*2.2. Nanoindentation Experiment*

Nanoindentation was carried out by utilizing the Agilent G200 Nanoindenter system (Santa Clara, CA, USA); it can also be used for nanoindentation and nanoscratch, and works as a nanomechanical microscope. This experimental process was carried out automatically, according to a set of procedures to improve the reliability and competition of the experimental data. Within a total displacement range of 1.5 mm, the indenter can generate indentations deeper than 500 μm with a displacement resolution of less than 0.01 nm and a load resolution of 50 nN, respectively, under maximum load (standard) higher than 500 mN.

In nanoindentation experiments, single crystal SiC substrates were fixed onto the carrier plate using hot melt adhesives. The authors conducted a series of static sti ffness experiments on Si and C faces of 6H-SiC under loads increasing from 1 mN to 500 mN using a triangular pyramid Berkovich indenter produced by the Agilent Company (Santa Clara, CA, USA). The edge plane and edge of the indenter showed 65.3◦ and 77.05◦ angles with the center line, and the equivalent cone angle was 70.32◦. In this experiment, nine points were tested under each load at 3 a.m. to avoid any influences from the surrounding environment; after which the hardness *H* and the elastic module *E* of the points with maximum depth under static sti ffness were obtained directly from the software and their average values were calculated as experimental results after excluding particular data.

## *2.3. Fixed Abrasive Machining*

The grinding experiments using fixed grinding abrasives were carried out on the DMG-6011V vertical, uniaxial and high-precision end face grinder manufactured by the Lapmaster SFT Corp. (Tokyo, Japan). On this machine tool, cup-shaped grinding wheels were used to cut and grind the workpieces axially. The contact length, contact area, and cutting angle of the grinding wheels and workpieces were constant and half of the machined workpieces were always outside the grinding wheels. Furthermore, by using online thickness measuring devices, crystal plates were ground precisely and were not a ffected by any wear in the grinding wheels (Figure 1). The movement of the spindle of this machine tool was set at 0.01 μm per unit for its minimum rate of movement is 0.01 μm/sec, while the grinding thickness was controlled at 0.1 μm. The experiments utilized #325 metal-bonded and #8000 ceramic-bonded diamond grinding wheels, with deionized water as a coolant. The experimental parameters are shown in Table 1.



**Figure 1.** Principle of workpiece rotation grinding.

#### *2.4. Free Abrasive Machining*

The lapping experiments using free abrasives were carried out on the KD15BX (Dongguan KIZI Precision Lapping Mechanical Manufacture Co., LTD., Dongguan, China) precision plane grinder. This machine includes precision lapping plates, a trim ring, ceramic plates, cushions, clump weights, a retaining hook structure, a precision dressing machine, and a supply system of slurry. After being dressed for flatness, the lapping plates had a total thickness variation (TTV) of less than 10 μm and the parallelism of the upper and lower sides of the ceramic plates was less than 2 μm. The workpiece was fixed onto the ceramic plates with paraffin and faced to the lapping plate, while the trim ring was fixed outside the round ceramic plates. Clump weights were placed over and isolated with the ceramic plates by cushions. The pulley of the retaining hook structure was tangential with the outer cylinder of the trim ring so that the trim ring and ceramic plates could rotate due to friction between the retaining hook structure and the lapping plates. The experiments were carried out on the cast iron lapping plate and ceramic lapping plates using W14 and W1.5 diamond abrasives for 5 min as the lapping plates were rotating at 80 rpm. The mass fraction of abrasives in the slurry, flows of slurry, and lapping pressure were 4 wt.%, 15 mL/min, and 30 kPa, respectively.

## *2.5. Semi-fixed Abrasive Machining*

The MR finishing experiments with semi-fixed abrasives were carried out using the experimental equipment for MR plane finishing that was developed by the laboratory (Figure 2). This equipment utilized the servo motion of CNC milling machines and was controlled in the four directions (X, Y, C1 and C2) needed for the polishing process via programming. To analyze the material removal mechanism of single crystal SiC substrates under MR processing, N35 cylindrical permanent magnets (20 mm × 15 mm) with a flat bottom were arranged circumferentially on the polishing plates in a 135 mm diameter in the same direction as the magnetic poles to form a cupped polishing circle. The single crystal 6H-SiC substrates were 2 inches in diameter and were faced directly in the center of the magnetic pole by adjusting movement in the X and Y directions. The distance between the lower surface of the substrates and the upper surface of the polishing plates was set at 0.8 mm by controlling the Z axis. The

workpiece was static, while the polishing plates rotated at C1 = 200 r/min to machine the workpiece for 35 min, thus producing arc polishing belts on the workpiece. In the experiment, water-based MR fluids mixed with certain proportions of abrasives were used as MR polishing fluids. These MR polishing fluids consisted mainly of W3.5 carbonyl iron powders (4 wt.%), deionized water (88 wt.%), W3.5 diamond powders (4 wt.%), and stabilizer (4 wt.%).

**Figure 2.** Experimental apparatus for cluster MR effect in plane polishing.

The surface roughness was measured with a MarWin XT20 (Mahr, Goettingen, Germany) surface roughometer, and the surface morphology was analyzed by OLS4000 (Olympus Corporation, Tokoy, Japan) laser confocal microscopy and a scanning electron microscope (SEM) (Hitachi, Ltd., Tokoy, Japan).

## **3. Results and Discussion**

#### *3.1. Results and Theoretical Analysis of this Nanoindentation Experiment*

The experiments were carried out on the C and Si faces of polished 6H-SiC substrates by utilizing the static stiffness method under 1 mN, 2 mN, 5 mN, 10 mN, 20 mN, 50 mN, 100 mN, 200 mN, 300 mN and 500 mN loads. Each experiment was repeated nine times in a 3 × 3 array. The surface morphologies of the 6H-SiC substrates obtained from the experiment are shown in Figure 3 and the hardness and elastic moduli of the C and Si faces are shown in Table 2.

The results indicate that single crystal SiC substrates showed the obvious effects of indentation size, whilst due to the influences of surface roughness, oxide layers on the surfaces and elastic-plastic deformation of the workpieces, the hardness and elastic modulus gradually increased as the loads and depth of indentation increased in a certain range. Afterwards, due to the soft base of the hot melt adhesives used as test patches, the hardness and elastic modulus gradually decreased as the loads and depth of indentation increased. The hardness of static stiffness of C and Si faces under 500 mN loads were 38.596 Gpa and 36.246 Gpa, respectively, which were basically consistent with the experimental result (38 Gpa) of Chen et al. [19]. Moreover, the elastic moduli of the static stiffness of C and Si faces under 500 mN loads were 563.019 Gpa and 524.839 Gpa, respectively, which were slightly larger than the result (448 Gpa) of Mehregany et al. [20]; this was probably due to the soft base of the hot melt adhesives. Therefore, the C and Si faces of single crystal 6H-SiC showed slightly different mechanical properties and demonstrated obvious anisotropy. Furthermore, the hardness and elastic moduli of the C face were larger than the Si face, which indicated that the C face was more difficult to machine than the Si face.

**Table 2.** Hardness and elastic modulus of Single crystal 6H-SiC substrates by quasi-static indentation.


**Figure 3.** Surface topography of nanoindentation.

By using the static stiffness method, the load-depth curves between loads and indentation depth of the indenter are shown in Figures 4 and 5. Here, at 1.941 mN, a slight pop-in appeared on the C surface of SiC (considering the effects of the soft base of hot melt adhesives, the indentation depth was not used as a reference standard), while a micro pop-in was found on Si surface of 2.710 mN. Therefore, pop-ins under small loads were considered to be the turning point where materials changed from elastic to plastic deformation. Similarly, under higher loads, micro pop-ins appeared on the C and Si surfaces of SiC in 385.362 mN and 448.217 mN, respectively, showing that obvious brittle fractures occurred to the SiC materials under these loads.

**Figure 4.** Test results of Single crystal 6H-SiC substrates under small loads (**a**) Load and depth curve in small loads of C face and (**b**) Load and depth curve in small loads of Si face.

In the nanoindentation experiment, a single ideal diamond grinding grain acted perpendicularly onto the single crystal SiC substrates. For brittle materials, as the load increased, they went through stages such as elastic and plastic deformation, and brittle fracture. Elastic deformation was mainly shown as an increase or decrease of the interatomic distance where elastic recovery occurred after unloading, while plastically deformed materials cannot recover to their original shapes after unloading. Under the effect of plastic deformation, plastic flows and plastic budges were generated on the materials which caused the easy formation and accumulation of defects such as fault and slippage, and because these stresses were beyond the minimum stress required by crack extension, brittle fractures caused the materials to produce microcracks which travelled inside the materials. The transition from elastic to plastic deformation and then to brittle fracture was attributed to the pop-ins of critical stress and

strain on the materials. Therefore, the effective evaluation and test of the critical stress and strain of the materials provided an important basis for analyzing the material removed.

**Figure 5.** Test results of Single crystal 6H-SiC substrates under large loads (**a**) Load and depth curve in large loads of C face and (**b**) Load and depth curve in large loads of Si face.

According to the simple contact load model proposed by Stevanovic for predicting elastic, elastic-plastic, and plastic deformations [21]:

$$F\_o = \frac{9\pi^3}{16} H^3(\frac{\rho}{E}) \, ^2. \tag{1}$$

When the force on surfaces of the material is *F* ≤ 115*Fo*, it is considered that pure elastic deformation occurred, and elastic-plastic deformation occurs when the force is 115*Fo* < *F* < 15*Fo*; moreover, pure plastic deformation appears when the force is *F* ≥ 15*Fo*. In the Equation, *H*, ρ, and *E* represent the hardness of the materials, the radius of curvature of grinding grains on the materials, and the elastic moduli of the materials, respectively.

In Equation (1), the hardness and elastic moduli of single crystal SiC were obtained through the nanoindentation experiment, while the radius ρ of the curvature of the Berkovich indenter was unknown. When *F* = 15*Fo* was used as the critical load for the plastic transition of single crystal 6H-SiC, then

$$F\_C = 15 \frac{9\pi^3}{16} 38.5^3 \left(\frac{\rho\_C}{560}\right)^2 = 1.941 \text{ mN} \tag{2}$$

$$F\_{\rm Si} = 15 \frac{9\pi^3}{16} 36^3 \left(\frac{\rho\_{\rm Si}}{524}\right)^2 = 2.71 \text{ mN}.\tag{3}$$

By solving Equations (2) and (3), ρ*C* = 0.2 mm and ρ*Si* = 0.3 mm. As a single crystal SiC is very hard, when the Berkovich indenter was pressed into the surfaces of carbon and silicon in batches, it was worn, which further influenced the precision of the measuring results. The silicon surface was trimmed and calculated according to ρ*C* value, so theoretically the critical load for the plastic transition of the silicon surface under the same conditions was

$$F\_{S'} = 15 \frac{9\pi^3}{16} 36^3 (\frac{0.2}{524})^2 = 1.77 \text{ mN}. \tag{4}$$

Therefore, when the forces on the surfaces of single crystal 6H-SiC materials were *Fc* ≤ 8.63 μN and *Fsi* ≤ 7.87 μN, pure elastic deformation occurred, but if the forces were 8.63 μN < *Fc* < 1.941 mN and 7.87 μN < *Fsi* < 1.77 mN, elastic-plastic deformation appeared, and if the forces were *FC* ≥ 1.941 mN and *Fsi* ≥ 1.77 mN, pure plastic deformation occurred.

When the materials were processed by plastic removal, plastic flows which did not result in cracks were observed on the surface layers, but because the materials hindered the dislocation guide, while the loads increased constantly, many dislocations accumulated at a point to form a dislocation pile-up group. Once the force reached a certain limited value, the dislocation pile-up group induced the generation and extension of cracks, which turned into brittle removal [22], and therefore the critical loads needed for the materials to produce cracks were those for the brittle transition of materials, and the critical loads were mainly determined by mechanical properties of the materials. These mechanical properties included the hardness H, the fracture toughness Kc, and elastic module E of the materials. Based on the model of indentation fracture mechanics [23,24], the critical load P\* for brittle materials changing from plastic deformation to brittle fracture was expressed as

$$P^\* = 54.5(\alpha/\eta^2\gamma^4)(\mathcal{K}\_c^{\;4}/H^3) \tag{5}$$

where α (for ordinary Victorinox indenter, α = 2/π), η and γ (η ≈ 0.6, γ ≈ 0.1) are constants. *Kc* and *H* represent fracture toughness (generally, *Kc* is 1.9 Mpa m1/2 for single crystal SiC [25]) and hardness of materials. The theoretical critical loads for the brittle fractures of the C and Si faces of single crystal SiC are defined below respectively.

$$P^\*\_{\mathbb{C}} = 54.5((2/\pi)/(0.6^2 0.1^4))(1.9^4/38.5^3) = 366.82 \text{ mN} \tag{6}$$

$$P\_{\text{Si}} = 54.5((2/\pi)/(0.6^2 0.1^4))(1.9^4/36^3) = 448.67 \text{ mN}.\tag{7}$$

Note that the results of this theoretical calculation were consistent with those obtained in the nanoindentation experiment and can be used as basic parameters for analyzing the removal of materials machined by different fixed abrasives.

#### *3.2. The Results of Grinding with Fixed Abrasives and Materials Removal Analysis*

Since the grains in the grinding experiment were always fixed onto the grinding wheels, it is known as machining with fixed grinding grains (Figure 6). Figure 7 presents the surface SEM morphologies of ground 6H-SiC substrates. Note also that the diameter, feed, and linear speed of the grinding wheels influences the surface finish, so the morphologies of the machined surfaces obviously changed and the methods for removing material were also different. Under the No.1 process condition, most single crystal SiC was removed through brittle fracture and small amounts of plastic removal. After decreasing the feed of the grinding wheels in the No.2 process, brittle removal dominated and there was more plastic removal. As process No.3 shows, the linear speed of the grinding wheels affected the material removal modes, and the faster the linear speed, the larger the proportion of plastic removal. Although same process parameters were used in the No.3 and No.4 conditions, No.4 resulted in completely different effects where materials were mainly removed via the plastic mode because the grains in the grinding wheels had different diameters and densities.

**Figure 6.** Schematic diagram of fixation abrasive grinding.

**Figure 7.** Morphology of Single crystal 6H-SiC substrates ground with di fferent grades of grit (**a**) No.1 (Ra 0.303 μm), (**b**) No.2 (Ra 0.029 μm), (**c**) No.3 (Ra 0.015 μm) and (**d**) No.4 (Ra 0.002 μm).

Many studies have shown that the amount of material removed by grinding is related to the cutting depth *dc* of the grit, which in turn is related to the properties of materials and the maximum thickness of undeformed substrate *h*max. When the maximum thickness of undeformed substrate *h*max is less than the critical grit cutting depth dc, the plastic domain of grinding is achieved [26].

When grinding takes place the brittleness *H*/*Kc* of materials is the principal factor influencing the brittle-plastic removal of materials.

Bifano et al. [26] put forward the critical grit cutting depth dc for the brittle-plastic transition of material removal modes in grinding.

$$d\_c = 0.15(\frac{E}{H})(\frac{K\_c}{H})\,^2. \tag{8}$$

In Equation (9), *E*, *H*, and *Kc* stand for the elastic module, hardness, and fracture roughness of materials, respectively, so by substituting the experimental results above into Equation (8), the authors found that the critical grit cutting depth for C and Si faces of single crystal 6H-SiC were 5.3 nm and 6.1 nm, respectively.

During grinding, the maximum thickness of undeformed substrate *h*max denotes the maximum depth of grinding grit cutting into the workpiece. Apart from the size of the grit and the feed rate of the grinding wheels passing over the workpiece, the maximum thickness of undeformed substrate was also related to factors such as the speed at which the workpieces and grinding wheels rotate and the size of grinding wheels. During grinding realized by the self-rotation of the workpieces, suppose that the speed at which the grinding wheel and workpieces rotate, and the feed and radius of the grinding wheels are represented by *ns*, *nw*, *f* and *R*, respectively. *Lw* and *W* stand for the circumference and tooth width of layers of grinding material of the cupped grinding wheels. Furthermore, *C*, Fv

(when the diamond density was 3.25 g/cm3, *Fv* = 0.25 C) and *rm* indicate the concentration of grinding materials of diamond grinding wheels, the volume fraction of the layers of material in the grinding wheels, and the radius at any point on the surface of the workpieces, respectively. According to the study by Shang [27], the maximum thickness of undeformed substrate *h*max was:

$$h\_{\text{max}} = 2.239 R (\frac{4 f r\_m n\_w}{L\_w W C n\_2^2}) \,\text{}^{0.4}.\tag{9}$$

In this experiment, the radius *R* of the grinding wheels and the tooth width *W* of the layers of grinding material in the cupped grinding wheels were 100 mm and 3 mm, respectively. The *rm* value is the position of the edges of the workpiece (for a 2-inch diameter workpiece, r m was 25.4 mm). Moreover, the value of concentration *C* in Equation (9) can be obtained through simple geometric relationships, as shown in the following equation [28]:

$$C = \frac{4f}{d\_{\mathcal{S}}^2 \left(\frac{4\pi}{\mathcal{S}v\_d}\right)^{2/3}}.\tag{10}$$

In Equation (10), *dg*, *vd*, and *f* represent the equivalent spherical diameter of diamond particles, the volume fraction of diamonds in the grinding wheel, and the fraction of diamond particles that actively cut when grinding, respectively. The grinding wheel used in the present study had a density of 100, or in other words, the volume fraction *vd* was 0.25. To obtain the value of C, it was assumed that only one-half of the diamond particles on the wheel surface was actively engaged in cutting [28], or the value of *f* was equal to 0.5.

By substituting the experimental conditions and parameters in Table 1 into Equations (9) and (10), the maximum thicknesses of undeformed substrates from processes No.1 to No.4 were 47.59 nm, 9.95 nm, 7.56 nm, and 0.43 nm, respectively. It is obvious that the maximum thicknesses of undeformed substrates from No.1 to No.3 were larger than the critical grit cutting depth dc, and therefore the materials were removed by brittle fractures. As the maximum thickness of undeformed substrates decreased the proportion of plastic removal on the surfaces gradually increased, however, the maximum thickness of the undeformed substrate in process No.4 was smaller than the critical grit cutting depth dc, so the material removal modes were completely transformed into plastic flows.

#### *3.3. The Lapping Results of Free Abrasives and Analysis of Material Removal*

Free abrasives in the form of slurry which covered the entire surface were used for lapping the workpieces. These abrasives created typical two-body and three-body friction, as shown in Figure 8. In three-body friction the abrasives are dispersed and move freely between the workpiece and the lapping plate so they simultaneously rub against the workpiece and the lapping plate; this meant that the surface materials of the workpiece are removed through surface rolling and scratching resulting in a uniform and lustrous surface consisting of countless micro broken pits. In two-body friction the abrasives become embedded into the lapping plates and only rubbed against the workpiece; this means the loading pressure is transferred directly onto the abrasives. In this process, the abrasives cut deeper into the workpiece than during three-body friction, and the workpiece materials are mainly removed through ploughing and micro-cutting modes. This indicates that the material removal rate is higher under two-body friction [29].

Figure 9 shows the surface morphologies of the cast iron and ceramic plates lapped using W14 and W1.5 diamond abrasives; the surfaces of the workpieces consist of countless micro broken pits whose size depends on whether W14 or W1.5 abrasives used. This means that material was mainly removed by a larger number of brittle fractures which caused debris to break away from the material and leave many broken pits on the surface of the workpiece; this is where three-body friction occurred. The larger the diameter of the abrasives, the less abrasives per unit area and therefore a greater force

was exerted on single abrasives under the same loads, and the larger the material removed, the rougher the surface was.

**Figure 8.** Three-Body and two-body abrasion.

(a)

(b)

**Figure 9.** Morphology of single crystal 6H-SiC substrates after lapping. (**a**) After lapping by W14 diamond. (**b**) After lapping by W1.5 diamond.

In lapping, the space *hwp* between the workpieces and the lapping plates was mainly determined by the maximum diameter *d*max of abrasives in the space, so abrasives with diameters larger than *hwp* did most of the lapping. The distribution laws of the diameters of abrasives met the normal probability density function [30]. Suppose that abrasives with diameters larger than D50 (D50 of W14 and W1.5 was 11 μm and 1.2 μm) did most of the lapping, then abrasives filled all of the space between the workpieces and lapping plates. Based on indentation theory, the contact deformation when abrasives were pressed into the workpieces and lapping plates is shown in Figure 10. Assume that *H* and Fw indicate the hardness of the workpieces and the average pressure on single abrasive, then [31],

$$h\_{\rm uv} = \sqrt{\frac{F\_{\rm uv} k\_{\beta}}{H}}.\tag{11}$$

*K*β is the coefficient relating to the shape of the grinding grains. Suppose that the grinding grains are ideal spheres with a radius *dw*, the actual contact area between a single abrasive and the workpieces may be calculated as:

$$S\_A = \pi \left(\frac{d\_w}{2}\right)^2 - \left(\frac{d\_W}{2} - h\_W\right)^2 \,. \tag{12}$$

**Figure 10.** Contact state diagram of the abrasives, workpiece, and lapping plate. (**a**) Ideal contact state. (**b**) Actual contact state.

In accordance with the equilibrium condition of forces, the following equation is obtained.

$$p\text{S} = F\kappa K\_b (4S/\pi d\_w^2) \pi \left(\frac{d\_w}{2}\right)^2 - \left(\frac{d\_W}{2} - h\nu\_W\right)^2 \tag{13}$$

where *p*, *S*, and *Kb* show the lapping pressure, the workpiece area, and the distribution coefficient of abrasives between the workpieces and lapping plates, respectively. When the abrasives were distributed closely, the coefficient is 1. Combining Equations (11) and (13), the following equation was obtained.

$$F\_W = \sqrt{pd\_W H / 4K\_b K\_\emptyset}.\tag{14}$$

When *K*β and *Kb* were 1 and the nanoindentation results and grinding parameters were substituted [31], the forces of effective W14 abrasives on C and Si faces were 1723 mN and 1728 mN, separately, and the forces of effective W1.5 abrasives on C and Si faces were 569 mN and 588 mN, respectively. Obviously, where the abrasives completely filled the gaps between workpieces and lapping plates, the forces for pressing abrasives in workpieces under two process conditions were beyond the theoretical critical load for brittle fractures of the single crystal SiC substrates. However, many abrasives between the workpieces and lapping plates could not be involved in lapping, thus causing larger actual forces for pressing abrasives into workpieces. Therefore, in two process conditions such as three-body friction and rolling of abrasives, pure brittle fractures occurred on the surface of single crystal SiC, and the rolling and fragile broken accumulation removed workpiece materials and formed uniform and smooth surfaces consisting of countless micro broken pits.

#### *3.4. The Cluster MR Finishing Results of Semi-fixed Abrasives and Analysis of Material Removal*

In cluster MR plane finishing, small magnetic bodies are embedded into polishing plates made of diamagnetic materials according to the cluster principles. When MR fluids mixed with abrasives are poured onto the polishing plates, the upper magnetic bodies of the MR fluids are lined as chain structures along the direction of magnetic lines of forces to solidify and form semi-fixed Bingham viscoelastic micro grinding heads. This array of micro grinding heads regularly formed as polishing pads. When the workpiece is close to the polishing plates and moves along it, positive pressures and shear forces are generated in the contact areas such that the surfaces materials of the workpieces are removed. Owing to the semi-fixed and soft-constraint of viscoelastic MR fluids for the abrasives, the material on the surfaces of the workpieces are removed flexibly without damaging and scratching, so the polishing process is very efficient and results in super smooth surfaces [32].

Figure 11 shows the surface SEM images at the entrance and exit of the micro grinding heads in the polishing belts on the 6H-SiC substrates obtained from the experiment. Note the deep elastic-plastic grooves where the arc polishing belts entered the surfaces of single crystal 6H-SiC; these grooves were deeper at the entry edge and then became shallower towards the interior, and since there were no grooves where the arc polishing belts exited, the surfaces were smoother and flatter. However, some pits and pores generated by the lapping process remained but even though the polishing belts left deep grooves, there were no brittle removal scratches.

**Figure 11.** Scanning electron microscope (SEM) morphology of Single crystal 6H-SiC wafer polishing belt. (**a**) Schematic diagram of polishing, (**b**) schematic diagram of the polishing belt, (**c**) SEM morphology of entrance and (**d**) SEM morphology of exit.

The removal rate of material during polishing is generally related to the mechanical properties of the polishing pads. Yong's modulus of MR elastic polishing pads was ~1 Mpa(~2G ), which was much smaller than traditional polishing pads (~50–100 Mpa) [33]. Therefore, the abrasives constrained by magnetic connection were dislocated and deformed by the cutting forces, causing the accommodated-sinking effect. This was why the forces exerted by the abrasives onto the workpieces was small and the materials were removed from the surface of workpieces under elastic-plastic deformation.

Furthermore, in accordance with the Preston equation, the rate of removing MR material was proportional to the pressure on the surface of workpieces and the relative speed. As Figure 11a shows, the polishing pressure *PF* of the MR micro grinding heads on the workpieces was a complex parameter which included hydrodynamic pressures, and pressures produced by the MR effects and liquid buoyancy. Moreover, the pressures produced by MR effects consisted of magnetizing and magnetostrictive pressures, and since MR fluids are non-compressible, the magnetostrictive pressures in the magnetic fields induced by changes in volume were approximately zero, the expression for polishing pressures is

$$P\_F = P\_d + P\_\mathcal{F} + P\_\mathcal{m} \tag{15}$$

where *Pd* and *Pm* represent the hydrodynamic pressure and the pressure produced by MR effects of the MR fluids, respectively. Furthermore, *Pg* stands for the buoyancy of MR fluids, but it was ignored in the calculation because it was much less than *Pd* and *Pm*.

According to the studies of Feng et al. [34], the pressure produced by the MR e ffects of spherical magnetic particles in MR fluids due to external magnetic fields was calculated using the following equation.

$$P\_m = \frac{3q\mu\_0(\mu - \mu\_0)}{\mu + 2\mu\_0} \int\_0^{H\_m} H\_m \text{d}H\_m \tag{16}$$

where μ0, *Hm*, μ and ϕ represent the magnetic inductivity of a vacuum, the strength of the external magnetic fields on the surface of the workpieces, the magnetic inductivity of magnetic particles, and the proportion of magnetic particles in the MR fluids, respectively. Obviously, the pressure produced by the MR e ffects was directly proportional to the magnetic field strength H of the external magnetic fields on the surface of the workpieces.

In cluster MR polishing, because the workpiece and the polishing plates are parallel, there is no wedge pop-in, so in theory, dynamic pressures could not be formed and there were only the pressures produced by the MR e ffects. However, where the micro grinding heads entered the workpieces from the bottom surface of tool heads, a pop-in of height existed, and the dynamic pressures existing at the edges of polishing belts will meet the following equation [35].

$$\frac{dP\_d}{d\mathbf{x}} = 6\eta\nu \frac{h - h\_0}{h^3} \tag{17}$$

where η and *h*0 denote the initial viscosity of the MR fluids and the distance (i.e., *h*0 is the machining gap Δ) from the polishing plates to the surfaces of workpieces, respectively, then h represents the height from the polishing plates to the surfaces of tool heads. If the thickness of workpieces is *t* then *h* = Δ + *t*.

In machining, the material removal modes are only related to the maximum force of the grinding grains, which means the forces of grinding grains in the machining gaps can be transmitted and the forces of each grinding grain and carbonyl iron powder grain are basically consistent. Moreover, the number of grinding grains and carbonyl iron powder grains in the machining gaps in the same chain is Δ/*dw* so the abrasive forces at the entrance and the exit of the polishing belts in contact with the workpieces can be calculated approximately as:

$$F\_{\rm in} = \frac{\Delta}{d\_w} (\frac{3\rho\mu\_0(\mu-\mu\_0)}{\mu+2\mu\_0} H\_m + 6\eta\nu \frac{t}{\left(\Delta+t\right)^3}) \; \pi \left(\frac{d\_w}{2}\right)^2\tag{18}$$

$$F\_{\rm out} = \frac{\Delta}{d\_{\rm w}} \frac{3q\mu\_0(\mu - \mu\_0)}{\mu + 2\mu\_0} H\_{\rm m} \,\pi(\frac{d\_{\rm w}}{2}) \,. \tag{19}$$

When the data in Table 3 were substituted into Equations (18) and (19), the theoretical pressures of abrasives at the entrance and exit of workpieces were computed as 1514.27 μN and 3.47 μN, respectively; this meant they were in the mechanical range for single crystal SiC producing elastic-plastic and elastic deformations and therefore single crystal SiC materials were removed in full elastic-plastic mode, thus realizing machining without any sub-surface damage.

**Table 3.** Calculating Parameters.

