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Abstract
Ultra-precision tool-servo-based diamond cutting (UTSDC) is a promising technology for fabricating true 3-dimensional optical microstructures. The diamond tool in UTSDC moves alternatively upward and downward along the thrust direction. However, most studies on the material removal mechanism are limited to the orthogonal cutting condition where the depth of cut is invariant. The effect caused by the tool motion in the thrust direction has been overlooked. In this paper, the indentation effect affected by the tool path, tool shape and cutting speed is systematically studied. It is found that the inclined angle between the tool path direction and the main cutting direction plays a key role in the determination of the material spring back and the formation of side burr. The characteristics of indentation force and material spring back indicates that the indentation mechanism is dominant in the cut-in process where the inclined angle is large, while the shearing mechanism is dominant in the cut-out process. A new theory is proposed to explain the tool indentation mechanism in UTSDC, and the simulation results show that the theory can well predict the indentation force under various cutting conditions.
© 2021 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The demand for precision microstructured functional surfaces is increasing in many advanced optical products and devices [1].The overall size of the surfaces typically ranges from hundreds of micrometers to tens of millimeters, whose feature size of a single microstructure is in micrometer or nanometer scale. The typical micro-structures have true three-dimensional shapes, such as micro-lens array, diffractive optical structures, pyramidal micro-structures, etc. Such demand has driven the development of ultra-precision tool-servo-based diamond cutting (UTSDC) technologies including slow/fast tool servo turning, fly-cutting and diamond micro chiseling.
The investigation of the material removal process is the groundwork for the optimization of operation parameters to reduce surface errors. Studies on the ploughing mechanism in orthogonal cutting condition in micro/nano scale give hints for the understanding in tool indentation mechanism in UTSDC. The ploughing mechanism is explained by material separation model and material spring back model. Previous works [2–4] have demonstrated that in orthogonal cutting condition, where the uncut chip thickness is comparable or less than the tool edge radius, the material flows to the tool separates at the point ${A_1}$ at the cutting edge and springs back to contact point ${B_1}$ at the tool’s clearance face, as shown in Fig. 1. The positions of the two points and the tool’s shape determine the volume of the ploughed material. Since it is of great difficulty to measure the positions directly, research on the ploughing mechanism focused on the theoretical or indirect determination methods. Wu [5] proposed that the stagnation height ${h_{stag}}$ was linear proportional to the instantaneous length of the shear plane, considering that the stagnation height was dependent on the temperature and the flow stress remained relatively constant. Son et al. [6] assumed that the material above the point ${A_1}$ was perfectly plastic and the material blow ${A_1}$ was perfectly elastic. The stagnation angle ${\theta _m}$ was further assumed to be almost equal to the shear angle $\phi $. However, this equality of ${\theta _m}$ and $\phi $ was not yet proved. Malekian et al. [7] pointed out the stagnation point ${A_1}$ located at the point where the power needed for the material removal was at its minimum value, indicating that the stagnation angle ${\theta _m}$ was very close to the friction angle between the workpiece and the rake face of the tool. Zhou et al. [8] provided a theoretical method to determine the stagnation height by assuming that the stress changed linearly in the tool edge-workpiece interface and the shear stress at ${A_1}$ was zero. The research on the material spring back in ultra-precision machining has also been attracted great attentions for many years. The correlation between the deep swelling and surface roughness was revealed in ultra-precision turning [9]. The nano-scratching test utilizing the atomic force microscopy-based method was conducted to reveal the elastic recovery behaviors of polycarbonate. It was reported that the elastic recovery rate was insensitive to the variation of the normal load but increased with the velocity increasing in the range of 0.3 to 12 mm/min [10]. Similar to cases of the material separation theories, the material spring back models are highly dependent on assumptions. The most common models take the assumption that the material’s spring back ${h_s}$ is a function of uncut chip thickness and the minimum chip thickness. Table 1 summaries the typical material’s spring back models developed in recent years. The material separation model and material spring back model provide essential information for the ploughing force prediction [11,12]. However, although various models were proposed to predict the ploughing force, it is still challenge to verify the models because of the difficulty to separate the ploughing force from the total thrust force [2].
Table 1. The spring back models.a
In UTSDC, as the diamond tool moves alternatively upward or downward along the thrust direction, the material deformation process is fundamentally different from that in orthogonal cutting condition. Figure 2 illustrates the tool indentation mechanism in UTSDC. Material under the diamond tool’s cutting edge is indented as the tool cut into the surface. The indented material is compressed elastically first and then plastically when the depth of cut increases, as shown in Fig. 2(b) and Fig. 2(c), respectively. The volume of the indented material in Fig. 2(c) is not only determined by the uncut chip thickness h, but also affected by the tool path and the m aterial spring back at previous cutting locations. Besides, the spring back models developed in the orthogonal cutting conditions may not be valid in UTSDC condition because the actual angle between the machined surface and tool’s clearance face is smaller than the nominal clearance angle. While the tool moves upwards in the cut-out process in Fig. 2(d), the indented volume decreases. The penetration depth ${h_{ct}}$ is smaller than the spring back height ${h_s}$. Due to the complexity of material deformation in UTSDC, the ploughing theory developed in orthogonal cutting condition is not valid in this cutting condition. The micro/nano indentation theory is also invalid in UTSDC because not all the material in the space occupied by the tool is indented, some of the material is removed as a chip. As a result, a new theory is needed to explain the tool indentation mechanism in UTSDC and its effect on the surface finish of the microstructured surface.
Fig. 2. Illustration on the tool indentation process in UTSDC. (a) The initial contact of the diamond tool and work material; (b) The diamond tool cuts into the surface in the total elastic region; (c) The volume of indented material in the cut-in process after the chip formation; (d) The volume of indented material in the cut-out process after the chip formation.
Henceforth, this paper focuses on the study of the tool indentation mechanism with the following two scopes. (i) To characterize the indentation effect on the surface generation and cutting forces in UTSDC of micro-structured surfaces; (ii) To establish a theoretical semi-static model for the indentation force by considering the elastic and plastic pressure at the workpiece and cutting-edge element interface which are affected by the tool path trajectory and material spring back. The content of the paper is organized as follows: the experiment design and setup are presented in Section 2, followed by the investigation on the material spring back and burr formation in UTSDC in Section 3. The cutting force and indentation force are analyzed in Section 4. A novel mechanistic model is built to predict the indentation force, which utilizes small discontinuous step functions in the simulation to replace the designed tool trajectory. The indentation force model and simulation results are presented in Section 5.
2. Experimental procedures
A set of cutting experiments were conducted to cut special symmetric microstructures on Aluminum 6061-T6 with a round-nose conical type diamond cutting tool as shown in Fig. 3. Both cut-in and cut-out processes are included to produce the symmetric microstructure. The experiments were designed based on Taguchi method [16] with four control factors: tool nose radius ${r_t}$, inclined angle $\gamma $, cutting speed v and maximum depth of cuts ${h_{max}}$. The orthogonal arrays (${L_9}$) are listed in Table 2. The experiments were conducted on a Moore 350FG ultra-precision machine, as shown in Fig. 4. Three natural single crystal diamond tools (Contour Fine Tooling Ltd) with different tool nose radii were used in the experiments. The rake angle and clearance angle of all the tools are 0 $^\circ $ and 10 $^\circ $, respectively. Before cutting the microstructures, the top surfaces of the workpieces were flattened by fine face turning operations and a straight line was cut for the reference of the cutting direction. The tool edge radii of the diamond tools were measured by a scanning electron microscope. The edge radii lie in the range of 0.2 - 0.3 µm, as shown in Fig. 5(a). During the machining process, the thrust force and the cutting force were measured by a Kistler Dynamometer 9256C1, which was mounted beneath the diamond tool. The surface topography was obtained by a Zygo NexviewTM 3D optical surface profiler and a scanning electron microscope (Tescan VEGA3). No lubricant was used in order to reduce the force signal noise.
Fig. 3. (a) Schematic diagram of the cutting process for the symmetric microstructures; (b) the measured surface profile and material spring back (c) the tool trajectory in the cross section of the microstructure.
Table 2. Orthogonal array of the experiments
3. Surface topography of the indentation effect
Figure 6 shows scanning electron microscopy (SEM) photos of the microstructures. Scratches along the cutting direction are found on the machined surface generated in the cut-in process in experiment A-2, A-3, A-8 and A-9, as shown in Fig. 6(b), 6(c), 6(h), and 6(i), respectively. The positions of scratches are the same for each microstructure. As a result, it is unlikely that the scratches were generated from cutting hard particles which are randomly distributed in the aluminum alloy. Since there are slightly tool wear on the flank face of diamond tools as shown in Fig. 5(b), the scratches are regarded as the evidence of the contact of the recovered work material and the tool flank face in the cut-in process. In addition, the scratches disappeared on the surface produced in the cut-out process. This infers that the contact area decreased when the cutting changed from cut-in state to cut-out state. In experiment A-1, A-4 and A-7 whose inclined angle γ were set to 1°, no obvious scratch is found on the surface. The contact area of the recovered work material and the tool was small so that the recovered material did not contact the wear part of the cutting tools. Apart from the scratches, the vibration marks which is perpendicular to the cutting direction are found on the machined surface. The vibration marks are not apparent in A-1, A-4 and A-7, but much more obvious in A-3 and A-6. This infers that the inclined angle affected the relative vibration between the tool and the workpiece. To improve the optical quality of the machined surface, the cutting tool should be carefully selected as the wear on the tool’s clearance face would generate scratches. Besides, the inclined angle cannot set to high value otherwise the relative tool-work vibration may be enhanced.
Fig. 6. SEM photos of the microstructures. (a) - (i) refer to the results of experiments (A-1) - (A-9) respectively.
Previous study shows that large instantaneous acceleration oscillations of the cutting slide in slow-tool-servo generates follow error [17]. However, the maximum cutting speed component in the thrust direction ${v_{thrust}}$ is 1.17 mm/min, which is only 15.64% of the total velocity v. No obvious relationship can be identified between ${v_{thrust}}$ and the root-mean-square ($RMS$) values of the measured surface profiles along the cutting direction, as shown in Fig. 7.
Fig. 7. The relationship between the cutting speed component in the thrust direction and surface finish.
3.1 Material spring back
The material spring back ${h_s}$ is considered as the difference between the ideal depth of the tool trajectory and the actual depth of the surface profile measured along the cutting direction, as shown in Fig. 3(b). To ensure the cutting tool is engaged with the workpiece at the beginning of cutting of the microstructure, the actual depths of cut at the starting point ${B_1}$ are in the range from tens to hundreds nanometers in the experiment. The reference line before cutting microstructures, as shown in Fig. 3(a), represents the cutting direction. The measured profile ${B_1}{B_2}$ is parallel to the reference line. The position of ${B_1}{B_2}$ is determined by the bottom point of ${A_1}{A_2}$, which is perpendicular to the reference line. Before determining the material spring back, the measured profile ${B_1}{B_2}$ is tilted such that the heights of ${B_1}$ and ${B_2}$ are the same. Figure 8 shows the estimated material spring back in different cutting conditions. The inclined angle γ is the dominant factor that determines the behavior of material spring back. In the case that $\mathrm{\gamma } = 1^\circ $, ${h_s}$ is close to zero during the whole cutting process, as depict in Fig. 8(a). When $\mathrm{\gamma } = 5^\circ $, the material recovers at the beginning of the cut-in process where ${h_s}$ jumps to about 0.05 $\mu m$. Hence, ${h_s}$ keeps the relative constant value until the end of cut-in process. When the cut-out process begins, ${h_s}$ suddenly drops to 0, as shown in Fig. 8(b). The material spring back in the experiments when $\mathrm{\gamma } = 9^\circ $ exhibits similar behavior as the case of $\mathrm{\gamma } = 5^\circ $. ${h_s}$ jumps to around 0.05 $\mu m$ as the cutter penetrates the surface and ${h_s}$ suddenly drops to 0 when the cutter moves upward in the thrust direction. But unlike the case of $\mathrm{\gamma } = 5^\circ $, ${h_s}$ in Fig. 8(c) gradually as the depth of cut increases in the cut-in process. When the cutter approaches to the bottom of the microstructure, the slope of spring back curves of A-6 and A-9 suddenly increases. Such increase probably results from the roundness of tool edges, not the actual material spring back. The relationship between ${h_s}$ and the uncut chip thickness h in the cut-in process when $\mathrm{\gamma } = 9^\circ $ is shown in Fig. 8(d). Thus, in UTSDC, ${h_s}$ is a function of h and $\mathrm{\gamma }$:
3.2 Burr formation
The cross-section profiles of the microstructures generated under different cutting conditions are extracted and plotted in Fig. 9. As shown in Fig. 9(a), Fig. 9(d), and Fig. 9(e), no side burrs are found in the profiles corresponding to both the cut-in and cut-out process when $\mathrm{\gamma } = 1^\circ $. However, burrs appear in the cut-out process when the $\mathrm{\gamma } = 5^\circ \; \textrm{and}\; 9^\circ $. The height of most burrs is around 0.2 $\mu m$. Such phenomenon indicates that the material side flow may not be severe in the cut-in process, while part of material tends to flow to the side to form burrs in the cut-out process. This is particularly significant when the inclined angle $\mathrm{\gamma }$ is large. The formation of the burrs in the cut-out process is probably caused by the highly negative effective rake angle [18].
Fig. 8. (a) Material spring back along the percentage cutting distance when $\mathrm{\gamma } = 0^\circ $; (b) Material spring back along the percentage cutting distance when $\mathrm{\gamma } = 5^\circ $; (c) Material spring back along the percentage cutting distance when $\mathrm{\gamma } = 9^\circ $; (d) The relationship between ${h_s}$ and the uncut chip thickness h in the cut-in process when $\mathrm{\gamma } = 9^\circ $.
Fig. 9. The cross-section profiles of the microstructures, (a) - (i) refer to the results of experiments (A-1) - (A-9) respectively.
4. Cutting forces of the indentation effect
The cutting force ${F_c}$ and the thrust force ${F_t}$ are measured during the cutting experiments. As the force signals contain noise from the environment, the measured force is processed by a low pass filter with the upper boundary of 200 Hz. The characteristics of ${F_c}$, ${F_t}$ and the force ratio ${F_t}/{F_c}$ in time domain are shown in Fig. 10. ${F_c}$ is nearly symmetrical as the tool path is symmetric for the designed microstructure. However, ${F_t}$ experiences a sudden increase at the beginning of the cut-in process and a sudden decrease at the start of the cut-out process, which is seldomly reported in the literature. The sudden increase in thrust force is not the result of the impact of tool and workpiece, otherwise the impact force would decrease to zero in a short time. ${F_t}/{F_c}$ decreases nonlinear with the increase of the depth of cut in the cut-in process. The force ratio keeps to a relative stable value (around 0.2) in the cut-out process, which is consistent with the friction coefficient obtained in orthogonal diamond cutting experiment [19]. The actual tool motion direction is different in the cut-in and cut-out process, which affects the effective shear angle and effective rake angle in the orthogonal cutting force model [20,21]. However, the change of cutting direction is not the major factor for the sudden change of thrust force. Otherwise, the following phenomena would have been observed: (i) the force ratio in the cut-in process would be a constant, since the effective rake angle and effective clearance angle did not change during the cut-in process; (ii) the cutting force increases significantly when the cutting process transfers from cut-in process to cut-out process as the effect rake angle changes from positive to negative. As a result, the tool indentation is the major factor for the sudden change of the thrust force.Based on the statistical analysis of the main cutting force ${F_c}$ and the cutting area ${A_c}$ in the 9 experimental conditions, it is found that the ratio of the ${F_c}$ and ${A_c}$ is almost constant. The averaged R-squared values in each cutting condition are found in the range of [0.90, 0.99]. Therefore, the tool indentation effect on ${F_c}$ is very limited. The main cutting force ${F_c}$ in UTSDC can be well estimated by the classical cutting force expression, i.e., Eq. (4) and Eq. (5), which are commonly used in the orthogonal cutting condition,
Fig. 10. Typical force and force ratio profiles in time domain (experimental result of trial A-3) (a) The cutting force ${F_c}$, (b) The thrust force ${F_t}$, (c) The thrust force to cutting force ratio ${F_t}/{F_c}$.
As discussed above, the thrust force ${F_t}$ shows different behavior compared with ${F_c}$. Figure 11 shows that ${F_t}$ is not only determined by the cutting area, but also the inclined angle $\mathrm{\gamma }$ in the cut-in process. Let ${h_1}$ presents the uncut chip thickness in the cut-in process and ${h_2}$ presents the corresponding uncut chip thickness in the cut-out process. And $\; {h_1} = {h_2}$. Based on the data in Fig. 11, the characteristics of ${F_t}$ can be summarized as follows:
- (i) When $\mathrm{\gamma } = 1^\circ $, ${F_t}({{h_1}} )$ ${\approx} $ ${F_t}({{h_2}} )$. That is, when the inclined angle is very small $\mathrm{\gamma } \le 1^\circ $, no obvious difference of thrust force is found between cut-in and cut-out process.
- (ii) When $\mathrm{\gamma } = 5^\circ $ or 9 $^\circ $, ${F_t}({{h_1}} )$ increases rapidly where $0 < {h_1} < {h_{critical}}$. Then, ${F_t}({{h_1}} )$ increases with a slower rate where ${h_1} > {h_{critical}}$. ${h_{critical}}$ is around $0.05\; \mu m$.
- (iii) For the same cutting tool, if ${h_1} \le {h_{critical}}$, ${F_t}({{h_1},\; \; \mathrm{\gamma } = 5^\circ } )\approx $ ${F_t}({{h_1},\; \; \mathrm{\gamma } = 9^\circ } )$; if ${h_1} > {h_{critical}}$, ${F_t}({{h_1},\; \; \mathrm{\gamma } = 5^\circ } )< $ ${F_t}({{h_1},\; \; \mathrm{\gamma } = 9^\circ } )$. Consider that the material spring back is also close to $0.05\; \mu m$ at the beginning of the cut-in process, it can be inferred that the material deformation transferred from elastic to plastic where ${h_1} \approx {h_{critical}}$.
- (iv) For the same cutting tool, ${F_t}({{h_2},\mathrm{\gamma } = 1^\circ } )\approx {F_t}({{h_2},\; \mathrm{\gamma } = 5^\circ } )\approx {F_t}({{h_2},\mathrm{\gamma } = 9^\circ } )$. This behavior reflects that the contact between the surface and the tool’s clearance face is much reduced. The indentation force in the cut-out process might be ignored.
Fig. 11. The relationship between the thrust force and inclined angle (a) Experiment A-1,2,3; (b) Experiment A-4,5,6; (c) Experiment A-7,8,9.
According to the characteristics of the cutting forces and the surface generation, it can be inferred that if the inclined angle is small, the material removal process is governed only by the cutting mechanism. The tool indentation effect is significant in the cut-in process and at the beginning of the cut-out process when the inclined angle is large. If the inclined angle is greater than 5 $^\circ $, the indented material fully recovers as the tool penetrates from 0 to the critical depth ${h_{critical}}$. Since ${F_t}({{h_1},\; \; \mathrm{\gamma } = 5^\circ } )\approx $ ${F_t}({{h_1},\; \; \mathrm{\gamma } = 9^\circ } )$ for the same tool at this stage, the value of thrust force is not influenced by the inclined angle. Compared with the thrust forces with different tools, the tool with greater tool nose radius generates higher thrust force. When the tool penetrates deeper, the growth rate of thrust force in terms of uncut chip thickness becomes smaller. The indented material is partially recovering and the surface spring back increases with the inclined angle. Thus, the growth rate of thrust force at this stage depends on the inclined angle. When the cutting process changes from cut-in state to cut-out state, the thrust force suddenly drops down as the contact of recovered workpiece material and the cutting tool reduces to zero. The thrust forces with different inclined angles and cutting speed are colinear at this stage, as shown in Fig. 11. As a result, the cutting mechanism retakes the control of the material removal process in the cut-out process.
The thrust force ${F_t}$ is considered as the summation of the force component due to shearing ${F_{ts}}$ and the indentation force ${F_p}$.
Since in the cut-out process, the indentation force ${F_p}$ can be ignored, the indentation force ${F_p}({{h_1}} )$ in the cut-in process can be approximated by the asymmetric force ${F_{as}}({{h_1}} )$:
where ${h_1} = {h_2}$, $\dot{h_1} > 0,\; \dot{h_2} < 0$.5. Theoretical modelling of the indentation force
As discussed above, the prediction of indentation force cannot be directly achieved by the conventional indentation model or orthogonal cutting model. As a result, in the presented work, a novel mechanistic model is built to estimate the indentation force. The model is used to predict the ‘static’ component of the indentation force. The cutting system is assumed to be rigid. The basic methodology of the proposed model is to utilize discontinuous step functions (Fig. 12(b)) to approximate the designed tool trajectory (Fig. 12(a)), so that in each step, the cutter moves along either the thrust direction or the main cutting direction. Discretization of tool path allows the prediction of forces based on the pure orthogonal cutting model and indentation model in each step. If the step size is small enough, the simulated force should be close to the true value.
Fig. 12. Modelling of the penetration depth in UTSDC (a) initial contact of the cutting tool and workpiece surface; (b) the first indentation step in the simulation; (c) the orthogonal cutting step in the simulation; (d) the determination of the penetration depth when the cutter moves from ${P_i}$ to ${P_{i + 1}}$
The modelling procedure is briefly described as follows:
- (i) ${P_0}$ is the first contact point between the cutting tool and workpiece on the original surface. The cutting tool moves from ${P_0}$ to ${P_1}$ by an indentation step as shown in Fig. 12(b). In this step, the material beneath the tool flank face is compressed into the surface. The compressed material is first elastic deformed until the pressure reaches the plastic limit. As the cutting tool has 3D shapes, part of material in the projected area may be plastically deformed and the other is elastically deformed. The indentation force ${F_p}$ increases but the main cutting force ${F_c}$ does not change in this step.
- (ii) The cutting tool moves from ${P_1}$ to ${P_2}$ by an orthogonal cutting step. In this step, the indentation force is constant but the main cutting force changes with the cutting area in the last indentation step. The elastic deformed part is recovered at the surface machined behind as shown in Fig. 12(c).
- (iii) The cutting tool moves from ${P_2}$ to ${P_3}$ by the next indentation step, which is like the tool motion from ${P_0}$ to ${P_1}$. The major difference is that the pressure on the tool is not zero at point ${P_2}$, because the material spring back at the last orthogonal cutting step. After an arbitrary indentation step, the cutting tool moves from ${P_i}$ to ${P_{i + 1}}$, the penetration ${h_{ct}}$ is determined by the contact between the tool flank face and recovered surface in the last few steps, as shown in Fig. 12(d).
The tool path for the triangular microstructure is expressed in Eq. (8) in the workpiece coordinate ${O_w} - {X_w}{Z_w}$ as defined in Fig. 12(a),
The tool path is then approximated by discontinuous step functions with a constant interval $dx$. The discretized path for simulation is described as $({{X_s},{Z_s}} )$:
The total number of $({{X_s},{Z_s}} )$ are M. $M = {L_{max}}/dx$.
The round-nose cutting tool is conical type as shown in $O - XYZ$ coordinate in Fig. 13(a). To analyze the complex pressure distribution on the conical type of diamond tool, the diamond tool is discretized along $Y - axis$ with the same interval $\Delta y$. A local frame ${O_{yj}} - {X_{yj}}{Y_{yj}}{Z_{yj}}$ is attached at the j-th point along the cutting edge as shown in Fig. 13(b).
Fig. 13. The discretized shape of round-nose conical type cutting tool, (a) the 3D shape of the diamond tool, (b) the j-th slice of the diamond tool.
In the j-th slice, the tool shape can be described by Eq. (10) in the $O - XYZ$ coordinate:
In the ${O_{yj}} - {X_{yj}}{Z_{yj}}$ plane, the contact in the elastic deformation area is assumed to be frictionless. The elastic contact pressure ${p_{elastic}}(x )$ at the cutting edge interface can be obtained based on the pressure model of superposing the indentation of inclined punch with rounded edges proposed by Goryacheva et al. [22] as shown in Eq. (12) to Eq. (14):
Three cases of the contact zone location are presented in Fig. 14. ${h_{i{n_y}}}$ is the effective contact depth in the ${O_{yj}} - {X_{yj}}{Z_{yj}}$ plane. Point A and Point B are the ends of contact. Point C is the intersect point of the tool edge round curve and the straight line representing the flank face. The height of point C can be determined by Eq. (15):
The solution of Eq. (12) which satisfied the condition $\; {p_{elastic}}(a )= {p_{elastic}}(b )= 0$ is given in the following cases [22] :
Case I: ${h_{i{n_y}}}[m ]< {h_c}$:
${h_{i{n_y}}}[m ]$ is the local penetration.Case II: ${h_c} \le {h_{i{n_y}}}[m ]\le {r_e}$:
Case III: ${h_{i{n_y}}}[m ]> {r_e}$:
The boundary of elastic zone is identified where the elastic pressure reaches the plastic limit ${p_{plastic}}$.
${C_1}$ is a material constant which depends on friction conditions and indenter geometry [23]; ${\sigma _Y}$ is the tensile yield strength of the workpiece material.The corrected pressure distribution can be obtained by:
The maximum penetration ${h_{ct}}$ is determined by the material spring back, tool shape and the inclined angle,
$n$ is the obtained such thatFor the m-th point $({{X_s}[m ],{Z_s}[m ]} )$ of the tool path $({{X_s},{Z_s}} )$, the maximum contact length of the tool and the workpiece in $\; O - YZ\; $ plane is denoted as $2{y_{max}}[m ]$, where
The half of the tool is then sliced into ${y_{max}}/\Delta y$ pieces. Thus, the indentation force ${F_{pj}}$ in the ${O_{yj}} - {X_{yj}}{Z_{yj}}$ plane when the tool moves to the point $({{X_s}[m ],{Z_s}[m ]} )$,
where $a,b$ is a function of ${h_{i{n_y}}}[j ]$, as shown in Eq. (16)-(18)The total indentation force ${F_p}[m ]$ is derived as
To verify the proposed indentation force model, the parameters for the model are reasonable set as shown in Table 3.
Table 3. Parameters for the indentation force model
The predicted indentation force and the measured asymmetric force is illustrated in Fig. 15. Although the measured thrust force is small and strongly coupled with vibration component and noise, the predicted force could reflect the trend of how the asymmetric force changes with the uncut chip thickness. The average percentage error between the simulated force and the regression force is 18%. The errors may mainly result from the approximation of the tool shape geometry and the material’s microstructure effect. The simulation results demonstrate the effectiveness of the proposed methodology which utilizes indentation theory and stepped tool path to determine the indentation force.
Fig. 15. Simulated indentation force and the measured asymmetric force (a) Experiment A-2, (b) Experiment A-3, (c) Experiment A-5, (d) Experiment A-6, (e) Experiment A-8, (f) Experiment A-9.
Considering that vibration marks found on the machined surface, the sudden increase and decrease of the thrust force may be the source to induce the relative displacement between the tool and the workpiece along the thrust direction with small amplitude. The proposed model pre-assumes that the cutting system is rigid. As a result, the model is limited to the condition that the cutting speed is below 7.5 mm/min. Further investigation will be carried out on how the indentation mechanism affects the dynamics of the cutting system and surface generation. Although the spring back model needs to be obtained from the experiment data, the method to develop the indentation force model can be extended to other workpiece materials. Hence, the proposed indentation model contributes to a better understanding of the indentation mechanism in the tool-servo-based diamond cutting operations. It can also be used to the development of the control algorithm for the promising force-controlled diamond machining in the further.
7. Conclusions
In this paper, the tool indentation mechanism in ultra-precision tool-servo-based diamond cutting (UTSDC) of optical microstructure is theoretically and experimentally investigated. The conclusions are summarized below:
- (1) The material spring back ${h_s}$ in UTSDC is determined not only by the uncut chip thickness, but also the inclined angle $\mathrm{\gamma }$ of tool path trajectory. When the $\mathrm{\gamma }$ is large, ${h_s}$ increases rapidly at the beginning of the cut-in process and drops to zero quickly in the cut-out process.
- (2) Burrs only appear in the cut-out process when the inclined angle $\mathrm{\gamma }$ is large. Such phenomenon indicates that side flow may not be severe in the cut-in process, while in the cut-out process, part of material tends to flow to the side to form burrs, especially when $\mathrm{\gamma }$ is large.
- (3) In UTSDC, the indentation mechanism is dominant in the cut-in process where the inclined angle is large, while the shearing mechanism is dominant in the cut-out process.
- (4) A mechanistic analytical model is built to estimate the indentation force with varying uncut chip thickness. The proposed model makes use of the discontinuous step functions to determine the designed tool trajectory so that the indentation theory can be utilized to estimate the force in the indentation steps. The simulation results show that the proposed model can predict for the main trend of the indentation force well under various cutting conditions.
Funding
Ministry of Science and Technology of the People's Republic of China (2017YFE0191300); PhD Studentship, Hong Kong Polytechnic University (RUK0).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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