Effect of processing parameters of rotary ultrasonic machining on surface integrity of potassium dihydrogen phosphate crystals

Abstract

Potassium dihydrogen phosphate is an important optical crystal. However, high-precision processing of large potassium dihydrogen phosphate crystal workpieces is difficult. In this article, surface roughness and subsurface damage characteristics of a (001) potassium dihydrogen phosphate crystal surface produced by traditional and rotary ultrasonic machining are studied. The influence of process parameters, including spindle speed, feed speed, type and size of sintered diamond wheel, ultrasonic power, and selection of cutting fluid on potassium dihydrogen phosphate crystal surface integrity, was analyzed. The surface integrity, especially the subsurface damage depth, was affected significantly by the ultrasonic power. Metal-sintered diamond tools with high granularity were most suitable for machining potassium dihydrogen phosphate crystal. Cutting fluid played a key role in potassium dihydrogen phosphate crystal machining. A more precise surface can be obtained in machining with a higher spindle speed, lower feed speed, and using kerosene as cutting fluid. Based on the provided optimized process parameters for machining potassium dihydrogen phosphate crystal, a processed surface quality with Ra value of 33 nm and subsurface damage depth value of 6.38 μm was achieved.

Keywords

Potassium dihydrogen phosphate crystal rotary ultrasonic machining surface roughness subsurface defect depth

Introduction

Potassium dihydrogen phosphate (KDP, with chemical formula of KH₂PO₄) crystals, with excellent optical characteristics such as of high nonlinearity and high laser damage threshold, are widely applied as optical frequency converters, electro-optical switching elements, and frequency harmonic generation lenses in the field of nonlinear optics. However, because KDP crystals are soft and brittle, easily deliquesced, temperature-sensitive, and highly anisotropic, significant difficulties are experienced in the precision processing of large KDP crystals.^1,2 This material presents unique challenges in the processes required to fabricate ultra-precision optical elements.³

Commonly used machining methods for ultra-precision machining process of KDP crystal include single-point diamond turning and magnetorheological finishing (MRF). In 1986, Fuchs et al.⁴ stated that single-point diamond turning could be used to produce finished KDP parts without additional surface polishing. The procedure and the equipment necessary for single-point diamond flycutting of KDP crystals were described by Montesanti and Thompson⁵ in 1995. In recent years, various experiments have been conducted to determine optimal machining conditions in single-point diamond turning under dry conditions or a spiral turning method based on the ductile cutting mechanism of brittle materials.^6–9 According to research results, single-point diamond turning is capable of yielding a surface roughness with root-mean-square value of 1.0–3.0 nm under some specific cutting parameters and conditions.³ Researchers, such as Menapace et al.³ and Jacobs¹⁰ from Lawrence Livermore National Laboratory, discussed the feasibility of finishing water-soluble single-crystal KDP with MRF by modifying the magnetorheological fluid chemistry and mechanics to match the unique physical properties of each workpiece. A MRF-polished KDP surface with microroughness of 0.65 nm root-mean-square value was achieved.

To improve the material removal rate and processing efficiency and to achieve the desired workpiece shape, precision machining methods, such as grinding and milling, are used before the last ultra-precision machining procedure. However, because the surface quality and subsurface damage depth (SSDD) of the KDP crystals after precise machining affect the machining quality and efficiency of the ultra-precise machining significantly, it is extremely important to control the surface roughness and SSDD of the KDP crystal workpiece throughout the precise machining process.

Recently, to solve difficult machining problems of brittle materials that are encountered with regular machining, many researchers have studied rotary ultrasonic machining, which is a combination of the hammering effect of ultrasonic vibration, the polishing effect of a diamond tool, and the effect of ultrasonic cavitation. Komaraiah and Reddy¹¹ studied the impacts on fracture toughness and hardness of brittle materials. Pei and Ferreira¹² investigated the ductile-mode material removal mechanism in rotary ultrasonic machining. Bertsche et al.¹³ presented an analytical model of rotary ultrasonic milling. Many researchers^14–17 have conducted machining experiments on glass, ceramic, titanium, and silicon carbide matrix composites. Wang et al.¹⁸ and Zhang et al.¹⁹ investigated the surface roughness of KDP crystal material in rotary ultrasonic machining. However, the best surface roughness value that was obtained is higher than 0.3 μm. Research on subsurface damage, which is a key factor in the subsequent ultra-precise machining, was not mentioned.

This article aimed to determine the effects of process parameters on surface integrity and to improve machining precision through experimental research on surface and subsurface properties with a (001) crystal surface produced by traditional and ultrasonic machining of KDP crystals. It analyzed the influence of five process parameters, including spindle speed, feed speed, type and size of sintered diamond wheel, ultrasonic power, and selection of cutting fluid, on a workpiece surface and subsurface quality. The experimental program including the test workpiece preparation, experimental system, machining conditions, and measurements are described in section “Experimental program.” The experimental results and discussion of the influence of process parameters on surface roughness and SSDD are discussed in section “Experimental Results.” Conclusions were drawn in section “Conclusion.”

Experimental program

Test workpiece

When the absolute temperature is greater than 123 K, the KDP crystal belongs to a tetragonal crystal family with negative crystal axis, and the ideal KDP crystal shape is an adduct of a square column and a square bipyramid. Because of this lattice structure, this crystal exhibits strong anisotropy in physical and mechanical properties. The KDP crystal was oriented using a YX-1 X-ray crystal orientation analyzer, and experimental pieces (15 × 10 × 5 mm) were cut out using an inner-circle cutter. As shown in Figure 1, surface ABCD is a (001) crystal surface. The [110] crystal direction was selected as the machining direction based on preliminary experimental results.

Figure 1.

Potassium dihydrogen phosphate crystal test piece: (a) (001) crystal surface, (b) [110] crystal direction, and (c) test piece.

Experimental system

An experimental system for rotary ultrasonic machining of the KDP (001) crystal surface was established using an Ultrasonic 50 ultrasonic machine from DMG Company, Germany, as shown in Figure 2. Rotary ultrasonic machining is a composite processing method that combines traditional ultrasonic machining with abrasive machining, and as a novel machining method for the precise processing of brittle materials, it allows for a machining process that is unconstrained by workpiece size. Because of the introduction of high-speed rotation and axial high-frequency vibration, the deformation behavior, processing mechanism, and tool stress state during material processing differ from traditional machining.

Figure 2.

Experimental setup: (a) schematic diagram and (b) cutting system.

Given that KDP crystals are sensitive to temperature change, the KDP test piece was pasted onto a fixture with low-melting-point binder using a micro-control electro-thermal board. A Kistler 9257B dynamometer was used to measure the cutting force during machining process. Because KDP crystals are deliquescent and soluble in water, a self-made cutting fluid circulation system was designed and fitted to the Ultrasonic 50 as shown in Figure 2(b). Based on the mechanical properties of KDP crystals, the impact force F_c and spouting velocity v_c of the cutting fluid were designed to meet the constraints in equation (1) and minimize the influence of cutting fluid on test piece surface

v_{c}^{'} \geq 20 m / s, F_{c} \leq 10 N

(1)

F_c and v_c were calculated and decided according to equation (2) based on the theorem of impulse. The designed experimental values of the impact force and spouting velocity were found to be ∼4.6 N and 34 m/s, respectively

(- F_{c} + mg) t_{c} = {mv}_{c}^{'} - {mv}_{c}

(2)

where F_c is the impact force of the cutting fluid, m is the mass of the cutting fluid, g is the gravitational acceleration, t_c is the spouting time of the cutting fluid, v_c is the spouting velocity of the cutting fluid, and $v_{c}^{'}$ is the cutting fluid velocity on the surface of test piece surface.

Machining conditions

Based on an orthogonal experiment method, 26 test pieces were designed. Five process parameters, including spindle speed, feed speed, three types of diamond tools with six kinds of granularity levels, ultrasonic power, and three types of cutting fluid, were considered to analyze their influence on the surface and subsurface characteristics of the KDP crystal process. From pre-research results of the brittle–ductile transition characteristics of KDP crystals, the cutting depth was selected as 10 μm. Values of the process parameters used in the experiments are shown in Table 1. Each experiment was conducted twice on one piece, and average test values were used as experimental results. In Table 1, special cutting oil (SCO) is a type of emulsified cutting oil that was provided by the Technical Institute of Physics and Chemistry, Chinese Academy of Sciences. The external diameters of all diamond tools were 50 mm. Figure 3 shows the dimensions and shapes of three diamond tool types.

Table 1.

Potassium dihydrogen phosphate crystal machining process parameters.

Process parameters	Values
Spindle speed (r/min)	2500, 4000, 5000, 6000, 7000, 8000
Feed speed (mm/min)	1, 5, 10, 30, 80, 100
Cutting depth (μm)	10
Diamond tool	V.D.100, V.D.200, V.D.270; M.D.100, M.D.140, M.D.200; B.D.30, B.D.70, B.D.100
Cutting fluid	None, SCO, kerosene
Ultrasonic power (W)	0, 6, 12, 18, 24, 30

V.D.: vitrified sintered diamond tool; M.D.: metal-sintered diamond tool; B.D.: brazed diamond tool. The numbers after these initials are the granularity values of the diamond machining wheel.

Figure 3.

Structures of diamond tools (unit: mm): (a) V.D., (b) M.D. and (c) B.D.

The surface quality of the test pieces after processing was monitored and detected using a three-dimensional white light interferometer surface morphology analyzer with a pixel resolution of 480 × 752 and an XYZ feeding distance of 100 × 100 × 100 mm. The morphology of three 3.5 mm lines on the test piece surface was measured to calculate the value of the surface roughness Ra for the test piece.

The SSDD of the test piece is an important indicator to evaluate the quality of optical crystal material processing, and it affects directly the material’s optical functionality. Common methods used to measure the SSDD include layer-by-layer polishing and etching, cross-section microscopy, the angle-polishing method, the ball dimpling method, and MRF technology.²⁰ The rheological finishing method can resolve the weakness of the other methods, such as additional damage, low measurement efficiency, and unfavorable measurements and observation,^20,21 and it was applied to measure the damage depth of the subsurface of the test piece subsurface. A KDMRF-1000 MRF machine, developed by the College of Mechatronics Engineering and Automation, National University of Defense Technology, was used to remove the test pieces surface and expose the subsurface. A surface morphology analyzer (Talysurf 5P-120, Taylor Hobson Corp., UK) was used to measure depth profiles of the centerline of the MRF spots. After being cleaned with ethyl alcohol, test pieces were placed on an inching platform to measure and determine the SSDD using an optical microscopy (KEYENCE, Japan).

Experimental results

Influence of process parameters on surface roughness

Experimental data on the influence of spindle speed, feed speed, diamond granularity level of the tools, cutting fluid, and ultrasonic power on KDP surface roughness Ra are shown in Figure 4(a)–(e).

Figure 4.

Influence of processing parameters on surface roughness: (a) influence of spindle speed on surface roughness, (b) influence of feed speed on surface roughness, (c) influence of diamond granularity level on surface roughness, (d) influence of cutting fluids on surface roughness, and (e) influence of ultrasonic power on surface roughness.

As shown in Figure 4(a) and (b), the surface roughness Ra decreased as the spindle speed increased and increased as the feed speed increased. When the spindle speed increased from 2500 to 8000 r/min, Ra decreased from 0.72 to 0.28 μm, with ∼60% reduction. When the feed speed increased from 1 to 80 mm/min, Ra increased by ∼130% from 0.24 to 0.56 μm. When the feed speed was increased to 100 mm/min, the test piece was crushed. The process with KDP crystal cannot be completed under such process parameters.

Under the same processing conditions with the same type of diamond tool as shown in Figure 4(c), the larger diamond granularity and smaller diamond particles yield a lower test piece surface roughness Ra value after processing. In addition, the KDP test piece processed with the metal-sintered diamond tool had the smallest surface roughness Ra value below 0.2 μm. For the brazed diamond machining wheel, because its granularity is relatively small, the surface roughness of the KDP crystal test piece after processing is expected to be somewhat less than the roughness results after processing with the vitrified sintered diamond machining wheel. When the granularity value was increased from 30 to 100, the surface roughness Ra decreased from 0.68 to 0.23 μm (∼60% reduction). For the vitrified sintered diamond machining wheel, when the granularity increased from 100 to 270, the surface roughness Ra value decreased from 0.76 to 0.39 μm (∼48% reduction).

Regarding the cutting fluid as shown in Figure 4(d), although SCO is safer than kerosene, the surface roughness Ra value of the test piece was largest after using SCO for machining. Ra with no cutting fluid was second, and that of 37 nm was the smallest when kerosene was used as the cutting fluid. However, pieces processed without cutting fluid showed obvious border cracks, and the processing quality and surface integrality of the test pieces were not guaranteed. In summary, kerosene is the best choice as cutting fluid in KDP crystal machining if good safety protection exists.

The surface roughness Ra value for the KDP crystal test piece after machining decreased with increase in ultrasonic power as shown in Figure 4(e). As the ultrasonic power increased from 0 to 30 W, the surface roughness of the test piece after processing decreased from 0.44 to 0.11 μm. When the ultrasonic power was increased from 6 to 12 W, the surface roughness of the test piece decreased significantly, whereas changes from alterations in other parameters were relatively minor.

Figure 5 shows the surface morphology after traditional processing for test piece No. 4 and ultrasonic machining for No. 24 with an ultrasonic power of 30 W, whereas the other processing parameters were similar. The surface of the test piece after the ultrasonic process, compared with the test piece surface without ultrasonic process, is relatively smooth and the distribution of surface morphology after rotary ultrasonic machining is smoother than that after traditional machining.

Figure 5.

Three-dimensional surface morphology of processed test pieces: (a) Piece No. 4 (traditional machining) and (b) Piece No. 24 (ultrasonic power = 30 W).

Influence of the process parameters on the SSDD

Experimental data on the influence of spindle speed, feed speed, diamond granularity level of the tools, cutting fluid, and ultrasonic power on the SSDD of the KDP crystal test pieces are shown in Figure 6.

Figure 6.

Influence of processing parameters on subsurface damage depth (SSDD): (a) influence of spindle speed on SSDD, (b) influence of feed speed on SSDD, (c) influence of diamond granularity on SSDD, (d) influence of cutting fluids on SSDD, and (e) influence of ultrasonic power on SSDD.

According to Figure 6(a) and (b), the SSDD decreased as the spindle speed increased and increased as the feed speed increased. When the spindle speed increased from 2500 to 8000 r/min, the SSDD decreased from 73.9 to 23.6 μm (∼60% reduction). When the feed speed increased from 1 to 80 mm/min, the SSDD increased from 22.7 to 70.6 μm (∼200% increase).

For the diamond tool used with the same manufacturing process as shown in Figure 6(c), a larger diamond grit granularity resulted in a smaller SSDD. Of the different tested machining wheels, the test piece processed with metal-sintered diamond tool had a minimum SSDD. When the granularity was increased from 100 to 200, the SSDD decreased by ∼29% from 30.4 to 21.4 μm. The test piece processed using the brazed diamond machining wheel had a maximum range of change in SSDD. When the granularity was increased from 30 to 100, the SSDD decreased from 55.4 to 27.6 μm (∼50% reduction). If the vitrified sintered diamond machining wheel was used for processing and the granularity was increased from 100 to 270, the SSDD decreased from 0.76 to 0.39 μm, which represents a 48% reduction.

Similar to the surface roughness characteristics, when SCO was used as cutting fluid as shown in Figure 6(d), the SSDD value after processing was the largest. Without cutting fluid, the SSDD was second largest. The SSDD value was smallest when kerosene was used as cutting fluid.

As shown in Figure 6(e), ultrasonic power had a significant impact on reducing the SSDD of the processed test pieces. A greater ultrasonic power resulted in a higher amount of power transfer to the test pieces via ultrasonic vibration. Ultrasonic hammering creates local stress forces on the KDP crystal test pieces that are much larger than the crystal’s fracture limit. Therefore, microcracks are generated and expanded prior to removal of the KDP crystal. However, cracked grits are not too large during the removal process, which is relatively stable. After machining, grits of the diamond tool are embedded onto the test piece surface. The grits scratch, grind, polish, and rip the test piece surface. A large part of material removal occurs in an area of the test piece surface that already had cracks generated by the process instead of achieving complete removal by the direct cutting force on the surface, which would reduce the level of subsurface damage significantly and lead to a significantly reduced SSDD.²² In this experiment, when the ultrasonic power was increased from 0 to 30 W, the SSDD for the processed test piece decreased from 38.8 to 15.3 μm (∼60% reduction).

Figure 7(a) and (b) shows the subsurface morphology of test piece No. 4 at a 33.1 μm depth and test piece No. 24 at a 15.3 μm depth. For No. 4 test piece, small pit-like damage exists even at 33.1 μm beneath the surface, whereas for No. 24 test piece, no damage existed at 15.3 μm beneath the surface.

Figure 7.

Subsurface morphology of processed test pieces: (a) Piece No. 4 (traditional machining) and (b) Piece No. 24 (ultrasonic powder: 30 W).

Discussion

For the discussed machining methods, ultrasonic vibration affects the surface quality of the KDP crystal test pieces. Rotary ultrasonic machining process is a compositional process with diamond tool rotational processing and ultrasonic vibration in the axial direction. A schematic of the single-diamond movement trajectory, presented in our early research, is shown in Figure 8.²³

Figure 8.

(a) Movement trajectory and (b) Kinematic characteristics of single-grit diamond.²³

During rotary ultrasonic machining, the vibration frequency is f = 19–20 kHz and diamond grit undergoes a periodic “grind–empty cut–grind” process. Therefore, the machining speed of rotary ultrasonic machining is significantly larger than that in traditional machining. Under the same spindle speed and feed speed within the same period, for rotary ultrasonic surface machining processing, the trajectory length and movement speed of a single-grit diamond are much greater than those for traditional processing, which leads to a better test piece surface quality. The ultrasonic frequency vibration in rotary ultrasonic machining can improve material equivalent microhardness, restrain crack generation and propagation, and reduce the number and size of cracks.

According to experimental data, the surface roughness Ra and SSDD of the test piece after processing decreased as the spindle speed increased, increased as the feed speed increased, and decreased as the ultrasonic power increased. For diamond machining wheels with the same manufacturing process, larger granularity values corresponded to smaller surface roughness values and SSDD of the test piece after processing. When kerosene was selected as the cutting fluid, the surface roughness Ra and SSDD were smallest, but the Ra and SSDD were largest when SCO was selected. Ra and SSDD existed between these two values when no cutting fluid was used, but the test pieces showed serious cracking.

When the vitrified sintered diamond machining wheel and SCO were applied as cutting fluid for KDP crystal surface machining, the greatest factor affecting surface roughness Ra and SSDD was the ultrasonic power. Ultrasonic vibration makes the machining speed of the diamond machining wheel grit within a given unit of time significantly larger than that of regular processing, which leads to test pieces with better surface and subsurface qualities. Secondary factors were diamond machining wheel granularity and feed speed, whereas the impact of spindle speed is less significant.

Based on the research results, optimal process parameters for the ultrasonic machining of a KDP crystal test piece are as follows: a spindle speed of 8000 r/min, a feed speed of 0.6 mm/min, a cutting depth of 6 μm, a metal-sintered diamond machining wheel chosen as the diamond machining wheel with a granularity value of 200, an ultrasonic power of 30 W, and kerosene as cutting fluid. An additional experiment with optimization processing parameters was conducted. Using this set of parameters to process the KDP crystal, a machined surface with Ra value of 33 nm and SSDD of 6.38 μm were obtained.

Conclusion

This article presented an experimental research on surface roughness and subsurface damage characteristics in traditional and rotary ultrasonic machining on the (001) surface of KDP crystals. The influence of process parameters on surface integrity was discussed. Ultrasonic power level has a significant effect on test piece surface roughness Ra and especially SSDD. It can effectively reduce the depth of subsurface damage and restrain crack growth. A metal-sintered diamond tool is better than other tool types for machining KDP crystals. The surface roughness Ra and SSDD decrease as the diamond granularity increases and increase as the feed speed increases. The surface roughness Ra and SSDD decrease with increase in spindle speed. The cutting fluid played a key role in machining because the KDP crystal is easily deliquesced and sensitive to temperature. Compared with SCO and no cutting fluid, kerosene is the best choice as cutting fluid to machine KDP crystals with good safety protection. Experimental results indicated that rotary ultrasonic machining can produce good surface integrity with a surface roughness of 33 nm and a SSDD of 6.38 μm by optimizing process parameters for the KDP crystal. Rotary ultrasonic machining method can be used in the precision machining of KDP crystals or as a former procedure for ultra-precise machining of KDP crystals.

Footnotes

Academic Editor: Pedro AR Rosa

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

This work was financially supported by the National Nature Science Foundation of China (Grant No. 51475260), Beijing Nature Science Foundation (Grant No. 3141001), and State Key Laboratory of Tribology Foundation (Grant No. SKLT2013B03).

References

Chen

Pang

Wang

. Analysis of 3D microtopography in machined KDP crystal surfaces based on fractal and wavelet methods. Int J Mach Tool Manu2008; 48: 905–913.

Barkauskas

Melninkaitis

Mikšys

. Characterization of KDP crystals used in large aperture doublers and triplers. Proc SPIE2006; 6403: 64031.

Menapace

Ehrmann

Bickel

. Magnetorheological finishing (MRF) of potassium dihydrogen phosphate (KDP) crystals: nonaqueous fluids development, optical finish, and laser damage performance at 1064 nm and 532 nm. In: Proc SPIE 7504: laser induced damage in optical materials, Boulder, CO, 21 September 2009, paper no. 750414. Bellingham, WA: SPIE.

Fuchs

Hed

Baker

. Fine diamond turning of KDP crystals. Appl Optics1986; 25: 1733–1735.

Montesanti

Thompson

. A procedure for diamond turning KDP crystals (No. UCRL-ID-121651). Washington, DC: Coal and Electricity Division, USDOE Economic Regulatory Administration, 1995.

Namba

Katagiri

Nakatsuka

. Single point diamond turning of KDP inorganic nonlinear crystal for laser fusion. Jpn Soc Precis Eng1998; 64: 1487–1491.

Wang

Gao

. Near-surface mechanical properties of KDP crystals in single point diamond turning. Mater Sci Tech Ser2007; 15: 133–136.

Zhao

Wang

. Investigation of anisotropic mechanisms in ultra-precision diamond machining of KDP crystal. J Mater Process Tech2009; 209: 4169–4177.

Tie

Dai

Guan

. Research on subsurface defects of potassium dihydrogen phosphate crystals fabricated by single point diamond turning technique. Opt Eng2013; 52: 033401-1–033401-6.

10.

Jacobs

. Manipulating mechanics and chemistry in precision optics finishing. Sci Technol Adv Mat2007; 8: 153–157.

11.

Komaraiah

Reddy

. Study on the influence of workpiece properties in ultrasonic machining. Int J Mach Tool Manu1993; 33: 495–505.

12.

Pei

Ferreira

. Modeling of ductile-mode material removal in rotary ultrasonic machining. Int J Mach Tool Manu1998; 38: 1399–1418.

13.

Bertsche

Ehmann

Malukhin

. An analytical model of rotary ultrasonic milling. Int J Adv Manuf Tech2013; 65: 1705–1720.

14.

Kumar

Khamba

Mohapatra

. Investigating and modeling tool-wear rate in the ultrasonic machining of titanium. Int J Adv Manuf Tech2009; 41: 1107–1117.

15.

Bertsche

Ehmann

Malukhin

. Ultrasonic slot machining of a silicon carbide matrix composite. Int J Adv Manuf Tech2013; 66: 1119–1134.

16.

Jiao

Liu

Pei

. Study on edge chipping in rotary ultrasonic machining of ceramics: an integration of designed experiments and finite element method analysis. J Manuf Sci E: T ASME2005; 127: 752–758.

17.

Huang

Wang

. Improvement effects of vibration on cutting force in rotary ultrasonic machining of BK7 glass. J Mater Process Tech2013; 213: 1548–1557.

18.

Wang

Cong

Pei

. Rotary ultrasonic machining of potassium dihydrogen phosphate (KDP) crystal: an experimental investigation on surface roughness. J Manuf Process2009; 11: 66–73.

19.

Zhang

Feng

. Experimental investigation on surface roughness of KDP crystal processed with rotary ultrasonic face milling. Key Eng Mat2012; 499: 223–228.

20.

Wang

Dai

. Research on measurement and prediction methods of subsurface damage depth of optical materials in machining process. Aviat Precis Manuf Technol2007; 43: 1–5.

21.

Randi

Lambropoulos

Jacobs

. Subsurface damage in some single crystalline optical materials. Appl Optics2005; 44: 2241–2249.

22.

Zhen

Feng

. Development trends of rotary ultrasonic machining technology. J Tsinghua Univ2009; 49: 1799–1804.

23.

Zhang

Feng

. Mathematical model for cutting force in rotary ultrasonic face milling of brittle materials. Int J Adv Manuf Tech2013; 69: 161–170.