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Strain distribution across an individual shear band in real and simulated metallic glasses Sergio Scudino, and Daniel Sopu Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.7b04816 • Publication Date (Web): 16 Jan 2018 Downloaded from http://pubs.acs.org on January 16, 2018
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Strain distribution across an individual shear band in real and simulated metallic glasses
Sergio Scudino1,*, Daniel Şopu2,3
1
Institute for Complex Materials, IFW Dresden, Helmholtzstraße 20, D-01069 Dresden, Germany
2
Institute of Materials Science, Technische Universität Darmstadt, Otto-Berndt-Strasse 3, D-64287
Darmstadt, Germany 3
Erich Schmid Institute of Materials Science, Austrian Academy of Sciences, Jahnstraße 12, A-8700
Leoben, Austria
*
Corresponding author. E-mail address:
[email protected]; Tel. +49 351 4659 838
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Abstract: Due to the fast dynamics of shear band formation and propagation along with the small size and transient character of the shear transformation zones (STZs), the elementary units of plasticity in metallic glasses, the description of the nanoscale mechanism of shear banding often relies on molecular dynamics (MD) simulations. However, the unrealistic parameters used in the simulations related to time constrains may raise questions about whether quantitative comparison between results from experimental and computational analyses is possible. Here, we have experimentally analyzed the strain field arising across an individual shear band by nanobeam x-ray diffraction and compared the results with the strain characterizing a shear band generated by MD simulations. Despite their largely different spatio-temporal scales, the characteristic features of real and simulated shear bands are strikingly similar: the magnitude of the strain across the shear band is discontinuous in both cases and the direction of the principal strain axes exhibits the same antisymmetric profile. This behavior can be explained by considering the mechanism of STZ activation and percolation at the nanoscale, indicating that the nanoscale effects of shear banding are not limited to the area within the band but they extend well into the surrounding elastic matrix. These findings not only demonstrate the reliability of MD simulations for explaining (also quantitatively) experimental observations of shear banding, but also suggest that designed experiments can be used the other way around to verify numerical predictions of the atomic rearrangements occurring within a band.
Keywords: Metallic glasses, shear band formation, x-ray diffraction, molecular dynamics simulation
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Room temperature plastic deformation in bulk metallic glasses (BMGs) is accommodated by highly-localized shear bands.1,2 Due to the autocatalytic strain softening within a shear band,2,3 only a few bands are formed in these materials under tensile loading, leading to premature fracture and effectively zero macroscopic tensile plasticity;3 a limitation that precludes their extensive use in engineering applications despite the high strength and large elastic limit. Early failure of metallic glasses under tension can be delayed by preventing plastic deformation along a single shear band and significant efforts have been made in recent years to improve the tensile ductility by inducing the formation of multiple shear bands and thus through a more homogeneous distribution of the strain.4-7 The proper control of shear band multiplication, however, requires the knowledge of the local atomistic mechanism governing shear band generation and propagation. Due to the combination of fast dynamics and highlylocalized character of shear banding,1 investigations of the atomic-level mechanism of shear band formation and propagation are often based on MD simulations. MD simulations can reproduce the liquid-like amorphous structure and the characteristic features of deformation in metallic glasses, providing the analytical and visualization tools to investigate the nanoscale phenomenon of shear banding.8,9 These studies have resulted in quite a number of important observations regarding the formation of shear bands in metallic glasses, including strain localization and cavitation within the shear bands,10 the variation of the arrangements of atoms and their spatial topology inside and outside the bands,11 the structural variations occurring upon cyclic loading12 and the atomic-level mechanism underlying the STZ percolation process.13 Considerable progress in the understanding of the main features characterizing shear bands and the surrounding (nominally) undeformed material (hereafter named elastic matrix) has been achieved also experimentally owing to the development of analytical tools with enhanced resolution. For example, calorimetry and nanoindentation measurements have 3 ACS Paragon Plus Environment
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revealed that significant softening and accumulation of free volume along with the reduction of the Young´s modulus occur in the matrix adjacent to the shear bands,14,15 whereas transmission electron microscopy investigations have shown that small deflections and large density variations occur within the shear bands,16 a behavior compatible with the alignment of Eshelby-like quadrupoles.17 Strain analysis by high-energy x-ray diffraction (XRD), originally applied to metallic glasses by Poulsen et al.,18 is among the optimal instruments to experimentally evaluate the atomic rearrangements occurring in BMGs.19-21 Although in recent years the size of the available x-ray beams has been continuously reducing down to the sub-micron range,22,23 the characteristic shear band thickness (~10-20 nm1) is still well-below the resolution limit of the state-of-the-art x-ray measurement tools, which thus precludes the analysis of the structural changes occurring exclusively within a shear band. The use of an XRD beam with size much smaller than the shear band spacing, however, permits to accurately study the strain characterizing the surrounding elastic matrix. For example, recent strain analysis performed by high-energy XRD have shown that shear banding induces significant shear strain (a fundamental parameter for the irreversible activation of STZs24) in the elastic matrix at distances of more than 15 µm from a shear band.22 Despite the recent research progress in the fields of experimental and computational characterization, only seldom experiments and simulations are combined to give a comprehensive description of the phenomena occurring in metallic glasses.25,26 MD simulations are an excellent tool to examine the process of shear banding at length- and timescales well below the resolution of the current experimental measurements; however, the stringent simulation parameters regarding cooling rate (~1010 K/s), sample size (~100 nm), simulation time (in the nanosecond range) and strain rate (~107 1/s)8 may raise questions about their reliability as mechanistic support to understand the features experimentally observed. Here, we combine experimental and computational analytical tools in order to study the strain generated across an individual shear band at different length scales and to clarify 4 ACS Paragon Plus Environment
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whether quantitative comparison between results from experiments and simulations are possible despite their largely different spatio-temporal scales. The spatial variation of the strain was experimentally investigated by nanobeam XRD, which allows to examine the recoverable elastic strain characterizing the elastic matrix at the µm-level, whereas MD simulations were used to investigate the strain distribution within and around the shear band at the nanoscale. Experimental strain analysis was performed on glassy samples with nominal composition Zr52.5Ti5Cu18Ni14.5Al10 (at.%) cold rolled at room temperature to a thickness reduction of about 5 %. Small reductions ensure the shear band spacing to be much larger than the beam size.27 Samples were cut along the cross-section of the samples and carefully polished to give a uniform thickness of 100 µm. The structure of the thin slices was investigated by XRD in transmission using a high-intensity high-energy monochromatic synchrotron beam (E = 63.32 keV) at the ID11 beamline of the European Synchrotron Radiation Facility (ESRF). The schematic representation of the XRD setup and the coordinate system used are shown in Figure 1a. The orientation of the sample was selected in order to have the X and Y axes respectively parallel and perpendicular to the shear band direction. Diffraction patterns were collected every 1.0 µm along the X-axis (i.e. parallel to the shear band) and every 0.5 µm along the Y-axis (i.e. perpendicular to the shear band) using a beam with size of 5x×0.15y µm2. The accurate position on the sample was ensured by the use of computer-controlled positioners with sub-micron precision encoders. The XRD patterns were collected using a twodimensional charge coupled device (CCD) Frelon camera.28 The two-dimensional patterns were integrated in 10° azimuthal slices using the Fit2D program29 to give the XRD intensity distributions I(q,φ) as a function of the scattering vector q and the integration azimuthal angle
φ. The position of the first diffuse maximum q1 (Figure 1b), which is a reliable indicator to evaluate the structural changes occurring in the medium-range order,30 was determined by 5 ACS Paragon Plus Environment
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fitting using a pseudo-Voigt function. The strain ε was measured through the shift of q1 with 0 respect of the reference value q1 as
q10 (φ , x, y) − q1 (φ , x, y) ε (φ , x, y) = q1 (φ , x, y)
,
(1)
which depends on the azimuthal angle and on the position (x,y) scanned on the sample. In order to minimize possible errors introduced by the uncertainty of the sample-to-detector 0 distance, q1 was selected within the investigated sample as suggested by Shakur Shahabi et
al.,22 by choosing a position far enough from the shear bands and, thus, less affected by shear banding. The three components of the strain tensor (εxx, εyy and εxy) for each point scanned on the samples were determined according to the method described in Poulsen et al..18 The error over the measurements presented later in this work (see Figures 2(c) and 2(d)) is related to the uncertainty of the fitting steps.
Figure 1. (a) Schematic illustration of the setup and coordinate system used in the XRD measurements and (b) representative XRD pattern showing the position of the main scattering peak q1 used for the strain evaluation. (c) Notched Cu64Zr36 sample and coordinate system used in the MD simulations. Note that, due to the different experimental setups, the coordinate system in (c) is rotated by 45° with respect to the reference frame in (a).
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The atomic-level mechanism of shear banding was analyzed by molecular dynamics (MD) simulations in a model Cu64Zr36 metallic glass using the code LAMMPS.31 No appropriate interatomic potentials exist for the Zr52.5Ti5Cu18Ni14.5Al10 metallic glass; the Cu64Zr36 composition was chosen because reliable Finnis-Sinclair type potential are available.32 Moreover, the Cu64Zr36 glass is characterized by a high fraction of fullicosahedral clusters,33 a requirement that guarantees strain localization. The simulated specimen was a rectangular sample with dimensions 46×5×106 nm3 containing about 1.5×106 atoms (Figure 1c) generated by replicating a unit glass cubic cell with 8000 atoms produced by quenching it from the melt to 50 K at a cooling rate of 1010 K/s. A small notch was created in the specimen in order to ensure strain localization and the formation of an individual shear band. The notched sample was loaded under uniaxial tension along the Y´-axis at 50 K using a strain rate of 4×107 1/s. Periodic boundary conditions were applied along the Y´ and Z´ directions, while free surface condition was used along the X´-axis. The parameters employed in this work are within the range typically used in MD simulations.8,9,34 The present results, therefore, represent properly the characteristic MD simulations usually used to describe the formation of shear bands in metallic glasses. The atomic Green-Lagrangian strain tensor was calculated and visualized using the OVITO software. This micromechanical quantity is calculated at each atomic site from the relative motion of the neighboring atoms within a cutoff range of 3.5 Å as follows. The atomic deformation gradient F is first computed using a least-squares fit to the atomic displacement vectors35 (with the atomic configuration at zero load being the reference configuration) and the Green-Lagrangian strain tensor is then calculated from F. Two stages of shear band propagation were investigated: (i) inserting shear band (i.e. shear band already initiated but not yet transecting the entire sample)36 and (ii) mature shear band cutting the entire specimen. The stage of the inserting shear band was selected where the band comes to an end approximately in the middle of the sample. The position on the simulated stress-strain curve corresponding to the inserting and mature shear 7 ACS Paragon Plus Environment
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bands are given in Figure S1 of the Supporting Information. The results plotted along the Y´axis displayed later in this work (see Figures 3 and 4) are average values calculated within volumes with size of about 6x×0.5y×1z nm3 with error calculated over 100 atoms. The variation of the average values for different cutoffs, test temperatures and positions across the inserting shear band are given in Figures S2-S4 of the Supporting Information.
Figure 2. Spatially-resolved maps obtained by XRD of the eigenvalues λ (color maps), indicating the magnitude of the principal strains, and of the eigenvectors η (unit-length vectors), representing the principal strain directions: (a) λ1,η1 and (b) λ2,η2. (c),(d) Variation along the Y-axis of the eigenvalues
λ1 and λ2 and of the angle formed by the eigenvectors η1 and η2 with the X-axis in (a),(b). The white dashed lines in (a),(b) mark the position of the shear band.
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For x-ray beam sizes larger than the typical shear band spacing, no distinction can be made between elastic and plastic strain, as the structural information would come from both shear band and surrounding elastic matrix. Here, the combination of large shear band spacing (50100 µm)27 and the extremely small x-ray beam size (150 nm) allows us to evaluate the strain distribution in the elastic matrix. The magnitude of the components of the strain tensor depends on the coordinate system chosen, which, in the present work, is different in the XRD experiments and MD simulations (Figure 1). In order to avoid the dependence of the selected reference frame, for each point scanned we determined the variation of the eigenvectors (η1 and η2) of the strain tensor, which give the (orthogonal) directions of the principal strain axes along which there are no shear strains, and their magnitude, the eigenvalues (λ1 and λ2). The resulting data are shown in Figures 2a and 2b, where the color maps indicate the magnitude of the strain (tensile for λ1 and compressive for λ2) and the unit-length vectors represent the principal strain directions, whereas Figures 2c and 2d display the variation along the Y-axis of the eigenvalues λ1 and λ2 and of the angle formed by the eigenvectors η1 and η2 with the Xaxis. The strain distribution across the shear band displays a local minimum for both λ1 and λ2 at positions corresponding to the shear band (marked by the white dashed lines in Figures 2a and 2b), which is surrounded by two maxima of different magnitudes. The strain then progressively decreases with increasing distance from the shear band. The size of the area comprised by the maxima, which can be considered as the shear band-affected zone (denoted by a shadowed area in Figures 2c and 2d), is about 3 µm, therefore larger than the thickness of the shear bands (10-20 nm)1. The formation of the shear band considerably affects the strain direction, as indicated by the variation of the angle formed by the eigenvectors η1 and η2 with the X-axis. The profile of the eigenvectors angle exhibits an antisymmetric sigmoidal shape with a sudden decrease within the affected zone, where the strain axes rotate by about 45°. A 9 ACS Paragon Plus Environment
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similar rotation has been detected in the local atomic displacement generated by an Eshelby’s inclusion in the surrounding elastic matrix,37 which displays quadrupolar geometry. For large deformations, the local Eshelby-like plastic events align along the direction of the imposed shear to form a shear band38 leading to rotational rearrangements at opposite sides of the shear band, in analogy with the angular variation shown in Figure 2. The observed affected zone might be an artifact of the measurement. The aspect ratio for the sampled volume in the present experiment (100 µm thick sample and beam size of 150 nm) is about 667:1. If the shear band and the x-ray beam are misaligned, the beam would average over the entire sample thickness containing both the shear band and surrounding elastic matrix, giving rise to an apparent affected zone. However, we can rule out that the affected zone is an experimental artefact resulting from misalignment, as the shear bands in the cold-rolled Zr52.5Ti5Cu18Ni14.5Al10 BMG are parallel to the beam direction (Z-axis) for about 100 µm.27 The formation of an individual shear band was studied at the nanoscale by MD simulations and the results for the eigenvector η1 at two stages of shear band propagation –(i) inserting shear band (i.e. shear band not yet transecting the entire sample)36 and (ii) mature shear band cutting the entire specimen– are shown in Figure 3. The shear band initiates at the notch and propagates along a plane of maximum resolved shear stress. Along this plane, the strain distribution is rather discontinuous for the inserting shear band, whereas it becomes continuous for the mature shear band (Figures 3a-3d). The nanoscaled characteristics of shear banding can be better analyzed by inspecting Figures 3e-3h, which compare the variation of the eigenvalue
λ1 and of the angle formed by the eigenvector η1 with the X´-axis occurring within the shadowed areas in Figures 3a and 3c at the two stages of shear band propagation. The strain magnitude λ1 varies rather irregularly in the inserting shear band, where two distinct strain maxima occur. In the same area, the strain is considerably higher for the mature band and the profile becomes more continuous and symmetric, displaying a single strain maximum. The two 10 ACS Paragon Plus Environment
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stages of shear band propagation are characterized by different directions of the principal strain axes (Figures 3g and 3h). The eigenvector angle for the inserting band exhibits an antisymmetric behavior with values varying between -20° and 20° and becomes rather symmetric in the mature shear band, where a maximum angle of about 40° corresponding to the maximum of strain in Figure 3f is observed.
Figure 3. Spatially-resolved maps obtained by MD simulations of the eigenvector η1 (a),(c) and eigenvalue λ1 (color maps in (b),(d)) for two stages of shear band propagation: (a),(b) inserting shear band not yet transecting the entire sample and (c),(d) mature shear band cutting the entire specimen. Variation within the shadowed area in (a),(c) of the eigenvalue λ1 and of the angle formed by the eigenvector η1 with the X´-axis for: (e),(g) inserting shear band and (f),(h) mature shear band. The
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magnitude of the eigenvectors is represented by both the length of the vectors in (a),(c) and the color maps in (b),(d). The red contours in (a),(c) indicate the deformation occurring in the specimen. Only the portion of sample displaying the shear band is shown in (a)-(d).
Figure 4 compares the experimental and computational results for the eigenvector η1 across the individual shear band evaluated in Figures 2 and 3. Despite the different length scales, the experimental findings from XRD are in excellent agreement with the MD data of the inserting band. The strain across the shear band is discontinuous in both cases and the direction of the eigenvector η1 displays a strikingly similar behavior with the angle varying between 20 and 70°. The shear band analyzed by XRD (Figure 2) is, therefore, an inserting shear band. Additionally, the coinciding behavior shown in Figure 4b confirms that the features observed by XRD are not due to sample misalignment.
Figure 4. Comparison of the experimental and simulated results for the eigenvector η1 across the individual shear band shown in Figures 2c, 3e and 3g. (a) Eigenvalue λ1 and (b) angle formed by the eigenvector η1 with the X´-axis. The coordinate system of the simulated specimen was rotated by 45°
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in order to match the direction of the shear band observed in the XRD experiments. The position of the shear band center for both experimental and simulated shear bands was selected at the inflection point of the data in (b).
Microscopically, a shear band is a planar defect that mediates plastic deformation across the sample, as schematically depicted in Figure 5 (upper panels). The shear band is formed at one side of the glass sample, where it generates the characteristic shear offset.27 Plastic deformation is then carried through the sample with increasing load by the propagating shear band and, finally, a corresponding shear offset is formed on the opposite side of the specimen, shearing the upper portion of material with respect to the bottom part. At intermediate stages, where the inserting shear band comes to an end within the material, strain compatibility with the surrounding unsheared material would generate compressive and tensile stresses (and strains) in the direction parallel to the band, as indeed reported by Shakur Shahabi et al..22 The variation of the normal and shear strains across the shear band22 would thus explain the rotation of the principal strain axes observed here at the µm-scale (Figure 2). These features, however, originate from the structural rearrangements occurring at the nanoscale. Here, shear band nucleation and propagation is based on the percolation of the STZs,39 which can be described by using a sequential two-unit STZ-vortex mechanism.13 This mechanism is schematically illustrated in Figure 5 (lower panels), where the non-activated STZs are represented by green circles and the activated STZs and their preferential direction by ellipses. For an inserting shear band, the activated STZ perturbs the neighboring (elastic) material by generating a strong antisymmetric compressive-tensile strain field, which in turn induces a collective vortex-like atomic motion. The energy progressively stored in this area is then able to activate the following STZ in an autocatalytic manner.13 The antisymmetric strain field, therefore, mediates the percolation of STZs at the nanoscale and, apparently, leaves a mark in the surrounding elastic matrix at a distance of several microns, explaining why the rotation of 13 ACS Paragon Plus Environment
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the principal strain axes across the band is observed at different length scales (Figure 4b). The strain distribution then becomes symmetric once all STZs percolate along the entire shear band, the strain is released and the mature band (and the corresponding shear offset) emerge on the opposite side of the specimen (Figure 3f), in agreement with the symmetric variation of hardness and elastic modulus reported for mature bands transecting the entire specimen.14,15
Figure 5. Schematic illustration of shear band nucleation and propagation at the micro- and nanoscales (upper and lower panels, respectively). The shear band propagates from left to right with increasing load. The non-activated STZs are represented by green circles in (a), which are progressively distorted and finally activated (ellipses in (b) and (c)).
To conclude, we have analyzed the strain across an individual shear band in real and simulated metallic glasses by using nanobeam XRD studies and MD simulations. Strikingly similar strain distributions have been observed for the inserting shear band in the experimental and computational methods despite their different characteristics regarding sample size (mm- vs. nm-scale), cooling rate (102 K/s vs. 1010 K/s) and strain rate (10-4 1/s vs. 107 1/s): the magnitude of the strain across the shear band is discontinuous in both cases, displaying two local maxima, and the direction of the principal strain axes exhibits the same antisymmetric sigmoidal profile consisting of an angular variation of about 45°. The occurrence of the antisymmetric profile and the subsequent development of a symmetric variation for the mature shear band transecting the entire specimen in the simulated glass can 14 ACS Paragon Plus Environment
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be explained by considering the mechanism of STZ activation and percolation at the nanoscale. This behavior indicates that the effect of the structural variations occurring within a shear band at the nanoscale are not limited to the area within the band but extend well into the surrounding elastic matrix. Our findings not only demonstrates the validity of MD simulations as a mechanistic support to experimental observations, but also indicates that quantitative comparison between results from experiments and simulations is indeed possible. Finally, we expect our analysis to open the possibility to use the structural changes in the elastic matrix as a diagnostic for experimentally verifying numerical predictions of the main atomic rearrangements occurring within a shear band.
Author information. The authors declare no competing financial interest.
Acknowledgements. The authors thank H. Shakur Shahabi for sample preparation, J. Eckert, S. Pauly and M. Stoica for stimulating discussions and U. Nitzsche for technical assistance concerning the computer simulations. D.Ş. acknowledges the financial support by the Deutsche Forschungsgemeinschaft (DFG) through the grant SO 1518/1-1. The authors gratefully acknowledge the computing time granted by the John von Neumann Institute for Computing (NIC) and provided on the supercomputer JURECA at the Juelich Supercomputing Centre (JSC). We thank the European Synchrotron Radiation Facility for provision of synchrotron radiation facilities and we would like to thank J. Wright for assistance in using beamline ID11.
Supporting information: variation of the average values for different cutoffs, test temperatures and positions across the inserting shear band.
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