First-principles atomic level stresses: application to a metallic glass under shear

Unlike crystalline alloys, metallic glasses (MGs) do not possess distinctive defects but exhibit a highly heterogeneous response to shear deformation. The difficulties in describing such non-uniform behaviour hamper the prediction of the mechanical properties of MGs. Using the first-principles athermal quasi-static shear simulation on a CuZr glass, we investigate the mechanical responses of various atomic-level parameters, such as the first-principles atomic stresses and electronic properties (an atomic charge, chemical bonds, etc), and their correlations. We find that the atomic von Mises stress is correlated with a Dmin2 parameter, which is commonly employed and also serves as a unique measure of the degree of non-uniform responses. We also show little correlation between the mechanical and electronic properties during the relaxation process, while we perceive a high correlation between the change in chemical and topological bonds. We discuss the physical insights behind these correlations.


Introduction
Unique properties of metallic glasses (MGs) [1,2], such as high mechanical strength [3], hardness [4], and excellent corrosion resistance [5], make them promising engineering materials. Nevertheless, difficulties in controlling those properties, especially brittle behaviour leading to catastrophic failure, limit the industrial use of MGs as structural materials. Plastic deformation of periodic crystals is well described by the motion of defects such as dislocation, while the MGs, in contrast, deform in a highly heterogeneous way even at elastic regimes, and the mechanical response involves complex activation processes with a cooperative atomic motion, usually referred to as the shear transformation zones (STZs) [6,7]. Many experiments and simulations show discrepancies regarding the size of STZs [8][9][10][11], which could be mainly attributed to the stability of samples with different thermal treatments, limitations dictated by experimental equipment, such as the size of the nanoindentation chip, and the choice of local structural parameters explored in simulations. Moreover, STZs intrinsically have a hierarchical nature ranging from initiation on atomic scales to propagation over larger scales [12,13], making it challenging to build a consistent picture of STZs. From a fundamental science perspective, whether an STZ is pre-existing as a structural defect or emergent also remains controversial [14,15]. Thus, revealing the atomistic origin of the mechanical properties of MGs has been a long-standing problem in physics and material science, so the gap in understanding between macroscopic mechanical properties and microscopic responses remains open.
Stress at the atomic level is a valuable concept because it allows us to directly relate macroscopically observed plastic phenomena, such as stress drop, to where they start. While it is still difficult to observe such stress at an atomic level, atomistic simulations have an advantage in capturing it. Atomic stress in virial form, employed in classical MD simulations, offers valuable insights into multiple mechanical aspects of materials, including surface tension [16], crack propagation [17], and spallation region nucleation [18]. Calculations of atomic stress Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
have been used extensively to describe disordered structures of liquids and glasses [19] or to characterise mechanical behaviours of the atomic environment, such as the occurrence of localized events and the growth of elastic fields through athermal quasi-static shear (AQS) simulations. Thus, atomic stress analysis is believed to be an effective technique for understanding the deformation of structurally disordered systems, including colloids [20] and granular materials [21]. Indeed, obtained microscopic findings, including localized plastic events or the shear band formation, have deepened our understanding of how glass deformation evolves with stress.
Recent advances in computer technology allow us to perform kinetic simulations of MGs with highaccuracy methods of the density functional theory (DFT) [22][23][24]. It may unveil the details of electronic structure and its connection with atomic stress state in MGs. However, in contrast to classical atomic stresses, the nature of the first-principles atomic stresses [25] is far from being fully understood, especially in glassy systems. Most previous studies have focused on the methodology for computing the first-principles atomic stress itself [26][27][28][29]. Despite its potential in materials design, even basic properties of atomic stress (such as statistical properties or correlation with other atomic parameters) have not been quantified yet. Furthermore, the first-principles AQS simulation has been investigated only in a few works [30,31], and it still has the capability of obtaining new insights into the mechanical properties of glasses based on quantum mechanics.
It has been known that in metallic glasses, the local structural relaxation, such as non-affine displacements and local plasticity, leads to a significant decrease in shear modulus compared to the crystalline counterpart [32]. This phenomenon is referred to as shear softening, which was recognised in early studies [33], and now the effect has been studied in more detail in connection to the process of beta relaxations in amorphous materials [34]. Owing to this essential inhomogeneity of metallic glasses, the details of the response to deformation at the atomic level are still unknown.
In the present study, we perform an AQS simulation of a CuZr glass and examine how the first-principles atomic stresses respond as the deformation proceeds. For atomic stress calculation, we employed an energy decomposition scheme that has been successfully applied to study the mechanical properties of single crystals [35], surfaces [36,37], grain boundaries [38][39][40], and multi-component random alloys [41][42][43]. Previously, we studied atomic stress in CuZr glasses under elastic deformation [44] with a DFT code, Open source package for Material eXplorer (OpenMX) [45], based on the linear combination of atomic orbitals (LCAO) method. The results revealed different responses between the Cu and Zr subsystems against shear deformation. By extending the findings, this study proposes an effective method for probing atomic shear stresses and discusses its correlations with the atomic structure and electronic bonding changes expressed by various parameters. This paper is organized as follows. Section 2 covers the setup of DFT simulations and the software used for the mechanical deformation of a metallic glass. Next, in section 3, we detail the process of atomic stress calculations in DFT and von Mises atomic stress formulation. In the following section 4, the parameters characterising the system change from affine to relaxed state are described. Finally, we discuss our results in section 5 before a conclusion.

Basic setup for ab initio calculations
We use Vienna Ab initio Simulation Package (VASP) [46] to simulate a metallic glass of Cu 50 Zr 50 under shear. In VASP, valence electrons are treated in the projector-augmented waves (PAW) formalism [47] with the exchange-correlation potential in the Perdew-Burke-Ernzerhof generalized gradient approximation (PBE-GGA) form [48]. The energy cutoff regulating the number of basis functions is set to 410eV. The convergence of the self-consistent calculations is enhanced by the Methfessel-Paxton method [49] with the smearing of 0.1 eV. The first-principles molecular dynamics (FPMD) is performed with Γ-point only for k-space sampling, while for calculations of the partial density of states (PDOS), the 3 × 3 × 3 k-points grid is used. To calculate inherent glass structures at zero temperature, structures are optimized using the conjugate gradient method [50] until the atomic forces become less than 0.01 eV Å −1 .
The DFT software for the first-principles atomic stress calculations, OpenMX, uses norm-conserving pseudo potentials [51,52]. In contrast to VASP, which is based on plane waves, OpenMX utilizes the orbitalbased energy decomposition scheme [35]. Here, we also use the PBE-GGA exchange-correlation functional for results compatibility. The pseudo-atomic orbitals were generated by the orbital optimization method [53]. For accuracy, we use the set of basis functions (shown in table 1), the total number of which is slightly more than that of the benchmark tests [54].
We analyze the charge density in two ways. Bader charges associated with atoms are calculated using the Henkelman group's package [55]. Also, the points in space where the charge field's gradient vanishes (as implemented in the software Critic2 [56]) allow the identification of chemical bonds between atoms. We refer to such points as bond critical points (BCP). Charge density files output from VASP are used for that purpose.

Sample preparation and deformation protocols for Cu 50 Zr 50 glass
Glassy structures of Cu 50 Zr 50 were obtained using FPMD in VASP, with a rapid cooling process, as shown in the following steps. Initially, atoms were randomly distributed in a cubic box with a number density of 57.1 nm −3 . Then, FPMD was performed under periodic boundary conditions at a high temperature of 3000 K for 2 ps. After the system melted, it was instantly cooled to 1200 K with an effective cooling rate of 1.8 × 10 18 K/s and thermalized at that temperature for 1 ps. Next, the system was cooled down from 1200 K to 700 K with a cooling rate of 0.5 K/fs before instant quenching to 300 K with an effective cooling rate of 4.0 × 10 17 K/s. After relaxing at 300 K for 1 ps, it froze instantly to 0 K, followed by energy minimization with the shape of a simulation cell and atomic positions changed to maintain zero pressure and shear stress. Estimating the glass transition temperature T g of our system precisely from the cooling protocol is challenging. However, based on experimental value [57] and results obtained by classical MD simulation [58], it is assumed to be around or above 700 K.
The total number of atoms is 96 (48 for Cu and 48 for Zr) in a cell. To improve statistics, we have generated four different glass structures by following the abovementioned procedure. Moreover, the final glass structures obtained in VASP are energy-minimized using OpenMX to check the consistency with VASP. Note that the glass structures optimized by different software are slightly different, but the difference in the structure is small.
For deformation, the AQS approach was conducted on the prepared glasses. Firstly, affine deformation (uniform deformation) with a shear strain of 0.5, 1.0, 2.0, 4.0, 6.0, and 8.0% was separately applied stepwise to the same glass structure. Then, the uniformly deformed structures were relaxed with the atomic position changed and with the cell shape fixed during energy minimization. For statistics, we independently impose all possible shear strain components (±xy, ± xz, ± yz) to four different structures and make a statistical ensemble over these data. Therefore, the total number of structures used for averaging macroscopic stress for each strain is 24. A similar ensemble average was made across the studied ensemble for the distribution of local parameters. Figure 1 shows stress-strain curves for a glass of Cu 50 Zr 50 under shear computed in VASP and OpenMX, respectively. We can confirm the consistency in VASP and OpenMX from almost identical stress-strain curves. Also, the decrease in shear stress during relaxation, so-called shear softening, is evident from figure 1. The stress is almost linearly proportional to small strain with roughly estimated shear moduli of 44 GPa and 21 GPa for affine and relaxed cases, respectively. Accordingly, the decrease in shear modulus amounts to about 50% for present samples. The shear modulus of the relaxed sample is slightly lower than the experimental value, around 30 GPa [59], which could be due to the fast quenching rate employed in the first principles calculation, making the system a more liquid-like state. Also, both the affine and relaxed shear moduli are in good agreement with the values obtained from classical MD simulation with similar composition [60,61].  The first-principle calculation also allows us to evaluate the electronic state. The density of states is a valuable tool for the electronic states analysis. Figure 2 shows the d-band partial density of states (PDOS) for each element of the CuZr glass. For comparison, we have also plotted it for the B2 structure (two simple cubic lattices of Cu and Zr shifted relative to each other by the half of [111] vector) of the CuZr alloys. In the B2 structure, the PDOS with a narrow shape, i.e., spatially localized d-band, are found around −3 eV and around the Fermi energy. The former and latter mainly originate from a fully occupied d-band of Cu and a partially occupied d-band of Zr, respectively. The PDOSs of the MG show significant differences from the B2 structure. In particular, the PDOS of Zr has a flattened shape near the Fermi energy, indicating delocalized electronic bonding. This nonlocality reflects the non-crystalline atomic structure of MGs with various coordination numbers and atomic distances, which may imply difficulty in gaining insights from the PDOS. Generally, the density of states near the Fermi energy contains essential information about electronic bonding [62,63], so we parameterise it to see its relationship with other quantities, as described later.

Local stresses
3.1. The first-principles atomic stress tensor In DFT, an atomic stress tensor for a k-th atom, s ij k , is obtained as the strain derivative of energy assigned to a single atom: where E k and V k are atomic energy and a volume associated with an atom, respectively, and geometrical components are expressed by i, j = (x, y, z). In this study, the LCAO method was used to decompose the total energy, which allows for a straightforward calculation of E k by collecting only the contributions of orbitals associated with a particular atom. Average volume (=V/N) was used as V k where V is the volume of the entire system, and N is the number of atoms. Then, the macroscopic stress tensor is given by the sum of s ij k over all atoms: DFT calculations lack pair forces acting between atoms, which makes atomic stress implementation fundamentally different from that in classical MD. In DFT, atomic stress cannot be calculated using a virial for each atom, and the symmetry of the atomic stress tensor is not guaranteed [64] (though the symmetry of the macroscopic stress tensor holds). We indeed find relatively small asymmetry in our calculations, but the asymmetry of shear stress was eliminated by applying the formula ( ) to each component of the stress matrix. This amendment allows one to compare the results obtained in first-principles calculations with the previous classical MD results.

Stress tensor invariants
In continuum mechanics, a stress tensor is expressed as a symmetrical matrix with six independent components to satisfy the balance of angular momentum. It is convenient to use invariants of that matrix, which are not where σ 1 , σ 2 , σ 3 are three eigenvalues of σ ij matrix. We negate stress when calculating pressure to maintain the standard thermodynamic notation of positive (negative) P corresponding to the compressive (tensile) state. At the same time, stress is associated with the derivative of the energy with respect to strain, so positive (negative) σ corresponds to the tensile (compressive) state.
Another invariant relevant to shear is von Mises stress, which consists of the principal components of the deviatoric stress tensor s ij = σ ij + Pδ ij (here, δ ij stands for the Kronecker symbol). The von Mises stress σ vM can be expressed using components of σ ij matrix as: If a simple shear strain is applied to a homogeneous and isotropic system, only the σ xy develops in a linear elastic regime, and σ vM is directly connected to σ xy through the following relation: The local mechanical responses of glassy materials can often be characterised by local parameters that are not altered by a rotation of the coordinate axis. From the atomic stress tensor s ij k computed for the k-th atom, the atomic pressure p k and von Mises stress s k vM can be constructed in the same manner as shown above for macroscopic characteristics. Figure 4 shows the distribution of p k , s xy k , and s k vM for undeformed glasses, each of which is averaged over all initial structures. The pressure indicates differences between Cu and Zr subsystems in the glass. The Cu atoms are under tension (negative pressure), while the Zr atoms are under compression (positive pressure). There is no surprise that, in an alloy, the pressure on atoms of different species bifurcates from zero into opposite directions due to the volume mismatch [19,65,66].
The s k vM is useful in probing local shear response to a shear strain γ, because s xy k is not stress invariant. There have been simulation studies on the structural transformation of homogeneous systems using von Mises strain [67][68][69][70]. Moreover, the von Mises stress was proven effective for identifying triggers of the so-called β-process in CuZr glass under thermal activation [71] and for understanding the origin of local shear relaxation in liquids [72]. To the best of our knowledge, the von Mises stress has not yet been studied in the first-principles calculations of MGs.

Indicators of local structural relaxation 4.1. Change in atomic von Mises stress
As already described in section 2.2, the AQS simulation consists of two processes: affine deformation followed by its relaxation processes for a given shear strain, which causes the local change from an affine to a relaxed state in the deformed glass. To probe such local changes in stress associated with shear, we use the change of atomic von Mises stress, s D k vM , which is defined as ,aff with s ij k,aff and s ij k,rlx being stress components under affine deformation and after relaxation, respectively. This parameter corresponds to the second invariant of the matrix expressing the deviatoric stress change on an atom and gives an insight into a local stress change in all shear directions. A similar equation has been proposed to detect thermal excitations at saddle points on the energy landscape in a metallic glass [71,73].
The distributions of the difference in atomic pressure and stress between affine and relaxed state under the shear strain of 4.0% are given in figure 5. As expected, atomic pressure does not change much, and the distribution of the change has a peak at the zero value for both Cu and Zr subsystems. On the other hand, the peak shift of s D xy k demonstrates a negative stress change on Zr atoms, leading to a stress drop and a positive stress change on Cu atoms, indicating an increase in s xy k during relaxation, which is not seen in classical simulations. This behaviour was reported previously (see figure 2 in [44]). The mechanism of such behaviour of s D xy k remains unclear, but a relationship between Δσ xy and charge transfer has been found in [44].      (7). We see a substantial deviation from the perfectly affine behaviour (solid lines) for Cu and Zr atoms. This is because the components other than s D xy k significantly change in response to the shear strain. This indicates that the local deformation of atoms is highly heterogeneous, and the symmetry of deformation is locally violated due to structural disorder, although macroscopic deformation remains homogeneous, as shown in figure 3. Accordingly, the s D k vM measures the degree of nonaffine deformation in terms of the local stress relaxation and provides valuable information about local shear relaxation that is independent of the rotation of the coordinate system.

Other atomic parameters
Various local parameters have been proposed in the literature to describe mechanical behaviour at the atomic level, and as shown later, we examine a correlation between s D k vM and the change in other parameters based on atomic displacements or electronic properties during relaxation from affine to relaxed state.
One of the common parameters characterising the local distortion due to applied strain is D min 2 [74]. In our case, that parameter measures the non-affine change of atomic coordinates r from the affine to the relaxed state for each atom: where n runs through the neighbours of the atom in question (which has the identifier 0), greek letters represent spatial components x, y, z, δ αβ is the Kronecker delta symbol, and the matrix ε αβ is found as the 'best-fit' of local strain for the atom in question. Neighbours are assigned according to the cut-off radii for Cu-Cu, Cu-Zr, and Zr-Zr, which are defined as the first minimum distances in the corresponding partial pair distribution functions [75] of the MG (see supplemental material of [44]). Using the same cut-off distances, the topological coordination number ( ( ) N c topo ) is calculated as the number of the nearest neighbours. In addition to the topological N c , we also calculate the chemical coordination number ( ( ) N c chem ) by counting the chemical bonds between atoms, which are found by analysing charge density. Then, we introduce the two parameters (chemical that account for a total change in connectivity due to relaxation by summing up the number of broken and newly formed bonds of a given atom. To characterise the change of a d-band PDOS near the Fermi level, we use the parameter ( ) DPDOS d E F , which is calculated as an integral of absolute differences between the PDOS values in the relaxed and affine states over the region near the Fermi level (area shown in figure 2 by the yellow rectangle): The character of ( ) DPDOS d E F is different for Zr and Cu, which comes from the larger values of the d-orbital PDOS at the Fermi level for Zr atoms compared to Cu. Consequently, a rearrangement of neighbours around Zr leads to a larger change in the d-orbital PDOS. On the other hand, the Cu subsystem undergoes considerable geometrical change, as shown before [44], yet the PDOS of Cu atoms changes less than that of Zr atoms, which indicates the more directional character of bonds formed by Zr atoms.
As explained in 2.1, the Bader analysis can estimate atomic charge in an atomic system. By using this, we estimate charge transfer, Q Bader , as the increase or decrease from the pseudo-valence charge in the bulk crystal. Similarly to the PDOS, we also estimate the difference of Q Bader between the affine and relaxed states and refer it to ΔQ Bader . . Here, the total macroscopic Δσ xy is negative, corresponding to the stress drop in the system. The parameters described above are collected for each atom for further analysis of correlations between them.

Results and discussion
Here, we examine a relationship between the first-principles atomic stresses and various local parameters, including atomic charge, coordination number, PDOS, and atomic displacements.

The correlations between local parameters in initially prepared undeformed glass
It is well-known that CuZr alloy, in both crystal and glass structures, exhibits a significant amount of charge transfer due to the difference in electronegativity between Cu and Zr atoms [76]. Previous studies have also indicated a correlation between charge transfer and atomic pressure in fcc or bcc random alloys [41,43]. Our Bader analysis reveals that the charge transfer (Q Bader ) strongly correlates with the atomic pressure (p k ) in the initial glass structure (refer to figure 7(a)). However, no correlation is observed between atomic shear stress and charge transfer in the initial state, as illustrated in figure 7(b). Additionally, no correlation is found for the pair Q Bader and s k vM . The coordination number (N c ) is another parameter that provides insights into the local atomic structure. Figure 8 illustrates the relationship between N c and p k . The Zr atoms possess a larger radius than Cu atoms. Consequently, the ( ) N c topo of Zr atoms is higher than that of Cu atoms, as figure 8 (a) illustrates. The disparity in N c between figure 8(a) and (b) highlights that a small distance between atoms does not necessarily indicate the formation of a chemical bond between them. In the CuZr glass, we observe four distinct partial coordination numbers: 'Cu-Cu' (the number of Cu atoms surrounding a Cu atom), 'Cu-Zr' (the number of Zr atoms surrounding a Cu atom), 'Zr-Cu' (the number of Cu atoms surrounding a Zr atom), and 'Zr-Zr' (the number of Zr atoms surrounding a Zr atom). Notably, figures 8(c) and (d) reveal that Zr atoms are more frequently surrounded by Cu atoms (average N c equals 4.99) than by other Zr atoms (average N c equals 2.32). This correlation indicates that Cu-Zr bonding is favoured over Zr-Zr bonding.
Turning our attention to the mechanical properties, we observe that the total chemical N c shows a limited correlation with p k , in contrast to the total topological N c (compare figures 8(b) and (a)). The latter exhibits a strong positive correlation, consistent with earlier studies on MGs [77]. However, figures 8(c) and (d) reveal a distinct trend: Zr atoms surrounded by multiple Cu atoms experience compression, while Cu atoms surrounded  by numerous Zr atoms undergo tension. This trend aligns with our physical intuition, as the space surrounded by small Cu atoms tends to be small, while the space by large Zr atoms tends to be large. We did not find any such correlation between the chemical coordination number and s xy k or s k vM , unlike the correlation observed with p k .

Relaxation under shear strain
Let us move now from the initial structures to the system transformation from affine to relaxed state under shear strain. We present in figure 9 the Pearson correlation coefficients (PCC) between two different parameters, each of which indicates the change in atomic parameter due to such transformation for Cu and Zr subsystems under 4% strain. Each value in the table represents a correlation coefficient for a pair of parameters, and colour highlights the magnitude of the corresponding PCC. Firstly, we describe the correlation between mechanical properties during the relaxation. It can be seen from figure 9 that s D k vM shows a relatively high PCC with the D min 2 (∼0.6) for Cu and Zr subsystems. Since the D min 2 quantifies the non-affine deformation [78] of the nearest neighbours of an atom, its correlation with s D k vM indicates that s D k vM partially reflects a local collective relaxation with nearest neighbours. Also, there is a correlation between s D k vM and s D xy k (with the PCC value of 0.59 for Cu and −0.61 for Zr), which is due to the major contribution to s D k vM originating from s D xy k in shear deformation. However, the s D xy k has a weak correlation with D min 2 . The reason is that, unlike s D k vM and D min 2 , s D xy k captures only the deformation mode projected on the shear plane and may be ineffective in detecting the non-affine deformation while owing to the fast cooling rate, our system represents more liquid-like regions of a glass [79], and therefore many atoms should undergo a non-affine deformation. Moreover, s D xy k has no correlation with other parameters. Now, let us discuss the importance of s D k vM in characterising non-uniform responses under shear. The D min 2 has been widely used to quantify the local distortion under deformation, and it is capable of capturing the spatial extent of non-elastic rearrangement in a wide range of amorphous materials [80]. However, the relationship with local shear stresses remains unclear because the D min 2 is calculated based on atomic displacements. The correlation of s D k vM with D min 2 indicates that s D k vM could serve as an alternative parameter to identify the local relaxation of the shear stress. As shown in figure 3, s D k vM also works as the measure of the non-affine relaxation by comparing s D k vM and s D xy k , which is highly advantageous in studying heterogeneous mechanical relaxation in glassy materials. Furthermore, the lack of correlation between Δp k and s D k vM demonstrates the orthogonality between hydrostatic and shear modes, in agreement with linear elasticity theory. In other words, Δp k and s D k vM are independent parameters describing a complex mechanical response of a system, and both are invariant under coordinates rotations. From these properties of s D k vM , we found the local shear relaxation easily quantifiable with s D k vM and practically useful. Next, we focus on the cross-correlation between electronic and mechanical properties during the relaxation. As shown in figure 9, we found that ( ) DPDOS d E F has no correlation with mechanical properties, and neither does ΔQ Bader . Also, there is no correlation between the change in chemical connectivity ( ) D  N b chem and mechanical properties. Therefore, these results indicate that the electronic properties explored in this study have little correlation with the first-principles atomic stresses in the relaxation process.
Finally, we discuss the lack of correlation between atomic stress changes and the parameters representing the electronic structure change in figure 9. The Bader analysis in figure 10 provides visual evidence of extensive bond fracture and recreation on a large scale and simultaneously under atomic relaxation. This observation is further supported by the correlation between  figure 9, indicating a significant change in the glass's structural characteristics. Moreover, even in its initial state, the atomic structure exhibits a glassy and heterogeneous nature, as evident from the PDOS results in figure 2. Through an extensive examination of the results, we have demonstrated the challenge of isolating the electronic structure change that corresponds to stress change in such a complex state. Nevertheless, it is essential to note that this complexity does not imply the absence of the electronic structure's contribution to stress. Figure 10 clearly illustrates the distinct bonding states between Cu and Zr, with Cu exhibiting s-p-d bonding and Zr showing d-bonding, as evident from the bond kinks. Investigating this relationship in future studies, particularly with more rigid metallic glasses where stress drop events are more localized than in this study, could provide valuable insights into the connection between electronic structure and stress changes.

Conclusion
We have studied how the mechanical and electronic properties of the Cu 50 Zr 50 MG locally change under shear with the first-principles AQS simulation and analysed the correlations between atomic parameters in the MG. The analysis showed that the change in the first-principles atomic von Mises stress ( s D k vM ), which is of rotation invariant nature, has a correlation with D min 2 parameter and also quantifies the local deviation from shear deformation, making it possible to probe the heterogeneous shear relaxation. The versatile and effective features of s D k vM provide an alternative approach to characterise the complex mechanical response of disordered materials in combination with the atomic pressure. Furthermore, we found that the change in the electronic Figure 10. Connectivity change in one structure under 4% strain. Red (blue) spheres represent Cu (Zr) atoms. Rods and spheres show chemical bonds between atoms: (a) demonstrates in cyan (yellow) colour only newly formed (broken) bonds due to the relaxation process, (b) shows all bonds in the relaxed structure coloured grey with highlighted by green colour the bonds with critical points far from the line connecting two bonded atoms.
properties, including atomic charge, chemical connectivity and PDOS, have no correlation with the mechanical properties. The visualization of chemical bonds demonstrated that a large number of bond formation and breaking events take place during mechanical relaxation. This could be due to the instability of rapidly quenched glasses, and more stable glasses may improve the correlation. The bond-switching events are considered to be the key elements in STZ concepts, and their conjunction with mechanical properties is left for future work. It should also be noted that determining how this correlation map would vary with different materials remains a task to be addressed in future.