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 ABSTRACT: Classical fracture mechanics as well as modern strain gradient plasticity theories assert the existence of stress concentration (or strain gradient) ahead of a notch tip, albeit somewhat relaxed in ductile materials. In this study, we present experimental evidence of extreme stress homogenization in nanocrystalline metals that result in immeasurable amount of stress concentration at a notch tip. We performed in situ uniaxial tension tests of 80 nm thick (50 nm average grain size) freestanding, single edge notched aluminum specimens inside a transmission electron microscope. The theoretical stress concentration for the given notch geometry was as high as 8, yet electron diffraction patterns unambiguously showed absence of any measurable stress concentration at the notch tip. To identify possible mechanisms behind such an anomaly, we performed molecular dynamics simulations on scaled down samples. Extensive grain rotation driven by grain boundary diffusion, exemplified by an AshbyVerrall type of grain switching process, was observed at the notch tip to relieve stress concentration. We conclude that in the absence of dislocations, grain realignment or rotation may have played 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, AshbyVerrall model
he existence of flaws, such as cracks or notches, in a load bearing solid is known to increase the magnitude of stresses in the immediate vicinity of a notch tip. Such stress concentration or gradient depends on the notch and specimen geometry (notch length, tip radius, specimen width) and can increase stress by as high as a factor of 10 for a U-shaped notch.1 Plastic flow mechanisms significantly reduce the stress concentration near a notch tip in ductile materials. For brittle materials, however, no such relief mechanisms exist and the concept of stress concentration is thought to hold up to the fracture strength. For metallic materials, the flow stress usually increases with decreasing grain size,2,3 which should increase the stress concentration according to conventional wisdom as well as strain gradient plasticity theories. Furthermore, Coble creep4 and grain boundary (GB) sliding5 are also known to develop stress concentrations at GBs and triple junctions.6,7 Therefore, the significance of stress concentration can be expected to become more prominent at the nanoscale, where metals exhibit brittle-like behavior due to confinement, scarcity, or starvation of dislocations.810 In this paper, we provide experimental evidence that questions the validity of the scaling down of stress concentration at the nanoscale. This is spurred by the indications in the literature that even brittle materials may exhibit flaw insensitivity at the nanoscale. For example, biomaterials such as bone and tooth contain ultrafine flaws that apparently do not affect their fracture toughness.11 More recently, prenotched nanocrystalline metal specimen was reported to show insensitivity to the flaw.12 These
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observations suggest that stress concentration may not be relevant at the nanoscale. This conclusion seems to be supported by fracture mechanics13 and computational14 analyses, which assume that the material around the flaw fails not by crack propagation, but by uniform rupture at the theoretical limiting strength. At the failure point, the classical singular stress field is replaced by a uniform stress distribution with no stress concentration near the flaw. However, the uniform rupture strength may not necessarily be the same as the theoretical strength.12 Also, a mechanism-based explanation is still missing in the literature on why the concept of stress concentration (or strain gradient) would cease to exist at the nanoscale. To investigate the role of stress-concentration at the nanoscale, here we adopt a combined experimentalcomputational approach. We performed single edge notch fracture experiments in situ inside a transmission electron microscope (TEM), which allows us to visualize the specimen microstructures, electron diffraction patterns and the deformation mechanisms in real time. Using nanofabrication techniques, we have developed a 3 5 mm2 size chip that integrates freestanding aluminum specimens with microactuators and sensors for force and displacement. This is shown in Figure 1a. Details of the actuation mechanism are given elsewhere.15 The specimen is cofabricated Received: March 31, 2011 Revised: May 11, 2011 Published: May 18, 2011 2510
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Figure 1. (a) Nanofabricated device with integrated actuator, sensors, and freestanding specimen for in situ testing inside a TEM. (b) Zoomed view of the freestanding specimen.
Figure 2. TEM bright-field micrograph of an 80 nm thick aluminum specimen with a focused ion beam milled notch. Inset shows the electron diffraction pattern.
with the actuators and sensors, so that the misalignment in specimen loading is virtually nonexistent.16 Co-fabrication of the specimen also ensures that specimen gripping is automatically achieved and the specimen does not experience any mishandling prior to the experiments. Figure 1a also shows silicon microbeam at the left of the specimen, whose deflection provides a measure of the force on the specimen. The deflection of this force sensor beam and the elongation in the specimen (measured as difference in displacements of the two ends of the specimen) are measured by a purely mechanical displacement sensor shown in Figure 1b. Details of the force and displacement measurement are given in the literature.16 After device fabrication, a focused ion beam is used to mill a U-shaped single edge notch on the specimen. The testing device is then mounted on a custom designed TEM specimen holder with electrical biasing capability. The single edge notch tension tests are carried out in situ inside a JEOL 2010 LaB6 TEM at 200 KV of accelerating voltage. The notched section of the specimen is kept under the TEM camera as the specimen is quasi-statically loaded by supplying current through electro-thermal actuator beams. For each step increase of 0.1 V in applied voltage, brightfield images, and video are recorded for deformation mechanism at the notch tip. Then selected-area electron diffraction (SAED) patterns at and away from the notched section is recorded together with the motion of strain sensor. Figure 2 shows a
bright-field image of the notched section of a specimen and the corresponding electron diffraction pattern. In order to visualize and quantify the strain gradient, SAED patterns are recorded at various positions with respect to the notch tip with an aperture size that is large enough to allow repeatable positioning of the TEM beam on the notch tip, yet small enough to cover an assembly of only a few grains. Once the diffraction data is acquired, best-fit circles were drawn for all the diffraction rings and the diameters were measured. The fit of the diameter is verified by measuring the peak-to-peak distance on intensity profile. This is shown in Figure 3 for the first ring [111], where the x-axis shows the distance (1/nm) in reciprocal space and y-axis shows the intensity variation in gray scale. The dspacing values for the rings are then compared for different locations from the notch tip and also with the unstrained values. This technique allows direct measurement of the strain in the assembly of grains with 0.5% strain resolution. The strain values calculated from the SAED are further corroborated from the values calculated from the strain sensor built in with the testing chip. As a secondary measurement, we overlaid both the SAED patterns (at and away from the notch) over each other and confirmed the absence of any measurable strain gradient. Single edge notched tension test experiments were performed in situ inside a TEM on 99.99% pure evaporated aluminum specimens that are about 100 μm long, 3.55 μm wide, and 80125 nm thick. The average grain size is found to be 50 nm. U-shaped notches (with length approximately 1530% of the specimen width and radius of 50 nm) were milled on the midspan of the specimens using focused ion beam. The notch geometry is designed to produce an effective stress at the notched section 1520% higher than that in the far field. With the notch length between 400 and 1000 nm and root radius of 50100 nm, stress concentration is expected to be around 58.1 The stressstrain data for the specimen (both notched and un-notched) showed brittle-like behavior with the fracture strain close to about 2%. This is also confirmed by the absence of dislocation activities during our in situ observation of the deformation in the TEM. From the specimen force and displacement data, the fracture stress was calculated to be about 450480 MPa, which implies that for an effective stress concentration factor of 58, the notch tip stress should be approximately 2.254 GPa. Such high-stress concentration is expected to result in very significant strain gradient around the notch tip, which should be clearly identified and quantified by the strain measurement technique using SAED. 2511
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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) at notch tip.
But diffraction patterns recorded in this study did not show presence of any large strain gradient as shown in Figure 3. At 2% strain for one of the specimens, precise measurement of SAED pattern suggests that strain experienced at and away (about 20 μm) from the notch tip is the same and equal to the far-field strain measured by strain sensor in the testing chip. It is important to note that most of the specimens did not fracture at the notch tip despite the high theoretical stress concentration expected there. Rather, these specimens fractured far away from the notch and at random locations. None of the fracture locations were near the specimen grips, which suggest that the specimen loading was not nonuniform and edge effects did not play any role in the specimen failure. Our repeated observation of the suppression of strain gradient at the notch tip suggests that a unique deformation mode for nanocrystalline metals may have resulted in extreme homogenization of the strain field across grains under different stresses. Evidently, such deformation mode needs to be fundamentally different from the conventional dislocation based macro- and microscale plasticity, where statically stored dislocations in the grain interiors allow different grains to accommodate different values of strains with strain gradients across the GBs accommodated by geometrically necessary dislocations. These experimental observations have spurred us to investigate the fundamental mechanisms behind such anomalous deformation and fracture through molecular dynamics (MD) simulations. To mimic the experiments, we considered a polycrystalline aluminum thin film sample with length, width, and thickness equal to 85, 60, and 5 nm, respectively. The sample consists of 200 grains and 1.45 million atoms with a mean grain size of 5 nm. A U-shaped edge notch is created with a tip radius of 5 nm and a length of 22 nm. Thus, all characteristic sizes of the simulated samples are roughly proportionally scaled down from the specimen used in experiments. The out-of-plane orientations of all grains are close to [111], while the in-plane orientations are random. To generate realistic GBs, we equilibrated the entire system at room temperature for 200 ps at the initial stage of
simulations. After equilibration, the sample was stretched in the length direction with a constant engineering strain rate of 108 s1 via the stepwise straining method.17 Such loading is accomplished by applying a strain increment of 0.01% every 1 ps. Atoms at both ends in the length direction of the sample were fixed, while all other surfaces were kept free. During simulations, we used the embedded-atom-method potential18 to calculate interatomic forces, and kept constant temperature via the NoseHoover thermostat.19 A multiple-time step algorithm21 was adopted with short and long time steps of 1 and 3 fs, respectively. To identify/visualize defects during plastic deformation, we colored face-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 in yellow. The rotation of each grain is measured by tracking changes in the orientation of specified marker lines within grain interior. Such marker lines are composed of two layers of atoms in the (110) plane, having a length of 3.2 nm in the (111) plane, and spanning over the sample thickness with midpoints located at the center of each grain. Because the TEM image acquisition is only 32 frames per second, we carried out the tensile testing experiments quasi-statically in order to track the deformation mechanism at the notch tip. We also carried out experiments with dynamic loading (fatigue) for 200 nm thick Al films with similar grain structure at effectively 220 Hz at low to very high stress levels.20 While the strain rate is not very high, the thicker specimens also showed extensive grain rotation at the notch tip as well as absence of any apparent stress concentration. The inherently low frequency of thermal actuation in our current experimental setup does not allow us to study notch tip behavior at higher strain rates, which we complement with the MD analysis. Figure 4a show the simulated average tensile stresses over two selected cross sections indicated in Figure 4b under different applied strains. When the sample was in the elastic regime, the stress level near the notch tip is much higher than that away from the tip, implying significant stress concentration at the tip. As the applied 2512
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Figure 4. MD simulations of a notched nanocrystalline Al sample directly scaled down from that used in experiments. (a) Average tensile stresses on the cross sections, which are illustrated in (b), are plotted at the different strains. Atomic tensile stress (unit, GPa) distributions on the midlongitudinal section 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 the midlongitudinal section plane of the sample under (g) 5.13% and (h) 10.52% strains. The scale bar is 10 nm.
strain increases, the stress concentration near the tip was gradually reduced. The stress distribution in front of the notch tip becomes nearly uniform while the sample plastically deformed up to 10.52% strain, as shown by the distribution of atomic tensile stress over the midlongitudinal section plane shown in Figure 4c, d. Therefore, MD simulation shows that the stress concentration is dramatically reduced even under very high strain rate. What is the underlying deformation mechanism that leads to such drastic reduction of stress concentration at the notch tip? To address this question, we noted significant grain rotation activities near the notch tip, while dislocations in grain interior were rare, as evidenced by atomic configurations shown in Figure 4e,f. As the applied strain increased, a few GBs vanished due to grain rotation. Figure 4g,h shows the distribution of grain rotation angles at 5.13 and 10.52% strains, respectively. Grain rotations up to 20° were observed near the notch tip as the effective stain reached 10.52%. According to previous theoretical,22,23
computational,24 and experimental25,26 studies, grain rotation is usually assisted by GB diffusion, especially when the grain size falls below 10 nm. Therefore, our study suggests that in the absence of statistically significant dislocation activities in ultrafine grains, conventional plasticity mechanisms that lead to stress or strain gradients are replaced by rotation of the grains assisted by alternative mechanisms such as GB diffusion or geometrically necessary dislocations. The direction and magnitude of such grain rotation is governed by the externally applied deformation and the magnitude of crystallographic orientation mismatch in neighboring grains. The grains ahead of a flaw therefore experience heterogeneous distribution of grain rotation to produce a homogeneous (gradient free) strain field. The net effect is the suppression of stress or strain gradient across the flaws and elsewhere in the specimen. Interestingly, we observed from MD simulations a specific grain switching process at the notch tip, which is very reminiscent 2513
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Nano Letters of the AshbyVerrall model27 of superplastic deformation in polycrystalline materials. Figure 5 shows the evolution of an assembly of four grains circled in Figure 4g,h. It is observed that, as lateral strain increased from 0 to 16.18%, two vertically aligned grains A and C approached each other while two horizontally aligned grains B and D drifted away from each other. In the AshbyVerrall model, the deformation is accomplished not through dislocation activities, but through configurational switching of an assembly of grains via GB sliding and GB diffusion. In the process, the shape of each grain remains the same before and after switching. In contrast, it is observed in Figure 5 that the four grains changed their shapes significantly during deformation. This suggests that both GB migration and grain rotation are important in the present switching process. Although the AshbyVerrall model has been related to possible superplastic deformation in nanocrystalline metals,28,29 to the best of authors’ knowledge this is the first time such deformation mode is reported in MD simulations, particularly at a notch tip. Previous investigations2226 suggested that GB migration and grain rotation are essentially assisted by GB diffusion. Still, it is somewhat surprising that such switching process occurred under such extreme strain rates. Undoubtedly, grain size may have played an important role in the observed switching process.
Figure 5. Sequences of snapshot atomic configurations of four grains near the notch tip at different strains. These snapshots capture the grain switching process accommodated by GB diffusion.
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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 d3 kB T
where σ is the stress, Ω is the atomic volume, δ is the grain boundary width, Dgb is the grain boundary diffusivity, kB is the Boltzmann constant, and T is the absolute temperature; R is a constant related to the grain geometry. On the basis of the above equation, if the grain size is reduced by 1 order of magnitude, the strain rate due to GB diffusion would be enhanced by 3 orders of magnitude. The very large stresses and very small grain sizes in a nanocrystalline metal could therefore greatly facilitate GB diffusion and promote GB sliding, GB migration, and grain rotation. The above observations indicate that GB-mediated mechanisms and grain rotation dominated the plastic deformation in our simulation samples. In particular, the AshbyVerrall type grain switching process may have been activated near the notch tip as a powerful deformation mechanism to relieve the local stress concentration. To compare the simulation results with experimental results, we carefully examined the bright-field TEM images and video captured in real time. Discrete dislocation activities were observed only in larger grains. Elsewhere, presence, regeneration, or motions of statistically significant dislocations were not observed even under quasi-static loading. The most prevalent event in the captured videos is contrast changes in individual grains indicating out of plane rotation of the grains, which appeared to be extremely sensitive to the applied strain. The rapid change in grain contrast makes it very difficult to locate the same grain under different values of the applied strain. The SAED patterns were recorded for each step increase in specimen loading and were therefore reliable for in-plane rotation measurement only. Since SAED patterns do not rotate due to any change in objective lens excitation, change in inclination will give the actual rotation of grain. This technique is extremely accurate as long as individual grain can be isolated using one of the selected area apertures. For a large grain (∼100 nm) at the notched section, inplane rotation is calculated to be about 1.2° as shown in Figure 6. Even though out of plane rotation cannot be measured, change in intensity of SAED spots clearly suggest significant out of plane rotation. Tracking individual grains for their rotation is possible
Figure 6. SAED patterns at (a) 0% and (b) 0.35% strains, showing extensive grain rotation. 2514
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Figure 7. Schematic illustration of deformation mechanisms in samples with different grain sizes. (a) Dislocation-based plasticity allows strain gradient at or near the notch tip. Orange and red points represent dislocation nucleation and pile-up sites, respectively. (b) As the grain size reduces to nanoscale, grain rotation and GB-mediated processes replace dislocation plasticity, thereby removing any strain inhomogeneity and the accompanying stress concentration. Pink, cyan, and brown arrows represent grain rotation, GB sliding, and GB diffusion, respectively.
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 grain size in this study. Our diffraction-based measurements made on >100 nm sized grains showed significant rotation and the scaling law of GB diffusion would suggest that the rotation would be even more significant for smaller grains. This is supported by our MD simulation results. The above results are suggesting a mechanism-based explanation for the suppression of stress or strain gradient in nanocrystalline materials. In the case of bulk deformation, dislocation-based deformation mechanisms dominate. If there is stress concentration due to any stress raiser, geometrically necessary dislocations (GNDs) can be introduced to facilitate plastic deformation and to help reduce stress while retaining the basic features of strain gradient and stress concentration. At the nanoscale, dislocationbased mechanisms are suppressed and the stress level is very high, leading to the prevalence of grain rotation as a dominant deformation mechanism replacing the conventional dislocationbased plasticity. Here, the grains rotate both in- and out-of-plane to reduce the heterogeneity of the crystallographic planes, which results in strain gradient suppression. This is illustrated in Figure 7. An important corollary of the hypothesis is that it is no longer necessary for the material to reach or exceed the theoretical limiting strength to exhibit flaw insensitivity. This is because the
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strain homogenization through grain rotation occurs at all stress values in direct proportion with the applied stress level. Similar flaw tolerance has been recently reported in hierarchical protein networks with structural defects.31,32 A series of molecular mechanisms, such as protein folding, alpha-to-beta transition and sliding, operate to reduce the stress concentration near the defects, leading to a large extensibility of the protein networks.31,32 The analogy between these studies on biological materials and the present investigations on crystalline materials suggests that nanoscale mechanisms could lead to flaw-tolerant behaviors at much larger dimension of structures that was conventionally believed. In conclusion, we performed in situ TEM fracture experiments on 80 nm thick (average grain size 50 nm) freestanding aluminum films with prefabricated deep U notches. Brittle-like behavior was observed in the specimens, which is consistent with the observed absence of dislocation-based plasticity at this length-scale. The very high theoretical stress concentration factor suggested large strain gradient at the notch tip, yet the electron diffraction analysis confirmed no appreciable strain gradient. MD simulations were performed to obtain insights for the observed stress concentration suppression. The combined experimentalcomputational investigation suggests that grain rotation assisted by GB-mediated deformation is the primary mechanism by which nanocrystalline metal thin films deform and minimize strain anisotropy and gradient in the deformation field. Since the direction and magnitude of grain rotation is governed by the externally applied deformation and the magnitude of crystallographic orientation mismatch in neighboring grains, the grains ahead of a flaw experience heterogeneous distribution of grain rotation to produce a homogeneous (gradient free) strain field. The net effect is the suppression of stress or strain gradient across the flaws and elsewhere in the specimen, which manifests in the phenomenon of flaw insensitivity.
’ AUTHOR INFORMATION Corresponding Author
*E-mail: (A.H.)
[email protected]; (H.G.) huajian_gao @brown.edu.
’ ACKNOWLEDGMENT A.H gratefully acknowledges the support from the Center for Nanoscale Mechatronics & Manufacturing of the Korea Institute of Machinery & Materials and the National Science Foundation, U.S.A. (CMMI No. 1029935). X.L. and H.G. acknowledge financial support by the NSF through the MRSEC Program (award number DMR-0520651) and Grant CMMI-0758535 to Brown University. The simulations reported were performed on NSF 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. 2515
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