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Modelling and analysis of micro scale milling considering size effect, micro cutter edge radius and minimum chip thickness
ARTICLE IN PRESS
International Journal of Machine Tools & Manufacture 48 (2008) 1–14 www.elsevier.com/locate/ijmactool
Modelling and analysis of micro scale milling considering size effect, micro cutter edge radius and minimum chip thickness Xinmin Laia,, Hongtao Lia, Chengfeng Lia, Zhongqin Lina, Jun Nib a
State Key laboratory of Vibration, Shock and Noise, Shanghai Jiaotong University, Shanghai 200030, China b Shien-Ming Wu Manufacturing Research Center, University of Michigan, MI 48109, USA Received 7 April 2007; received in revised form 2 August 2007; accepted 8 August 2007 Available online 22 August 2007
Abstract This paper presents mechanisms studies of micro scale milling operation focusing on its characteristics, size effect, micro cutter edge radius and minimum chip thickness. Firstly, a modified Johnson–Cook constitutive equation is formulated to model the material strengthening behaviours at micron level using strain gradient plasticity. A finite element model for micro scale orthogonal machining process is developed considering the material strengthening behaviours, micro cutter edge radius and fracture behaviour of the work material. Then, an analytical micro scale milling force model is developed based on the FE simulations using the cutting principles and the slip-line theory. Extensive experiments of OFHC copper micro scale milling using 0.1 mm diameter micro tool were performed with miniaturized machine tool, and good agreements were achieved between the predicted and the experimental results. Finally, chip formation and size effect of micro scale milling are investigated using the proposed model, and the effects of material strengthening behaviours and minimum chip thickness are discussed as well. Some research findings can be drawn: (1) from the chip formation studies, minimum chip thickness is proposed to be 0.25 times of cutter edge radius for OFHC copper when rake angle is 101 and the cutting edge radius is 2 mm; (2) material strengthening behaviours are found to be the main cause of the size effect of micro scale machining, and the proposed constitutive equation can be used to explain it accurately. (3) That the specific shear energy increases greatly when the uncut chip thickness is smaller than minimum chip thickness is due to the ploughing phenomenon and the accumulation of the actual chip thickness. r 2007 Elsevier Ltd. All rights reserved. Keywords: Micro scale milling process; Size effect; Minimum chip thickness; Cutter edge radius; Strain gradient
1. Introduction Recent years, the production of miniaturized components with complex small features is gaining increasing importance due to the trend of miniaturization which is determining the development of products for various industries, such as biomedical instruments, electronic products, defence industry and so on. Most of these components fall into the scales from 10 mm to 1 mm known as micro/meso scale in mechanical engineering. Considered as one of the most effective techniques, micro scale milling process can be used to fabricate these components with complex micro features over a wide range of material types. Corresponding author.
E-mail address: [email protected] (X. Lai). 0890-6955/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmachtools.2007.08.011
However, further advances in both the efficiency and the quality are limited by the incomplete understanding of its basic mechanisms. To satisfy the increasing need of miniaturized manufacturing, the mechanisms studies are becoming more and more important. Micro scale milling is not simply downsized from the conventional operation but has its own characteristics, such as size effect, cutter edge radius and minimum chip thickness. Shaw [1] studied the effect of round edges on the chip formation in micro scale machining and stated that the plastic deformations would be prevented when the cutter edge radius is relatively larger than the uncut chip thickness. Kim [2] investigated the effect of static tool deflection on the micro milling and proposed a static chip model based on the attainable micro scale machining force data. Ni [3] investigated the chip formation using molecule
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Nomenclature s plastic equivalent stress (MPa) t shear strength (MPa) e effective strain _ effective strain rate reference strain rate _ 0 pl _¯ plastic strain rate A, B, C, m, n coefficients of the Johnson–Cook model p static pressure (MPa) q Von mises stress (MPa) ¯ pl plastic strain at failure f D¯pl the increment of the equivalent plastic strain Tmelt melting temperature of the workpiece T0 transition temperature of the workpiece w damage criterion d1 d2 d3 d4 d5 coefficients of Johnson–Cook material shear failure criterion E Young’s module n Poisson ratio r density of work material rs density of statistically store dislocation rg density of geometry necessary dislocation Z strain gradient l scale variable G shear modulus (MPa) b Burgers vector (nm)
dynamic (MD) simulations and presented an approach to calculate the minimum chip thickness by identifying a local maximum in the radial thrust forces in the micro milling. Vogler [4] determined the minimum chip thickness for steel using finite element (FE) simulations with regard of the microstructure properties of work material. According to their research, the critical chip thickness is 0.2 and 0.3 times of the edge radius for pearlite and ferrite, respectively. In addition, several analytical cutting force models ([5,22,23,26]) had been developed to investigate the micro scale machining process. Chae [6] surveyed the state of art of micro scale machining and reported that current researches had made valuable attempts in this field though most of them were carried out by means of nano-level and macro-level approaches. From the discussions above, it is obviously that the processing system for micro scale milling is far from established. Its characteristics need to be studied and the related mechanisms need to be revealed through experiments and theoretical modelling. One of the most significant characteristics of the micro scale milling operation is the size effect. Some efforts have been carried out to explain it. Lucca [7] investigated the size effect of cutting energy of micro scale machining process through the experiments; they found nonlinear increase in specific cutting energy or cutting forces as the uncut chip thickness was decreased. Kopalinsky and Oxley [8] studied the size effect with sharp tools by turning tests. They
h, hmin uncut chip thickness and minimum chip thickness L length of primary machining deformation zone f the coefficient of friction between the workpiece and chip V rotation speed of the spindle j shear angle a rake angle R cutter edge radius g clear angle k FE predicted stress along the primary shear zone AB, CA, Rsl length variables in slip line model [21] wsl width of the workpiece in slip line model [21] asl, ysl, gsl, Zsl, rsl angle variables in slip line model [21] Scut, Pcut, Sthr, Pthr shear force and ploughing force in slip line model [21] Fc, FT, cutting force and thrust force Fx, Fy, milling forces in x- and y-direction KC, KT, wc2, wt2 coefficients of force model when h is less than hmin PC, PT, wc1, wt1 coefficients of force model when h is greater than hmin y angle of the milling tool position ft feed per tooth ap depth of cut
concluded that the cause was the decrease in the tool chip interface temperature. Nakayama and Tamura [9] analyzed the size effect through experiments performed at a very low cutting speed to minimize the temperature and strain rate effects. They attributed this effect to plastic flow in the workpiece subsurface. The experiments of the literature (Oxley, [8]; Nakayama, [9]) implied that there should be other underlying mechanisms for the size effect besides cutter edge radius, temperatures and strain rate. On the other hand, a similar size effect in micro indentation tests was found in the mechanics studies, which was shown as remarkable material strengthening behaviours at the micron level. Fleck [10] found that the torsion strengthened three times when the diameter of the sample was decreased from 170 to 12 mm. Stolken and Evans [11] found the similar trend in bending material tests. The bending strength increased significantly when the thickness of thin beam was decreased from 100 to 12.5 mm. This size effect in material properties has turned to be the focus of mechanics research recent years. Strain gradient (SG) plasticity is the most effective method to interpret this phenomenon. Fleck [12] developed SG theory by introducing the material inner variable. Then, Nix and Gao [13] improved it by describing the concept of inner variable and mechanism-based strain gradient (MSG) theory was developed. From this point, it is supposed that there might be some similarities between
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the two size effect phenomena. The size effect of material behaviours may be the cause of size effect in micro scale machining. Melkote [14] attempted to analyze the size effect using SG plasticity based on analytical solutions. The results showed that it is capable of explaining the size effect. However, this model could not consider many other characteristics of micro scale machining process, such as large strain, high strain rate, cutter edge radius, minimum chip thickness and so on. Therefore, further studies are needed and more precise FE models should be developed. The goal of this research is to provide deeper understanding of micro scale milling process. Firstly, a modified Johnson–Cook (JC) constitutive equation is formulated using SG plasticity. A FE model for micro scale orthogonal machining process is developed considering material strengthening behaviours, tool geometry and fracture behaviour. Then, a milling force model is developed based on the FE simulations using the cutting principles and slip-line theory. Finally, chip formation and size effect of micro scale milling are investigated by applying the model, and the effects of the main characteristics of micro scale process, size effect of material behaviours, micro mill cutter radius and minimum chip thickness are discussed as well. The results demonstrate that the proposed approach for micro scale milling is capable of the analysis of micro scale milling process and provides the academic foundation for further process design and optimization studies.
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2. Micro scale milling process modelling Milling operation is a very complex process as shown in Fig. 1a. Its performance is affected by so many parameters that it is not possible to take all the factors into account at the same time. According to the analysis of micro scale milling process and its characteristics, two assumptions are made: (a) The objective of the first assumption is to simplify the complicated 3D milling process (Fig. 1a) to a 2D process shown as Fig. 1b. It is reasonable because the depth of cut (DOC) of the micro scale milling is very small, sometimes only several microns. Under these conditions, the helix angle of the milling process has so little effect that could be ignored and DOC could be considered constant during milling process. Therefore, the constant DOC in milling is represented by the thickness of the workpiece in the simplified 2D process. (b) The other assumption is made to build the relationship between the simplified 2D milling process and orthogonal machining process. As Fig. 1b shows, the deformation area is much smaller when compared with the tool and the workpiece since the diameter of the tool is usually several hundreds microns while the dimension of the machining zone is at the micron level. In this study, the feed per tooth is less than 2 mm. Therefore, the deformed area can be considered as kind of orthogonal machining process shown as Fig. 1c and
Fig. 1. Simplifications for micro scale milling process.
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d. For the orthogonal machining, uncut chip thickness is equivalent to the chip thickness of the milling process. Based on these two assumptions, modelling of micro scale milling process could be divided into three steps: the first is the material strengthening behaviours modelling. A modified JC model is formulated by introducing the SG theory. Then, based on the proposed material model, a FE model for orthogonal machining is developed with respect to micro cutter edge radius, chip separation criterion, and the friction between the tool and the chip. The last step is micro scale milling force modelling. The model is calibrated based on the FE simulates using the cutting principles and slip-line approach. 2.1. Material strengthening behaviours modelling 2.1.1. Guideline for the development of the constitutive equation Material model is the fundamental of FE simulations and is required to represent the features of material behaviours during the deformation process. Machining operations are associated with large strains, high strain rates and high temperatures. For micro scale machining process, size effect is also a dominant factor. Therefore, a material model capable of capturing these effects is highly necessary. Semi-empirical constitutive equations are widely applied in present FE simulations of cutting process [18]. Among them, the constitutive model proposed by JC [15] is one of the most widely used models. It provides good description of metal material behaviours undertaking large strains, high strain rate and temperatures. It describes the flow stress of work material with the product of strain, strain rate and temperature effects that are individually determined as T T0 m _ n s¯ JC ¼ ½A þ BðÞ 1 þ C ln 1 . T melt T 0 _ 0 (1) However, the conventional FE analysis is a non-dimensional process, where flow stress is independent to the scale variable. Therefore, it cannot be used to describe the size effect. In this research, a material model of micro scale machining is to be developed by introducing a size variable l into the conventional material model (JC model). The framework of the proposed constitutive equation is expressed as s ¼ f ðsconventional ; ZÞ,
(2)
where sconventional ¼ sJC, Z ¼ gl (l is the size variable, when l reaches to macro level: s ¼ sJC) The objective is to establish the relationship between the size variable l and flow stress, and keep the advantages of JC model as well. Next, the deduction of the constitutive equation will be presented in detail.
2.1.2. Deduction of the material model In the present approach, the work material is considered as a plastically isotropic material. The total strain rate tensor _ij , is given by the sum of the elastic strain rate _ eij , and the plastic strain rate _pij , as _ij ¼ _eij þ _pij .
(3)
The elastic part is given by Hooke’s law, while the plastic part is expressed by the Levy-von Mises equation: _pij ¼
3 _p S ij . 2s
(4)
Relating it to the deviatoric stress tensor Sij in terms of the von Mises equivalent plastic strain rate: rffiffiffiffiffiffiffiffiffiffiffiffi 2 p p p (5) _ _ . _ ¼ 3 ij ij Then, the Von Mises equivalent stress can be obtained: rffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 S ij Sij . s¼ (6) 2 From the SG plasticity, Taylor’s dislocation model gives the flow stress in terms of the dislocation density, as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi s ¼ 3aGb r ¼ 3aGb rs þ rg . (7) rs can be determined by the material test in the absence of strain gradient, pffiffiffiffiffi (8) s ¼ 3aGb rs ¼ sconventional . In order to properly estimate the total dislocation density, m is introduced: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (9) s ¼ 3aGb rms þ rmg . Substituting Eq. (8) into Eq. (9), the flow stress can be written as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m rg s ¼ sconventional 1 þ . (10) rs The density of geometrically necessary dislocation (GND) is given by rg ¼
2Z . b
(11)
Thus, the constitutive equation turns to be as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m 18 a2 G 2 bZ s ¼ sconventional 1 þ . (12) s2conventional As discussed above, sJC is selected to be sconventional, then, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m 18 a2 G 2 bZ . (13) s ¼ sJC 1 þ s2JC From the works of Melkote [14], the SG is obtained through the dislocation analysis of primary shear zone for
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micro scale machining, as follows: 1 . L So, the constitutive equation can be expressed as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mffi 18 a2 G2 b s ¼ sJC 1 þ . s2JC L Z¼
(14)
(15)
So far, the constitutive equation has been developed. Next, the work comes to the calculation of the length of the primary shear zone, L. The minimum chip thickness has significant impact on the micro scale machining operation, especially when the uncut chip thickness is close to micro cutter edge radius. On these occasions, the machining process can be divided into two situations: chip forms and no chip forms. Kim [19] studied this phenomenon using MD simulations, as shown in Fig. 2. For the situation that chip forms when h^hmin, as shown in Fig. 2a, the length of shear zone can be obtained by the cutting principles [17]: L¼
h sin ðfÞ
f¼
r cos an 1 r sin an
h r¼ . t
(16)
For the situation that no chip forms as shown in Fig. 2b, when hohmin, the shear angle becomes to be very small and chip thickness t does not exist. Therefore, Eq. (16) is not applicable any more. Through the analysis of shear zone, the arc length of contact part is proposed to be the length of primary shear zone. arccos Rh R pR L¼ . (17) 180 Hence, the constitutive equation can be expressed as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mffi 18 a2 G 2 b s ¼ sð; _; T; L; hmin Þ ¼ sJC 1 þ s2JC L T T0 m _ n sJC ¼ ½A þ BðÞ 1 þ C ln 1 . T melt T 0 _ 0 ð18Þ For aforementioned formulation, the scale variable is expressed as the shear zone length L. Since the SG is the
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reciprocal of L, it will decrease with the increase of uncut chip thickness. Therefore, the GND will disappear for the large test piece and macro scale machining process. This fully satisfies the constraint condition that the flow stress is equal to the results by JC model when size variable reaches to macro level (Eq. (2)). From the aforementioned deduction and analysis, the proposed constitutive equation has two advantages: (a) This material model can describe the characteristics of micro machining, including the material strengthening behaviours and the minimum chip thickness. (b) It is based on JC model, which provides good fit for strain hardening behaviours of metals in macro scale machining process, such as large strains, high strain rates and high temperatures. 2.2. FE modelling for micro scale orthogonal machining process After establishing the constitutive equation, a FE model for orthogonal micro scale machining process is developed using a commercial FE analysis code, ABAQUS/EXPLICIT. In this study, the work material is OFHC copper, and the micro tool used is tungsten carbide micro end miller with the diameter of 0.1 mm. The cutting speed is 104.72 mm/s, which is calculated from the spindle rotation speed of 20,000 r/min. The parameters of material properties are listed in Table 1. Main considerations of micro scale machining FE modelling will be presented as follows. 2.2.1. Material behaviours Based on the constitutive equations deduced in Section 2.1, stress–strain curves of work material at different uncut chip thickness are calculated and shown as Fig. 3. The left one show the stress–strain curves when h is 1, 10 and 100 mm and when h reaches macro level (JC model). Size effect can be observed obviously. From this figure, we can also see that the stress approaches to that of JC model with the increase of uncut chip thickness. The right figure shows the stress–strain curves when h is 1, 4, 10 and 20 mm. Same trend can be found from this figure with the left one. The
Fig. 2. Chip formation with regard of cutter edge radius. (a) Chip formed; (b) no chip formed.
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Table 1 Parameters for work material OFHC copper Parameter
Value
Mechanical properties [15]
r (kg/m3) E (MPa) u Tmelt (1C) T0 (1C)
8960 124 0.34 1356 293
Johnson Cook model [15]
A (MPa) B (MPa) n C m
90 292 0.31 0.025 1.09
direction of decrease in h
Value
Fracture [16]
d1 d2 d3 d4 d5
0.54 4.89 3.03 0.014 1.12
Plastic strain gradient [14]
a G (GPa) b (nm) m
0.5 39 0.256 0.38
direction of decrease in h
600
500 400 300 Johnson-Cook h=1um h=10um h=100um
200 100 0
Stress (MPa)
Stress (MPa)
600
Parameter
500 400 300 Johnson Cook h=1um h=4um h=10um h=20um
200 100 0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Strain (%)
Strain (%) Fig. 3. Material properties.
Fig. 4. SEM picture of micro mill cutter.
material model is integrated into solution algorithm using ABAQUS/EXPLICIT VUMAT. 2.2.2. Micro cutter edge radius Edge radius of micro cutter is one of the most significant features of micro scale machining process. It is considered to be the main cause of minimum chip thickness by some researches. The edge radius of 0.1 mm diameter 2-flute micro-end-miller is measured by SEM shown as Fig. 4. The cutter edge radius is obtained to be about 2 mm.
2.2.3. Chip separation criterion Chip separation criterion is another critical issues in FE simulations of cutting process. Two approaches were usually used in precious researches: pre-defined separation path and adaptive remeshing technique. However, both of them obviate the fracture mechanics behaviours of the workpiece. In this model, the chip separation is modelled with the shear failure of the material. JC shear failure criterion is utilized shown as Eq. (19). During the solving process, once the equivalent plastic strain exceeds the
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Input the material parameters and state variables (On-1, n-1, d n, A, B, C, m, n, d1, d2, d3, d4, d5)
Calculate the trial stress increment d On = D dn Calculate the trial stress O ntrial = On-1+ d On Calculate the yield stress according to the proposed material model
Deform state
O ntrial < O yield Elastic deform
O ntrial > O yield Plastic deform Calculate equivalent strain increment d pl
Calculate the total stress
Calculate the equivalent strain
Calculate the critical strain according to the shear failure criterion
n> The shear failure state
n-1
<
pl
Set the element state variable to zero element deletion
pl
Update the strain, stress, inelastic specific energy and the state variables Fig. 5. Flow chart of the ABAQUS/VUMAT program.
critical value, the element is considered to be failed and will be deleted in the next solver step. The criterion is applied using ABAQUS/ EXPLICIT VUMAT. The flow chart is shown as Fig. 5. h pl ih
i 8 pl d 3 pq TT 0 _¯ > ¼ ½d þ d e 1 þ d ln 1 þ d ¯ > 1 2 4 5 T melt T 0 _ 0 < f . P pl D¯ > > ¼1 pl :o ¼ ¯ f
(19) 2.2.4. Friction at the tool–chip interface The contact and the friction between the chip and the tool are influenced by factors such as cutting speed, feed rate, rake angle, etc. Due to very limited research has been carried out for micro friction at the tool chip interface, coefficient of friction f of 0.3 was used in this model [25]. 2.3. Micro scale milling force modelling As outlined at the beginning of Section 2, the last step of the modelling is to build the analytical micro scale milling
force model. It could be divided into two parts: first is to calculate the machining force using slip line approach. Second is to calibrate the analytical milling force model based on these results. 2.3.1. Micro scale machining force calculation Slip line methods are widely used for the calculation of the machining forces [20]. The slip line model developed by Waldort [21] is one of the most accurate models and can take the ploughing phenomenon and cutter radius into account. This model is used to calculate the machining force from the FE results. The descriptions of the model and its parameters can be found in the Ref. [21]. According to this model, the deformation zone of the orthogonal cutting process is shown in Fig. 6. The cutting forces and ploughing forces respectively in the cutting and thrust direction can be calculated by ( S cut ¼ kwsl ½cos ðf þ ð1 þ 2ysl Þ sin ðfÞAB , (20) S thr ¼ kwsl ½ð1 þ 2ysl Þ cos ðfÞ sin ðfÞAB
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8 Pcut ¼ kwsl ½cos ð2Zsl Þ cos ðf gsl þ Zsl Þ > > > > < þð1 þ 2ysl þ 2gsl þ sin ð2Zsl ÞÞ sin ðf gsl þ Zsl ÞCA , Pthr ¼ kwsl ½1 þ 2ysl þ 2gsl þ sin ð2Zsl Þ > > > > : cos ðf g þ Z Þ cos ð2Z Þ sin ðf g þ Z ÞCA sl
sl
sl
sl
Based on aforementioned calculation, the relationship between the machining forces and the FE simulations is established.
sl
(21) where Rsl , sin ðZsl Þ Rsl ¼ sin ðZsl Þ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! u p a pffiffi2ffiR sin ðr Þ 2 u sl sl t sl þ þ 2½Rsl sin ðrsl Þ2 . þ re tan 2 4 tan p2 þ asl
CA ¼
Hence, the total forces in cutting and thrust direction could be obtained by ( F C ¼ S cut þ Pcut . (22) F T ¼ S thr þ Pthr
2.3.2. Micro scale milling force model calibration The second part is micro scale milling force model calibration. For micro scale operation, the minimum chip thickness has a significant influence on the milling force. Therefore, the analytical force model should be formulated considering when there is chip formed and no chip formed respectively, as Eq. (23): (1) When hphmin, (
F T ¼ PT hwt1 ðyÞap
Fc
t chip
R Pthr
F
A
Pcut C
Scut
Rsl
B
Sthr
V
Y
ha
FT Fig. 6. Slip line model [21].
.
(2) When h4hmin, (
Tool
F C ¼ PC hwc1 ðyÞap
F C ¼ K C hwc2 ðyÞap . F T ¼ K T hwt2 ðyÞap
(23)
Where chip thickness h(y) ¼ ft sin y, PC, PT, wc1, wt1, KC, KT, wc2, wt2 are coefficients for the force model, hmin is obtained from FE simulations, and is proposed to be 0.5 mm, 0.25 times of the micro cutter edge radius for OFHC copper. The chip formation will be discussed in detail in Section 5.1. The parameters for the micro scale milling force model calibration are seen in Table 2. In the table, Fc and Ft were calculated using Waldorf’s method presented in Section 2.3.1. Then, PC, PT, wc1, wt1 were calibrated to be 741.6, 88.5, 0.7536 and 0.3459, respectively, when h is smaller than hmin, and KC, KT, wc2, wt2 are 553.5, 360.4, 0.5879 and 0.4217, respectively, when h is larger than hmin. The transforms of cutting force and
Table 2 Parameters for calibration of micro scale milling force Parameters of micro tool
Process parameters
Hohmin H4hmin
Diameter (mm)
Number of teeth
Rake angle (deg)
0.1
2
10
Work material
RPM (r/min)
ap (mm)
OFHC copper
20,000
10
Feed per tooth (mm)
FC (N)
FT (N)
0.2 0.4 0.6 0.8 1 1.4
0.0121 0.0204 0.0697 0.0837 0.0971 0.1154
0.0465 0.0591 0.1449 0.1870 0.2087 0.2165
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W ) HC OF e( iec kp or
Rotation
Fy
Fx Fy Milling tool Fx
Ft
Fz Fc
n tio ec dir ed Fe
Fz
Feed direction Fig. 7. Micro scale milling force.
thrust force to the forces in X- and Y-direction is shown as Fig. 7. Therefore, the machining forces can be transformed to the milling forces in X- and Y-direction as given by ( F x ¼ F C cos y F T sin y . (24) F y ¼ F C sin y þ F T cos y
3. Experimental details
Fig. 8. The miniaturized machine tool.
Cutting force dynamometer (Kistler 9256 C1) Workpiece Micro end miller Spindle Y-stage
Adapter
To perform the experiments needed for the research, a miniaturized machine tool is established, as shown in Fig. 8. The overall volume of the machine tool is 270 190 220 mm, and the working volume is 30 30 30 mm. Fig. 9 shows the components of the miniaturized machine tool system. It consists of five subsystems: the first is the positioning subsystem, consisting of 3 high precision stages with the resolution of 50 nm; the second is the NSK electronic spindle subsystem with maximal rotation speed of 80,000 r/min; the third is motion control subsystem. A PMAC motion controller is used to control the motion of the stages; the fourth is the micro end miller; and the fifth is the cutting force measurement subsystem, including the force dynamometer and its data acquiring instruments. The work material used in the experiments was OFHC copper 101, which is widely applied for optics, laser, and micro-electronics field. The tools used were all 0.1 mm diameter micro end millers. For the micro scale milling process, the cutting forces are usually as small as several hundreds of micro Newton [24]. Moreover, its frequency is extremely high due to the high rotation speed of the spindle. For example, it is 667 Hz when the rotation speed of the spindle is 20,000 r/min and 2667 Hz for the maximum rotation speed 80,000 r/min. Both of the magnitude and frequency require a precise multi-component cutting force sensor with very high precision. A 5component Kistler dynamometer 9256C1 with a 2 mN
X-stage
Spindle support Z-stage Base plate
X Y Z Motor drivers
PMAC motion control card
Unigraphics/CAM
Amplifier
NI data inquiring card
Data inquiring software
Fig. 9. Components of the miniaturized machine tool system.
threshold was used to measure the milling forces. Its natural frequency is 5.1, 5.5and 5.6 kHz in X-, Y-, Zdirection, respectively.
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4. Model validation To use the proposed model to predict the forces and analyze micro scale milling process, extensive experiments are needed to validate the model over a widely range of conditions. In this section, experiments and simulations were performed to examine DOC and ft in the applicable conditions range of the miniaturized machine tool and the micro cutters. Four sets of correlations were done when DOC was 6, 10, 15 and 20 mm, respectively. For each set, ft was selected to be 0.2, 0.4, 0.6, 0.8, 1 and 1.4 mm. The experimental conditions are shown in Table 3. Fig. 10 shows the comparisons between the simulated and experimental milling forces in X- and Y-direction. The experimental data were measured with the Kistler dynamTable 3 Experimental conditions for model validation Parameter
Value
Micro-end-mill tool Material Teeth number Diameter (mm) Workpiece material OFHC copper 101 Milling conditions RPM of spindle (rev/min) ft (mm) Depth of cut (mm)
Peak to Valley Force (N)
0.20 0.15
ometer and the data sample rate was 10,000 Hz. Average peak to valley forces are chosen to examine the proposed model. The experimental peak to valley forces were calculated for each revolution and averaged over the milling process for every experiment. Figs. 10a–d show the comparisons of the simulated and experimental results when DOC is 6, 10, 15 and 20 mm, respectively. In general, the micro scale forces are very small, only several hundreds of micro Newton. The forces increase with feed per tooth, and the force in Y-direction is a little larger than that in X-direction. In contrast, good agreements both in the magnitudes and trends can be seen from the comparisons between the extensive predicted and the measured milling force at all ft and DOC. In addition, the following experimental trends are also well predicted by the model. The peak to valley forces are increased with nearly proportional to the DOC. Based on these analysis, the model is considered to be satisfactory validated and can be used to analyze the micro scale milling process.
Tungsten carbide 2 0.1
5. Results and discussions 5.1. Chip formation analysis of micro scale machining
20,000 0.2, 0.4, 0.6, 0.8, 1, 1.4 6, 10, 15, 20
In this section, chip formation of micro scale machining was investigated through the FE simulations. The goal is to
0.35 Fx-Exp. Fy-Exp. Fx-Sim. Fy-Sim.
0.10 0.05
Peak to Valley Force (N)
10
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
0.20 0.15 0.10 0.05
0.7 Fx-Exp. Fy-Exp. Fx-Sim. Fy-Sim.
0.3 0.2 0.1
Peak to Valley Force (N)
Peak to Valley Force (N)
feed per tooth (um)
0.4
0.25
Fx-Exp. Fy-Exp. Fx-Sim. Fy-Sim.
0.00 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 feed per tooth (um)
0.00
0.5
0.30
0.6
Fx-Exp. Fy-Exp. Fx-Sim. Fy-Sim.
0.5 0.4 0.3 0.2
0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 feed per tooth (um)
0.2 0.4 0.6 0.8 1.0 1.2 1.4 feed per tooth (um)
Fig. 10. Comparisons of milling forces between the experimental and predicted results. (a) DOC ¼ 6 mm; (b) DOC ¼ 10 mm; (c) DOC ¼ 15 mm; (d) DOC ¼ 20 mm.
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find out hmin and its relationship with the micro cutter edge radius. The uncut chip thickness for micro scale orthogonal machining was selected to be 0.1 R (0.2 mm), 0.2 R (0.4 mm), 0.3 R (0.6 mm). As Fig. 11 shows, it can be observed that there is no chip formed when h is 0.1 and 0.2 R, while chip forms when the h is 0.3 R. From these results, hmin is proposed to be 0.25 R (0.5 mm) for OFHC copper when cutter edge radius is 2 mm and rake angle is 101. The results are consistent with some literatures. For example, Vogler [4] estimated the minimum chip thickness to edge radius ratios of pearlite and ferrite to be 0.20 and 0.3, respectively. Kim [2] proposed that the minimum chip
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thickness to be 30% of the cutting edge radius by means of MD simulations and experiments. It illuminates that the cutting edge radius is the main cause of the minimum chip thickness. The results demonstrate that the proposed model is capable of the chip formation studies of micro machining by applying the proposed material model and the shear failure fracture criterion. 5.2. Size effect analysis of micro scale machining Many simulations were conducted applying the proposed FE model to examine the size effect of micro scale
Fig. 11. Chip formation process. (a) h ¼ 0.1 R (0.2 mm); (b) h ¼ 0.2 R (0.4 mm); (c) h ¼ 0.3 R (0.6 mm).
Fig. 12. Results when uncut chip thickness is 10 mm. (a) Without consideration of size effect; (b) with consideration of size effect.
Fig. 13. Size effect at different uncut chip thickness. (a) h ¼ 1 mm; (b) h ¼ 4 mm (c) h ¼ 20 mm.
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milling process. Fig. 12 shows the comparisons between the results with and without consideration of material strengthening effect when uncut chip thickness is 10 mm. The left figure shows the predicted result which material strengthening behaviour was not considered, while the right one was the result considering the size effect. From the comparisons, it can be seen that the stress distribution is similar to each other. However, the maximum effective stress with regard of size effect (764 MPa) is much higher than the result without considering size effect (544 MPa). It illuminates that the size effect is considerably remarkable when uncut chip thickness is 10 mm. Moreover, many simulations were also performed to examine the size effect under different uncut chip thickness, 1, 4 and 20 mm. It can be seen from Fig. 13 that maximum effective stress increase as the uncut chip thickness is decreased, which is 1009, 888, 764 and 704 MPa when uncut chip thickness is 1, 4, 10 and 20 mm, respectively. From the analysis, it is indicated that the size effect in micro scale machining could be well explained by SG plasticity.
hmin
Exp. Sim.
40 35 30 25 20 15 0.2
0.4
0.6
0.8
1.0
1.2
1.4
feed per tooth (um) Fig. 14. Size effect in specific shear energy at different ft.
Peak to Valley Force (N)
Specifics hear energy (GPa)
45
0.30
FxExp FxSim Fx-Sim w/o Size effect
0.25 0.20 0.15 0.10 0.05 0.00 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 feed per tooth (um)
From the experimental data, size effect is also observed. As shown in Fig. 14, the SSE is calculated by dividing amplitude of the in-plane milling force by the product of ft and DOC. From the results, the higher SSE is observed for the smaller feed per tooth. These results correspond to the well-known size effect. In addition, another trend is also found that the SSE increases greatly with the decreasing of the feed per tooth when the milling force is less than hmin. Next, detail discussions of the size effect will be presented. The effect of material strengthening behaviours and hmin on the milling force will be studied.
5.3. Effect of material strengthening behaviours on the size effect To reveal the influences of size effect of material behaviours, simulations with and without consideration of material strengthening behaviours were performed. Fig. 15 shows plots of the milling forces versus feed per tooth with and without material strengthening behaviours when DOC is 10 mm. It can be seen that the milling force predicted by the model with material strengthening behaviours matches well with the experimental data and can captures the size effect as discussed above. While for the results without the strengthening effect, the predicted milling force is much smaller than the experimental data. Fig. 16 shows the comparisons of the SSE between the results with and without considering size effect of material behaviours. It is observed that the SSE considering material strengthening behaviours is much larger. For the results without the strengthening effect, the SSE is nearly keeping constant with the decrease of ft. Therefore, it is concluded that the size effect of micro scale milling process is caused by the material strengthen behaviours at the micron level. However, the larger increase in the SSE when ft is smaller than hmin is still observed without considering the size effect. So there might be some other reasons for the larger increase at this condition.
Peak to Valley Force (N)
12
0.35 0.30
FyExp FySim Fy-Sim. w/o Size effect
0.25 0.20 0.15 0.10 0.05 0.00 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 feed per tooth (um)
Fig. 15. Comparisons of milling force between results with and without considering size effect.
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5.4. Effect of the minimum chip thickness on the size effect of the micro scale milling Similarly, in order to examine the effect of hmin, simulations without consideration of hmin were carried out. Fig. 17 shows plots of the milling forces versus feed per tooth with and without consideration of hmin. It can be seen that the milling forces predicted by the model with hmin agree well with the experimental data. For the results without hmin, the predicted milling force has a similar trend to the results with consideration of hmin, and is very close to the experimental data when ft is larger than hmin. From these comparisons, the conclusion is confirmed that the size effect of micro scale milling process is due to the material strengthen behaviours at the micron level. Fig. 18 shows the comparisons of the SSE with and without consideration of hmin. It can be seen that the increase in the SSE without considering hmin is much smaller than the experimental data. Therefore, the large
13
increase in the SSE when h is less than hmin is mainly contributed by the ploughing phenomenon and the accumulation of the actual chip thickness. Under the condition when ft is 0.2 or 0.4 mm, the chip is not formed after the first tool pass, and then the chip thickness is doubled for the second tool pass. The chip thickness would be accumulated until the actual chip thickness is larger that hmin, which will in turn causes the increase in the milling force and the SSE. 6. Conclusions 1. By applying SG plasticity, a constitutive equation was formulated to model the material strengthening behaviours. This flow stress can describe not only the characteristics of micro scale machining, size effect and minimum chip thickness, but the features of the macro scale process, including large strains, high strain rates and temperatures.
SSE-Exp.
SSE-Exp. SSE-Sim. SSE-Sim. w/o hmin
SSE-Sim.
45
SSE-Sim. w/o size effect
45
hmin
Specific Shear Energy (GPa)
Specific Shear Energy (GPa)
40 35 30 25 20 15 10
40
hmin
35 30 25 20
5 15 0 0.2
0.4
0.6 0.8 1.0 feed per tooth (um)
1.2
1.4
Peak to Valley Force (N)
Fig. 16. Comparisons of SSE between results with and without considering size effect.
0.30
Fx-Exp. Fx-Sim. Fx-Sim. w/o hmin
0.25 0.20 0.15 0.10 0.05 0.2 0.4 0.6 0.8 1.0 1.2 1.4 feed per tooth (um)
0.0
0.2
0.4
0.6 0.8 1.0 feed per tooth (um)
1.2
1.4
Fig. 18. Comparisons of SSE between results with and without considering hmin.
0.35 Peak to Valley Force (N)
0.0
0.30
Fy-Exp. Fy-Sim. Fy-Sim. w/o hmin
0.25 0.20 0.15 0.10 0.05 0.2 0.4 0.6 0.8 1.0 1.2 1.4 feed per tooth (um)
Fig. 17. Comparisons of milling force between results with and without considering hmin.
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2. An analytical micro scale milling force model is developed based on FE simulations using cutting principles and slip line theory. The model is well validated through the micro scale milling experiments with the miniaturized machine tool system. 3. Chip formation is investigated using FE simulations. The micro cutter edge radius is concluded to be the cause of the minimum chip thickness and is proposed to be 0.25 times of cutter edge radius for OFHC copper when the edge radius is 2 mm and tool rake angle is 101. 4. From the analysis of size effect, it is indicated that size effect in micro scale machining is caused by the size effect of material behaviours at the micron level and could be well formulated by SG plasticity. 5. SSE will increase greatly when the chip thickness is smaller than minimum chip thickness due to the ploughing phenomenon and the accumulation of the actual chip thickness.
Acknowledgements The support of National Natural Science Foundation of China under Grant #50575134, National Basic Research Program of China under Grant no.2005CB724100 and the Programme of Introducing Talents of Discipline to Universities under Grant no. B06012 are gratefully acknowledged. References [1] M.C. Shaw, Precision finishing, CIRP Annals 44 (1) (1995) 343–348. [2] C.-J. Kim, J. Rhett Mayor, J. Ni, A static model of chip formation in micro scale milling, ASME Journal of Manufacturing Science and Engineering 126 (2004) 710–718. [3] X. Liu, M.P. Vogler, S.G. Kapoor, R.E. DeVor, K.F. Ehmann, R. Mayor, C.-J. Kim, J. Ni, Micro-endmilling With meso-machine-tool system, NSF Design, Service and Manufacturing Grantees and Research Conference Proc., Dallas, TX 1 (2004) 1–9. [4] M.P. Vogler, R.E. Devor, S.G. Kapoor, Microstructure-level force prediction model for micro-milling of multi- phase materials, ASME Journal of Manufacturing Science and Engineering 125 (2004) 202–209. [5] N. Fang, Slip-line modelling of machining with a rounded-edge tool—Part I: new model and theory, Journal of Mechanics and Physics of Solids 51 (2003) 715–742. [6] J. chae, S.S. Park, T. Freiheit, Investigation of micro-cutting operations, International Journal of Machine Tools and Manufacture 46 (2006) 313–332. [7] D.A. Lucca, R.L. Rhorer, R. Komanduri, Energy dissipation in the ultraprecision machining of copper, CIRP Annals 40 (1991) 69–72. [8] E.M. Kopalinsky, P.L.B. Oxley, Size effects in metal removal processes, Institute of Physics Conference Series 70 (1984) 389–396.
[9] K. Nakayama, K. Tamura, Size effect in metal-cutting force, ASME Journal of Engineering for Industry 90 (1968) 119. [10] N.A. Fleck, G.M. Muller, M.F. Ashby, J.W. Hutchinson, Strain gradient plasticity: theory and experiments, Acta Metallurgica et Materialia 42 (2) (1994) 475–487. [11] J.S. Stolken, A.G. Evans, A microbend test method for measuring the plasticity length scale, Acta Materialia 46 (14) (1998) 5109–5115. [12] N.A. Fleck, J.W. Hutchinson, A phenomenological theory for strain gradient effects in plasticity, Journal of Mechanics and Physics Solids 41 (1993) 1825–1857. [13] C.-S. Han, H.J. Gao, Y.G. Huang, W.D. Nix, Mechanism-based strain gradient crystal plasticity—I. Theory, Journal of the Mechanics and Physics of Solids 53 (2005) 1188–1203. [14] S.S. Joshi, S.N. Melkote, An explanation for the size effect in machining using strain gradient plasticity, ASME Journal of Manufacturing Science and Engineering 126 (2004) 679–684. [15] G.R. Johnson, W.H. Cook, A constitutive model and data for metals subjected to large strains, high strain rates and high temperature, in: Proceedings of the Seventh International Symposium on Ballistics, The Haque, The Netherlands, 1983, pp. 541–547. [16] G.R. Johnson, Fracture characteristics of three metals subjected to various strains, strain rates, temperatures and pressures, Engineering Fracture Mechanics 20 (1) (1985) 31–48. [17] M.C. Shaw, Metal Cutting Principles, Clarendon Press, 1984. [18] T. Ozel, E. Zeren, Determination of work material flow stress and friction for FEA of machining using orthogonal cutting tests, Journal of Materials Processing Technology 153–154 (2004) 1019–1025. [19] C.-J. Kim, Mechanism of the chip formation and cutting dynamic of the micro scale milling process, Ph.D. Dissertation of the University of Michigan, 2004. [20] W.F. Hastings, P. Mathew, P.L. Oxley, Machining theory for predicting chip geometry, cutting forces, etc. from work materials properties and cutting conditions, Proceedings of The Royal Society of London, Series A: mathematical and Physical Sciences 371 (1980) 569–587. [21] D.J. Waldort, R.E. DeVor, S.G. Kapoor, A slip line field for ploughing during orthogonal cutting, Journal of Manufacturing Science and Engineering 120 (1998) 693–699. [22] M.T. Zaman, A. Senthil Kumar, M. Rahman, S. Sreeram, A threedimensional analytical cutting force model for micro end milling operation, International Journal of Machine Tools and Manufacture 46 (3–4) (2006) 353–366. [23] W.Y. Bao, I.N. Tansel, Modelling micro-end-milling operations. Part I: analytical cutting force model, International Journal of Machine Tools and Manufacture 40 (15) (2000) 2155–2173. [24] S. Filiz, C.M. Conley, M.B. Wasserman, O.B. Ozdoganlar, An experimental investigation of micro machinability of copper 101 using tungsten carbide micro endmill, International Journal of Machine Tools and Manufacture 47 (2007) 1088–1100. [25] K.W. Kim, W.Y. Lee, H.C. Sin, A finite element analysis for the characteristics of temperature and stress in micro-machining considering the size effect, International Journal of Machine Tools and Manufacture 39 (1999) 1507–1524. [26] T.A. Dow, E.L. Miller, K. Garrard, Tool force and deflection compensation for small milling tools, 2004, Precision Engineering 28 (2004) 31–45.
International Journal of Machine Tools & Manufacture 48 (2008) 1–14 www.elsevier.com/locate/ijmactool
Modelling and analysis of micro scale milling considering size effect, micro cutter edge radius and minimum chip thickness Xinmin Laia,, Hongtao Lia, Chengfeng Lia, Zhongqin Lina, Jun Nib a
State Key laboratory of Vibration, Shock and Noise, Shanghai Jiaotong University, Shanghai 200030, China b Shien-Ming Wu Manufacturing Research Center, University of Michigan, MI 48109, USA Received 7 April 2007; received in revised form 2 August 2007; accepted 8 August 2007 Available online 22 August 2007
Abstract This paper presents mechanisms studies of micro scale milling operation focusing on its characteristics, size effect, micro cutter edge radius and minimum chip thickness. Firstly, a modified Johnson–Cook constitutive equation is formulated to model the material strengthening behaviours at micron level using strain gradient plasticity. A finite element model for micro scale orthogonal machining process is developed considering the material strengthening behaviours, micro cutter edge radius and fracture behaviour of the work material. Then, an analytical micro scale milling force model is developed based on the FE simulations using the cutting principles and the slip-line theory. Extensive experiments of OFHC copper micro scale milling using 0.1 mm diameter micro tool were performed with miniaturized machine tool, and good agreements were achieved between the predicted and the experimental results. Finally, chip formation and size effect of micro scale milling are investigated using the proposed model, and the effects of material strengthening behaviours and minimum chip thickness are discussed as well. Some research findings can be drawn: (1) from the chip formation studies, minimum chip thickness is proposed to be 0.25 times of cutter edge radius for OFHC copper when rake angle is 101 and the cutting edge radius is 2 mm; (2) material strengthening behaviours are found to be the main cause of the size effect of micro scale machining, and the proposed constitutive equation can be used to explain it accurately. (3) That the specific shear energy increases greatly when the uncut chip thickness is smaller than minimum chip thickness is due to the ploughing phenomenon and the accumulation of the actual chip thickness. r 2007 Elsevier Ltd. All rights reserved. Keywords: Micro scale milling process; Size effect; Minimum chip thickness; Cutter edge radius; Strain gradient
1. Introduction Recent years, the production of miniaturized components with complex small features is gaining increasing importance due to the trend of miniaturization which is determining the development of products for various industries, such as biomedical instruments, electronic products, defence industry and so on. Most of these components fall into the scales from 10 mm to 1 mm known as micro/meso scale in mechanical engineering. Considered as one of the most effective techniques, micro scale milling process can be used to fabricate these components with complex micro features over a wide range of material types. Corresponding author.
E-mail address: [email protected] (X. Lai). 0890-6955/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmachtools.2007.08.011
However, further advances in both the efficiency and the quality are limited by the incomplete understanding of its basic mechanisms. To satisfy the increasing need of miniaturized manufacturing, the mechanisms studies are becoming more and more important. Micro scale milling is not simply downsized from the conventional operation but has its own characteristics, such as size effect, cutter edge radius and minimum chip thickness. Shaw [1] studied the effect of round edges on the chip formation in micro scale machining and stated that the plastic deformations would be prevented when the cutter edge radius is relatively larger than the uncut chip thickness. Kim [2] investigated the effect of static tool deflection on the micro milling and proposed a static chip model based on the attainable micro scale machining force data. Ni [3] investigated the chip formation using molecule
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Nomenclature s plastic equivalent stress (MPa) t shear strength (MPa) e effective strain _ effective strain rate reference strain rate _ 0 pl _¯ plastic strain rate A, B, C, m, n coefficients of the Johnson–Cook model p static pressure (MPa) q Von mises stress (MPa) ¯ pl plastic strain at failure f D¯pl the increment of the equivalent plastic strain Tmelt melting temperature of the workpiece T0 transition temperature of the workpiece w damage criterion d1 d2 d3 d4 d5 coefficients of Johnson–Cook material shear failure criterion E Young’s module n Poisson ratio r density of work material rs density of statistically store dislocation rg density of geometry necessary dislocation Z strain gradient l scale variable G shear modulus (MPa) b Burgers vector (nm)
dynamic (MD) simulations and presented an approach to calculate the minimum chip thickness by identifying a local maximum in the radial thrust forces in the micro milling. Vogler [4] determined the minimum chip thickness for steel using finite element (FE) simulations with regard of the microstructure properties of work material. According to their research, the critical chip thickness is 0.2 and 0.3 times of the edge radius for pearlite and ferrite, respectively. In addition, several analytical cutting force models ([5,22,23,26]) had been developed to investigate the micro scale machining process. Chae [6] surveyed the state of art of micro scale machining and reported that current researches had made valuable attempts in this field though most of them were carried out by means of nano-level and macro-level approaches. From the discussions above, it is obviously that the processing system for micro scale milling is far from established. Its characteristics need to be studied and the related mechanisms need to be revealed through experiments and theoretical modelling. One of the most significant characteristics of the micro scale milling operation is the size effect. Some efforts have been carried out to explain it. Lucca [7] investigated the size effect of cutting energy of micro scale machining process through the experiments; they found nonlinear increase in specific cutting energy or cutting forces as the uncut chip thickness was decreased. Kopalinsky and Oxley [8] studied the size effect with sharp tools by turning tests. They
h, hmin uncut chip thickness and minimum chip thickness L length of primary machining deformation zone f the coefficient of friction between the workpiece and chip V rotation speed of the spindle j shear angle a rake angle R cutter edge radius g clear angle k FE predicted stress along the primary shear zone AB, CA, Rsl length variables in slip line model [21] wsl width of the workpiece in slip line model [21] asl, ysl, gsl, Zsl, rsl angle variables in slip line model [21] Scut, Pcut, Sthr, Pthr shear force and ploughing force in slip line model [21] Fc, FT, cutting force and thrust force Fx, Fy, milling forces in x- and y-direction KC, KT, wc2, wt2 coefficients of force model when h is less than hmin PC, PT, wc1, wt1 coefficients of force model when h is greater than hmin y angle of the milling tool position ft feed per tooth ap depth of cut
concluded that the cause was the decrease in the tool chip interface temperature. Nakayama and Tamura [9] analyzed the size effect through experiments performed at a very low cutting speed to minimize the temperature and strain rate effects. They attributed this effect to plastic flow in the workpiece subsurface. The experiments of the literature (Oxley, [8]; Nakayama, [9]) implied that there should be other underlying mechanisms for the size effect besides cutter edge radius, temperatures and strain rate. On the other hand, a similar size effect in micro indentation tests was found in the mechanics studies, which was shown as remarkable material strengthening behaviours at the micron level. Fleck [10] found that the torsion strengthened three times when the diameter of the sample was decreased from 170 to 12 mm. Stolken and Evans [11] found the similar trend in bending material tests. The bending strength increased significantly when the thickness of thin beam was decreased from 100 to 12.5 mm. This size effect in material properties has turned to be the focus of mechanics research recent years. Strain gradient (SG) plasticity is the most effective method to interpret this phenomenon. Fleck [12] developed SG theory by introducing the material inner variable. Then, Nix and Gao [13] improved it by describing the concept of inner variable and mechanism-based strain gradient (MSG) theory was developed. From this point, it is supposed that there might be some similarities between
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the two size effect phenomena. The size effect of material behaviours may be the cause of size effect in micro scale machining. Melkote [14] attempted to analyze the size effect using SG plasticity based on analytical solutions. The results showed that it is capable of explaining the size effect. However, this model could not consider many other characteristics of micro scale machining process, such as large strain, high strain rate, cutter edge radius, minimum chip thickness and so on. Therefore, further studies are needed and more precise FE models should be developed. The goal of this research is to provide deeper understanding of micro scale milling process. Firstly, a modified Johnson–Cook (JC) constitutive equation is formulated using SG plasticity. A FE model for micro scale orthogonal machining process is developed considering material strengthening behaviours, tool geometry and fracture behaviour. Then, a milling force model is developed based on the FE simulations using the cutting principles and slip-line theory. Finally, chip formation and size effect of micro scale milling are investigated by applying the model, and the effects of the main characteristics of micro scale process, size effect of material behaviours, micro mill cutter radius and minimum chip thickness are discussed as well. The results demonstrate that the proposed approach for micro scale milling is capable of the analysis of micro scale milling process and provides the academic foundation for further process design and optimization studies.
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2. Micro scale milling process modelling Milling operation is a very complex process as shown in Fig. 1a. Its performance is affected by so many parameters that it is not possible to take all the factors into account at the same time. According to the analysis of micro scale milling process and its characteristics, two assumptions are made: (a) The objective of the first assumption is to simplify the complicated 3D milling process (Fig. 1a) to a 2D process shown as Fig. 1b. It is reasonable because the depth of cut (DOC) of the micro scale milling is very small, sometimes only several microns. Under these conditions, the helix angle of the milling process has so little effect that could be ignored and DOC could be considered constant during milling process. Therefore, the constant DOC in milling is represented by the thickness of the workpiece in the simplified 2D process. (b) The other assumption is made to build the relationship between the simplified 2D milling process and orthogonal machining process. As Fig. 1b shows, the deformation area is much smaller when compared with the tool and the workpiece since the diameter of the tool is usually several hundreds microns while the dimension of the machining zone is at the micron level. In this study, the feed per tooth is less than 2 mm. Therefore, the deformed area can be considered as kind of orthogonal machining process shown as Fig. 1c and
Fig. 1. Simplifications for micro scale milling process.
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d. For the orthogonal machining, uncut chip thickness is equivalent to the chip thickness of the milling process. Based on these two assumptions, modelling of micro scale milling process could be divided into three steps: the first is the material strengthening behaviours modelling. A modified JC model is formulated by introducing the SG theory. Then, based on the proposed material model, a FE model for orthogonal machining is developed with respect to micro cutter edge radius, chip separation criterion, and the friction between the tool and the chip. The last step is micro scale milling force modelling. The model is calibrated based on the FE simulates using the cutting principles and slip-line approach. 2.1. Material strengthening behaviours modelling 2.1.1. Guideline for the development of the constitutive equation Material model is the fundamental of FE simulations and is required to represent the features of material behaviours during the deformation process. Machining operations are associated with large strains, high strain rates and high temperatures. For micro scale machining process, size effect is also a dominant factor. Therefore, a material model capable of capturing these effects is highly necessary. Semi-empirical constitutive equations are widely applied in present FE simulations of cutting process [18]. Among them, the constitutive model proposed by JC [15] is one of the most widely used models. It provides good description of metal material behaviours undertaking large strains, high strain rate and temperatures. It describes the flow stress of work material with the product of strain, strain rate and temperature effects that are individually determined as T T0 m _ n s¯ JC ¼ ½A þ BðÞ 1 þ C ln 1 . T melt T 0 _ 0 (1) However, the conventional FE analysis is a non-dimensional process, where flow stress is independent to the scale variable. Therefore, it cannot be used to describe the size effect. In this research, a material model of micro scale machining is to be developed by introducing a size variable l into the conventional material model (JC model). The framework of the proposed constitutive equation is expressed as s ¼ f ðsconventional ; ZÞ,
(2)
where sconventional ¼ sJC, Z ¼ gl (l is the size variable, when l reaches to macro level: s ¼ sJC) The objective is to establish the relationship between the size variable l and flow stress, and keep the advantages of JC model as well. Next, the deduction of the constitutive equation will be presented in detail.
2.1.2. Deduction of the material model In the present approach, the work material is considered as a plastically isotropic material. The total strain rate tensor _ij , is given by the sum of the elastic strain rate _ eij , and the plastic strain rate _pij , as _ij ¼ _eij þ _pij .
(3)
The elastic part is given by Hooke’s law, while the plastic part is expressed by the Levy-von Mises equation: _pij ¼
3 _p S ij . 2s
(4)
Relating it to the deviatoric stress tensor Sij in terms of the von Mises equivalent plastic strain rate: rffiffiffiffiffiffiffiffiffiffiffiffi 2 p p p (5) _ _ . _ ¼ 3 ij ij Then, the Von Mises equivalent stress can be obtained: rffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 S ij Sij . s¼ (6) 2 From the SG plasticity, Taylor’s dislocation model gives the flow stress in terms of the dislocation density, as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi s ¼ 3aGb r ¼ 3aGb rs þ rg . (7) rs can be determined by the material test in the absence of strain gradient, pffiffiffiffiffi (8) s ¼ 3aGb rs ¼ sconventional . In order to properly estimate the total dislocation density, m is introduced: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (9) s ¼ 3aGb rms þ rmg . Substituting Eq. (8) into Eq. (9), the flow stress can be written as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m rg s ¼ sconventional 1 þ . (10) rs The density of geometrically necessary dislocation (GND) is given by rg ¼
2Z . b
(11)
Thus, the constitutive equation turns to be as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m 18 a2 G 2 bZ s ¼ sconventional 1 þ . (12) s2conventional As discussed above, sJC is selected to be sconventional, then, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m 18 a2 G 2 bZ . (13) s ¼ sJC 1 þ s2JC From the works of Melkote [14], the SG is obtained through the dislocation analysis of primary shear zone for
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micro scale machining, as follows: 1 . L So, the constitutive equation can be expressed as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mffi 18 a2 G2 b s ¼ sJC 1 þ . s2JC L Z¼
(14)
(15)
So far, the constitutive equation has been developed. Next, the work comes to the calculation of the length of the primary shear zone, L. The minimum chip thickness has significant impact on the micro scale machining operation, especially when the uncut chip thickness is close to micro cutter edge radius. On these occasions, the machining process can be divided into two situations: chip forms and no chip forms. Kim [19] studied this phenomenon using MD simulations, as shown in Fig. 2. For the situation that chip forms when h^hmin, as shown in Fig. 2a, the length of shear zone can be obtained by the cutting principles [17]: L¼
h sin ðfÞ
f¼
r cos an 1 r sin an
h r¼ . t
(16)
For the situation that no chip forms as shown in Fig. 2b, when hohmin, the shear angle becomes to be very small and chip thickness t does not exist. Therefore, Eq. (16) is not applicable any more. Through the analysis of shear zone, the arc length of contact part is proposed to be the length of primary shear zone. arccos Rh R pR L¼ . (17) 180 Hence, the constitutive equation can be expressed as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mffi 18 a2 G 2 b s ¼ sð; _; T; L; hmin Þ ¼ sJC 1 þ s2JC L T T0 m _ n sJC ¼ ½A þ BðÞ 1 þ C ln 1 . T melt T 0 _ 0 ð18Þ For aforementioned formulation, the scale variable is expressed as the shear zone length L. Since the SG is the
5
reciprocal of L, it will decrease with the increase of uncut chip thickness. Therefore, the GND will disappear for the large test piece and macro scale machining process. This fully satisfies the constraint condition that the flow stress is equal to the results by JC model when size variable reaches to macro level (Eq. (2)). From the aforementioned deduction and analysis, the proposed constitutive equation has two advantages: (a) This material model can describe the characteristics of micro machining, including the material strengthening behaviours and the minimum chip thickness. (b) It is based on JC model, which provides good fit for strain hardening behaviours of metals in macro scale machining process, such as large strains, high strain rates and high temperatures. 2.2. FE modelling for micro scale orthogonal machining process After establishing the constitutive equation, a FE model for orthogonal micro scale machining process is developed using a commercial FE analysis code, ABAQUS/EXPLICIT. In this study, the work material is OFHC copper, and the micro tool used is tungsten carbide micro end miller with the diameter of 0.1 mm. The cutting speed is 104.72 mm/s, which is calculated from the spindle rotation speed of 20,000 r/min. The parameters of material properties are listed in Table 1. Main considerations of micro scale machining FE modelling will be presented as follows. 2.2.1. Material behaviours Based on the constitutive equations deduced in Section 2.1, stress–strain curves of work material at different uncut chip thickness are calculated and shown as Fig. 3. The left one show the stress–strain curves when h is 1, 10 and 100 mm and when h reaches macro level (JC model). Size effect can be observed obviously. From this figure, we can also see that the stress approaches to that of JC model with the increase of uncut chip thickness. The right figure shows the stress–strain curves when h is 1, 4, 10 and 20 mm. Same trend can be found from this figure with the left one. The
Fig. 2. Chip formation with regard of cutter edge radius. (a) Chip formed; (b) no chip formed.
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Table 1 Parameters for work material OFHC copper Parameter
Value
Mechanical properties [15]
r (kg/m3) E (MPa) u Tmelt (1C) T0 (1C)
8960 124 0.34 1356 293
Johnson Cook model [15]
A (MPa) B (MPa) n C m
90 292 0.31 0.025 1.09
direction of decrease in h
Value
Fracture [16]
d1 d2 d3 d4 d5
0.54 4.89 3.03 0.014 1.12
Plastic strain gradient [14]
a G (GPa) b (nm) m
0.5 39 0.256 0.38
direction of decrease in h
600
500 400 300 Johnson-Cook h=1um h=10um h=100um
200 100 0
Stress (MPa)
Stress (MPa)
600
Parameter
500 400 300 Johnson Cook h=1um h=4um h=10um h=20um
200 100 0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Strain (%)
Strain (%) Fig. 3. Material properties.
Fig. 4. SEM picture of micro mill cutter.
material model is integrated into solution algorithm using ABAQUS/EXPLICIT VUMAT. 2.2.2. Micro cutter edge radius Edge radius of micro cutter is one of the most significant features of micro scale machining process. It is considered to be the main cause of minimum chip thickness by some researches. The edge radius of 0.1 mm diameter 2-flute micro-end-miller is measured by SEM shown as Fig. 4. The cutter edge radius is obtained to be about 2 mm.
2.2.3. Chip separation criterion Chip separation criterion is another critical issues in FE simulations of cutting process. Two approaches were usually used in precious researches: pre-defined separation path and adaptive remeshing technique. However, both of them obviate the fracture mechanics behaviours of the workpiece. In this model, the chip separation is modelled with the shear failure of the material. JC shear failure criterion is utilized shown as Eq. (19). During the solving process, once the equivalent plastic strain exceeds the
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Input the material parameters and state variables (On-1, n-1, d n, A, B, C, m, n, d1, d2, d3, d4, d5)
Calculate the trial stress increment d On = D dn Calculate the trial stress O ntrial = On-1+ d On Calculate the yield stress according to the proposed material model
Deform state
O ntrial < O yield Elastic deform
O ntrial > O yield Plastic deform Calculate equivalent strain increment d pl
Calculate the total stress
Calculate the equivalent strain
Calculate the critical strain according to the shear failure criterion
n> The shear failure state
n-1
<
pl
Set the element state variable to zero element deletion
pl
Update the strain, stress, inelastic specific energy and the state variables Fig. 5. Flow chart of the ABAQUS/VUMAT program.
critical value, the element is considered to be failed and will be deleted in the next solver step. The criterion is applied using ABAQUS/ EXPLICIT VUMAT. The flow chart is shown as Fig. 5. h pl ih
i 8 pl d 3 pq TT 0 _¯ > ¼ ½d þ d e 1 þ d ln 1 þ d ¯ > 1 2 4 5 T melt T 0 _ 0 < f . P pl D¯ > > ¼1 pl :o ¼ ¯ f
(19) 2.2.4. Friction at the tool–chip interface The contact and the friction between the chip and the tool are influenced by factors such as cutting speed, feed rate, rake angle, etc. Due to very limited research has been carried out for micro friction at the tool chip interface, coefficient of friction f of 0.3 was used in this model [25]. 2.3. Micro scale milling force modelling As outlined at the beginning of Section 2, the last step of the modelling is to build the analytical micro scale milling
force model. It could be divided into two parts: first is to calculate the machining force using slip line approach. Second is to calibrate the analytical milling force model based on these results. 2.3.1. Micro scale machining force calculation Slip line methods are widely used for the calculation of the machining forces [20]. The slip line model developed by Waldort [21] is one of the most accurate models and can take the ploughing phenomenon and cutter radius into account. This model is used to calculate the machining force from the FE results. The descriptions of the model and its parameters can be found in the Ref. [21]. According to this model, the deformation zone of the orthogonal cutting process is shown in Fig. 6. The cutting forces and ploughing forces respectively in the cutting and thrust direction can be calculated by ( S cut ¼ kwsl ½cos ðf þ ð1 þ 2ysl Þ sin ðfÞAB , (20) S thr ¼ kwsl ½ð1 þ 2ysl Þ cos ðfÞ sin ðfÞAB
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8 Pcut ¼ kwsl ½cos ð2Zsl Þ cos ðf gsl þ Zsl Þ > > > > < þð1 þ 2ysl þ 2gsl þ sin ð2Zsl ÞÞ sin ðf gsl þ Zsl ÞCA , Pthr ¼ kwsl ½1 þ 2ysl þ 2gsl þ sin ð2Zsl Þ > > > > : cos ðf g þ Z Þ cos ð2Z Þ sin ðf g þ Z ÞCA sl
sl
sl
sl
Based on aforementioned calculation, the relationship between the machining forces and the FE simulations is established.
sl
(21) where Rsl , sin ðZsl Þ Rsl ¼ sin ðZsl Þ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! u p a pffiffi2ffiR sin ðr Þ 2 u sl sl t sl þ þ 2½Rsl sin ðrsl Þ2 . þ re tan 2 4 tan p2 þ asl
CA ¼
Hence, the total forces in cutting and thrust direction could be obtained by ( F C ¼ S cut þ Pcut . (22) F T ¼ S thr þ Pthr
2.3.2. Micro scale milling force model calibration The second part is micro scale milling force model calibration. For micro scale operation, the minimum chip thickness has a significant influence on the milling force. Therefore, the analytical force model should be formulated considering when there is chip formed and no chip formed respectively, as Eq. (23): (1) When hphmin, (
F T ¼ PT hwt1 ðyÞap
Fc
t chip
R Pthr
F
A
Pcut C
Scut
Rsl
B
Sthr
V
Y
ha
FT Fig. 6. Slip line model [21].
.
(2) When h4hmin, (
Tool
F C ¼ PC hwc1 ðyÞap
F C ¼ K C hwc2 ðyÞap . F T ¼ K T hwt2 ðyÞap
(23)
Where chip thickness h(y) ¼ ft sin y, PC, PT, wc1, wt1, KC, KT, wc2, wt2 are coefficients for the force model, hmin is obtained from FE simulations, and is proposed to be 0.5 mm, 0.25 times of the micro cutter edge radius for OFHC copper. The chip formation will be discussed in detail in Section 5.1. The parameters for the micro scale milling force model calibration are seen in Table 2. In the table, Fc and Ft were calculated using Waldorf’s method presented in Section 2.3.1. Then, PC, PT, wc1, wt1 were calibrated to be 741.6, 88.5, 0.7536 and 0.3459, respectively, when h is smaller than hmin, and KC, KT, wc2, wt2 are 553.5, 360.4, 0.5879 and 0.4217, respectively, when h is larger than hmin. The transforms of cutting force and
Table 2 Parameters for calibration of micro scale milling force Parameters of micro tool
Process parameters
Hohmin H4hmin
Diameter (mm)
Number of teeth
Rake angle (deg)
0.1
2
10
Work material
RPM (r/min)
ap (mm)
OFHC copper
20,000
10
Feed per tooth (mm)
FC (N)
FT (N)
0.2 0.4 0.6 0.8 1 1.4
0.0121 0.0204 0.0697 0.0837 0.0971 0.1154
0.0465 0.0591 0.1449 0.1870 0.2087 0.2165
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W ) HC OF e( iec kp or
Rotation
Fy
Fx Fy Milling tool Fx
Ft
Fz Fc
n tio ec dir ed Fe
Fz
Feed direction Fig. 7. Micro scale milling force.
thrust force to the forces in X- and Y-direction is shown as Fig. 7. Therefore, the machining forces can be transformed to the milling forces in X- and Y-direction as given by ( F x ¼ F C cos y F T sin y . (24) F y ¼ F C sin y þ F T cos y
3. Experimental details
Fig. 8. The miniaturized machine tool.
Cutting force dynamometer (Kistler 9256 C1) Workpiece Micro end miller Spindle Y-stage
Adapter
To perform the experiments needed for the research, a miniaturized machine tool is established, as shown in Fig. 8. The overall volume of the machine tool is 270 190 220 mm, and the working volume is 30 30 30 mm. Fig. 9 shows the components of the miniaturized machine tool system. It consists of five subsystems: the first is the positioning subsystem, consisting of 3 high precision stages with the resolution of 50 nm; the second is the NSK electronic spindle subsystem with maximal rotation speed of 80,000 r/min; the third is motion control subsystem. A PMAC motion controller is used to control the motion of the stages; the fourth is the micro end miller; and the fifth is the cutting force measurement subsystem, including the force dynamometer and its data acquiring instruments. The work material used in the experiments was OFHC copper 101, which is widely applied for optics, laser, and micro-electronics field. The tools used were all 0.1 mm diameter micro end millers. For the micro scale milling process, the cutting forces are usually as small as several hundreds of micro Newton [24]. Moreover, its frequency is extremely high due to the high rotation speed of the spindle. For example, it is 667 Hz when the rotation speed of the spindle is 20,000 r/min and 2667 Hz for the maximum rotation speed 80,000 r/min. Both of the magnitude and frequency require a precise multi-component cutting force sensor with very high precision. A 5component Kistler dynamometer 9256C1 with a 2 mN
X-stage
Spindle support Z-stage Base plate
X Y Z Motor drivers
PMAC motion control card
Unigraphics/CAM
Amplifier
NI data inquiring card
Data inquiring software
Fig. 9. Components of the miniaturized machine tool system.
threshold was used to measure the milling forces. Its natural frequency is 5.1, 5.5and 5.6 kHz in X-, Y-, Zdirection, respectively.
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4. Model validation To use the proposed model to predict the forces and analyze micro scale milling process, extensive experiments are needed to validate the model over a widely range of conditions. In this section, experiments and simulations were performed to examine DOC and ft in the applicable conditions range of the miniaturized machine tool and the micro cutters. Four sets of correlations were done when DOC was 6, 10, 15 and 20 mm, respectively. For each set, ft was selected to be 0.2, 0.4, 0.6, 0.8, 1 and 1.4 mm. The experimental conditions are shown in Table 3. Fig. 10 shows the comparisons between the simulated and experimental milling forces in X- and Y-direction. The experimental data were measured with the Kistler dynamTable 3 Experimental conditions for model validation Parameter
Value
Micro-end-mill tool Material Teeth number Diameter (mm) Workpiece material OFHC copper 101 Milling conditions RPM of spindle (rev/min) ft (mm) Depth of cut (mm)
Peak to Valley Force (N)
0.20 0.15
ometer and the data sample rate was 10,000 Hz. Average peak to valley forces are chosen to examine the proposed model. The experimental peak to valley forces were calculated for each revolution and averaged over the milling process for every experiment. Figs. 10a–d show the comparisons of the simulated and experimental results when DOC is 6, 10, 15 and 20 mm, respectively. In general, the micro scale forces are very small, only several hundreds of micro Newton. The forces increase with feed per tooth, and the force in Y-direction is a little larger than that in X-direction. In contrast, good agreements both in the magnitudes and trends can be seen from the comparisons between the extensive predicted and the measured milling force at all ft and DOC. In addition, the following experimental trends are also well predicted by the model. The peak to valley forces are increased with nearly proportional to the DOC. Based on these analysis, the model is considered to be satisfactory validated and can be used to analyze the micro scale milling process.
Tungsten carbide 2 0.1
5. Results and discussions 5.1. Chip formation analysis of micro scale machining
20,000 0.2, 0.4, 0.6, 0.8, 1, 1.4 6, 10, 15, 20
In this section, chip formation of micro scale machining was investigated through the FE simulations. The goal is to
0.35 Fx-Exp. Fy-Exp. Fx-Sim. Fy-Sim.
0.10 0.05
Peak to Valley Force (N)
10
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
0.20 0.15 0.10 0.05
0.7 Fx-Exp. Fy-Exp. Fx-Sim. Fy-Sim.
0.3 0.2 0.1
Peak to Valley Force (N)
Peak to Valley Force (N)
feed per tooth (um)
0.4
0.25
Fx-Exp. Fy-Exp. Fx-Sim. Fy-Sim.
0.00 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 feed per tooth (um)
0.00
0.5
0.30
0.6
Fx-Exp. Fy-Exp. Fx-Sim. Fy-Sim.
0.5 0.4 0.3 0.2
0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 feed per tooth (um)
0.2 0.4 0.6 0.8 1.0 1.2 1.4 feed per tooth (um)
Fig. 10. Comparisons of milling forces between the experimental and predicted results. (a) DOC ¼ 6 mm; (b) DOC ¼ 10 mm; (c) DOC ¼ 15 mm; (d) DOC ¼ 20 mm.
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find out hmin and its relationship with the micro cutter edge radius. The uncut chip thickness for micro scale orthogonal machining was selected to be 0.1 R (0.2 mm), 0.2 R (0.4 mm), 0.3 R (0.6 mm). As Fig. 11 shows, it can be observed that there is no chip formed when h is 0.1 and 0.2 R, while chip forms when the h is 0.3 R. From these results, hmin is proposed to be 0.25 R (0.5 mm) for OFHC copper when cutter edge radius is 2 mm and rake angle is 101. The results are consistent with some literatures. For example, Vogler [4] estimated the minimum chip thickness to edge radius ratios of pearlite and ferrite to be 0.20 and 0.3, respectively. Kim [2] proposed that the minimum chip
11
thickness to be 30% of the cutting edge radius by means of MD simulations and experiments. It illuminates that the cutting edge radius is the main cause of the minimum chip thickness. The results demonstrate that the proposed model is capable of the chip formation studies of micro machining by applying the proposed material model and the shear failure fracture criterion. 5.2. Size effect analysis of micro scale machining Many simulations were conducted applying the proposed FE model to examine the size effect of micro scale
Fig. 11. Chip formation process. (a) h ¼ 0.1 R (0.2 mm); (b) h ¼ 0.2 R (0.4 mm); (c) h ¼ 0.3 R (0.6 mm).
Fig. 12. Results when uncut chip thickness is 10 mm. (a) Without consideration of size effect; (b) with consideration of size effect.
Fig. 13. Size effect at different uncut chip thickness. (a) h ¼ 1 mm; (b) h ¼ 4 mm (c) h ¼ 20 mm.
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milling process. Fig. 12 shows the comparisons between the results with and without consideration of material strengthening effect when uncut chip thickness is 10 mm. The left figure shows the predicted result which material strengthening behaviour was not considered, while the right one was the result considering the size effect. From the comparisons, it can be seen that the stress distribution is similar to each other. However, the maximum effective stress with regard of size effect (764 MPa) is much higher than the result without considering size effect (544 MPa). It illuminates that the size effect is considerably remarkable when uncut chip thickness is 10 mm. Moreover, many simulations were also performed to examine the size effect under different uncut chip thickness, 1, 4 and 20 mm. It can be seen from Fig. 13 that maximum effective stress increase as the uncut chip thickness is decreased, which is 1009, 888, 764 and 704 MPa when uncut chip thickness is 1, 4, 10 and 20 mm, respectively. From the analysis, it is indicated that the size effect in micro scale machining could be well explained by SG plasticity.
hmin
Exp. Sim.
40 35 30 25 20 15 0.2
0.4
0.6
0.8
1.0
1.2
1.4
feed per tooth (um) Fig. 14. Size effect in specific shear energy at different ft.
Peak to Valley Force (N)
Specifics hear energy (GPa)
45
0.30
FxExp FxSim Fx-Sim w/o Size effect
0.25 0.20 0.15 0.10 0.05 0.00 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 feed per tooth (um)
From the experimental data, size effect is also observed. As shown in Fig. 14, the SSE is calculated by dividing amplitude of the in-plane milling force by the product of ft and DOC. From the results, the higher SSE is observed for the smaller feed per tooth. These results correspond to the well-known size effect. In addition, another trend is also found that the SSE increases greatly with the decreasing of the feed per tooth when the milling force is less than hmin. Next, detail discussions of the size effect will be presented. The effect of material strengthening behaviours and hmin on the milling force will be studied.
5.3. Effect of material strengthening behaviours on the size effect To reveal the influences of size effect of material behaviours, simulations with and without consideration of material strengthening behaviours were performed. Fig. 15 shows plots of the milling forces versus feed per tooth with and without material strengthening behaviours when DOC is 10 mm. It can be seen that the milling force predicted by the model with material strengthening behaviours matches well with the experimental data and can captures the size effect as discussed above. While for the results without the strengthening effect, the predicted milling force is much smaller than the experimental data. Fig. 16 shows the comparisons of the SSE between the results with and without considering size effect of material behaviours. It is observed that the SSE considering material strengthening behaviours is much larger. For the results without the strengthening effect, the SSE is nearly keeping constant with the decrease of ft. Therefore, it is concluded that the size effect of micro scale milling process is caused by the material strengthen behaviours at the micron level. However, the larger increase in the SSE when ft is smaller than hmin is still observed without considering the size effect. So there might be some other reasons for the larger increase at this condition.
Peak to Valley Force (N)
12
0.35 0.30
FyExp FySim Fy-Sim. w/o Size effect
0.25 0.20 0.15 0.10 0.05 0.00 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 feed per tooth (um)
Fig. 15. Comparisons of milling force between results with and without considering size effect.
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5.4. Effect of the minimum chip thickness on the size effect of the micro scale milling Similarly, in order to examine the effect of hmin, simulations without consideration of hmin were carried out. Fig. 17 shows plots of the milling forces versus feed per tooth with and without consideration of hmin. It can be seen that the milling forces predicted by the model with hmin agree well with the experimental data. For the results without hmin, the predicted milling force has a similar trend to the results with consideration of hmin, and is very close to the experimental data when ft is larger than hmin. From these comparisons, the conclusion is confirmed that the size effect of micro scale milling process is due to the material strengthen behaviours at the micron level. Fig. 18 shows the comparisons of the SSE with and without consideration of hmin. It can be seen that the increase in the SSE without considering hmin is much smaller than the experimental data. Therefore, the large
13
increase in the SSE when h is less than hmin is mainly contributed by the ploughing phenomenon and the accumulation of the actual chip thickness. Under the condition when ft is 0.2 or 0.4 mm, the chip is not formed after the first tool pass, and then the chip thickness is doubled for the second tool pass. The chip thickness would be accumulated until the actual chip thickness is larger that hmin, which will in turn causes the increase in the milling force and the SSE. 6. Conclusions 1. By applying SG plasticity, a constitutive equation was formulated to model the material strengthening behaviours. This flow stress can describe not only the characteristics of micro scale machining, size effect and minimum chip thickness, but the features of the macro scale process, including large strains, high strain rates and temperatures.
SSE-Exp.
SSE-Exp. SSE-Sim. SSE-Sim. w/o hmin
SSE-Sim.
45
SSE-Sim. w/o size effect
45
hmin
Specific Shear Energy (GPa)
Specific Shear Energy (GPa)
40 35 30 25 20 15 10
40
hmin
35 30 25 20
5 15 0 0.2
0.4
0.6 0.8 1.0 feed per tooth (um)
1.2
1.4
Peak to Valley Force (N)
Fig. 16. Comparisons of SSE between results with and without considering size effect.
0.30
Fx-Exp. Fx-Sim. Fx-Sim. w/o hmin
0.25 0.20 0.15 0.10 0.05 0.2 0.4 0.6 0.8 1.0 1.2 1.4 feed per tooth (um)
0.0
0.2
0.4
0.6 0.8 1.0 feed per tooth (um)
1.2
1.4
Fig. 18. Comparisons of SSE between results with and without considering hmin.
0.35 Peak to Valley Force (N)
0.0
0.30
Fy-Exp. Fy-Sim. Fy-Sim. w/o hmin
0.25 0.20 0.15 0.10 0.05 0.2 0.4 0.6 0.8 1.0 1.2 1.4 feed per tooth (um)
Fig. 17. Comparisons of milling force between results with and without considering hmin.
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2. An analytical micro scale milling force model is developed based on FE simulations using cutting principles and slip line theory. The model is well validated through the micro scale milling experiments with the miniaturized machine tool system. 3. Chip formation is investigated using FE simulations. The micro cutter edge radius is concluded to be the cause of the minimum chip thickness and is proposed to be 0.25 times of cutter edge radius for OFHC copper when the edge radius is 2 mm and tool rake angle is 101. 4. From the analysis of size effect, it is indicated that size effect in micro scale machining is caused by the size effect of material behaviours at the micron level and could be well formulated by SG plasticity. 5. SSE will increase greatly when the chip thickness is smaller than minimum chip thickness due to the ploughing phenomenon and the accumulation of the actual chip thickness.
Acknowledgements The support of National Natural Science Foundation of China under Grant #50575134, National Basic Research Program of China under Grant no.2005CB724100 and the Programme of Introducing Talents of Discipline to Universities under Grant no. B06012 are gratefully acknowledged. References [1] M.C. Shaw, Precision finishing, CIRP Annals 44 (1) (1995) 343–348. [2] C.-J. Kim, J. Rhett Mayor, J. Ni, A static model of chip formation in micro scale milling, ASME Journal of Manufacturing Science and Engineering 126 (2004) 710–718. [3] X. Liu, M.P. Vogler, S.G. Kapoor, R.E. DeVor, K.F. Ehmann, R. Mayor, C.-J. Kim, J. Ni, Micro-endmilling With meso-machine-tool system, NSF Design, Service and Manufacturing Grantees and Research Conference Proc., Dallas, TX 1 (2004) 1–9. [4] M.P. Vogler, R.E. Devor, S.G. Kapoor, Microstructure-level force prediction model for micro-milling of multi- phase materials, ASME Journal of Manufacturing Science and Engineering 125 (2004) 202–209. [5] N. Fang, Slip-line modelling of machining with a rounded-edge tool—Part I: new model and theory, Journal of Mechanics and Physics of Solids 51 (2003) 715–742. [6] J. chae, S.S. Park, T. Freiheit, Investigation of micro-cutting operations, International Journal of Machine Tools and Manufacture 46 (2006) 313–332. [7] D.A. Lucca, R.L. Rhorer, R. Komanduri, Energy dissipation in the ultraprecision machining of copper, CIRP Annals 40 (1991) 69–72. [8] E.M. Kopalinsky, P.L.B. Oxley, Size effects in metal removal processes, Institute of Physics Conference Series 70 (1984) 389–396.
[9] K. Nakayama, K. Tamura, Size effect in metal-cutting force, ASME Journal of Engineering for Industry 90 (1968) 119. [10] N.A. Fleck, G.M. Muller, M.F. Ashby, J.W. Hutchinson, Strain gradient plasticity: theory and experiments, Acta Metallurgica et Materialia 42 (2) (1994) 475–487. [11] J.S. Stolken, A.G. Evans, A microbend test method for measuring the plasticity length scale, Acta Materialia 46 (14) (1998) 5109–5115. [12] N.A. Fleck, J.W. Hutchinson, A phenomenological theory for strain gradient effects in plasticity, Journal of Mechanics and Physics Solids 41 (1993) 1825–1857. [13] C.-S. Han, H.J. Gao, Y.G. Huang, W.D. Nix, Mechanism-based strain gradient crystal plasticity—I. Theory, Journal of the Mechanics and Physics of Solids 53 (2005) 1188–1203. [14] S.S. Joshi, S.N. Melkote, An explanation for the size effect in machining using strain gradient plasticity, ASME Journal of Manufacturing Science and Engineering 126 (2004) 679–684. [15] G.R. Johnson, W.H. Cook, A constitutive model and data for metals subjected to large strains, high strain rates and high temperature, in: Proceedings of the Seventh International Symposium on Ballistics, The Haque, The Netherlands, 1983, pp. 541–547. [16] G.R. Johnson, Fracture characteristics of three metals subjected to various strains, strain rates, temperatures and pressures, Engineering Fracture Mechanics 20 (1) (1985) 31–48. [17] M.C. Shaw, Metal Cutting Principles, Clarendon Press, 1984. [18] T. Ozel, E. Zeren, Determination of work material flow stress and friction for FEA of machining using orthogonal cutting tests, Journal of Materials Processing Technology 153–154 (2004) 1019–1025. [19] C.-J. Kim, Mechanism of the chip formation and cutting dynamic of the micro scale milling process, Ph.D. Dissertation of the University of Michigan, 2004. [20] W.F. Hastings, P. Mathew, P.L. Oxley, Machining theory for predicting chip geometry, cutting forces, etc. from work materials properties and cutting conditions, Proceedings of The Royal Society of London, Series A: mathematical and Physical Sciences 371 (1980) 569–587. [21] D.J. Waldort, R.E. DeVor, S.G. Kapoor, A slip line field for ploughing during orthogonal cutting, Journal of Manufacturing Science and Engineering 120 (1998) 693–699. [22] M.T. Zaman, A. Senthil Kumar, M. Rahman, S. Sreeram, A threedimensional analytical cutting force model for micro end milling operation, International Journal of Machine Tools and Manufacture 46 (3–4) (2006) 353–366. [23] W.Y. Bao, I.N. Tansel, Modelling micro-end-milling operations. Part I: analytical cutting force model, International Journal of Machine Tools and Manufacture 40 (15) (2000) 2155–2173. [24] S. Filiz, C.M. Conley, M.B. Wasserman, O.B. Ozdoganlar, An experimental investigation of micro machinability of copper 101 using tungsten carbide micro endmill, International Journal of Machine Tools and Manufacture 47 (2007) 1088–1100. [25] K.W. Kim, W.Y. Lee, H.C. Sin, A finite element analysis for the characteristics of temperature and stress in micro-machining considering the size effect, International Journal of Machine Tools and Manufacture 39 (1999) 1507–1524. [26] T.A. Dow, E.L. Miller, K. Garrard, Tool force and deflection compensation for small milling tools, 2004, Precision Engineering 28 (2004) 31–45.