quantifying the early stages of plasticity through nanoscale...

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PHYSICAL REVIEW B 67, 104105 ~2003!

Quantifying the early stages of plasticity through nanoscale experiments and simulations

Krystyn J. Van Vliet,1,* Ju Li,2,† Ting Zhu,2 Sidney Yip,1,2 and Subra Suresh1,‡

1Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 0212Department of Nuclear Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

~Received 14 June 2002; published 25 March 2003!

Nucleation and kinetics of defects at the atomic scale provide the most fundamental information about themechanical response of materials and surfaces. Recent advances in experimental and computational analysesallow us to study this phenomenon in the context of nanoindentation and localized mechanical probing ofsurfaces. Here, we present an analytical formulation of the elastic limit that predicts the location and slipcharacter of a homogeneously nucleated defect in crystalline metals, and extend this formulation to the atomicscale in the form of an energy-based, local elastic stability criterion, termed theL criterion. We demonstratethat this approach can be incorporated efficiently into computational methods such as molecular dynamics andfinite-element models. Furthermore, we validate and calibrate theL criterion directly through nanoindentationexperiments and two-dimensional experimental analogs such as the bubble raft model. We outline explicitly acompact and efficient application of theL criterion within the context of a nonlinear, interatomic potentialfinite-element model~IPFEM!. Further, we report three-dimensional molecular dynamics simulations in severalface-centered cubic systems that elucidate the transition from the initiation to the early stages of plasticityduring nanoindentation of metals, as characterized by homogeneous and heterogeneous nucleation of up tohundreds of dislocations. Correlation of these simulations with direct observations from nanoindentation ex-periments provides atomistic insights into the early stages of plasticity.

DOI: 10.1103/PhysRevB.67.104105 PACS number~s!: 62.25.1g, 02.70.2c, 81.07.2b, 81.40.2z

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I. INTRODUCTION

A fundamental transition in the mechanical behaviormaterials is the shift from elastic~reversible! to plastic~irre-versible! deformation. In crystalline metals at room tempeture, this process is mediated predominantly by the nuation and motion of dislocations. Theoretical frameworhave been established for dislocation nucleation atatomic scale,1–4 but currently no such framework exists fothe so-called homogeneous nucleation mechanism, wherfects are nucleated in the crystal in the absence of onearby, preexisting defects. Indeed, although we havelytical formulations of the self-energy of a single dislocatiothe conditions necessary tonucleatedislocations in the bulkand in thin films are unresolved.5 Currently we rely on con-tinuum concepts such as critical resolved shear stressyield strength to quantify the onset of plasticity in metaeven though this transition is initiated at the atomic scale

The need for nanoscale interpretation of defect nucleais even more pressing now that a sophisticated range ofperimental and computational tools are available to expmaterial response near the atomic level. In fact, recent deopments in both multiscale computational modeling ananoscale experiments allow these two distinct methodmaterial analysis to explore mechanical behavior at lenscales comparable to one another. Nanoscale experimennanoindentation, atomic force microscopy, and other teniques can probe mechanically the elastic stability of crysat subnanometer displacements and large local constrains. Results from these and other related experimhave indicated that such contact can induce elastic instaties that have been deposited as evidence of homogendislocation nucleation. The chief obstacle to further validtion of such experiments is the considerable difficulty in dtermining atomic scale activities in three dimensional~3D!,

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opaque crystals, as well as the challenges associatedconducting sufficiently clean experiments at this length scin initially defect-free materials.

Atomic scale simulations such as molecular dynam~MD! can provide the required visualization and precise ctrol over characterization of the evolving defect microstruture over a very well defined, though limited, time perioIndeed, several researchers have modeled indentation offcc crystals and observed dislocation activity.6–10 However,the length scales in such studies are limited by computatioresources, resulting in indenter tip dimensions and strrates that are incongruent with those attainable in expments. Even when computational analyses incorporate mtiple length scales to approach realistic dimensions, sucin the quasicontinuum method of Tadmoret al.,11 the obser-vations have not been sufficiently well calibrated againstperiments and do not culminate in predictive capabilitiThus, it appears that although both experiments and comtational modeling have the potential to explore the same pnomenon, neither has been constructed in such a way tharesults truly address the hypotheses and quantitative capaties of the other.

In this work, we outline experimental, analytical, ancomputational analyses of nanoindentation that weresigned together to elucidate the mechanisms by which loized contact can induce the transition from elastic to pladeformation in an initially defect-free crystal, and the mechnisms by which that plasticity proceeds at early stages,mechanisms of incipient plasticity. This discussion buildsour earlier results, reported by Liet al.12 First, we summarizeexperimental results of nominally sharp nanoindentationboth 3D fcc crystals and a two-dimensional~2D! experimen-tal analog. Next, we present an analytical description of etic stability at the atomic scale in the form of theL criterion.

©2003 The American Physical Society05-1

VAN VLIET, LI, ZHU, YIP, AND SURESH PHYSICAL REVIEW B 67, 104105 ~2003!

TABLE I. Experimental measurements of critical indentation load in fcc crystals.

Element Orientation Indenter radius Pc E* a tmaxb G/2p Reference

~nm! (mN! ~GPa! ~GPa! ~GPa!

Aluminum ~110! 50 30 72.6 7.2 4.36 13~111! 50 8 76.0 4.7 4.6 13~133! 50 15 63.6 5.7 4.46 13~113! NAd 11.3 65.8 4.0 47

Copper ~111!c 50 35 123 10.7 7.6 46Gold ~110! 205 7.9 117 2.5 4.6 49

~111! 205 31.3 81.8 3.1 6.6 49~111! 70 14.5 117 6.2 6.6 8~100! 205 43.8 43 2.2 2.46 49

Iridium ~100! 50 58.5 370 26.4 34.3 50

aEquation~16! for Ei51100 GPa andn i50.07.bEquation~15!.c60–80 %~111! texture.dReference 48; not available.

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We verify the corresponding predictions of defect nucleatand slip character with experiments and MD simulations2D indentation, as well as with MD and nonlinear finite eement modeling~FEM! of cylindrical indentation in elasti-cally anisotropic crystals. Finally, we present experimenresults and 3D MD simulations of nominally sharp nanodentation that reveal that, subsequent to the initial elainstability, the development of incipient plasticity under cotact loading is not due tocontinued, homogeneous nucleatioof dislocations. Rather, instabilities observed beyond theevent are due to the formation of heterogeneous desources that operate at local stresses well below thosequired for homogeneous nucleation. Although our expemental and computational correlations focus on nanoscontact, the analytical and computational tools presenherein are applicable to the general phenomenon of lastrain deformation, a process relevant to a wide rangedisciplines.

II. EXPERIMENTAL OBSERVATIONS

A. Nanoindentation

The results of load-controlled, nominally sharp nanodentation experiments on several different fcc crystals hbeen reported, as summarized in Table I. Although theseperiments were conducted on various instruments andelastic moduliE of the crystals ranged from 69 to 528 GPall of the indentation responses exhibited displacementcursions beyond some critical indentation loadPc ~see Fig.1!. In fact, as suggested by Gouldstoneet al.,13 a continuumapproximation of the maximum shear stress correspondinPc for each crystal indicates that the first displacementcursion occurs when the Hertzian maximum shear stresstmaxis on the order of the theoretical shear strength of the crysestimated asG/2p whereG is the resolved shear moduluThis observation suggests that the local stresses prior to tdiscontinuities are sufficient to induce homogeneous nuation in crystals of initially low defect density. However, th

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rough correlation cannot be used to rationalize subseqdisplacement bursts, or to determine the initial location astructure of such homogeneously nucleated defects.

Using reliable electronic structure methods, Roundyet al.calculated the ideal shear strengths on$111% planes of Al andCu to be 1.85 and 2.65 GPa, respectively, when the reming five stress components are fully relaxed.14 The nominaldifference between the above theoretical predictionsTable I is significant, but direct comparison may notmeaningful due to the following facts. As Roundyet al.noted, the tangential modulus is softened significantly jprior to yielding of a crystal, causing the linear elastici

FIG. 1. Schematic of indentation load-displacement respoindicating initial defect nucleation at a critical loadPc , and subse-quent defect activity at increased loads and displacements.~a! Dis-placement control, showing sharp decreases in load~dips! corre-sponding to defect activity.~b! Load control, showing largeincreases in indenter displacement~bursts! corresponding to defecactivity.

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QUANTIFYING THE EARLY STAGES OF PLASTICITY . . . PHYSICAL REVIEW B67, 104105 ~2003!

extrapolation15 that Table I uses tooverestimatethe localstress at the site of homogeneous nucleation. On the ohand, the stress condition at the defect nucleation siteneath an indenter is far more complicated than the pure scondition that Roundyet al. studied. Our MD calculationsshow that the mean pressure at the defect nucleation sion the order of tens of GPa; therefore significant pressinduced shear hardening is possible. These two countefactors make the comparison between the idealized etronic structure method results and the particular respounder indentation-induced strains a difficult task. The relution of this discrepancy can be mediated by a rigorolocalized elastic stability criterion that is applicable undgeneral loading conditions.

B. Bubble raft analog

In order to visualize readily the response of atoms tocalized nanoscale contact, Gouldstoneet al.16 employed theBragg-Nye bubble raft model, a two-dimensional experimtal analog to close-packed crystals.17 The bubble raft pro-vides a perfectly arranged, defect-free, 2D lattice comprisseveral thousands of bubbles arranged analogously toclosest-packed plane in an fcc crystal. The surface tensiothe solution and the bubble diameter can be designedthat the interaction among bubbles reflects a simple desction of the interatomic potential of Cu.18 By relating theindentation-induced motion of millimeter-scale bubbles inclose-packed raft to that of nanometer-scale atoms in a clpacked atomic plane, it was shown that homogeneous dcation dipole nucleation occurs subsurface, at the locatiomaximum shear stress predicted by 2D continuum conmechanics. Further experiments confirmed that, althoughdepth of dislocation nucleation scales with the width of cotact, the dipole structure and subsequent formation of sursteps are independent of length scales such as the indradius~see Fig. 2!. One inherent limitation of these experments is that the load–depth response does not converg2D indentation. That is, the indentation depth correspond

FIG. 2. ~Color online! Nanoindentation normal to the terminaedge of a close-packed plane of a 2D bubble raft produces subface dislocation dipole nucleation. Nucleation depthzc scales withindenter tip radiusR, but the dislocation dipole structure and motioupon nucleation is invariant. The scale bar is 10 mm, or 3 nm inatomistic size scale analogy of Gouldstoneet al. ~Ref. 16!.

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to a given applied load depends directly on the dimensionthe indented substrate. Thus, although it can be inferred fcontinuum analysis that the position of nucleation corsponds to that of the shear stress maximum, the appstress prior to nucleation cannot be measured experimentFor this reason, the quantitative results from nanoindenta(Pc) and bubble raft experiments~nucleation position anddislocation structure! cannot be compared directly.

III. ANALYTICAL CRITERION OF ELASTIC STABILITY

Unequivocal validation of hypotheses regarding contamediated instabilities requires not only precise correlationatomic scale motion and the far-field load-depth responbut also a theoretical framework that offers predictive cabilities from which further experiments can be designed.this end, we first review the work of Hill19 and Rice20 oncontinuum formulations of elastic stability, and then extethese concepts to the atomic scale in the Cauchy-Born frawork for crystalline metals.

A. Derivation of the L criterion

Hill 19 developed the general concept of continuum mdium stability in terms of ‘‘acceleration waves,’’ which in thcontext of this paper can be understood simply as lowavelength phonons or elastic waves. He derived a tenrepresenting the connection between wave frequency andcorresponding wave vectork and displacement vectorw:19

L~k,w![~Ci jkl wiwk1t j l !kjkl , ~1!

where t j l is the Cauchy stress andCi jkl is the isothermalelastic constant, defined as,21

F~Y![F~X!1V~X!H t i j ~X!h i j 11

2Ci jkl ~X!h i j hkl1•••J ,

~2!

where F(X) and F(Y) are the Helmholtz free energies ohomogeneous configurationsX andY andh is the Lagrang-ian strain ofY with respect toX. For a givenk, the value ofw that gives rise to an extremum inL(k,w) is the phononeigendisplacement, and in this contextL is proportional tothe vibrational frequency squared. In the long-wavelenlimit, uku anduwu are immaterial, and hence we stipulate th

uku51, uwu51, ~3!

and hereafterk andw are used only to denote directions.If there exists a pair ofk and w such thatL(k,w) is

negative, then the elastic stability of the homogeneousdium is lost: negativeL signifies negative concavity of thfree energy surface, and thus the frequency of the cosponding elastic wave is imaginary and its amplitudecreases exponentially with time. This would ultimately resin the destruction of the homogeneous medium and theation of defect singularities within. The growth of the ‘‘somode’’ wave proceeds in four stages:~i! linear growth of theunstable elastic wave, well described by continuum;~ii ! non-linear evolution at larger amplitudes, over which the wa

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profile steepens;~iii ! progressive steepening of the wafront until its length scale approaches atomic spacing, uwhich its description must be transferred from a continuto an atomistic basis;~iv! arrest of the atomistically sharwave front in a low-dimensional atomic energy landscapewhich point a defect is nucleated. If the wave is transvethe instability will likely result in the formation of a dislocation or twin of slip plane normalk and Burgers vector direction w; if the wave is longitudinal, a microcrack would likelresult. The exact character of the resulting defect is a fution of both the continuum-scale linear instability causedexternal loading and the atomic scale interactions withinmaterial. This dynamic process is studied in detail in anotpaper22 and is, in general, complicated. Acknowledging thcomplexity, we make the following observations for momonatomic crystals:

~1! If the angle betweenk andw is greater than 60°, i.e.the initial soft mode is chieflytransverse, then the final de-fect created is likely to be a dislocation loop or a twinniembryo. The structure of the final defect should correlstrongly with whetherk is the plane normal of a crystallographic dislocation slip system or twinning slip system.

~2! If the the angle betweenk andw is less than 30°, i.e.the initial soft mode is mostlylongitudinal, then the finaldefect created is likely to be a microcrack.

Rice20 derived the same stability criterion in the conteof a particular continuum phenomenon, that of shear bformation. From this viewpoint, one considers an interfacenormal k between a plastically deforming shear band aelastically deforming material, across which there is a dplacement discontinuityw. When L(k,w) becomes negative, Rice showed that traction equilibrium across the intface can no longer be maintained, and the shear bandgrows catastrophically. Although the nomenclature andticulation of the criterion formulated by Hill and Rice contrast slightly, they are entirely equivalent in content. Maematically, this criterion can be summarized as

Lmin[ minuku51, uwu51

~Ci jkl wiwk1t j l !kjkl , ~4!

and if Lmin is negative, the specific material point or reprsentative volume element~RVE! becomes elastically unstable.

A full phonon spectrum analysis of a strained crystal cbe carried out; this has been performed for binary copounds such as SiC.23 Such calculations show the connectiobetween Brillouin zone boundary soft phonons and the idstrength of these systems. Indeed, because the phononsstitute a complete basis set for all atomistic excitations icrystal, such analysis isnecessary and sufficientto ascertainthe stability of any crystal atT50. The present elastic wavanalysis, however, does not guarantee stability: It is a nesary condition. Despite this lack of sufficiency, the concand implementation of theL criterion are extremely powerful and cover the majority of instabilities that occur in moatomic crystals. The fact that Eq.~1! is a tractable analyticaformula involving only well-known mechanical parametet i j and Ci jkl , which are also well studied in atomistisimulations,24 makes it an ideal means to connect continu

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models such as FEM with atomistic simulations. Additioally, as theL criterion relies on the long-wavelength limand thermoelastic material response, it continues to funcat finite temperatures, when the phonon description loserigorous definition. We will demonstrate that in realistloading configurations such as nanoindentation, Eq.~1! cor-rectly predicts the critical load and the defect nucleation sIn addition, this energetic criterion reasonably predictscharacter of the defect when we combine information abthe soft modek andw with crystallographic structure. Thais, we can predict whether the linear instability will lead todislocation loop or a twinning embryo, specifying the slplane and the Burgers vector. This allows true predictpower in finite-element models, afforded by adefect nucle-ation criterion with no adjustable parameters for a givedescription of atomic interactions.

B. Incorporation of the L criterioninto computational analyses

Two practical difficulties prevented theL criterion, for-mulated at the continuum scale several decades ago,being widely used in practice. The first challenge is the laof accurate constitutive models to estimatet i j and Ci jkl atvery large strain, near elastic instabilities. The parametriztion of the strain energy function or stress functions in sdimensional general strain space is by no means a simtask, either conceptually or numerically. Prior to the advof modern, reliable electronic struture codes and their dertive empirical interatomic potentials, few data existed. Aother obstacle is the accurate calculation ofLmin . Eventhough four-dimensional minimization~since uku51,uwu51) is not an especially difficult computation to perfora single time, a reliable and efficient algorithm is necessdue to the iteration of this procedure for each material poFor example, in the context of MD, the calculation ofLmin isperformed for each of up to 23106 atoms at each time step

The first challenge can be resolved naturally in an atoistic calculation because one can conceptualize and evaatomistic local stresst i j and atomistic local elastic constanCi jkl by partitioning the virial and Ray sums,25,24 respec-tively, down to individual atoms. For pair and embeddeatom type interatomic potentials, such partitioningstraightforward because the total virial and Ray sumscomprised algebraically of contributions from atom paiand one can readily attribute half of every pair contributito each of the two participating atoms. More complicatinteratomic formulations and partitioning schemes26 wouldwork equally well, provided that basic symmetry and surule requirements are satisfied. This partitioning schemjustifiable insofar as the atom under consideration is aatomic spacings away from any topological defects, evenis under a large elastic strain. In other words, as longseveral nearest-neighbor levels of an atom retain their~bulk!crystalline coordination, the concept of atomistic local streand elastic constant is well posed and one can evaluate tdirectly from an interatomic potential, or even fromab initiocalculations.

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To overcome the second difficulty, we observe thL(k,w) is a biquadratic function inw and k, and can bealternatively written as

L~k,w![~Ci jkl kjkl !wiwk1t j l kjkl , ~5!

where the order of thek,w summation is swapped from Eq~1!. If we assumew is a constant in Eq.~1! and minimizeagainstk only, then the solution for that simple quadratform is straightforward: Diagonalize the 333 symmetricmatrix @A# j l [Ci jkl wiwk1t j l and choose the softest eigevector to bek. Alternatively, if we assume thatk is a con-stant in Eq.~5! and minimize againstw, then the solution isalso straightforward: Diagonalize the 33 3 symmetric ma-trix @B# ik5Ci jkl kjkl and choose the softest eigenvector tow. Therefore, we propose the following iterative algorithto minimizeL(k,w):

~a! Choose an arbitraryw.~b! Minimize Eq. ~1! for a fixedw, identifying an initial

value ofk that minimizesL(k,wfixed).~c! Minimize Eq. ~5! for a fixedk identified in the previ-

ous step, identifying an initial value ofw that minimizesL(kfixed,w).

~d! Repeat until mutual convergence is reached.The above uses only 333 symmetric matrix diagonaliza

tion, which can be implemented efficiently using the analycal, Cardano cubic equation solution.27

Using the approaches outlined above, a complete refor L criterion implementation in atomistic simulationsproposed:

~1! Select all atoms sufficiently separated from topolocal defects, even if they are withstanding large elastic str

~2! Evaluate the local atomistic stress and elastic conson an atom-by-atom basis by distributing the virial and Rsums.25,24 This allows L(k,w) to be properly defined foreach atom.

~3! Minimize L(k,w) for each atom, and recordLmin , k,andw for each atom.Lmin can be interpreted as themicros-tiffnessof the atom, and its sign can be interpreted asconcavity of the free energyF.

~4! Among all atoms in the configuration, identify thatom with the smallestLmin as the ‘‘softest atom’’ for theparticular displacement configuration.

Despite the large number of atoms in a typical 3D Msimulation of nanoindentation (;23106), the efficiency ofthe above minimization procedure is sufficient to allowmultaneous simulation and calculation ofLmin . At the onsetof homogeneous nucleation,Lmin of the ‘‘softest atom’’ tendsto zero. Note that theL criterion does not forcew andk tocorrespond to close-packed directions and planes. Racorrespondence ofw andk to ^110& and^111&, respectively,in the 3D simulations discussed below is a natural outcoof the interatomic potential within the fcc crystal structure

Because instabilities predicted by theL criterion are elas-tic in nature,21 and since all atomic information is incorporated throughCi jkl and t i j , this defect nucleation criterionalso can be readily incorporated within finite-element anasis. Such implementation requires that~i! the large strainformalism be used~e.g., the nonlinear geometry option in

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general purpose finite-element package such asABAQUS28!and ~ii ! an accurate constitutive relation be provided to tfinite-element code. The latter is facilitated by the modelCauchy-Born elasticity by which the local stress state amaterial point is dictated by the movements of underlyilattices.29,30 Specifically, FEM-imposed boundary conditioncause displacement of the nodes. The stress at each nocalculated in a subroutine comprising a small atomic lattfor which the lattice interaction is determined by an inteatomic potential~i.e., molecular statics atT50 and ensembleaveraging at finiteT). When this lattice is strained accordinto the FEM nodal displacement, interatomic interactions dtate stresses on each atom according to the potential, givilocal ~nodal! value of t i j . Both Ci jkl and t i j can be calcu-lated through manipulation of Eq.~2!. The quantityL(k,w)can also be calculated and minimized for each node, accing to the procedure outlined above. In this manner, theformation of large-scale models can be dictated by an estially atomic scale strain-to-failure criterion; this criterionparameter-free for a given interatomic potential. As the cstitutive response of this FEM approach is dictated by intatomic potential~IP!, we will refer to it hereafter as IPFEMAlthough IPFEM is not the first computational approach thallows observation of defect nucleation, this model repsents the first framework that incorporates apredictive, localelastic stability criterion that is directly linked to defenucleation.

IV. COMPUTATIONAL MODELINGOF ELASTIC STABILITY

Computational simulations such as MD are particulawell suited to quantify the relationship between internatomic activities and external loading sustained by a crysThe only assumption of material behavior in MD is thatcertain interatomic potential correctly describes the intertion among atoms. However, due to the computational cassociated with keeping track of large numbers of atoms,physical time scales and/or length scales are often unretically small. By contrast, continuum-based computatioapproaches such as FEM are, by definition, designedsimulate behavior over any length scale for which the marial response can be described by continuum parame~e.g., Young’s modulusE and yield strengthsy). The objec-tive of IPFEM, then, is to incorporate the accuracy aparameter-free nature of MD and related simulations intorealistic length and time scales of FEM.

IPFEM is distinct from the quasicontinuum~QC!method31 in that it doesnot transfer the basis of the materiadescription from continuum-based elements to atomisbased MD when the elemental response approaches thetic limit. Rather, IPFEM is a fully continuum finite-elemenmodel atall levels of strain, for which the nodal responsecrystalline RVEs is governed by an atomistic~Cauchy-Born!constitutive relation.29 In contrast, QC hybridizes the use olinear and Cauchy-Born nonlinear elasticity with MD, tranferring the constitutive relations according to the levellocal strain, and thus utilizes a numerical, rather than anacal, nucleation criterion. The fields calculable via IPFE

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are, by design, easily integrated with theL(k,w) analysistool. As such, IPFEM is conceptually less general thanquasicontinuum method and can handle only a subset oproblems the quasicontinuum method is theoretically capaof treating, for which the results of the two methods shobe largely the same. That is, although the quasicontinumethod can be used to identify and validate a defect nuation criterion, to our knowledge the quasicontinuum methhas not been applied to this task. Three considerationsus to think that IPFEM is sufficiently important and uniquto be identified separately from the general quasicontinumethod:

~1! Difference in perspective: We do not regard the smatomic lattice in question as being embedded in a larinhomogeneously strained atomic configuration. Instead,consider the small atomic lattice to be embedded in an inite, but homogeneously strained lattice, representing aterial point in the FEM calculation. This more ‘‘primitive’viewpoint enables modular design. That is, we can couFEM to either interatomic potential or to more accurateabinitio periodic boundary condition ~PBC! unit-cellcalculations.32 Although the incorporation ofab initio consti-tutive descriptions is not a large advance conceptually,ability would have great practical impact with respect toliability and accuracy of observations and predictions maat the atomic scale.

~2! Ease of implementation: One may utilize any generpurpose FEM package such asABAQUS to implementIPFEM, simply by supplying the appropriate constitutive rlation. Thus, IPFEM requires no modification of the mesing, visualization, and analysis tools, whereas the genquasicontinuum method requires significant effort to coand implement.

~3! Dedication to an important phenomenon: The speset of problems that IPFEM can study, e.g., the site and cacter of homogeneous nucleation occurring in nanoindetion, is sufficiently important and underdeveloped thatsimple method that achieves this goal with minimal effshould be promoted. As one cannot truly separate a mefrom the problems it can solve, IPFEM is, by design, linkto application of theL criterion and the general phenomenof homogeneous defect nucleation.

In this section, we discuss prediction of elastic instabilvia theL criterion for both MD and IPFEM simulations onanoindentation in two- and three-dimensional crystals.

A. MD simulation of indentation on a 2D half plane

In order to verify the assumptions and implementationthe L criterion, we conducted 2D MD simulations of thbubble raft experiments outlined in Gouldstoneet al.16 Aswe are not interested in the mechanical response of an asoap bubble raft, but instead regard it as a generic modeclose-packed 2D crystals, we arbitrarily choose the followpotential with a smooth, short-range cutoff for ease of impmentation,

V~r !5H a~r 21!42b~r 21!2, r ,1

0, r>1~6!

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which is plotted in Fig. 3. Here, the primary units of lengand energy are the cutoff distancer c at which the potential isvalued zero and potential well depthe, respectively, and areboth scaled to 1. Consequently, a single parameterr 0, theequilibrium interbubble distance, controls the shape ofpotential for any interbubble distancer<r c . We normalizethis distance byr c valued unity, so thatr 0 is a dimensionlessquantity less than unity. In order to convert from these nmalized values to physical units in actual experiments, omultiplies normalized quantities by physical measures oeand r c ~e.g., critical load of valueP, multiplied by $e/r c%5$1.55 J/0.003 m%, to obtain force in N!. We express thequantities that modulate the potential,a andb, as

a5~12r 0!24, b52~12r 0!22. ~7!

We observe more brittle behavior at largerr 0. In all thefollowing results,r 0 is taken to be 0.85, and plots of potetial and depth are shown in these reduced units. At equirium, the potential energy per bubble is23, the volume perbubble isV05A3r 0

2/2. At equilibrium, the raft is an isotro-pic elastic medium with elastic constant matrix,

C5S l12m l 0

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0 0 mD , ~8!

whereby the fourth orderCi jkl tensor is reduced to a 333matrix due to symmetry in an isotropic elastic systemwhich Voigt indices 1,2,3 denotexx,yy,xy stress/strain com-ponents, respectively. The strain energy per bubble atform Lagrangian strainh is

FIG. 3. ~Color online! Generic short-ranged interbubble potetial with equilibrium separation distancer 050.85. Both the poten-tial well depth and cutoff distance are set to unity and servenormalizations of energy and length, respectively. This potenwas used in molecular dynamics simulations of the bubble raft,model 2D system.

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E531V$r 0A@1,0#~112h!@1,0#T%

3V$r 0A@1/2,A3/2#~112h!@1/2,A3/2#T%

1V$r 0A@21/2,A3/2#~112h!@21/2,A3/2#T%. ~9!

Using the fact thatV8(r 0)50, we may expand the above a

E'V9r 0

2

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41

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from which we can derive that

C115C2259V9r 0

2

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8V0, C335

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~11!

or if we substitute

V9~r 0!512a~r 021!222b58

~12r 0!2, ~12!

we obtain

l5m52A3

~12r 0!2, ~13!

and the Poisson’s ratio is therefore 1/3. When we defineequilibrium separation distancer 050.85, we obtainl5m5153.96; this value ofm under zero far-field applied loading is the value ofLmin . Any shear wave withk'w mini-mizesL(k,w) equally well, with no preference along crystallographic directions,prior to the application of far-fieldloading.

The indenter is implemented as an external potentialthe bubbles,

Vext~r !5H expS R2r

d2

d

R2r D , r ,R

0, r>R,

~14!

wherer is the distance between any bubble to the originthe indenter,R is the indenter radius, andd is the indenterhardness, representing the distance over which a bubble~numerically! penetrate the indenter potential. As such, this zero friction between the indenter and the substrVext(r ) increases from 0 infinitely smoothly upon contabut rises precipitously when the bubble is;d inside theindenter, representing practically a very hard indenter.

Two simulation algorithms have been used: molecularnamics atT50 K and conjugate gradient minimization. Imolecular dynamics simulations, the bubble mass is atrarily chosen to be 1. For equilibrium interbubble distanr 050.85, Eq.~12! shows that the stiffness constant is on t

10410

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order of 400. Thereforenmax, the maximal oscillation fre-quency, is;3. We use the six-value Gear predictor-correcalgorithm33 to integrate the trajectory of atoms moving othe interatomic potential surface, with an integration timstep of 0.003, corresponding to about 100 force evaluatiper period for the hardest oscillation, ensuring numerical sbility. The thermostat scheme34 of Berendsenet al. is used,with the temperature-coupling time constant chosen to bmaking the system strongly dissipative such that vibratiodue to kinetic energy are quickly restored to the level desnated by the specified temperature; this minimizes reflecof oscillations from boundary conditions. At the beginninga simulation, the indenter is placed such that its perimeexactly touches the first row of bubbles, the indenter forceinitially zero but it increases immediately upon penetratioThe bottom layer of bubbles is fixed, constraining the raftbe deformed by the indenter. PBCs are used in thex direction~in-plane!. In the MD simulation, the indenter proceedsdisplacement control, moving toward the substrate bysmall, fixed increment every time step. In the conjugate gdient minimization, the indenter advances by 0.02 every safter which the entire raft of bubbles is relaxed except forbottom layer, so the net force on each bubble is zero. In bsimulations, the forceVext(r ) exerted on the bubbles isummed and is correlated with the indenter displacement.found it more advantageous computationally to exploregeneral phenomenon using molecular dynamics with smabubble rafts of thousands of bubbles. However, in orderverify the L criterion with maximum accuracy, we utilizeconjugate gradient minimization on larger bubble rafts coprising >100 000 bubbles.

Figure 4 shows the change in far-field indentation loandLmin as a function of imposed indenter displacementa 2D MD simulation comprising 112 400 bubbles. The identer is cylindrical with radiusR of 160, d of 0.8, and theinteratomic potential is given by Eq.~6!. The critical inden-tation depth for the first homogeneous defect nucleatevent is 14.84, in reduced units; conversion to physical urequires multiplication of this value by a real cutoff lengr c . The temporal correspondence of discontinuous loadlaxations and the vanishing ofLmin is excellent for the firstload discontinuity, and the same trend is observed for fisubsequent discontinuities. Through visualization ofatomic coordination number, we have verified that each lorelaxation corresponds exactly with the nucleation of odislocation dipole. This result indicates that theL criterioncan predict accurately the far-field load and displacemconditions that will induce defect nucleation in a perfecrystal. If atoms that comprise the nucleated defects areremoved numerically from the minimization procedureequivalent to removing the dislocation core from an elasanalysis—Lmin attains negative values. Of course, this trasition to negative values does not indicate a new materesponse or a negative microstiffness within the crystal.fact, this merely reflects the fact that one fundamentalsumption of the above elastic stability analysis, namely, tthe indentation strain field is homogeneous, breaks dowthe neighborhood of under- and overcoordinated atomscomprise defects. In other words, theL criterion predicts the

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conditions forhomogeneousdefect nucleation. Thus, the applicability of this stability analysis is restricted to regionsperfect crystallinity under elastic deformation.

Figure 5~a! shows the von Mises stress invariant distribtion in the bubble raft just prior to the elastic instability. Wfind that the maximum von Mises stress does not occur althe central loading axis but is distributed off-axis by 16.5both sides, at which point the stress is 20.97. In contrast,5~b! shows that the minimumLmin does indeed occur alongthe loading axis, at a depth greater than that correspondinmaximal von Mises stress. Figure 5~c! showstMises, t resolved,tcontinuum, andLmin in reduced units, along the central loading axis. Astmax

Mises520.26, this maximum shear stress donot deviate significantly from the absolute maximum thoccurs off-axis. More importantly, the homogeneous dislotion site corresponds exactly with the position of minimLmin , not with the position of maximum shear stress. Thuthemicrostiffnessprediction of the elastic limit embodied bthe L criterion describes the defect nucleation site morecurately than a stress-based criterion such astmax

Mises.The softk calculated by theL criterion agrees perfectly

with the nucleated dislocation glide plane normal, whichaligned to a crystallography orientation. However, the callated softw differs from the actual Burgers vector by;12°.This results from an activation volume effect, whereby tinstantaneous slip direction at the saddle differs from theslip direction due to the in-plane, corrugated potential proover which a migrating atom must slide over.

B. Comparison of MD and IPFEM for cylindrical indentation

We conducted simulations of cylindrical nanoindentativia MD and IPFEM in order to compare these approachand verify the slip character of the homogeneously nuclea

FIG. 4. ~Color! MD simulation of indentation of an initiallyelastic-isotropic close-packed plane, using the interbubble potenIndentation loadP and Lmin are shown as a function of imposeindentation depthh. Lmin is identically zero for each load relaxation, each of which corresponds exactly to the homogenenucleation of a dislocation dipole at the site within the crystalminimum Lmin .

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dislocations predicted by theL criterion. Here, we discussresults for single-crystal copper, a significantly elastanisotropic fcc metal. The Ackland interatomic potentia35

for Cu was used in both simulations, and displacemecontrolled nanoindentation was simulated with an indenradius of 20 nm. The Cu single crystal constructed for Msimulations, comprising 290 304 atoms and in-plane perio

al.

usf

FIG. 5. ~Color! Distribution of ~a! von Mises stress invariant inthe MD simulation of the 2D bubble raft, just prior to defect nuclation. The position of maximumt ~dark red! does not correspondexactly with the defect nucleation position.~b! Lmin in the samesimulation shows the global minimum of the energetic criterialong the loading axis~dark blue!, at a position corresponding exactly with the point of defect nucleation.~c! Stress andLmin alongthe central loading axis. As the interparticle potential in Eq.~6! is anarbitrary and scaled model, the magnitudes oft and L are notshown.

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boundary conditions, is illustrated schematically in Fig.The crystal, comprised of 16837234 unit cells, was free todisplace along the loading axis in they direction, as well asin the x and the out-of-planez directions. This freedom omovement in they direction was restrained by requiring ththe loadP imposed by the indenter be countered by anposite response of2P/N body force distributed equallyamong all atoms, whereN is the total number of atoms in thsimulation.

Figure 7~a! shows the position and slip character of tfirst dislocation nucleation event, as indicated by a decrein atomic coordination number of the interior atoms. Athough the defect is similar in structure to the edge dislotion dipoles observed in true 2D bubble raft experiments,position of nucleation is displaced from the loading axis, aoccurs nearer the surface than predicted by continuumlyis of cylindrical indentation in isotropic media (z50.78a).15 This shift of the dislocation nucleation positiois due to the elastic anisotropy of Cu. Using the recendeveloped linear elastic Stroh anisotropic analytical solutof the cylindrical punch indentation problem,36 we haveverified30 that the site of the maximum von Mises stressindeed off-center by 20% and nearer to the surfacez50.64a) than the isotropic solution, corresponding qualitively with the position identified via MD.

1. Interatomic potential finite element modeling

Figures 7~b! and 7~d! show the point of elastic instabilityin an IPFEM simulation, using the above indentation cofiguration and interatomic potential. The positions of mamum effective~von Mises! stress and maximum resolveshear stress, both based on linear elastic analyses, docoincide exactly with the position of minimumLmin at thepoint of defect nucleation. Not surprisingly, the positionmaximum resolved shear stress correlates more closely

FIG. 6. Schematic of MD simulation setup used for both cyldrical and spherical indentation. For quasi-2D cylindrical indention of Cu: 74.4365.131.02 nm, 16837234 unit cells, 290 304

atoms,D540 nm, h527 m/s; for 3D spherical indentation of Cu20.7319.1320.5 nm, 42318372 unit cells, 326,592 atoms,D

513 nm, h53.5 m/s. In both cases,x5^112&, y5^111&, z

5^110&.

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the position of minimumLmin . More importantly, defectnucleation occurs at the point whereLmin is exactly zero,indicating that the energetic criterion represented byLmincaptures the elastic instability much more accurately thastress-based approach. In other words, this result showsthe L criterion is a more accurate predictor of defect nucation position, and the only predictor of defect character.

In this model, each node represents an infinite latticeder homogeneous strain; nodal points need not be concuwith crystallographic lattice positions. Rather, the crystalgraphic information describing the simulated material is cotained in the constitutive relation input by the user. ThIPFEM does not resolve a single dislocation, but insteashear band indicating the slip orientation of such a dislotion. The shear band shown in Fig. 7~b! is identical in posi-tion and slip character to the individual dislocation observvia MD. The resulting lack of symmetry in the nucleatioevent indicates that the elastic anisotropy of the interatopotential was accurately reflected for otherwise symmeboundary conditions. In addition, the critical load at whithe elastic instability occurs is identical in MD and IPFEMThese correlations verify the application of theL criterion topredict the conditions for and slip character of elastic insbilities in computational methods heretofore limited to cotinuum analysis.

C. Three-dimensional analysis of the elastic limit

We conducted 3D MD simulations of spherical indention in Al ~Ercolessi-Adamssi potential37! comprising326 592~and also up to 23106) atoms, for a sphere of 13nm radius indented normal to the$111% plane in displacemencontrol. In all simulations, the initiation of dislocation activity was concurrent with far-field indentation load relaxatioand the dislocation structure comprised a dislocation gloop on a$111% slip plane.

The initial load relaxation indicated in Fig. 8~a! was dueto the subsurface, homogeneous nucleation of a defectbryo on a$111% plane. The nucleation site shown in Fig. 8~b!was located at a depthz equal to 51% of the contact radiusa,in good agreement with the depth of 0.48a predicted by Hert-zian ~continuum! analysis of spherical contact,15 but was lo-cated at a significant distance from the loading axis. Tlinear elastic solution for maximum shear stress correspoing to this load can be calculated as15

tmax50.31S 6PE* 2

p3R2 D 1/3

, ~15!

whereP is load,E* is the reduced Young’s modulus, whicaccounts for elastic deformation of both the indenter andsample:

1

E*5H S 12n indenter

2

EindenterD 1S 12nspecimen

2

EspecimenD J , ~16!

and is equivalent to the Young’s modulus of Al in this paticular case of a completely rigid indenter, andR is the in-denter radius. Just prior to dislocation nucleation, the lo

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FIG. 7. ~Color! Computational simulations ocylindrical indentation in Cu.~a! MD shows thedislocation dipole nucleation displaced off thloading axis. The atom color indicates coordintion numberN: yellow and pink denote perfecbulk and surfaceN, respectively; blue denoteimperfect bulk N, the defect site.~b! IPFEMshows a shear band of the same position andentation indicated by MD.~c! MD shows the dis-location dipole core displaced off the centrloading axis. The atom color indicatesN: Greendenotes imperfect bulkN. ~d! Coding of IPFEMoutput such that each visible node correspondsan atomic lattice site showing that the positioand orientation of defect nucleation are identicto those observed in MD simulations. Atom coloindicatesN: Green denotes imperfect bulkN.

siahentic

t toontelys ofipi-

imposed by the 13 nm diameter indenter was 20mN, indi-cating a corresponding Hertzian maximum shear stres3.41 GPa. Interestingly, this calculated value of Hertzshear stress is in excellent agreement with the ideal sstrength of 3.4 GPa obtained via pseudopotential electrostructure calculations for Al in the absence of elasrelaxation.38

10410

ofnaric

V. MECHANISMS OF INCIPIENT PLASTICITY

Next, we discuss the atomic scale activity subsequenthe onset of elastic instability. When such a dislocatinucleation and interaction process comprises approximahundreds of dislocations, it represents the earliest stagegross plasticity. This process is hereafter referred to as inc

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FIG. 8. ~Color! Homogeneous defect nucleation in Al. ~a! Indentation response to the poinof initial load relaxation. ~b! Correspondingatomic structure, indicating subsurface homogneous dislocation nucleation at a depthz50.51a, where z and a are depth and contacradius, respectively.~c! Plan view from withinthe crystal, oriented along the loading axis towathe free surface, showing significant displacemeof the nucleation site with respect to the centrloading axis. Atom color indicates coordinationumberN: pink denotes perfect surfaceN; bluedenotes imperfects bulkN ~defect site!; invisiblebulk atoms denote perfect bulkN.

5-10

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FIG. 9. ~Color! Correlation ofthe MD indentation response anatomistic activity in Al. At severaldisplacement increments, atomwith coordination number differ-ent from 12 are drawn.

ohaingri

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ent plasticity, a term suggested in this context by Tadmet al.39 First, we relate the dislocation structures and mecnisms observed in 3D MD simulations of spherical nanodentation in Al and Cu. Then, we correlate these findinwith results from nominally sharp nanoindentation expements in Al single crystals.

A. MD observations

Figure 9 correlates atomistic activity with external loaing in simulations of nanoindentation for Al. Figure 9~a!shows the internal defect structure subsequent to the hogeneous nucleation event, at the conclusion of the first lrelaxation. Immediately following this homogeneous nuc

10410

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ation event, further load relaxation was mediated by expsion of glide loops on the$111% plane@Fig. 9~b!#. Upon con-tinued loading, the intersection of these glide loops withsurface formed a110& slip step from which cross-slip ontonew $111% plane ensued@Fig. 9~c!#. A sessile lock would beformed when two loops on different$111% planes intersect.Several subsequent cross-slip events produced two parsets of dislocation segments, each having two partials srated by a stacking fault and pinched together at four vertiforming a prismatic loop, which mediates further load relaation via easy motion along its glide prism@Figs. 9~c! and9~d!#. Continued loading resulted in the formation of sevemore prismatic loops emanating from the same source,then finally the formation of a prismatic loop from a differe

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surface source with its own glide prism oriented 120° frothe first source@Fig. 9~e!#. The observed interaction of suface steps and subsurface dislocation is similar to the conof a defect complex hypothesized by Cleriet al. in the con-text of crack-tip plasticity.4

The initial stage of dislocation activity following the elatic instability in Al is likely to result in the formation of asessile dislocation structure. We have confirmed with simlations of up to 23106 atoms that the above-said structuand associated surface steps tend to be threefold symmconsistent with the threefold symmetry of available$111%^110& slip systems of the fcc crystal for the^111& indentationdirection. Because such atomic-scale inhomogeneitiesunableto move away from the high local stress region, susequent dislocation nucleations are likely to be thermallytivated and heterogeneous process. Additionally, our obvations indicate that these secondary products—the prismloops—tend to be much more mobile than the first setglide dislocation loops since they move outward and dointerfere with one other, and are able to move all the wdown to the substrate without impedance.

Identical 3D MD simulations of nanoindentation in Cindicated that partial dislocation loops are nucleated at a lcorresponding to the theoretical shear strength of Cu.glide loops formed near the surface, but their distributwas not symmetric with respect to the loading axis. Glloops nucleated at different indenter displacements~i.e., atdifferent times!, grew in-plane with little interaction. Andunlike in Al, plastic deformation tends to localize in aasymmetric fashion despite the threefold symmetry ofsetup. The immediate lock formation observed in Al is aabsent, perhaps due to the elastic anisotropy of Cu andinability of its widely dissociated dislocations to cross-slWe have also observed significant twin nucleation agrowth behavior.

The progression of the indentation response for both ctals can be viewed online.40 These results show that, forgiven crystal structure and indentation configuration,critical load at which dislocation glide loops are nucleatindeed correlates with the theoretical shear strength ofmaterial, indicating homogeneous nucleation. Further,development of incipient plasticity subsequent to the iniload relaxation is influenced strongly by the degree of elaisotropy, the ease of cross-slip as indicated by the intrinstacking fault energy, and the orientation of the loading awith respect to the preferred slip systems of the crystal.example, Fig. 10 illustrates the symmetry of the availaslip systems for 111& and^110& indentation directions. Theformer configuration promotes the intersection of glide looupon expansion, whereas the latter orientation promoglide loop repulsion. In the next section, we explore hthese elastic isotropy and orientation effects observed aatomic scale affect the far-field indentation response.

B. Correlation of local stress and external loading

Figure 11 illustrates the difference between the lostress developed in the crystal and the external load~far-fieldstress!. For a dislocation to be nucleated homogeneously,

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local stress in the vicinity of the nucleation site must aproach the theoretical strength of the crystal, on the ordeseveral GPa. Once a dislocation is nucleated and movesthe local stress decreases by a certain amount. In ordenucleate homogeneously anadditional dislocation, the localstress must climb back to the same level. However, sinceprevious dislocation—now supposedly piled up at tsubstrate—asserts a back stress on the critical region,external loading needs to be larger than before to achthis. Therefore weshouldobserve hardening in theP-h re-sponse in a quasi-uniform sequence associated with eachmogeneously nucleated dislocation,if indeed this happens. Itis exactly the sort of hardening behavior observed in thebubble raft simulations, whereby easy dislocation migratenables a sequence of homogeneous nucleation of distion dipoles, with an external load-displacement responsesteady and quasi-uniform hardening~see Fig. 4!.

FIG. 10. Indentation direction affects the symmetry of availaslip systems, either~a! promoting intersection of homogeneousnucleated glide loops on several slip planes or~b! promoting repul-sion of such loops.

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The above indentation responses arenot observed in ac-tual, 3D indentation experiments. Rather, such data typicshow a combination of small and large displacement reations, with largely varying degrees of hardening. Thishavior can be rationalized by the atomic scale observatfrom our MD simulations. Namely, the far-field indentatioload increases up to a critical value whereby dislocationshomogeneously nucleated and load relaxation is obserUpon dislocation reaction and lock formation, indentatiload again increases, indicating hardening. Then, uponcreased loading, such sessile structures form sources forerogeneous dislocation nucleation, which operate at stlevels far below the ideal strength, yielding tens of seconddislocations that propagate to the substrate, until the lostress drops to MPa levels. In the following section,verify this hypothesis with nominally sharp nanoindentatiexperiments in single-crystal Al of111& orientation.

FIG. 11. ~Color online! Schematic comparisons of external ainternal stress evolution during incipient plasticity in crystalstwo hypothetical situations.~a! All dislocations are homogeneouslnucleated;~b! the first and second dislocations are homogeneounucleated, but subsequent dislocations are heterogeneously nated from a source operating at a much lower stress.

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C. Experimental correlations

Bulk single-crystal $110% Al was indented in the asreceived~polished! condition. Experiments were conducteon a Hysitron TriboIndenter~Hysitron Inc., Minneapolis,MN! in load control to a maximum load of approximate100 mN, with an indenter tip radius of;150 nm. Figure12~a! shows a representative indentation response, indicaseveral displacement bursts of varying lengths. All expements showed at least three displacement bursts for thisrange. The load corresponding to the first displacement bwas very consistent among experiments (P510.80mN),corresponding to a Hertzian~linear! maximum shear stress o2.38 GPa. This value agrees well with that calculated frMD simulations for ^111& indentation, and withab initiocalculations of ideal shear strength.38 However, as indicatedin Fig. 12~b!, the first displacement burst length ranged fro2 to 11 nm, with a typical length of 2–3 nm. As the Burge

lycle-

FIG. 12. ~Color online! Nanoindention of$111% Al with an in-denter tip radius of 150 nm.~a! Representative load-depth responsshowing displacement bursts of varying length.~b! Histogram ofdisplacement burst length~solid! and the fraction of load carried foa given burst length~dashed!. Although the number frequency oshort bursts is greater, the majority of stress relaxation is carriedthe long bursts, corresponding to operation of defect sources.

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vector of Al is 0.29 nm, this burst length would correspoto almost 10 slip events and represents too large a relaxato be explained by a single dislocation nucleation eveThus, although the local shear stress is sufficient to indhomogeneous dislocation nucleation, the current resoluof the instrumentation may be insufficient to distinguish sua displacement burst from electronic noise.

Clearly, the distribution of displacement burst lengchanges with burst number. As the number of slip eveincreases, the burst length tends toward a bimodal distrtion, centered about 1.5 nm~5 slip events! and 5 nm~17 slipevents!. This distribution of minor and major stress relaations is consistent with the evolution of dislocation structobserved in the above MD simulations. Namely, dislocatinteraction subsequent to an initial homogeneous nucleaevent creates locks that inhibit dislocation motion and, upcontinued loading, act as sources that accommodate mheterogeneously nucleated slip events for marginal increin local stress. Back stresses from thus nucleated dislocaeventually reduce the local stress sufficiently to exhausparticular dislocation source~minor bursts!, and then furtherstress relaxation can be accommodated by the activationnew source~major bursts!.

As indicated in Fig. 12~b!, the number of minor burstsexceeds that of major bursts in a given experiment or seexperiments. However, we can also interpret these burscontributions toward stress relaxation~CSR!,

CSR[f l

L, ~17!

wheref is frequency,l is the length of an individual burst anL is the sum total of all burst lengths for a given experimeor set of experiments. From this viewpoint, we show thatmajority of stress relaxation is carried by major burstsfinding consistent with our MD observations that indicamajor stress relaxation events corresponding to continuoperation of defect sources.

VI. DISCUSSION

In this work we have performed a study of nucleationsingle dislocations and the initial stages of dislocation loformation in fcc metals by combining molecular dynamiand finite-element simulations with nanoindentation expments. An energetic, atomic-scale elastic stability criterithe L criterion, is shown to accurately predict the site aslip character of homogeneous dislocation emission, wthe simulations capture the effects of nonlinearity, crysanisotropy, and lattice periodicity in describing local yieprocesses on the microscopic scale. Correlation of the atistic mechanisms revealed by molecular dynamics withperimentally observed load-displacement responses indicthat incipient plasticity proceeds via heterogeneous dislotion sources such as sessile locks operating locally at grereduced stress thresholds.

TheL criterion, first introduced by Hill,19 is similar to theelastic stability criterion for homogeneous deformation, cosidered by Wallace41 and later refined, in that both may bderived from free energy considerations. However, these

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criteria describe different conditions with respect to theternal loading. For the local instability represented by theLcriterion, the ‘‘weak’’ point in the material is assumed to bisolated from the external boundary, so the Hemholtz fenergyF is the appropriate thermodynamic potential for tprocess. In contrast, for the elastic instability under homoneous deformation,21,42,43the Gibbs free energy is the appropriate potential because the externally applied work isdirect contact with the material volume that becomes elacally unstable. Just as Eq.~1! is the curvature of the Hemholtz free energy change, the elastic stiffness coefficient gerns structural stability under homogeneous deformationfinite strain.41

Another general observation that deserves mention isthe application of Eq.~4! provides a quantitative measurethe weakest spot in the system, with regard to both spalocation and vibrational instability. In this respect, it woube interesting to cast theL criterion in the formalism of areal-space Green’s function that has been applied to anathe onset of fracture in a homogeneously strained lattice.21,44

Beyond the theoretical significance of theL criterion, ourpractical implementation through MD and FEM has immdiate applications. First, one can imagine determining Eq.~4!using electronic structure methods rather than relying on Mwith classical potentials. This approach could improveaccuracy of computational simulations in materials for wha classical potential is believed to be less reliable. Secothe viability and relative simplicity of IPFEM suggest ththis approach will find many applications, wherever locconstitutive relations are needed in FEM calculations.

We conclude by noting that a theme that underlies a snificant portion of the present study is the competitive roledislocation nucleation relative to dislocation migration. Thissue is central to other long standing problems in deformtion and plasticity. In geoscience, a well-known challengeto understand the phenomenon of weakening of quartz inpresence of water.45 Here, the hydrolytic effects on the emission of glissile dislocations versus the gliding of existindislocations still remain to be resolved. Application of thlarge strain formalism and implementation proposed incurrent study may serve to advance this and related topicwhich localized deformation processes are poorly undstood.

ACKNOWLEDGMENTS

This work was supported by the Defense University Rsearch Initiative on NanoTechnology~DURINT! on ‘‘Dam-age and Failure Resistant Nanostructured Materials andterfacial Coatings,’’ which is funded at MIT by the Office oNaval Research, Grant No. N00014-01-0808. We thank AArgon for insightful comments. K.J.V.V. gratefully acknowedges the support of the National Defense Science andgineering Graduate program. J.L., T.Z. and S.Y. acknowlesupport by Honda R&D Co., Ltd., AFOSR~Grant No.F49620-00-10082!, NSF ~Grant No. DMR-9980015!, andLawrence Livermore National Laboratory under an ASCLevel 2 grant.

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*Present address: Department of Surgery, Children’s HospBoston, Massachusetts 02115.

†Present address: Department of Materials Science and Engiing, The Ohio State University, Columbus, Ohio 43210.

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