On the creation of structured abrasive tools via multiple-pass rotary wire EDM: A geometrical model

Structured abrasive tools (SATs) are considered as one of the next-generation abrasive tool solutions due to their superior ability to transport cutting fluids into grinding zones to lower grinding temperature and therefore enable high-quality machined surfaces. There are several SAT fabrication methods including mechanical, electroplating, brazing, and laser-based methods. Mechanical methods cannot produce SATs with small-sized structures due to significant contact forces, while electroplating has poor controllability of abrasive grain allocations. Brazing requires special machines with high-precision motion control, while laser-based methods need significant efforts on laser parameter selection and optimization. With this, here, we present a multiple-pass rotary wire electrical discharge machining (MPRWEDM) method to address the aforementioned limitations. We also develop a theoretical model of the created kerf profile during the MPRWEDM so as to enable controllable fabrication of SATs. The model was experimentally validated, showing a decent relative error of 9.8%. The nonlinear multiple-pass effect was studied both analytically and experimentally. Based on MPRWEDM, not only the SAT with designed grooves but also the structured surface (having an array of pyramid geometries) generated by the SAT were successfully created, proving the great potential of MPRWEDM in controllable production of even more advanced tools.

Wire feed distance in Y direction (m) R c Single pulse discharge center radius (m) R d Plasma channel radius (m) r Wire electrode radius (m) t num Duration time of num discharge sparks (s) t off The pulse interval (s) t on The pulse width (s) v m Wire moving speed (m*s −1 ) v s Relative wire cut speed (m*s −1 ) w Wire feed distance in Y direction (m) ∆B n Increment of kerf depth BB N (m) ∆C n Increment of kerf depth CC N (m) θ The angle of extension line OC 0 and line OA (m) ω Workpiece rotation speed (rad*s −1 )

Introduction
As a new class of abrasive tools, structured abrasive tools (SATs) are establishing themselves as one of the most promising tool solutions in the fields of mechatronics, automotive, energy, and aerospace [1]. Li et al. [2] reported that SATs were increasingly employed in the grinding operations of difficult-to-machine materials, especially the materials having not only good strength at high temperatures but also poor thermal conductivity such as ceramics [3], glass, Ti/Ni superalloys [5], and carbon fiber-reinforced polymers [6,7]. At present, typical SAT applications can include (i) improvement of machined surface quality and reduction of grinding forces/temperature by enhancing cutting fluid transportation into the grinding zone [8], and (ii) the possibility to generate the structured surface in a fast and controllable way. In aspect (i), the grinding force of SATs was reduced by 25-35% in the grinding of hardened steel (100Cr6/ AISI52100 60HRC) in comparison to the non-structured tool based on Walter et al. [9], 35.9-43.3% for bearing steel (A534) based on Xiao et al. [10], 46.7% for titanium alloy (Ti-6Al-4 V) based on Ding et al. [11], 50.2% for carbon steel (A135) based on Tawakoli [12], and more than 61.5% for carbon fiber-reinforced ceramics (CFRC-I (1,200 HV0.1)) based on Azarhoushang [13]. And they all proved that the reduction of grinding force reduces tool wear, thus improving the service life of the SATs. Moreover, SATs were found to be effective to reduce grinding temperature and improve machined surface quality as well [14]. Sun et al. [15] established the grinding temperature model of SATs, and the results showed that the grinding temperature was reduced by 110 °C compared with the non-structured tool, and the reported machined surface roughness Ra and Rz were reduced by 20% and 15%. A similar roughness reduction of 5-20%, together with the subsurface damage depth reduction of 25%, was reported by Guo et al. [16] as well. Besides, Zhang et al. [17], Dabrowski and Marciniak [18] experimentally proved that low grinding temperature caused by SATs not only reduces the occurrence of grinding burns but also improves the microhardness of the machined surface/subsurface. Moreover, Cao et al. [19] claimed that the grinding temperature and grinding force were significantly reduced by 40% and 41%, respectively, using SAT ultrasonic vibration-assisted contour grinding, in comparison with conventional grinding, ultrasonic vibration-assisted contour grinding decreased wear by about 22% during the steady wear stage [20][21][22].
In aspect (ii), the introduction of SATs into grinding is considered as a fast, cheap, and repeatable way to generate structured surfaces by replicating their structured patterns on the machined surface [23] and has wide applications in biology, medicine, energy, and even daily life [24][25][26]. The array of structured surfaces produced by SATs showed not only the promising hydrophobic function by the wetting angle of 103° [27] but also drag reduction proved to skin friction reduction of 4% [28]. These special surfaces have a broad application prospect, such as glass surface anti-fogging, traffic light color self-cleaning, and ship surface lubrication/anti-fouling. The produced structures can also serve as lubricant reservoirs and have application in hydrodynamic bearings, based on which the lubrication performance can be enhanced, and therefore, bearing load capacity can be increased [29].
Although the above advantages of SATs, the fabrication, however, is quite challenging due to the good hardness of abrasive grains. To address this challenge, substantial academic and engineering efforts have been paid so far, and the proposed SAT generation strategies can be, in general, divided into (i) mechanical, (ii) chemical, and (iii) thermal methods.
Mechanical methods were naturally the first solution because of low cost and no need for special machines. Aurich and Kirsch [30] employed a milling operation to generate seventy shallow slots on the abrasive wheel surface, based on which the hydrostatic pressure of the cutting fluid at the grinding zone was doubled, and therefore, the boiling point of the fluid was increased and the fluid evaporation was minimized. However, this method has low efficiency and high process forces, so generating small-size structures is inconvenient. Kim et al. [31] designed a textured diamond crushing roller and then duplicated the roller texture pattern onto a conventional grinding wheel surface. In comparison with conventional tools, the machined surface roughness was separately reduced by 4-5 times for copper, 1.5 times for brass, and 3 times for Al6061 based on the produced SATs. Although the use of textured diamond crushing roller improves the efficiency, it limits the diversity of structure geometry and size. To fix this, some novel dressing techniques were used to engrave patterns or textures on the SATs, by employing the direct drive dresser based on Dewar et al. [32], electro-mechanical exciter based on Oliveira et al. [33], fly-cutting kinematic based on Denkena et al. [34], and multiple cutting edge tools based on Aurich et al. [35]. They all produced a variety of texture geometries on the abrasive tool surface, such as white noise texture, triangular wave texture, linear texture, and cross-linear texture. Moreover, Gavas et al. [36] provided a more in-depth understanding of the effects of texture geometry on grinding performances. The SAT with helical grooves having angles of 30° resulted in high machined surface quality and geometrical accuracy for low-hardness metals such as AISI 1040, while 45° for high-hardness steel such as AISI 52,100. However, these grooves only cover a portion of the tool's circumferential surface. Mohamed et al. [37] generated SAT with a shallow circumferential groove and proved that a grooved SAT can improve grinding efficiency by reducing the consumed power by up to 61%. However, the above study shows that in the process of preparing SATs using mechanical methods, tool wear and strong contact force are unavoidable.
In order to avoid mechanical contact forces during SAT fabrication, chemical methods were proposed as well and mainly included electroplating and brazing. Aurich et al. [38] introduced the masks onto the steel wheel hub and then electroplated abrasive grains onto the hub with a pre-arranged pattern and measured low grinding force/ temperature and grinding power of up to 40% during the grinding experiment. However, Yu et al. [39] and Lyu et al. [40] claimed that the random allocation of the grains on the tool surface based on the above technology would result in reduced machined surface quality and therefore introduced a bio-inspired phyllotactic pattern so that the machined surface roughness was reduced by 19%. Ding et al. [41] concluded that electroplating easily resulted in weak grain retainment and uncontrollable abrasive grain distribution and therefore proposed the brazing-based SAT fabrication method. The proposed Cu-Sn alloy powders with the addition of Ti, TiN, TiB2, TiB, and TiAl3 enabled the highstrength bonding between grains and wheel body, achieving the compressive strength of 879.3 MPa. To further improved the bond strength, Huang et al. [42] introduced ultrasonic vibration-assisted brazing, and the results showed that the residual stress at the bond bridges was reduced by 35.4%.
The concern of chemical methods might be low efficiency, poor controllability, and significant environmental issue, according to Deng and Xu [43]. To address this, a large number of laser-based thermal methods were explored in depth, and the basic principle was the same. A laser beam ablated a specific region of abrasive tools, where materials were removed by gasification, oxidation, melting, and decomposition [44,45]. The continuous-wave Nd:YAG laser was employed to successfully generate a series of designed slots on not only superabrasive tools with bronze [46], copper [47], and resin bonds [48,49] but also conventional tools such as Al 2 O 3 wheel [50,51], although obvious heat-affected zone (HAZ) can be recognized. To minimize HAZ, a commercial picosecond laser was employed to generate SATs. Thanks to the limited thermal effect, a series of small-sized structures with a geometrical accuracy of 0.5 mm and the profiled geometrical error of 9.5 μm were generated separately in the literature [52][53][54]. In addition, ultrashort pulsed laser ablation enables high geometry flexibility of abrasive grain, which is hardly possible with any other process. Guo et al. [55] further exploited the above idea and produced micro-groove arrays even on individual abrasives so that the machining ability of SATs was highly improved thanks to the fact that the rake angle of each ablated grain was changed from a negative value to a positive one. Unfortunately, the abrasive grains on the produced SAT surface tend to fall off. To fix this, Li et al. [56] introduced a carbon dioxide laser to structure resin-bonded diamond abrasive tool surface in order to reduce the mismatch of the thermal expansion between the abrasive and the bond agent in laser ablation process. Based on systematic experiments, SATs with variable texture patterns including tilt line, parallelogram, hexagon, triangles, and rectangular were successfully produced.
Based on the above, it might conclude that there are still substantial challenges for each SAT fabrication method. For mechanical methods, the significant contact force limited the generation of SATs having small-sized structures and resulted in intensive tool wear. For electroplating, the poor controllability of abrasive grain allocation led to an unstable grinding performance. For brazing, the need for special machines and a deep understanding of brazing filler materials hindered its wide applications. For laser-based methods, the proper selection of laser ablation parameters (such as laser wavelength, power, and scanning speed) for a multiplematerial abrasive-bond system still lacks effective solutions.
To fill this gap, this paper aims to suggest a new multiple-pass rotary wire electrical discharge machining (MPRWEDM) and propose a geometrical model which enables the controllable generation of SATs. In comparison with previous methods, the proposed electrical dischargebased method is non-contact and, therefore, ideally has no tool wear regardless of material hardness or strength. More importantly, MPRWEDM allows the creation of small-sized structures onto abrasive tool surfaces so that more superior abrasive tools with more complicated geometrical shapes can be produced [57]. The principle of MPRWEDM and the comparison with conventional wire EDM are clarified in the Section 2. In order to provide an understanding of the process kinematics, the geometrical model of the proposed MPRWEDM is established in the Section 3, based on which the geometrical profile of the proposed SATs can be accurately predicted. In the Section 4, the proposed model is experimentally validated, and a typical SAT application produced by the proposed method is given in the Section 5. The possible key findings of this paper are given at the end. Considering the proposed method is, based on the best knowledge of the authors, the first electrical discharge-based SAT fabrication method, this investigation is anticipated to provide both academic and engineering references for the development of nextgeneration abrasive tools.

Machining principle of the MPRWEDM method
In comparison with conventional WEDM, the uniqueness of the proposed multiple-pass rotary wire EDM (MPRWEDM) is that the material removal behaviors are based on a dedicated 4-axis motion system. As shown in Fig. 1a-c, the X, Y, and Z linear motion modules enable the three-directional relative motion between the wire electrode and the workpiece, while the A-axis motion module enables the workpiece rotation so that the simultaneous 4-axis relative motion can be generated by using the standard G-code. Although the physics of the MPRWEDM process (plasma discharge and material removal) still remain the same in comparison with conventional WEDM, MPRWEDM has uniqueness including the following.
(i) Kinematics. In conventional WEDM, the relative motion between the workpiece and any point on the wire electrode is the same (see Fig. 1e, where points B and C have the same moving speed relative to the workpiece), while in MPRWEDM, the relative speed is varied depending on the selected point position on the wire electrode (see Fig. 1g, where points E and F have the different moving speed relative to the workpiece). (ii) Kerf profile. Due to the above kinematics, the kerf profile shapes are also different for both conventional WEDM and MPRWEDM. In conventional WEDM, the relative speed between the workpiece and any point on the wire electrode remains constant. In MPRWEDM, however, the workpieceelectrode relative speed varies at different positions on the wire electrode when the workpiece is rotating. As shown in Fig. 1f, the distance ( R F ) between point F and the rotation axis is larger the distance ( R E ) between point E and the rotation axis ( R F > R E ). Therefore, the workpiece-electrode relative speed at point F is greater than that at E. Therefore, the workpiece-electrode relative speed at point F is greater than that at E. Therefore, more pulse discharge channels in unit time are formed at the lower relative speed position (see front point E in Fig. 1g), resulting in a greater number of nonrepeated pulse discharges. The plasma channel is more prone to be deformed and early to be ruptured at the higher relative speed position (see point F in Fig. 1g). This leads to more material removed at a lower relative speed position (front point E in Fig. 1g) than that at higher relative speed position (point F in Fig. 1g) according to Haddad and Fadaei Tehrani [58] and Wang et al. [59]. This results in an elliptical-shaped kerf profile in MPRWEDM (a similar observation was also identified by Qu et al. [60]) rather than the circular-shaped kerf profile in conventional WEDM.
(iii) Discharge area. Due to the unique kinematics, the discharge area in conventional WEDM is a part of the cylindrical surface, while the intersecting surface between the wire electrode volume and the workpiece volume in MPRWEDM. Moreover, the discharge area in conventional WEDM would keep constant regardless of the instantaneous cut depth a p , while the discharge area in MPRWEDM varies (see the constant a p1 in Fig. 1d and the varied a p2 in Fig. 1f). (iv) Multiple-pass effect. Please also note another uniqueness of MPRWEDM is the multiple-pass effect, i.e., an increasing number of workpiece revolutions (namely pass number) would result in continuous material removal even when the wire electrode is not fed in but fixed at a specific position (see Fig. 2b).
More interestingly, the increment of the material removal depth is neither linearly related to the increasing pass number along a certain direction (i.e., in Fig. 2b, BB 3 ≠ BB 5 ≠ BB 7 ≠ BB 9 when the wire pass number is increased from 1 to 9), nor isotopically equal along different directions for a certain pass number (i.e., in Fig. 2b, BB 3 ≠ CC 3 , BB 5 ≠ CC 5 , BB 7 ≠ CC 7 , and BB 9 ≠ CC 9 when the pass number was separately 3, 5, 7, and 9). This process complexity indicated the strong and nonlinear effect of pass number on the produced structure shapes and sizes, which was rarely reported in previous investigations.

Modeling of the machined kerf profile in MPRWEDM
Based on the above analysis of the unique process kinematics and physical details in MPRWEDM, it might be necessary to propose a theoretical model of the full kerf profile of the obtained structure in this section so as to guide The model can be elucidated by two subchapters including the kerf profile calculation considering the wire is fed (i) along only the radial (see the Section 3.1) and (ii) along both the radial and axial directions of the workpiece (see the Section 3.2).

Kerf profile calculation when the electrode is fed along the radial direction of the workpiece
The kerf profile calculation when the electrode is fed along the radial direction of the workpiece should be divided into two situations depending on the relationship between the wire feed distance w and the wire electrode radius r. Please note that this paper aims to generate structured tools by MPRWEDM. In this process, the wire electrode is fed to the specific position and then stays at this position to discharge and remove material until the rotary workpiece rotates a number of turns. Therefore, in our model, the feed rate is not the input parameter.

Condition 1: w < r (see Fig. 3a, b)
When w < r is satisfied, the discharge process that happened on the wire electrode is part of a half-cylindrical surface with a central angle of less than 180 degrees. The kerf profile, therefore, in this case, would be related to the wire feed distance w in Y direction. Considering the different relative speed between any point on the wire electrode and the rotating workpiece and consequently the varied material removal depths (see EE' and FF' in Fig. 1g), the generic shape of the kerf profile is defined as a standard ellipse as Eq. (1). where the coefficients m and n are the unknown variables.
To solve m and n , at least two special points on the kerf ellipse should be fully defined, and they can be (i) the point B N in Fig. 3b, referring to the front point along the workpiece radial direction after N th ( N ∈ N * ) pass, and (ii) the point C N , referring to the intersecting point between the kerf profile after N th pass and the extension line of OC 0 , where the point C 0 is the intersecting point between the wire electrode and the original workpiece surface before MPRWEDM.
Based on Fig. 3b, the coordinates of the points B N ( x B N , y B N ) and C N ( x C N , y C N ) can be expressed by where points O and B 0 are separately the current wire electrode center and the front point on the wire electrode, the angle can be obtained by = arccos(1 − w∕r) , ΔB N and ΔC N are separately the material removal depths after the N th pass in comparison with the 1st pass profile along the directions of BB N and CC N and can be obtained by performing calibration tests, as shown in Fig. 2c, and k B and k C are separately the gap distance between the kerf profile and the wire electrode points B 0 and C 0 after the 1st pass in MPRWEDM.
In order to solve k B and k C , Tosun et al. [61] performed a series of experimental trials according to the Taguchi method and concluded the gap distance between the kerf profile and any wire electrode point can be expressed as where MRR is the material removal rate; h is the equivalent workpiece thickness, i.e., the contact height between the wire electrode front edge along Y-axis and the workpiece and be calculated by h = 2 2 ; v sB and v sC are separately the relative moving speed between the wire electrode points B 0 and C 0 and the workpiece and be calculated by v sB = * L B 0 , v sC = * L C 0 , where is the workpiece rotation speed, L B 0 and L C 0 are separately the distance between the wire electrode points B 0 and C 0 and the workpiece rotation axis.
where V is the volume removed from the workpiece material after the 1st pass when the wire electrode was fed to the fixed point O and discharged, t is the time required for one rotation of the workpiece, which is related to the rotational speed of the workpiece and can be obtained by Eq. (7).
Volume removed from the workpiece material after the 1st pass V when w < r can be approximately calculated as integrating the area of material removal groove section AMB 1 N (as shown in Fig. 3a), expressed as Eq. (8).
The area of a part of an ellipse OC 1 B 1 C 1 ′ and triangle OC 0 C 0 ′ can be expressed as Eqs. (9) and (10).

Fig. 4
The kerf profile in the RWEDM if the electrode is fed along both the radial and the axial directions of the workpiece when the previous kerf profile (a) does and overlaps with the current profile ( p ≤ K current + K next ∕2 ) and (b) does not overlap with the current profile ( p > K current + K next ∕2 ). The wire feed w in the Y direction in each groove generation was random ( 3.1.2 Condition 2: w ≥ r (see Fig. 3c, d) In the situation where w ≥ r is satisfied, the electrical discharge process that happened on the wire electrode is the halfcylindrical surface facing to the workpiece. Therefore, the kerf profile, in this case, would be a half ellipse as Eq. (1) and two straight lines tangential to the half ellipse (see the blue solid line in Fig. 3c, d).
The full profile in this case, therefore, can be defined by using a similar approach; however, there are something differences that need to be solved: (1) two special points on the kerf ellipse and (2) the volume removed from the workpiece material after the 1st pass V.
(1) Defining two special points on the kerf ellipse and then solving unknown parameters in Eq. (1). These   Table 2 two points can be (i) the point B N in Fig. 3d, referring to the front point along the workpiece radial direction after N th ( N ∈ N * ) pass, and (ii) the point C N , referring to the intersecting point between the elliptic-shaped kerf profile after N th pass and the left straight edge profile.
Based on Fig. 3d, the coordinates of these two points B N ( x B N , y B N ) and C N ( x C N , y C N ) can be expressed as where the points O , B 0 , and C 0 are separately the current wire electrode center, the front, and the left points on the wire electrode.
(2) The volume removed from the workpiece material after the 1st pass V in condition 2 when w ≥ r can be calculated as integrate the area of material removal groove section MC 1 B 1 N (as shown in Fig. 3c), expressed as Eq. (15) The area of the half-standard ellipse C 1 B 1 N ′ and rectangle MC 1 N ′ N can be calculated as Eqs. (16) and (17).
Similar to the above derivation, the kerf profile when w ≥ r can be expressed as Eq. (18).

Full kerf profile calculation when the electrode is fed along the radial and the axial directions of the workpiece
The application of the proposed MPRWEDM is to create a complex structured abrasive tool based on the following motion kinematics.
(1) The wire electrode is set to be tangent to the workpiece periphery surface; (2) the workpiece begins to rotate with the angular velocity ; meanwhile, the wire electrode is fed along Y axis with the distance of w 1 (see point are repeated several times until the wire electrode completely cuts through the whole length of the workpiece along the X axis, during which several grooves with the profile A 0 A 1 A 2 A 3 A 4 are generated (see Fig. 4a). The full kerf profile calculation when the electrode is fed along both the radial and the axial directions of the workpiece should be divided into two situations depending on the relationship between the wire feed distance p in X direction and the kerf width of current (denoted as K current ) and the next groove (denoted as K next ).

Condition 1: p ≤ K current + K next ∕2 (see Fig. 4a)
When p ≤ K current + K next ∕2 is satisfied, the previous kerf profile overlaps with the current one at the intersecting points (see A 1,2,… in Fig. 4a). The full kerf profile, therefore, is a piecewise curve as Eq. (19).
⋮ where x 1,2… are the X coordinate values of the intersecting points between two adjacent kerf profiles.

Condition 2: p > K current + K next ∕2 (see Fig. 4b)
When p > K current + K next ∕2 is satisfied, no kerf profile overlap was generated. The full kerf profile, therefore, is a piecewise curve as Eq. (20).

Experiments
In order to validate the proposed model for MPRWEDM, a series of validation experiments were performed in this study.
The cylindrical tungsten carbide rods having a diameter of 12 mm and a length of 80 mm were used as the raw material of SATs (see the detailed properties in Table 1), and the brass wire with a diameter of 200 μm was employed as the wire electrode. Before the experiments, WC-Co samples were properly polished until the rod surface roughness Ra achieved 0.5 μm, and the total runout was reduced to less than 2 μm so as to avoid any potential factors influencing the results.
During the experiments, all the trials were performed on a high-performance slow-speed WEDM machine (MV2400S, Mitsubishi) equipped with a dedicated 4-axis motion control system (see Fig. 5a), and deionized water was used as the dielectric liquid. The WC-Co workpiece was fixed on the chuck of the motion system and rotated around the A axis, and in the meantime, the wire electrode was fed along X and/ or Y directions by the 4-axis system. In order to validate the (20) proposed model with a focus on the multiple-pass effect, different pass numbers including 1, 3, and 5 were employed for each set of machining parameters, and three different sets of parameters were used to validate the model for the three cases with and without overlapping in X direction (see detailed parameters in Table 2 and schematic illustration in Fig. 5).
After the trials, the samples were ultrasonically cleaned in pure water for ten minutes and dried out in the air. For each sample, the experimental kerf profiles were measured at three different positions at 120 degrees apart by using a confocal laser scanning microscope (LSM 700, Zeiss AG). The three measured profiles were recorded, and the envelope region was plotted as a result. The morphology was observed by an optical microscope (NSZ-810, NOVEL) and a scanning electron microscope (SIGMA VP, ZEISS) so that the kerf profiles can be observed.

Model validation
In order to validate the proposed model, experimental kerf profiles and the ones calculated by the proposed model were compared in detail in three different cases depending on (i) whether the Y-direction feed w can be larger than the wire electrode radius r and (ii) whether the X-direction feed p can generate kerf profile overlap. The results showed that the experimental profiles in all three cases were, in general, in a good match with the calculated ones, no matter the pass numbers were 1, 3, or 5, to a large extent proving the proposed model's feasibility and accuracy.
Case 1 in Fig. 6 is the comparison when the wire feed in the radial direction of the workpiece was larger than the wire radius (i.e., w > r ), and in the meantime, the wire feed in the axial direction was small enough to generate kerf profile overlap (i.e., p < K current + K next ∕2 ). No matter the pass number was 1, 3, or 5, the large errors between the calculated and the experimental profiles were, in most cases, at the profile peaks and the maximum error was 17.44 μm (see Table 2 The machining parameters employed in the experimental trials  . 6 The comparison between theoretical and experimental kerf profiles in three different cases depending on whether the Y-direction feed w is larger than the wire electrode radius r and whether the X-direction feed p can generate kerf profile over-lap, where in case 1 w > r and p < K current + K next ∕2 , in case 2 w < r and p < K current + K next ∕2 , and in case 3 w > r and p > K current + K next ∕2 d p1,max in Fig. 6). These errors might attribute to the obvious material loss at the experimental profile peaks where the combination of thermal stress concentration at the peaks, flushing forces of the flowing dielectric liquid, and micro discharge spark force resulted in the insufficient support of the melted materials at the peaks (see the yellow area in case 1 in Fig. 6). Encouragingly, the maximum error rate was 7.0% in case 1 when comparing the error value with the kerf dimension. Case 2 in Fig. 6 is the comparison when w < r and p < K current + K next ∕2 . Similarly, no matter the pass number was 1, 3, or 5, the maximum relative error was at the profile peaks as well. The maximum error value was 9.44 μm (see d p2,max in Fig. 6) but accounted for only 6.7% in comparison with the kerf dimension.
An interesting observation of the experimental kerf profile in both cases 1 and 2 can be the obvious fluctuation at the profile valleys. This might be because the relative speed between the rotating workpiece and the wire electrode was the lowest at the valley, leading to more intensive discharge density during a unit period of time and, therefore, the generation of the recast layers with more spherical melted droplets and micro pores at the valleys. Encouragingly, the maximum and minimum relative errors at the valleys in case 2 were separately 4.1 μm and 2.5 μm (see d v,max and d v,min ) which can be considered acceptable, although the proposed model was based on kinematics and therefore can not describe these material micro behaviors.
Unlike cases 1 and 2, case 3 in Fig. 6 is the comparison in the case of w > r and p > K current + K next ∕2 , where no kerf profile overlap was generated. The calculated kerf profile was consistent with the experimental one, showing a maximum absolute error of 16.2 μm and a relative error of 4% in comparison with the profile dimension. Please note it should ideally have several vertical lines on the kerf profile (see lines E 1 e 1 , E 2 e 2 , E 3 e 3 , and E 4 e 4 in case 3 in Fig. 6) due to the employed machining parameters in case 3. However, no vertical lines were observed in the experiments, leading to the error of the slot wall thickness of 17.8 μm (see d w,exp. and d w,theo. ). Moreover, burrs were observed at the corners between the original sample surface and the kerf edge, leading to an error of 9.9 μm (see d b,max ). The occurrence of these two phenomena might attribute to the limited material removal thanks to the dragging force during the wire electrode retraction. Considering the slot wall thickness error only accounted for 9.8% in comparison with the experimental wall thickness while the burr-induced error accounted for 2.3%, it might still conclude that the proposed model can, to a large extent, describe the kerf profile in MPRWEDM.
Except for the profile shape, the calculated and experimental dimensions of the kerf profiles such as the peak-tovalley (PV) distance were also in line with each other. In cases 1 and 2, the maximum and minimum relative errors were separately 7.0% and 5.3%, and 6.7% and 4.2%.

Interesting multiple-pass effect in MPRWEDM
In comparison with other SAT generation methods, the uniqueness of MPRWEDM is the interesting multiple-pass effect, which has rarely been reported in previous studies.
For the geometrical aspect, the curious phenomena can include the following (as seen in Fig. 6).
Phenomenon (i). In case 1, with the increasing pass number, the PV value was reduced, i.e., H N 1 > H N 3 > H N 5 , and the reduction rate was also decreasing, i.e., Phenomenon (ii). In case 2, with the increasing pass number, the PV value was increased, i.e., h N 1 < h N 3 < h N 5 , but the increasing rate was decreasing, i.e., The above two phenomena can be explained by mathematically solving the residual heights between two adjacent overlapped grooves (see Eqs. (21) and (22)), their first derivatives in Eqs. (23) and (24), and their plots in Fig. 7). The PV distance decreased in case 1 and increased in case 2, while their first derivatives gradually approached 0 in both cases 1 and 2. This means that the greater the pass number was, the larger the kerf profile overlap rate was for a fixed X feed distance, according to Eqs. (25) and (26). In case 1, the Fig. 7 The PV distance and its derivative in (a) case 1 and (b) case 2; (c) the kerf profile overlap rate in case 1 and case 2 Fig. 8 (a, f) Global and (r, s) amplification of the cross-sectional morphologies of the machined kerfs when the pass numbers were separately 1 and 5, (b, g) the machined surface topography at peaks when the pass numbers were separately 1 and 5, (c, h) the detailed morphologies at peaks when the pass numbers were separately 1 and 5, (d, i) the machined surface topography at valleys when the pass numbers were separately 1 and 5, (e, j) the detailed morphologies at valleys when the pass numbers were separately 1 and 5, (k-o) the EDS mapping at valleys when the pass number was 5, and (p, q) the EBSD analysis at valleys when the pass numbers were separately 1 and 5 kerf profile overlap rate increased not only faster than that in case 2 (see Fig. 7c) but also faster than the growth rate of the kerf depth in the Y direction, resulting in the phenomena (i). On the contrary, the kerf profile overlap rate in case 2 increased slower than the growth rate of kerf depth in the Y direction and therefore resulted in the phenomena (ii).
Except for the geometrical aspect, the multiple-pass effect can be clearly identified from the morphological aspect as well (see Fig. 8).
Obvious metallurgical variance from the machined surface to the bulk material can be identified and generally classified into a white recast layer and a dark heat-affected layer (see Fig. 8a, f). The white layer was formed due to the rapid cooling of melted material by the dielectric fluid, while the formation of the dark heataffected layer was due to the change of microstructure and properties caused by the high temperature of spark discharge. Compared with the grooves obtained by different pass numbers in case 1, it can be seen that the overall surface roughness and recast layer and dark heataffected layer thickness of the groove obtained under a large pass number were separately smaller and thinner than that under a small pass number. This might be because (i) the greater the pass number was, the greater the distance between the wire electrode (cathode) and the workpiece (anode) was, which increased the energy consumed by the plasma region in the discharge channel so that the energy allocated to the workpiece electrode (anode) decreased (see Eq. (27)). The energy released by the electric spark discharge was mainly distributed in the form of heat energy at the anode and cathode. An instantaneous high-temperature heat source was generated, which affects the workpiece surface quality and the heat-affected layer thickness. As suggested by Eq. (28) (according to Yadav et al. [62] and Kansal et al. [63]), the heat flux model with Gaussian distribution was established to estimate the machining temperature field and thermal stress of the workpiece. Due to the reduction of the energy allocated to the workpiece ( F w in Eq. (29)), it led to the decrease of the heat flux acting on the workpiece according to Eq. (28). Therefore, the etched molten metal layer and the recast layer were relatively thinner. (ii) In the process of WEDM, the pulse voltage applied between the wire electrode and the workpiece formed an electric field. The intensity of this electric field decreased with the increase of the gap distance between the wire electrode and the workpiece. The breakdown voltage was proportional to the electric field intensity. Therefore, the greater the pass number was, the smaller the breakdown voltage ( U in Eq. 28) between the two electrodes was. According to Eq. (28), the decrease of breakdown voltage ( U ) can reduce the heat flux acting on the workpiece electrode, thus forming a thin heat-affected layer. (iii) Dielectric fluid might flow more fiercely in a large discharge gap, leading to most of the melted metal was washed away and therefore a part of the dispersed fine metal droplets was retained. The fine spherical droplets protruding from the machined surface under a large pass number enabled smoother surfaces than the ones under a small pass number.
where W plasmachannell , W anode , and W cathode are separately the energy consumed by the plasma, anode, and cathode.
According to Refs. [30,31], it has where q w (r) is the heat flux acting on the workpiece, F w is the energy partition to the workpiece, U is the breakdown (discharge) voltage, I is current, r is the distance from a point to the center of discharge, and R pc is the anode equivalent heat input radius.

Other unique phenomena in MPRWEDM
Except for the multiple-pass effect, other unique phenomena were observed in MPRWEDM as well.

(i) Surface roughness at peaks and valleys
Both the surface roughness and the recast layer thickness at valleys were larger than those at peaks for the same pass number (see Fig. 8b, d, g, i). This might be because (i) the relative speed between the wire electrode and workpiece at valleys was smaller than that at peak edges, which resulted in a smaller spark discharge removal area at valleys during the same period of time. Therefore, valley regions might be repeatedly discharged when the discharge energy was the same, and the continuous superposition of discharge pits caused the large valley surface roughness. (ii) More spherical droplets were solidified at valleys (see the label (IV) in Fig. 8c, e, h, j) because molten metal close to the rotation axis was difficult to be washed away by the dielectric fluid, leading to a thick and coarse recast layer at valleys, which can also be evidenced by a significant fluctuation at valleys, rather than peaks, in all three cases in Fig. 6.
(ii) Surface morphologies Significant microcracks were observed on the machined surface (see label (I) in Fig. 8c, e, h, j), and most of them were distributed at micro-hole edges (see label (II) in Fig. 8c, e, h, j). This might be because of (i) large thermal expansion differences between tungsten carbide and bonded phase cobalt at high temperatures, (ii) unevenly distributed stresses, and (iii) tensile strength reduction due to micropores.
Besides, material debris was also found on the machined kerf surface (see the label (III) in Fig. 8c, e, h, j), and the EDS results proved this debris was elemental carbon (see Fig. 8k-n). This might be explained by the fact that tungsten carbide with poor high-temperature oxidation resistance was heated and decomposed into W2C and C in the oxidizing atmosphere formed by the vaporization of deionized fluid at high temperatures. Two typical elements O (oxygen) and Cu (copper) were observed as well (as shown in Fig. 8o), where Cu might have come from the brass wire electrode while O was due to the oxidation reaction between the molten metal and the vaporized deionized fluid. The compounds including W2C, CuO, CoO, and Co3O4 therefore might be produced on the machined surface.

(iii) Microhardness
The increasing microhardness separately on the bulk material, the heat-affected layer, and the top recast layer was measured by performing a microhardness test. In a comparison of the bulk material hardness of 1300 HV, the recast layer had an increased microhardness of ca. 1500 HV, while the heat-affected layer had a hardness of 1420 HV (see Fig. 8r, s). These increasing microhardness phenomena might also be explained by the grain map from EBSD analysis showing a clear stratified pattern (see Fig. 8p, q). The grain sizes from large to small were observed separately in the bulk material, the heat-affected layer, and the top recast layer. It might be because the molten metal was cooled and instantly solidified by the flowing dielectric fluid, increasing the condensate depression and refining grains. The heat-affected layer had finer grains than the ones in the substrate. This is because, although this layer was not in contact with the dielectric fluid, the heat was transmitted from the top recast layer to this layer, reaching the quenching temperature. Both the refined recast layer and the heat-affected layer, therefore, might be beneficial to wear resistance performance, surface quality, and high-temperature stability if MPRWEDM is employed to fabricate SATs.

Model application on controllable generation of SATs
With the assistance of the validated model, the proposed MPRWEDM can be used in the generation of structured abrasive tools with accurate control of the cutting element/ feature position, shape, and size. Figure 9a shows a typical example of a formed abrasive tool by strategically changing the wire electrode pass number. On the peripheral tool surface, nine grinding edges were uniformly distributed along the tool axis direction with a fixed interval of 425 μm, while the edge tips were in a tilted line having an angle of 3 degrees in relation to the horizontal line. These edge tips were sharp, where the nose radius was less than 7 μm in size, showing the potentially good grinding performance of the abrasive tool. It is easy to imagine that small structured tools with such sharp edges might not be machined by using any other contact-based machining methods such as turning/milling because contact force might easily lead to the potential damage of either the sharp tips of SATs or the turning/milling cutters [64]. More importantly, the absence of the proposed geometrical model might also lead to failure in the accurate control of the texture geometries.
With the produced SATs, the functionally structured surfaces can be produced by successively performing two grinding passes along two perpendicular directions, as shown in Fig. 9b. Figure 9c-e shows the structured surface, where an array of pyramids having the incrementally changed shapes was generated. Please note this structured surface (having not only the macro geometrical feature such as the valley tips in the line having an angle of 3 degrees relative to the horizontal line but also the micro pyramid structures) might only be achieved by using the dedicated formed abrasive tools. The good consistency between the abrasive tool profile and the cross-section profile of the structured surfaces (see Fig. 9e) proved the strong fabrication ability of the formed abrasive tool in the generation of structured surfaces, especially on the special-shaped parts. This strong ability might be even more superior when considering the high machining efficiency. Such a structured surface having an area of 4 mm * 4 mm can be easily produced in no more than 10 s, showing the potential applications in the fabrication of large-scale-structured surfaces.
The application scenarios of SATs produced by MPRWEDM might at least include the fabrication of micro oil reservoirs used for lubrication on the inner ring of the bearings [65] in mechanical engineering, the generation of microlens array used for light concentration in optics [66], the production of hydrophobic/superhydrophobic surfaces used for waterproofing, drag reduction, and self-cleaning in biomimetic engineering [4], and the creation of micro-velcro mechanisms in MEMS.

Conclusion
In this study, an electrical-based fabrication technology of SATs, MPRWEDM, was suggested, theoretically modeled, and experimentally validated, followed by several applications. The key findings of this work include the following.
(i) The proposed MPRWEDM is an electric-based and contactless method, which is feasible and has advantages in the preparation of SATs. The created smallsized groove array structures have sharp edge tips with a nose radius of less than 7 μm in size. Such a structure is difficult to be produced by other existing methods. (ii) A theoretical model was developed which can enable the accurate prediction of the cut kerf profile. The model was verified by experiments, and the maximum error was only 9.8%, proving the feasibility and accuracy. (iii) Interesting multiple-pass effect was recognized and discussed in depth. From the geometrical aspect, the increasing pass number resulted in decreased or increased PV value depending on whether the wire electrode feeding distance was larger than the wire radius. From the morphological aspect, a large pass number resulted in a fine machined surface with smaller roughness values and a thinner recast and heat-affected layer. (iv) One formed SAT fabricated by the proposed method, together with the structured surface machined by this SAT, was given in the end, showing not only a large potential of the proposed MPRWEDM in producing specially structured abrasive tools but also the strong ability of the created SATs in generating structurally functionalized surfaces.
Author contribution All authors contributed to the material preparation, data collection, and experimental study. All authors commented on previous versions of the manuscript. The first draft of the manuscript was written by Bixuan Wang, and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.
Funding This work was supported by the National Natural Science Data availability Not applicable.
Code availability Not applicable.

Declarations
Ethics approval This paper is the original research that has not been published previously nor is under consideration for publication elsewhere, in whole or in part.

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Conflicts of interest
The authors declare no competing interests.