Chenwei SHAN,Xu ZHANG,Bin SHEN,Dinghu ZHANG

aKey Laboratory of Modern Design and Integrated Manufacturing Technology of Ministry of Education,Northwestern Polytechnical University,Xi'an 710072,China

b206 Institution,North Electronics Research Institute Co.,Ltd,Xi'an 710100,China

KEYWORDS Cutting temperature;Moving heat source method;Orthogonal cutting;Relief angle;Titanium alloy

Abstract Cutting heat has significant effects on the machined surface integrity of titanium alloys in the aerospace field.Many unwanted problems such as surface burning,work hardening,and tool wear can be induced by high cutting temperatures.Therefore,it is necessary to accurately predict the cutting temperature of titanium alloys.In this paper,an improved analytical model of the cutting temperature in orthogonal cutting of titanium alloys is proposed based on the Komanduri-Hou model and the Huang-Liang model.The temperatures at points in a cutting tool,chip,and workpiece are calculated by using the moving heat source method.The tool relief angle is introduced into the proposed model,and imaginary mirrored heat sources of the shear plane heat source and the frictional heat source are applied to calculate the temperature rise in a semi-in finite medium.The heat partition ratio along the tool-chip interface is determined by the discretization method.For validation purpose,orthogonal cutting of titanium alloy Ti6Al4V is performed on a lathe by using a sharp tool.Experimental results show to be consistent well with those of the proposed model,yielding a relative difference of predicted temperature from 0.49%to 9.00%.The model demonstrates its ability of predicting cutting temperature in orthogonal cutting of Ti6Al4V.

1.Introduction

Most of the mechanical energy during metal cutting is converted into thermal energy.Hence,many serious problems such as thermal stress distribution,surface burning,work hardening,and tool wear can be induced by the excessive cutting heat generated during the titanium alloys cutting process in the aerospace field.Moreover,the cutting heat has potential effects on the fatigue life and machining distortion of thinwalled titanium alloy parts,especially those used in the aerospace field.Obviously,an accurate analysis and measurement of the cutting temperature is a possible basis of predicting the residual stress in machining,which is the key to improve the machined surface integrity1.Therefore,it is of great significance to effectively predict the cutting temperature in machining of titanium alloys.

Techniques like numerical simulation,analytical,and experimental methods have been used for prediction of cutting temperatures accurately2.Compared with numerical simulation methods,analytical models have the advantage of being able to provide a clear description of the cutting mechanism.Mathematical analytical methods used in the study of the cutting temperature include the moving heat source method,the partial differential equations(PDE)method,and the empirical formula method3.Among them,the moving heat source method has been widely used by many researchers to study the cutting temperature4.The moving heat source method was proposed by Jaeger5and introduced into the study of cutting temperature by Rosenthal6,and then the Hahn model7,the Chao-Trigger model8-9,the Leone model10,the Loewen-Shaw model11,the Weiner model12,the Boothroyd model13,the Dutt-Brewer model14,the Dawson-Malkin model15,the Komanduri-Hou model16-18,and the Huang-Liang model19have emerged one after another.Among them,the Komanduri-Hou model16-18and the Huang-Liang model19have been widely adopted by many researchers,and these two models have also been improved by subsequent theoretical models.For example,Li and Liang20introduced a cold source for micro-lubrication cutting conditions,under which an analytical model of the cutting temperature in the case of tool wear was established.Liang,et al.21developed an improved cutting temperature model based on the Jaeger model to predict the temperature when machining with a nearly-sharp tool.The shear plane heat source with non-uniform heat liberation intensity was determined based on the slip line field theory.Zhou,et al.22presented a cutting temperature distribution model in the parallel and non-equidistant primary shear zone.The thickness of the primary shear zone was considered,and the primary shear zone was regarded as a non-uniform volume moving heat source in their model.Karpat and ?zel23-24,Kang25,and Huang and Yang26considered the thermal conductivity and diffusivity as a piecewise function,a fourthorder polynomial function,and an exponential function of the cutting temperature,respectively,and then the best values for the thermal conductivity and diffusivity were determined through updating the values by continuous iteration.Huang and Yang26introduced the heating time into the integral upper limit of the zero-order second modified Bessel function,so that the calculated isotherm which extended from the shear plane returned to the machined surface.Karpat and ?zel23-24,Chou and Song27,and Li28combined the analytical model of cutting forces with the cutting temperature,and then the thermomechanical coupling analytical modeling of the cutting process was realized effectively.

In the last few years,many researchers have studied the cutting temperature from other ways.Yan,et al.29proposed a model which considered the effect of fl ank wear in dry milling of nickel-based superalloys for a turbine blade to determine the variation in the temperature of a coated tool.Wu,et al.30proposed a new cutting temperature prediction method based on an analytical model for a cutting tool in end milling.The real friction state between the chip and the tool was taken into account in this method,and the heat flux and tool-chip contact length were worked out by finite element simulation.Karaguzel,et al.31proposed an analytical model of the cutting temperature based on Green's functions to solve the 3D transient heat conduction problem,and to verify their model,they presented an experimental method to measure transient tool temperatures in dry face milling operation with a K-type thermocouple.Sun,et al.32presented an analytical model which considered a continuously-varying chip thickness in intermittent end milling of Ti6Al4V with a solid carbide tool.The frictional heat between the tool and the workpiece was ignored,and the cutting temperature was measured by using a semiartificial thermocouple.Avevor,et al.33presented a hybrid analytical-numerical model which considered the coupling between the material flow in the primary shear zone and the thermo-mechanical load along the tool rake face.The finite element model of their model was only used to solve the transient nonlinear thermal problem in the tool-chip-workpiece system.Chen,et al.34presented a non-uniform moving heat source model to analyze the heat transfer problem during ultrasonic vibration-assisted machining of Ti6Al4V.

The analytical modeling method needs a few assumptions to simplify the problem.The most common assumptions used in the models are adiabatic boundaries,assumptions of heat partitions,and ignoring the contribution of the relief angle.From the literature,it can be noticed that among the previous analytical models,only the Huang-Liang model mentioned that in order to simplify the problem,the cutting wedge angle was considered as 90°.However,the Komanduri-Hou model and the most subsequent models did not mention the problem of the tool relief angle.In fact,many studies35-38about cutting tests or simulations have proven that the tool relief angle affects the cutting temperature and force more or less,especially during nanoscale cutting processes38.Therefore,whether the influence of the tool relief angle on the cutting temperature is important or not,for ensuring the integrity and rationality of an analytical model in terms of cutting principles,it is necessary to consider the tool relief angle in the analytical model.Therefore,in this paper,an improved analytical model of the cutting temperature is proposed based on the Komanduri-Hou model and Huang-Liang model.The tool relief angle is taken into account, and the desirable parts of the Komanduri-Hou model and the Huang-Liang model are adopted.Cutting temperatures are calculated by using the moving heat source method.Finally,model validation is performed through an orthogonal cutting of titanium alloy Ti6Al4V.

2.Analytical modeling of the cutting temperature

As mentioned above,analytical modeling needs a few simplifying assumptions.Hence,the following basic assumptions are made in this paper:

a)The orthogonal cutting process is taken as a steady-state cutting process.As a consequence,the cutting temperature field can be regard as being in a stable state.

b)All deformation work is converted into thermal energy,and the heat escaping to the air through the boundary is negligible.Therefore,boundaries 1,2,3,and 4 in contact with the air can be regarded as adiabatic boundaries,as shown in Fig.1.

c)The cutting edge of the tool is perfectly sharp,namely there is no tool wear.Thus,the influence of the rubbing heat source on the tool fl ank face is considered to be negligible.Consequently,only the shear plane heat source OA and the frictional heat source OB on the tool-chip interface are taken into account in the proposed model,as shown in Fig.1.

d)The cutting tool is regarded as a semi-in finite medium that is relative to chips,and the cutting velocity is V.

2.1.Temperature rise induced by the shear plane heat source

Fig.2 shows diagrams of the undeformed layer and the deformed layer which were presented by Komanduri and Hou16.In Fig.2,t is the uncut chip thickness,α is the rake angle,γ is the relief angle,φ is the shear angle,and the shear plane heat source moves inside them at the cutting velocity V and the chip flow velocity Vc,respectively.The shear plane heat source is described as having uniform heat intensity in the Komanduri-Hou model,and its heat intensity qscan be expressed as

where Fsis the shear force,Vsis the shear velocity,apis the back engagement,and L is the shear plane length.

2.1.1.Workpiece side temperature modeling

A local coordinate system XOZ is de fined to analyze the cutting temperature on the workpiece side,as shown in Fig.3.

Fig.3 Temperature modeling of shear plane heat source on workpiece side.

As mentioned above,the upper surface of the undeformed layer is regarded as an adiabatic boundary.Based on the Komanduri-Hou model16,an imaginary mirrored heat source O1A of the shear plane heat source,which has the same heat intensity,is introduced to obtain the temperature rise Twork-shearcaused by the shear plane heat source on the workpiece side,and then the temperature rise Twork-shearat an arbitrary point M(x,z)in the workpiece can be obtained as

where x and z are the coordinate values of point M(x,z)on the workpiece,λwis the thermal conductivity of the workpiece,awis the thermal diffusion coefficient of the workpiece,K0(x)is the modified Bessel function of second kind of order zero,dliis the differential small segment of the shear band heat source,and liis the location of dlirelative to its upper end and along its width.

2.1.2.Chip side temperature modeling

Another local coordinate system X'BZ'is established as shown in Fig.4.According to the Komanduri-Hou model16,the toolchip interface cannot be considered as an adiabatic boundary for the shear plane heat source,but the upper surface of the chip can be.Therefore,in a similar way,an imaginary mirrored heat source O'A of the shear plane heat source,which has the same heat intensity,is introduced to obtain the temperature rise Tchip-shearcaused by the shear plane heat source on the chip side,and then the temperature rise Tchip-shearat an arbitrary point M′(x',z')in the chip can be obtained as

Fig.2 Diagrams of undeformed layer and deformed layer16.

Fig.4 Temperature modeling of shear plane heat source on chip side.

where x′and z′are the coordinate values of point M′(x′,z′)on the chip,λchis the thermal conductivity of the chip,achis the thermal diffusion coefficient of the chip,tchis the deformed chip thickness, Lcis the tool-chip contact length,

2.2.Temperature rise induced by the frictional heat source

According to the Komanduri-Hou model16,the frictional heat source along the tool-chip interface is also described as having uniform heat intensity.Then,the uniform heat liberation intensity of the frictional heat source qfcan be expressed as

where Fuis the friction force.

However,According to the Huang-Liang model19,the frictional heat source along the tool-chip interface should be described as having non-uniform heat intensity qf(x),as shown in Fig.5,where lstand lslare the lengths of the sticking friction region and the sliding friction region,respectively.It can be seen that qf(x)increases linearly from 0 to qf1in the sliding friction region,and then remains a constant value qf1in the sticking friction region.

Setting that lst=aLc,then lsl=(1-a)Lc,in which a is the ratio of the sticking length to the tool-chip contact length,and its valve is in the range of 0＜a＜1.According to the law of conservation of energy,qf(x)can be expressed as

Fig.5 Non-uniform heat intensity of frictional heat source.

Calculating the lengths of the sticking friction region and the sliding friction region on the tool-chip contact surface during a cutting process has always become an important issue in the field of metal cutting.Two detailed calculating methods of ratio a can be found in the references by Karpat and ?zel24and Atkins39,respectively.Because the calculation process of ratio a is very complicated and not the focus of this paper,to simplify the problem for facilitating a fast analytical solution of the cutting temperature,it is assumed that the sticking length and the sliding length are equal to each other in this paper,i.e.,ratio a is set to 0.5,which is the same as the calculating example of the Huang-Liang model19.

In addition,according to the Komanduri-Hou model17,the heat partition ratio is also non-uniform.Assuming that the ratio of the frictional heat transferred into the chip is B(x),then that of the others transferred into the tool is 1-B(x).As a consequence,the heat intensity acting on the chip along the tool-chip interface can be expressed as B(x)qf(x),and that acting on the tool is[1-B(x]qf(x).

2.2.1.Temperature modeling on the chip side

As shown in Fig.6,the local coordinate system X′BZ′which has already been de fined in Fig.4 is also used in the temperature modeling of the frictional heat source on the chip side.According to the Komanduri-Hou model17,both the toolchip interface and the upper surface of the chip are regarded as adiabatic boundaries for the frictional heat source.Three imaginary heat sources 1,2,and 3 which have the same heat intensity as that of the frictional heat source are introduced to calculate the temperature rise Tchip-frictioncaused by the frictional heat source on the chip side.Then,the temperature rise Tchip-frictionat point M'(x',z')in the chip can be expressed as

Fig.6 Temperature modeling of frictional heat source on chip side.

2.2.2.Temperature modeling on the tool side

A local three-dimensional coordinate system X′′Y′′Z′′is de fined to analyze the cutting temperature on the tool side,as shown in Fig.7.In this paper,the relief angle is taken into consideration for the first time since the boundary effect tends to vary with different relief angles.For example,the approaching angle will go down with an increase of the relief angle,which makes it difficult to dissipate the cutting heat,and as a result,more cutting heat may accumulate near the tool nose.

Based on the Komanduri-Hou model17,both the tool-chip interface and the tool fl ank face are regarded as adiabatic boundaries,and in a similar way,three imaginary heat sources 1′,2′,and 3′that have the same heat intensity as that of the frictional heat source are introduced to calculate the temperature rise Ttool-frictioncaused by the frictional heat source on the tool side,the distribution of which is shown in Fig.7.Then,the temperature rise Ttool-frictionat an arbitrary point M′′(x′′,y′′,z′′)in the tool can be expressed as

where x′′,y′′,and z′′are the coordinate values of point M′′(x′′,y′′,z′′)in the tool,λtis the thermal conductivity of the tool,dxiand dyiare the differential small segments of the heat source,xiis the component of the distance between the heat source small segment and the origin in the x direction,yiis the component of the distance between the heat source small segment and the origin in the Y direction,and

2.3.Determination of the temperature model

Fig.8 shows a schematic diagram of all the three local coordinate systems.Based on the superposition principle of heat sources,the temperatures in the workpiece,chip,and tool can be calculated.

2.3.1.Temperatures in the workpiece,chip,and tool

According to Eq.(2),the workpiece temperature Tworkat point M(x,z)in the workpiece can be expressed as

where Tambientis the ambient temperature.

According to Eq.(3)and Eq.(6),the chip temperature Tchipat point M′(x′,z′)in the chip can be expressed as

Fig.8 Schematic diagram of all three coordinate systems.

Fig.7 Temperature modeling of frictional heat source on tool side.

According to Eq.(7),the tool temperature Ttoolat point M′′(x′′,y′′,z′′)in the tool can be expressed as

2.3.2.Determination of the heat partition ratio

So far,the unknown parameter in the model is the heat partition ratio in Eq.(9)and Eq.(10).In this paper,the heat partition ratio is determined by the idea of discretization that was proposed by Huang and Liang19.As shown in Fig.9,the toolchip interface is evenly divided into n intervals along the coordinate axes.As a result,there are n+1 points from x0to xn.

According to the continuity of the temperature field,the temperatures on the chip side and the tool side along the tool-chip interface should be equal to each other,which can be expressed as

Then,an equation set about the heat partition ratio can be obtained from Eq.(11),and its matrix form can be written as

Matrix［ A］n×ncan be expanded as

Fig.9 Discretization of heat partition ratio.

Matrix［ B］n×1can be described as

Matrix［ C］n×1can be expanded as

Finally,the values of the heat partition ratio of n different intervals can be obtained by solving Eq.(12),and then the temperature in the chip or tool can be calculated by substituting the heat partition ratio into Eq.(9)and Eq.(10).

3.Model validation

3.1.Cutting conditions and experimental parameters

Cutting experiments are conducted on a horizontal lathe,equipped with an FANUC-Oi-Mate-TC NC unit.The workpiece is a thin-walled titanium alloy Ti6Al4V tube,with external and internal diameters of 71 and 67 mm,respectively.As a result,the back engagement apis 2 mm.The ratio of the wall thickness to the external diameter is approximately 1:35,and hence face cutting can be considered as orthogonal cutting.Uncoated carbide tool inserts(ZIMF608N-1)provided by Zhuzhou Cemented Carbide Cutting ToolsCo.,Ltd.(ZCCCT)are adopted in the experiments,with rake and relief angles of 0°and 7°,respectively.In addition,the thermophysical properties of a tool insert and the workpiece are listed in Table 1.

Fig.10 shows the experimental setup for orthogonal cutting of titanium alloy Ti6Al4V.Cutting force signals during turning are measured using a Kistler dynamometer 9257B.The dynamometer is charged,and signals are collected by a data acquisition system which includes a Kistler multi-channel charge amplifier 5080A,a Dewesoft data collector,and Kistler DynoWare software.

Cutting temperature signals are collected by a standard K-type armored thermocouple that is embedded at a specific location of a tool insert.As shown in Fig.11,a straight slot on the tool insert with a width of 0.55 mm is made by wire electricaldischarge machining.The probe of the K-type armored thermocouple with a diameter of 0.5 mm is placed in it and stuffed with a good ductile copper sheet for fixing.The gap between the thermocouple and the straight slot is fi lled with thermal paste to ensure the performance of temperature measurement.As a result,the measuring point is located at the centerline below the cutting edge with a distance o f 1.8 mm.The cold junction of the K-type armored thermocouple is connected to a temperature transmitter,and temperature signals are collected by the data collector of the data acquisition system.

As shown in Table 2,9 groups of tests are designed.The cutting velocity V is in the range of 22.3-44.6 m/min,the feed rate f is in the range of 0.05-0.15 mm/r,and the back engagement aphas a constant value of 2 mm.The cutting force Fcand the feed force Ffmeasured in the tests for each group are also listed in Table 2.To avoid the effects of tool wear,a new sharp tool insert is replaced for each group of tests.

3.2.Determination of the model input parameters

The input parameters of the temperature model can be calculated according to the cutting force model that was proposed by Altintas35.The shear angle φ can be calculated as

Table 1 Thermophysical properties of a tool insert and the workpiece.

The friction angle β in Eq.(16)can be expressed as

Meanwhile,the other input parameters of the cutting temperature model can be calculated based on the shear angle φ,the friction angle β,the cutting parameters used in the tests,and the measured cutting forces Fcand Ffas follows:

3.3.Results and discussion

In order to verify the modified cutting temperature model proposed in Section 2,the prediction temperature field of the tool is obtained by programing in MATLAB?.The programing flow chart is shown in Fig.12.

Fig.10 Experimental setup for orthogonal cutting of titanium alloy Ti6Al4V.

Fig.11 Thermocouple fixation.

Table 2 Orthogonal cutting tests of titanium alloy Ti6Al4V.

Fig.12 Flow chart of cutting temperature prediction programed in MATLAB?.

Fig.13(a),(b),and(c)are the temperature field cloud maps of the cutting tool obtained by the proposed analytical model in the cases of 4th,5th,and 6th group tests,respectively.It can be seen that the high temperature region is mainly concentrated in the region near the tool nose,which is also the region with the largest temperature gradient.The maximum temperature on the rake face is located above the tool nose,which corresponds to the distribution characteristics of the actual cutting temperature field.In addition,it can be seen from Fig.13 that the high temperature region is moving away from the tool nose with an increase of the feed rate.The reason is that the lengths of both the tool-chip interface and the corresponding sticking friction region will increase as the feed rate increases.

Fig.14(a),(b),and(c)are the prediction temperature curves of the discrete points along the tool-chip interface in the cases of 4th,5th,and 6th group tests,respectively.The length values of the tool-chip contact interface calculated by the presented model in the cases of 4th,5th,and 6th group tests are 0.128 mm,0.202 mm,and 0.282 mm,respectively,and the distances from the tool-chip separation point to the maximum-temperature point on the tool-chip contact interface for the three cases are 0.084 mm,0.120 mm,and 0.153 mm accordingly.Asa result,it can be obtained that the maximum-temperature points are right above the tool nose by distances of 0.044 mm,0.082 mm,and 0.129 mm in the cases of 4th,5th,and 6th group tests,respectively.

Fig.15(a),(b),and(c)are the heat partition ratios Biof the discrete points along the tool-chip interface in the cases of 4th,5th,and 6th group tests,respectively.It can be seen that some values of the heat partition ratio Biare outside the range of values between 0 and 1.The reason is that the shear plane heat source affects not only the temperature directly on the chip side,but also the temperature indirectly on the tool side.Therefore,the heat partition ratio calculated here is not for the frictional heat source,instead it is for the combination effect of the shear plane heat source and the frictional heat source.

Fig.13 Temperature field cloud maps obtained by the proposed analytical model.

Fig.14 Prediction temperature curves of discrete points along tool-chip interface.

The predicted values of the temperature by the proposed model and the Huang-Liang model,as well as the experimental values at the same measuring points under the same cutting conditions,are listed in Table 3 for 9 groups of tests.For a more visual representation,the predicted and experimental values of the temperature are plotted as a histogram in Fig.16.It can be clearly seen that the predicted values of the proposed model agree well with the experimental results.The relative differences between the predicted values of the proposed model and the experimental results are from 0.49%to 9.00%,and those of up to 7 groups are less than 3.75%.In addition,by comparing the predicted values of the proposed model with those of the Huang-Liang model,it can be seen that the proposed model is more accurate in predicting the cutting temperature in orthogonal cutting of Ti6Al4V.

Fig.15 Heat partition ratios Biof discrete points along tool-chip interface.

It can be seen from Table 3 and Fig.16 that the predicted values are greater than the experimental ones at the same cutting condition.The main reason is that there are some assumptions in the modeling of the cutting temperature.It is assumed that all the mechanical energy during metal cutting is converted into thermal energy.However,in an actual cutting process,a small part of mechanical energy can be indeed converted into potential energy or other forms of energy.Moreover,a small amount of heat generated in the cutting process is dissipated into the surrounding environment due to convection heat transfer,which is not taken into accountin the proposed model.Hence,it is reasonable that the predicted temperature values at the measuring point are greater than the experimental ones.

Table 3 Predicted and experimental values of temperature at measuring points.

Fig.16 Histogram of predicted and experimental values of temperature.

4.Conclusions

In this paper,an improved analytical model based on the moving heat source method is developed for predicting the distribution of the cutting temperature in orthogonal cutting of titanium alloy Ti6Al4V.The proposed model adopts the desirable parts of the Komanduri-Hou model and the Huang-Liang model,and the tool relief angle is also taken into account.The frictional heat source along the tool-chip interface is processed by mirroring accordingly for obtaining the temperature rise.Validation experiments are carried out by cutting a thinwalled titanium alloy Ti6Al4V tube on a lathe with a sharp tool.The cutting tool temperature is measured by a standard K-type armored thermocouple that is embedded in the tool insert.The following conclusions are found from the verification experience:

1.The characteristics of the temperature field cloud map calculated by the presented model are consistent with the distribution characteristics of the actual cutting temperature field,and it is found that the high temperature region is moving away from the tool nose with an increase of the feed rate.

2.The relative differences between the temperature values predicted by the proposed model and those measured in the experiments are within 0.49%-9.00%.Moreover,the proposed model is more accurate than the Huang-Liang model in orthogonal cutting of Ti6Al4V in the ranges of the selected experimental cutting parameters.Hence,the improved analytical model demonstrates its ability of predicting the cutting temperature in orthogonal cutting of Ti6Al4V.

Acknowledgements

The authors wish to thankfully acknowledge Zhuzhou Cemented Carbide Cutting Tools Co.,Ltd.(ZCCCT)for providing cutting tools for our experiments.This work was cosupported by National Science and Technology Major Project of China(No.2015ZX04004001),National Natural Science Foundation of China(No.51875473),Natural Science Foundation of Shaanxi province of China(No.2017JM5027),and Fundamental Research Funds for the Central Universities of China(No.3102017gx06007).