Research in minimum undeformed chip thickness and size effect in micro end-milling of potassium dihydrogen phosphate crystal

Accepted Manuscript

Research in minimum undeformed chip thickness and size effect in micro end-milling of potassium dihydrogen phosphate crystal Ni. Chen , Mingjun Chen , Chunya Wu , Xudong Pei , Jun Qian , Dominiek Reynaerts PII: DOI: Reference:

S0020-7403(17)32092-1 10.1016/j.ijmecsci.2017.10.025 MS 3991

To appear in:

International Journal of Mechanical Sciences

Received date: Revised date: Accepted date:

7 August 2017 21 September 2017 16 October 2017

Please cite this article as: Ni. Chen , Mingjun Chen , Chunya Wu , Xudong Pei , Jun Qian , Dominiek Reynaerts , Research in minimum undeformed chip thickness and size effect in micro end-milling of potassium dihydrogen phosphate crystal, International Journal of Mechanical Sciences (2017), doi: 10.1016/j.ijmecsci.2017.10.025

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Highlights 

A model of chip formation is developed to estimate the minimum undeformed chip thickness in micro-milling of KDP crystal.

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The difference of normalized minimum undeformed chip thickness between KDP crystal and

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metallic materials is analyzed. Severe size effect appears when feed per tooth is less than minimum undeformed chip thickness.

The feed per tooth which is slightly larger than minimum undeformed chip thickness and

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smaller than the cutting edge radius is recommended in micro-milling of KDP crystal or other

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soft-brittle crystal.

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Research in minimum undeformed chip thickness and size effect in micro end-milling of potassium dihydrogen

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phosphate crystal

Ni Chen1,2,3, Mingjun Chen*1,2, Chunya Wu*1,2, Xudong Pei1,2, Jun Qian3, Dominiek Reynaerts3 State Key Laboratory of Robotics and System, Harbin Institute of Technology, Harbin 150001, China 2

Center for Precision Engineering, Harbin Institute of Technology, Harbin 150001, China Department of Mechanical Engineering, KU Leuven, Leuven 3000, Belgium

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*Corresponding author:

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Dr. MJ Chen, P.O. Box 413, Harbin 150001, P.R. China.

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Tel.: +86(0)451-86403252. Fax: +86(0)451-86403252. E-mail: [email protected].

Dr. CY Wu, P.O. Box 413, Harbin 150001, P.R. China. E-mail: [email protected].

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Abstract: Micro-milling is a promising approach to repair the micro-defects on the surface of potassium dihydrogen phosphate (KDP) crystal. However, KDP crystal is difficult to machine due to its properties of soft-brittle and easy deliquescence. This study investigates the minimum undeformed chip thickness hm and the size effect in micro end-milling of KDP crystal by

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comprehensively analyzing cutting force, specific cutting force and machined surface quality. A model of the chip formation, which is capable of connecting the minimum undeformed chip thickness, the undeformed chip thickness and the periodicity of cutting force together, is developed

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to predict the value of minimum undeformed chip thickness. The normalized minimum undeformed chip thickness λe in terms of the ratio of the minimum undeformed chip thickness to the cutting edge radius re, is estimated to be 0.43 ≤ λe ≤ 0.48. The significantly non-proportional increase of specific

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cutting force indicates the existence of size effects when the ratio of feed per tooth to cutting edge radius ft/re is less than 0.7. The machined surface quality also reflects severe size effect by the

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phenomenon that the micro cracks and brittle pits appear on the groove base, and the value of surface

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roughness Ra is large when the ratio ft/re is less than 0.5. Furthermore, the surface quality deteriorates and the brittle cutting appears when the ratio ft/re is much larger than 1 which seems similar to

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macro-milling. The perfect machined surface with almost no ploughing effect and brittle cutting is

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achieved at a ratio ft/re of 0.7. Therefore, a feed per tooth, slightly larger than the minimum undeformed chip thickness but smaller than cutting edge radius, is recommended for micro-milling of KDP crystal or soft-brittle crystal.

Keywords: KDP crystal; Micro-milling; Minimum undeformed chip thickness; Size effect; Specific cutting force

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1. Introduction Micro-milling has been applied in industry for the fabrication of micro features. Due to its processing flexibility, it has been used in producing component of different sizes and shapes in a wide variety of metallic and non-metallic materials [1-3]. In micro-milling, the cutting chip will not

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be formed if the undeformed chip thickness h is less than a critical value, which is normally defined as the minimum undeformed chip thickness hm [4]. Namely, when the undeformed chip thickness is less than the minimum undeformed chip thickness, the chip would not be generated due to the elastic

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deformation (ploughing); on the contrary, when the undeformed chip thickness is larger than the minimum undeformed chip thickness, the chips will be generated, and cutting process seems similar to the macro-cutting [5, 6], as shown in Fig. 1. This chip formation mechanism causes the so-called

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size effect, which is characterized by the significant increase of surface roughness and specific

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cutting force with the decrease of the undeformed chip thickness on the premise of h
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Fig. 1 Chip formation relative to the minimum undeformed chip thickness in micro-machining, a. h < hm, b. h ≥ hm.

Therefore, many experts had devoted themselves to study the minimum undeformed chip

thickness and size effect in micro-machining utilizing both theoretical and experimental approaches. In the perspective of research with an analytical and numerical approach, Malekian et al. [7] found that the average ratio of minimum undeformed chip thickness hm to cutting edge radius re, termed as 4

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the normalized minimum undeformed chip thickness λe, was approximately 0.23 in micro-milling of Al6061 by establishing analytical models based on the minimum energy principle and the infinite shear strain method. Ramos et al. [8] developed a model on the basis of the experimental results to determine the minimum undeformed chip thickness of AISI 1045, and figured out that the minimum

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undeformed chip thickness decreased significantly with the increasing of cutting velocity but increased moderately with the increasing of cutting edge radii. Son et al. [9] proposed an ultra-precision cutting model to estimate the minimum undeformed chip thickness for metals, and

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found that the tool edge radius and friction coefficient were the principal factors determining the minimum undeformed chip thickness. Yuan et al. [10] established the relationship between the cutting edge radius and minimum undeformed chip thickness, and the normalized minimum

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undeformed chip thickness was finally determined to be about 0.2-0.4. Vogler et al. [11] predicted that the normalized minimum undeformed chip thickness was within the range of 0.14-0.25 and

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0.29-0.43 for pearlite and ferrite steels respectively by using the finite element model developed by

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Chuzhoy et al. [12].

The experimental approach to investigate the minimum undeformed chip thickness and the size

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effect has also been employed by many researchers. Oliveira et al. [5] studied the size effect and

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minimum undeformed chip thickness by associating two quantitative and sensitive variables (i.e., specific cutting force and surface roughness) with the qualitative analysis of the chip formation and machined surface topography. The normalized minimum undeformed chip thickness was found to vary practically from 0.25 to 0.33, regardless of workpiece material, tool geometry, specific mechanical machining process or the adopted measuring/estimating techniques. Lai et al. [13] clearly showed that the normalized minimum undeformed chip thickness was around 0.25 for OFHC copper 5

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by finite element analysis. Aramcharoen et al. [1] conducted an investigation of the size effect in micro-milling of H13 hardened tool steel, and concluded that the ratio of undeformed chip thickness to the cutting edge radius was a critical control parameter to influence the size effect in micro-milling, especially when the ratio was less than 1. Kim et al. [6] presented a method for estimating the

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minimum undeformed chip thickness of various combinations of tools and workpiece materials based on easily attainable cutting force data, finding that the normalized minimum undeformed chip thickness was between 0.16 and 0.25. Sooraj et al. [14] revealed that the size effect was indicated by

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the drastic variation of specific cutting pressure and cutting forces at lower feed per tooth in micro-milling, and the minimum undeformed chip thickness during micro end-milling of brass was evaluated as 0.97 μm.

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As noted above, the minimum undeformed chip thickness and size effect have been studied mainly by three methods, namely, 1) analysis based on theoretical models, 2) finite element models, and 3)

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experimentally analyzing the specific cutting force and the quality of the machined surface. The

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workpiece materials used in above researches can almost be classified as metals, and only extremely few papers investigated the size effect and minimum undeformed chip thickness in machining of

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brittle materials. Arif et al. [15] studied the critical conditions for the modes of material removal in

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the milling process of hard-brittle materials, and found that size effect appeared which performed as the increasing of surface roughness Ra value when radial depth of cut and feed rate are small. Amin et al. [16] reported that there was a ploughing phase at all the applied combinations of speed and feed rates in end-milling when they carried out an experimental approach to determine the critical depth of cut in brittle-to-ductile transition during end-milling of soda-lime glass. Although Arif et al. and Amin et al. have done some researches on the size effect in machining of brittle materials, the above 6

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size effect studies in end-milling of brittle materials were carried out just by observing the abnormal changes of surface roughness Ra value when cutting parameters are small, and the boundaries of cutting parameters or minimum chip thickness to avoid the size effect have not been given out. In this research, potassium dihydrogen phosphate (KH2PO4, KDP) crystal, a soft-brittle optical

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crystal, is the target of study. KDP crystal has been widely used as the frequency multiplication elements in inertial confinement fusion (ICF) project. Till now, it is still the most suitable nonlinear material due to its good quality of growth and large electro-optic coefficient [17-19]. Furthermore,

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KDP crystal is widely recognized as a typical difficult-to-machine material because of its unique physical property. In order to achieve the desired machined surface quality in micro-milling of KDP crystal, the minimum undeformed chip thickness and size effect will be studied in detail, helping to

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understand the difference in micro-milling mechanism of metal and soft-brittle optical crystal. The

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optimal ratio of the feed per tooth to the cutting edge radius is established in this investigation.

2. Calculation of the undeformed chip thickness and the specific

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cutting force in micro end-milling

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During the cutting process of micro end-milling, if the feed per tooth is considered to be much smaller than the tool radius, the cutting trajectory could be simplified as a circle. The cutting process

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in micro end-milling illustrated in Fig. 2a is similar to the diamond turning [20]. ft, R, and h are the feed per tooth, the radius of tool and the undeformed chip thickness, respectively. θ is the position angle at the cutting point A, γ is the complement angle of θ, and β is the intersection angle between two lines which connected the cutting points (A, B) and tool centers (O1,O2) on (j-1)th and jth tool pass.

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Fig. 2 The schematic illustration of cutting process in micro-milling, a. 2D model, b. 3D model by CBN tipped micro end mill. (w in the figures represents the rotation of micro end mill)

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As shown in Fig.2a, in the triangle O1O2A, the following Eq. 1 and Eq. 2 can be derived based on the triangular sine theorem:

sin[𝜋 − (𝛽 + 𝛾)]⁄R = sin(β+γ)⁄R =sin β⁄ft

(1)

sin(π/2-θ)⁄(R-h) =sin β⁄ft

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(2)

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After some transformation, the equations Eq. 3 and Eq. 4 can be obtained from Eq. 1 and Eq. 2: tanβ= (ft /R)cosθ⁄[1-(ft /R)sinθ]

(3)

h=R- ft sin(π/2-θ)⁄sinβ = R- ft cos𝜃⁄sinβ

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(4)

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From Eq. 3 and Eq. 4, the undeformed chip thickness can be expressed by ft, R and θ: (5)

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1h=R- ft cosθ√1+ft 2 /R2 -2(ft /R)sinθ⁄Xcosθ =R-R√1+ft 2 /R2 -2(ft /R)sin θ

If the tool diameter R is considered to be much larger than the feed per tooth ft, (i.e., ft/R«1), the

undeformed chip thickness can be expressed as: 2

h=R-R(1+(1/2)(-2sinθft /R)+ (1/2)(1/2-1)(-2sinθft /R) ⁄2! + …)

(6)

In micro-milling of KDP crystal, the actual state corresponding to the assumption of ft/R «1 exists, therefore the undeformed chip thickness h can be expressed as: 8

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h=ft sinθ

(7)

Normally, the specific cutting force kc is defined by the ratio of the mean cutting force to the mean cutting section. Considering following two aspects, the peak-valley (P-V) cutting force (FP-V), which refers to the amplitude difference between the maximum and minimum cutting force within a tool

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pass, is used in the study of specific cutting force [13,21]. Firstly, the tangential and radial cutting forces are changed with position angle on every tool pass, and some experts used root mean square of cutting force to study the specific cutting force, however, the whole cutting force may have a drift

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in the measurement, so it is not accurate enough. Secondly, P-V value of cutting forces would represent the change of cutting force in the cutting process, and the severe change of cutting force would cause the shake of micro end mill. If the shake of micro end mill coupling with the machine

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tool vibration, the machined surface quality will be extremely deteriorated. The mean cutting section

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in this research is defined as the ratio between the volume cut in a half circle V (intersection volume between the (j-1)th and the jth tool pass) and semicircle perimeter l. So, the specific cutting force can

kc= FP-V⁄( V⁄l )

(8)

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be expressed as:

The volume V equals to the volume of the area A, since area A is removed by the jth tool pass, as

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shown in Fig. 2b. The cubic boron nitride (CBN) tipped micro end mill and the cemented carbide micro end mill have been used in the experiments. Because the CBN tipped micro end mill presents a certain tool tip with the radius of r, the area A can be divided into two parts, namely A1 and A2 (volumes corresponding to VCBN, V1 and V2, respectively): VCBN=V1+V2

(9)

Area A1 is machined by tool radius R′, which equals to the difference between total tool radius R 9

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and radius of tool tip r. Area A2 is machined by radius of tool tip r. The tool center at the (j-1)th tool pass on the machined surface is defined as coordinate origin O1, and the coordinate system was shown in Fig. 2b. The curved surface generated by tool tip on the area A2 can be expressed as: 2

The volume of area A1, A2 can be expressed as: V1=2R'ft ap R'+√2rap -ap 2

∫R'

dy ∫

ap

r-√r2 -(y-R')

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dz=ft [(ap -r)√2rap -ap 2 +r2 arcsin √2rap -ap 2 ⁄r]

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f V2 =2 ∫0 t dx

(10)

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(𝑦 − 𝑅′)2 + (z-r) = r2 { 0 ≤ z ≤ ap 0 ≤ x ≤ ft

(11) (12)

The semicircle perimeter l for CBN tool can be expressed as:

𝑙CBN = π R'+ π √2rap -ap 2

(13)

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Since the tip of carbide micro end mill can be regarded as absolutely sharp, the volume cut in a

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half circle Vcarbide and semicircle perimeter lcarbide can be represented by the equation similar to the

𝑉carbide =2Rft ap

(14)

𝑙carbide = π R

(15)

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volume of area A1 for CBN tipped micro ball end mill:

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3. Experimental procedure 3.1 Experimental set-up The micro end-milling experiments have been carried out on the CNC machining center (KERN MMP2522, KERN Microtechnik GmbH, Germany) shown in Fig. 3. The maximum spindle speed is 40,000 r/min and the resolution of position and repetition is ±1 m. The workpiece is made of KDP crystal with a dimension of 95 mm × 27 mm × 14 mm. Because the CBN tipped micro end mill and 10

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carbide micro end mill have been commonly used in micro end-milling of KDP crystal, both these two two-tooth micro end mills of 1 mm diameter are used in the experiments. The tool tip radius r of the CBN tipped micro end mill is 50 m. The cutting edge radii of both micro end mills are measured by atomic force microscope and fitted by Matlab® (R2011a)™. CBN tipped tool has a

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cutting edge radius of around 2.3 m and carbide tool has a cutting edge radius of around 3 m. The

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micro end mills used in the experiments and the fitting of cutting edge radius are shown in Fig. 4.

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Fig. 3 Experimental setup for CNC machining center, a. hardware platform, b. schematic illustration of working

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process.

Fig. 4 The micro end mills used in the experiments and cutting edge radius measurement, a. CBN tipped micro end mill, b. carbide micro end mill, c. the fitting of cutting edge radius for CBN tipped micro end mill.

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3.2 Design of cutting experiments In order to study the size effect and the minimum undeformed chip thickness which are closely related to the feed per tooth ft and the cutting edge radius re in micro-milling of KDP crystal, the ratio of feed per tooth to cutting edge radius (ft/re) is set from 0.025 to 3. Since the cutting edge radii of

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two micro end mills used in the experiments are different, the feed per tooth is set 0.06~6.9 m/z for CBN tool, and 0.08~9 m/z for carbide tool. To study the effect of spindle speed on the size effect in micro-milling of KDP crystal, two different spindle speeds are also set for the two tools. The uniform

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depth of cut of 5 m is set for both tools by considering depth of cut is unrelated with undeformed chip thickness. The micro grooves have the uniform length of 3 mm, but only one micro groove is milled in each experiment, therefore one micro groove corresponds to only one series of cutting

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parameters. Detailed cutting parameters are listed in Table 1. The cutting force signals are acquired using a charge amplifier (5233A) and piezo electric dynamometer (9256C2 from Kistler ®) in the

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experiments. Matlab® (R2011a)™ and Labview V.7.1™ from National Instruments are used for

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post-processing and signal acquisition with a 10 kHz sampling frequency.

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Table 1 The cutting edge radii of tools and cutting parameters in the experiments Spindle speed

Tool

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n (10 r/min)

CBN

Depth of cut

Cutting edge

Cutting length

radius re (m)

l (mm)

2.3

3

3

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Feed per tooth ft (m/z) ap (m) 0.06, 0.12, 0.23, 0.46,

2, 4

5 1.15, 1.61, 2.3, 4.6, 6.9 0.08, 0.15, 0.3, 0.6, 1.5,

Carbide

2, 4

5 2.1, 3, 6, 9

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3.3 Evaluation for micro structure of the machined surfaces The surface morphology of workpiece is examined in a scanning electron microscope (SEM, from FEI). The Form Talysurf® PGI1240 was used to evaluate the roughness value Ra of the micro grooves. Since eight different areas are examined for each micro groove, the mean and standard deviation of

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these measured values are taken as the surface roughness Ra of the machined groove and the measurement error, respectively.

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4. Results and discussion 4.1 Analysis of minimum undeformed chip thickness

The model of a stable cutting process is established with the dynamic response of the tool and

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workpiece not being considered. The reasons are as follow: firstly, the minimum undeformed chip thickness can be more easily explained and studied under the assumption of a stable cutting process;

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secondly, the KERN MMP2522 used in the experiments is a micro-machining center with stable

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cutting performance; thirdly, the minimum undeformed chip thickness in micro-milling of metals has

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already investigated successfully by many experts under the assumption of stable cutting [5, 6, 11]. Since the workpiece material—KDP crystal is soft-brittle and the maximal strain on the tool is less

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than 10 to minus 5 of magnitude when micro-milling of KDP crystal [19], the deflection of the micro end mill can be neglected in the experiments. The effect of the minimum undeformed chip thickness in micro end-milling can be explained by

the graphical description given in Fig. 5. The two-tooth micro end mill used in Fig. 5 is assumed as perfect symmetry, and the cutting process is assumed as stable. It can be seen that during slot milling the instantaneous undeformed chip thickness h varies from zero at the initial engagement, to a 13

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maximum value (almost equals to the feed per tooth) at the approximate center of the channel, and back to zero at the exit of the cutting path. Fig. 5 gives an typical example that the maximum instantaneous undeformed chip thickness during one tool pass is less than the minimum undeformed chip thickness hm, therefore there is no chip formed on the jth tool pass, and the chip cannot be

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generated until the instantaneous undeformed chip thickness exceeds the minimum undeformed chip thickness on the (j+1)th tool pass. If the undeformed chip thickness keeps below the minimum undeformed chip thickness, the material of workpiece can recover elastically after the tool passes.

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However, once the instantaneous undeformed chip thickness exceeds the minimum undeformed chip thickness, the chip starts to form (Fig. 5b), but the chip generation will stop when the instantaneous undeformed chip thickness becomes less than the minimum undeformed chip thickness (Fig. 5c).

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During the (j+2)th tool pass, the cutting edge firstly encounters the residual material from the previous tool pass, and the initial point of chip creation recedes to an earlier angular position,

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because the current tool pass experiences a larger undeformed chip thickness for the region where the

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chip was not generated in the previous tool pass (Fig. 5d). Then the cutting edge passes the residual material (Fig. 5e), but the chip cannot be formed due to the previous assumption that the

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instantaneous undeformed chip thickness is less than the minimum undeformed chip thickness. After

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that, the chip continues to form when the (j+2)th tool pass encounters the residual material from the previous tool pass again (Fig. 5f). Therefore, if the instantaneous undeformed chip thickness is larger than the minimum undeformed chip thickness, the chip will be produced on every pass of the tooth as expected in macro-scale. On the contrary, the chip cannot be generated on every tool pass if the instantaneous undeformed chip thickness remains less than the minimum undeformed chip thickness, but the undeformed chip thickness would increase with the progression of tool passes until it is 14

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beyond the minimum undeformed chip thickness, resulting in the generation of chips [6, 11, 13, 22].

Fig. 5 The process of chip formation in micro end-milling by considering the effect of the minimum undeformed

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chip thickness, a. b. c. (j+1)th tool pass, d. e. f. (j+2)th tool pass. (w in the figures represents the rotation of micro

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end mill)

On the basis of the above analysis, the minimum undeformed chip thickness hm and instantaneous

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undeformed chip thickness h at a certain cutting angle θ can be inferred to possess periodicity, which

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is denoted by p in Eq. 16, and the minimum undeformed chip thickness could be predicted as below: p = roundoff ( hm ⁄h )

(16)

h*(p-1) ≤ ℎ𝑚 ≤ h*p

(17)

ft sinθ (p-1)≤ ℎ𝑚 ≤ ft sinθ p

(18)

Where the function roundoff(x) gives the smallest integer bigger than or equal to x. The most important thing is to determine the periodicity p. In micro-milling, both Malekian et al. [23] and Zhang et al. [24] have proved that the radial and tangential forces acting on a discretized disk element, dFt and dFr, can be expressed as: 15

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(Ktc h+Kte ) dz dFt = { (K A +K ) dz

when h≥hm (Chip formation) when h
(19)

(Krc h+Kre ) dz dFr = {(K A +K ) dz

when h≥hm (Chip formation) when h
(20)

tp p

rp p

te

re

Where Ktc and Krc are the tangential and radial cutting coefficients in chip formation process,

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respectively; Ktp and Krp are the tangential and radial cutting coefficients in ploughing process, respectively; Kte and Kre are the tangential and radial edge coefficients representing the tool–workpiece friction, and both of them remain constant in ploughing and chip formation process;

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Ap is the ploughed area; dz is the height of the differential flute element.

As shown above, when all other coefficients keep constant, the cutting force in Eq. 19 and Eq. 20 can be considered as only being related with the ploughed area Ap or undeformed chip thickness h.

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Since the undeformed chip thickness (in ploughed area) at a certain cutting angle of θ changes periodically due to the existence of minimum undeformed chip thickness, the cutting force at the

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same cutting angle on different tool passes should also exhibit an obvious periodicity.

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Taking the micro-milling of KDP crystal with a CBN tipped micro end mill as an example, a spindle speed of 20, 000 r/min, depth of cut of 5 m, and an actual feed per tooth (the word of actual

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will be explained in this section later) of 0.12, 0.23, 0.46 and 2.3 m/z are selected. Considering the

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following two reasons, one is that the feed and cross feed force obtained from the dynamometer are radial and tangential cutting force at the position angle of 90° without needing to be transformed; the other is that the cutting force will reach the maximum value at the position angle of 90° because the undeformed chip thickness is the biggest at this point, and the cross feed force accounts for the largest part of the total cutting force [19]. So, the cross feed force at the position angle of 90° is chosen to study the minimum undeformed chip thickness. 16

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Fig. 6 Cross feed force analysis at the actual feed per tooth of 0.23 m/z, a. evolution of cross feed force with global position angles, b. evolution of cross feed force at position angle of 90° with number of tool passes, c. FFT

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analysis of cross feed force at position angle of 90°.

In the case of actual feed per tooth being 0.23 m/z (Fig. 6), the cross feed force within an entire cutting circle of 360° only shows one maximum point, therefore it can be inferred that mainly one

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tooth is involved in. Above phenomenon has been also observed by Malekian et al. [23], and they explained that it is influenced by the run-out and tool imperfections when the feed per tooth is at a

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relatively low value by using experimental and theoretical approach. Hence, the feed per tooth for

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the small feed rate is actually twice as much as nominal feed rate, and it is defined as actual feed per

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tooth in the following analysis.

As shown in Fig. 6b, the cutting force at the position angle of 90° on every tool pass has clear

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regularity for the actual feed per tooth of 0.23 m/z, and the periodicity is estimated to be in the range of 4~5, whereas the same force at the same position angle for the feed per tooth of 0.12 m/z changes randomly with the increase of number of tool passes (Fig. 7a). This phenomenon is caused by significant size effect in the cutting process, and it will be verified by the following analysis on the specific cutting force (Section 4.2) and machined groove quality (Section 4.3). When the feed per tooth reaches 0.46 m/z (Fig. 7b), the cutting force at the position angle of 90° on every tool pass 17

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varies regularly, and the periodicity is estimated between 2 and 3. If the feed per tooth is increased to 2.3 m/z (Fig. 7c), equals to the cutting edge radius, the cutting force at the position angle of 90° on

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every tool pass varies quite slightly, indicating a stable cutting process.

Fig. 7 Evolution of cross feed force at position angle of 90° with the number of tool passes at different actual feed per tooth, a. 0.12 m/z, b. 0.46 m/z, c. 2.3 m/z.

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The Fast Fourier Transform (FFT) algorithm is adopted to study the periodicity of extracted

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cutting force at the position angle of 90° (Fig. 6c and Fig. 8). The result showed in Fig. 6c indicate a characteristic frequency of 0.22 at case of feed per tooth of 0.23 m/z, and Fig. 8b suggests

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characteristic frequencies of 0.32 and 0.42 at the feed per tooth of 0.46 m/z; however, no

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characteristic frequency is presented in the curves of FFT results corresponding to the feed per tooth of 0.12 m/z and 2.3 m/z (Fig. 8a and c). The periodicity of the cutting force at the position angle

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of 90° can be calculated by Eq. 21 (p=4.55 for feed per tooth of 0.23 m/z; p=3.13, 2.38 for feed per tooth of 0.46 m/z), but the periodicity for feed per tooth of 2.3 m/z can be regarded as 1 because of its consistency. After calculating the values of p, the minimum undeformed chip thickness hm, which is estimated by the intersection among different following inequations from Eq. 18 (0.82 ≤ hm ≤ 1.05 m, (0.63, 0.98) ≤ hm ≤ (1.09, 1.44) m, and 0 ≤ hm ≤ 2.3 m under the feed per tooth of 0.23 m/z, 0.46 m/z, and 2.3 m/z, respectively), is 0.98 ≤ hm ≤ 1.05 m, and the normalized minimum 18

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undeformed chip thickness of 0.43 ≤ λe ≤ 0.46 can also be consequently obtained. The detailed information for the estimation of minimum undeformed chip thickness is listed in Table 2. p = 1⁄f0

(21)

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Where f0 is the characteristic frequency of cutting force at the position angle of θ.

Fig. 8 FFT analysis of the cross feed force at position angle of 90° at different actual feed per tooth, a. 0.12 m/z, b.

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0.46 m/z, c. 2.3 m/z.

For the carbide tool with the cutting edge radius re of 3 m, the same analysis is also carried out,

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and the normalized minimum undeformed chip thickness is 0.45 ≤ λe ≤ 0.48, as shown in Table 2. So

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it can be seen that the normalized minimum undeformed chip thickness increases with the increase of

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the cutting edge radius, which shows same changing trend with the carbon steels [4]. This phenomenon can be explained by the increase of ploughing effect, which causes a higher cutting

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force, thereby resulting in higher energy dissipation and higher cutting temperatures with the tool edge radius increased [19]. As a result of the increased cutting temperature, the material becomes more ductile, finally leading to a larger normalized minimum undeformed chip thickness. Many experts have researched the normalized minimum undeformed chip thickness λe of metallic materials, and Oliveira et al. [5] summarized that it is between 0.25 and 0.33, which is much smaller than the normalized minimum undeformed chip thickness λe of KDP crystal obtained in this paper 19

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(0.43 ≤ λe ≤ 0.48). The results can be attributed to two aspects. On one hand, since the effective rake angle γe could be calculated by Eq. 22 [25] when the undeformed chip thickness is smaller than the cutting edge radius, the effective rake angle is between -47.9 and -49.7 for these two micro mills in this work. As stated in reference of Chen et al. [19], the optimal rake angle of the tool is within the

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range of -45~-50 for micro-milling of KDP crystal because the P–V values of cross feed force and maximal tensile stress on the crystal are relatively small under this special condition. On the other hand, KDP crystal is a typical soft-brittle material with the melting point of 252.6C, which is

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significantly lower than the metallic materials. Therefore, the thermal softening effect will make the KDP crystal more ductile than the metallic materials at the cutting temperatures, when the undeformed chip thickness is relatively small. Further analysis on the minimum undeformed chip

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thickness will be carried out by analyzing the specific cutting force, the machined groove morphology and roughness value Ra in Section 4.2 and 4.3. γe =-sin-1 [(2re -h)⁄2re ]

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(22)

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Table 2 Estimation of the normalized minimum undeformed chip thickness λe Cutting edge radius re of 2.3 m

Cutting edge radius re of 3 m

0.1

0.2

1

0.1

0.2

1

Characteristic frequencies

0.22

0.32, 0.42

--

0.21

0.31

--

4.55

2.38, 3.13

1

4.76

3.23

1

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ft /re

Period at position angle of 90° Estimated hm (m)

0.98~1.04

1.34~1.43

λe(hm /re)

0.43 ≤ λe ≤0.46

0.45 ≤ λe ≤0.48

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4.2 Analysis of size effect based on specific cutting force As shown in Eq. 7, the maximum undeformed chip thickness equals to the feed per tooth at the position angle of 90°, and the specific cutting force is a critical factor to research the size effect, so the effect of the ratio of feed per tooth to cutting edge radius ft/re on the specific cutting force is

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studied in this section. It can be seen clearly that the specific cutting force increases significantly non-linear with the decrease of the ratio ft/re (Fig. 9), when the feed per tooth is less than the cutting edge radius (i.e., the ratio ft/re is less than 1). Once the feed per tooth is larger than the cutting edge

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radius (i.e., the ratio ft/re is larger than 1), the specific cutting force increases almost linearly with the decrease of the ratio ft/re. This result seems similar behavior to the evolution of the specific cutting force for micro-milling of metals, which was presented by Aramcharoen et al. [1]. The above trend is

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mainly derived from the size effect in micro-milling. The process of chip formation is interrupted by incomplete material removal when the cutting edge radius is larger than the undeformed chip

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decrease of the ratio ft/re.

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thickness, and hence the ploughing effect and elastic deformation become more obvious with the

Compared with the carbide tool (re=3 m), the specific cutting force in micro-milling using a CBN

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tool (re=2.3 m) is smaller, but the difference decreases with the increase of the ratio ft/re, especially

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when it is larger than 1. This is because ploughing effect fades away with the increase of ratio ft/re. Compared with the states corresponding to the spindle speed of 20, 000 and 40, 000 r/min, the

specific cutting force with the spindle speed of 40, 000 r/min is smaller, and the difference increases with the augment of the ratio ft/re. Especially when the ratio ft/re is larger than 1, the specific cutting force with the spindle speed of 40, 000 r/min becomes only half of the value of the specific cutting force at the spindle speed of 20, 000 r/min. This phenomenon can be explained by the law presented 21

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in Eq. 23 that the cutting speed v increases in proportion to the spindle speed. As documented in the published works [4, 22], the increased cutting speed influences the material behavior in two ways. Firstly, the cutting temperature increases as the cutting speed increased, which enhances the thermal softening effect and increases the ductility. Secondly, as the cutting speed increased, the effective

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strain and strain rate increase, which enhances the strain-hardening effect and reduce the ductility. Therefore, it can be deduced that the thermal softening effect dominates the micro-milling process of KDP crystal. When the ratio ft/re is less than 1, the difference in specific cutting force between two

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different cutting velocities is extremely minor, thus the ploughing effect becomes dominant rather than thermal softening or strain-hardening effect. However, the ploughing effect fades away with the increase of ratio ft/re (especially when it exceeds 1), and the thermal softening effect becomes notable, which results in an increasing difference in specific cutting force between two spindle speeds.

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v=nπD/1000

(23)

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Where v is cutting speed, n spindle speed and D the diameter of micro end mill.

Fig. 9 Specific cutting force in cross feed direction with the change of the ratio of feed rate to cutting edge radius.

4.3 Analysis of size effect based on machined surface quality 22

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The morphologies of the machined grooves under different feed per tooth are displayed on Figs. 10 and 11, and the feed per tooth (ratios ft/re) of 0.12 m/z (0.05), 0.46 m/z (0.2), 1.15 m/z (0.5), 1.61 m/z (0.7), 2.3 m/z (1) and 6.9 m/z (3) are selected, respectively. From Fig. 10a, it can be found that a few micro cracks and brittle pits assuming white areas are

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presented on the surface of groove. To get a better view, the interesting part is magnified in Fig. 11a, where micro cracks formed by excessive ploughing effect can be identified clearly, and many particles left on the machined surface are also obviously visible. The particles may be extracted from

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the workpiece by the micro mill in the primary shear zone, but they cannot be classified as chip-like due to the crushing and extruding processes caused by the lowest feed per tooth of 0.12 μm/z. Compared with the surface morphology under the feed rate of 0.12 μm/z, the machined surface

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quality under the feed rate of 0.46 μm/z (Fig. 10b) seems much better with less micro cracks and white areas. Since there are only smaller micro cracks and scarcely visible particles on the surface

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(Fig. 11b), it can be concluded that softer ploughing occurs in the cutting process, indicating less

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difficulty in removing materials. Meanwhile, only very few particles are left on the machined surface, demonstrating that the particles withdrawn by micro-milling may be more chip-like.

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As shown in Fig. 10c and 11c, for feed rate of 1.15 μm/z (around the minimum undeformed chip

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thickness), only slight ploughing effects still remain, which are manifested as the small slits and heaves on the machined surface but this pattern of surface defects distribution disappears completely when feed rate becomes greater than 1.61 μm/z (Fig. 10d and 11d). However, the slits and heaves start emerging again on the surface under the feed rate of 2.3 μm/z (Fig. 10e and 11e), making the surface quality even a little worse than the state for the feed rate of 1.61 μm/z. When the feed rate reaches 6.9 μm/z (Fig. 10f and 11f), many cracks and white areas are presented on the surface, 23

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indicating that the cutting is almost in the brittle mode as in macro-milling.

Fig. 10 SEM images for morphology of machined grooves under different feed rates, (a) ft=0.12 m/z, ft/re =0.05, (b) ft=0.46 m/z, ft/re=0.2, (c) ft=1.15 m/z, ft/re =0.5, (d) ft=1.61 m/z, ft/re =0.7, (e) ft=2.3 m/z, ft/re =1, (f) ft=6.9 m/z, ft/re =3.

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Fig. 11 Partly enlargement SEM images for morphology of machined grooves under different feed rates, (a) ft=0.12 m/z, ft/re =0.05, (b) ft=0.46 m/z, ft/re=0.2, (c) ft=1.15 m/z, ft/re =0.5, (d) ft=1.61 m/z, ft/re =0.7, (e) ft=2.3 m/z, ft/re =1, (f) ft=6.9 m/z, ft/re =3.

The effect of the ratio of feed rate to the cutting edge radius ft/re on the surface roughness Ra is depicted in Fig. 12. The curves of surface roughness Ra decrease gradually with the augment of the 25

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ratio ft/re (ft/re<1), reaching the minimum value region at ft/re=0.7~1, then the curves turn to increase with the augment of the ratio ft/re. The decrease of surface roughness Ra with the augment of ratio ft/re can be attributed to the abating ploughing effect for higher feed per tooth, which will result in less elastic recovery [15, 26] and less brittle cutting. Once the ratio ft/re becomes larger than 1, the

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cutting process seems more likely macro milling, and the brittle cutting will appear with the surface roughness Ra increasing significantly [27]. These phenomena supports the observation from the machining process of sintered tungsten carbide with the ratio ft/re changed from around 0.1 to 4 [15].

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Meanwhile, as shown in Fig.9, the specific cutting force reduces sharply from 624 GPa to 155 GPa when the ratio ft/re increases from 0.05 to 0.7, whereas the decrease of specific cutting force continues moderately from 155 GPa to 61 GPa when the ratio ft/re changes from 0.7 to 3. Therefore,

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it can be inferred from the analysis of the specific cutting force and the machined surface quality that the severe size effect plays an important role in micro-milling of KDP crystal, when the undeformed

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chip thickness is less than the minimum undeformed chip thickness. Aramcharoen and Mativenga

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determined that the optimum surface finish can be received in micro-milling of tool steel when the ratio ft/re is 1 [1]. However, the optimum surface finish in this study appears when the ratio is around

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0.7. This may be explained by the fact that the brittle material is more readily to be removed than the

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metallic materials under effective negative rake angle. Considering the relationship between the machined surface quality and the ratio ft/re, we observed

that the roughness value Ra of surface machined by CBN tool (re=2.3 m) with the spindle speed of 20, 000 r/min is larger than the Ra of the surface machined by the same tool with a higher spindle speed (40, 000 r/min), but the value is smaller than the Ra of surface machined by carbide tool (re=3 m) with the same spindle speed (20, 000 r/min). 26

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Fig. 12 Surface roughness Ra with respect to the ratio of feed rate to cutting edge radius

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To validate the distinct sensibilities of the cutting edge radius and the spindle speed upon surface roughness value Ra, the analysis of variance (ANOVA) data with the ratio ft/re of 0.5 and 3 is carried out (Table 3). The ANOVA results show that the cutting edge radius and the spindle speed have a

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significant effect on surface roughness value Ra under the two specified ratio values, since the Prob

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＞F is less than the adopted significance level (α=0.05). In a word, the cutting edge radius has a more significant impact on the surface roughness value Ra than the spindle speed because the Prob＞

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F for spindle speed is 20 times larger than the cutting edge radius under the ratio ft/re of 0.5; while

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the spindle speed has a more significant impact on the surface roughness value Ra than the cutting

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edge radius because the Prob＞F for spindle speed is almost zero under the ratio ft/re of 3. Fig. 13 presents the comprehensive results of the specific cutting force, surface roughness Ra as

well as the corresponding surface morphology with respect to the ratio between the feed rate and the tool edge radius ft/re under the spindle speed of 20, 000 r/min. It can be found that the ratio of the specific cutting force and the surface roughness value (kc/Ra) decreases almost linearly with the augment of the ratio ft/re (ft/re < 0.5), indicating severe size effect and heavy ploughing in 27

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micro-milling of KDP crystal. However, when the ratio ft/re is raised to the range of 0.5~1, the cutting process will be under transitional phase with the ratio kc/Ra changing relatively slower. According to the machined surface morphologies under different ratio ft/re displayed in Fig. 13, the slight size effect can still be observed when the ratio ft/re is between 0.5 and 0.7, and it has all but

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disappears when the ratio ft/re is between 0.7 and 1, so the perfect surface can be obtained with the ratio ft/re around 0.7. When the ratio ft/re is larger than 1, the ratio kc/Ra decreases slowly to approaching zero, because with the augment of ratio ft/re, the specific cutting force decreases but the

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roughness value Ra increases, which indicates that the cutting is likely to shift to macro milling, resulting in the degeneration of surface quality. When the ratio ft/re reaches 3, the cutting process is almost in brittle mode.

ft/re =0.5

df SS

n

1

Interaction

1

Error

28

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408.33

31

F

Prob＞F

SS

MS

F

Prob＞F

408.33

34.27

0.0004

259.2

259.2

12.67

0.0026

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1

147

147

12.34

0.0079

7527.2

7527.2

368.08

0

33.33

33.33

2.8

0.133

480.2

480.2

23.48

0.0002

95.33

11.92

327.2

20.45

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re

ft/re =3

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Factor

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Table 3 ANOVA of the cutting edge radius and spindle speed upon surface roughness value Ra (α=0.05)

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8593.8

From the analysis above, it can be concluded that the surface quality will get even worse when the feed per tooth is much smaller (causing severe size effect) or larger than the minimum undeformed chip thickness (entering the brittle cutting mode). Therefore, a feed per tooth, slightly larger than the minimum undeformed chip thickness (avoiding severe size effect) and smaller than the cutting edge 28

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radius (limiting the effective rake angle in the range of negative, which is more suitable for machining brittle material) is recommended for micro-milling of KDP crystal or any other soft-brittle

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crystal.

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Fig. 13 The comprehensive results of specific cutting force, surface roughness Ra as well as the corresponding

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5. Conclusions

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surface morphology with respect to the ratio between feed rate and tool edge radius ft/re.

This study investigates in depth the minimum undeformed chip thickness and the size effect in

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micro-milling of KDP crystal with a soft-brittle property. Two micro end mills with two different cutting edge radii and several cutting parameters have been used in this study. The minimum undeformed chip thickness is estimated by analyzing the periodicity of the cutting force obtained during the experiments. The size effect in micro-milling of KDP crystal is studied by correlating the specific cutting force, the morphology of the machined surface and surface roughness Ra. The following conclusions can be drawn based on the research: 29

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(1) A model of chip formation, which combines the minimum undeformed chip thickness, the undeformed chip thickness and the periodicity of cutting force at a certain cutting angle θ together, is developed to estimate the minimum undeformed chip thickness. (2) The normalized minimum undeformed chip thicknesses are 0.43 ≤ λe ≤ 0.46 and 0.45 ≤ λe ≤ 0.48

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for cutting edge radii of 2.3 m and 3 m, respectively. The normalized minimum undeformed chip thickness increases with an augment of the cutting edge radius, because the material becomes more ductile caused by the increasing ploughing effect.

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(3) The normalized minimum undeformed chip thickness of the KDP crystal (0.43~0.48) is larger when compared to the metallic materials (0.25~0.33). There are two possible contributions to the results: 1) the effective rake angle is almost equal to the optimal rake angle in micro-milling of

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KDP crystal when the undeformed chip thickness is around normalized minimum undeformed

become more ductile.

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chip thickness; 2) the KDP is softer and the thermal softening effect could make the KDP crystal

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(4) Correlating the specific cutting force, the morphology of the machined surface and surface roughness Ra together, the severe size effect and heavy ploughing in micro-milling of KDP

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crystal are found when the ratio ft/re less than 0.5, and the size effect is almostly disappeared with

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the ratio ft/re increasing to 0.7. When the ratio ft/re larger than 3, cracks and brittle areas start emerging again on the surface which indicates that the cutting seems more likely in macro milling.

(5) The cutting edge radius has a more significant impact on the surface roughness value Ra than the spindle speed when the ratio ft/re is smaller than 0.5. The impact of spindle speed on the surface roughness value Ra value increases with the augment of the ratio ft/re. 30

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(6) The surface quality will get even worse when the feed per tooth is much smaller (causing severe size effect) or larger than the minimum undeformed chip thickness (entering the brittle cutting mode). A feed per tooth, which is a few larger than the minimum undeformed chip thickness and smaller than the cutting edge radius, is recommended for the micro-milling of KDP crystal or

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other soft-brittle crystals. (7) The research method of this study could be a reference to the similar study of other brittle materials. The obtained results could potentially guide the selection of the cutting parameters and

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cutting edge radius to improve the integrity and quality of machined surface in micro-milling of

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other brittle materials.

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Acknowledgments This work was sponsored by Science Challenge Project (No. JCKY2016212A506-0503), National 863 Program (No. 2015AA043301), Self-Planned Task (No. SKLRS201718A) of State Key Laboratory of Robotics and System (HIT), the Foundation for Innovative Research Groups of the

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National Natural Science Foundation of China (No. 51521003) and the scholarship from China Scholarship Council. The authors would like to thank colleagues at KU Leuven, Dr. Benjamin

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Gorissen, Dr. Marius Nabuurs, Mr. Jian Wang, Mr. Jun Tang and Mr. Cheng Guo for their assistance.

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Mach Tools Manuf 2016; 107:21–40. [25] Fang FZ, Zhang GX. An experimental study of edge radius effect on cutting single crystal silicon. Int J Adv Manuf Tech 2003; 22:703–707.

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[27] Chen N, Chen MJ, Guo YQ, Wang XB. Effect of cutting parameters on surface quality in ductile cutting of KDP crystal using self-developed micro PCD ball end mill. Int J Adv Manuf

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Tech 2015; 78:221–229.

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Graphical Abstract

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