nanocrystals & nanomechanics: mechanisms &...

1Nanocrystals & nanomechanics: mechanisms & models. A selective review

t XjUbWYXKhiXm YbhYfcDhX

Rev. Adv. Mater. Sci. 35 (2013) 1-24

Corresponding author: I.A. Ovid'ko, e-mail: [email protected]

NANOCRYSTALS & NANOMECHANICS:MECHANISMS & MODELS. A SELECTIVE REVIEW

OPT1,2,3 and Elias C. Aifantis4,5,6

1Department of Mathematics and Mechanics, St. Petersburg State University, St. Petersburg 198504, Russia2Institute of Problems of Mechanical Engineering, Russian Academy of Sciences,

St. Petersburg 199178, Russia3St. Petersburg State Polytechnical University, St. Petersburg 195251, Russia

4Laboratory of Mechanics and Materials, Polytechnic School, Aristotle University of Thessaloniki,Thessaloniki GR-54124, Greece

5Michigan Technological University, Houghton, MI 49931,USA6King Abdulaziz University, Jeddah 21589, SA

Received: October 16, 2013

Abstract. LYdUdYfZ]fghfYj]YkggcaYwdUfhmibbckbw]b]h]UcVgYfjUh]cbgaUXYVmhYsecond author and his Michigan Tech colleagues in an effort to capture the basic deformationmechanisms at the nanoscale for developing corresponding scale-dependent gradient elasticity,[fUX]YbhdUgh]W]hmUbX[fUX]YbhZfUWhifYacXYgLY]fhkcaU]bWcbWig]cbgw]YhYYaYf[YbWYof two co-existing phases (bulk and grain boundary) and the dominance of nanovoid formationUbXaUhYf]UfchUh]cbUgcddcgYXhcaUhYf]Ug]dwUfYhYbigYXhcach]jUhYgcaYaYhUphysics and strength of materials models for interpreting the associated experimental findings,as well as the construction of gradient continuum mechanics models (nanoelasticity,nanoplasticity) to be used in relation to the formulation and solution of related boundary valuedfcVYagUhhYbUbcgWUY ]bUmhYUZcfYaYbh]cbYXcVgYfjUh]cbgcb[fU]bfchUh]cbw]bconjunction with the standard theory of defects, as employed recently by the first author and hisWckcfYfgw]gigYXhcXYgWf]VYfchUh]cbXYdYbXYbhghfYb[hYb]b[gcZhYb]b[aYWUb]gag]bnanopolycrystals weakened by cracks.

1. INTRODUCTION

We begin with a brief but self-contained summaryof the basic experimental observations, as firstrecorded by the second author and his MichiganTech (MTU) coworkers, pertaining to characteristicmaterial flow processes in nanocrystalline (NC) andultrafine grain (UFG) materials. This is done mainlyfor historical reasons and partly because these earlyTEM in situ straining and tensile experiments seemto remain unnoticed by current researchers in thefield, especially those performing computerexperiments and advancing MD simulations. Next,we focus on certain of the aforementionedexperimental observations and attempt to interpret

them by employing metal physics and standardcontinuum mechanics tools also used forconventional materials. These include the nucleationand stability of nanopores in triple grain boundaryibWh]cbgcZF {ghYUgmaaYhfmcZm]YX]b[]bhYbsion/compression, as well as the orientation ofnanoshear bands in UFG. These considerationsmake apparent the need for non-standardapproaches in order to analyze a variety of othercrucial phenomena (inverse Hall-Petch behavior,thickness of shear bands, elimination of singularitiesin nanodefects and nanocrystals) which arerepeatedly recorded in the NC and UFG materialsbut are commonly not observed or remained

2 6 WKFM and E.C. Aifantis

unimportant in traditional polycrystals. The non-stan-dard approach pursued herein is a proposition for aphenomenological but physically-based gradientframework for continuum nanomechanics. Its rami-fications to elasticity and plasticity at the nanoscaleUfYX]gWiggYXUbX[YbYf]WacXYgcZxbUbcYUgh]W]hmyUbXxbUbcdUgh]W]hmyUdd]WUVYhcF UbXMmaterials are proposed. Among the most notablegeneric results are the modeling of shear band thick-ness and its dependence on the grain size, the in-verse Hall-Petch relation, as well as the eliminationof singularities from dislocation cores and crack tips.It is expected that such generic results could havemany other applications in NC and UFG modelingand, thus, the point of view is advanced here thatgradient theory may be used as an efficient tool forbridging macro/meso and micro/nano configurations.

LYhYfaxbUbcaYWUb]WgykUg]bhfcXiWYXVmElias Aifantis in two sequential symposia in Osakaand Beijing in the summer of 1995 [1] in an effort tointerpret related experimental observations obtained]bhYYUfmb]bYh]YgVm]gWcwkcfYfgUhELMS4]. These observations pertained to the identificationof the deformation and damage mechanismsoccurring at severely deforming regions (near anadvancing crack) of nanograined (NC) thin filmsstrained inside the transmission electronmicroscope (TEM). Nanovoid nucleation/growth andcoalescence, as well as nanograin rotation andsliding were recorded for grain sizes at the regimeof 10 nm, while intragranular dislocation activity wasonly noted for grain sizes ten times larger. Theyalso measured large grain rotations exceeding 6o

sometimes reaching up to 15s. By fixing sights ona material triangle (strain nanorosette) of the in situdeforming planar film, they measured thecorresponding strain tensor giving an effective strainof the order of 20-30%, indicating that plasticitytakes place by grain sliding and rotation, rather thanby the classical intragranular dislocation motiondominating plastic deformation of conventionalpolycrystals. This nanoscale deformation transitionaYWUb]gaZfcaxaUgg]jYy]bhfU[fUbiUfX]gcWUh]cbach]cbhc]bhfU[fUbiUfX]gcWUh]cbxXYdYh]cbyreplaced by grain translation and rotation leading tooverall plastic flow of the material, was subsequentlyargued by several groups on the basis of moleculardynamics simulations [5,6] or other grain boundarydislocation processes [5-11]. A different type of de-formation transition mechanism was alsocVgYfjYXVmhYELM[fcidS TZcfxViyihfUfine grained (UFG) polycrystals with grain size ofUVcihU ibXfYXh]aYgUf[YfhUbhYcbYg]bxh]bZ]ayWcbZ][ifUh]cbg [U]b hYWcbhfc]b[

mechanism here was not dislocation motion butxaUgg]jYyaih]dYgYUfVUbXZcfaUh]cbYUX]b[hcperfect plasticity deformation curves for the testedUFG materials, in contrast to their conventionalcounterparts exhibiting the usual work (dislocation)hardening behavior. Such multiple shear bandformation has also been reported afterwards for sev-eral deforming nanopolycrystals ([16]; see also re-cent papers in the literature).

All attempts made so far to address materialand process behavior at the nanoscale were basedeither on a direct and straightforward adoption ofcorresponding models at macro and micro scalesor on elaborate but also conceptually straightforwardcomputer simulations involving ab initio and MDprocedures based on empirical potentials. A notableexample is the case of carbon nanotubes whereclassical elasticity theory was used to describe theirdeformation under simple loading configurations,leading to predictions remarkably consistent withcorresponding MD simulations [17,18]. As soon asstructural defects were involved, however, thedeformation/flow and strength/fracture propertiescould not be captured by any of the twoaforementioned approaches, in accordance withanalogous experience from metal physics. Thissuggests that for realistic material and deformationconfigurations relevant to a variety of nanoengineeringapplications, the standard continuum mechanicsmodel needs to be revised. Such revision shouldtake into account not only the occurrence ofnanostructural defects, and grain boundaries andother planar inhomogeneities (internal surfaces), butalso the small volumes available for their nucleationand interactions, as well as the external surfacebounding the material micro/nano domain underconsideration, where boundary conditions are as-signed.

The point of view advanced in this article (partlybased on the aforementioned experimentalobservations) is that in NC and UFG polycrystals,deformation evolves through two distinct butoverlapping material regimes: the bulk (no disloca-tion slip) and grain boundary (dislocation-inducedrotation) space. A simple but sufficiently general con-tinuum nanomechanics framework may then becVhU]bYXVmUck]b[YUWxViydc]bhhcYlWUb[Y[19] mass, momentum, energy and entropy with thesurrounding internal (grain boundary) or externalxgifZUWYy bchYfZcfaUmYei]jUYbhdc]bhcZj]Ykis to assume that NC and UFG polycrystals con-g]ghcZUa]lhifYcZhkc]bhYfUWh]b[dUgYghYxViyUbXhYx[fU]bVcibXUfmydUgYgLYhkcdUgYgcan exchange mass, momentum, energy and en-


tropy but they are constrained to obey the overallbalance equations of a standard continuum. Sepa-rate constitutive equations may then be assumedfor each phase which, in conjunction with the bal-ance laws and appropriate boundary conditions, candescribe the material at hand. As a result of suchconsiderations, it can be shown that a continuummechanics framework is generated where the clas-sical constitutive equations for elasticity and plas-ticity are replaced by corresponding ones enhancedwith the Laplacian of strain multiplied by an internallength parameter accounting for the aforementionedtwo-state interactions.

As an alternative to the above continuum/gradi-ent nanomechanics framework, an approach basedon the theory of defects in NC and UFG polycrys-tals will be considered in this review. This approachexploits the standard methods of mechanics ofdefects in solids but, in addition, it takes intoaccount the dominant nanoscale and interface ef-fects. This is done by considering the role of spe-cific defect configurations (e.g. defect configurationsat grain boundaries) on the plastic flow and fractureprocesses in NC and UFG polycrystals; i.e. poly-crystals characterized by ultra small grain sizes andextremely large amounts of grain boundaries. Inparticular, grain rotations and nanocrack generationnear tips of microcracks growing in NC and UFGpolycrystalline films will be discussed.

The plan of the paper is as follows: Section 2provides a summary of the main experimentalobservations pertaining to the deformation ofnanograined films and UFG polycrystals as theywere first recorded by the MTU group. Somequantitative metal physics and standard mechanicsarguments are given in Section 3. Section 4 outlinesthe mathematical structure of the proposedcontinuum gradient nanomechanics framework andXYf]jYghYWcffYgdcbX]b[[fUX]YbhxbUbcYUgh]W]hmyUbXxbUbcdUgh]W]hmyacXYg]YhYbYWYggUfm

modifications of the classical elasticity and plastic-ity equations to account for small (nano) volumeeffects through the introduction of spatial gradientsof strain. Finally, Sections 5 and 6 are focused onanalytical models (based on the theory of defects)describing grain rotation and nanocrack generationnear microcrack tips, respectively.

2. SUMMARY OF THE MTUEXPERIMENTAL OBSERVATIONS

2.1. Deformation mechanisms innanograined (NC) thin films

In conventional metals, deformation mechanismsobserved by conducting in situ studies inside a high-resolution transmission electron microscope(HRTEM) may not represent bulk behavior as thefilm thickness is quite smaller than the grain size.This is not the case for nanograined thin filmsstudied by the second author and his collaborators(MTU group) in the early and mid nineties [16]. De-formation mechanisms were observed directly in anatomic resolution TEM, fitted with a straining stage.The tested materials were gold and silvernanograined thin films coated on aluminum, poly-mer or carbon substrates. The film thickness wasin the range of about ~100 nm and the average grainsize varied from ~8-150 nm. By loading in steps,the sequence of deformation can be ascertained withvery high resolution and the deformation mecha-nisms in the coating can be viewed directly eachtime that the loading is interrupted.

Fig. 1 shows schematically the region (plasticzone wake of a macrocrack) of the nanograined (8nm grain size) gold film on aluminum substrate anda corresponding bright field TEM micrographindicating the relevant plastic deformationmechanisms: nanopore growth and link-up leadinghc[fU]bVcibXUfmx[fccj]b[yY[fY[]cb UgkY

Fig. 1 FUbcdUgh]W]hmaYWUb]gag lhYbg]jYbUbcdcfYbiWYUh]cb[fckhUbXWcUYgWYbWY]bhYxdUgh]WncbYkUYycZUaUWfcWfUW YUX]b[hcbUbc[fU]bVcibXUfm[fccj]b[UbXbUbc[fU]bVcibXUfmg]X]b[ ]ZZig]cbwinduced deformation in the absence of dislocation activity: 8 nm Au on Al substrate.

(a) (b)


as ligament formation leading to ductile crack bridg-ing (e.g. region B). This type of nanopore or nanovoidgrowth and coalescence, leading to extensivexbUbcXUaU[YyaU]bmhfci[bUbc[fU]bVcibXUfmx[fccj]b[yUddUfYbhmcWWifghfci[ aUggX]ZZision in the absence of any dislocation activity. Itshould be pointed out that the aforementioned MTUobservations were the first to suggest and confirmdislocation-free plasticity of nanopolycrystalls, atopic elaborated upon extensively through molecular

(a)

(b)

Fig. 2. Nanoplasticity mechanisms: Single nanopore nucleation and growth at a triple grain boundaryjunction ahead of a crack tip. The material between the nanopore and the crack tip is thinned down throughmass diffusion and localized necking/ligament formation leading to crack advance and crack tip blunting :ba icb giVghfUhYSK]a]Uf@JLEd]WhifYgkYfYcVhU]bYXfYWYbhmVmOY]QUb[{g]b ]bUUg

reported in his plenary lecture in ICF 13 in Beijing, June 2013 - without being awared of the previous MTUwork.]

Fig. 3. (a) Schematic illustration of grain rotation induced by unbalanced shear stresses due toinhomogeneous grain boundary sliding. (b) Lattice-image TEM micrographs of in situ deformation in 10 nmgrain diameter nanocrystalline Au film on polymer substrate, revealing 6s relative rotation between neighboringgrains. For clarity, one set of lattice fringes is highlighted in the two grains.

dynamic simulations of Swygenhoven and cowork-YfgS Tk]hcihdfcdYffYZYfYbWYhcELM{gZ]bX]b[g(see also [9-11]).

Fig. 2 shows schematically the nucleation andgrowth of a nanopore at a triple boundary junction,as well as corresponding lattice-image micrographs]bX]WUh]b[UfYUxbUbcdcfYybiWYUh]b[UYUXcZUbintergranular crack tip. The film is again 8 nm grainsize gold but now deposited on carbon. Thisexperiment also suggests diffusion-assisteddeformation at low homologous temperatures.

(a)

(b)


(a)

(b)(b)

Fig. 4. (a) Relative grain rotation between different grains in the neighborhood of a growing crack tip. A nano-strain gage rosette is shown. (b) Relative grain rotation between different grains as a function of imposedoverall displacement.

The nanodamage development associated withnanopore formation and linkage (Fig. 1), as well asthe ligament thinning process and the associatedshape changes (Fig. 2), seem to be unrelated todislocation activity within individual grains, as suchactivity was not recorded. This suggests that othermechanics may be responsible for ductile behaviorat the nanoscale. This suggestion is partially sup-ported by the results presented in Figs. 3 and 4.

These figures show such a different deformationmechanism observed in ~10 nm average grain sizenanocrystalline gold films with thickness of about~20 nm deposited on polymer substrates withthickness of about ~50 nm. The induced grainrotation is schematically illustrated in Fig. 3a, andexperimentally revealed by the lattice imagemicrographs of Fig. 3b. The method of observationhere is to image a set of grains in atomic resolutionmode, observing actual crystallographic planes. Thefilm is then strained and by measuring the relative

angles between planes in different grains, the rota-tion may be calculated. A relative rotation of 6o fortwo neighboring grains is shown in Fig. 3b (81o and75o angles before and after straining; one set of lat-tice fringes in each grain is shown for clarity), whilerelative grain rotations up to 15o have been observedahead of a crack tip as illustrated in Fig. 4a. In thesame figure a set of 3 material points is identifiedwhich are visible before and after straining. By mea-suring changes in length and angles of the resultingtriangle before and after straining, the in-plane strainscan be calculated by such in situ nano-strain gagerosette. A typical strain tensor measured this waywas found to have components

11 = -0.09,

12 =

21

=0.10,22

= 0.16, 33

= -0.01 (assuming incompress-ibility) with all other

ij components being zero. This

gives a value of about ~20% for the second invariantof the strain tensor or effective strain, suggestingthat the material is well into the plastic deformationregime. Grain rotation is inhomogeneous as indi-


cated in Fig. 4b where the relative rotation betweendifferent grains as a function of imposed macroscopicdisplacement is plotted [4,16]. The maximum ef-fective strain calculated in these experiments was~30%. Such measurements of the in-plane straintensor should only be viewed as approximate dueto the difficulty involved in identifying three materi-als points which can be unambiguously trackedduring displacement of the straining stage. Anotherdifficulty is the fact that small strain analysis is usedand also that the deformations are highly inhomo-geneous. In this set of experiments, intragranulardislocation activity was not observed. This suggeststhat the strain necessary for grain rotation was pro-vided by grain boundary sliding and other grainboundary processes.

As a result, the point of view may be advancedthat in a nanopolycrystal, the grain boundary spacebehaves like an interconnected continuum wherematerial flow occurs by plastic deformation processes(most likely induced by intergranular dislocationactivity), while the embedded in the grain boundaryspace nanograins slide and rotate in a highlyinhomogeneous mode (in the absence of observableintragranular dislocation activity) in a manner con-

Fig. 5. (a) Stress-strain curves obtained in compression for varying grain size nanostructured Fe-10% Cualloys. (b) Hall-Petch plot of compressive yield data of Fe-10% Cu. (c) Corresponding Hall-Petch plot ofVickers hardness data.

trolled by the aforementioned grain boundary pro-cesses. Such view may be extended to larger ultrafine-grained polycrystals where grain rotations pro-cesses lead to multiple shear banding as discussedbelow [16,20].

2.2. Deformation mechanisms in ultrafine grained (UFG) bulknanopolycrystals

While the thin film experiments described in theprevious section are of fundamental interest fordetermining deformation mechanisms, anddescribing mechanical behavior of coatings andmicroelectronics components, they are not structuralmaterials. Fe-10%Cu bulk samples with grain sizesfrom about 45 nm to 1.7 m were prepared at MTUby first ball milling with limited contamination andthen by rapid forging with minimal grain growth.cmg@AH{YXUh s and 700s for 30 minutes at

170 MPa were fully dense. As expected, the grainsize coarsened substantially from the original ~18nm size. The final average grain sizes wereapproximately 100 nm after a 600 sC HIP and 130nm after a 700 sC HIP. While significantly coarser

(a)

(b) (c)


Fig. 6. Optical micrographs showing deformation and fracture behavior under compression of nanostructuredFe-10% Cu alloy for different grain sizes. Shear fracture occurred rapidly at the finest grain sizes. Shearbands and shear offsets occur at coarser grain sizes. Buckling, instead of fracture, is noted at a larger grainsize. Wavy lines are intersections of shear bands coming out of the plane of the paper at an angle. Note theincrease in width of shear bands at increasing grain sizes. The compression axis was vertical.

than the 10 nm grain-size films discussed in theprevious section, these alloys are still much finerthan traditional steels, and may be considered toVY]bhYiddYffUb[YcZxbUbcghfiWhifYXaUhYf]Ugy

The mechanical and deformation behavior ofthese materials were quite surprising; essentially,they behaved in an elastic-perfectly plastic manner.This is demonstrated in the stress-strain curves ofFig. 5. In this figure, the iron material had muchcoarser grain size than the alloys, and it exhibitedtraditional strain-hardening behavior. Thenanostructured alloys, however, exhibited either nostrain hardening or slight softening upon yield. Theeffect of grain size upon the yield response is alsoclear in this figure, and is shown to obey the classicalHall-Petch equation. More on the Hall-Petch relationand its inverse behavior will be given in a companionarticle, where some mathematical models motivatedby the deformation mechanisms discussed here willbe developed. Nevertheless, we also provide in Fig.5 experimental data for the hardness (H) and theyield strength (

y) of these materials. The data are

fitted by the equations H = 0.94 + 32d-1/2 and

y = 0.31 + 12d-1/2 where H and s

y are measured in

GPa and d]gaYUgifYX]bbaAh]gbchYXhUhLUVcf{grelation (H ~ 3

y) is followed reasonably well, and it

is also noted that the constants are close to thosefor pure iron.

The deformation mechanisms observed were alsoquite surprising. The first mechanism of plasticdeformation was intense shear banding. Shearbanding at the yield point was observed at all grain

sizes investigated, from about 100 nm to around1700 nm. As the grain size increased, the strengthdecreased and the shear band width increased.Typical shear bands from representative specimensare shown in Fig. 6.

The observation of shear banding as the firstmechanism of permanent deformation, at staticstrain rates, appears to be unique among metals.Shear banding is normally observed at very highstrain rates, and/or after large amounts of cold work,but not in annealed metals at slow strain rates. Thisbehavior is observed, however, in amorphouspolymers and metallic glasses. This commonalityof behavior between the nanostructured metals andamorphous materials is intriguing, and deservesfurther study. In any case, it may be argued that atthe finest grain sizes below 100 nm thenanostructured metal can be considered as aWcadcg]hYWcbg]gh]b[cZWfmghU]bYx[fU]bycfxViydUgYUbXUbUacfdcigxVcibXUfmydUgY

iYhchYZUWhhUhxaUgg]jYygYUfVUbX]b[]gthe main observable mode of deformation in the bulkultra fine-grained alloys considered in this study (aswell as related subsequent studies for bulknanopolycrystals [21]), the behavior of shear bandsunder various type of loading is of substantialinterest. The shear bands did not occur on the planeof maximum shear stress or on the plane of zeroextension, as would be predicted by continuummodels, i.e. pressure independent classicalplasticity. In uniaxial compression the shear bandplanes were reproducibly inclined at 49s to the stress


axis, instead of at 45s; while in plane strain tensionthe angles were around 52s, instead of 55s. Further,the strength in compression was about 30% higherthan the strength in tension, i.e. a significant tension-compression asymmetry was observed. While thistype of behavior has not been observed in metals, ithas been observed in amorphous polymers and somemetallic glasses. In such materials, the angle ofthe shear band and the tension-compressionanisotropy have been ascribed to a dependence ofyielding on the hydrostatic pressure or the normalstress. This is also the point of view adopted here,i.e. a modified von Mises yield criterion with pressuresensitivity is proposed. Then, in conjunction with azero extension shear instability condition, this yieldcriterion is able to correctly predict both the angleof the shear band planes and the tension-compression anisotropy of the yield strength. Moredetails can be found in [12,13,16,22], and asummary of these results along with a furthermodification of the yield criterion to include the effectof strain gradients are included in the next section.The aforementioned MTU experimental resultsclearly show, for bulk UFG and nanopolycrystallinematerials, a strong dependence of the yield stresson the hydrostatic pressure, as well as a perfectlyplastic behavior (no hardening or very slight softening)which has not previously been observed for metals.They confirm, moreover, that excessive spatiallydistributed shear banding is the mechanismresponsible for the observed perfectly plasticbehavior. In addition, the measured shear bandangles and widths were in good agreement with thepredictions of the gradient theory, as explained inthe next section.

Such type of behavior has also beensubsequently observed in other classes of novelmaterials including bulk nanocrystalline materialsproduced by severe plastic deformation [23-25]. Inthis latter class of materials it has been observedthat a bi-modal grain size distribution allows for thepossibility to optimize mechanical properties byadjusting the ductility-to-strength ratio, while at thegUaYh]aYhYdYbcaYbcbcZaih]dYcfxaUgg]jYyshear banding has also been noted [21]. ThexgaUYfyg]nY[fU]bg YfYaUmdUmhYfcYcZhYsmall Cu precipitates in the case of the Fe-10%Cualloys.

3. STANDARD METAL PHYSICS ANDMATERIAL MECHANICSMODELING

From the experimental observations listed in thedfYj]ciggYWh]cbhkcX]gh]bWhxdUgh]W]hmhfUbg]h]cby

aYWUb]gagghUbXcih]hYxbUbcdUgh]W]hmyhfUbsition observed in ~10nm thin nanograined films fromdiffusion-induced deformation and grain boundarysliding/rotation processes to intragranular disloca-h]cbUWh]j]hmUbX]]hYxM dUgh]W]hmyhfUbg]h]cbfrom homogeneous intergranular dislocation motionhcxaUgg]jYycfxaih]dYygYUfVUbX]b[ YckkYprovide simple metal physics and standard mechan-ics arguments to interpret this behavior.

3.1. The nanoplasticity transitionmechanism

A simplified, discrete metal physics model is de-scribed below for the critical grain size and strainrate necessary to obtain dislocation-based plasticityin nanograined thin films. As discussed in Section2.1 at the ~10 nm grained films intragranulardislocations were not present, while at the ~100nm grained films dislocation activity, leading toplastic thinning, ligament formation and ductilefracture within individual grains, was observed. Thestrain rates in both cases were ~10-4 s-1. Instead ofdislocations, small pores (nanopores ~1 nm) areobserved to nucleate and grow at the grain boundarytriple junctions located just ahead (~50-100 nm)ahead of a crack tip. In traditional materials suchtype of pores at triple grain boundary junctions areknown as r-type cavities and they commonly occurat elevated temperatures as a result of grainboundary sliding. Based on standard metal phys-ics arguments, the local stress required to openup a nanopore of size r is given by the relation

n

r, (3.1)

where is the interfacial energy, r is the pore radiusand the numerical constant n for cylindrical poresmay be taken as 0.5-0.7 depending on the contactangle. The local stress is therefore determined tobe on the order of ~1 GPa or ~10-2 G, where G isthe shear modulus for the materials considered.Although this is well beyond the yield point ofxbcfaUyicf[bcX]gcWUh]cbVUgYXdUgh]W]hmwas observed and the individual crystals in thenanograin material appear to have a significantlyhigher yield point than the corresponding normalgrain size material. It may thus be argued that thisxUVbcfaUyVYUj]cf]gXiYhchYUW cZX]gcWUtions and, thus, the lack of dislocation sources inthe interior of nanocrystalline grains. This is remi-niscent to the mechanical behavior of defect-freewhiskers and it should not be, therefore, surprisingthat a grain in the nanocrystalline material has a


high yield point for dislocation-based plasticity ifdislocation production is difficult.

A possible quantitative explanation of this ob-servation and the resulting implications for themechanical behavior of the nanocrystalline materi-als may be sought to the image forces on disloca-tions due to the close proximity of grain boundaries.Attractive forces between dislocations and grainboundaries arise because there are no long rangestresses associated with the low energy dislocationstructure at a grain boundary. The elastic energy ofa dislocation is therefore less in the organization ofa grain boundary structure than a dislocation isolatedin the grain interior [16]. As a result, there will be areduction in energy as the dislocation approachesthe grain boundary, provided there is sufficientthermal energy to allow for rearrangement of the grainboundary structure. This type of thermal energy islikely present during the deposition process. Thestrain energy per unit length W of a dislocation canbe written in the following general form applicable tothe two limiting cases to be considered in this model

Gb x DW

b

2 / 2ln .

4

(3.2)

Eq. (3.2) is applicable when x>>D and whenx = 0, where x is the distance from the grain bound-ary and D is the spacing of dislocations in the grainboundary. For these two limiting cases, Eq. (3.2)reverts to the relevant expressions given by standarttexts in dislocation theory. The magnitude of theimage force per unit length ( b) acting to move adislocation in the center of a grain of diameter d tothe grain boundary is thus determined as the de-rivative of Eq. (3.2) evaluated at x = d/2>>D, or

Gbb

d

2

.2

(3.3)

Here is the image stress. For a dislocation to bestable in the vicinity of the grain boundary duringprocessing, the image stress must be less thanthe stress necessary to move dislocations from theiroriginal (grown in) positions. It is not straightforwardto determine this stress. In the case of glissiledislocations on the {111} glide planes, and ignoringd]bb]b[YZZYWhgcZhY[fU]bVcibXUf]YgcbhYxYbXgyof the dislocations, as well as line tension forces(as the dislocation bows towards the grain bound-ary) and solute strengthening, this resisting stresswould be the {111} CRSS of pure gold; i.e.about 1-3 MPa depending on purity. However, grown-in dislocations often thread crystals on planes other

than the glide planes, and so the resisting stressmay be dominated by the CRSS on non-glide planes.Further, thin film deposition is an energetic process,and so climb may be important. As a first approxi-mation, we assume here that the resisting stress isUVcihYeiUhchYxm]YXghfYggycZWcaaYfW]Udirity gold or silver; i.e. about 35 MPa. For d = 10 nm,the image stress is on the order of 4x10-3 G oraround 300 MPa; i.e., substantially greater than theyield point of normal Au or Ag (~35 MPa). This sug-gests that any dislocations which might be presentin the grain interior during processing when theaUhYf]U]gxchykciXVYZcfWYXhchY[fU]bVcibXary, resulting in dislocation free nanocrystallites.However, if the 100 nm grains are considered, Eq.(3.3) determines an image stress on a dislocationat the grain center which is less than the yield stress.This suggests that the dislocations are stable atthe grain center and may act as sources for dislo-cation reproduction.

The image force approach can be extended toconsider the stress required to activate a grainboundary source by evaluating the derivative of Eq.(3.2) with respect to x at x = 0 and assuming thatD = b. This approximates the stress required to pulla grain boundary dislocation into the grain interioras

gb

G12GPa.

2 (3.4)

This appears to be well above the stress deter-mined by the pore diameter measurements. In fact,at low strain rates grain boundary sliding may actto keep the stress lower than

gb. Application of the

Mukherjee model [26] of grain boundary sliding con-trolled plasticity to our case where d~10 nm, thestrain rate is ~10-4 s-1, and the grain boundary diffu-sion coefficient is ~10-15 cm2/s gives a stresswhich is consistent with the pore diameter mea-surements. For this diffusion coefficient, theMukherjee model has the following form for Au/Ag

b

s d G

2410; s- seconds. (3.5)

Using Eqs (3.4) and (3.5), it is possible to deter-mine a critical strain rate at which dislocation basedplasticity due to grain boundary sources might oc-cur. In other words, when the stress in Eq. (3.5)approaches the value of

gb in Eq. (3.4), grain

boundary sources may become active. Carrying outthis algebraic manipulation determines the criticalstrain rate as


b

s d

24

2

10 1.

4 (3.6)

Eq. (3.6) agrees with the experimental observa-tions in that the strain rate present in our experi-ments (10-4 s-1) is too low to observe grain boundarydislocation sources in the 10 nm material. But thisstrain rate is also just below what is required toobserve active grain boundary dislocation sourcesin the 100 nm grain size material. However,dislocation sources may be present in the interiorsof these larger grains. Another limiting factor as tothe operation of grain boundary sources is theconsideration that the large stresses required toactivate dislocation sources in the 10 nm grains willnot occur at the head of a pile-up, for there are nodislocations in the grain interior. In this case,dislocation structures cannot induce a line sourceof stress at the grain boundary. Instead, the stresswill be applied more uniformly to the grain boundary.This may cause a loss of cohesion at the grainboundary and failure by brittle fracture before theactivation of grain boundary sources.

3.2. The UFG plasticity transitionmechanism

In order to describe the tension-compressionasymmetry and the shear band characteristicsobserved in Fe-10%Cu UFG polycrystals, one mayresort to a modified von-Mises yield criterionincorporating the effect of high-stress triaxility orhydrostatic pressure upon yield. Another standardaYWUb]WgdfcWYXifYVUgYXcbhYxnYfcYlhYbg]cbWf]hYf]cby]bWcbibWh]cbk]hEcf{gW]fWYaUmUgcbe used to obtain the orientation of shear bands atthe onset of yielding, i.e. at the emergence ofinhomogeneous plastic flow.

The pressure-dependent yield condition reads[12,22]

ijf f J p

2( ) 0, (3.7)

where p = kk

/3 is the pressure and J2 =

ij ij/2

(ij =

ij - p

ij) is the second invariant of the deviatoric

stress. The coefficient is a material parameterdenoting the effect of hydrostatic stress on the yieldstrength . Moreover, Eq. (3.7) indicates that plasticdeformation must be accompanied by a change involume proportional to the magnitude of which

must be less than 3 / 2 because a greater valuewould imply expansion in three dimensions underuniaxial tension. This property, commonly knownas dilatancy, was the initial motivation for postulating

on a phenomenological basis a yield criterion of theform of Eq. (3.7) for soils (Drucker-Prager yieldcondition, see also [27]). A motivation for adoptingEq. (3.7) for the present ultra fine grained alloysmay be sought to our earlier mentioned observationsof locally inhomogeneous deformation on bothnanograined thin films and ultra-fine grained bulksamples revealing the possibility of plasticdeformation via the mechanisms of void formationand growth at triple grain boundary junctions, aswell as by grain sliding and rotation in addition tothe traditional mechanism of conservative disloca-tion glide, which is the dominant mechanism of ho-mogeneous-like plastic flow at larger grain size.Moreover, it is not unreasonable to expect that yield-ing in these materials would depend on hydrostaticpressure, as this has a direct effect on vacancy gen-eration and motion which, in turn, facilitate disloca-tion climbing processes as well as nanovoid nucle-ation and growth. In fact, the pressure-dependentyield condition given by Eq. (3.7) was readily de-rived in [27] on the basis of dislocation glide andclimbing processes.

The zero-extension criterion [28] for determiningshear band orientation is expressed by the condition

t = 0 which should be utilized in conjunction with

the pressure-dependent yield condition of Eq. (3.7)UbXEcf{gW]fWYhcXYhYfa]bYhYaUhYf]UdUbYgon which this component of tangential strain

tacts

upon [29]. These are the planes determining theinitial planar shear band development wheresubsequent macroscopic plastic flow evolves underhomogeneous loading conditions. The details of suchstrength of materials type of analysis can be foundin [12,16] and the results obtained with this simplemethod are comparable to those deduced by a morerigorous but a lot more involved bifurcation orinstability approach [27,30]. Here, we onlysummarize the main results as follows:

Yield strength in tension :

ys

ysJ p 12 1

1,

33 (3.8)

Yield strength in compression :

ys

ysJ p 32 3

1,

33 (3.9)

Shear band angle in plane strain tension :

1

2

1 3sin ,

4 2 3 (3.10)


Shear band angle in uniaxial compression :

11 1 4cos .2 2 3 3 3

(3.11)

The above relations resulting from standardmechanics or strength of material considerationsare used in the sequel to interpret the mainexperimental observations for the UFG Fe-10%Cualloys. The most consistent and reproducible dataobtained from the experiments was themeasurement of the shear band angle from theuniaxial compression test samples. The plane ofthe shear band was oriented at 49s to the load axisand was independent of the scale of themicrostructure. Based on this shear band angle,the pressure-sensitive material parameter wascalculated with Eq. (3.11) to be 0.25. Shear bandingin the tensile samples occurred in the fillet areabetween the gage and grip sections which isundergoing plane strain deformation. The anglebetween the load axis and the plane of the shearbands was measured to be approximately 54s. Thesedata were used to verify the calculation of thepressure-sensitive parameter. For = 0.25,Eq. (3.10) was used to predict a shear band anglefor plane strain tension of 52.3s which compareswell with the above experimental value.

The observed strength differences in tension andcompression (see [12,16]) provided anotheropportunity to confirm the pressure dependence ofthis material. For a given grain size, the yield strengthin tension was ~70% of the yield strength incompression as shown in Table 1. The yield con-stant was estimated from Eqs (3.8) and (3.9) forboth uniaxial compression and for plane strain ten-sion with a pressure-sensitivity coefficient = 0.25.In actuality, the yield constant is independent of themode of testing. The calculated results for , listed

Grain Tensile yield Compressive yield (tensile) (compressive)size(nm) strength (Mpa) strength (MPa) (MPa) (MPa)

1170 526 721 348 3561370 484 694 320 343

Table 1. Verification of the yield condition for = 0.25.

Grain size(nm) (MPa) sWcadfYgg]cb shYbg]cb

1170 351 0.27 48.6 53.91370 329 0.308 47.8 54.1

Table 2. Verification of the pressure-sensitivity coefficient.

in Table 1, indicate that the value used for the pres-sure-sensitivity coefficient ( = 0.25) models suffi-ciently well the measured yield strengths ys

1 and

ys

2 entering Eqs (3.8) and (3.9).The yield strengths were used to verify the mag-

nitude of the pressure-sensitivity coefficient via adifferent approach. For a given grain size, the yieldconstant and the pressure sensitivity coefficient

were solved for simultaneously using Eqs. (3.8)and (3.9) and the measured yield strengths in ten-sion and compression. The calculated parametersare listed in Table 2. The results indicate that a hasa value near 0.25 as determined above. The valuesof and were subsequently used to calculate theshear band angles shown in Table 2 with Eqs. (3.10)and (3.11). Again, the obtained values for the shearband angles agree very well with the experimentaldata.

This close agreement among the determinedparameters using two separate approaches (asshown in Tables 1 and 2), provides further confidencein the model used.

The width of the shear bands was observed tobroaden with increasing structural scale (or grainsize) of the compression test samples. The widthsof the most prominent shear bands were comparedto the grain size of the alloys but this aspect wasdiscussed in the next section in conjunction with amore refined continuum nanoplasticity theory[31,32] developed to capture the evolution of locallyunstable inhomogeneous plastic flow and its mani-festation to the overall macroscopic mechanicalbehavior.

It is concluded that the ultra fine grained iron-copper alloys investigated herein, exhibited amechanical behavior which suggested a strong pres-sure-dependence of plastic flow. Shear banding oc-curred as the only mode of plastic deformation from


the yield point with nearly perfectly plastic behaviorin both tension and compression. The experimentalobservations on tension/compression yield strengthdifferences and shear band orientations were con-sistently interpreted on the basis of a pressure-de-pendent yield condition with the use of only onenew material parameter. The spatial characteristicscZhYgYUfVUbXgUbXhY]ad]WUh]cbgcZhYxaUgg]jYycfxaih]dYygYUfVUbX]b[dYbcaYbcbcbthe macroscopic stress-strain response, will be dis-cussed elsewhere and some preliminary resultshave already been reported in [32].

4. GRADIENT NANOMECHANICS

An extended or generalized continuum mechanicsframework is outlined here for addressing themechanical response of NC and UFG polycrystals.This extension is based on generalizing the standardcontinuum mechanics structure by viewing NC andM dcmWfmghUgUgUa]lhifYcZxViyUbXx[fU]bVcibXUfmygdUWY]bUWWcfXUbWYk]hhYX]gWigg]cbprovided in earlier sections. The two phases caninteract mechanically by exchanging mass andacaYbhiaVihhYcjYfUxWcadcg]hYygciXcVYmthe standard balance laws of continuum mechanicsand each phase should obey its own constitutiveequations. We present this discussion, forconvenience, separately for elastic and separatelyfor plastic deformations. The resultant governingdifferential equations are proposed to be used inconnections with the determination of themechanical response of polycrystals at thesubmicron and nano regimes for both elastic (micro/nanoelasticity) and plastic (micro/nanoplasticity)deformations.

4.1. Micro/Nanoelasticity

Here a two-state material model is adopted forderiving quantitative expressions for the elasticdeformation field in a nanograined polycrystal. It isassumed that bulk and grain boundary phasesoccupy the same material point but locally interactvia an internal body force f . The differential equationsof equilibrium are then expressed in the form (seealso [22])

1 2div ; div ; div 0L L (4.1)

with 1 2

( , ) denoting the partial stress tensors foreach individual phase and

1 2 being the

stress tensor for the nanostructured materialconsidered as a whole. By further assuming thatYUW dUgYcVYmg@ccY{gUkUbXhUhhY]bhYf

action force is proportional to the difference of theindividual displacements, we have the relationships(for k = 1,2)

k k k

T

k k k

1 2;

; div; ,

Y L Y Y

L G G I (4.2)

where (k,

k, ) are phenomenological coefficients, I

is the identity tensor, and (div, ) denote thedivergence and gradient operators respectively, withT denoting transposition. Uncoupling, leads to thefollowing differential equation for the totaldisplacement u = 1/2(u

1 + u

2) associated with the

nanostructured material

c

2

2 2

div

div 0

u u

u u (4.3)

where the coefficients ( , ,c) are related explicitlyto those appearing in Eq. (4.2) which, in turn, shouldsatisfy certain special conditions in order that onlyone gradient coefficient c to appear in the final formof the governing differential equations for the totaldisplacement. As a direct consequence of Eq. (4.3),hYZcck]b[[fUX]YbhacX]Z]WUh]cbcZ@ccY{gUkmay be suggested at a first approximation to modelelastic deformation at the nanoscale (nanoelasticity)

c 2(tr ) 2 (tr ) 2 . (4.4)

4.2. Micro/Nanoplasticity

Along the same lines, as for the elastic case, it isassumed that the flow stress of a nanograinedpolycrystal is made up of two parts. The flow stress

1cZhYxViyUbXhYZckghfYgg

2cZhYx[fU]b

VcibXUfmydUgYggiWhUh

1 2. (4.5)

In the case of simple shear the applied stressappl is carried out by both phases, each of which

obeys its own equilibrium equation

x x x1 2f; f 0, (4.6)

where f denotes, as before, the exchange of mo-mentum between the two simultaneously deform-]b[xdUgYgyibXYfg]adYgYUf]bUX]fYWh]cbbcfmal to the x-axis. We further assume that

k k k 1 2( ); f ( ), (4.7)

where k and

k denote the flow stress and corre-

sponding shear strain in each one of the two phases(k = 1,2). If both the strains

k and the gradients


x k are of order (


ever, these dislocation processes are suppressedin NC polycrystals with fine grains due to nanoscaleand interface effects [44]. Recently, theoreticalmodels [46,47] have been suggested describingnanoscale rotational deformation occurring throughcooperative grain boundary sliding and climb of grainboundary dislocations in crack-free and pre-crackedNC materials. Another possibility is to describenanoscale rotational deformation as a fast processcarried by groups of nanodisturbances [48]. Below,we discuss the basic points and results of thesetheoretical schemes taking into account small vol-ume constraint inherent to NC polycrystals.

5.1. Diffusion-controlled grainrotations near crack tips in NCs

First, let us examine, following [46,47], crystal latticerotations associated with cooperative grain boundarysliding and climb of grain boundary dislocations ina NC specimen, weakened by a flat (Mode I) crack,under a tensile stress

0 (Fig. 7). For simplicity, we

Fig. 7. Diffusion-controlled rotational deformation mode in a model square grain of a pre-crackednanocrystalline specimen (schematically) [47]. (a) Tensile deformation of a pre-cracked nanocrystallinegdYW]aYb YbYfUj]Yk VwYKdYW]UfchUh]cbUXYZcfaUh]cbcWWifg]bUbUbc[fU]bhfci[ ZcfaUh]cbcZimmobile disclinations (triangles) whose strengths gradually increase during the formation process con-ducted through grain boundary sliding and diffusion-controlled climb of grain boundary dislocations. Grainboundary sliding occurs through local shear events (grey ellipses) in grain boundaries AB and CD. Grainboundary sliding results in the formation of grain boundary dislocations at junctions A, B, C and D. Diffu-sion-controlled climb of dislocations along grain boundaries AC and BD provides special rotational deforma-tion accompanied by formation and evolution of a quadrupole of wedge disclinations (triangles) at junctionsA, B, C, and D.

consider a two-dimensional grain structure thatserves as a good model for columnar nanoscalestructures of films and a first-approximation modelfor bulk NC materials. High stresses operating indeformed nanocrystalline solids and stressconcentrations near crack tips can initiate thespecial rotational deformation of grains, as shownin the figure.

To fix ideas, let us consider a rectangular grainin a crack tip which reaches a junction of grainboundaries. In terms of the theory of defects, thecorresponding rotational deformation in such a graincan be described through the formation of aquadrupole of immobile wedge disclinations whosestrengths gradually increase during the formationprocess [46,47]. This special rotational deformationoccurs through grain boundary sliding along grainboundaries AB and CD and diffusion-controlled climbof grain boundary dislocations along grain bound-aries AC and BD (Figs. 7b-7e). Adopting the con-cept of local shear events - shear transformationsof local atomic clusters - as carriers of plastic flow

0

0


Fig. 8. Dependence of the normalized fracture tough-ness on the angle characterizing the orientationof the deformed grain near the crack tip, for variousvalues of the ratio t of the sizes of grain facets [47].

in grain boundaries in metals [49,50] and covalentsolids [51], we assume that local shear events in-duce sliding along grain boundaries AB and CD. Thissliding results in the formation of grain boundarydislocations at junctions A, B, C, and D [52,53], asgWYaUh]WUmgckb]b ][g W X ]ZZig]cbWcbtrolled climb of the dislocations along grain bound-aries AC and BD yields a special rotational defor-mation mechanism accompanied by the formationof quadrupole of wedge disclinations at grain bound-UfmibWh]cbg UbX ][g XUbX Y LYdisclination quadrupole creates stresses that influ-ence crack growth.

Within a macroscopic mechanical description,the effect of local plastic flow - the special rotationaldeformation resulting in the formation of a disclinationquadrupole - on crack growth can be accounted forthrough the introduction of a critical stress intensityfactor K

IC. In this case, the crack is considered as

propagating under the action of a tensile load per-pendicular to the crack growth direction, while thepresence of the disclination quadrupole simplychanges the value of K

IC, as compared to the case

of brittle crack propagation. Such a critical stressintensity factor K

IC was calculated in terms of the

disclination quadrupole characteristics [47]. TheX]gW]bUh]cbghfYb[hgUfYXYbchYXVmv , the qua-drupole arms (distances between the disclinationsof the quadrupole) by k and m, and the angle be-tween the crack plane and one of the quadrupolearms by . The quadrupole arms were assumed tobe small compared to the crack length l (k,m

ICK ) if -2


respectively. Thus, within the proposed model [47],the increase in fracture toughness due to the con-sidered special rotational deformation near crackh]dg]bF aUhYf]Ug]gUfcibX w

To summarize, rotational deformation can serveas a special toughening mechanism in NCpolycrystals due to the following two factors: (i) NCmaterials are characterized by large volume fractionsoccupied by grain boundaries, and (ii) plasticdeformation occurs at very high stresses in thesematerials. At the same time, the special rotationaldeformation hardly contributes to toughening ofcoarse-grained polycrystalline metals where thetoughening mechanisms associated with plasticflow are realized through conventional dislocationemission from crack tips (e.g. [54-56]). The rate ofthe aforementioned special rotational deformationmechanism is controlled by diffusion, facilitatinggrain boundary dislocation climb that accommo-dates grain boundary sliding [46] (Figs. 7d and 7e).In similar situations with grain rotations controlledby diffusion and driven by the sensitivity of the grainboundary energy on the associated misorientationparameters [57], the rate of grain rotations in NCmaterials is by several orders larger than that incoarse-grained polycrystals. This is related to thefact that diffusion is greatly accelerated with de-creasing grain size due to corresponding increasein the volume fraction occupied by grain boundarieshaving enhanced diffusivity. Therefore, as with grainrotations, diffusion-controlled processes inducingspecial rotational deformation in NC materials arehighly enhanced, as compared to those occurringin coarse-grained polycrystals. It follows that rota-tional deformation is expected to significantly con-tribute to stress relaxation near crack tips in NCmaterials, while it is hardly effective in coarse-grained polycrystals.

5.2. Shear-induced fast grain rotationsin NCs

Specific mechanisms of plastic deformation comeinto play in NC solids due to their specific structuralfeatures such as nanometer sizes of grains and largeamounts of grain boundaries [25,44,58]. As alreadydiscussed, experiments [2-4,38-41] and computersimulations [42,43] provided convincing evidence forthe important role of rotational deformation in plas-tic flow processes in NC metals and ceramics. Ithas been found that rotational deformation stronglyinfluences the unique mechanical characteristics ofNC solids. Besides, similar to the change in defor-mation mechanisms due to decrease of grain size

down to the nanometer range [25,44,58], the physi-cal mechanisms of plastic deformation can changein NC polycrystals when the external stress levelincreases from conventional levels up to very highvalues (realized, for example, during dynamic load-ing [59]). In the context discussed, it is critical toidentify the pertinent rotational deformation mecha-nisms operating in solids which have a NC struc-ture and/or are deformed at extremely high stresses.In this subsection, following [48], we discuss a spe-cific mode of rotational plastic deformation in NCnanopolycrystals deformed at high stresses. Thismode manifests as a result of the fast nanoscalerotational deformation occurring through collectiveevents of local ideal shear processes.

Let us consider the geometry of the fastnanoscale rotational deformation in a NC solidconsisting of grains divided by grain boundaries. Atwo-dimensional section of the solid is schematicallyshown in Fig. 9a. The solid is under a mechanicalload producing a shear stress in the region ABCDas shown in the figure. It is suggested that thegeneration of walls of dislocation dipoles by meansof nanoscale ideal shears can serve as a mecha-nism of rotational deformation in NC solids. Moreprecisely, nanoscale ideal shears simultaneouslyoccur under the shear stress in several (n) parallelslip planes (Figs. 9b-9e). Such shears are charac-terized by a tiny shear magnitude s and produce awall of n generalized stacking faults havingnanoscopic sizes (Fig. 9c). (In the theory of crys-tals, a generalized stacking fault is defined as aplanar defect resulting from a cut of a perfect crys-tal across a single plane into two parts which arethen subjected to a relative displacement by an ar-bitrary vector s (lying in the cut plane) and rejoined[60-62].) Such generalized stacking faults areVcibXYX VmzbcbWfmghUc[fUd]W{dUfh]UX]gcWUh]cbgWUfUWhYf]nYXVmbcbeiUbh]nYXzbcbWfmghUc[fUd]W{ if[YfgjYWhcfgvs with a quitesmall magnitude s


tional dipoles of perfect dislocations, in which case generalized stacking faults disappear (Fig. 9e).The resultant finite walls of perfect dislocations bound a nanoscale region with the crystal lattice misoriented

relative to that of the neighboring material (Fig. 9e). That is, plastic deformation carried by the walls ofnanodisturbances is accompanied by crystal lattice rotation and this is another type of a special kind ofrotational deformation. It was quantified in [48] which gives the following formula for the energy change W(per unit dislocation line and per one dipole) that characterizes the fast nanoscale rotational deformationoccurring through the formation of a regular array of n nanodisturbances (dipoles of non-crystallographicX]gcWUh]cbgk]hhY if[YfgjYWhcfgvs and generalized stacking faults):

n

k

Gs d p k d dW s n n k sd s d

s p k p k d

2 2 2 2 21

2 2 2 2 21

2( ) ln 1 1/ ln ( ) .

2 1

(5.4)

Here p denotes the interspacing between the neighboring nanodisturbances, d the nanodisturbance length][ W G the shear modulus, the Poisson ratio, and (s) is the specific energy of a generalized stacking

fault. The first term on the right hand side of Eq. (5.4) describes the total energy of the dislocation dipole,including its self energy (the first two terms in the curly brackets) and the total energy of the dipole-dipoleinteraction per dipole (the last two terms in the curly brackets). The second term (- sd) is the work of theexternal stress, spent for dislocation formation. And the third term ( (s)d) characterizes the energy of ageneralized stacking fault associated with a nanodisturbance (Fig. 9c). The energy density (s) is effec-h]jYmUddfcl]aUhYXVmUxhkc iadYXyXYdYbXYbWYS T

m

m m

m

u u

s u u

u u

0 0

sin 2 , 0 1/ 4

( ) cos 4 , 1/ 4 3 / 4,2 2

sin 2 , 3 / 4 1

(5.5)

where u = s/b, whereas m and

0 are the maximum and minimum values of (s), respectively.

Corresponding calculations [48] suggest that when the values of the shear stress are not very high,W increases at small s, in which case there is an energy barrier for the fast nanoscale rotational deforma-

tion. When the values of are very high, W always decreases with an increase in s LUh]gUxbcbVUff]Yfynanoscale rotational deformation takes place. The critical stress =

c - the minimum stress at which NRD

cWWifg]bUxbcbVUff]YfyZUg]cbWcffYgdcbXghchYWcbX]h]cb W/ s = 0 at s = 0 and W . This conditionyields

c mb2 / . (5.6)

O]hh]gZcfaiUGj]X{cUbXKY]bYfaUbS TZcibXhYjUiYc 4.3 GPa (G/17) of the critical shear

stress in the case of NC Ni; i.e. a very high value. Such high values c > G/20- based on computer

simulations [60-62,67] of the maximum value m of the energy density function (s) - were also found for

many other materials, such as Pd, Al, Ir, Pt, Cu, Au, SiC, Pb, Ag, Fe. It is noted that the fast nanoscalerotational deformation hardly occurs in crack-free solids at conventional conditions of quasistatic loading. Infact, for comparatively low yield stresses, other deformation modes operate at conventional conditions. Onthe contrary, the fast nanoscale rotational deformation can come into play at extreme conditions characterizedby very high values of both applied (or local) stress and plastic strain rate. At such extreme conditions,WcbjYbh]cbUXYZcfaUh]cbacXYgwwY[[]XYcZdfYYl]gh]b[Uhh]WYX]gcWUh]cbgUbX[YbYfUh]cbcZ]bX]j]XiUdislocations by conventional mechanisms like dislocation cross-slip and Frank-Read sources - cannotgenerate high plastic strain rates. Extreme conditions are realized, in particular, during shock loading andball milling, and it is natural to expect that the fast nanoscale rotational mechanism dominates at thesedeformation regimes. High plastic strain rates at extreme conditions are effectively generated by the fastnanoscale rotational deformation involving intensive collective generation of dislocation groups (Fig. 9), butnot dislocation motion and generation of individual dislocations. In particular, the fast nanoscale rotationaldeformation can play a significant role in the formation of high-density ensembles of dislocations experi-mentally observed [59] in shock-loaded polycrystals.


6. NANOCRACK GENERATION NEARMICROCRACK TIPS IN NCs

In recent decades, growing attention has been paidto crack nucleation and related crack growthprocesses in NC solids; see, for example, reviews[68-72]. This was motivated by the fact that theseprocesses crucially influence key mechanicalcharacteristics (strength, ductility, fracturetoughness) of NC materials [25,58,68-73] . Inparticular, research in this area has shown that grainboundaries and their triple junctions in NC solidsserve as preferable places for crack nucleation andgrowth. This is because the atomic density is low,and interatomic bonds are weak at grain boundaries

as compared to the bulk. As discussed in Sectionx]bg]hiyhfUbga]gg]cbYYWhfcba]WfcgWcdmcV

servations [3,4] of plastically deforming NC Aufilms (with average grain sizes of around 8-10 nm)revealed the absence of bulk dislocation activity;instead, nanovoid nucleation and growth at triplejunctions of grain boundaries was observed aheadof a blunt crack tip. More recently, Kumar et al. [74]UgcfYdcfhYXcbx]bg]hiyYldYf]aYbhU L Eobservations of nanocracks generated at triplejunctions of grain boundaries near the tip of a bluntedcrack growing in NC Ni during tensile deformation.The crack induces the generation of triple junctionbUbcjc]XgbYUf]hgh]dUbXgiVgYeiYbhm]hxUVgcfVgythem during its growth [74]. Besides, molecular

Fig. 9. Fast nanoscale rotational deformation (through nanoscale ideal shears) in nanocrystalline solidsSTULkcX]aYbg]cbU[YbYfUj]YkcbbUbcWfmghU]bYgdYW]aYbVwYLkcX]aYbg]cbUj]YkcZbUbcgWUYrotational deformation in a crystallographic plane of a grain with a cubic crystalline lattice. (b) Initial state ofa nanoscale grain. (c) A wall of nanodisturbances is generated. Each nanodisturbance consists of a dipoleof non-crystallographic dislocations with tiny Burgers vectors . Generalized stacking faults (wavy lines) areformed between the dislocations composing the nanodisturbances. (d) The Burgers vector magnitude s(characterizing the nanodisturbances) gradually increases, and generalized stacking faults evolve in paral-lel with the growth of s. (e) The non-crystalographic dislocations transform into conventional perfect disloca-tions (when s reaches the Burgers vector magnitude b of a perfect dislocation), and generalized stackingfaults disappear.


dynamics simulations [75,76] show nanocracksgenerated at triple junctions of grain boundariesahead of the tips of pre-existent large cracks in NCNi with grain size ranging from 5 to 10 nm. Thesenanocracks then join the growing main crack[75,76]. Similar processes have also been observedin computer simulations [77] for -Fe NC materialsunder a mechanical load. According to theseexperimental observations and related computersimulations, nanocracks at triple junctions of grainboundaries serve as typical carriers of fracture inNC solids, and their formation is enhanced near pre-existing/growing cracks.

Generation of cracks in coarse-grainedpolycrystals under a mechanical load isconventionally described as a process induced bythe superposition of the external stress and stressescreated by lattice dislocation pile-ups stopped ateither grain boundaries or other obstacles [78].Nanometer size grains in NC solids prevent theformation of lattice dislocation pile-ups [58,79], inwhich case such pile-ups do not play any role incrack generation. With such inability for crackinitiation at lattice dislocation pile-ups in NCmaterials, there is an increasing interest to identifyand describe alternative micromechanisms for crackgeneration in these materials. The generation ofnanocracks at triple junctions of grain boundariesin deformed NC solids (free of pre-existing cracks)

was theoretically described as a process inducedby the superposition of the external stress andstresses created by grain boundary defects suchas dislocations [80], disclination dipoles [81] anddislocation-disclination configurations [82]. Suchdefects are effectively produced during grainboundary deformation modes typical for NC solids.For instance, grain boundary sliding produces triplejunction dislocations whose stress fields are capableof initiating triple junction nanocracks [80]. Acorresponding theoretical model for such NC solidswas developed [80] without assuming a pre-existingcrack, whereas the experiments [2-4,74] showgeneration of nanocracks and nanovoids in the vicinityof large blunted cracks. Recently, a theoretical modelhas also been suggested to describe the generationof nanocracks near the tip of a pre-existing/growingblunted microcrack in NC solids [83], as discussedbelow.

Let us consider a NC solid consisting of grainsdivided by grain boundaries and containing a longblunted crack, under a remote tensile load

0. A two-

dimensional section of the pre-cracked NC solid isgWYaUh]WUmgckb]b ][ UDYhhYWfUWintersect the boundary at a distance r

0 from the

nearest triple junction (Fig. 10b). The local stressesnear the crack tip in the solid under the tensile loadinitiate grain boundary sliding through local shearevents along the grain boundary AB (Fig. 10b) and

Fig. 10. Evolution of a nanocrystalline structure near a tip of a blunt crack of elliptic shape in a deformednanocrystalline solid [83]. (a) General view. (b) The magnified inset highlights generation of an edge dislocationat the triple junction B due to sliding along grain boundary AB near the tip of a long crack that intersects theboundary. (c) A nanocrack forms at the triple junction dislocation B.


other grain boundaries in the vicinity of the cracktip. Since grain boundaries end at triple junctions,such junctions serve as natural geometric obstaclesfor grain boundary sliding in nanocrystalline andmicrocrystalline materials [52,53,84,85]. Inparticular, the triple junction B obstructs grainboundary sliding along the boundary AB (Fig. 10b).This is because the conditions for lattice disloca-tion slip - slip plane orientation, Burgers vector mag-nitude for lattice dislocations, stress characterizingresistance to plastic shear (the Peierls stress),YhWww]bhY[fU]b]bhYf]cfUfYX]ZZYfYbhZfcahcgYfor grain boundary sliding along the grain boundaryAB. In these circumstances, following standard ar-guments on grain boundary sliding [52,53,84,85],hYxibWcadYhYXydUgh]WgYUfUggcW]UhYXk]hgrain boundary sliding is accumulated at and nearthe triple junction B. In terms of the theory of de-fects in solids, the triple junction B contains a dis-location (Fig. 10b) whose Burgers vector magnitudegradually grows with the continuing plastic shearassociated with grain boundary sliding [53]. The lim-iting value b

c of the Burgers vector magnitude b of

the triple junction dislocation near the crack tip iscontrolled by both the magnitude of the externalstress and the associated crack growth configura-tion under the external stress.

Attention is next focused [83] on the situationwhere the magnitude b of the Burgers vector of atriple junction dislocation is large enough to initiatethe formation of a nanocrack with length l in thestress field of a triple junction dislocation near apre-existing crack (Fig. 10c). Previous estimates[86] have shown that if the crack is flat, the stressesnear the crack tip are too small to produce a triplejunction dislocation whose Burgers vector b is largeenough to generate a nanocrack. This is related tothe fact that growth of a flat crack occurs at acomparatively low level of external stress, and thislevel is not sufficiently high to produce a triple junctiondislocation with a sufficiently large Burgers vector.However, the situation can change if the crack hasbeen blunted due to previous events of latticedislocation emission from the crack tip or grainboundary sliding at the crack tip. A blunted crackrequires a much higher applied load to propagateand, as a result, the stresses near its tip can bemuch higher than those in the case of a similar sharpcrack. By following [56, 87], the blunted crack mayeffectively be modeled as an elongated ellipse witha curvature radius at the crack tip, which is muchgaUYfhUbhYY]dh]WU{gWfUW UZYb[h (Fig.11c). The curvature radius is related to the ellipsesemi-axes a and p through the relation: = p2/a.

Fig. 11. System state diagrams in the coordinates(d, l) for (a) nanocrystalline Al with = 2 nm, and (b)nanocrystalline -Fe with =1.3 nm [83].

UgYXcbgiW WcbZ][ifUh]cb Gj]X{cUbXSheinerman [83] calculated the conditions fornanocrack generation/growth in the case of anintragrain nanocrack in NC Al and -Fe. It was foundthat nanocrack generation and growth in NC Al anda-Fe are energetically favorable for a rather widerange of the relevant material parameters. The stressconcentration near blunted cracks induces grainboundary sliding which, in turn, leads to the formationof dislocations at triple junctions of grain boundaries.The superposition of the external stress (elevatednear crack tips) and stresses created by thesedislocations is capable of initiating generation andgrowth of nanocracks. It was also found thatnanocrack generation and growth are enhanced innanocrystalline solids with increasing the crack tipcurvature radius . This model confirms both theYldYf]aYbhUx]bg]hiyLEcVgYfjUh]cbgS TcZnanocracks and nanovoids generated at the triple


junctions of grain boundaries near the tips of bluntedcracks (growing in NC Ni and Au), and the com-puter simulations [75-78] showing that nanocracksare generated at the triple junctions of grain bound-aries ahead of the tips of pre-existing large cracksin NC Ni and Y ifhYfacfYGj]X{cUbXSheinerman [83] have constructed related maps orsystem state diagrams in the coordinates (d, l wwwhere d denotes grain size and l nanocrack length.The diagrams are presented in Figs. 11a and 11b,for nanocrystalline Al and a-Fe, respectively, whena new nanocrack grows in the direction character-ized by = /2. It follows from these diagrams thatwhen the grain size d is small enough (see the re-gions left to the left dashed vertical lines in Fig. 11),the nanocrack growth is favored at any nanocracklength, that is, the nanocrack is able not only tonucleate but also to reach a large length or eventransform into a catastrophic macrocrack. At largergrain size (see the regions between the dashedvertical lines in Fig. 11), the nanocrack growth isfavored in the regions l < l

e and l > l

c, while crack

growth in the intermediate nanocrack length regionle < l < l

c requires overcoming an energy barrier. As

the grain size increases further (see the regions rightto the right dashed vertical lines in Fig. 11), thecritical nanocrack length l

c becomes infinite, while

the equilibrium nanocrack length le becomes

vanishingly small (i.e. smaller than 0.8 nm), so thatthe nanocrack is not formed at all. Thus, Fig. 11demonstrates that, for a specified value of the cracktip curvature radius , there is a rather fast transitionfrom an absence of nanocracks state to theformation of sufficiently large nanocracks (near thetips of large cracks) with decreasing grain size d.

It is concluded that the following two tendencieswere theoretically established. First, nanocrackgeneration and growth are enhanced in NC solidswith increasing crack tip curvature radius . Thismeans that the blunting of cracks ww themicromechanism typically responsible forenhancement of ductility and toughness inconventional coarse-grained polycrystals[54-56] ww is not effective for tougheningnanocrystalline solids. Second, nanocrack genera-tion and growth are enhanced in NC solids with de-creasing grain size. This means that NC materialstend to have lower ductility and toughness whenthe grain size decreases. These tendencies are ingood agreement with experiments. The first ten-XYbWmwwWcbjYbh]cbUhci[Yb]b[a]WfcaYWUb]gathrough blunting of cracks is not effective in NC sol-]Xgww]ggiddcfhYXVmbiaYfcigYldYf]aYbhUXUhUon low toughness and ductility exhibited by most

NC solids; see reviews [25,58,79,88]. The secondhYbXYbWmwwbUbcWfUW[YbYfUh]cbUbX[fckhUfYenhanced in NC solids with decreasing grain size][ ww]g]bUWWcfXUbWYk]hhYYldYf]aYbhU

fact that some NC metals with fcc lattice exhibit aductile-to-brittle transition with decreasing grain size[89-91]. More precisely, as shown in these experi-ments [89-91], ductile fracture occurs in NC Ni witha mean grain size around 44 nm, while brittle inter-granular fracture comes into play in NC Ni-15%Fealloy with a mean grain size around 9 nm.

It should be noted that the generation ofnanocracks near blunted cracks (Fig. 10) is not thesole underlying reason for the brittle behavior of NCsolids. There are other factors (e.g. suppression oflattice dislocation slip in nanograins) that are ofcrucial importance for the brittleness of NC materials.At the same time, experimental data [2-4,74] andcomputer simulations [75-77] demonstrate that thegeneration of nanocracks in the vicinity of crack tipsis a rather typical process in NC materials.Therefore, in-line with the results of the theoreticalanalysis presented in [83], it is reasonable to expectthat the generation of nanocracks near blunt cracks(Fig. 10) can play a significant (or even dominant)role in the brittle behavior of nanocrystalline materialsunder a wide range of conditions.

7. CONCLUDING REMARKS

The paper first reviews some early pioneering ex-periments on nanoscale deformation at MichiganTech, by the second author and his co-authors,which have not been sufficiently acknowledged insome experimental and simulations works performedsubsequently by other investigators. It is shown howthis initial experimental work has motivated con-tinuum models on nanoelasticity and nanoplasticitythat are successfully used today. These models arebased on treating the material as a superposition ofhkcdUgYghYxVi dUgYyk]h ]a]hYXcfbcbYl]gh]b[X]gcWUh]cbUWh]j]hmUbXhYx[fU]bVcibXUfmdUgYyk]hX]gcWUh]cb]bXiWYX[fU]bgfchUh]cbsliding. A most recent account for such continuumnanomechanics can be found in [42].

Then, the paper considers flow and fracture pro-cesses in NC and UFG polycrystals by exploitingthe nanoscopic configuration of a polycrystal, asproposed by the first author and his co-authors, atthe nanoscale; in particular, its large volume frac-tion of grain boundaries and the defect (dislocation/disclination) processes occurring there. Two phe-nomena are discussed and quantified: nanoscalerotational plastic deformation and nanocrack nucle-


ation near the tips of blunt microcracks. Nanoscalematerial rotation proceeds along two paths: a slowxX]ZZig]cbWcbhfcYXyacXYWUff]YXVm[fU]bVcibXUfmXYZYWhgUbXUZUghxg]dWcbhfcYXyacXYhfci[ideal shearing within grain interiors. The experimen-tally observed formation of nanocracks near tips ofblunt microcracks was described as a process ini-tiated by grain boundary sliding in NCs. It revealstwo very interesting features: First, crack blunting,i.e. the micromechanism typically responsible forenhancement of ductility and toughness in conven-tional coarse-grained polycrystals, is not effectivefor the toughening of NCs. Second, nanocrack gen-eration and growth are enhanced with decreasinggrain size.

ACKNOWLEDGEMENTS

ECA was supported by the Greek Ministry ofEducation / Greek Secretariat of Research andTechnology (GSRT) through a major grant ERC-13:Internal Length Gradient Mechanics Across Scalesand Materials: Theory, Experiments and Applications- Project No. 88257, also with modest support byThales/Intermonu: Conservation and Restoration ofMonuments of Cultural Heritage - Project No. 68/1117. IAO was supported by the Russian Ministryof Education and Science (Grant 14.B25.31.0017and Contract 8025), as well as the St. PetersburgState University research grant 6.37.671.2013.

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