*3.4. Shear O*ff*set Fluctuations*

Considering the shear propagation mechanism proposed in Section 3.3, the nano-scale heterogeneities of the SB [17,18] should somehow accumulate with each segmental length increment. The regions of low density along the SB evolve to from nanovoids and microcracks, which leads to fracture [9,10,19,48]. This density fluctuation growth should then influence the observed SB step morphology. The SWLI technique allows us to detect such an influence, which manifests itself through the wavy fluctuation of the SB step height (Figure 4). The "old" SBs (i.e., bands formed by several shear events) exhibit these fluctuations, which are clearly visible on the tilted map (Figure 4a). Isometric view of the SB part zooms in the morphology and shows an inhomogeneous nature of BMG shear. The observed SB fluctuates both along its path up to ± 1.5 μm (Figure 4c) and the step height around ± 200 μm (Figure 4d).

**Figure 4.** Fluctuations of the shear band (SB) steps on the polished surface of the Pd-based bulk metallic glass (BMG) specimen tested in compression. The 3D surface map obtained with the scanning white-light interferometry (SWLI) technique is represented as a slope map, clearly showing a wavy character of SB steps (**a**). A magnified view of the area in a black rectangular (**a**) is shown in the isometric view below (**b**), where colour scale represents height Z in the range of 750 nm. Profiles of the SB step in XY and XZ planes are shown qualitatively with the dashed lines (**b**) and quantitatively on graphs (**<sup>c</sup>**,**d**), respectively, including the shear strain introduced by shear step deviations (**e**).

As suggested by Gilman, the observed path deviations up to 30% would produce the elastic energy increase of less than 10% [49]. The shear stress τxz introduced by the offset fluctuation can be estimated via the Cauchy's infinitesimal strain γxz tensor as:

$$
\pi\_{XZ} = G \cdot \gamma\_{XZ} = G \frac{1}{2} \left( \frac{d\mathbf{x}}{dz} + \frac{dz}{d\mathbf{x}} \right) = \frac{G}{2} \frac{dz}{d\mathbf{x}} \tag{1}
$$

where *G* = 35.5 GPa is the shear modulus of the Pd40Cu30Ni10P20 alloy [2]. The calculated local shear stress value varies in the range of ± 600 MPa (Figure 4e). It is worth mentioning that a real stress range is likely even larger, considering the of 45◦ angle between a shear plane and the scanned surface. Such large stress values up to 1 GPa result from the accumulation of several (up to 10 in observed "old" SB) successive shear events on one plane. This means that the stress oscillations along the SB, introduced by one mode III shear event, should be of an order of magnitude smaller, i.e., around 100 MPa. However, the calculation of strains performed by Binkowski et al. [17] shows that the strain variations occur in the range from +0.30 to −0.10 alongside an individual mode II shear band, which yields the local shear stress values up to several GPa.

Nevertheless, the above described quantitative estimations strongly sugges<sup>t</sup> the inherently fluctuating nature of the shearing processes occurring in BMGs. Local stress values along the SB are comparable with those at the SB tip, as shown in Section 3.1. This means that the SB stress interaction can occur not only around the SB tip but also along a whole SB length.

### *3.5. Shear Bands' Behaviour during Indentation*

Several indentation experiments were conducted on the surface of Zr-based BMG, as described in Section 2. Examples of morphology of indented surfaces are presented in Figure 5. A typical triangular pyramidal indentation footprint is surrounded by SBs forming stepwise pile-ups with the height variations observable on 3D maps (Figure 5d,e). The 20–30 μm wide pile-ups form near each side of the indent. To reveal the influence of shear deformation prehistory on the formation of SBs, the distance between the indentation centers was chosen to be close enough to ensure that the shear zones created by two neighbouring indents overlap. The first indentation in the non-deformed material is labelled #1 and the subsequent indentation is marked #2. Three pairs of indents were made so that the edges of their footprints were aligned parallel to each other at 43 μm, 35 μm, and 28 μm apart, respectively (Figure 5a–c). The second indentation was also performed with the indenter tip perpendicular to the edge of the first imprint (Figure 5f).

**Figure 5.** The topology of the surface in the vicinity of the indenter footprint reflecting the different behaviour of shear bands (SBs) formed by the indentation of the polished surface of the Zr-based bulk metallic glass (BMG). The optical images (**<sup>a</sup>**–**d**) and 3D surface maps (**d**,e) were obtained with the scanning white-light interferometry (SWLI) technique. The shear bands form regular arc-shaped steps when indentation is performed on a "fresh" non-deformed surface (marked as "1" on (**<sup>a</sup>**–**c**,**f**)). Indenting the area near the previous indents forms the shear bands deviated from arc-shape (marked as "2" on (**<sup>a</sup>**–**f)**). The scale bar refers to 50 μm on (**<sup>a</sup>**–**c,f)**. 3D surfaces of images on (**a**) and (**b**) are coloured according to the height magnitude (**d**,**<sup>e</sup>**). The schematics of SB deflection due to the stress field gradient observed in (**f**) is shown in the subfigure (**g**).

Indentation of the non-deformed BMG forms regular concentric arcs of shear o ffsets around the pyramidal imprint, indentations #1 in Figure 5a–c. The monotonic stress field gradient form accurately curved shear traces, which can be considered as iso-stress lines (Figure 5a–f, indentation #1). The picture changes significantly when the plastic shearing is enforced in the pre-deformed area. In this case, the formation SBs caused by the second indentation is driven by the superposition of their stress fields with the external stress field created by the indenting pyramid tip and the stress field existing in the specimen from the previously formed SBs around indent #1. The most clear e ffect is seen as shear branching and the dramatic change in the SBs morphology form a regular circle line to a set of segmented, significantly distorted lines (Figure 5a–f). Such a complex branched morphology cannot be obtained as a geometrical sum of slip events occurring in two-slip systems [50]. A new SB can swap its arc curvature to the opposite one from the previous indent (Figure 5a). This e ffect can be related to the reactivation of the "old" preexisting SBs, as it was observed in Reference [34]. The significant residual compressive stresses from the first indentation can even suppress the formation of new SBs when the edges of the indents are aligned to each other (Figure 5c). In the case of the normal orientation of imprints (Figure 5f), one can notice the deflection of the second set of SBs from the ideal circumferential shape in the vicinity of the previously formed SBs. The possible mechanism of this e ffect is sketched schematically in Figure 5g. Although the exact initiation point of the SB is not known, it is likely to be located on the plane perpendicular to the centre of the indent side as the point of the maximum stress. The propagating SB reaches the pre-deformed area where the stresses of both indents superimpose (Figure 5g, blue), and the resulting stress field at the SB tip shifts significantly toward the compressive direction. This forces the SB to turn in the direction of the compressive stress maximum.
