**3. Results and Discussion**

### *3.1. Measurement of the Long-Range Stress Fields Produced by Shear Bands*

The first brief report on the direct observations of the long-range elastic strain and stress fields created by a shear band terminated inside the BMG was made by the same authors in Reference [26]. For consistency with what follows, we repeat the same experiment and provide new additional insight into this topic. The specially designed sample geometry promotes obtaining SBs with either the dominating shear mode (mode II, by analogy with that of the shear crack), c.f., the inset in Figure 1g, or the tearing mode (mode III out-of-plane shear), c.f., the inset in Figure 1c, during compression. After in-situ imaging of emerging SBs in PdCuNiP specimens, the displacement fields around the SB tips were calculated using the DIC algorithm (Figure 1a,d).

Notice that the presented in Figure 1a,d displacements *<sup>U</sup>*y and *U*x are "raw", i.e., they are calculated using only the correlation coe fficient method without any smoothing or filtering. The displacement of the mode II shear was calculated directly from the XY-plane. The mode III shear o ffset was inclined to the observed surface at 45◦, which projects itself on the XY-plane. This made it possible to quantify the out-of-plane shear o ffset via its XY-projection. Using the well-known expressions for the strain and stress fields created by perfect screw and shear dislocations in elastic continuum [39]

with account of the actual shear geometry in the present experiments, the expected displacement fields have been calculated [26] and plotted in Figure 1b,f for comparison with experimental data, (Figure 1a,e) respectively. Even a simple juxtaposition of these respective figures suggests good qualitative agreemen<sup>t</sup> between the experimental findings and the predictions from the dislocation theory both for mode III (Figure 1a,b) and mode II shear (Figure 1e,f). The quantitative comparison is represented in Figure 1d,h using a circular contour around the shear band tip (with the tip in the center and the arbitrarily chosen radius). The dislocation-based elastic model predictions exhibit impressive numerical agreemen<sup>t</sup> with the experimental data within the error of less than 10%, which is a remarkable result.

Both experimental data and theoretical calculations show that elastic fields around the SB associated with the Volterra-type macro-dislocation [40] are large enough to increase the local stresses significantly, i.e., of 10–100 MPa stress hundreds of micrometres apart from the shear tip (Figure 1c,g).

**Figure 1.** Examples of type III (Left) and type II (Right) shear bands (SBs) in the amorphous Pd40Cu30Ni10P20 deformed in compression. The displacement fields *<sup>U</sup>*y (**<sup>a</sup>**,**b**) and *U*x (**<sup>e</sup>**,**f**) were measured experimentally (**<sup>a</sup>**,**<sup>e</sup>**) and modelled as Volterra's screw (**b**) and edge (**f**) dislocations [26]. The σxz and σxx components were calculated for the modelled screw (**c**) and edge (**g**) dislocations, respectively. The comparison of experimental (rounds) and model (red lines) displacements as well as

stresses (blue lines) is presented in (**d**,**h**) for the depicted circular paths around the SB tips. Excellent agreemen<sup>t</sup> can be seen between the experimental data and model predictions within a ± 20 nm band (red dash lines). The insets on (**<sup>c</sup>**,**g**) represent the schemes of used samples geometry: the red line indicates the SB with a tip surrounded by an investigated circle cross-section.

The experimental evidence for the existing Volterra dislocation-type long-range elastic fields around the SB front offers a new insight into the interpretation of previously reported experimental results regarding the SB-related properties of metallic glasses, such as apparent work hardening [11–13], internal friction relaxation peak [41–43], reactivation and suppression of the SB activity [34], etc. While there are multiple similarities in the shear behaviour in crystalline and amorphous solids, there are also major differences. Due to the absence of long-range order within the atomic structure of a glass, BMGs do not have crystallographically defined slip planes and directions. Therefore, (i) the shear slip in the amorphous structure requires dilatation [44], which is notably more pronounced than that in the core of the crystal dislocations [45], and (ii) the SB's offset and propagation direction are not constrained by a crystal structure. These factors collectively lead to considerable topological differences between the mode II and mode III SBs, which is further investigated in Sections 3.2 and 3.3, respectively.

## *3.2. Mode II Shear Morphology*

The offset of the mode II SB aligns with the shear propagation direction. This promotes the straight path of the shear front through the specimen. Therefore, under the uniform far-field stress (e.g., during compression of an unnotched specimen), the SB appears commonly as a straight line. Since such an SB does not produce a step at the lateral surface, it is challenging to observe it by any topology-sensitive microscopic imaging technique. The offsets of plastic shear along the SB can be, however, easily noted on the lightly scratched surface, as exemplified by Figure 2d. This figure shows the topology map of the surface fragment represented in Figure 2a (the red line indicates the scratch intersecting the SB and the red arrows indicate 1μm plastic shear offsets associated with the bands).

The pure mode II shear is rarely observed. Many, if not most, SBs of this mode contain some contribution from the mode III component, which manifests itself as a surface step (Figure 2b). This step is nicely resolvable with aid from SWLI (Figure 2) where three shear bands marked #1, #2, and #3, respectively, are shown (SBs #1 and #2 are parallel to each other while SB #3 intersects SB #1 at the right angle). It has been shown that a mode II shear offset changes linearly along the SB from the maximum at the surface step to zero at the shear tip [27]. However, the mode III offset component within the primary mode II SB (Figure 2a) exhibits different types of behaviour, which can be seen as

(1) the monotonic decrease in a shear step height to a zero (SB #1),

(2) the non-monotonic decrease in the shear step height alongside the shear band (SB #2, notice the altering step direction in the middle of the figure),

(3) propagation blocking due to the SB-SB interaction (SB #3).

The offset of the off-plane component of the predominantly mode II SB can linearly decrease from maximum (step 7 ≈ 250 nm, Figure 2c) to zero at the tip of SB #1, similarly to the mode II offset reported in Reference [27]. The step direction can change the sign and becomes negative, c.f., SB #2, step 6 ≈ −10 nm (Figure 2c). Figure 2 illustrates the known fact [46] that one SB can block the propagation of another, c.f., SB #3 is terminated at SB #1 (Figure 2). One can see that SB #3 exhibits the step height 150 nm (step 1) at the initiation site at the specimen edge, and it ends sharply when meeting SB #1 with the step height of 42 nm (step 2). The site is featured by a local plastic pile-up of 90 nm (step 4) visible in the inset of Figure 2a between steps 3 and 5, being of 50–60 nm each. On the other hand, SB #1 propagates straight by 0.5 mm into the specimen with the gradually vanishing shear offset.

Considering the macroscopic homogeneity of the glassy microstructure, the fact that one SB can block the other, which has been frequently observed in abundant literature, is ye<sup>t</sup> to be understood. One of the plausible explanations to this blockage effect can be given based on the possible SB-SB interaction via their elastic stress fields. As mentioned in the introduction, the experimentally revealed nano-scale SBs heterogeneities create alternating tension/compression regions along the shear path [17,18]. Thus, the elastic field of the tip of SB #3 can interact with SB #1, which results in the effective blockage of shear bands.

If the SB-induced elastic stresses extend across the scales from nano-range to macro-range, one should be able to observe the following related effects.

(1) The long-range elastic stress field around the SB-tip (c.f., Figure 1) should influence the SB step morphology if the shear offset is out-of-plane (mode III).

(2) The local deviations should manifest themselves not only along the SB path but also in the shear offset.

(3) The SB propagation should be sensitive to the superposition of the local stresses and the self-stresses arising from the SB tip.

These effects are overviewed in Sections 3.3–3.5, respectively.

**Figure 2.** Typical behaviour of primarily mode II shear bands (SBs) in the compressed Pd-based bulk metallic glass (BMG). The 3D surface map (330 × 300 μm2) obtained with the scanning white-light interferometry (SWLI) technique (**a**) shows three SBs (#1, #2, #3) and their topology features. The site of the collision of SB #1 and SB #3 is shown on the inset (**a**) and obtained with scanning electron microscopy (SEM) (**b**) for comparison. Shear steps No. 1 to 7 heights, measured by SWLI and pointed on (**a**) are represented on the bar chart (**c**). The slope map reveals the breaks in the fine scratch (marked with red arrows) by witnessing the mode II shear (**d**).
