growth and faceting of voids isshown in Figure 7 for the different loading orientationsexamined herein. In agreement with Stevens et al.[5] and

Seppala et al.,[7] Figure 7 shows a strong dependence ofvoid growth and faceting (loss of spheroidicity) on theloading orientation for low to intermediate strains(~8 pct strain). Because of the different resolved stressesand active slip systems caused by the loading orienta-tion, the nucleation, propagation, and interaction ofdislocations results in shape distortion of the initiallyspherical void. A faceted void shape is most prominentfor the [110] loading orientation and was less pro-nounced for the other loading directions. Interestingly,the overall vvf (Figure 8) is relatively insensitive to thedifferent loading orientations. For instance, the voidvolume fraction as a function of strain falls within acertain band and a trend does not exist with respect toloading orientation (i.e., the [210] orientation is largestat ~5 pct strain, then the [110] orientation, then the [100]orientation). This finding suggests that the plasticanisotropy caused by the initial lattice orientation hasonly a minor role in influencing the void growth rate.To ascertain the applicability of continuum models at

the nanoscale, Figure 8 compares the nanoscale simu-lation results for void growth to a continuum damagemechanics model. The development of the continuumdamage mechanics begins with introducing a scalardamage variable, / (Kachanov[57]), which represents thedegradation of strength in a one-dimensional tensile barbecause of creep. The concept of describing materialdegradation using damage variables of different orderhas resulted in several mathematical formulations forthese variables. For the evolution equation of isotropic

Fig. 5—Stress as a function of strain with different orientation for16-nm cell size, an initial vvf of 0.4 pct, and an applied strain rate of 108 s�1.This illustrates that the initial dislocation loop nucleation from the void surface has a stronger dependence on the crystallographic orientation.

Table II. Yield Stress at Different Orientationwith and without Void

LoadingDirection

Yield Stresswithout Void

(GPa)

Yield Stresswith 0.4 pct vvf

(GPa)

Percent Changein Yield Stress

(pct)

[1 0 0] 8.89 5.85 34.19[1 1 0] 7.59 5.16 32.02[1 1 1] 7.12 5.08 28.68[2 1 0] 6.59 4.98 24.43[2 1 1] 6.63 5.22 21.23[2 2 1] 6.32 5.02 20.51[3 2 1] 6.57 5.13 21.82


damage, we consider nonassociative form of Cocks andAshby[58] and Gurson[59] to compare with our atomisticresults, as follows:

_/ ¼ dp sinh 2 2m� 1ð Þ

2mþ 1

p

r

�1� 1� /½ �mþ1

1� /½ �m

!½3�

where m is the material rate sensitivity parameter, theratio p

r is the stress triaxiality, and dp is the effective

plastic strain. We fit the nanoscale strain rate sensitiveparameter m, using Eq. [3] and vvf data from Figure 8,and we found it to be close to 1 compared with anexperimentally measured value of 0.02 for aluminumalloy AA5182-O.[29] The large difference in the strain

rate sensitivity parameter is caused by the inherentnature of length and time scales associated with nano-scale simulations and provides a natural bridging scaleparameter for the continuum scale. The vvf trendobtained at the nanoscale agrees qualitatively with thetrend from the continuum model for strains < ~11 pct.At higher strains, it is expected that there will be greaterdeviation between nanoscale results and the continuummodel because of interactions through the periodicboundaries. Nevertheless, nanoscale simulation resultssuch as these can indeed inform continuum void growthmodels regarding the initial dislocation nucleationevents and the subsequent void growth for applicationin multiscale models.

Fig. 6—The stress–strain behavior with and without a void for the [210] loading direction for different initial vvf with 16-nm cell size at anapplied strain rate of 108 s�1 for 300 K (27 �C).

Fig. 7—The evolution of void shape and faceting for different loading directions shows the influence of loading orientation. The surface atomsof voids are shown. Green atoms are surface atoms with lower centrosymmetry value and red atoms are surface atoms with higher centrosymme-try value (Color figure online).


IV. CONCLUSIONS

Nanovoid growth in the fcc Al was investigated usingMD to reveal its dependence on the void size, strain rate,crystallographic loading orientation, initial nanovoidvolume fraction, and simulation cell size. A sphericalnanovoid was embedded into a periodic fcc Al latticeand a remote tensile load was applied to elucidatedislocation nucleation and shear loop formation fromthe void surface as well as the subsequent void growthmechanisms. The nucleation stresses and void growthmechanisms were compared for four different strainrates (107 to 1010 seconds�1), five different simulationcell sizes (4-nm to 28-nm lengths), four different initialnanovoid volume fractions, and seven different tensileloading orientations representative of the variabilitywithin the stereographic triangle. The following conclu-sions can be drawn from this work:

1. The simulation cell size, crystallographic loadingorientation, initial void volume fraction, and strainrate affect the yield stress for simulations of single-crystal Al lattice without a void (single-crystal bulkmaterial in Table II). Whereas the crystallographicorientation does affect dislocation nucleation interms of the slip systems activated and the evolvingvoid morphology (Figure 7), the crystallographicorientation dependence on yield stress was less pro-nounced for simulations of single-crystal Al latticecontaining a void (Figure 6, Table II).

2. The evolution of the nanovoid volume fraction withincreasing strain is relatively insensitive to loadingorientations (Figure 4), which suggests that thenanoscale plastic anisotropy, caused by the initiallattice orientation, has only a minor role in influ-encing the nanovoid growth rate.

3. The initial nanovoid volume fraction does have asignificant effect on the yield stress (Figure 6, Tables Iand II), i.e., a ~35 pct decrease in yield stress wascaused by introducing a 0.4 pct nanovoid volumefraction.

4. A continuum scale bridging parameter, m—a mate-rial rate sensitivity parameter in continuum scaledamage mechanics—was calculated and found to beclose to 1 compared with an experimentally mea-sured value of 0.02 for aluminum alloy AA5182-O.[29]

The large difference in the strain rate sensitivityparameter is because of the inherent nature of thelength and time scales associated with the nanoscalesimulations, and it provides a natural bridging scaleparameter for the continuum scale.

5. The vvf trend (Figure 8) obtained at the nanoscaleagrees qualitatively with the trend from the contin-uum model for strains<~11 pct. At higher strains, itis expected that greater deviation will exist betweennanoscale results and the continuum model becauseof interactions through the periodic boundaries.

Consequently, atomistic simulations of this type canindeed inform continuum void growth models forapplication in multiscale models.

ACKNOWLEDGMENTS

This material is based on work supported by theDepartment of Energy and the National Energy Tech-nology Laboratory under award number DE-FC26-02OR22910 and the Office of Naval Research undercontract number N00014-09-1-0661.

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0 0.05 0.1 0.150

10

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100

Strain

Voi

d V

olum

e F

ract

ion

(%)

Continuum Model[100] Orientation[110] Orientation[111] Orientation[210] Orientation[211] Orientation[321] Orientation

Fig. 8—The vvf evolution for different loading orientations. Interest-ingly, the vvf is relatively insensitive to loading orientations, whichsuggests that the plastic anisotropy caused by the initial lattice orien-tation has only a minor role in influencing the void growth rate.


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Investigating Damage Evolution at the Nanoscale: Molecular Dynamics Simulations of Nanovoid Growth...

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