Analytical investigation of workpiece internal energy generation in peripheral milling of titanium alloy Ti–6Al–4V

International Journal of Mechanical Sciences 161–162 (2019) 105063

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Analytical investigation of workpiece internal energy generation in peripheral milling of titanium alloy Ti–6Al–4V Dong Yang a,∗, Yulei Liu a, Feng Xie a, Xiao Xiao b a b

Department of Mechanical Engineering, Anhui University, Hefei 230601, China Hefei Science and Technology College, Hefei 231201, China

a r t i c l e

i n f o

Keyword: Cutting energy Workpiece internal energy Energy conversion Peripheral milling Ti–6Al–4V

a b s t r a c t Energy management is crucial to the cutting process where generation and conversion of various forms of energy is occurring. Reducing the energy that stored in the machined surface layer can significantly improve the quality of processing and dimensional stability of machined structures. To achieve this, an accurate energy conversion model is required. In the present study, energy generation and conversion process during peripheral milling titanium alloy Ti–6Al–4V are investigated. Analytical model of cutting energy that acted on the machined surface is first established. And then, energy criteria for machining-induced residual stress field are proposed, and the energy that stored in the machined surface layer is calculated on the basis of the measured residual stress. Finally, a Q-factor is proposed to relate cutting energy to workpiece internal energy. By experimental analysis, there is about 10.86% of the energy that acted on the machined surface converted to the workpiece internal energy. On the basis of the proposed energy conversion model, the influences of milling processing parameters on the machining-induced material internal energy are revealed. According to the research results, the increase in cutting speed, feed rate and radial depth of cut increase the machining-induced material internal energy, and it is the cutting speed and feed per tooth that turn out to be the chief reasons for energy conversion in peripheral milling.

1. Background and research outline 1.1. Research background Peripheral milling is always used in making casing, compressor blades and other aeronautical parts. Most of these peripheral milled parts are made of titanium alloys for their excellent properties in fracture toughness, corrosion resistance, and so on [1]. The unique service environment of titanium alloy parts set a higher request to peripheral milling quality, which is directly related to the machining-induced surface integrity. As one of the foremost characteristics of surface integrity, residual stress is a major cause of dimensional instability and fatigue failure for peripheral milling part structures [2,3]. Therefore, accurate characterization and prediction of residual stress are vital for the effectively control of peripheral milling quality. It is widely accepted that the main sources of residual stresses are the non-uniform plastic deformation of materials caused by mechanical and thermal loads [4]. As a consequence, the method of thermal mechanical coupling is often used in the theoretical analysis of machining-induced residual stresses. Liang and Su [5] defined a thermo-mechanical loading experienced by the workpiece on the basis of the predictions of cutting force and cutting temperature in orthogonal cutting. And in the study ∗

of Lazoglu et al. [6], thermal and mechanical stresses those acted on the workpiece were coupled firstly, and a relaxation procedure was applied then. Yao et al. [7] investigated the influences of cutting forces and temperature on residual stress in high-speed milling of titanium alloy TB6. By measuring the cutting force and cutting temperature with different tool wear, Tang et al. [8] investigated the relationship between tool wear and residual stress in milling aluminum alloy. For the reason that the precise measurement of cutting force and cutting temperature is not easy to implement, accuracy of the above conclusion is difficult to guarantee. For most of studies in the previous decades, prediction models for machining-induced residual stress were built on the basis of experimental design and analysis methods, and these methods including multi-objective optimization method [9], response surface methodology (RSM) [10] and Taguchi methodology, etc., provide high precision ways to optimize the machining-induced residual stress. After years of accumulation, the related researches have basically covered various important parameters in the cutting process, including cutting parameters (cutting speed, feed rate and depth of cut) [7,11], tool parameters (material and geometric parameters) [12], lubrication and cooling conditions [13,14] or the properties of the workpiece material [15]. With regard to the machining-induced residual stress of titanium alloy, Sun and Guo

Corresponding author. E-mail address: [email protected] (D. Yang).

https://doi.org/10.1016/j.ijmecsci.2019.105063 Received 11 April 2019; Received in revised form 5 July 2019; Accepted 4 August 2019 Available online 5 August 2019 0020-7403/© 2019 Elsevier Ltd. All rights reserved.

D. Yang, Y. Liu and F. Xie et al.

[16] claim for milling Ti–6Al–4V that tensile stress is formed on the machined surface under large feed rate condition, while an increase in cutting speed makes the surface residual stress more compressive. The same conclusion was reached by Sridhar et al. [17] in milling IMI-834. However, Guerville et al. [18] claimed the opposite to Sun and Sridhar´s conclusion that the increase cutting speed produces more tensile stress on the machined surface. Mantle and Aspinwall [19] also declared the same conclusion with Guerville when milling gamma-titanium aluminide. In consequence, a unified conclusion of process parameters on residual stress is hard to obtain by using the current evaluation criteria of residual stress, such as the value of surface residual stress, the peak value of the tensile/compressive stress, the depth of the peak compressive value, and the influencing range of machining-induced residual stresses. There are two main reasons for this problem. On one hand, the specific parameters and their numerical ranges are not the same for different researches. On the other hand, and the most fundamental, the units of the studied parameters in the cutting process and machininginduced residual stress are different in the international system of units. Machining-induced residual stress field is actually the energy field that stores strain energy input by cutting system and dissipated by various forms, including plastic deformation, etc. [20]. Since energy is a scalar variable, energy approach should be successfully used in solving complex problems which are difficult to solve with the stress method. As a consequence, an accurate surface energy model is urgent and necessary for the precise management of the processing quality induced by peripheral milling. Energy in cutting operation mainly consists of the energy consumed during machine setup and tool change, cutting energy and energy that used to produce cutting tool per cutting edge [21,22]. Among these components, cutting energy can be directly relate to the machining-induced residual stress, because of it is normalized and is more sensitive to various conditions specially applied in cutting process, such as cutting parameters, tool geometry and cooling environment, etc. Rodrigues and Coelho [23] found that the specific cutting energy decreases with the increase of chamfer angle when cutting the quenched steel ASTM H13. Velchev et al. [24] investigated the influences of the insert grade, cutting speed, the feed and depth of cut on the minimum energy consumption. Balogun and Mativenga [25] investigated the machining efficiency through surface finish of machined component. It was found that during machining of titanium alloy Ti–6Al–4V, the specific energy decreases as the material removal rate increases. In addition, the relationship of specific ploughing energy to cutter swept angle is also mentioned in Balogun’s work [26]. The above literature review shows that most of studies are focused on optimization of cutting variables for minimizing energy consumption during the machining process, while the energy converted to the interior of the workpiece material has received little coverage. According to Khludkova [27] that the specific cutting energy in cutting process can be put as the sum of the energy involved in changing the kinetic energy of the chip, the work done against the frictional forces at the leading edge of the cutter, and the specific deformation energy of the metal in the shear zone. For the fact that the radius of the tool tip exists, both chip formation energy and frictional energy contribute to the energy absorbed in the workpiece material [28]. In consequence, minimizing energy consumption during machining process does not guarantee the minimum value of the energy converted to the workpiece material at the same time. It makes great sense for the reduction of the energy converted to the workpiece material to investigate the energy conversion process in peripheral milling process and energy-storing properties of titanium alloy Ti–6Al–4V. 1.2. Research outline In the present study, energy generation and conversion process during peripheral milling titanium alloy Ti–6Al–4V are investigated. Fig. 1 shows the research outline of the present study. Analytical models of cut-

International Journal of Mechanical Sciences 161–162 (2019) 105063

Fig. 1. Research outline of energy conversion in peripheral milling.

Fig. 2. Geometric model of oblique cutting.

ting energy that acted on the machined surface and machining-induced material internal energy are established. Secondly, analytical expressions for energy conversion are presented and the coefficients of energy conversion model for peripheral milling of Ti–6Al–4V are estimated. Finally, cutting parameters those affect the machining-induced material internal energy are investigated on the basis of the proposed energy conversion model. The research results can be used to optimize the process parameters of cutting to lower the machining-induced material internal energy, which is beneficial to the realization of the precise management of the processing quality induced by peripheral milling. 2. Modeling of energy conversion during peripheral milling 2.1. Cutting energy acted on the machined surface Cutting process is a procedure by which a cutting tool does work on a workpiece. Tangential and radial cutting powers can be calculated by peripheral milling force model. Several models based on theoretical assumptions and experimental observations have been developed to predict the cutting forces, such as average rigid force model, instantaneous rigid force model, instantaneous force with static deflection feedback model and regenerative force model, etc. [29]. For ease of calculation, instantaneous rigid force model was adopted in the present study. This model does not consider the force produced on the cutter to be simply proportional to the average power, but rather computes the instantaneous force on incremental sections of the helical cutting edge. In the cutting force model of peripheral milling, each flute of peripheral milling cutter can be treated as a combination of a series of oblique cutting elements. As illustrated in Fig. 2 that the geometric model for oblique cutting element, there is an angle 𝛽 between the direction of cutting vc and workpiece, where vc is the resultant velocity of tangential cutting speed vt and radial cutting speed vr . 𝛼 n is the normal rake angle of tool, and 𝜂 c is the chip flow angle. bc , t are the cutting width and undeformed chip thickness, respectively. Through discretization along the axis of the end mill (as shown in Fig. 3), the differential tangential cutting power dPti (𝜑i ) and radial cutting power dPri (𝜑i ) done by the ith flute edge on any floor can be drawn

D. Yang, Y. Liu and F. Xie et al.

International Journal of Mechanical Sciences 161–162 (2019) 105063

Fig. 4. Curvilinear coordinate system of the ith tooth. Fig. 3. Discrete form of the end mill.

according to the oblique cutting model. d𝑃𝑡𝑖 (𝜑𝑖 ) = 𝐾𝑐 𝑡𝑖 (𝜑𝑖 )𝑣𝑡 dz

(1)

d𝑃𝑟𝑖 (𝜑𝑖 ) = 𝑐 𝐾𝑐 𝑡𝑖 (𝜑𝑖 )𝑣𝑟 dz

(2)

where Kc and c are the coefficient of cutting-force and cutting-force ratio, respectively. dz is the width of each floor, which can be calculated as follow, dz = R cot (𝛽)𝑑𝜑

In Eqs. (1) and (2), ti (𝜑i ) is the undeformed chip thickness when the position angle of the flute tip is equal to 𝜑i , R and 𝛽 are the radius and helix (inclination) angle of the end mill, respectively. 𝜑i is the position angle of a point on the cutting edge in the tool coordinate system, which can be calculated by Eq. (4). 𝜑𝑖 = 𝜑 + 𝜃 + (𝑖 − 1)

2𝜋 𝑁𝑓

(

0 ≤ 𝜑 ≤ 𝜓, 1 ≤ 𝑖 ≤ 𝑁𝑓

)

(4)

where 𝜑 is the lag angle, Nf is the teeth number of the end mill. 𝜃 is the instantaneous rotation angle of the flute tip, which is a function of the angular velocity of the spindle 𝜔. 𝜓 is the axial immersion angle of a tooth within the axial depth of cut ba , and its expression is as follow. ψ=

𝑏𝑎 tan 𝛽 𝑅

(5)

According to the kinematics of peripheral milling, the undeformed chip thickness removed by a point on the ith helical flute can be calculated as follows. For down-milling, { ( ) 𝑓𝑡 sin(𝜑𝑖 ) 0 ≤ 𝜑𝑖 ≤ Λ 𝑡𝑖 𝜑𝑖 = (6) else 0 For up-milling, { ( ) 𝑓𝑡 sin(−𝜑𝑖 ) 𝑡𝑖 𝜑𝑖 = 0

−Λ ≤ 𝜑𝑖 ≤ 0 else

(7)

where ft is the feed per tooth, and Λ is the tool radial immersion angle within the radial depth of cut ae , where 𝑎 Λ = arccos(1 − 𝑒 ) 𝑅

Fig. 5. Schematic diagram of cutting process.

(3)

(8)

In oblique cutting, the tangential cutting speed vt and the radial cutting speed vr are the components of feed speed vf and spindle speed vs in tangential and radial directions, respectively. As shown in Fig. 4, a curvilinear coordinate system (x1 , y1 , z1 ) is established to relate the tangential and radial cutting speeds to the feed and spindle speeds, in

which ( ) 𝑣𝑡 = 𝜔𝑅 − 𝑣𝑓 cos 𝜑𝑖

(9)

( ) 𝑣𝑟 = 𝑣𝑓 sin 𝜑𝑖

(10)

By applying Eq. (9), Eq. (1) becomes [ ] d𝑃𝑡𝑖 (𝜑𝑖 ) = 𝐾𝑐 𝑡𝑖 (𝜑𝑖 ) ωR − vf cos(φi) Rcot(β)dφ

(11)

Based on the force model of orthogonal cutting, the differential radial cutting power done by the ith flute edge on any discrete layer floor along the tool axis is given by d𝑃𝑟𝑖 (𝜑𝑖 ) = 𝑐 𝐾𝑐 𝑡𝑖 (𝜑𝑖 )vfsin(φi)Rcot(β)dφ

(12)

Summing the above two equations gives the differential cutting power in cutting as d𝑃𝑖 = d𝑃𝑡𝑖 (𝜑𝑖 ) + d𝑃𝑟𝑖 (𝜑𝑖 )

(13)

As shown in the schematic diagram of cutting process (Fig. 5), part of the uncut material becomes chip at the point of separation P, and the other part of the material flows into the machined surface. As a consequence, there is only part of the cutting power that acts on the machined surface, and the other part is mainly used to form the chip. According to the geometric characteristic of the cutting tool tip, the differential cutting power that acts on the machined surface is given by Eq. (14). d𝑃𝑒 =

𝑡0 d𝑃 𝑡 𝑖

(14)

where t0 is the height of the separation point P, and its value can be calculated by Eq. (15) [30]. 𝑡0 = (1 + sin 𝛼𝑟 )𝑟𝑒

(15)

where re is the radius of the cutting edge. ar is the effective rake angle of the end mill.

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International Journal of Mechanical Sciences 161–162 (2019) 105063

Therefore, the distortional strain energy density Pd can be expressed as follows, )2 ( )2 ( )2 ] 1 + 𝜇 [( 𝑃𝑑 = 𝑃𝜀 − 𝑃𝑣 = 𝜎1 − 𝜎2 + 𝜎2 − 𝜎3 + 𝜎1 − 𝜎3 (24) 6𝐸 Considering that the machining-induced internal stress (residual stress) is usually distributed only in the shallow part of the workpiece, the normal stress 𝜎 2 in the direction perpendicular to the machined surface is approximately zero [9]. Therefore, the equation of strain energy density is updated as follows, 𝑃𝜀 = 𝑃𝑑 = Fig. 6. Trajectory of peripheral milling.

Beside of the partial cutting power, the other power that acts on the machined surface is the friction power dPf , and the differential friction power can be calculated by Eq. (16). [ ] d𝑃𝑓 = −𝜇𝑓 𝑐 𝐾𝑐 𝑡𝑖 (𝜑𝑖 ) ωR − vf cos(φi) Rcot(β)dφ (16) where 𝜇 f is the friction coefficient. Thus, the total cutting power acted by the i th flute edge is given by 𝑃𝑐𝑢𝑡,𝑖 =

𝜑𝑒

∫𝜑𝑠

𝑑 𝑃𝑐𝑢𝑡,𝑖 d𝜑𝑖

𝑃𝜀 =

] 1 +𝜇[ 2 𝜎1 (𝑥) + 𝜎32 (𝑥) − 𝜎1 (𝑥)𝜎3 (𝑥) 6𝐸

(26)

Therefore, the strain energy stored in unit time during the machining process can be drawn as follow, 𝑃𝑠𝑢𝑟 =

𝑣𝑓 𝑏𝑎 (1 + 𝜇) 6𝐸

ℎ𝑟𝑦 [

] ∫ 𝜎12 (x) + σ23 (x)−𝜎1 (x)𝜎3 (x) dx

(27)

0

(18)

2.3. Q-factor of energy conversion in machining

Thus, Eq. (17) changed to Eq. (20). 𝜑𝑒

As the value of the machining-induced workpiece internal stress is not the same for different position beneath the machined surface, the strain energy density for different position is different. At a given depth x to the machined surface, the strain energy density P𝜀 is given.

where hry is the depth of the deformation layer of the material induced by machining. vf is the feed speed.

Hypothesis is made that there was no runout of the tool in the process of milling, and the plastic deformation amount of workpiece material was ignored. In addition, the track of the cutting edge is regarded as a circle. According to the kinematics of peripheral milling (as shown in Fig. 6), 𝜑e and 𝜑s are given as follows. ( ) { 𝜑𝑠 = −arcsin 𝑓𝑡 ∕2𝑅 ( ) (19) 𝜑𝑒 = arcsin 𝑓𝑡 ∕2𝑅

𝑃𝑐𝑢𝑡,𝑖 = 2 ∫ d𝑃𝑐𝑢𝑡,𝑖 d𝜑𝑖

(25)

(17)

where, d𝑃𝑐𝑢𝑡,𝑖 = d𝑃𝑒 + d𝑃𝑓

) 1 +𝜇( 2 𝜎1 + 𝜎32 − 𝜎1 𝜎3 6𝐸

(20)

During the process of cutting, the stored procedure of strain energy is accompanied by the generation of cutting heat, and part of the cutting energy is converted to the machined material. On the basis of the assumption that the distribution of material internal energy is uniform beneath the machined surface, energy conversion model in peripheral milling is given by Eq. (28). ( ) 𝑃𝑠𝑢𝑟 = 𝑄 𝑃cut,avg (28) where Pcut,avg is the mean value of Pcut , and Q is the quality factor (Qfactor), which is related to material properties and cutting conditions.

0

Summing up the cutting powers that acted by all of the cutter teeth, the total cutting power applied on the machined surface is 𝑁

𝑃𝑐𝑢𝑡 =

𝑓 ∑

𝑖=1

𝑃𝑐𝑢𝑡,𝑖

(21)

2.2. Machining-induced workpiece internal energy Machining-induced material internal energy exists in the form of strain energy and been stored in the machined surface layer. Assumptions are made that the studied material is an isotropic material and the material internal energy is in the state of self-balance. According to the elastic-plasticity theory, the strain energy density P𝜀 of one certain point beneath the machined surface can be expressed by Eq. (22). 1 1 1 𝜎 𝜀 + 𝜎 𝜀 + 𝜎 𝜀 (22) 2 1 1 2 2 2 2 3 3 where 𝜎 j , 𝜀j (j = 1, 2 and 3) are the principal stress and strain in threedimensional space, respectively. The strain energy density P𝜀 is the sum of distortional strain energy density Pd and dilatational strain energy density Pv , where Pv can be expressed as follows, )2 1 − 2𝜇 ( 3 𝑃 𝑣 = 𝜎𝑚 𝜀 𝑚 = (23) 𝜎1 + 𝜎2 + 𝜎3 2 6𝐸 where 𝜎 m , 𝜀m are the mean values of principal stresses and principal strains. E and 𝜇 are the Young’s modulus and the Poisson ratio of the workpiece material, respectively. 𝑃𝜀 =

3. Estimation of the coefficients of energy conversion model Known from the modeling process of the energy conversion, there are three coefficients that need to be estimated, and they are Kc , c and Q-factor, respectively. 3.1. Kc and c Integrating between the extreme values of the parametric angle 𝜑e and 𝜑s gives the total cutting force applied on the ith tooth. According to Ref. [31], the total cutting force applied on the whole cutting edge is given as follow. For down milling: ⎧ [ ( )]0.2 [ ( )]||𝜑 𝑓𝑡 𝑅 cot (𝛽) 0.5𝜑𝑖 − 0.25 sin(2𝜑𝑖 ) + 0.5556𝑐 sin1.8 𝜑𝑖 | 𝑒 ⎪𝐹𝑖𝑥 = −𝐾𝑐 sin 𝜑𝑖 |𝜑 𝑠 ⎪ | ⎨ | [ ] [ ( ) ( )] 0.2 ⎪ |𝜑𝑒 1.8 ⎪ 𝐹𝑖𝑦 = 𝐾𝑐 sin(𝜑𝑖 ) 𝑓𝑡 𝑅 cot (𝛽) 0.5556sin 𝜑𝑖 − 0.5𝑐 𝜑𝑖 + 0.25𝑐 sin 2𝜑𝑖 ||𝜑 ⎩ | 𝑠

(29) where

[ ] 2𝜋 𝜑𝑠 = max 0, − 𝜔𝑡𝑖𝑚𝑒 +(𝑖 − 1) 𝑁𝑓

(30)

[ ] 2𝜋 𝜑𝑒 = min Λ,𝜓 − 𝜔𝑡𝑖𝑚𝑒 +(𝑖 − 1) 𝑁𝑓

(31)

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International Journal of Mechanical Sciences 161–162 (2019) 105063

Table 1 L16 (34 ) orthogonal experimental parameters. Level/factors

Cutting speed vc (m/min)

Feed per tooth ft (mm/tooth)

Radial depth of cut ae (mm)

Level Level Level Level

20 50 80 110

0.02 0.03 0.04 0.05

0.5 1.0 1.5 2.0

1 2 3 4

And for up milling: ⎧ [ ]0 . 2 [ ( ) ( )]||𝜑 ⎪ 𝐹𝑖𝑥 = −𝐾𝑐 sin(𝜑𝑖 ) 𝑓𝑡 𝑅 cot (𝛽) 0.5𝜑𝑖 − 0.25 sin 2𝜑𝑖 − 0.5556𝑐 sin1.8 𝜑𝑖 | 𝑒 |𝜑 𝑠 ⎪ | ⎨ [ ]0 . 2 [ ( ) ( )]||𝜑𝑒 ⎪ 1.8 ⎪𝐹𝑖𝑦 = −𝐾𝑐 sin(𝜑𝑖 ) 𝑓𝑡 𝑅 cot (𝛽) 0.5556sin 𝜑𝑖 + 0.5𝑐 𝜑𝑖 − 0.25𝑐 sin 2𝜑𝑖 ||𝜑 ⎩ | 𝑠

(32) where

[ ] 2𝜋 𝜑𝑠 = max 0, −𝜓 + 𝜔𝑡𝑖𝑚𝑒 − (𝑖 − 1) 𝑁𝑓 [ ] 2𝜋 𝜑𝑒 = min Λ, 𝜔𝑡𝑖𝑚𝑒 − (𝑖 − 1) 𝑁𝑓

(33) (34)

As a consequence, the total force applied on the whole end mill is 𝑁𝑓 ⎧ ∑ ⎪𝐹 𝑥 = 𝐹𝑖𝑥 ⎪ 𝑖=1 ⎨ 𝑁𝑓 ⎪ ∑ 𝐹𝑖𝑦 ⎪ 𝐹𝑦 = ⎩ 𝑖=1

(35)

To obtain values of the coefficient Kc and the ratio c, peripheral milling tests were set up. The material studied was Ti–6Al–4V, and its

microstructures are equiaxed shapes. The chemical compositions of Ti– 6Al–4V in weight are 4.83–6.8% Al, 3.5–4.5% V, <0.25% Fe, <0.05% N, <0.2% O and balance Ti. The machining experiments were carried out on a vertical type machining center (DAEWOOACE-V500). There are 4 flutes with variable helix angles (𝛽 1 = 𝛽 3 = 38° and 𝛽 2 = 𝛽 4 = 41°) of the cutter, and the tool diameter was 6.0 mm. In addition, another geometric parameter named the effective rake angle of the cutter is characterized. The effective rake angle ar is determined by Eq. (36) [32]. And by calculation the value of the effective rake angle ar is approximately equal to 5°. The experiments were performed with dry machining and the milling mode was down milling. Taguchi’s L16 (34 ) orthogonal array was applied, the cutting parameters and their levels for experimental design are given in Table 1. Axial depth of cut was set to 5.0 mm for all cutting experiments in the present study. All experiments were performed three repetitions to ensure the accuracy of data. ( ) ( ) ( ) ( ) (36) sin 𝑎𝑟 = sin (𝛽) sin 𝜂𝑐 + cos 𝜂𝑐 cos (𝛽) sin 𝑎𝑛 where 𝛽 is the helix angle, 𝜂 c is the chip flow angle and an is the normal angle. The cutting forces in peripheral milling were measured using a Kistler 9129AA dynamometer, and the sampling rate is 1000 Hz. The measuring system of cutting force is shown in Fig. 7(b). Fig. 7. Set-up of the peripheral milling test. (a) Experimental setup, and (b) Measuring system of cutting force.

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International Journal of Mechanical Sciences 161–162 (2019) 105063

Fig. 8. Measured cutting forces under the conditions of (a) A02: vc = 20 m/min, ft = 0.03 mm/tooth, ae = 1.0 mm, (b) A08:vc = 50 m/min, ft = 0.05 mm/tooth, ae = 1.5 mm, (c) A11:vc = 80 m/min, ft = 0.04 mm/tooth, ae = 0.5 mm and (d) A13: vc = 110 m/min, ft = 0.02 mm/tooth, ae = 2.0 mm.

Measured cutting forces under different cutting conditions (partial) are shown in Fig. 8. As a multi-objective optimization method, particle swarm optimization (PSO) method is applied to get the coefficients of the cutting force model. PSO is an evolutionary method which is solved by iterative method. In each iteration of the method, the location of particles is updated by determining the optimal solution Pbest and global optimal solution Gbest of a single particle. And when the two solutions are de-

termined, the position and velocity of each particle are updated by Eqs. (37) and (38). ( ) ( ) 𝑣𝑝 = 𝛿𝑣𝑝 + 𝑐1 rand(0, 1) ∗ 𝑃𝑏𝑒𝑠𝑡 − 𝐿𝑝 + 𝑐2 rand(0, 1) ∗ 𝐺𝑏𝑒𝑠𝑡 − 𝐿𝑝 (37) 𝐿𝑝 = 𝐿𝑝 + 𝑉𝑝

(38)

where Lp , vp are the current velocity and position of a particle. Rand (0,1) represents a random value between 0 and 1. c1 , c2 are learning factors. 𝛿 is weighting coefficient.

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International Journal of Mechanical Sciences 161–162 (2019) 105063

Table 2 Results of Kc and c. Test no. A01 A02 A03 A04 A05 A06 A07 A08 A09 A10 A11 A12 A13 A14 A15 A16

vc (m/min) 20 20 20 20 50 50 50 50 80 80 80 80 110 110 110 110

ft (mm/tooth) 0.02 0.03 0.04 0.05 0.02 0.03 0.04 0.05 0.02 0.03 0.04 0.05 0.02 0.03 0.04 0.05

ae (mm) 0.5 1.0 1.5 2.0 1.0 0.5 2.0 1.5 1.5 2.0 0.5 1.0 2.0 1.5 1.0 0.5

Kc (J•m − 3 ) 9

1.21 × 10 1.26 × 109 1.29 × 109 1.16 × 109 1.21 × 109 1.25 × 109 1.23 × 109 1.27 × 109 1.20 × 109 1.21 × 109 1.20 × 109 1.29 × 109 1.19 × 109 1.26 × 109 1.27 × 109 1.24 × 109

c

R(𝜏)

0.46 0.46 0.51 0.48 0.54 0.55 0.47 0.50 0.54 0.51 0.46 0.48 0.53 0.50 0.46 0.46

0.87 0.83 0.89 0.82 0.85 0.87 0.81 0.81 0.81 0.89 0.83 0.82 0.85 0.84 0.84 0.85

The measured and predicted cutting forces are regarded as time domain signals Fm (t) and Fp (t), and their cross-correlation R(𝜏), which is a measure of similarity of two series, is used as a target function. The cross-correlation is defined as ∞

R(τ) =

∫−∞

𝐹𝑚∗ (𝑡)𝐹𝑝 (𝑡 + 𝜏)d𝑡

(39)

where F∗𝑚 (t) denotes the complex conjugate of Fm (t), and 𝜏 is the displacement that be determined on the basis of the sampling rate of the dynamometer. The value of R(𝜏) lies in the range [0, 1], with “1” indicating a perfect correlation and “0” indicating an imperfect one. Kc is set in the range of [105 , 1010 ] J m−3 , and the ratio c is in the range [0, 1]. Results of Kc and c are listed in Table 2. As can be seen from Table 2, there is a certain error between the measured and predicted cutting force. It is mainly because of non-thought of the ploughing force. Nevertheless, the cross-correlation between the measured and predicted cutting force is greater than 0.8, which means that the two coefficients can effectively characterize the cutting energy model that be established based on cutting force model. In the present study, the average values of Kc and c are used to predict the cutting power that acted on the machined surface, and their values are 1.23 × 109 J m−3 and 0.49, respectively.

Fig. 9. Xstress-3000 residual stress measurement system. Table 3 Parameters of the electro-chemically polished method [33]. Chemical compositions

HClO4 N‑butyl alcohol Methanol

30 ml 175 ml 300 ml

Electrolytic parameters

Voltage Current density Polishing rate

17 V 0.05–0.10 A/cm2 25 nm/s

Environment

Room temperature (25 °C)

3.2. Q-factor To obtain the cutting energy that acted on the machined surface and the machining-induced workpiece internal energy, cutting experiments were performed under the same cutting conditions shown in Section 3.1. Peripheral milling induced surface layer residual stress profile was characterized firstly to model the machining induced material internal energy. As shown in Fig. 9, a Xstress-3000 diffractometer with Cu K𝛼 radiation tube was applied. In addition, electro-chemically polished method was applied to get the residual stress profile in machined surface layer, and the test surface layers are 2 𝜇m, 4 𝜇m, 6 𝜇m, 8 𝜇m, 10 𝜇m, 20 𝜇m, 40 𝜇m, 60 𝜇m, 80 𝜇m beneath the machined surface. Parameters of the electro-chemically polished method are listed in Table 3. As shown in Fig. 10 that a similar distribution pattern of the residual stress was formed for peripheral milling Ti–6Al–4V under different cutting conditions. It can be found that there is a tensile (or small compressive) peak at the machined surface, and a compressive peak occurred at the place of approximately 2–10 𝜇m beneath the machined surface, which value can reach 600 MPa. The residual stress profile settled at a distance that less than 100 𝜇m. Furthermore, the standard deviation lays in the range of [−20, 20] Mpa. The parameters of residual stress were taken at three locations and repeated twice at each location on the tested surface layers, and then the

average values were gained. Based on the mean values of residual stress, material internal energy was calculated. The values of the strain energy stored in unit time during the machining process were in the range of 0.24–33.65 W, and the maximum value was obtained under the cutting condition of A16: vc = 110 m/min, ft = 0.05 mm/tooth, ae = 0.5 mm, while the minimum value for A02: vc = 20 m/min, ft = 0.03 mm/tooth, ae = 1.0 mm. According to Eqs. (15) and (16), the cutting edges obtuse radius re and the friction coefficient uf between the cutting tool and workpiece should be determined to calculate the cutting power acted on the machined surface. A new tool was used to assess the influence of cutting parameters on the cutting force (and energy conversion) independently of tool wear. Tool edge radius re was measured by a confocal laser scanning microscope, and as shown in Fig. 11 that the measured value was approximately 8.9 μm. Based on the measurements of cutting force components, the value of friction coefficient uf can be calculated by

D. Yang, Y. Liu and F. Xie et al.

International Journal of Mechanical Sciences 161–162 (2019) 105063

Fig. 11. Measurement of the tool edge radius.

Fig. 12. Relationship between Pcut,avg and Psur .

Eq. (40) [34], and the average value of the calculated results is 0.58. ( ) ( ) 𝐹𝑥 sin 𝛼𝑟 +𝐹𝑦 cos 𝛼𝑟 𝜇𝑓 = (40) ( ) ( ) 𝐹𝑥 cos 𝛼𝑟 − 𝐹𝑦 sin 𝛼𝑟 Based on the measured tool edge radius re and the calculated friction coefficient uf , the cutting power acted on the machined surface was calculated and its relationship with the strain energy stored in unit time during the machining process, which is represented by the Q-factor, is established. As shown in Fig. 12 that it shows a certain linear relation between the cutting power acted on the machined surface Pcut, avg and the machininginduced material internal energy Psur . The relationship between Pcut,avg and Psur can be fitted as Psur = 0.1086 Pcut,avg , namely, Q = 0.1086. And their linear correlation coefficient R2 = 0.8377, which means that the Q-factor can effectively characterize the energy conversion in peripheral milling of titanium alloy Ti–6Al–4V. 4. Effects of cutting parameters on Psur

Fig. 10. Residual stress profiles of (a) A02, (b) A06, (c) A10 and (d) A14.

The cutting parameter has an important effect on the machininginduced material internal energy because it directly determines the input energy of the cutting system. Based on the established model of energy conversion during peripheral milling, cutting parameters those

D. Yang, Y. Liu and F. Xie et al.

International Journal of Mechanical Sciences 161–162 (2019) 105063

Fig. 13. Predicted Psur under different cutting parameters.

Table 4 Tested cutting parameters. Case no.

vc (m/min)

ft (mm/tooth)

ae (mm)

Case 1 Case 2 Case 3

50/100/150 100 100

0.03 0.03/0.04/0.05 0.03

1.0 1.0 0.5/1.0/1.5

affect Psur in peripheral milling of titanium alloy Ti–6Al–4V were investigated under the condition of down milling. To investigate the effect of cutting parameters on Psur , tool parameters are fixed and their details are: R = 3 mm, m = 4, 𝛽 = 30°, 𝛼 r = 5°. In addition, the axial depth of cut ba is set to 5 mm for all cases. Test parameters are cutting speed vc , feed per tooth ft and radial depth of cut ae , their detailed are in Table 4. Fig. 13 shows the predicted Psur when the cutting speed vc , feed per tooth ft and radial depth of cut ae changes, respectively. Seen from Fig. 13(a), the increase in cutting speed increase the machining-induced material internal energy Psur , low cutting speed is preferred to obtain the low Psur in peripheral milling titanium alloy Ti– 6Al–4V. And Eqs. (11)–(13) can be used to explain this phenomenon. It also can be seen from Fig. 13(b) that the high feed rate will lead to the high Psur , due to the size effect of undeformed chip thickness. According to Fig. 13(c), the increase in radial depth of cut increases Psur by increasing the material removal of cutting edge. Meanwhile, the difference was compared with the analysis of variance (ANOVA). In this study, ANOVA analyses were carried out with a confidence level of 95% (significance level of 0.05). Variance values are 10.39, 8.03 and 1.25 for cutting speed, feed per tooth and radial depth of cut, which means that it is the cutting speed and feed per tooth that turn out to be the chief reasons for energy conversion in peripheral milling. 5. Conclusions Energy conversion in peripheral milling of titanium alloy Ti–6Al– 4V has been investigated in this study whereby two types of energy are combined (cutting energy that acted on the machined surface and machining-induced workpiece internal energy). A detailed analysis including cutting energy modeling in peripheral milling, energy evaluation for machining-induced residual stress field, energy conversion modeling and sensitivity study of cutting parameters have been carried out. The conclusions from this research are drawn as follows. The particle swarm optimization algorithm (PSO) was applied to predict the coefficients Kc and c in cutting energy model of peripheral milling of titanium alloy Ti–6Al–4V. On the basis of the measured cutting force, the average values of Kc and c are obtained as 1.23 × 109 J•m − 3 and 0.49, respectively. The cross-correlation

between the measured and predicted cutting force is greater than 0.8, and it means that the two coefficients can effectively characterize the cutting energy model that be established on the basis of cutting force model. Energy criteria for residual stress field was proposed, and the workpiece internal energy properties were discussed based on the measured residual stress profile beneath the machined surface. It was found that peripheral milling produces a hook-shaped residual stress profile and the value of the maximum compressive stress can reach 600 MPa in the place of approximately 2–10 𝜇m beneath the machined surface, and the response depth of residual stress is less than 100 𝜇m. The cutting power acted on the machined surface and the machininginduced material internal energy showed a certain linear relation. And their linear correlation coefficient R2 = 0.8377, which means that the Q-factor can effectively characterize the energy conversion in peripheral milling of titanium alloy Ti–6Al–4V. The increase in cutting speed, feed rate and radial depth of cut increase the machining-induced material internal energy, and it is the cutting speed and feed per tooth that turn out to be the chief reasons for energy conversion in peripheral milling. Acknowledgments The authors would like to acknowledge the financial support from the Key Natural Science Project of Anhui Provincial Education Department (KJ2018A0021), Natural Science Foundation of Anhui Province (1908085QE230) and High-level talent fund of Anhui University. References [1] Been J, Grauman JS. Titanium and titanium alloys. Uhlig’s Corrosion Handbook; 2011. p. 861–78. [2] Masoudi S, Amini S, Saeidi E, Saeidi E, Eslami-Chalander H. Effect of machining-induced residual stress on the distortion of thin-walled parts. Int J Adv Manuf Technol 2015;76(1–4):597–608. [3] Yakovlev M G. Improving fatigue strength by producing residual stresses on surface of parts of gas-turbine engines using processing treatments. J Mach Manuf Reliab 2014;43(4):283–6. [4] Schajer GS, Ruud CO. Overview of residual stresses and their measurement. In: Practical residual stress measurement methods. John Wiley & Sons; 2013. p. 1–27. [5] Liang SY, Su JC. Residual stress modeling in orthogonal machining. CIRP Ann 2007;56(1):65–8. [6] Lazoglu I, Ulutan D, Alaca B E, Engin S, Kaftanoglu B. An enhanced analytical model for residual stress prediction in machining. CIRP Ann 2008;57(1):81–4. [7] Yao CF, Wu DX, Tan L, Ren JX, Shi KM, Yang ZC. Effects of cutting parameters on surface residual stress and its mechanism in high-speed milling of TB6. Proc Inst Mech Eng Part B 2013;227(4):483–93. [8] Tang ZT, Liu ZQ, Pan YZ, Wan Y, Ai X. The influence of tool flank wear on residual stresses induced by milling aluminum alloy. J Mater Process Technol 2009;209(9):4502–8. [9] Yang D, Liu Z, Ren X, Zhuang P. Hybrid modeling with finite element and statistical methods for residual stress prediction in peripheral milling of titanium alloy Ti–6Al–4V. Int J Mech Sci 2016;108:29–38.

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