Is Stress Concentration Relevant for Nanocrystalline Metals?

Published: May 18, 2011

r 2011 American Chemical Society 2510 dx.doi.org/10.1021/nl201083t |Nano Lett. 2011, 11, 2510–2516

LETTER

pubs.acs.org/NanoLett

Is Stress Concentration Relevant for Nanocrystalline Metals?Sandeep Kumar,† Xiaoyan Li,‡ Aman Haque,*,† and Huajian Gao*,‡

†Department of Mechanical and Nuclear Engineering, Penn State University, University Park, Pennsylvania 16802, United States‡School of Engineering, Brown University, Providence, Rhode Island 02912, United States

The existence of flaws, such as cracks or notches, in a loadbearing solid is known to increase the magnitude of stresses

in the immediate vicinity of a notch tip. Such stress concentrationor gradient depends on the notch and specimen geometry (notchlength, tip radius, specimen width) and can increase stress by ashigh as a factor of 10 for a U-shaped notch.1 Plastic flowmechanisms significantly reduce the stress concentration near anotch tip in ductile materials. For brittle materials, however, nosuch relief mechanisms exist and the concept of stress concen-tration is thought to hold up to the fracture strength. For metallicmaterials, the flow stress usually increases with decreasing grainsize,2,3 which should increase the stress concentration accordingto conventional wisdom as well as strain gradient plasticitytheories. Furthermore, Coble creep4 and grain boundary (GB)sliding5 are also known to develop stress concentrations at GBsand triple junctions.6,7 Therefore, the significance of stressconcentration can be expected to become more prominent atthe nanoscale, where metals exhibit brittle-like behavior due toconfinement, scarcity, or starvation of dislocations.8�10

In this paper, we provide experimental evidence that questionsthe validity of the scaling down of stress concentration at thenanoscale. This is spurred by the indications in the literature thateven brittle materials may exhibit flaw insensitivity at thenanoscale. For example, biomaterials such as bone and toothcontain ultrafine flaws that apparently do not affect their fracturetoughness.11 More recently, prenotched nanocrystalline metalspecimen was reported to show insensitivity to the flaw.12 These

observations suggest that stress concentration may not berelevant at the nanoscale. This conclusion seems to be supportedby fracture mechanics13 and computational14 analyses, whichassume that the material around the flaw fails not by crackpropagation, but by uniform rupture at the theoretical limitingstrength. At the failure point, the classical singular stress field isreplaced by a uniform stress distribution with no stress concen-tration near the flaw. However, the uniform rupture strength maynot necessarily be the same as the theoretical strength.12 Also, amechanism-based explanation is still missing in the literature onwhy the concept of stress concentration (or strain gradient)would cease to exist at the nanoscale.

To investigate the role of stress-concentration at the nano-scale, here we adopt a combined experimental�computationalapproach. We performed single edge notch fracture experimentsin situ inside a transmission electron microscope (TEM), whichallows us to visualize the specimen microstructures, electrondiffraction patterns and the deformation mechanisms in realtime. Using nanofabrication techniques, we have developed a 3�5 mm2 size chip that integrates freestanding aluminum speci-mens with microactuators and sensors for force and displace-ment. This is shown in Figure 1a. Details of the actuationmechanism are given elsewhere.15 The specimen is cofabricated

Received: March 31, 2011Revised: May 11, 2011

ABSTRACT: Classical fracture mechanics as well as modernstrain gradient plasticity theories assert the existence of stressconcentration (or strain gradient) ahead of a notch tip, albeitsomewhat relaxed in ductile materials. In this study, we presentexperimental evidence of extreme stress homogenization innanocrystalline metals that result in immeasurable amount ofstress concentration at a notch tip. We performed in situ uniaxialtension tests of 80 nm thick (50 nm average grain size) free-standing, single edge notched aluminum specimens inside atransmission electron microscope. The theoretical stress con-centration for the given notch geometry was as high as 8, yetelectron diffraction patterns unambiguously showed absence ofany measurable stress concentration at the notch tip. To identifypossible mechanisms behind such an anomaly, we performed molecular dynamics simulations on scaled down samples. Extensivegrain rotation driven by grain boundary diffusion, exemplified by an Ashby�Verrall type of grain switching process, was observed atthe notch tip to relieve stress concentration.We conclude that in the absence of dislocations, grain realignment or rotation may haveplayed a critical role in accommodating externally applied strain and neutralizes any stress concentration during the process.

KEYWORDS: Stress concentration, nanocrystalline metal thin film, notch tip, grain rotation, grain boundary diffusion,Ashby�Verrall model

2511 dx.doi.org/10.1021/nl201083t |Nano Lett. 2011, 11, 2510–2516

Nano Letters LETTER

with the actuators and sensors, so that the misalignment inspecimen loading is virtually nonexistent.16 Co-fabrication of thespecimen also ensures that specimen gripping is automaticallyachieved and the specimen does not experience any mishandlingprior to the experiments. Figure 1a also shows silicon microbeamat the left of the specimen, whose deflection provides a measureof the force on the specimen. The deflection of this force sensorbeam and the elongation in the specimen (measured as differencein displacements of the two ends of the specimen) are measuredby a purely mechanical displacement sensor shown in Figure 1b.Details of the force and displacement measurement are given inthe literature.16

After device fabrication, a focused ion beam is used to mill aU-shaped single edge notch on the specimen. The testing deviceis then mounted on a custom designed TEM specimen holderwith electrical biasing capability. The single edge notch tensiontests are carried out in situ inside a JEOL 2010 LaB6 TEM at 200KV of accelerating voltage. The notched section of the specimenis kept under the TEM camera as the specimen is quasi-staticallyloaded by supplying current through electro-thermal actuatorbeams. For each step increase of 0.1 V in applied voltage, bright-field images, and video are recorded for deformation mechanismat the notch tip. Then selected-area electron diffraction (SAED)patterns at and away from the notched section is recordedtogether with the motion of strain sensor. Figure 2 shows a

bright-field image of the notched section of a specimen and thecorresponding electron diffraction pattern.

In order to visualize and quantify the strain gradient, SAEDpatterns are recorded at various positions with respect to thenotch tip with an aperture size that is large enough to allowrepeatable positioning of the TEM beam on the notch tip, yetsmall enough to cover an assembly of only a few grains. Once thediffraction data is acquired, best-fit circles were drawn for all thediffraction rings and the diameters were measured. The fit of thediameter is verified by measuring the peak-to-peak distance onintensity profile. This is shown in Figure 3 for the first ring [111],where the x-axis shows the distance (1/nm) in reciprocal spaceand y-axis shows the intensity variation in gray scale. The d-spacing values for the rings are then compared for differentlocations from the notch tip and also with the unstrained values.This technique allows direct measurement of the strain in theassembly of grains with 0.5% strain resolution. The strain valuescalculated from the SAED are further corroborated from thevalues calculated from the strain sensor built in with the testingchip. As a secondary measurement, we overlaid both the SAEDpatterns (at and away from the notch) over each other andconfirmed the absence of any measurable strain gradient.

Single edge notched tension test experiments were performedin situ inside a TEM on 99.99% pure evaporated aluminumspecimens that are about 100 μm long, 3.5�5 μm wide, and80�125 nm thick. The average grain size is found to be 50 nm.U-shaped notches (with length approximately 15�30% of thespecimenwidth and radius of 50 nm)weremilled on themidspanof the specimens using focused ion beam. The notch geometry isdesigned to produce an effective stress at the notched section15�20% higher than that in the far field. With the notch lengthbetween 400 and 1000 nm and root radius of 50�100 nm, stressconcentration is expected to be around 5�8.1 The stress�straindata for the specimen (both notched and un-notched) showedbrittle-like behavior with the fracture strain close to about 2%.This is also confirmed by the absence of dislocation activitiesduring our in situ observation of the deformation in the TEM.From the specimen force and displacement data, the fracturestress was calculated to be about 450�480 MPa, which impliesthat for an effective stress concentration factor of 5�8, the notchtip stress should be approximately 2.25�4 GPa. Such high-stressconcentration is expected to result in very significant straingradient around the notch tip, which should be clearly identifiedand quantified by the strainmeasurement technique using SAED.

Figure 1. (a) Nanofabricated device with integrated actuator, sensors, and freestanding specimen for in situ testing inside a TEM. (b) Zoomed view ofthe freestanding specimen.

Figure 2. TEM bright-field micrograph of an 80 nm thick aluminumspecimenwith a focused ion beammilled notch. Inset shows the electrondiffraction pattern.


Nano Letters LETTER

But diffraction patterns recorded in this study did not showpresence of any large strain gradient as shown in Figure 3. At 2%strain for one of the specimens, precise measurement of SAEDpattern suggests that strain experienced at and away (about20 μm) from the notch tip is the same and equal to the far-fieldstrain measured by strain sensor in the testing chip. It isimportant to note that most of the specimens did not fractureat the notch tip despite the high theoretical stress concentrationexpected there. Rather, these specimens fractured far away fromthe notch and at random locations. None of the fracture locationswere near the specimen grips, which suggest that the specimenloading was not nonuniform and edge effects did not play anyrole in the specimen failure.

Our repeated observation of the suppression of strain gradientat the notch tip suggests that a unique deformation mode fornanocrystalline metals may have resulted in extreme homogeni-zation of the strain field across grains under different stresses.Evidently, such deformation mode needs to be fundamentallydifferent from the conventional dislocation based macro- andmicroscale plasticity, where statically stored dislocations in thegrain interiors allow different grains to accommodate differentvalues of strains with strain gradients across the GBs accommo-dated by geometrically necessary dislocations.

These experimental observations have spurred us to investi-gate the fundamental mechanisms behind such anomalousdeformation and fracture through molecular dynamics (MD)simulations. To mimic the experiments, we considered a poly-crystalline aluminum thin film sample with length, width, andthickness equal to 85, 60, and 5 nm, respectively. The sampleconsists of 200 grains and 1.45 million atoms with a mean grainsize of 5 nm. AU-shaped edge notch is created with a tip radius of5 nm and a length of 22 nm. Thus, all characteristic sizes of thesimulated samples are roughly proportionally scaled down fromthe specimen used in experiments. The out-of-plane orientationsof all grains are close to [111], while the in-plane orientations arerandom. To generate realistic GBs, we equilibrated the entiresystem at room temperature for 200 ps at the initial stage of

simulations. After equilibration, the sample was stretched in thelength direction with a constant engineering strain rate of 108 s�1

via the stepwise straining method.17 Such loading is accom-plished by applying a strain increment of 0.01% every 1 ps. Atomsat both ends in the length direction of the sample were fixed,while all other surfaces were kept free. During simulations, weused the embedded-atom-method potential18 to calculate intera-tomic forces, and kept constant temperature via the Nose�Ho-over thermostat.19 A multiple-time step algorithm21 was adoptedwith short and long time steps of 1 and 3 fs, respectively. Toidentify/visualize defects during plastic deformation, we coloredface-centered cubic (fcc) atoms in gray, hexagonal close-packed(hcp) atoms in red, atoms in dislocation cores and GBs in green,atoms near vacancies in blue, and fully disordered atoms inyellow. The rotation of each grain is measured by trackingchanges in the orientation of specified marker lines within graininterior. Such marker lines are composed of two layers of atomsin the (110) plane, having a length of 3.2 nm in the (111) plane,and spanning over the sample thickness with midpoints locatedat the center of each grain.

Because the TEM image acquisition is only 32 framesper second, we carried out the tensile testing experimentsquasi-statically in order to track the deformation mechanism atthe notch tip. We also carried out experiments with dynamicloading (fatigue) for 200 nm thick Al films with similar grainstructure at effectively 2�20Hz at low to very high stress levels.20

While the strain rate is not very high, the thicker specimens alsoshowed extensive grain rotation at the notch tip as well as absenceof any apparent stress concentration. The inherently low fre-quency of thermal actuation in our current experimental setupdoes not allow us to study notch tip behavior at higher strainrates, which we complement with the MD analysis. Figure 4ashow the simulated average tensile stresses over two selectedcross sections indicated in Figure 4b under different appliedstrains. When the sample was in the elastic regime, the stress levelnear the notch tip is much higher than that away from the tip,implying significant stress concentration at the tip. As the applied

Figure 3. Diffraction pattern and intensity profile for diameter just before fracture at ∼2% far field strain (a) away from notch tip/section and (b) atnotch tip.


Nano Letters LETTER

strain increases, the stress concentration near the tip was graduallyreduced. The stress distribution in front of the notch tip becomesnearly uniform while the sample plastically deformed up to10.52% strain, as shown by the distribution of atomic tensilestress over the midlongitudinal section plane shown in Figure 4c,d. Therefore, MD simulation shows that the stress concentrationis dramatically reduced even under very high strain rate.

What is the underlying deformation mechanism that leads tosuch drastic reduction of stress concentration at the notch tip?To address this question, we noted significant grain rotationactivities near the notch tip, while dislocations in grain interiorwere rare, as evidenced by atomic configurations shown inFigure 4e,f. As the applied strain increased, a few GBs vanisheddue to grain rotation. Figure 4g,h shows the distribution of grainrotation angles at 5.13 and 10.52% strains, respectively. Grain rota-tions up to 20� were observed near the notch tip as the effectivestain reached 10.52%. According to previous theoretical,22,23

computational,24 and experimental25,26 studies, grain rotation isusually assisted by GB diffusion, especially when the grain sizefalls below 10 nm. Therefore, our study suggests that in theabsence of statistically significant dislocation activities in ultrafinegrains, conventional plasticity mechanisms that lead to stress orstrain gradients are replaced by rotation of the grains assisted byalternative mechanisms such as GB diffusion or geometricallynecessary dislocations. The direction and magnitude of suchgrain rotation is governed by the externally applied deformationand the magnitude of crystallographic orientation mismatch inneighboring grains. The grains ahead of a flaw therefore experi-ence heterogeneous distribution of grain rotation to produce ahomogeneous (gradient free) strain field. The net effect is thesuppression of stress or strain gradient across the flaws andelsewhere in the specimen.

Interestingly, we observed from MD simulations a specificgrain switching process at the notch tip, which is very reminiscent

Figure 4. MD simulations of a notched nanocrystalline Al sample directly scaled down from that used in experiments. (a) Average tensile stresses on thecross sections, which are illustrated in (b), are plotted at the different strains. Atomic tensile stress (unit, GPa) distributions on the midlongitudinalsection plane of the sample under (c) 5.13% and (d) 10.52% strains. Atomic configurations of the simulated sample under (e) 5.13% and (f) 10.52%strains. Here we removed all fcc atoms and atoms on front and rear free surfaces for clarity. Grain rotation (unit, degree) mappings on themidlongitudinal section plane of the sample under (g) 5.13% and (h) 10.52% strains. The scale bar is 10 nm.


Nano Letters LETTER

of the Ashby�Verrall model27 of superplastic deformation inpolycrystalline materials. Figure 5 shows the evolution of anassembly of four grains circled in Figure 4g,h. It is observed that,as lateral strain increased from 0 to 16.18%, two vertically alignedgrains A and C approached each other while two horizontallyaligned grains B and D drifted away from each other. In theAshby�Verrall model, the deformation is accomplished notthrough dislocation activities, but through configurationalswitching of an assembly of grains via GB sliding and GBdiffusion. In the process, the shape of each grain remains thesame before and after switching. In contrast, it is observed inFigure 5 that the four grains changed their shapes significantlyduring deformation. This suggests that both GB migrationand grain rotation are important in the present switching pro-cess. Although the Ashby�Verrall model has been related topossible superplastic deformation in nanocrystalline metals,28,29

to the best of authors’ knowledge this is the first time suchdeformation mode is reported in MD simulations, particularlyat a notch tip. Previous investigations22�26 suggested that GBmigration and grain rotation are essentially assisted by GBdiffusion. Still, it is somewhat surprising that such switchingprocess occurred under such extreme strain rates. Undoubtedly,grain size may have played an important role in the observedswitching process.

The well-known scaling law of Coble creep30 provides a rela-tionship between the creep strain rate ε

·and the grain size d,

_ε ¼ RσΩδDgb

d3kBT

where σ is the stress, Ω is the atomic volume, δ is the grainboundary width, Dgb is the grain boundary diffusivity, kB is theBoltzmann constant, and T is the absolute temperature; R is aconstant related to the grain geometry. On the basis of the aboveequation, if the grain size is reduced by 1 order of magnitude, thestrain rate due to GB diffusion would be enhanced by 3 orders ofmagnitude. The very large stresses and very small grain sizes in ananocrystalline metal could therefore greatly facilitate GB diffu-sion and promote GB sliding, GB migration, and grain rotation.The above observations indicate that GB-mediated mechanismsand grain rotation dominated the plastic deformation in oursimulation samples. In particular, the Ashby�Verrall type grainswitching process may have been activated near the notch tip as apowerful deformation mechanism to relieve the local stressconcentration.

To compare the simulation results with experimental results,we carefully examined the bright-field TEM images and videocaptured in real time. Discrete dislocation activities were ob-served only in larger grains. Elsewhere, presence, regeneration, ormotions of statistically significant dislocations were not observedeven under quasi-static loading. The most prevalent event in thecaptured videos is contrast changes in individual grains indicatingout of plane rotation of the grains, which appeared to beextremely sensitive to the applied strain. The rapid change ingrain contrast makes it very difficult to locate the same grainunder different values of the applied strain. The SAED patternswere recorded for each step increase in specimen loading andwere therefore reliable for in-plane rotation measurement only.Since SAED patterns do not rotate due to any change in objectivelens excitation, change in inclination will give the actual rotationof grain. This technique is extremely accurate as long asindividual grain can be isolated using one of the selected areaapertures. For a large grain (∼100 nm) at the notched section, in-plane rotation is calculated to be about 1.2� as shown in Figure 6.Even though out of plane rotation cannot bemeasured, change inintensity of SAED spots clearly suggest significant out of planerotation. Tracking individual grains for their rotation is possible

Figure 5. Sequences of snapshot atomic configurations of four grainsnear the notch tip at different strains. These snapshots capture the grainswitching process accommodated by GB diffusion.

Figure 6. SAED patterns at (a) 0% and (b) 0.35% strains, showing extensive grain rotation.


Nano Letters LETTER

using selected area diffraction, provided that the grain size is large(>200 nm) enough. We have used the nanobeam diffraction(NBD) but the technique is not effective for the average grainsize in this study. Our diffraction-based measurements made on>100 nm sized grains showed significant rotation and the scalinglaw of GB diffusion would suggest that the rotation would beeven more significant for smaller grains. This is supported by ourMD simulation results.

The above results are suggesting a mechanism-based explana-tion for the suppression of stress or strain gradient in nanocrystal-line materials. In the case of bulk deformation, dislocation-baseddeformation mechanisms dominate. If there is stress concentra-tion due to any stress raiser, geometrically necessary dislocations(GNDs) can be introduced to facilitate plastic deformation andto help reduce stress while retaining the basic features of straingradient and stress concentration. At the nanoscale, dislocation-based mechanisms are suppressed and the stress level is veryhigh, leading to the prevalence of grain rotation as a dominantdeformation mechanism replacing the conventional dislocation-based plasticity. Here, the grains rotate both in- and out-of-planeto reduce the heterogeneity of the crystallographic planes, whichresults in strain gradient suppression. This is illustrated in Figure 7.An important corollary of the hypothesis is that it is no longernecessary for the material to reach or exceed the theoreticallimiting strength to exhibit flaw insensitivity. This is because the

strain homogenization through grain rotation occurs at all stressvalues in direct proportion with the applied stress level.

Similar flaw tolerance has been recently reported in hierarch-ical protein networks with structural defects.31,32 A series ofmolecular mechanisms, such as protein folding, alpha-to-betatransition and sliding, operate to reduce the stress concentrationnear the defects, leading to a large extensibility of the proteinnetworks.31,32 The analogy between these studies on biologicalmaterials and the present investigations on crystalline materialssuggests that nanoscale mechanisms could lead to flaw-tolerantbehaviors at much larger dimension of structures that wasconventionally believed.

In conclusion, we performed in situ TEM fracture experimentson 80 nm thick (average grain size 50 nm) freestandingaluminum films with prefabricated deep U notches. Brittle-likebehavior was observed in the specimens, which is consistent withthe observed absence of dislocation-based plasticity at thislength-scale. The very high theoretical stress concentration factorsuggested large strain gradient at the notch tip, yet the electrondiffraction analysis confirmed no appreciable strain gradient. MDsimulations were performed to obtain insights for the observedstress concentration suppression. The combined experimen-tal�computational investigation suggests that grain rotationassisted by GB-mediated deformation is the primary mechanismby which nanocrystalline metal thin films deform and minimizestrain anisotropy and gradient in the deformation field. Since thedirection and magnitude of grain rotation is governed by theexternally applied deformation and the magnitude of crystal-lographic orientation mismatch in neighboring grains, the grainsahead of a flaw experience heterogeneous distribution of grainrotation to produce a homogeneous (gradient free) strain field.The net effect is the suppression of stress or strain gradient acrossthe flaws and elsewhere in the specimen, which manifests in thephenomenon of flaw insensitivity.

’AUTHOR INFORMATION

Corresponding Author*E-mail: (A.H.) [email protected]; (H.G.) [email protected].

’ACKNOWLEDGMENT

A.H gratefully acknowledges the support from the Center forNanoscale Mechatronics &Manufacturing of the Korea Instituteof Machinery & Materials and the National Science Foundation,U.S.A. (CMMI No. 1029935). X.L. and H.G. acknowledgefinancial support by the NSF through the MRSEC Program(award number DMR-0520651) and Grant CMMI-0758535 toBrown University. The simulations reported were performed onNSF TeraGrid resources provided by NICS under MSS090046.

’REFERENCES

(1) Barrata, F. I.; Neal, D. M. J. Strain Anal. Eng. Des. 1970,5, 121–127.

(2) Schiotz, J.; Di Tolla, F. D.; Jacobsen, K. W. Nature 1998,391, 561–563.

(3) Meyers, M. A.; Mishra, A.; Benson, D. J. Prog. Mater. Sci. 2006,51, 427–556.

(4) Wolf, D.; Yamakov, V.; Phillpot, S. R.; Mukherjee, A. K. Z.Metallkd. 2003, 94, 1091–1097.

(5) Pande, C. S.; Cooper, K. P. Prog. Mater. Sci. 2009, 54, 689–706.

Figure 7. Schematic illustration of deformation mechanisms in sampleswith different grain sizes. (a) Dislocation-based plasticity allows straingradient at or near the notch tip. Orange and red points representdislocation nucleation and pile-up sites, respectively. (b) As the grainsize reduces to nanoscale, grain rotation and GB-mediated processesreplace dislocation plasticity, thereby removing any strain inhomogene-ity and the accompanying stress concentration. Pink, cyan, and brownarrows represent grain rotation, GB sliding, and GB diffusion,respectively.


Nano Letters LETTER

(6) Moldovan, D.; Wolf, D.; Phillpot, S. R.; Mukherjee, A. K.;Gleiter, H. Philos. Mag. Lett. 2003, 83, 29–38.(7) Watanabe, T. Metal. Mater. Trans. A 1983, 14, 531–545.(8) Voyiadjis, G. Z.; Al-Rub, R. K. A. Int. J. Solids Struct. 2005,

42, 3998–4029.(9) Haque, M. A.; Saif, M. T. A. Proc. Natl. Acad. Sci. U.S.A. 2004,

101, 6335–6340.(10) Nix, W. D.; Greer, J. R.; Feng, G.; Lilleodden, E. T. Thin Solid

Films 2007, 515, 3152–3157.(11) Ji, B.; Gao, H. Annual Rev. Mater. Res. 2010, 40, 77–100.(12) Kumar, S.; Haque, M. A.; Gao, H. Appl. Phys. Lett. 2009,

94, 253104.(13) Gao, H. Int. J. Fract. 2006, 138, 101–137.(14) Buehler, M. J.; Yao, H.; Ji, B.; Gao, H.Model. Simul. Mater. Sci.

Eng. 2006, 14, 799–816.(15) Zhu, Y.; Espinosa, H. D. Proc. Natl. Acad. Sci. U.S.A. 2005,

102, 14503–14508.(16) Haque, M. A.; Saif, M. T. A. Exp. Mech. 2002, 42, 123–128.(17) Li, X.; Wei, Y.; Yang, W.; Gao, H. Proc. Natl. Acad. Sci. U.S.A.

2009, 106, 16108–16113.(18) Mishin,M.; Farkas, D.;Mehl, M. J.; Papaconstantopoulos, D. A.

Phys. Rev. B 1999, 59, 3393–3407.(19) Nose, S. A. J. Chem. Phys. 1984, 81, 511–519.(20) Kumar, S.; Alam, M. T.; Haque, M. A. J. Microelectromech. Syst.

2010, 20, 53–58.(21) Tuckerman,M.; Berne, B. J.; Mantyna, G. J. J. Chem. Phys. 1992,

97, 1990–2001.(22) Harris, K. E.; Singh, V. V.; King, A. H. Acta Mater. 1998,

46, 2623–2633.(23) Moldovan, D.; Wolf, D.; Phillpot, S. R. Acta Mater. 2001,

49, 3521–3532.(24) Upmanyu, M.; Srolovitz, D. J; Lobkovsky, A. E.; Warren, J. A.;

Carter, W. C. Acta Mater. 2006, 54, 1707–1719.(25) Ke, M.; Hackney, S. A.; Milligan, W. W.; Aifantis, E. C.

Nanostruct. Mater. 1995, 689–697.(26) Shan, Z.; Stach, E. A.; Wiezorek, J. M. K.; Knapp, J. A.;

Follstaedt, D. M.; Mao, S. X. Science 2004, 305, 654–657.(27) Ashby, M. F.; Verrall, R. A. Acta Metall. 1973, 21, 149–163.(28) Yang, W.; Wang, H. J. Mech. Phys. Solids 2004, 52, 875–889.(29) Wang, H.; Yang, W. J. Mech. Phys. Solids 2004, 52, 1151–1173.(30) Coble, R. L. J. Appl. Phys. 1963, 24, 1679–1682.(31) Ackbarow, T.; Sen, D.; Thaulow, C.; Buehler, M. J. PLoS One

2009, 4, e6015.(32) Qin, Z.; Buehler, M. J. ACS Nano 2011, 5, 3034–3042.

Is Stress Concentration Relevant for Nanocrystalline Metals?

Documents

Transcript of Is Stress Concentration Relevant for Nanocrystalline Metals?