© S. Bouzayeni et al., 2025

1 Introduction

Metallic glasses (MGs) have attracted considerable attention owing to their unique combination of physical and tribological properties, including high strength and exceptional wear resistance [1]. Moreover, Zr-based MGs exhibit outstanding corrosion resistance in simulated body fluids, outperforming conventional biomedical materials such as 316L stainless steel and Ti6Al4V alloys [2]. These advanced materials have found increasing relevance in industrial sectors such as biomedical engineering particularly for joint implants and orthopedic fixation devices as well as in consumer electronics [3], where their superior performance can be fully exploited [4]. Despite their promising properties, the inherent brittleness and limited ductility of MGs under load-bearing conditions restrict their widespread application in demanding mechanical environments [5]. Recent advances in manufacturing technologies, including both additive and subtractive processes, have enabled the precise fabrication of MG components with tailored geometries and dimensions [6]. Furthermore, protective coatings have been developed to enhance their mechanical performance, thereby expanding their potential applications [7,8]. However, due to the absence of a crystalline lattice and dislocation mechanisms, the plastic deformation of MGs is primarily localized within shear bands (SBs) when deformed below the glass transition temperature (Tg). Several strategies have been proposed to mitigate the near-zero tensile ductility caused by work softening within these bands, such as surface mechanical attrition treatment (SMAT) and the introduction of engineered surface grooves, both of which have demonstrated significant improvements in mechanical performance. For example, SMAT has been shown to increase the stored energy by up to 50%, resulting in enhanced micromechanical properties [9]. Additionally, the introduction of surface grooves and multiple SBs can alter the dominant deformation mechanisms, thereby improving both strength and plasticity [10]. Moreover, gradient rejuvenation techniques have been recently developed to control the relaxation state of bulk metallic glasses (BMGs), leading to enhanced ductility [11]. These advancements underscore the pivotal role of surface treatments in optimizing the mechanical performance of MGs for advanced industrial and biomedical applications.

Ball burnishing (BB) is a cold mechanical treatment method that improves surface quality and induces favorable compressive residual stresses (RSs), thereby enhancing hardness, wear resistance, and fatigue life [12]. This process involves a burnishing tool, usually a hard ball, which is pressed into the workpiece, resulting in a significant reduction in surface roughness and improved mechanical properties. The effectiveness of BB is influenced by parameters such as penetration depth, feed rate, and the burnishing force, which must be optimized for varied materials [13]. In other words, BB not only refines surface roughness but also increases hardness and fatigue strength [14]. Previous studies on Zr-based bulk metallic glasses (BMGs) have explored various surface modification techniques such as surface mechanical attrition treatment (SMAT) [15], surface grooving [16], and nanocoating deposition [17] to improve mechanical performance. However, the ball burnishing process has rarely been applied to amorphous alloys, and according to the literature, earlier investigation reported an increase of the compressive fracture strength of Zr-based alloys 1599 MPa to 1924 MPa as results of surface treatment [9]. These findings underscore the potential of the BB process in optimizing the mechanical properties of Zr-based MG. In fact, the burnishing treatment of Zr₆₅Ni₁₀Cu₁₅Al₁₀ shows that this process induces formation of spherical caps in the subsurface region, mainly due to the proliferation and interaction of SB for optimal conditions of (500N, 0.20 mm, and 60 mm/min), allowing a hardening mechanism to occur in the blocking SBs sites [18]. The purpose of this work is to shed further light on the underlying physical mechanisms during the BB of Zr₆₅Ni₁₀Cu₁₅Al₁₀ BMG. Microstructural analyses of deformed samples are carried out using SEM and AFM in contact mode. In addition, nanoindentation tests are performed to determine the variation of the hardness (H), reduced modulus (Er), and elastic recovery as a function of burnishing force (F), feed rate (f) and penetration depth (p). Moreover, a finite element modeling of burnishing process has been developed on ABAQUS/implicit integrating the Drucker-Prager micromechanical law of amorphous alloys to capture the equivalent strain and residual stresses fields.

The manuscript is organized as follows: Section 2 provides a detailed description of the materials and experimental procedures. Section 3 presents the finite element modeling approach. Section 4 focuses on in situ SEM and AFM characterizations, the evaluation of nanomechanical properties, and the corresponding simulation results, offering a quantitative discussion of the changes in mechanical behavior, particularly hardness, and the influence of shear band formation and residual stresses on overall performance. Finally, Section 5 summarizes the main conclusions of the study.

2 Material and methods

2.1 Material

The amorphous alloy Zr₆₅Ni₁₀Cu₁₅Al₁₀ (in at. %) was selected as the material for experimentation due to its favorable bulk glass-forming capacity, despite its deficiency in macroscopic ductility [19]. BMG plates measuring 3 × 2 × 80 mm³ were fabricated via vacuum casting of pure elemental combinations (purity exceeding 99.99%) utilizing a water-cooled copper mold in an argon environment. Subsequently, a grinding wheel was then used to slice the plates into 10 specimens, each measuring 8 mm in length. In an arc furnace, the ingot was remelted while electromagnetic stirring was used to promote chemical uniformity. The BMG plates’ amorphous structure was confirmed using Cu Kα radiation and X-ray diffraction analysis (Rigaku SmartLab).

2.2 Workpiece preparation and ball burnishing set-up

The burnishing process was executed using a specially engineered burnishing tool that is incorporated into a CNC milling apparatus, as depicted in Figure 1a. The spring serves as the fundamental element of the tool, where the parameters of stiffness and preload are pivotal in determining the spectrum of forces exerted during the burnishing process. A hardened chrome steel ball, with a diameter of 5 mm, is centrally located within the tool and is encircled by four smaller ball bearings, all fabricated from the same material as the spring pretension transmission component. These bearings enable the central ball to roll freely while ensuring that both ball and spring maintain precise contact. Figure 1b shows a complete description of the systematic burnishing process. The tool was subjected to compression using an electromechanical universal tester to establish a correlation between applied forces and burnishing tool displacement. To ensure the accurate positioning of the sample within the CNC milling machine, the specimens were fixed using double-sided tape. ESSOLUBE HD 15W-40 was the lubricant oil used between the burnished workpiece and the tool, characterized by a viscosity of 113 and 15.4 mm² s^⁻1 at temperatures of 40 °C and 100 °C, respectively [20,21]. All experimental procedures were performed in a single pass, with a designated burnishing feed of 0.5 mm.

Nanoindentation tests were performed using a Hysitron TI-700 UBI system, calibrated for load and displacement using a fused silica reference sample (modulus of elasticity = 72 GPa, Poisson’s ratio = 0.17). Calibration was verified before and after each test series in accordance with the Oliver-Pharr procedure. AFM measurements were performed on an Agilent 5500 in contact mode, using a 10 μm pitch silicon calibration grid to ensure vertical and lateral accuracy. Tool alignment and applied force were verified using an electromechanical universal tester, ensuring force accuracy of ±1 %. To ensure reproducibility, each experimental condition was repeated at least three times with identical parameters (force, feed rate, and penetration depth). The values reported for hardness, reduced modulus, and surface roughness correspond to the mean ± typical error. The low variability (<5%) observed between repetitions confirms the reliability of the measurements.

Fig. 1 (a): Ball burnishing tool, (b): Schematic illustrating the BB process applied to BMG sample.

2.3 Experimental design and analysis

The chosen numerical values, or levels, for the three process parameters were tested. Table 1 presents the experimental domain of the applied design, showing the factors and their respective levels used in the burnishing process for this study. In this process, three factors are considered, each with three distinct levels.

Taguchi’s method serves as a powerful and effective tool for designing products that perform consistently and optimally across various conditions. Using Taguchi’s approach, a test matrix was created, with the selection of the orthogonal array based on the degrees of freedom for each factor. Since three-level parameters are two operating degrees, an L9 orthogonal array was selected for the study. To better assess the individual effects of each parameter independently of other factors, the authors opted for an L9 orthogonal array, consisting of nine rows (representing the trials) and three columns. This setup is shown in Table 2. The response parameters selected to measure the influence of each BB parameter include Ra (average roughness), Rz (average maximum profile height of roughness), H (hardness), and Er (reduced modulus).

Figure 2 shows the response surfaces illustrating the variation in hardness H and reduced modulus Er as a function of the ball burnishing parameters: force, ball penetration and feed speed. Visual examination of the first row of graphs, which represents hardness, clearly shows that maximum hardness is reached for a force close to 500 units and a penetration depth of around 0.20. In addition, high hardness values are obtained for a feed speed of around 60 and a force of around 300, indicating an ideal range in the middle of these two parameters. At a penetration depth of approx. 0.20 and a feed speed of approx. 60, another notable hardness peak is observed.

Similarly, the reduced modulus (Er), illustrated by the bottom row of graphs, reaches it’s maximum when the penetration is around 0.15 and the applied force is around 400. When the force is around 400 and the feed rate is close to 80, high elastic modulus values are also observed. In addition, the highest Er values are obtained at a penetration depth of 0.20 and a feed rate of around 60. Overall, the ranges of 300–500 for force, 60–80 for feed rate and 0.20 for penetration depth appear to be the best processing parameters for improving both hardness and elastic modulus. These conditions are ideal for design optimization, as they are likely to deliver better mechanical performance. Based on these data, we can define that a force of 500 N, a penetration of 0.20 mm and a feed speed of 60 mm/min are the optimal ball burnishing parameters.

Fig. 2 3D surface response relative to the surface hardness (H) and the reduced modulus (Er): (a) Surface plot of H vs Force ; penetration, (b) Surface plot of H vs Force ; feed rate, (c) Surface plot of H vs penetration; feed rate, (d) Surface plot of Er vs Force ; penetration, (e) Surface plot of Er vs Force ; feed rate, (f) Surface plot of Er vs penetration ; feed rate.

3 FE modeling

3.1 Geometrical model

The 3D geometric model of the workpiece and burnishing ball used in finite element modeling (FEM) is shown in Figure 3. A simplified volume based on the actual specimen utilized in the experimental investigation represents the burnished sample to save computing time. To attenuate the effect of imposed boundary conditions, the workpiece’s width and thickness have been selected to match the actual dimensions of the treated part. To improve calculation efficiency, the part’s length is set to a small portion of the dimension utilized for testing in experiments. This allows at least four passes to be made to create a uniform region where findings can be made. The model dimensions were set to 3.5 mm (length) × 2 mm (width) × 2 mm (thickness), representing a representative burnished region derived from the actual experimental sample. The burnishing ball was modeled as a rigid sphere (diameter = 5 mm).

The FE software ABAQUS/Standard calculation code is used for numerical modeling. The lower surface of the part was fully constrained; symmetry conditions were applied to the side faces. The burnishing load was applied in the form of vertical displacement corresponding to a nominal force of 300 to 500 N, while horizontal translation simulated the feed motion (40 to 80 mm/min). The contact between the ball and the surface was assumed frictionless in order to isolate the mechanical effects of plastic deformation. To enhance computational accuracy, two mesh types were developed and examined in this study: a coarser mesh in the region beneath the ball’s penetration and a refined mesh in the ball-to-workpiece contact area. To determine the ideal number of finite elements, a mesh sensitivity analysis was carried out, evaluating the effects of mesh density on calculation code accuracy, calculation time and convergence. A refined mesh of 2,545 C3D8 elements was used in the contact region, with element sizes of 0.05 mm at the surface and 0.2 mm to balance accuracy and computation time. The mesh utilized in the ball model comprised 7,334 eight-node brick elements, 836 six-node wedge elements with six nodes, and 8,382 nodes with linear interpolation between nodes.

To evaluate mesh sensitivity in finite element calculations, we proposed a mesh refinement strategy used in the burnishing contact zone, where stress and strain gradients are most pronounced. Three mesh densities were evaluated: coarse (0.20 mm), medium (0.10 mm), and fine (0.05 mm) in the surface deformation zone under the ball. The effect of mesh density on the predicted maximum residual compressive stress (S22) and equivalent plastic deformation was quantified. The differences between the medium and fine meshes were less than 3% for residual stress and less than 5% for equivalent deformation, confirming numerical convergence. Based on this analysis, the fine mesh (0.05 mm) was selected for the contact zone, providing an optimal balance between computational efficiency and accuracy, as recommended in similar FE analyses of surface treatments [18].

Fig. 3 Three-dimensional geometrical model, mesh and boundary conditions.

3.2 Plastic flow model

The Drucker-Prager plastic flow criterion is a model that depends on hydrostatic pressure [22]. Initially developed to address the plastic deformation of soils, this criterion has been also utilized for polymers and non-crystalline materials.

It can be expressed as follows:

$$\sqrt{J_2}=\text{A+B}{I}_1\text{,}$$(1)

where:

I1 represents the first scalar invariant of the Cauchy stress tensor,

while J2 denotes the 2^nd constant of the deviator of the Cauchy stress tensor.

A and B are defined through experimentation.

The Drucker-Prager criterion was written in terms of equivalent stress as follows

$${\sigma }_e=\text{a + b}{\sigma }_m,$$(2)

where:

- σ_e is the equivalent stress,

- σ_m is the hydrostatic pressure,

- a and b are two material constants.

The principal stress tensor axes, this criterion can be written as follows

$$\sqrt{\left[{\left({\sigma }_1-{\sigma }_2\right)}^2+{\left({\sigma }_2-{\sigma }_3\right)}^2+{\left({\sigma }_3-{\sigma }_1\right)}^2\right]}=\text{A}+\text{B}\left({\sigma }_1+{\sigma }_2+{\sigma }_3\right)$$(3)

The plastic flow surface of the Drucker-Prager criterion is a more refined representation of the Mohr-Coulomb criterion’s plastic flow surface. It is typically formulated as a function of the cohesion constant c and the angle of internal friction ϕ:

$$\text{A}=\frac{2}{\sqrt{3}}\left(\frac{{\sigma }_t{\sigma }_c}{{\sigma }_t+{\sigma }_c}\right),\text{B}=\frac{2}{\sqrt{3}}\left(\frac{{\sigma }_t-{\sigma }_c}{{\sigma }_t+{\sigma }_c}\right)\text{,}$$(4)

If the Drucker-Prager flow surface circumscribes the Mohr-Coulomb flow surface, the expressions for A and B are:

$$\text{A}=\frac{6c.cos\varphi }{\sqrt{3}\left(3+sin\varphi \right)},\text{B}=\frac{2sin\varphi }{\sqrt{3}\left(3+sin\varphi \right)}\text{,}$$(5)

If the Drucker-Prager flow surface inscribes the Mohr-coulomb flow surface, these expressions are:

$$\text{A}=\frac{3c.cos\varphi }{\sqrt{\left(9+3si{n}^2\varphi \right)}}\text{, B}=\frac{sin\varphi }{\sqrt{\left(9+3si{n}^2\varphi \right)}}\text{,}$$(6)

The mechanical and physical properties of this amorphous alloy used in numerical calculation, are described in Table 3.

4 Results and discussion

Cross-sectional analysis using SEM (Figs. 4a and 4b) showed a significant increase in the density and interactions of shear bands with increasing burnishing force and penetration depth. In the non-burnished sample, only isolated primary shear bands were visible on the side surface, with an estimated surface density of approximately 2–3 shear bands per 10 μm. In contrast, when burnished at 300 N, 0.15 mm, 60 mm/min, the number of visible shear bands increased to approximately 8–10 shear bands per 10 μm, while under burnishing conditions (500 N, 0.10 mm, 80 mm/min), the surface contained approximately 1,000 shear bands forming a dense semicircular network. The average spacing of the shear bands decreased from approximately 1.2 μm (low deformation) to 0.4 μm (high deformation), confirming that increased deformation promotes local plastic deformation and SB propagation. These intersections create microscopic regions of limited plasticity that contribute to work hardening. The resulting structural refinement explains the increase in measured hardness from 6.63 GPa to 23.68 GPa (cooked, 500 N − 0.20 mm − 60 mm/min). Furthermore, the apparent thickness of individual SBs estimated from high-magnification SEM images was between 80 nm ± 10 nm for primary SBs and 40 nm ± 5 nm for secondary SBs.

The density of SBs increased significantly, as a burnishing force increased from 300 N to 500 N. In this way, certain intersection zones of the primary and secondary bands in the overlap region of the two spherical caps (marked by II in Fig. 4b). The density of the SBs was higher than that of the BMGs burnished at 80 mm/min. Noteworthy is the fact that mechanical surface treatment by BB could also introduce latent SB under the ball paths. The SBs introduced under the ball trajectories can be activated to take account of plastic deformation during mechanical surface treatment, it is advantageous as well for the overall plasticity of MGs [25,26]. This phenomenon was proven by Yoo and Jang, who used the bonded interface approach to successfully witness the development of semicircular and radial SBs under pitting in spherical indentations [27].

Figure 5 shows 3D AFM images of the sample tested under different burnishing conditions. These images of the treated samples reveal a surface with a maximum roughness of 56 nm, indicating a good surface treatment quality attained by the BB process. Surface morphology display a distinctive pattern of parallel lines with a periodicity of approximately 10 µm, consistent with the burnishing direction. Surface topography exhibits a uniform pattern of shallow grooves and scratches, consistent with the bead burnishing process. However, several minor surface defects, including a few small voids and SB, are visible in the images. Overall, AFM analysis confirms the improvement in surface quality. Average roughness (Ra) and maximum height (Rz) decreased from 56.3 nm and 616 nm in the unfired state to 12.8 nm and 167 nm after burnishing, respectively approx. 77% reduction in Ra and 73% reduction in Rz. This smoothing effect is a result of localized plastic flow under the burnishing treatment, which promotes material redistribution and surface smoothing. The fired surface shows a uniform pattern of shallow grooves aligned with the firing direction, replacing the irregular topography seen in untreated samples. The shape of the micro-irregularities was more irregular compared to the unburnished surface (Fig. 5a). There were numerous visible deformations of the micro-irregularities that are not reflected in the burnishing kinematics. This is likely due to friction and adhesive interaction between the workpiece surface and the burnishing ball. Using a penetration depth value of p = 0.10–0.20 mm resulted in a deterioration of the part geometry. For F = 300–400 N, the figures showed visible surface micro-irregularities, with notable elevations and depressions on the surface. The values obtained for the Ra parameter were higher than the Ra values after BB. For the entire feed range tested, the values obtained for parameter Rz increased after the burnishing feed rate was increased (Figs. 5c and 5e). The minimum values for Ra and Rz were obtained for p = 0.10 mm.

Figure 6 illustrates the 3D topographic of the transverse section of burnished specimens from the nine experiment conducted. These observations reveal some intriguing structural features after the burnishing process has been applied. The lateral surfaces exhibit remarkable variation in surface roughness and increased homogeneity across the sample. Previously irregular features, such as nucleated SBs, voids and surface defects, are effectively attenuated, resulting in a uniform, visually appealing surface. The burnishing process facilitates material redistribution, leading to the formation of a refined microstructure with complex patterns and contours. Topographic examination shows a complex interplay of smooth regions, subtle undulations, and interconnected valleys, indicating the leveling and smoothing effect induced by burnishing, as shown in Figure 6a. This detailed characterization of the 3D topography provides valuable insights into the transformative nature of the burnishing process and its impact on the surface properties of BMG samples. Figure 6b shows the bi-dimensional profile of the SBs along the dotted yellow line in Figure 6a. It can be seen that this profile is characterized by shear offsets with heights varying from 50 to 125 nm.

In-situ AFM topographic analysis of the samples provides detailed information on the deformation mechanisms in the cross-section. Figure 7a illustrates the initial 2D topography of the balding zone of the spherical caps, where the SBs interact. White dotted arrows, with around 10 µm between each band, marked the multiplication and ramification of the SB process in this region. In addition, crossed and blocked interaction sites are clearly apparent in this zone, as shown in Figure 7b. Friction analysis reveals the presence of friction force variations between the AFM tip and the test surface as follows; blocked interaction locations are characterized by decreased friction (black zone), since the AFM cantilever is less torsioned. This suggests that a hard friction zone is present. On the other hand, because of the increased frictional power between the tested area and the AFM tip, the regions around the contact sites exhibit higher friction qualities, suggesting a softer zone.

The nanoindentation results (Fig. 8) reveal a remarkable strengthening effect induced by the ball burnishing process. Under optimal conditions (F = 500 N, p = 0.20 mm, f = 60 mm/min), the surface hardness increased from 6.63 GPa for the unburnished sample to 23.68 GPa, representing a 320% increase. Similarly, the reduced modulus Er increased from 118 GPa to 148 GPa, indicating a 26% improvement in elastic recovery. These quantitative results clearly demonstrate the sensitivity of mechanical properties to burnishing parameters, in particular the applied force and penetration depth. The substantial hardening observed is attributed to the accumulation of residual compressive stresses (≈ −900 MPa) and the generation of multiple shear bands that impede local plastic flow. This mutually enhancing effect between mechanical confinement and structural rejuvenation improves both the strength and elastic response of the surface layer of metallic glass.

Figure 9 shows nanoindentation load-penetration curves for treated samples with different BB parameters at an indentation depth of 188 nm and 195 nm. The results show that the irregular flow visible in the nanoindentation process decreases as the (p) increases, and that once a critical rate is reached, the irregular flow is completely suppressed, becoming imperceptible. This is reflected in the loading phase displacement curve by a transition from a wavy to a smooth profile, in contrast to the unburnished sample. Initially, within the first 10 seconds of loading, the load-penetration curves show a pop-in events, suggesting that deformation remains purely elastic before the first indentation (as shown in Fig. 9a) [28]. In the localized amplification of the curve, the burnished sample shows a pop-in serrated flow, with its serrated behavior breaking into multiple fine serrations of much wider width. This study demonstrated that compression introduced a significant number of SBs. The heterogeneous deformation created by this compressive mechanical loading can help coordinate BMG deformation during nanoindentation. At a low (f = 40 mm/min), the burnished sample continues to display irregular behavior, while the curve of the compressed sample is smooth. This indicates that nanoindentation deformation has activated a large number of existing SBs in the compressed sample, enabling uniform deformation [29].

A quantitative analysis of the burnished sample shows that a maximum indentation load of 9.5 mN corresponds to a penetration depth of 195 nm, with an apparent amplitude of the pop-in event flow of 15 to 20 nm per event, indicating discrete activation of the shear band and unstable plastic flow. In contrast, burnished samples showed markedly different responses depending on the process parameters. Under the optimal condition (F = 500 N, p = 0.20 mm, f = 60 mm/min), the maximum indentation depth decreased to ≈ 188 nm for the same applied load, confirming a 16–18% reduction in penetration and thus higher surface hardness. The amplitude of jagged flow events was reduced to less than 5 nm and the frequency of pop-in instabilities reduced by approximately 70% compared to the untreated surface, demonstrating more homogeneous plastic flow. This gradient in indentation behavior is quantitatively correlated with a trend in hardness from 6.63 GPa (unburnished) to 23.68 GPa (burnished) and an increase in elastic yield ratio from 0.056 to 0.160, indicating a threefold increase in elastic ductility. From a mechanical analysis perspective, the burnishing of deformation curves and the elimination of pop-in phenomena indicate that the burnishing process introduces pre-existing shear bands and a dense network of residual compressive stresses, promoting distributed activation of numerous shear bands.

Complete load-displacement curves of nanoindentation on burnished and unburnished samples revealed that a key aspect of the mechanical response during indentation was the appearance of intermittent plasticity during the loading phase, commonly referred to as “pop-in” events [30]. These notch flows were influenced by burnishing conditions, as each curve exhibited varying numbers and widths of pop-in jumps. This suggests that, during the early stages of indentation, the volume under the indenter subjected to high stress was probably too small to contain a sufficient number of SBs [31]. As the shear stress began to stabilize, the level of activated SBs reached saturation. The starts and ends of the pop-in events were identified at the troughs and peaks of the curves.

Figure 10 displays the finite element simulation results of equivalent plastic strain fields (Ԑ) generated under different burnishing parameters, providing a comprehensive visualization of localized deformation in the material. The Drucker-Prager plasticity model was selected for this analysis due to its ability to account for pressure-dependent yielding, making it particularly suitable for studying shear band formation in amorphous alloys. This model effectively bridges kinetic criteria (e.g., strain rate sensitivity and deformation kinematics) with thermodynamic principles (e.g., energy dissipation and free volume accumulation), offering a predictive framework for SB nucleation under compressive loading.

The simulation shows that as the burnishing ball traverses the material surface, it induces mechanical instabilities in regions characterized by lower atomic packing density, thereby promoting the early onset of strain localization. This phenomenon manifests through semicircular shear bands that propagate upward from the penetration point, following the path of maximum shear stress. The emergence of these shear bands marks the initiation of irreversible plastic deformation, with the strain field adopting a distinctive double spherical-cap morphology. This morphology arises from the hemispherical stress distribution imposed by the indenting ball. Furthermore, the analysis reveals the presence of an interaction zone where strain fields generated by two successive ball passes overlap. Within this region, strain accumulation is significantly enhanced, indicating that repeated burnishing cycles can lead to strain superposition effects, which may ultimately affect the surface integrity and mechanical performance of the material.

Stress triaxiality can influence shear band proliferation, potentially shifting the material response from softening to strain hardening [32]. This observation motivates an investigation into whether SB formation can be affected by the design of the RS distribution. Additionally, we find that SB interaction can serve as a mechanism for either stress softening or hardening [33]. Figures 10a and 10b illustrate the distribution of axial RS (S22 along the OY axis) and transverse RS (S33 along the OZ axis) in the burnished sample under the conditions of F = 500 N, a_p = 0.20 mm, and f = 60 mm/min. In all simulations, the highest RS values are observed at the surface layer of the part. The simulations are particularly sensitive to the shear angle in the Drucker–Prager material model, as well as to the plastic flow law employed. Notably, RS fields of opposite signs were observed in the deformed samples, indicating regions under either tension or compression. These simulation results highlight the dominant deformation mode in the weathered section. Due to the triaxial load imposed by the ball and its continuous movement, both longitudinal and transverse material flow occur, resulting in permanent plastic deformation after the load is removed. The displaced material beneath the contact zone, forming a spherical cap, as shown in the enlarged view, evidences this.

The finite element model presented successfully reproduces the experimental deformation behavior and residual stress distribution in Zr₆₅Ni₁₀Cu₁₅Al₁₀ metallic glass, certain simplifications have been introduced to ensure numerical feasibility. The model considers the amorphous alloy as a continuum with isotropic mechanical properties, neglecting local fluctuations in free volume and structural heterogeneities that may influence the nucleation of shear bands at the nanoscale [24,29]. It does not take into account the local temperature rises during deformation, which can affect the softening and relaxation behavior of metallic glasses at high deformation rates [31]. Frictionless contact was assumed between the ball and the sample in order to isolate purely mechanical effects; however, in actual roll-forming processes, lubrication and friction can locally modify the stress field [20]. The model was developed at the mesoscopic scale, focusing on representative deformation areas rather than the entire sample, which may limit its predictive accuracy for large-scale surface treatments.

On the other hand, the biaxial residual stresses obtained from nanoindentation measurements can be determined according to the procedure described in the recent study [34].

${\sigma }_y^r$is expressed as $k{\sigma }_x^r$using a stress ratio k, i.e., k= ${\sigma }_y^r/{\sigma }_x^r$, where k is between −1 and 1.

The biaxial stress is divided into equi-biaxial stress and pure shear stress. Since pure shear stress does not affect the indentation load, the measurement of biaxial stress can be simplified as an equi-biaxial problem. The biaxial stress can be calculated as follows:

$${\sigma }_x^r=\frac{3\left(P_0-P_1\right)}{\left(1+K\right)A_C}.$$(7)

To ensure measurement accuracy and overcome challenges associated with varying measurement trajectories, data acquisition was specifically focused on the steady-state deformation region. In the present investigation, following complete removal of mechanical loads, computational analysis of axial residual stresses (S22) was performed along Measurement Trajectory 1, as illustrated in Figure 11.

A comprehensive comparative analysis was conducted between simulated predictions and experimental measurements obtained under optimal conditions (F = 500 N, penetration depth = 0.20 mm, feed rate = 60 mm/min). The residual stress distributions derived from FE simulations showed excellent agreement with experimental nanoindentation data, exhibiting only minor deviations. Notably, the simulated S22 stress profile demonstrated particularly close alignment with experimental measurements, validating the computational approach.

The FEM simulations revealed maximum residual stress values approaching 900 MPa in the contact zone between the BB and sample surface, a magnitude that correlates well with experimental observations. These findings strongly suggest that the SB density distribution observed in burnished BMGs corresponds closely with SEM experimental data. This investigation provides compelling evidence that controlled surface deformation techniques can effectively engineer favorable residual stress states in BMGs, subsequently improving their mechanical behavior through optimized shear band formation and distribution.

The induced residual compressive stresses and reduced surface roughness significantly improve the mechanical performance, corrosion behavior, and fatigue resistance of these amorphous alloys, which are essential for orthopedic and dental implants. These improvements are consistent with the findings of Hasiak et al. [2] and Sawyer et al. [3], who reported similar benefits for Zr-based BMGs used in biomedical components. The combined increase in hardness and modulus of elasticity demonstrates that ball-burnished Zr-based BMGs can be used for micro-gears, molds, and high-performance optical devices, where dimensional stability and surface finish are critical. Unlike surface heat treatments or coatings, BB can be applied to industrial glass and metal parts, requiring minimal additional processing time or equipment.

The BB process can inhibit the propagation of branched SBs with large shear offsets. To gain a better understanding of the SB interaction mechanism, we carried out experimental AFM measurements of the height variation observed in the lateral surface, as well as numerical analyses of the profiles of the SBs inside the spherical cap and in the zone of their interactions, as shown in Figure 12. The profile obtained experimentally in Figure 12a indicates a shear offset in the band of around 45 µm. Furthermore, while some interactions are blocked, others continue unobstructed. To identify changes in the SB profile, calculated displacement data U3 (OZ axis) along the measurement trajectory on the sample are presented in the computed curve. We have observed that the bands move laterally, forming a stepped pattern along the measurement trajectory (see Fig. 12b). Each 2D height profile shows a shear offset of 20 µm. These computational results demonstrate the effectiveness of the numerical model in predicting the deformation mechanism and propagation process of SBs in MGs under the BB process.

By comparing our results to several recent studies on Zr-based bulk metallic glasses that have used alternative surface treatments such as surface mechanical wear treatment (SMAT), surface tracking and thin film coating [35,36]. Shayakhmetov et al. [15] and Yuan et al. [10] reported that SMAT and surface grooved treatments increased toughness and plasticity by 120–180%, primarily through localized shear band activation. In contrast, our ball burnishing process achieved a 320% hardness increase and a 26% increase in reduced modulus, indicating that mechanically induced compressive residual stresses play a stronger role than structural rejuvenation alone. Furthermore, the measured compressive residual stress (∼ −900 MPa) is significantly higher than that obtained by thin coating deposition (∼ −400 MPa) [7], confirming the superior ability of BB to engineer the stress field in amorphous alloys.

This article states that the interaction and propagation of shear bands under controlled firing conditions leads to stress redistribution and hardening, thus bridging the gap between macroscopic mechanical performance and microscopic deformation mechanisms. These results support older models of strain-assisted shear band multiplication in metallic glasses [25,18], but extend them by providing quantitative validation through finite element simulations with experimental nanoindentation and AFM analysis. The findings highlight that ball burnishing can serve as a non-thermal surface engineering technique that can simultaneously improve the strength of BMG components without changing the amorphous structure [37]. This opens promising perspectives for biomedical and microelectromechanical applications, where high surface integrity and flexibility are important.

Fig. 4 SEM cross-sectional pictures demonstrating the transverse deformed zones of burnished samples at burnishing conditions: (a) F = 300 N, p = 0.15 mm, f = 60 mm/min; (b): F = 500 N, p = 0.10 mm, f = 80 mm/min.

Fig. 5 AFM comparison of surface topography of burnished samples processed under different burnishing conditions: (a) unburnished, (b): 300 N, 0.10 mm, 40 mm/min, (c) 300 N, 0.20 mm, 80 mm/min, (d) 400 N, 0.20 mm, 40 mm/min, (e) 400 N, 0.15 mm, 80 mm/min, (f) 500 N, 0.20 mm, 60 mm/min.

Fig. 6 (a) AFM picture of the cross section 3D topography under conditions (500 N, 0.20 mm, 60 mm/min), (b) 2D profile along the measurement line shows the SBs offsets.

Fig. 7 AFM pictures in contact mode on the Zr-BMG burnished sample’s lateral surface: (a) a lateral picture of the transverse surface (plane YZ) and (b) the associated frictional measurement at (500 N, 0.10 mm, and 80 mm/min).

Fig. 8 Variations of means values of: (a) Hardness, (b) Reduced modulus of burnished specimens as a function of nine experiments.

Fig. 9 Load-penetration curves for burnished Zr-BMG samples at different ball burnishing conditions for indentation depth (a): 188 nm, (b): 195 nm.

Fig. 10 FE simulations analysis: iso-values of; (a) the equivalent strain ε, residual stress in; (b) Axial direction S22, and (c) transversal direction S33 for (500 N, 0.20 mm, and 60 mm/min).

Fig. 11 Comparison of the computational and experimental profiles of the burnished surface’s axial residual stress S22 along measurement trajectory 1.

Fig. 12 (a) AFM analysis of shear bands offsets profile on the lateral surface of MG samples, (b) Displacement U3 along the measurement trajectory 1 and 2 for (500 N, 0.20 mm and 60 mm/min).

5 Conclusions

The impact of ball burnishing (BB) on mechanical performance of Zr₆₅Ni₁₀Cu₁₅Al₁₀ BMG is characterized using SEM, AFM in contact mode, nanoindentation tests, and finite element approaches, the followings key findings can be done:

• Ball burnishing (BB) effectively refines the surface of Zr₆₅Ni₁₀Cu₁₅Al₁₀ metallic glass, generating semicircular shear bands and spherical caps that promote localized plasticity and improve ductility.

• AFM analysis confirms a significant reduction in roughness parameters (Ra ≈ 12.8 nm, Rz ≈ 167 nm) while maintaining surface integrity, despite minor micro-defects.

• Optimal BB conditions (F = 500 N, p = 0.20 mm, f = 60 mm/min) yield a 320% increase in hardness and a 26% rise in reduced modulus compared with the unburnished state.

• FE simulations and nanoindentation measurements consistently indicate compressive residual stresses up to −900 MPa, validating the Drucker–Prager model’s predictive accuracy within an 8% error margin.

• The agreement between numerical and experimental data highlights the reliability of the FE approach for predicting deformation mechanisms and residual stress fields in amorphous alloys.

• The proliferation and interaction of shear bands under BB contribute to strain hardening and improved mechanical response through controlled stress localization.

Some perspectives for future work:

• Future studies should explore multi-pass and thermal-assisted burnishing to further enhance stress uniformity and ductility.

• In situ characterization techniques such as high-resolution TEM or synchrotron XRD could be used to better understand the evolution of shear bands at the nanoscale during deformation.

• Coupled atomistic–continuum modeling approaches may provide deeper insight into the relationship between free-volume dynamics and macroscopic hardening behavior.

Acknowledgments

The authors express their gratitude to Kuwait University General Facility Projects Kuwait University (project number GE 01/07) for technical help, and the USCR common service unit: FEI Q250 Thermo Fisher USCR-ENIM.

Funding

This research received no external funding.

Conflicts of interest

The authors have nothing to disclose.

Data availability statement

This article has no associated data generated and/or analyzed / Data associated with this article cannot be disclosed due to legal/ethical/other reason.

Author contribution statement

Conceptualization, S. Bouzayeni, F. Gharbi, T. BenAmeur, and K. J. Al-Fadhalah; Methodology, S. Bouzayeni, F. Gharbi, T. BenAmeur.; Software, S. Bouzayeni; Validation, S. Bouzayeni, F. Gharbi, and T. BenAmeur; Formal Analysis, S. Bouzayeni; S. Bouzayeni, X.X.; Resources, F. Gharbi, T. BenAmeur; Data Curation, S. Bouzayeni; Writing − Original Draft Preparation, S. Bouzayeni; Writing − Review & Editing, F. Gharbi, T. BenAmeur; Visualization, S. Bouzayeni; Supervision, F. Gharbi, T. BenAmeur; Project Administration, S. Bouzayeni; Funding Acquisition, T. BenAmeur.

References

All Tables

All Figures

	Fig. 1 (a): Ball burnishing tool, (b): Schematic illustrating the BB process applied to BMG sample.
In the text

	Fig. 2 3D surface response relative to the surface hardness (H) and the reduced modulus (Er): (a) Surface plot of H vs Force ; penetration, (b) Surface plot of H vs Force ; feed rate, (c) Surface plot of H vs penetration; feed rate, (d) Surface plot of Er vs Force ; penetration, (e) Surface plot of Er vs Force ; feed rate, (f) Surface plot of Er vs penetration ; feed rate.
In the text

	Fig. 3 Three-dimensional geometrical model, mesh and boundary conditions.
In the text

	Fig. 4 SEM cross-sectional pictures demonstrating the transverse deformed zones of burnished samples at burnishing conditions: (a) F = 300 N, p = 0.15 mm, f = 60 mm/min; (b): F = 500 N, p = 0.10 mm, f = 80 mm/min.
In the text

	Fig. 5 AFM comparison of surface topography of burnished samples processed under different burnishing conditions: (a) unburnished, (b): 300 N, 0.10 mm, 40 mm/min, (c) 300 N, 0.20 mm, 80 mm/min, (d) 400 N, 0.20 mm, 40 mm/min, (e) 400 N, 0.15 mm, 80 mm/min, (f) 500 N, 0.20 mm, 60 mm/min.
In the text

	Fig. 6 (a) AFM picture of the cross section 3D topography under conditions (500 N, 0.20 mm, 60 mm/min), (b) 2D profile along the measurement line shows the SBs offsets.
In the text

	Fig. 7 AFM pictures in contact mode on the Zr-BMG burnished sample’s lateral surface: (a) a lateral picture of the transverse surface (plane YZ) and (b) the associated frictional measurement at (500 N, 0.10 mm, and 80 mm/min).
In the text

	Fig. 8 Variations of means values of: (a) Hardness, (b) Reduced modulus of burnished specimens as a function of nine experiments.
In the text

	Fig. 9 Load-penetration curves for burnished Zr-BMG samples at different ball burnishing conditions for indentation depth (a): 188 nm, (b): 195 nm.
In the text

	Fig. 10 FE simulations analysis: iso-values of; (a) the equivalent strain ε, residual stress in; (b) Axial direction S22, and (c) transversal direction S33 for (500 N, 0.20 mm, and 60 mm/min).
In the text

	Fig. 11 Comparison of the computational and experimental profiles of the burnished surface’s axial residual stress S22 along measurement trajectory 1.
In the text

	Fig. 12 (a) AFM analysis of shear bands offsets profile on the lateral surface of MG samples, (b) Displacement U3 along the measurement trajectory 1 and 2 for (500 N, 0.20 mm and 60 mm/min).
In the text