The International Journal of Advanced Manufacturing Technology https://doi.org/10.1007/s00170-018-2622-5
ORIGINAL ARTICLE
Effect of tool inclination on surface quality of KDP crystal processed by micro ball-end milling Qi Liu 1,2 & Jian Cheng 1,2 & Yong Xiao 2 & Mingjun Chen 1,2 & Hao Yang 1,2 & Jinghe Wang 2 Received: 18 February 2018 / Accepted: 19 August 2018 # Springer-Verlag London Ltd., part of Springer Nature 2018
Abstract Micro-milling has been considered as the most promising method to repair the micro-defects on the surface of KH2PO4 (KDP) crystal. However, acquiring an ultra-smooth repaired surface by ball-end milling remains a longstanding challenge for KDP crystal due to its soft-brittle properties. In micro ball-end milling of KDP crystal, tool inclination angle has a remarkable effect on the quality of machined surface. Therefore, picking out an optimal tool inclination angle plays a great role in guaranteeing the ductile-mode machining and improving the surface quality of brittle KDP crystal. In this work, the effect of tool inclination on the brittle–ductile transition and surface quality of micro-milled KDP crystal were investigated. A theoretical model considering the tool inclination direction and angle was proposed to calculate the undeformed chip thickness (UCT) and cutting speed involved in the micro ball-end milling process. Besides, micro groove experiments were conducted to evaluate the change rule of the brittle– ductile transition and surface quality related to the tool inclination. The experimental results agree well with the theoretical results, which shows that the evolution of surface quality with respect to the tool inclination depends on the competitive mechanisms between UCT and cutting speed. Inclining the cutter in the tool feed direction (positive inclination angle) and increasing the tool inclination angle both contribute to the ductile cutting of KDP crystal. A + 45° inclination angle is the optimal angle for the micro ball-end milling of KDP crystal and the best surface roughness value achieved could be up to 35.3 nm. Keywords Tool inclination angle . Micro ball-end milling . Brittle–ductile transition . Surface quality
1 Introduction Owing to the excellent nonlinear optical and electro-optical properties, KDP crystal is currently the unique candidate for frequency doubling and optical switching in the laser-driven inertial confinement fusion (ICF) facilities [1]. However, for this kind of optical material, it is prone to generate some micro-defects such as cracks, pits, and ablation during both
Qi Liu and Jian Cheng contributed equally to this work. * Jian Cheng
[email protected] * Mingjun Chen
[email protected] 1
State Key Laboratory of Robotics and System, Harbin Institute of Technology, P.O. Box 413, Harbin 150001, People’s Republic of China
2
School of Mechatronics Engineering, Harbin Institute of Technology, P.O. Box 413, Harbin 150001, People’s Republic of China
the diamond ultra-precision machining and laser preirradiating processes [2]. These micro-defects would grow rapidly under the subsequent high-power laser irradiation and hence severely downgrade the optical performance and service life of KDP crystal components. Therefore, considering the time-consuming and costly process of crystal growth, the most economical way is to repair the optical component by replacing those original defects with predesigned smooth Gaussian contours, which is termed as “optical recycle loop strategy” that was first proposed by Lawrence Livermore National Laboratory [3]. According to the state of the art in mitigating the growth of surface micro-defects on KDP crystal, micro machining has been deemed to be the most promising method to completely remove the micro-defects on crystal surface [4]. Nevertheless, it is very difficult to achieve a fracture-free surface on KDP crystal due to its soft and brittle properties. Being very fragile, brittle fracture can be introduced by even very small cutting forces, resulting in the formation of surface and subsurface damage [5]. Therefore, the major challenge in repairing KDP crystal is to prevent the occurrence of brittle fracture and achieve a fracture-free
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surface. It means that the material is expected to be removed by plastic deformation rather than crack propagation, which is known as ductile-mode machining [6]. During the past decades, considerable efforts on fundamental understanding of the brittle–ductile transition of brittle materials have been done. By performing the indentation test, Lawn [7] stated that no matter how brittle the material is, it could be machined in ductile mode if the undeformed chip thickness (UCT) is below a critical value. Cai [8] utilized molecular dynamic method to simulate the machining process of silicon and reported that ductile-mode machining is achievable when the UCT is smaller than the cutting edge radius (sub-micron scale). Arif [9] explained the brittle–ductile transition mechanism from the view of specific cutting energy. By considering the whole cutting conditions, such as workmaterial intrinsic properties, tool geometry, and process parameters, it was found that the energy required to induce brittle fracture does exceed the energy required for plastic deformation and hence the ductile-mode machining can be realized. As for KDP crystal, Baruch [10] made the first attempt to use single-point diamond turning (SPDT) to directly produce finished surface without additional polishing process. Then, Vickers indentation experiments were performed by Chen [11] with different loads, and the results manifested that if the cutting parameters are appropriately selected, it would be machined in the ductile mode. Afterwards, Tie [12] employed the spiral turning technique to explore the ductile cutting mechanism of KDP crystal and eventually obtained an ultrasmooth surface with the root-mean-square (RMS) value fluctuating from 1.3 to 1.7 nm. Apart from experimental researches, Zong [13] utilized finite element model to analyze the effect of diamond tool geometries on the surface quality of KDP crystal in flycutting process. He revealed that diamond tool with negative rake angle is conducive to the plastic deformation by generating large compressive stress and finally reduces the surface roughness. Besides, by considering the impacts of elastic and pressure-dependent plastic behaviors of this material, Wang [14] proposed a new constitutive model and adopted tensile stress as the fracture criterion for KDP crystal. The 3D cutting simulations applying this new model with LS-DYNA software showed that the hydrostatic pressure was generated in ductile cutting zone and it suppressed the increment of tensile stress, indicating that high hydrostatic pressure could facilitate the plastic deformation. In order to meet the ICF requirements of full-aperture (415 × 415 mm) KDP optics with high-quality surfaces, Liang [15] designed an ultra-precision diamond flycutting machine tool, which could produce an ultra-smooth surface with 1.3 μm flatness and 2.4 nm RMS roughness. Although the works reviewed above have made great progresses in understanding the ductile-mode machining mechanisms of KDP crystal using SPDT method, this machining method is only applicable to achieve large flat surfaces. It
means that SPDT method is not suitable for fabricating complex shapes on KDP surface owing to the limitation of its own configuration. Thus, an alternative process in ductile-mode machining is highly desired for the ductile cutting of KDP crystal with free-form shapes, such as Spherical and Gaussian contours [2, 16]. Currently, micro-milling has been deemed as the most promising method to repair the microdefects on KDP crystal surfaces into the pre-designed complex shapes [17]. In recent years, significant research has been published in relation to the enhancement of surface quality machined by milling processes. Many processing parameters such as tool geometry parameters, spindle speed, depth of cut, and feed per tooth have been proven to be strongly linked with the machining performance [18–20]. It is also found that the selection of tool inclination angle exerts a critical impact on the machined surface quality in milling process [21, 22]. In fact, the optimal tool inclination angle is a prerequisite to determine the scientific combination of other cutting parameters. A comparison of reported researches with respect to the optimization of tool inclination angle in micro ball-end milling processes is shown in Table 1, from the perspectives of workpiece materials, tool inclination direction, research method, and the optimal lead angle. It can be concluded that, as for metal materials, a small tool inclination angle is conducive to obtain a small Ra-value surface and reduce tool wear in micro ball-end milling processes [21–23, 29, 30]. When it comes to brittle materials, Ono [25, 31, 32] firstly carried out the micro-milling tests of crown glass with a ball-end mill and investigated the critical feed rate with various cutting direction. Liu [28] further pointed out the tool inclination angle has a remarkable effect on the machining mechanics of soda-line glass. Afterwards, Qiu [24] obtained a fracture-free quartz glass surface by inclining the micro ball-end milling cutter along the feed direction with an angle of 50°. Arif [27] reported the highest critical feed rate for brittle–ductile transition of silicon can be achieved at a tool inclination angle of 30° in feed direction, while Mass [26] achieved a minimum Ra-value surface of single crystal sapphire at a small tool inclination angle (15°). From the results listed above, it is clear that the optimal tool inclination angles for different materials to obtain high-quality surfaces are quite different from each other. Thus, to achieve smooth repairing surface of KDP optics, it is essential to explore the most suitable tool inclination angle for KDP crystal in micro-milling repairing process. Furthermore, these researches reviewed above [24–28, 31, 32] with respect to micro-milling of brittle materials only focused on the case of tool inclining parallel to the feed direction (namely, positive lead angle) and do not pay any effort in the case of tool inclining opposite to the feed direction (namely, negative lead angle). Besides, the majority of researches above optimized the tool inclination angle totally by micromilling experiments and do not give depth to theoretically
Int J Adv Manuf Technol Table 1 Comparison of reported researches with respect to the optimization of tool inclination angle in micro ball-end milling processes
Material
Lead angle direction
Research method
Optimal lead angle
+, − +, −
Cutting speed Experiment
± 5° for small Ra + 15° for small Ra
Brittle
H13 steel [23] Brass [22] SKD11 steel [21]
+, −
Envelop condition
+ 15° for less wear
Ductile
Quartz glass [24] Crown glass [25]
+ +
Experiment Experiment
+ 50° for small Ra + 45° for small Ra
Sapphire [26]
+
Experiment
+ 15° for small Ra
Silicon [27] Soda-line glass [28]
+ +
Velocity gradient Experiment
+30° for high feed + 45° for small Ra
(+ indicates positive lead angle; − indicates negative lead angle)
investigating how the tool inclination angle influences the brittle–ductile transition in micro ball-end milling process. At the same time, few attentions have been paid on the cutting of KDP crystal with micro ball-end mills. Therefore, it is necessary to investigate which tool inclination lead angle is suitable for brittle KDP crystal to achieve a fracture-free surface and can be applied to micro ball-end milling repairing process of KDP optics in the future. In fact, although micro-milling has been regarded as the most promising method for repairing surface defects on KDP surface, the dedicated micro-milling machine tool for repairing large-aperture (430 mm × 430 mm) KDP optics is still under development. Meanwhile, the tool inclination angle is considered as the premise parameter of the design of machine tool because it is closely related to the spindle postures. Therefore, exploring the effect of tool inclination angle on the surface quality of KDP optics is very beneficial to the design of the dedicated micro-milling machine tool and is of great significance for the future engineering repairing of large-aperture KDP components. In this work, the effect of tool inclination on the brittle– ductile transition involved in the micro-milling of KDP crystal is theoretically and experimentally investigated. The theoretical models for calculating the UCT and cutting speed characteristic during the micro-milling process of KDP crystal are presented in Section 2. The experimental procedure and settings are described in detail in Section 3. And then, the detailed discussions on the effect of tool inclination angle on the brittle–ductile transition and surface quality of KDP crystal are performed in Section 4 with a conclusion finally drawn in Section 5.
milling process is actually more complicated than the SPDT due to its ball type periphery, there is still less literature uncovering the relationship between the UCT and tool inclination angle in micro ball-end milling of brittle material. In this section, a theoretical model for calculating the UCT with various tool inclination angles is deduced in detail. The schematic diagrams for micro ball-end milling of KDP crystal with various inclination directions are presented in Fig. 1. It can be seen that the spindle rotates clockwise and the workpiece moves in a horizontal direction. The cutter inclines clockwise with the spindle and the angle of the tool axis against the vertical line is regarded as the lead angle. From the perspective of tool feed movement, the values of lead angles are positive when the tool inclines along the tool feed direction, as shown in Fig. 1b. The values of lead angles are negative when the tool inclines along the opposite direction, as shown in Fig. 1c. Figure 2 shows the micro-milling schematic when the lead angle is positive without the consideration of tool run-out. The inclination angle is marked as α. The symbols O1 and O2 represent the centers of tool at the adjacent tool passes. The highlighted region indicates the intersection zone of the adjacent tool passes. The point Pm1 on curve I denotes crossover point between the cutting edge and upper machined surface. The points of Pu, Pm2, Pc on curve II are the upper, middle,
2 Theoretical analyses in micro ball-end milling of KDP crystal 2.1 Model for UCT calculation with various lead angles Since the UCT can be used to characterize the ductile–brittle transition, it is indispensable to know the UCT under various micro-milling conditions. Even though the micro ball-end
Fig. 1 Schematic diagram for micro-milling of KDP crystal with various inclination directions: a actual tool system with inclination angle; b positive lead angle case; c negative lead angle case
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Fig. 2 Schematic of the micro-milling process with positive lead angle
lower points of the cutting edge involved in the next-step milling process, respectively. The line segments Pu Tu, Pm2Tm, PcTc represent the radial distances perpendicular to the tool axis with respect to points Pu, Pm2, Pc, respectively. The origin coordinate system X′O1Y′ is set as the reference coordinate system, and the point Pm1 on curve I can be written as: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ( 0 x m1 ¼ 2Rap þ ap 2 ð1Þ 0 y m1 ¼ R−ap The tool coordinate system X1O1Y1 in the former milling process is obtained by positively rotating the reference coordinate system X′O1Y′ with an angle of α. Consequently, the coordinate of Pm1 in X1OY1 can be acquired through the transformation of coordinates. And the transformation matrix can be expressed as: 0 cosα sinα xm1 x m1 ¼ ð2Þ 0 ym1 −sinα cosα y m1 Considering the translation transformation between the adjacent tool passes due to feed motion, the coordinate of point Pm2 on curve II can be obtained by solving the following equation: 2 2 xm2 − f z cosα þ ym2 þ f z sinα ¼ R2 ð3Þ
As for the case of negative lead angle, the micro-milling schematic is shown in Fig. 3. If the lead angle is too small, the tool tip is likely to engage in the chip formation area and scratches the workpiece surface. The cutting process is absolutely in brittle machining because of the zeroth cutting speed at the tool tip. Here, we do not discuss the UCT under this condition. Hence, the critical negative lead angle can be calculated as follows: that is to say, the influence of tool inclination on UCT is effective only when the value of lead angle exceeds the αcritical. Although the chip formation area in the negative milling is the same as that in the positive milling, the UCT is totally different from each other. For the case of negative lead angles, the UCT can be calculated on basis of Eqs. (2) and (5–7) when the tool tip is away from the top surface of KDP crystal in the milling process. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ( 0 x m1 ¼ − 2Rap −a2p ð6Þ 0 y m1 ¼ R−ap 2 2 ð7Þ xm2 þ f z cosα þ ym2 − f z sinα ¼ R2
2.2 Model for cutting speed calculation with various lead angles During the micro-milling of KDP crystal, the cutting speed varies greatly along the cutting region where a distinct cutting velocity gradient exists. This is attributed to the fact that the effective cutting radius between the rotational axes and cutting edge changes in the whole contact area. The cutting speed is zero at the tool tip, and it increases along the cutting edge from the bottom as a function of the effective cutting radius. The velocity gradient is regarded as the gap between the maximum and minimum velocities. It is interesting to note that when the lead angle becomes smaller, the engaged cutting edge gets closer to the tool tip. Therefore, the corresponding velocity gradient would be larger due to its ball-type periphery.
where fz is the feed per tooth, and f and n denote the number of tool teeth, tool feed speed, and spindle speed, respectively. It can be seen that the vertical coordinates of Pm1 and Pm2 are the same in the X1O1Y1 coordinate system: ym1 ¼ ym2
ð4Þ
As indicated in Fig. 2, the UCT in the case of positive milling with various inclination angles can be determined as follows: hmax ¼ xm1 −xm2
ð5Þ
Fig. 3 Schematic of the micro-milling process with negative lead angle
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When the tool inclines at various angles along the tool feed direction, the different parts of the cutting edge are engaged in the milling process. As a result, the produced effective cutting speed and velocity gradient are different as well. This would produce a direct impact on the mechanical and thermal loads generated in the high-speed milling process, which could eventually affect the ductile-mode machining and machined surface quality of KDP crystal. Hence, it is essential to analyze the effective cutting speed and velocity gradient during the milling process. For various inclination angles of ball-end mill, the contact areas between the cutting edge and workpiece would be different, making it difficult to analyze the cutting speed and velocity gradient. In this work, the average speed is adopted as the effective cutting speed for investigating the relationship between cutting speed and tool inclination angle in the micro ball-end milling process. The schematic diagram for calculating the average speed with positive and negative lead angles is presented in Fig. 4. The point of O stands for the ball center. The points of Pu, Pl, Pm are the upper, lower, middle points of the cutting edge engaged in the milling process, respectively. The line segment TuPu, perpendicular to the axis with respect to Pu, presents the effective cutting radius. Tu1 is the intersection point between the vertical line normal through the point O and the horizontal line through the point Pu. As seen from Fig. 4a, there are specific geometrical relationships in the cutting area described by the following equations. 8 R−ap > < ∠Pu OTu1 ¼ arccos R ð8Þ f > z : ∠Pl OTu1 ¼ arcsin 2R where R is the radius of ball end mill, ap is the depth of cut, and fz is the feed rate per tooth.
Fig. 4 Schematic diagram for calculating the average cutting speed: a positive lead angle case; b the case of critical negative lead angle with tool tip located in the top surface of workpiece
The effective cutting radius of the upper and lower points engaged in the cutting process can be expressed as: (
Tu Pu ¼ Rsinðα þ ∠Pu OTu1 Þ Tl Pl ¼ Rsinðα−∠Pl OTu1 Þ
ð9Þ
Considering the inclination angle of ball-end mill and spindle speed N, the maximum, minimum, and effective cutting speeds can be derived from Eqs. (8) and (9) as follows: 8 R−ap > > ¼ 2πNRsin a þ arccos V > max > > R > > > f > z < V min ¼ 2πNRsin a−arcsin 2R 1 0 > R−ap fz > > −arcsin arccos > > B > R 2R C > V eff ¼ 2πN sin@a þ A > > 2 :
ð10Þ
For the case of the negative lead angle, we only need to analyze the cutting speed characteristic when the angle value exceeds the critical angle, which is illustrated in Fig. 4b. The detailed calculating steps are similar to that under positive lead angles as discussed previously. In this condition, the final effective cutting speed can be obtained according to following equations: 8 fz > > V ¼ 2πNRsin a þ arcsin max > > 2R > > > > R−a p > < V min ¼ 2πNRsin a−arccos R 1 0 > R−ap fz > > −arccos arcsin > > B > 2R R C > V eff ¼ 2πN sin@a þ A > > 2 :
ð11Þ
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3 Experimental setup and procedure
3.3 Experimental procedure
3.1 Machine tool
With the aim to systematically evaluate the effect of tool inclination on the surface quality of micro-milled KDP crystal, the tool inclination angles were divided into two categories: positive and negative values, as aforementioned. The absolute angle value was varied from 5° to 55° (in steps of 5°) in the experiments. Meanwhile, the spindle rotation speed was set as 30,000 rpm, and the depth of cut was set as 2 and 8 μm. The detailed cutting parameters are given in Table 2. The morphology of machined grooves was observed with an ultra-depth 3D microscopy system (VHX 1000E; Keyence) and an AFM (Dimension 3100; Veeco). A surface profilometer (PGI 1420; Taylor Hobson) was employed to measure the surface toughness. For each machined groove, the surface roughness was tested three times from the groove center along the feed direction and the reported surface roughness was the average value.
As shown in Fig. 5, a homebuilt miniature five-axis vertical spindle machine tool [33] was used for performing the microgroove milling experiments. The high-speed motorized spindle used in this machine tool is NAKANISHI with the highest rotational speed of 80,000 rpm. Meanwhile, the spindle is attached to a rotating unit using Akribis with high motion accuracy at low speed. Further, the micro-milling feed motion is provided using Parker company’s linear motion units with 0.1-μm resolution and ± 0.35 μm/10 mm straight-line positional precision. In addition, the micro-milling workstation is placed on a vibration-isolated table made of granite materials to reduce the negative effect of vibration.
3.2 Workpiece material and milling tool A well fly-cut KDP crystal was used as workpiece. In order to ensure the same depths of the machined micro-grooves, a 16 mm × 30 mm planar area was firstly machined on the surface by flat-end milling. The cubic boron nitride (CBN) micro ball-end mill with a radius of 0.25 mm was utilized in the micro-milling experiments (SSBL 200, NS Tool). The micro-milling tool featured a helix angle of zero degree and possesses two cutting edges, and the corresponding cutting edge radius is around 1.1 μm.
Fig. 5 Images of the homebuilt five-axis vertical spindle machine tool
4 Results and discussion 4.1 Effect of tool inclination direction on the machined surface quality One objective of this work is to assess the effect of tool inclination direction on the machined surface quality. Therefore, the cutting process and corresponding experiment results are discussed as follows. As seen in Fig. 6, two milling modes (up-milling and down-milling) exist in the micro-milling processes with positive or negative lead angles. In the up-milling mode, with the rotation of tool, the cutting direction is opposite to the feed motion. While in the down-milling mode, the cutting direction becomes the same to the feed motion direction. The UCT starts from zero when the cutting edge contacts with the workpiece and increases to be the maximum value (dmax) in the middle of the groove. And then it falls back to zero when the cutting edge departs from the workpiece. In other words, the cutting process manifests as the ductilemode machining in the up-milling mode, while the brittlemode fracture is more likely to occur in the down-milling mode if the feed rate is sufficiently high. In addition, owing to the size effect [34], the plowing would take place instead of chip formation if the UCT is smaller than the minimum chip thickness required for forming chips. Under this circumstance, the plowing could play the dominant role in micro-milling process both at the beginning and end of the cut. Figure 7 shows the typical grooves machined under the depth of cut of 8 μm with the lead angle of + 30° and − 30°, respectively. It can be observed that the surface machined in the up-milling mode clearly manifested a smoother finish than that in the down-milling mode for both negative and positive milling processes. There were continuous periodic feed marks
Int J Adv Manuf Technol Table 2 The cutting parameters applied in the micro-milling experiments of KDP crystal
Inclination direction
Lead angles α (°)
Spindle speed N (× 104 rpm)
Depth of cut ap (μm)
Feed per tooth fz (μm)
+
5, 10, 15, 25, 30, 35, 40, 45, 50, 55
3.0
2.8
0.5
−
left by ball-end mill at the bottom of the grooves. Some typical brittle defects, such as micro pits, were found on the groove surface in the latter half of the cut. This is primarily because these areas undergo the down-milling process as aforementioned. Thus, the lower parts of the groove surfaces along the exit edges are more likely to have defects as shown in Fig. 7. The comparison of milling processes with positive and negative lead angles shows a significant difference only in terms of surface quality. As seen from Fig. 7, the difference of the peak and valley (PV) on the bottom surface decreased quickly from 274.698 to 116.738 nm. The maximum height of the profile Rz decreased from 94.744 to 72.937 nm when the lead angle changed from − 30° to + 30°. Interior edge chipping only appeared on the surface produced with negative lead angles. This phenomenon could be explained that smaller undeformed chip thickness with positive lead angles brings about smaller equivalent shear angle, which is very conducive to the ductile-mode machining and correspondingly improves the final surface quality. Meanwhile, it is recalled that micro pits would take place on the groove surface along the entry edge due to the plowing effect [34]. When the cutting edge touches the workpiece at the beginning of the cut, the UCT is too small to form chips. Therefore, the plowing is predominant and would finally result in the formation of micro cracks. This scenario becomes more apparent in the case of smaller lead angles. The engaged cutting edge possesses lower cutting speed on account of smaller positive lead angle, which could exacerbate the plowing effect. Besides, as shown in Fig. 7, regular tool marks measured by AFM occur on the groove bottom, meaning the cutting process is in ductile regime. In other words, the
Fig. 6 Schematic of two different milling modes involved in the microgroove milling
removal of material is by shear cutting. Figure 8a shows the morphology of machined grooves with + 5° lead angle. It was found that some characteristic marks with the maximum height of 144.296 nm were produced on the groove surface. This special morphology may be attributed to the excessive elastic and plastic deformations induced by the plowing mechanism. Conversely, a large number of brittle fractures appeared in the middle part of the groove surface machined with − 5° lead angle as shown in Fig. 8b. These fractures were caused by the tool tip engaged into the cutting area and the maximum fracture depth was 2.115 μm. It means that the scratching of tool tip eventually results in the formation and propagation of cracks. All in all, the positive milling process is proven to be a better strategy during the defect micro-milling repairing of KDP crystal optics in terms of surface defects.
4.2 Effect of tool inclination angle on the ductile–brittle transition In order to investigate the effect of tool inclination on the ductile–brittle transition, a set of micro-groove milling experiments were performed. Figure 9 shows the morphology of the machined groove surfaces with the depth of cut of 2 μm. One can see that the tool inclination direction and angle have remarkable influence on the ductile–brittle transition. As mentioned in Section 1, the UCT can be used to determine whether the cutting is in ductile or brittle mode. Only when the UCT is smaller than a critical value can the material be removed by plastic deformation rather than crack propagation. According to Bifano’ study [35], the critical undeformed chip thickness in the ductile–brittle transition can be defined as a function of the material properties. The corresponding equation can be described as follows: 2 E Kc dc ¼ 0:15 ð12Þ H H where E, H, and Kc denote the Young’s modulus, hardness, and fracture toughness of the workpiece, respectively. All the material parameters of KDP crystal are listed in Table 3, and the theoretically calculated critical UCT value for KDP crystal is 77 nm. Meanwhile, with the aim to investigate the relationship between the tool inclination angle and brittle–ductile transition, the UCT with various lead angles has been calculated and the corresponding result is depicted in Fig. 10.
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Fig. 7 Morphology of the machined grooves with different inclination angles: a α = + 30° lead angle; b α = − 30° lead angle
Figure 9a depicts the micro straight groove produced with lead angle α = + 15°. It can be observed that the groove was completely machined in the brittle mode and the groove
surface comprised obvious micro cracks and pits. This is primarily because the UCT value with α = + 15° lead angle is much larger than the critical one, as shown in Fig. 10. It means
Fig. 8 Morphology of machined grooves and surface morphology parameters with different inclination angles: a α = + 5° lead angle; b α = − 5° lead angle
Int J Adv Manuf Technol Fig. 9 The microscopic images of machined groove surfaces with different tool inclination angles: a α = + 15°; b α = + 30°; c α = + 45°; b α = + 55°; e α = − 15°; f α = − 30°; g α = −45°; h α = − 55°
that the crack propagation would occur in the cutting area for this case. Besides, some piled chips appeared along the entry edge due to the extrusion effect in the beginning cutting stages [13]. For the case of α = + 30° lead angle in Fig. 9b, similar surface defects were observed as well. As seen from Fig. 9a and b, there was no appearance of piled chips along the exit edge, which always occurred along the entry edge. This is because the cutting process is in down-milling mode. Figure 9c shows the occurrence of ductile cutting with lead angle of α = + 45°. Although slight piled chips also took place Table 3
Material properties of KDP crystal [18, 36]
Density ρ (kg/cm3) Poisson’s ratio ν Young’ modulus E (GPa) Hardness H (GPa) Fracture toughness Kc (MPa·m1/2)
2.344 × 103 0.24 44 1.7 0.24
in the beginning of the cut owing to the plowing effect, there was no visible brittle crack on the entire machined surface. Besides, the regular tool marks further indicate the smooth fracture-free surface that has been obtained. This experimental result is consistent with the theoretical result where the UCT with the lead angle α = + 45° is 79 nm. This calculated value approximately equals the critical UCT value of 77 nm. It means that by increasing the tool inclination angle, the UCT decreases and comes near the critical UCT. The crack propagation and brittle-mode cutting could be prevented when the tool inclination angle exceeds + 45°. However, the interior edge chippings appeared on the groove surface with the lead angle α = +55°, as shown in Fig. 9d. Even though the tool inclination angle was within a reasonable range where the UCT was smaller than the critical value, the brittle defects still existed on the groove surface. One potential reason is that owing to the strain rate hardening effect, the workpiece would be more difficult to be cut off when the cutting speed along the cutting edge is very high
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Fig. 10 The calculated undeformed chip thickness for various tool inclination angles
Fig. 11 The tested surface roughness with respect to the tool inclination angle
[32]. On the other hand, the stiffness of the milling system would decrease in this situation. As a result, the stability of the milling process declines, which could lead to the damage of surface integrity [26]. This can be also proved by the ragged edges due to the actual mill tool run-out. For the cases of negative lead angles shown in Fig. 9e–g, it can be seen that the amount of brittle cracks decreased gradually with the increase of tool inclination angle. There were a few micro pits on the surface produced with the lead angle of α = − 45°. However, the brittle fracture would take place once the tool inclination angle exceeds 45°, as shown in Fig. 9h. This remarkable trend is exactly the same as that in the positive milling process. As mentioned above, this observation was attributed to the variation of the tool inclination angle. Even when comparing the surface defects produced by positive and negative milling processes with the same lead angle, the number and size of brittle defect produced by negative milling are larger than those produced by the positive milling. The calculated UCT should be responsible for the differences in machined surface integrity. A generally larger UCT indeed exists in the negative milling as shown in Fig. 10, which would give rise to the deterioration of machined surface integrity. It should be noted that these micro cracks and pits would not only deteriorate the quality of machined surface but also be more likely to induce local light intensification, which would incur potential laser damage to the optical components that are applied in the high-power lasers [2]. For this reason, these surface defects must be avoided on the repaired optical surface in the “optical recycle loop strategy” of KDP crystal optics. It can be concluded from the results and discussion above that the positive milling process is more applicable than the negative milling process to cut KDP crystals when repairing the KDP optics by micro ball-end milling method. Moreover, a + 45° lead angle in positive milling is found to be the most favorable for removing the crystal material in the ductile mode.
4.3 Effect of tool inclination angle on the machined surface roughness Figure 11 displays the influence of tool inclination angle α on the surface roughness produced by micro ball-end milling. It is illustrated that with the increase of lead angle, the machined surface quality became better and surface roughness values decrease when the lead angle was less than 45°. This phenomenon was attributed to the increase of effective cutting speed and decrease of velocity gradient. The former is usually considered to be beneficial for the material removal owing to the thermal softening, and the latter is conducive to reduce cutting force fluctuation. The calculation results of effective cutting speed and velocity gradient with various tool inclination angles are presented in Fig. 12. It is proved that inclining the ball-end mill at large lead angles can significantly improve the effective cutting speed and also decrease the velocity gradient. However, there is a sharp increase of surface roughness when the lead angle arrives at 55° in Fig. 11. One potential explanation is that the excessive effective cutting speed could result in the material removal difficulty owing to the strain rate
Fig. 12 The effective cutting speed and velocity gradient with various lead angles
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Fig. 13 The measured surface roughness with the lead angle of α = + 45°
hardening effect as mentioned previously [5]. Besides, coupled with the instability of micro-milling system at large lead angles, brittle-mode machining prevails under this condition and eventually causes higher surface roughness values. Therefore, the evolution of surface quality with respect to the tool inclination angle depends on the competitive mechanisms between UCT and cutting speed, and the results demonstrate that the + 45° lead angle is the optimal angle for micro ballend milling of KDP crystal. The best surface roughness value achieved could be up to 35.3 nm, as shown in Fig. 13. It is also found that the machined surface roughness is extremely sensitive to the tool inclination direction. As illustrated in Fig. 11, the smoother surface can be achieved if the tool inclination direction is the same as the feed direction (positive lead angle). This is probably because micro-milling with positive lead angles generally leads to larger effective cutting speed and smaller velocity gradient than those in the micro-milling with negative leading angles as shown in Fig. 12. As a consequence, the more consistent shear stress is generated in the chip formation area when machining with positive lead angles. It means that the ductile cutting can be more easily attained and ultimately these factors (larger effective cutting speed and smaller velocity gradient) can jointly facilitate the ductile mode machining and the reduction of surface profile peaks.
5 Conclusion In this paper, the influence of tool inclination on the surface quality of KDP crystal machined by micro ball-end milling was thoroughly investigated. The micro ball-end milling experiments were carried out and the machined surface quality
was measured and analyzed. The conclusions are drawn as follows: 1. The theoretical models for calculating the UCT and cutting speed in the micro ball-end milling process of KDP crystal have been deduced. These models could be applied to account for the experimental results of the effect of tool inclination direction and angle on the surface quality of KDP crystal machined by micro ball-end milling. 2. It is found that inclining the ball-end mill in the tool feed direction (positive angle) possesses the smaller UCT than that in the opposite direction (negative angle) and the former should be preferred when repairing the microdefects on KDP surface by micro ball-end milling method. 3. Increasing the tool inclination angle causes the increase of effective cutting speed and decrease of velocity gradient along the engaged cutting edge, which could jointly result in smaller equivalent shear angle and consequently reduce the surface roughness. 4. The evolution of surface quality with respect to the tool inclination angle depends on the competitive mechanisms between the UCT and cutting speed characteristic. The results demonstrate that the + 45° lead angle is suggested to be the optimal angle for the micro ball-end mill cutting of KDP crystal and the best surface roughness value achieved could be up to 35.3 nm. Acknowledgments The authors gratefully acknowledge the financial support from the National Natural Science Foundation of China (no. 51775147, no. 51705105), Science Challenge Project (no. TZ20160060503-01), China Postdoctoral Science Foundation funded project (no. 2017 M621260), Heilongjiang Postdoctoral Fund (no. LBH-Z17090), and Self-Planned Task (no. SKLRS201718A) of State Key Laboratory of Robotics and System (HIT).
Int J Adv Manuf Technol Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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