Micro-Grinding Parameter Control of Hard and Brittle Materials Based on Kinematic Analysis of Material Removal

Abstract

This article explores the intricacies of micro-grinding parameter control for hard and brittle materials, with a specific focus on Zirconia ceramics (ZrO₂) and Optical Glass (BK7). Given the increasing demand and application of these materials in various high-precision industries, this study aims to provide a comprehensive kinematic analysis of material removal during the micro-grinding process. According to the grinding parameters selected to be analyzed in this study, the a_c-max values are between (9.55 nm ~ 67.58 nm). Theoretical modeling of the grinding force considering the brittle and ductile removal phase, frictional effects, the possibility of grit to cut materials, and grinding conditions is very important in order to control and optimize the surface grinding process. This research introduces novel models for predicting and optimizing micro-grinding forces effectively. The primary objective is to establish a micro-grinding force model that facilitates the easy manipulation of micro-grinding parameters, thereby optimizing the machining process for these challenging materials. Through experimental investigations conducted on Zirconia ceramics, the paper evaluates a mathematical model of the grinding force, highlighting its significance in predicting and controlling the forces involved in micro-grinding. The suggested model underwent thorough testing to assess its validity, revealing an accuracy with average variances of 6.616% for the normal force and 5.752% for the tangential force. Additionally, the study delves into the coefficient of friction within the grinding process, suggesting a novel frictional force model. This model is assessed through a series of experiments on Optical Glass BK7, aiming to accurately characterize the frictional forces at play during grinding. The empirical results obtained from both sets of experiments—on Zirconia ceramics and Optical Glass BK7—substantiate the efficacy of the proposed models. These findings confirm the models’ capability to accurately describe the force dynamics in the micro-grinding of hard and brittle materials. The research not only contributes to the theoretical understanding of micro-grinding processes but also offers practical insights for enhancing the efficiency and effectiveness of machining operations involving hard and brittle materials.

1. Introduction

Recently, the demand for hard and brittle materials, including Zirconia (ZrO₂) and BK7 glass, has increased significantly, capturing heightened attention [1,2,3]. This interest is attributed to their desirable mechanical properties, such as their high bending strength, metallic luster, effective conductivity, elevated hardness, robust corrosion resistance, substantial modulus of elasticity, superior resistance to crack propagation, and minimal thermal expansion [4,5]. The exceptional qualities of Zirconia have led to its widespread use in precision applications, particularly in the fields of medical implants and cell phone bodies [6]. The material’s aesthetic appeal, exceptional hardness surpassing that of glass or plastic, near-complete resistance to corrosion, ability to be highly polished after shaping, and lighter weight compared to metals are not its only advantages [6]. Additionally, it facilitates the easy penetration of radio waves, resulting in stronger signals for Wi-Fi, LTE, or Bluetooth without the need for lengthy antenna cutouts [6]. Unfortunately, the hard and brittle properties of Zirconia ceramics have redoubled the machining difficulties used in these applications, and surface finishing and accuracy are of the utmost importance [7,8]. The grinding process is almost the only suitable process to duly machine Zirconia and is the proficient choice for machining hard and brittle ceramics [9]. The grinding wheel generates a small cutting force because the grinding wheel employs many small cutting edges instead of one or a few large cutting edges like those used in milling. However, the small cutting edges cause dramatic forces that mitigate crack propagation in grinding. Optimal control of the grinding conditions can effectively mitigate subsurface damage. The grinding process is primarily influenced by elements like the grinding feed rate, grinding wheel velocity, depth of cut, tool size, qualities of the workpiece, and grinding wheel. Among these factors, the grinding force is the most crucial determinant [8,10]. In order to effectively reduce the surface and sub-surface damage in the surface grinding process, the grinding force during processing should be identified before the actual machining [11]. Therefore, building an arithmetical model of the cutting force in the grinding process depending on the material and mechanical properties and grinding factors is indispensable [12] in order to explore the mechanism of grinding and prognosticate the grinding process [13]. As a result, studying the control of the grinding force model at an in-depth level is paramount in the grinding of brittle and hard materials [14], as it can ensure optimum surface and subsurface integrity and enhance the efficiency of machining at the same time [15].

Contemporary studies in grinding focus on the interaction between the workpiece and the grain edges, which in turn evolves the grinding forces [16]. Accordingly, a lot of research has been conducted to improve arithmetical models of grinding forces by considering the cutting force as a reaction between the grain of the grinding wheel and the workpiece. For instance, Zhang et al. [17] presented a theoretical model for the force of surface grinding depending on the mechanisms of material elimination and ductile accumulation and considering the effect of the lubrication conditions; moreover, they investigated the component of the frictional force by adding the cutting face to the grain wear plane. Bingyao Zhao et al. [18] used an extended finite element method to construct a model correlating the prefabricated crack depth with the final crack depth. This model elucidates the impact of diverse subsurface damages on the propagation of cracks. Li et al. [19] created a theoretical model of grinding force considering that the grinding force is acquired through the following three cutting stages between the grits and workpiece: rubbing, ploughing, and cutting stages. This model can provide a detailed understanding of the representation of the grinding force at the microscale. Wang et al. [20] developed a model for predicting the grinding force in multi-grain abrasive operations. This model facilitates the accurate prediction of grinding force, thereby reducing the error between theoretical and experimental values. Cheng et al. [21] developed three different orientations in micro-slot grinding for predicting the grinding force on single cutting grain from the wheel surface topography. This model canceled the effects of the mechanical properties of the grinding tool in modeling the grinding force. Chengzu Ren et al. [22] established a grinding force model for electrolytic in-process dressing (ELID) groove grinding, augmented by an arc-track feeding unit. The modeling process entailed the determination of swing speeds, swing angles, and duty ratios of the arc-track feeding unit, serving as foundational elements for predicting tangential and normal forces. Hui Fu et al. [1] introduced a novel predictive model for surface roughness (Ra) in silicon nitride ceramics grinding, leveraging the dynamic grinding force model. In a study by Longxu Yao et al. [23], the mechanical properties of a 2.5D woven SiO₂f/SiO₂ ceramic matrix composite were assessed utilizing the stiffness averaging method. Additionally, a theoretical predictive model was formulated for the axial and transverse components of the grinding force in rotary ultrasonic elliptical grinding (RUEG) of the 2.5D SiO₂f/SiO₂ composite.

Many studies have been conducted that consider the effect of the brittle removal regime in the grinding force model without also considering the influence of the ductile removal mode. For instance, Binhua Gao et al. [24] formulated an innovative analytical force model that incorporates the actual wheel-workpiece contact geometry (WWCG). In the modeling procedure, the cup grinding wheel is segmented into numerous micro-cutting layers along the wheel axis, and expressions for geometrical–kinematic parameters were derived to delineate material removal across various positions within the primary grinding zone (PGZ). Liu et al. [25] derived a model, based on Vickers indentation theory, for the machining force in ultrasonic rotary machining of brittle materials, in which it was determined that the brittle rip of material removal is the main mechanism. Li et al. [26] used the Hertz contact theory to calculate the a_c-cut for SiC ceramics at the transition point from the elastic to plastic region while relying on AFM and SEM during the transference from the brittle to ductile phase. However, the force models have neglected the influence of the ductile regime and the frictional force.

Furthermore, Yang et al. [27] experimentally verified a force model based on a single abrasive grit scratch trial. This study mainly concentrated on optimizing the parameters of the grinding process according to tangential UVAG, whereas the frictional effects were ignored in this work. Gan Li et al. [28] established a flow stress model for a tungsten-heavy alloy (WHA) that considered strain hardening, strain rate hardening, and thermal softening effects. This model facilitates comprehension of the variances induced by the disparity between the two phases during the grinding process, thereby furnishing a theoretical foundation for achieving efficient and minimally detrimental grinding of WHA and analogous composite materials. Li et al. [29] considered the random allocation of abrasive and strain averages in an arithmetical model of grinding force; however, the model neglected the influence of brittle materials Sun et al. [30] suggested an improved model for a grinding force that considered the grinding force in the ductile and fragile regions, and compliance of the friction force to both ductile and fragile modes, including the impacts of the friction process. Nevertheless, the model did not include the properties of the grinding tool material, and the ability of the unique grain to pass through the workpiece material, thereby varying the depth of the cuts.

Previous studies have shown that existing studies in the literature are insufficient in fully elucidating the effective removal and disposal of brittle materials during the grinding process. In light of this, the present investigation proposes an arithmetic model of machining force that takes into account the removal mechanisms of both ductile and brittle materials, frictional disposal, and that also considers factors such as the abrasive contact state and the random distribution of abrasive radii. Additionally, a novel technique is introduced to determine the slipping velocity of an individual grain and the dynamic contact line between the abrasive grit and the workpiece, while also addressing the limitations of the a_c-max calculation, which is considered here for the first time.

Variations in grinding forces are contingent upon several factors: the material properties of the grinding tool, the effective contact length between the workpiece surface and the grinding tool, and the penetrative capability of the abrasive grit into the workpiece material. The thickness reduction function, denoted as a_c-cr, is articulated in terms of the material properties, encompassing the modulus of elasticity, fracture toughness, and hardness of the material. Conversely, the maximum undeformed chip thickness, denoted as a_c-max, is delineated based on the grinding parameters for two consecutive cutting grits. As a result, analytical models have been developed to ascertain the total grinding forces in both ductile and brittle removal modes, which correspond to the respective regions of ductile and brittle material removal mechanisms. This study presents a new method for analyzing the interaction between a single grinding grain and workpiece materials. Through the analysis of the contact and projection areas, parameters such as the contact length between the active grinding grains and the machined surface, as well as the cone angle of each individual grain, are considered. In addition, the study analyzes the frictional coefficient that arises from the interaction between the binder and workpiece surface, the grinding chip, and abrasion with the machined surface. In addition, the research examines the transition from the ductile to the brittle mode in hard and brittle materials and assesses the amount of active grinding grains that participate in the cutting action throughout the grinding process. This complete study offers a more in-depth comprehension of the intricate dynamics associated with material removal during grinding, hence facilitating enhanced efficiency and accuracy in machining processes.

2. Kinematics Analysis of the Grinding Process

Identification of the grinding process parameters is critically important for accurately characterizing grinding forces within both brittle and ductile regimes. This paper elucidates the fundamental principles of the surface grinding process, along with the parameters associated with the grinding zone, as is schematically illustrated in Figure 1. In particular, the grinding wheel is subject to rotation at an angular velocity, denoted as N, while concurrently advancing toward the inner surface of the workpiece in the direction of the x-axis, at a feed speed represented by v_p.

2.1. Displacement and Relative Motion

As demonstrated in Figure 1 and relative to the (x − y) coordinate system, the relative displacement of the single grain to the workpiece is defined as:

$${\mathit{S}}_{\mathit{r}}=(r_g\sin \left(wt\right)+v_{P}t)\mathit{i}+(r_g–r_g\cos (wt\left)\right)\mathit{j}+\left(0\right)\mathit{k}$$

(1)

where $r_g$ is the mean radius of the grinding tool, $w$ is the angular speed of a single grain, $t$ is the grinding time, $v_P$ is the linear speed of the workpiece or the feed rate velocity. Equation (1) describes the relative displacement of the contact grit relative to the workpiece as a function of time, whereas Equation (2) symbolizes the velocity of the grain relative to the workpiece velocity (feed rate) as a function of time in (x − y) coordinates, where the magnitude value of this velocity at any time can be calculated by Equation (3) as follows:

$${\mathit{v}}_r=\frac{\partial S}{\partial t}=(r_{g}w cos(wt)+v_p)\mathit{i}+r_{g}wsin\left(wt\right)\mathit{j}$$

(2)

$$v_r=\sqrt{{v_P}^2+{\left(2r_{g}w v_{p}cos\left(wt\right)\right)}^2+{\left(r_{g}w\right)}^2}$$

(3)

where v_r is the relative speed of a single grain to the workpiece.

2.2. Touch Arc Length between a Single Grit and Workpiece

The touch lengths between the grinding tool and the workpiece are classified as the geometric touch length l_s, and the dynamic touch length in both brittle and ductile stages l_D−T, which are separately clarified in Equations (4) and (5), as follows, and displayed in Figure 2:

$$l_S=r_g{\mathit{cos}}^{-1}\left(\frac{r_g-a}{r_g}\right)$$

(4)

$$l_{D-T}={\int }_0^{t_T}\sqrt{{v_P}^2+{\left(2r_g w v_P cos\left(wt\right)\right)}^2+{\left(r_g w\right)}^2}dt$$

(5)

$$t_T=\frac{{\mathit{cos}}^{-1}\left(\frac{r_g-a}{r_g}\right)}{2\pi N}$$

(6)

where $a$ is the depth of cut as clarified in Figure 1, N is the rotational speed of the grinding tool in rpm, $t_T$ is the total factual time of the grinding wheel to intersect the total dynamic length of one cycle $l_{D-T}$.

Due to the randomization distribution of the abrasive grains on the surface of the grinding wheel and the difference in protrusion height, the movement trajectory of any abrasive grain involved in material removal is different during the actual machining process. Therefore, the current study considers the outermost abrasive particles of the grinding wheel as the active grain.

2.3. Identifying the Undeformed Chip Thickness

Figure 2 displays the geometric model of the removal material by a single diamond grit during the micro-grinding process. The material removal modes are classified into two regions, a ductile region and a brittle fracture region, specified according to the chip thickness of the material removal [31]. The UDCT of a single grit changes gradually during the grinding process. The transformation point from ductile to brittle rip is identified according to the value of the chip thickness [32]. When the amount of the a_c-cr is larger than the value of the a_c-max, the material is removed under the ductile region. However, the materials removed will behave in ductile and brittle blend manners when the amount of a_c−max is larger than a_c-cr [33].

2.3.1. Critical Chip Thickness Calculations

By assuming that the material properties will not be affected by the grinding parameters, the critical transformation point from ductile to brittle rip is calculated through the properties of the material as Young’s modulus $E_p$, hardness $H_P$, and fracture toughness $K_{IC}$. Thus, the critical depth of cut can be calculated through Equation (7) [34,35,36]:

$$a_{c-cr}=\epsilon \left(\frac{E_p }{H_P}\right){\left(\frac{K_{IC} }{H_P}\right)}^2$$

(7)

where ε is a constant factor taken as ε = 0.15 [37]. The variables of Equation (7) are determined as material parameters out of the material properties of the workpiece. Therefore, the brittle region of material removal can be determined by selecting the appropriate parameters.

2.3.2. Maximum Uncut Chip Thickness

The maximum UDCT of a single-grain a_c-max is located at the last touchpoint between the grinding tool and the workpiece. The a_c-max depends on the grinding wheel speed N, feed rate v_p, and the radius of the grinding wheel r_g, which can be expressed by Equation (8) [38,39], as follows:

$$a_{c-max}={\left( \lambda \frac{{60v}_P}{\pi N{r}_g}\sqrt{\frac{a}{2r_g}}\right)}^{\frac{1}{2}}$$

(8)

where λ defined as the distance among the active cutting grains, depending on the grain diameter D_G and grain concentration fraction ${\mathrm{V}}_{\mathrm{G}}$, and can be classified as $\lambda ={\mathrm{D}}_{\mathrm{G}}\left(\frac{1}{2}\sqrt{\frac{\mathsf{\pi }}{{\mathrm{V}}_{\mathrm{G}}}}-1\right)$ [17].

The grits are randomly distributed, thus the a_c-max is a dynamic concept in the grinding and every single active grit creates a different value based on the wheel properties and grinding conditions. Therefore, controlling the a_c-max is important. Consequently, maintaining the value of a_c-max as smaller than a_c-cr leads to the elimination of cracks and brittle fractures [40].

However, the a_c-max not only depends on the grinding wheel speed N, feed rate v_p, and the radius of the grinding wheel r_g, but also relies on the depth of cut a, and the actual space between the active cutting grains of the grinding tool. By reconsidering this, it can be seen that the a_c-max cannot be greater than the value resulting from the analysis of the maximum displacement S_max in the normal direction as $\left(\frac{\left(r_g-a\right)}{2\pi r_{g}N}{\mathit{cos}}^{-1}\left(\frac{r_g-a}{r_g}\right)\right)$. Consequently, the S_max limits the magnitude of a_c-max.

2.4. Cross-Sectional Area of Grain Cutting into Workpiece

Depending on the directions of the cutting forces that act through the workpiece, the cross-sectional areas are divided relative to those directions. As displayed in Figure 3, the actual cutting area A_cut is the total area of the UDCT that is expressed in the next section. In the (n − t) coordinates system, NT forces are acting on the workpiece during the grinding process. Consequently, identifying the projection area in both NT axes is very important when calculating the grinding forces in those directions.

2.4.1. The Actual Cutting Area of a Single Diamond Grit

As shown in Figure 3 and Figure 4, the actual cutting area A_cut depends on the abrasive diameter of a single grit D_G, semi-apex angle β, which is classified as $\left(\beta ={\mathit{cos}}^{-1}\left(\frac{D_G-{2a}_c}{D_G}\right)\right)$, the undeformed chip thickness a_c, and the dynamic touch length between the grinding tool and material $l_{D-T}$. As a result, the actual cutting area A_cut has resulted through $\left(A_{cut }=l_{D-T}{D}_{G }\beta \right)$ and is represented in Equation (9), as follows:

$$A_{cut}=D_G{\mathit{cos}}^{-1}\left(\frac{r_G-a_c}{r_G}\right)·{\int }_0^{t_T}\sqrt{{v_P}^2+{\left(2r_g w v_P cos\left(wt\right)\right)}^2+{\left(r_g w\right)}^2}dt$$

(9)

It is confirmed that the value of the semi-apex angle β is placed in the range $0\le \beta \le \frac{\pi }{2}$; therefore, any possible increase will be neglected. a_c is the chip thickness of the removal that can be taken according to the grinding phase, which is calculated in Equations (7) and (8).

2.4.2. The Projection Contact Area in NT Axes

The projected area of a single abrasive grain in both NT axes A_ct, as clarified in Figure 5, is defined in Equation (10), which will be analyzed in both NT axes. Equation (11) exhibits the projection contact area in the tangential axis ($A_{t-ct}$), while the projection contact area in the normal axis ($A_{n-ct}$) is represented by Equation (12):

$$A_{ct}=\underset{0}{\overset{r_G}{\int }}2\beta r_G d{r}_G-\left(r_G \mathit{sin}\left(\beta \right)\right)\times \left(r_G \mathit{cos}\left(\beta \right)\right)=\frac{1}{8}{D_G}^2\left(2\beta -\mathit{sin}\left(2\beta \right)\right)$$

(10)

$$A_{t-ct}=\frac{1}{8}{D_G}^2\left(2\beta -\mathit{sin}\left(2\beta \right)\right)sin(\beta -\theta )$$

(11)

$$A_{n-ct}=\frac{1}{8}{D_G}^2\left(2\beta -\mathit{sin}\left(2\beta \right)\right)cos(\beta -\theta )$$

(12)

where β is the semi-apex angle that varies with the chip thickness a_c, r_G is the mean radius of a single grain, and θ is the angular displacement of a single grit (rotational angle) expressed as $\left({\theta =\mathit{cos}}^{-1}\left(\frac{r_g-a}{r_g}\right)\right)$ (see Figure 2).

3. Modeling of Grinding Force

The determination of the grinding force poses a significant challenge owing to the intricate interdependence among various arbitrary grinding factors associated with the grinding wheel. Consequently, the present investigation has chosen to concentrate on establishing the grinding force specifically for a solitary diamond grain. Simultaneously, Equation (13) serves to quantify the collective contribution of active grains engaged in the cutting process within the contact region during grinding, and can be expressed as follows:

$$N_d=\frac{a_w{D}_G\acute{C}}{2\pi }$$

(13)

where N_d is the number of effective diamond grits, $\acute{C}$ is the abrasive concentration calculated as $\acute{C}=\frac{2\pi }{{\left(D_G+\mathsf{\lambda }\right)}^2}$, a_w is the grinding width, and D_G is the grain diameter.

The grinding force is identified as relative to (n − t) axes. As mentioned in the previous section, the chip removal mechanism is categorized as a ductile stage and a combination of brittle and ductile stages. Consequently, the machining force is fundamentally and separately created out of the elastoplastic deformation through the ductile manner or combination of the elastoplastic deformation and brittle fracture. Equations (14) and (15) describe the components of the NT grinding forces, respectively, that portray the sources of the grinding force at the ductile stage:

$$F_t=F_{t-ct}+F_{t-fr}$$

(14)

$$F_n={F_{n-cut}+F}_{n-ct}+F_{n-fr}$$

(15)

where $F_t$ is the overall force in the tangential coordinate, $F_{t-ct}$ is the tangential force in the contact plane due to the compression stress on the contact area, $F_{t-fr}$ is the friction force in the tangential plane, $F_n$ is the aggregate force in the normal axis, $F_{n-cut}$ is the cutting force in the normal axis, $F_{n-ct}$ is the normal force acting on the contact plane due to the compression stress on the contact area, and $F_{n-fr}$ is the friction force acting on the normal direction.

The grinding force is a complex phenomenon owing to several challenges faced during the cutting process, such as the disconnection of ductile material, brittle fracture, and diffusion, as well as the slipping friction. As a result, machining forces can be categorized by gathering the ductile material removal component in the ductile cutting stage; the brittle fracture component in the brittle cutting stage; and the friction factor. Equations (16) and (17) are utilized to quantify the brittle grinding forces, as follows:

$$F_t=F_{t-ct}+F_{t-fr}+F_{t-b}$$

(16)

$$F_n={F_{n-cut}+F}_{n-ct}+F_{n-fr}+F_{n-b}$$

(17)

where F_t-b is the brittle cutting force component in the tangential axis, and F_n-b is the brittle cutting force component in the normal axis.

3.1. Grinding Force Model in Ductile Region

In the elastic removal within the ductile region, the material is removed by the particle forming uninterrupted chips generated through the rubbing and ploughing labor of the diamond grain [41].

3.1.1. Cutting Force Equation Acting on the Cutting Area

The ability of the grinding wheel grain edges to pass through the workpiece material is one of the grinding forces that act in the UDCT of the elastic region. Equation (18) describes the cutting force model as expressed in Figure 6a:

$$F_{n-cut}=N_d {\sigma }_y A_{cut }$$

(18)

where ${\sigma }_y$ represents the amount of the compression yield stress of the workpiece at the touch area, which can be calculated by Equation (19) [42], as follows:

$${\sigma }_y={A\left(\frac{{H_p}^4}{E_p}\right)}^{1/3}$$

(19)

where A is a dimensionless constant taken as (A = 1.5) for hard and brittle materials [43]. Therefore, the resulting total cutting force directly acts on the cutting area due to the shearing force, as stated in Equation (20):

$$\begin{gathered} F_{n-cut}={A{N}_{d}D}_G{\left(\frac{{H_p}^4}{E_p}\right)}^{\frac{1}{3}}{\mathit{cos}}^{-1}\left(\frac{r_G-a_c}{r_G}\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}·{\int }_0^{t_T}\sqrt{{v_P}^2+{\left(2 r_g w v_P cos\left(wt\right)\right)}^2+{\left(r_g w\right)}^2}dt \end{gathered}$$

(20)

3.1.2. Normal and Tangential Contact Forces

According to Figure 6b, the contact forces in the contact area are produced by the pressure of the grinding tool. Equations (21) and (22) simply classify the contact forces in both NT coordinates, respectively, as follows:

$$F_{t-ct}=N_{d}P{A}_{t-ct}$$

(21)

$$F_{n-ct}=N_{d}P{A}_{n-ct}$$

(22)

where P is the pressure in the touch area that varies with the variation of the grain contact radius touched by the workpiece R_c, and where R_c is defined according to the UDCT. Subsequently, and based on Hertz’s theory, the pressure in the touch area is represented by Equation (23) [44], as follows:

$$P=\frac{2E_e{R}_c}{{\pi R}_e}$$

(23)

and the effective modulus of elasticity $E_e$ is classified as [45]:

$$\frac{1}{E_e}=\frac{1-{{\nu }_p}^2}{E_p}+\frac{1-{{\nu }_g}^2}{E_g}$$

(24)

where $E_p$, ${\nu }_p$ are the elasticity modulus and Poisson’s ratio of workpiece material, respectively. $E_g$, ${\nu }_g$ are the elasticity modulus and Poisson’s ratio of the diamond indenter material, respectively. R_c is the contact radius between single grit and workpiece calculated in Equation (25) as follows:

$$R_c=2\sqrt{a_c{D}_G-{a_c}^2}$$

(25)

R_e is the effective curvature radius for a single grit calculated as:

$$\frac{1}{R_e}=\frac{2}{l_S}+\frac{2}{l_D}$$

(26)

By substituting Equations (24)–(26) into Equations (21) and (22), F_n-ct and F_t-ct can be classified as:

$$F_{n-ct}=N_d\frac{2E_e{r_G}^2{R}_c}{{\pi R}_e}\left(2\beta -\mathit{sin}\left(2\beta \right)\right)\mathit{cos}(\beta -\theta )$$

(27)

$$F_{t-ct}=N_d\frac{2E_e{r_G}^2{R}_c}{{\pi R}_e}\left(2\beta -\sin \left(2\beta \right)\right)\mathit{sin}(\beta -\theta )$$

(28)

3.2. Grinding Force Model in Brittle Zone

In the brittle elimination zone, cracks are created and proliferated by a single grit in the dynamic touchline between the tool and the workpiece, clarifying that the fragile fracture governs the elementary abstraction mechanism. The grinding forces in the brittle phase are established relative to (n − t) coordinates in Equations (29) and (30), respectively [46], as follows:

$$F_{n-b}=N_d{F}_{max}$$

(29)

$$F_{t-b}=\mu N_d{F}_{max}$$

(30)

As shown in Figure 7, F_max represents the maximum contact force spent on the single grain via the maximum depth according to Hertz’s equation [47],, where µ is the friction coefficient. Thus, the maximum contact force is identified in Equation (31), as follows:

$$F_{max}=\frac{2E_p}{3(1-{v_p}^2)}\sqrt{2D_G{a_{c-max}}^3}$$

(31)

$$F_{n-b}=\frac{2{N_{d}E}_p}{3(1-{v_p}^2)}\sqrt{2D_G{a_{c-max}}^3}$$

(32)

$$F_{t-b}=\frac{2\mu N_d{E}_p}{3(1-{v_p}^2)}\sqrt{2D_G{a_{c-max}}^3}$$

(33)

3.3. Frictional Force Model

In general, the abrasive frictional force is produced due to the rubbing among the abrasive grits and the surface of the workpiece materials [48]. Based on Hertz’s theory [47], the components of the frictional forces in (n − t) coordinates are represented in the following equations [30]. In the surface grinding of hard and brittle materials, the frictional effects are created due to two effects between the abrasives and the grinding surface. The first one is produced by the flux of the workpiece over the touch face, whereas the other one is generated by the friction between the wear protrusion of grits and the grinding face (see Figure 8). The influence of grinding heat causes the coefficient of friction to vary with the varying conditions and behavior of the surface grinding. Therefore, modeling the coefficient of friction considering all these factors is a highly significant task.

$$F_{n-fr}=\frac{4}{3}{N}_d{E}_e\sqrt{{{R_{e}a}_c}^3}$$

(34)

$$F_{t-fr}=\mu F_{n-cut}+{\mu F}_{n-fr}$$

(35)

$$\begin{gathered} F_{t-fr}=\mu {A{N}_{d}D}_G{\left(\frac{{H_p}^4}{E_p}\right)}^{\frac{1}{3}}{\mathit{cos}}^{-1}\left(\frac{r_G-a_c}{r_G}\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}·{\int }_0^{t_T}\sqrt{{v_P}^2+{\left(2 r_g w v_P cos\left(wt\right)\right)}^2+{\left(r_g w\right)}^2}dt+\frac{4}{3}\mu N_d E_e\sqrt{{{R_{e}a}_c}^3} \end{gathered}$$

(36)

4. Results and Discussion

4.1. Verification of the Grinding Variables within the Estimated Grinding Force

After confirming the efficacy of the grinding force model, it is imperative to conduct further analysis to explore the impact of process parameters on changes in grinding force and the correlation between different parameters to optimize the process parameters. To this end, the spindle speed, feed rate, grinding depth, and grinding width were considered as experimental variables and various levels of each variable were selected in order to analyze their impact on the established grinding force model. The grinding forces were calculated by using the proposed model, where the input values were the grinding parameters, as described in previous sections. Figure 9 is generated from theoretical models throughout the grinding force model framework, illustrating the anticipated correlation between different parameters that affect grinding forces in micro-grinding processes on hard and brittle materials. The figure illustrates the variation in NT forces with the grinding parameters, including grinding width, grinding depth, grinding tool velocity, and feed rate. Figure 9a,b shows that both NT grinding forces nonlinearly increase with the increases in grinding depth at various feed rate values. Conversely, an increase in grinding width leads to an approximately linear increase in grinding force, as shown in Figure 9c,d. Furthermore, Figure 9e,f demonstrates that the grinding forces (normal and tangential) decrease nonlinearly with the increases in spindle speed at all feed rate values. However, an increase in feed rates results in a nonlinear increase in grinding force at various spindle speeds, as shown in Figure 9g,h.

In conclusion, an extensive analysis of cutting forces and their impacts on grinding parameters has revealed that the grinding process is directly affected by the following four factors: grinding depth, grinding width, grinding wheel speed, and feed rate. The increase in grinding depth, grinding width, and feed rate results in greater removal of material from the workpiece while reducing the grinding force. However, such a reduction in grinding force may lead to a degradation of grinding quality. Conversely, an increase in spindle speed results in a reduction in grinding force and a larger radial runout for the spindle.

Table 1. Grinding parameters for determining predicted grinding force.

NO	Feed Rate vp (mm/min)	Spindle Speed N (rpm) ×103	Grinding Depth, a (µm)	Grinding Width aw (mm)
1	6, 15, 30, 45, 60	10	5, 7.5, 10, 12.5, 15, 17.5, 20, 22.5, 25	3
2	6, 15, 30, 45, 60	10	10	1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5, 5
3	6, 15, 30, 45, 60	5, 7.5, 10, 12.5, 15	10	3
4	6, 15, 30, 45, 60	5, 7.5, 10, 12.5, 15	10	3

outlines the specific grinding conditions employed to validate the predicted grinding force model.

4.2. Experimental Set-Up

To validate the mathematical model of the grinding force predicted in Figure 9 and determine the experimental factors, surface grinding experiments were implemented through a hybrid ultra-precision micromachine (Carver PMS23-A8) (Dongguan Xiesheng Industrial Co., Ltd., Guandong, China) under a dry cutting process. The setup of the experiment is shown in Figure 10, where the apparatus used to measure the surface grinding force is the Kistler 9257B. The results of the grinding force were taken as the average value of two repeated experiments for every set of grinding parameters.

4.3. Grinding Tool and Workpiece Materials Properties

The workpiece material used in the current study is yttria-stabilized Zirconia ceramics ZrO₂, which essentially consists of 94 (wt)% of ZrO₂, 5.31 (wt)% of Y₂O₃, 0.2300 (wt)% of Al₂O₃, 0.0113 (wt)% of SiO₂, 0.0008 (wt)% of Fe₂O₃, and 0.0013 (wt)% of TiO₂. By examining the grinding force model discussed in the previous sections, it can be confirmed that the grinding force is primarily affected by the material properties of the workpiece, which are dependent on the physical and chemical properties of ZrO₂. Wang et al. [49] studied the effects of chemical composition on the physical and mechanical properties of ZrO₂ that used Y₂O₃ as a stabilizer. Flexural strength, fracture toughness, Vickers hardness, elastic modulus, and Poisson’s ratio changed with changes in the percentages of the material’s chemical composition, and the grinding force changed accordingly. The mechanical and material properties of ZrO₂ are listed in

Table 2. Properties of the workpiece material “ZrO₂”.

Property	Value
Density (g/cm3)	6.05
Young’s elastic modulus Ep (Gpa)	210
Poisson’s ratio ${\nu }_p$	0.3
Bending stress (Mpa)	950
Fracture toughness KIC (Mpa.m1/2)	10
Hardness Hp (Mpa)	1200

, where the dimensions of the workpiece are 30 × 20 × 5 mm³.

Where the grinding tool used in the present study is CBN with a mesh number of #S1000. The mean rate of abrasive tip radii is obtained through surface grooves, which are gained by using SEM as 7.5 µm approximately.

Table 3. Specifications of the grinding wheel used in the experiments.

Property	Value
Type	CBN
Young’s elastic modulus Eg	800 Gpa
Poisson’s ratio ${\nu }_p$	0.1
Abrasive concentration C	#S1000
Wheel diameter Dg	4 mm
Grain Diameter	15 µm

lists the material properties of the grinding wheel.

4.4. Grinding Force Model Verification

The grinding force model was verified at the macro level. According to the experimental grinding parameters listed in

Table 4. Grinding parameters of the experiments.

Exp. Group No	Feed Rate vp (mm/min)	Spindle Speed N (rpm) ×103	Grinding Depth, a (µm)	Grinding Width aw (mm)
1	30	10	5, 7.5, 10, 15, 20	3
2	30	10	10	1, 2, 3, 4, 5
3	30	5, 7.5, 10, 12.5, 15	10	3
4	6, 15, 30, 45, 60	10	10	3

, the a_c-max values are between (9.55 nm ~ 67.58 nm) which is below the chip thickness value at the transition point from ductile-to-brittle as a_cr = 145.2 nm; therefore, the ductile mode is the dominant mode for material removal. Experiments were executed to verify the magnitude of the grinding force that was predicted throughout the mathematical models. Through the statistics of the grinding experiment data and the theoretical data predicted by the model, a comparison histogram of the experimental value and the theoretical value of the grinding normal force F_n and the tangential force F_t were, respectively, drawn. Figure 11 illustrates a comparison between the experimental and theoretical values of the grinding parameters listed in Table 4, confirming the validity of the mathematical model. The results show that the grinding forces increase with the increases in grinding depth during surface grinding, with average deviations between the predicted and experimental forces of 6.616% and 5.752% for both NT forces, respectively. However, both NT grinding forces decreased significantly as the spindle speed increased.

The accuracy of the arithmetical models, the condition of the grinding wheel, the location of the workpiece, and the ability to purify the grinding force are critical factors that can affect the accuracy of the predictive force model based on the experimental outputs. These factors contribute to the discrepancies between the established force model and the experimental results. The study also highlights the sensitivity of the grinding force to the conditions of the grinding wheel and local surface morphology, which cannot be fully restored by the simulation model of the grinding wheel.

Furthermore, the results indicate that the experimental values of the grinding force are generally higher than the predicted values of the model due to the simplified shape of the abrasive particles in the modeling process and the neglect of the abrasive wear and generated heat during grinding. Despite these limitations, the established grinding force model has a certain degree of reliability.

4.5. Experimental Setup

In order to verify the effectiveness of the above method, a grinding experiment was carried out on the controlled peripheral grinding machine, CarverPMS23_A8. The workpiece material was BK7 optical glass, and the grinding wheel was PCD with a mesh of #700. The grinding wheel was trimmed before each test. No grinding fluid was added during the grinding process. The grinding force passed the Kistler 9257B dynamic force measurement. The instrument performed the measurement, and the experimental device is shown in Figure 12.

The hard and brittle material selected for use in the experiments was optical glass BK7, with dimensions of 50 × 20 × 3 mm³. The chemical ingredients and material properties of BK7 glass are listed in

Table 5. Chemical ingredients of BK7 glass (wt%).

Chemical Element	SiO2	B2O3	BaO	Na2O	K2O	As2O3
Ingredient	69.13	10.75	3.07	10.40	6.29	0.36

and

Table 6. Material properties of BK7 glass.

Material	E (GPa)	K (MPa.m1/2)	H (GPa)	ν
BK7	88.5	2.63	7.8	0.203

, respectively. The selected grinding wheel is was PCD with a mesh of #700; its specifications are listed in

Table 7. Grinding tool specifications.

Type	Mesh	Dt	Dd	Ed	ῦ
PCD	#700	4 mm	21.8 µm	800 GP	0.07

. Figure 13 displays the statistical analysis for tip diameter D_d obtained by calculating the mean value of the tip radii of the ground grooves in the grinding surface measured through AFM.

4.6. Experimental Force

Table 8. Experimental conditions.

Exp. No	vp (mm/min)	N (min−1) × 103	a (µm)	aw (mm)	Dg (mm)
1	60	10	10	1, 5, 10	4
2	150	5, 10, 15	10	5	4
3	60	10	5, 10, 15, 20	5	4
4	60, 100, 150	5	10	5	4

presents the grinding conditions for the experiments. A Kistler 9257B grinding dynamometer was used to gauge the grinding forces.

The grinding process is the proper process for machining hard and brittle materials to successfully achieve the required product quality [27]. However, there are many problems with respect to controlling and optimizing the grinding conditions during the grinding process. The essential factor is the grinding force generated due to the fractures and friction between the grinding tool and the workpiece surface [50]. During grinding, the grinding force varies with the variations in the frictional coefficient (COF). The laboratory experiments prove that the frictional coefficient changes with grinding conditions; therefore, considering the frictional coefficient as a dynamic value to optimize the grinding conditions is a primary task. Figure 14 briefly and experimentally presents the variation of the coefficient of friction with grinding distance during the micro-grinding process for hard and brittle materials.

The experiments were performed under dry grinding conditions, focusing on the grinding process. Each experimental group was used to investigate the influence of a particular process parameter on the normal and tangential grinding force in Zirconia ceramics (ZrO₂) and optical glass (BK7). The differences in COF values, presented in Figure 14, appeared due to the decrease in the grinding wheel diameter during grinding because of the dislocation of the abrasive grains during machining. The grinding temperature also affects the frictional coefficient, especially during dry grinding, but was not considered in the current study.

5. Conclusions

The current study developed a mathematical model of the micro-grinding forces by providing a comprehensive analysis of the parameters influencing grinding for hard and brittle materials, specifically Zirconia ceramics (ZrO₂) and Optical Glass (BK7). Through a detailed kinematic analysis of material removal, the research not only advanced the theoretical models but also substantiated them with empirical evidence, thereby bridging the gap between theoretical predictions and practical, experimental outcomes.

Key findings included the observation that material removal occurs via both ductile and brittle mechanisms, significantly influencing the grinding forces. This led to the theoretical model that effectively describes the grinding forces during these phases. The analysis showed that the normal grinding force is more significant than the tangential grinding force, and plays a crucial role in affecting both the surface and subsurface integrity of the material.

The research indicates that enhancing the surface quality can be realized by reducing the grinding forces, which can be accomplished by increasing the linear velocity of the grinding wheel. Furthermore, the study establishes that the grinding force exhibits a direct proportionality to both the feed rate and the depth of grinding. These findings offer critical insights for optimizing grinding conditions, thereby facilitating the achievement of superior surface finishes.

The validity of the proposed model was rigorously tested, showing a high level of accuracy, with average deviations of 6.616% for normal force and 5.752% for tangential force, thereby confirming the model’s reliability in predicting grinding forces under varying conditions. This research contributes towards a better understanding of surface grinding dynamics and offers a validated theoretical framework for predicting and optimizing grinding forces, thereby improving the surface integrity and quality of Yttria-stabilized Zirconia ceramics.

Future research could examine the suitability of the suggested mathematical model for different materials and grinding parameters, analyze the impact of supplementary variables on grinding forces, and further verify the model’s precision considering machining temperatures and lubricants.