Introduction

For decades, bulk metallic glasses (BMGs) have garnered significant attention due to their remarkable mechanical properties1,2,3,4. When subjected to external stresses, unlike in crystalline metals, localized shear phenomena are anticipated to predominate in BMGs, resulting in the formation of shear bands — instances of plastic instability characterized by the localized occurrence of extensive shear strains within a relatively narrow band during material deformation5,6,7. Shear bands hold considerable significance and garner widespread attention as they play a crucial role in unraveling the mechanics of deformation in BMGs8,9. However, the formation of shear bands is not thermodynamically governed — the dynamic process and the inhomogeneity in space complicate attempts at quantitative and accurate description of shear bands with experiments.

Molecular dynamics simulation emerges as an ideal tool for replicating and analyzing the shear banding that occurs during deformation. Unfortunately, Molecular Dynamics (MD) simulation comes with inherent limitations. Despite its capability to operate on a scale of millions of atoms and time steps, its scope still remains relatively modest compared to real-world processes, preventing the perfect reproduction of shear banding process in BMGs. Some previous MD work successfully reproduce the shear bands in BMGs, however, they either require significantly higher strains (ϵ > 0.2)10 or lower temperatures (T ~50 K)11 to induce the shear bands, or creates voids/notches within the BMG structure to facilitate their formation12,13,14,15.

Previous studies have investigated shear band formation in BMGs from multiple perspectives, including the collective organization and percolation of shear transformation zones (STZs)16,17 and the role of structural heterogeneities such as medium-range order18, fertile sites19 and density variations20. Given the difficulty of forming shear bands in large-scale MD simulations, mechanistic insights from these theoretical studies provide valuable guidance.

In light of experimental observations and theoretical mechanisms that shear-band formation is accompanied by volumetric dilation, which manifests as a local density reduction or fluctuation6,11,20,21,22, we systematically evaluate the effect of density fluctuation by purposely introducing low-density regions in BMGs to facilitate shear band formation in MD simulations. The low-density regions exhibit a realistic nature, potentially arising from impurities such as vacancies or non-metallic atoms, or originating from intrinsic density fluctuations naturally present in real metallic glasses. This controlled density reduction can also be viewed as a special form of localized rejuvenation, which has been reported in experiments to promote shear-band formation14,23,24,25. Unlike conventional thermally or mechanically induced rejuvenation in experiments, our approach provides a simple, efficient, and computationally accessible means of introducing such rejuvenated states in atomic-scale simulations while preserving the overall structural integrity of the glass.

In this work, we use the well-known Zr50Cu50 BMG for our analysis of shear bands. We also define the critical strain for shear band initiation and analyze how low-density areas affect the formation of shear bands.

Results

Shear band formation and local shear-strain characterization

In this work, we use the Zr50Cu50 metallic glass as an example. The atomic structure of metallic glass is generated through MD simulation of cooling, after which two rigid diamond carbon slabs are added on both sides of the sample to impose uniaxial loading, as shown in Fig. 1a. Figure 1b shows the spatial distribution of local shear strain under a uniaxial strain of 10% applied along the x direction, where no pronounced shear-band localization is observed. Under a much higher uniaxial strain of 20% as suggested in ref. 10, we examine the average density as a function of the distance to the \((\overline{1}01)\) plane, as shown in Fig. 2a. A broad low-density feature is observed in the vicinity of the shear-localization region, indicating that the density reduction is closely associated with the shear-banding process. Motivated by the observation, we introduce a localized low-density region in a controlled manner by randomly removing a small fraction of atoms near the \((\overline{1}01)\) plane (as marked in Fig. 1a, with the aim of facilitating shear-band nucleation under more experimentally relevant conditions. Specifically, we provide an example with only 3% of the atoms within 5 Å of the \((\overline{1}01)\) plane removed from the original structure. This treatment produces a more localized density fluctuation, and correspondingly a clearer and more localized shear band is observed, as shown in Fig. 1c and Fig. 2b.

Fig. 1: Atomic structures and local shear of BMG.
Fig. 1: Atomic structures and local shear of BMG.The alternative text for this image may have been generated using AI.
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a Atomic structure with two diamond layers added on both sides of the ZrCu BMG to apply stress. The red dashed line marks the region near the \((\overline{1}01)\) plane, where a fraction of atoms is removed in subsequent simulations; (b) the local shear of atoms under strain ϵx = 10% with perfect BMG Zr50Cu50; (c) the local shear of atoms under strain ϵx = 10% with 3% of the atoms within 5 Å of the \((\overline{1}01)\) plane removed from the original structure to help forming the shear band.

Fig. 2: The average density of the Zr50Cu50 BMG as a function of distance to the \((\overline{1}01)\) plane.
Fig. 2: The average density of the Zr50Cu50 BMG as a function of distance to the $$(\overline{1}01)$$ ( 1 ¯ 01 ) plane.The alternative text for this image may have been generated using AI.
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a Perfect BMG Zr50Cu50 under strain of 20%; (b) 3% of the atoms within 5 Å of the \((\overline{1}01)\) plane are removed from the original structure under different strain.

To clarify the role of density reduction on shear band formation, additional simulations are performed by removing atoms either homogeneously or locally along different planes, as shown in Fig. 3. When 3% of atoms are removed homogeneously throughout the entire sample, no distinct shear band is observed at either 5% (Fig. 3a or 10% strain Fig. 3b), despite the overall softening of the material. This indicates that density reduction or homogeneous rejuvenation alone is insufficient to induce shear localization. We further examine the effect of localized atom removal along planes with different orientations. When atoms are removed along a \((\overline{1}02)\) plane, which is not aligned with the direction of maximum resolved shear stress under uniaxial loading, local stress concentration is observed at low strain (5%) due to local rejuvenation (Fig. 3c). However, at larger strain (10%), deformation preferentially localizes along directions closer to 45 degrees relative to the loading axis, while the initially softened \((\overline{1}02)\) plane no longer dominates the shear band formation (Fig. 3d). This behavior is consistent with some established theories of shear banding in amorphous solids, which predict that shear localization arises from a collective instability of interacting Eshelby-like quadrupolar plastic events and preferentially develops along planes experiencing the maximum resolved shear stress16. In contrast, when atoms are removed along the \((001)\) plane, the uniaxial strain on x axis contains no shear component along the density reduction region. As a result, it makes no contribution to the formation of shear band, as shown in Fig. 3e, f. Finally, two parallel low-density regions are introduced along the \((\overline{1}01)\) plane. At 5% strain, strain localization develops on both softened planes (Fig. 3g). At 10% strain, although some local strain concentrations remain along the 45 degrees direction, clear interaction among STZs associated with the two low-density regions is observed (Fig. 3h). Overall, these results demonstrate that density reduction or fluctuation–whether homogeneous or localized–does not solely determine shear band formation, but instead needs to be coupled with other shear banding mechanisms. Nevertheless, from a practical MD simulation perspective, introducing a moderate density reduction along a physically motivated direction provides an efficient and controlled way to generate shear band structures with a well-defined and experimentally relevant atomic configuration, serving as a suitable starting point for subsequent theoretical investigations of shear band properties.

Fig. 3: Atomic structures and local shear of Zr50Cu50 BMG, with different density reduction regions.
Fig. 3: Atomic structures and local shear of Zr50Cu50 BMG, with different density reduction regions.The alternative text for this image may have been generated using AI.
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a 3% of atoms homogeneously removed through the entire sample, ϵx = 5%; (b) 3% of atoms homogeneously removed through the entire sample, ϵx = 10%; (c) 3% of atoms removed within 5 Å of the \((\overline{1}02)\) plane, ϵx = 5%; (d) 3% of atoms removed within 5 Å of the \((\overline{1}02)\) plane, ϵx = 10%; (e) 3% of atoms removed within 5 Å of the \((001)\) plane, ϵx = 5%; (f) 3% of atoms removed within 5 Å of the \((001)\) plane, ϵx = 10%; (g) 3% of atoms removed within 5 Å of two parallel \((\overline{1}01)\) planes, ϵx = 5%; (h) 3% of atoms removed within 5 Å of two parallel \((\overline{1}01)\) planes, ϵx = 10%. Red dashed lines indicate the region where a fraction of atoms is removed.

To better characterize the shear band region, some prior studies of shear bands employed the concept of von Mises local shear strain by marking atoms with μvM > 0.226 or 0.2827. In this work, we employ a refined and precise methodology to delineate the shear bands by utilizing local shear strain data. In Fig. 4, we plot the average of local shear strains of atoms as a function of the distance to the \((\overline{1}01)\) plane. From Fig. 4a, we observe that the perfect MG shows no pronounced inhomogeneity along the \([\overline{1}01]\) direction. Therefore, although Fig. 1b exhibits some non-uniformity in the spatial distribution of local shear strain, this variation is not sufficiently concentrated to form a localized shear band, in contrast to the clear localization observed in Fig. 1c and Fig. 4b. It is worth emphasizing that the observation in Fig. 1b may not fully reflect the experimental deformation behavior of metallic glasses, as the cooling rate and deformation strain rate employed in our MD simulations are much higher than those typically accessible in experiments. Such rates are, however, commonly adopted in large-scale MD simulations of metallic glasses due to the prohibitive computational cost associated with more realistic conditions. These computationally feasible simulation settings are known to bias the system toward more homogeneous plastic deformation rather than pronounced shear-band localization. In this context, the introduction of density fluctuations in our simulations serves as an effective means to facilitate shear-band formation even under conditions that are computationally tractable but otherwise unfavorable for shear localization.

Fig. 4: The average of local shear strains of atoms as a function of distance to the \((\overline{1}01)\) plane.
Fig. 4: The average of local shear strains of atoms as a function of distance to the $$(\overline{1}01)$$ ( 1 ¯ 01 ) plane.The alternative text for this image may have been generated using AI.
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a Perfect bulk metallic glass Zr50Cu50. Strain from 0.02% to 10.5% is added on the X direction. b 2% of the atoms within 5 Å of the \((\overline{1}01)\) plane are removed from the original structure to help forming the shear band. Strain from 0.05% to 11.25% is added on the X direction.

In Fig. 4b, we can see that the local shear strain as a function of the distance to the \((\overline{1}01)\) plane (or the shear band center) exhibits shifted Gaussian-like profile:

$${\mu }_{vM}=A{e}^{-\frac{{x}^{2}}{2{c}^{2}}}+b$$
(1)

where A is the height of the peak, c is the standard deviation and b is the baseline. We remove 0.5% to 3% of the atoms uniformly within the area of 1 Å, 3 Å, and 5 Å of the \((\overline{1}01)\) plane, applying strain along the X direction and plotting the A, c and b to describe the shear band, as shown in Fig. 5.

Fig. 5: Parameters fitted as function of strain.
Fig. 5: Parameters fitted as function of strain.The alternative text for this image may have been generated using AI.
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A, Standard deviation and b fitted as a function of strain with different percentage of atoms removed within 1 Å, 3 Å, and 5 Å of the \((\overline{1}01)\) plane. a A with different percentage of atoms removed within 1 Å; (b) standard deviation with different percentage of atoms removed within 1 Å; (c) b with different percentage of atoms removed within 1 Å; (d) A with different percentage of atoms removed within 3 Å; (e) standard deviation with different percentage of atoms removed within 3 Å; (f) b with different percentage of atoms removed within 3 Å; (g) A with different percentage of atoms removed within 5 Å; (h) standard deviation with different percentage of atoms removed within 5 Å; (i) b with different percentage of atoms removed within 5 Å.

Influence of density fluctuation on Shear-Banding Behavior

It is observed that A, the height of the Gaussian peak consistently exhibits an “S-shaped” relationship with respect to strain. (except in some cases with only 0.5% atoms removed where no clear shear bands form) In other words, the second derivative of A to strain monotonically decreases from positive to negative in the strain range 0 to 0.18. We define the critical strain ϵSB for the shear band as the zero of the second derivative:

$$\left(\frac{{\partial }^{2}A}{\partial {\epsilon }^{2}}\right){| }_{\epsilon ={\epsilon }_{{\rm{S}}{\rm{B}}}}=0$$
(2)

In Fig. 6, we illustrate the relationship between the critical strain and the percentage of atoms removed in the region. Figure 6 shows that the critical strain exhibits a nonlinear dependence on the fraction of atoms removed, with a clear turning point near 1.5%. When the removal fraction is below 1.5%, the critical strain is significantly higher and decreases rapidly with additional atom removal. Once the removal fraction exceeds 1.5%, however, the critical strain saturates and remains within the typical shear-band nucleation range of 7–10% as observed experimentally in metallic glasses28.

Fig. 6: Critical strain for shear-band formation with different removal fractions.
Fig. 6: Critical strain for shear-band formation with different removal fractions.The alternative text for this image may have been generated using AI.
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The critical shear-banding strain for BMG with 0.5% to 3% atoms removed within 3 Å from the \((\overline{1}01)\) plane are shown. When the removal fraction is below 1.5%, the critical strain decreases rapidly with additional atom removal. When removal fraction exceeds 1.5%, the critical strain saturates and remains within the typical shear-band nucleation range of 7−10%.

This nonlinear relationship between removal fraction and the critical strain can be explained by a threshold-like effect, in the degree of local rejuvenation or increase in energy state introduced by atom removal. When the removal fraction is below 1.5%, the density fluctuation is insufficient to fully activate the low-density region as a preferred site for shear-band nucleation. In this regime, the region remains relatively stiff, and shear-band formation still requires energy from substantial external strain. Consequently, the critical strain is relatively high and decreases rapidly with increasing removal fraction, since each additional small reduction in density significantly increases the energy and lowers the external strain needed to compensate for the insufficient local activation energy. Once the removal fraction exceeds 1.5%, the low-density region acquires enough rejuvenation–a sufficiently elevated local energy state–to serve as an energetically favorable nucleation site for shear localization. In this regime, the external strain required to initiate a shear band becomes similar to the typical critical strain for shear-band formation in metallic glasses (around 7–10%, as observed in experiments). Thus, the turning point at 1.5% marks the transition from a “rejuvenation-limited” regime, where the perturbation is too weak, and shear-band initiation depends mainly on the input of external strain, to a “rejuvenation-enabled” regime, in which the low-density region itself provides sufficient intrinsic softening to promote shear-band formation.

Discussion

In this work, we demonstrate a new method that promotes shear-band formation by intentionally introducing a localized low-density region in the BMG. We observe the presence of low-density regions within the shear band and successfully simulate shear band formation using EAM-MD at relatively low strain by removing no more than 3% of atoms from a specific planar region of 10 Å thickness. Our simulation setup does not require non-physical parameters, such as large uniaxial strain, low temperature, or the introduction of large voids in the structure.

This method enables a computational investigation of the physical behavior of shear bands in BMGs by introducing localized rejuvenation in a physically plausible and computationally accessible manner. The shear banding process is non-steady in time and non-uniform in space. With the shear band atomic structures we generate, we develop an accurate and quantitative definition of shear bands and identify the critical strain for their formation in BMGs by utilizing a local shear parameter. Through the calculation of critical strain based on our MD simulations, we confirm the occurrence of shear banding at low strain. In addition, our method offers precise and quantitative control over the degree of localized rejuvenation. By adjusting both the thickness and removal fraction of low-density regions, we can deliberately tune the width and the degree of strain localization of the resulting shear band under a given external loading condition. The tunability of our method, together with the quantitative definition of shear-band critical strain, reveals a clear nonlinear relation between the critical strain and the removal fraction, indicating a threshold-like activation behavior and providing a direct and quantitative insight into how soft/low-density regions and the associated localized rejuvenation promote shear-band formation. It is worth mentioning that the density fluctuation in this work should not be regarded as an independent fundamental mechanism of shear-band formation, but rather as a realistic structural heterogeneity used to introduce local softening or rejuvenation, thereby facilitating or promoting shear-band formation in theoretical simulations.

In future work, our method can be extended to generate more complex multi-shear band structures, such as systems with groups of parallel shear bands, to further investigate their influence on the mechanical properties of BMGs. Additionally, we can apply our approach to systems with multiple elements and impurities, such as Zr-Cu-Al BMGs containing carbon or oxygen impurities (if reliable interatomic potentials are available), to explore how these impurities affect shear banding behavior in BMGs.

Methods

Molecular dynamics simulations

For all MD calculations, we use LAMMPS29,30 in conjunction with the EAM potentials from the group of Sheng31. For the unstrained Cu50Zr50 BMG structure with 540,000 atoms in a cubic simulation cell, we first equilibrate it at 2000 K using NVT and then NPT ensembles, and subsequently quenched to 300 K at a cooling rate of 1012 K/s under the NPT ensemble. A final NPT relaxation at 300 K was performed to obtain a fully stabilized amorphous configuration (cell length 211.70 Å). For the strained structures, two rigid diamond carbon slabs of thickness 13 Å are added on both sides, with 10 nm vacuum layers applied in both the y (front-back) and z (top-bottom) directions, as shown in Fig. 1a. The interactions between carbon atoms and the BMG are simulated with Lennard-Jones potentials32. The carbon layers are forced to move into the middle along the X-axis to apply uniaxial strain upon the glass, with constant velocities of ± 0.1 Å/ps. This corresponds to an effective macroscopic strain rate of 9.45 × 108 s−1. This geometry is better suited to study shear band formation than an homogenous simulation with periodic boundary conditions, because it induces a non-uniform stress, reflecting the fact, in a real material, shear bands form in regions of non-homogenous stress.

Local Shear strain calculations

The manifestation of shear bands within the metallic glass is elucidated through the assessment of the von Mises parameter, or the local shear strain experienced by each atom6. For each atom i in the system, a transformation matrix Ji is determined by minimizing the mapping error between the current (dji) and reference (\({{\boldsymbol{d}}}_{ji}^{0}\)) configurations:

$$\mathop{\sum }\limits_{j\in {N}_{i}^{0}}| {{\boldsymbol{d}}}_{ji}^{0}{{\boldsymbol{J}}}_{i}-{{\boldsymbol{d}}}_{ji}{| }^{2},$$
(3)

in which the configuration are described by atomic coordinates and where the reference is the system’s configuration before strain is applied. Then the local Lagrangian strain matrix can be computed as

$${{\boldsymbol{\eta }}}_{i}=\frac{1}{2}\left({{\boldsymbol{J}}}_{i}{{\boldsymbol{J}}}_{i}^{{\rm{T}}}-{\boldsymbol{I}}\right)$$
(4)

and the local shear (or the von Mises parameter) is defined as

$$\begin{array}{rcl}{\mu }_{i}^{vM} & = & \sqrt{{\eta }_{yz}^{2}+{\eta }_{xz}^{2}+{\eta }_{xy}^{2}}\\ & & +\frac{{({\eta }_{yy}-{\eta }_{zz})}^{2}+{({\eta }_{xx}-{\eta }_{zz})}^{2}+{({\eta }_{xx}-{\eta }_{yy})}^{2}}{6}\end{array}$$
(5)

The local shear calculations and atomic structure visualizations are performed using Open Visualization Tool (OVITO)33. Fitting for parameters in Equation(1) are performed using MATLAB34.