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# Dynamical theory of shear bands in structural glasses

Contributed by Peter G. Wolynes, December 15, 2016 (sent for review December 6, 2016; reviewed by Randall Hall and Jörg Schmalian)

## Significance

When glasses are under imposed stresses or strains, they are subject to plastic deformation. Unlike their crystal counterparts, shear within the glasses localizes in thin bands, known as shear bands. Forming the shear bands can lead to structural failure of the whole sample and prevent using glasses as structural materials. In this work, we show how shear bands arise dynamically by the coupling of activated dynamics of configurationally rearranging regions with elastic strain transport. This result also explains the non-Newtonian flow of glasses.

## Abstract

The heterogeneous elastoplastic deformation of structural glasses is explored using the framework of the random first-order transition theory of the glass transition along with an extended mode-coupling theory that includes activated events. The theory involves coupling the continuum elastic theory of strain transport with mobility generation and transport as described in the theory of glass aging and rejuvenation. Fluctuations that arise from the generation and transport of mobility, fictive temperature, and stress are treated explicitly. We examine the nonlinear flow of a glass under deformation at finite strain rate. The interplay among the fluctuating fields leads to the spatially heterogeneous dislocation of the particles in the glass, i.e., the appearance of shear bands of the type observed in metallic glasses deforming under mechanical stress.

Whether and how glasses flow have been fascinating questions for a long time. In the absence of stress, a glass seems to be static on human timescales, the molecules being arranged like a frozen snapshot of the liquid state. In fact, molecules in the glass are constantly moving and the glass itself is not in a state of equilibrium. Even without applied stress, molecules do change their locations through rare, activated events. These events occur at a rate that is both spatially and temporally heterogeneous. Glasses therefore continue to evolve, albeit slowly, as they approach equilibrium and age (1). These activated events are accelerated by applied stresses and typically will act to reduce the stress so that under sufficient stress the glass will not just deform elastically but visibly flow and possibly break. The deformations caused by stress are not uniform in the glass, but appear to concentrate in shear bands (2⇓–4). In this work, we show how shear bands arise dynamically by the coupling of the activated dynamics of configurationally rearranging regions with elastic strain transport. The heterogeneous activated dynamics of glasses under mechanical deformation are described using a first-principle framework based on the random first-order transition (RFOT) theory along with an extended mode-coupling theory that describes how activated events are coupled in space and time (5⇓⇓⇓⇓–10).

The random first-order transition theory of glasses is a microscopic theory that has already provided a unified quantitative description of large number of aspects of the behavior of supercooled liquids and structural glasses (11). The theory brings together two seemingly disparate aspects of glass formation: the breaking of replica symmetry that occurs in mean-field models and a theory of the activated events that tend to locally restore replica symmetry in systems with short-range interactions (7, 8, 12⇓–14). In mean-field models, there is a special temperature

Within RFOT theory the heterogeneous dynamics of the glass can be understood in terms of a mobility field

To describe shear bands we must first understand how stress affects glass mobility (19). RFOT theory describes this effect by first noting that when a sample of glass is put under a uniform shear stress

Several phenomenological models have been developed to explain how amorphous solids deform (3, 4, 21, 22). The free-volume model proposed by Spaepen and colleagues (23⇓–25) nearly three decades ago inspired the notion of a shear transformation zone (STZ). That model postulates that plastic flows are caused by a series of driven creation events of free volume via individual jumps of particles. The shear transformation zone theory argues that the plastic deformation involves irreversible rearrangements of small clusters of particles whose detailed nature is, however, left unspecified (26). It was later realized that particle rearrangements in a shear transformation zone might be triggered by neighboring particle jumps that again are described by the free-volume model (27⇓⇓–30). Impressionistically this model has many points in common with the present framework but the STZ theory introduces many phenomenological parameters. In contrast, the RFOT theory allows first-principle calculation of all of the elements of elastoplastic deformation, using only input thermodynamic and elastic data.

## Theoretical Framework

To account for the dynamics of a deformed glass, we first recount the continuum mechanics for a glass, emphasizing the role of spatiotemporal fluctuations. Initially the fluctuations of the local properties of the glass are inherited from the liquid state from which it is formed. When the liquid is cooled at a rate faster than the basic relaxation time, persistent disordered structures of the liquid state are to a first approximation frozen in at the glass transition temperature. The structure, however, actually continues to evolve, proceeding to lower-energy states. This protocol gives rise to frozen internal stresses within the sample of a glass. At the continuum level the equation of motion for these stresses can be found by equating the internal stress force

Glass dynamics occur on a range of length scales and timescales and strongly depend on the glass’s history. When a stress is applied for times short compared with the structural relaxation time

Owing to the aperiodic structure of glasses, we must complete the description of the stress evolution by introducing stochastic sources for the stress fluctuations as is familiar in the continuum theory due to Landau (32):

The local mobility field is defined as the longtime rate at which particles in a glass reconfigure. This field is formulated in real space through the low-frequency Fourier transform of the memory kernel of the mode-coupling theory (MCT). As discussed earlier by Bhattacharyya, Bagchi, and Wolynes (BBW) (15, 33), activated processes provide an extra decay channel for structural correlations beyond conventional mode-coupling theory, which ultimately restores the ergodicity of the glass. In the BBW treatment the total mobility field has two distinct contributions, one coming directly from the activated dynamics and another part coming from idealized mode coupling (18):

The completely microscopic theory of the mode-coupling memory kernel leads to a rather complicated mathematical form involving coupling to density fluctuation modes with other wave vectors. For inhomogeneous systems the relevant correlations must be rewritten in terms of multipoint spectral quantities. Expanding the resulting extended MCT with activated processes in a Taylor series in spatial and temporal derivatives of the mobility field leads to continuum equations that again must be supplemented with local random forces to restore the local fluctuation–dissipation theorem, giving the equation

Simply adding the relieved strain energy to the reconfigurational driving force allows one to write the barrier for the flow of a glass under stress in terms of the same function that gives the activation free energy for an equilibrium liquid in terms of its configurational entropy *J*-point scenario where barrierless reconfiguration can be approached by tuning either the temperature

The fictive temperature determines the local energy density of a region (1). Following the arguments of Lubchenko and Wolynes, the fictive temperature relaxes following an ultralocal relation with the local decay rate also given by the mobility

## Results and Discussion

In this work, we carried out for illustration numerical calculations of the mechanical responses of a sample mimicking the bulk metallic glass *x* direction. This essentially describes a thin-layer slab, but where surface mobility changes are neglected (39). We solve the system of Eqs. **1**, **3**–**5**, and **8** numerically by a finite-difference method that takes account of the stiffness of the equations, thereby neglecting sound waves (40, 41). To treat the stochastic terms in the numerical calculations, we used the Euler–Maruyama method in which random numbers are normally distributed (41). Consequently the mobility may rarely become negative locally. To avoid such unrealistic situations, the negative values of the mobility when they occur are set to zero. The computational domain used in this study is an 80 *x* direction and to be free, respectively. At the same time the boundary conditions on the bottom

To account for the fluctuations we note that we are essentially studying a thin slab roughly of width ^{−1} to ^{−1} and the stress and strain were measured. The preparation procedure used by us follows the experimental setups as described in ref. 38.

Fig. 2 shows the predicted stress and strain histories for the 2D Vitreloy 1 glass slab at different applied strain rates. The experimental results from ref. 38 for the same system in 3D are shown as blue symbols for comparison. As seen from the graph, both in our numerical calculations and in the laboratory the stress initially linearly increases as the strain increases. The linear behavior of the glass in this regime indicates the glass acts simply as an elastic solid. Increasing stress allows the glass to relax at a faster rate, leading to the apparently non-Newtonian fluid behavior in the late regime of the stress–strain curve where the stresses reach a plateau regime. Before transiting to the plateau regime an overshoot of the stress occurs at a high strain rate whose magnitude strongly depends on the strain rate. For strain rates below ^{−1}, no stress overshoot is observed.

It is worth mentioning that, in studies of some metallic glasses such as ^{−1}, the stress–strain curve exhibits a more intricate nonlinear oscillation in which the stress overshoot is followed by undershoot (22, 46). In our 2D calculation we have never seen this behavior. This issue may require full 3D simulation of a larger system.

Apart from these global measurements the numerical calculations resolve how the flows develop in space and time. Fig. 3 shows various stages of the Vitreloy 1 flow after the sample was compressed with a strain rate of 0.01 s^{−1}. The colored contour plots show the level of the equivalent stress (47), which is defined as *A*–*D* is shown when the strains (*A*. The stress increases quickly and reaches a maximum value when the strain is approximately equal to 0.03. Shear bands start developing at this stage.

Snapshots of the mobility field and the fictive temperature pattern are presented in Fig. 4 *A* and *B* for the case when the strain is equal to 0.2. Both plots also show the strain fields. The corresponding stress pattern at the same time is shown in Fig. 3*D*. The regions that develop shear bands also have higher mobility and higher fictive temperature than their neighboring regions. Experimental observations of similar results have been found in many other metallic glasses (3, 48).

The plots in Figs. 3 and 4 emphasize the interplay among the stress, strain, mobility, and fictive temperature fields in which these fluctuations lead to a very heterogeneous flow of the sample. The heterogeneity of the glass reflects the fact that the rates of reconfiguration events vary throughout the sample. These fluctuation are not completely independent from their neighbors, but become correlated over short length scales that then grow as the flow proceeds much like a rejuvenation front (16). This correlation allows the glass to differentially deform different regions to develop shear banding patterns like those in the laboratory (3, 4, 49, 50). We have marked the shear bands observed in our simulation in Fig. 3*D* with dashed lines. Note that neither band is perpendicular to the sides of the domain. The tilted angle of the band is 45° with respect to the horizontal axis and the width of the bands is ∼10 nm.

We also studied the temperature dependence of shear deformation and show the results for this in *Effect of Temperature on Stress-Strain Behavior*, *Effect of Strain Rate and Temperature on Apparent Viscosity*, and Figs. S1–S3. At fixed strain rates, the ambient temperature strongly influences the magnitude of the overshoot stress and the steady-state stress values.

## Effect of Temperature on Stress–Strain Behavior

The influence of temperature on the flow behavior is shown in Fig. S1. In this plot, all specimens were compressed at the strain rate of ^{−1}, using the numerical scheme described in the main text. The graph clearly shows that the temperature strongly influences the magnitude of the overshoot stress and the steady-state stress values. When the temperature is high (643 K) and the strain rate is low (^{−1}), the stress monotonically increases as the strain. All of the stresses finally reach their steady-state values and subsequently level off after the strain increases beyond 0.2.

## Effect of Strain Rate and Temperature on Apparent Viscosity

The values of the steady-state stress with respect to the strain rate are plotted in Fig. S2 at various temperatures: 603 K, 613 K, 623 K (the glass transition temperature), 633 K, and 643 K. The steady-state stress for each temperature linearly increases with the strain rate at low strain rates. At higher strain rates, a nonlinear relation between the stress and the strain rate appears, and the curves tend to bend.

Fig. S3 shows that a master curve can be constructed in terms of the normalized structural relaxation time,

## Summary

In the present work we have shown how RFOT theory naturally leads to shear bands in a 2D flow situation mimicking experiments on real metallic glasses. Extensions of the calculations to 3D are conceptually straightforward but computationally demanding. The shear bands correspond to regions of transiently high mobility that will also allow local crystallization and cavitation to occur in them, ultimately leading to material failure.

## Acknowledgments

This work was supported by the Center for Theoretical Biological Physics sponsored by the National Science Foundation(Grant PHY-1427654). Additional support was also provided by the D.R. Bullard-Welch Chair at Rice University (Grant C-0016). A.W. was supported by the Thai government research budget through the Theoretical and Computational Science Center under the Computational and Applied Science for Smart Innovation Research Cluster, Faculty of Science, King Mongkut’s University of Technology Thonburi, and the Thailand Research Fund (Grant TRG5880254).

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: pwolynes{at}rice.edu.

Author contributions: A.W. and P.G.W. performed research and wrote the paper.

Reviewers: R.H., Dominican University of California; and J.S., Institute for Theoretical Condensed Matter Physics, Karlsruhe Institute for Technology.

The authors declare no conflict of interest.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1620399114/-/DCSupplemental.

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