Prediction of Temperature Distribution in Orthogonal Machining Based on the Mechanics of the Cutting Process Using a Constitutive Model

: This paper presents an original method of predicting temperature distribution in orthogonal machining based on a constitutive model of various materials and the mechanics of their cutting process. Currently, temperature distribution is commonly investigated using arduous experiments, computationally inefﬁcient numerical analyses, and complex analytical models. In the method proposed herein, the average temperatures at the primary shear zone (PSZ) and the secondary shear zone (SSZ) were determined for various materials, based on a constitutive model and a chip-formation model using measurements of cutting force and chip thickness. The temperatures were determined when differences between predicted shear stresses using the Johnson–Cook constitutive model (J–C model) and those using a chip-formation model were minimal. J–C model constants from split Hopkinson pressure bar (SHPB) tests were adopted from the literature. Cutting conditions, experimental cutting force, and chip thickness were used to predict the shear stresses. The temperature predictions were compared to documented results in the literature for AISI 1045 steel and Al 6082-T6 aluminum in multiple tests in an effort to validate this methodology. Good agreement was observed for the tests with each material. Thanks to the reliable and easily measurable cutting forces and chip thicknesses, and the simple forms of the employed models, the presented methodology has less experimental complexity, less mathematical


Introduction
Determination of the temperature distribution is needed in the machining process because of its controlling influence on tool performance, and the quality of the machined part.Elevated temperatures have a negative impact on tool performance due to the softening of tool materials, and increasing diffusion, which also affect the quality of the machined part.It is well known that the increasing temperature in machining is caused by large plastic deformation.The shear plane heat source at the primary shear zone (PSZ) and the frictional heat source at the tool-chip interface (secondary shear zone, SSZ) are the two principal heat sources in machining.However, an accurate and convenient method for determining temperature distribution remains challenging due to the complexity of the contact condition, and the restricted accessibility during the cutting process.
Previous researchers made considerable progress in determining temperature distribution in machining.Experimental approaches, numerical approaches, and analytical approaches were developed in the past.
The heat fluxes in the tools, chips, and workpieces were determined from the energy balance in the controlled volume of the chip formation area [25].In order to reduce computing time and to simplify input requirements, Stephenson et al. developed a similar approach using insulated boundary conditions to predict transient temperature [26].Komanduri et al. developed a temperature prediction model that combined the effects of the shear plane heat source at the PSZ and the frictional heat source at the SSZ [27].The temperature increases in the chip and in the workpiece due to the heat source in the PSZ were solved using a modification of Hahn's moving oblique band heat source model [28].The temperature increase due to the heat source in the SSZ was solved using a modified Jaeger's moving band heat source model in the chip, and a stationary rectangular heat source model in the tool [29].Liang et al. further developed this model to predict temperature distribution with considerations of the tool thermal properties and the tool wear effects [30,31].Liang et al. also developed a cutting temperature model with an assumption of non-uniform heat intensity and partition ratio, and reported improved accuracy upon validation [32].Korkut et al. employed regression analysis and neural network analysis to predict the temperature of the tool-chip interface [33].The abovementioned analytical models have high mathematical complexity, which reduces the computational efficiency.
In this work, the authors present an original method of predicting temperature distribution in machining, specifically the average temperatures at the PSZ and at the SSZ, using a J-C model and a chip formation model.Experimental measurements of cutting forces and chip thicknesses, cutting conditions, and J-C model constants were used as inputs.The cutting conditions, and the reliable and easily measurable cutting forces and chip thicknesses were adopted from the literature, in which simple orthogonal cutting tests were conducted.The simple form of the J-C model, and the simple calculations in the chip formation model reduced the mathematical complexity and the computational cost of the proposed method.The J-C model constants are readily available for common metal materials in machining such as steels, aluminum alloys, and titanium alloys.Therefore, the proposed methodology has the advantages of less experimental complexity, less mathematical complexity, and high computational efficiency.

Johnson-Cook Constitutive Model
Constitutive relationships describe material behavior under various mechanical and thermal loading conditions.The J-C model is one of the most widely used constitutive relationships for the analytical modeling of force, temperature, and residual stress in machining because it is effective, simple, and easy to use.The J-C model is a semi-empirical constitutive model that predicts the flow stress of materials at high strains, high strain rates, and elevated temperatures.The J-C model can be expressed as the following equation: where A, B, C, m, and n are five materials constants.A is the yield strength, B is the strength coefficient, C is the strain rate coefficient, n is the strain hardening coefficient, and m is the thermal softening coefficient.σ is the flow stress, ε is the plastic strain, .
ε is the plastic strain rate, .ε 0 is the reference plastic strain rate, T is the temperature of the workpiece material, T r is the reference temperature, and T m is the melting temperature of the material.

Chip Formation Model
The chip formation model, as originally proposed by Oxley [34], based on orthogonal cutting, is widely used in the predictions of machining force, temperature, and residual stress with the available properties of workpiece materials, and cutting conditions.The chip formation model is illustrated in Figure 1, where α is the rake angle, φ is the shear angle, λ is average friction angle at the tool-chip interface, and θ is the angle between the resultant cutting force (R) and the primary shear zone (AB).t 1 and t 2 are the depth of the cut, and the chip thickness, respectively.V, V s , and V c are the cutting velocity, the shear velocity, and the chip velocity respectively.w is the cutting width, which is not shown in Figure 1.The assumptions made in the chip formation model are a perfectly sharp cutting tool, plain strain, steady-state conditions, a straight-line shape of shear zones, and uniform stress and temperature at shear zones.The PSZ and SSZ are characterized as the region encompassing line AB, and the region encompassing the tool-chip interface, respectively, as illustrated in Figure 2. Strain rate constants (C 0 and δ) are defined as the ratio of the shear plane length to the thickness of the PSZ (l/∆s 2 ), and the ratio of the thickness of the SSZ to the chip thickness (∆s 1 /t 2 ), respectively.uniform stress and temperature at shear zones.The PSZ and SSZ are characterized as the region encompassing line AB, and the region encompassing the tool-chip interface, respectively, as illustrated in Figure 2. Strain rate constants ( and ) are defined as the ratio of the shear plane length to the thickness of the PSZ (l/Δs2), and the ratio of the thickness of the SSZ to the chip thickness (Δs1/t2), respectively.

Experimental Measurements
The experimental data used in this paper were adopted from previous works found in the literature [36,37].The experimental forces were measured using a piezoelectric dynamometer in three mutually perpendicular directions.The cutting force and thrust force were then obtained based on cutting geometry [36,37].The chip thicknesses were measured using a micrometer [37].The temperature measurements at the PSZ and the SSZ were achieved using an IR camera with uniform stress and temperature at shear zones.The PSZ and SSZ are characterized as the region encompassing line AB, and the region encompassing the tool-chip interface, respectively, as illustrated in Figure 2. Strain rate constants ( and ) are defined as the ratio of the shear plane length to the thickness of the PSZ (l/Δs2), and the ratio of the thickness of the SSZ to the chip thickness (Δs1/t2), respectively.

Experimental Measurements
The experimental data used in this paper were adopted from previous works found in the literature [36,37].The experimental forces were measured using a piezoelectric dynamometer in three mutually perpendicular directions.The cutting force and thrust force were then obtained based on cutting geometry [36,37].The chip thicknesses were measured using a micrometer [37].The temperature measurements at the PSZ and the SSZ were achieved using an IR camera with

Experimental Measurements
The experimental data used in this paper were adopted from previous works found in the literature [36,37].The experimental forces were measured using a piezoelectric dynamometer in three mutually perpendicular directions.The cutting force and thrust force were then obtained based on cutting geometry [36,37].The chip thicknesses were measured using a micrometer [37].The temperature measurements at the PSZ and the SSZ were achieved using an IR camera with high spatial resolution, and a tool-work thermocouple technique, respectively.This paper focused on the determination methodology.The techniques for experimental measurements, especially those for temperature distribution, are briefly explained below.Further information can be found in References [7,38].
To measure the temperature at the PSZ, an IR camera was placed straight above the rake face of the tool so as to measure the temperature on the chip's free side, as shown in Figure 3.Only the contact zone between the tool and the workpiece was drawn, in an effort to clearly show the cutting edge.Measurements for each cutting condition were made at least in triplicate.The discernable temperature at the cutting edge was considered as the temperature at the PSZ once it had become stable.high spatial resolution, and a tool-work thermocouple technique, respectively.This paper focused on the determination methodology.The techniques for experimental measurements, especially those for temperature distribution, are briefly explained below.Further information can be found in References [7,38].
To measure the temperature at the PSZ, an IR camera was placed straight above the rake face of the tool so as to measure the temperature on the chip's free side, as shown in Figure 3.Only the contact zone between the tool and the workpiece was drawn, in an effort to clearly show the cutting edge.Measurements for each cutting condition were made at least in triplicate.The discernable temperature at the cutting edge was considered as the temperature at the PSZ once it had become stable.To measure the temperature at the SSZ, a tool-work thermocouple technique was employed as illustrated in Figure 4, in which the tool and the workpiece were connected by lead wires, forming a closed circuit.The contact area between the tool and the workpiece formed a hot junction, while the remote sections of the tool and the workpiece formed a cold junction.A copper brush was used to maintain connection during machining.Orthogonal cutting was enforced by turning a tubular workpiece with the cutting edge of the tool perpendicular to the cutting direction, as illustrated in the side view in Figure 4.  [38].To enforce orthogonal cutting, a tubular workpiece with uniform wall thickness was used [39].The cutting force was in the tangential direction of the tubular workpiece, and perpendicular to the tool's cutting edge, as shown in the side-view drawing (blue box).To measure the temperature at the SSZ, a tool-work thermocouple technique was employed as illustrated in Figure 4, in which the tool and the workpiece were connected by lead wires, forming a closed circuit.The contact area between the tool and the workpiece formed a hot junction, while the remote sections of the tool and the workpiece formed a cold junction.A copper brush was used to maintain connection during machining.Orthogonal cutting was enforced by turning a tubular workpiece with the cutting edge of the tool perpendicular to the cutting direction, as illustrated in the side view in Figure 4.
high spatial resolution, and a tool-work thermocouple technique, respectively.This paper focused on the determination methodology.The techniques for experimental measurements, especially those for temperature distribution, are briefly explained below.Further information can be found in References [7,38].
To measure the temperature at the PSZ, an IR camera was placed straight above the rake face of the tool so as to measure the temperature on the chip's free side, as shown in Figure 3.Only the contact zone between the tool and the workpiece was drawn, in an effort to clearly show the cutting edge.Measurements for each cutting condition were made at least in triplicate.The discernable temperature at the cutting edge was considered as the temperature at the PSZ once it had become stable.To measure the temperature at the SSZ, a tool-work thermocouple technique was employed as illustrated in Figure 4, in which the tool and the workpiece were connected by lead wires, forming a closed circuit.The contact area between the tool and the workpiece formed a hot junction, while the remote sections of the tool and the workpiece formed a cold junction.A copper brush was used to maintain connection during machining.Orthogonal cutting was enforced by turning a tubular workpiece with the cutting edge of the tool perpendicular to the cutting direction, as illustrated in the side view in Figure 4.  [38].To enforce orthogonal cutting, a tubular workpiece with uniform wall thickness was used [39].The cutting force was in the tangential direction of the tubular workpiece, and perpendicular to the tool's cutting edge, as shown in the side-view drawing (blue box).[38].To enforce orthogonal cutting, a tubular workpiece with uniform wall thickness was used [39].The cutting force was in the tangential direction of the tubular workpiece, and perpendicular to the tool's cutting edge, as shown in the side-view drawing (blue box).

Methodology and Validation
The presented methodology predicted the average temperature at the PSZ, and the average temperature at the SSZ using experimental cutting forces, and experimental chip thicknesses.The PSZ and the SSZ are the two major heat sources in orthogonal cutting, and the average temperatures at these locations are needed to further predict machining forces and tool wear.Cutting conditions, including cutting velocity, depth of cut, width of cut, rake angle, and J-C model constants, were also used as inputs.This methodology was developed based on the J-C model and the chip formation model.The temperatures were determined when the difference between the predicted shear stress using the J-C model and that using the chip formation model reached an acceptably low value, as shown in Figure 5.The temperature at the PSZ and the temperature at the SSZ were determined with calculations of primary shear stress and of secondary shear stress, respectively.

Methodology and Validation
The presented methodology predicted the average temperature at the PSZ, and the average temperature at the SSZ using experimental cutting forces, and experimental chip thicknesses.The PSZ and the SSZ are the two major heat sources in orthogonal cutting, and the average temperatures at these locations are needed to further predict machining forces and tool wear.Cutting conditions, including cutting velocity, depth of cut, width of cut, rake angle, and J-C model constants, were also used as inputs.This methodology was developed based on the J-C model and the chip formation model.The temperatures were determined when the difference between the predicted shear stress using the J-C model and that using the chip formation model reached an acceptably low value, as shown in Figure 5.The temperature at the PSZ and the temperature at the SSZ were determined with calculations of primary shear stress and of secondary shear stress, respectively.The chip compression ratio was expressed as the equations below, with an assumption of constant material flow rate.The shear angle (φ) was then calculated as The length of the shear zone (l AB ), and the strain (ε AB ) on the shear plane AB with the von Mises yield criterion were calculated as A strain hardening constant (n eq ) was expressed as The angles (θ and λ) in the chip formation model were expressed as The resultant force (R), the normal force at the tool-chip interface (N), the shear force at the primary shear zone AB (F s ), and the frictional force at the tool-chip interface (F) were calculated as The tool-chip contact length (h) at the tool-chip interface was expressed as The normal stress (σ N ) at the tool-chip interface was calculated as The normal stress was found with the J-C model, with stress boundary conditions on the primary shear zone defined as The Oxley constant (C 0 ) was determined when the difference between the normal stress (σ N ) and the normal stress σ N was minimal.Then, the strain rate on the shear plane AB with the von Mises yield criterion was calculated as .
The shear flow stress (k AB ) was calculated based on the chip formation model as The shear flow stress on the shear plane AB with the von Mises yield criterion was also calculated with the J-C model as The average temperature at the PSZ (T AB ) was determined when the difference between the shear flow stress (k AB ) and the shear flow stress (k AB ) was minimal.
The strain and the strain rate at the tool-chip interface were expressed with the von Mises yield criterion as .
The flow stress at the SSZ was calculated with the J-C model as The shear stress at the SSZ was also calculated as The average temperature at the SSZ (T int ) was determined when the difference between the shear flow stress (τ int ) and the shear flow stress (k int ) was minimal.The strain rate constant (δ) was determined by minimizing the cutting force (F c ), which was achieved by minimizing k int due to the positive correlation.F c was inversely calculated based on the equality between τ int and k int , and the chip formation model.
To test the proposed methodology, AISI 1045 steel and Al 6082-T6 aluminum were chosen for multiple tests for temperature predictions.The predicted temperatures were validated by comparing them to documented results found in the literature.

Results and Discussion
A computer program in MATLAB (2017) was developed to carry out the proposed methodology.The predictions of temperature distribution in machining AISI 1045 steel and Al 6082-T6 aluminum were made in multiple tests under various cutting conditions.As given in Table 1, the J-C model constants of AISI 1045 steel and Al 6082-T6 aluminum were separately adopted from the literature, in which split Hopkinson pressure bar (SHPB) tests were conducted to determine the model constants [37,40].The cutting conditions, including cutting velocity, depth of cut, width of cut, rake angle, and experimental measurements of force and chip thickness, are given in Table 2 [36,37].The values of chip thickness in machining Al 6082-T6 aluminum were adopted from predictions in the literature due to the unavailability of experimental data, and were validated using machining force because the chip thickness was utilized as an intermediate variable in the force prediction in each test.The machining forces were validated by comparing them to experimental forces.Detailed information regarding the validation of documented values is given in Appendix A, Table A1.Note: * represents values adopted from predictions in the literature [41].
The temperature distributions were predicted in four tests for each material.The predicted temperature at the PSZ (T AB ), the temperature at the SSZ (T int ), and other variables using the proposed method are listed in Table 3.To validate this methodology, the predicted values were individually compared to documented values.The documented temperatures were adopted from the literature [41,42], and validated by force comparisons because the chip thicknesses were intermediate variables in predicting machining forces.Good agreement was observed, as shown in Table A1.Force validation errors for AISI 1045 steel were found to be generally larger than those for Al 6082-T6 aluminum because of the differing prediction methods.The differing prediction methods are discussed further below.The error bars were added to the documented values due to the deviations of the inputs.The documented temperatures in the literature were determined using the predicted force, and the predicted chip thickness.The predictions using the proposed method utilized experimental force, and experimental chip thickness.Each error bar was calculated as the difference between the temperature using the adopted predicted values, and the temperature using the adopted experimental values in the proposed methodology.The comparisons of the temperatures in machining AISI 1045 steel and Al 6082-T6 aluminum are illustrated in Figures 6 and 7, respectively.The deviations between the predicted temperatures and the documented values are discussed below, regarding the accuracy of the input force and the input chip thickness, and the methodology for calculating the documented values.As shown in Figure 6, the predicted temperatures in the PSZ were larger than their corresponding documented values because the documented values were calculated using a heat partition equation ( = + Δ ), in which the heat partition ratio ( ) was taken as 1, based on the best performance of the predictions [40].This assumption gave acceptable prediction accuracy, as shown in Table A1, but it was not true in real cutting tests.On the other hand, the proposed method did not need this assumption for the temperature prediction.The other heat partition ratio ( , = + ∆ + ∆ ), used in calculating the temperature in the SSZ, was taken as 0.75, based on Xiong's study [43].This value was reported for their successful temperature prediction in the SSZ.The predicted temperatures in the SSZ were in good agreement with the documented values.As shown in Figure 6, the predicted temperatures in the PSZ were larger than their corresponding documented values because the documented values were calculated using a heat partition equation ( = + Δ ), in which the heat partition ratio ( ) was taken as 1, based on the best performance of the predictions [40].This assumption gave acceptable prediction accuracy, as shown in Table A1, but it was not true in real cutting tests.On the other hand, the proposed method did not need this assumption for the temperature prediction.The other heat partition ratio ( , = + ∆ + ∆ ), used in calculating the temperature in the SSZ, was taken as 0.75, based on Xiong's study [43].This value was reported for their successful temperature prediction in the SSZ.The predicted temperatures in the SSZ were in good agreement with the documented values.
(a) As shown in Figure 7, the documented temperatures were calculated using mean value calculations from a moving band heat source [41].The predicted temperatures were in good agreement with the documented values; however, they were smaller than the documented values due to the assumption of a perfectly sharp cutting tool edge used in the proposed method.Smaller temperature and force predictions with an assumption of a perfectly sharp tool were also reported in previously published works [21,40,44].
The proposed method predicted temperatures using the J-C model and the chip formation model.The availability of J-C constants for metal materials commonly used in machining is increasing; however, it is still a major limitation.For materials such as AISI 316L steel and the Ti6Al4V titanium alloy, the J-C constants must be chosen from a large number of J-C constants available for the same material [45,46].The error bars of documented temperatures in machining AISI 1045 steel were found to be generally larger than the error bars of documented temperatures in machining Al 6082-T6 aluminum because of the large input force errors in predicting temperatures for AISI 1045 steel.Therefore, the accurate measurements of cutting force and chip thickness were required for the temperature prediction.

Conclusions
In this work, an original methodology was presented for the prediction of temperature distribution in orthogonal cutting.Currently, the temperature distribution in machining is investigated with either arduous experiments using IR cameras, tool-chip interfaces, inserted thermocouples, etc., or inefficient numerical simulations, or complex analytical models, in which the determination of intermediate parameters remains difficult.This method was developed based on the J-C model and the chip formation model.The cutting conditions, the J-C model constants, the experimental cutting force, and the experimental chip thickness were used as inputs.The reliable and easily measurable cutting force and chip thickness were utilized in predicting the temperatures at the PSZ and at the SSZ.The temperatures were determined when the differences between the calculated shear stresses using the chip formation model and those using the J-C model reached acceptably low values.To validate the proposed methodology, machining data for AISI 1045 steel and Al 6082-T6 aluminum under various cutting conditions were adopted from the literature to predict the corresponding temperatures.The predicted temperatures were then compared to the documented results.Good agreement was observed for the tests with AISI 1045 As shown in Figure 6, the predicted temperatures in the PSZ were larger than their corresponding documented values because the documented values were calculated using a heat partition equation (T AB = T r + η∆T SZ ), in which the heat partition ratio (η) was taken as 1, based on the best performance of the predictions [40].This assumption gave acceptable prediction accuracy, as shown in Table A1, but it was not true in real cutting tests.On the other hand, the proposed method did not need this assumption for the temperature prediction.The other heat partition ratio (ψ, T int = T r + ∆T SZ + ψ∆T M ), used in calculating the temperature in the SSZ, was taken as 0.75, based on Xiong's study [43].This value was reported for their successful temperature prediction in the SSZ.The predicted temperatures in the SSZ were in good agreement with the documented values.
As shown in Figure 7, the documented temperatures were calculated using mean value calculations from a moving band heat source [41].The predicted temperatures were in good agreement with the documented values; however, they were smaller than the documented values due to the assumption of a perfectly sharp cutting tool edge used in the proposed method.Smaller temperature and force predictions with an assumption of a perfectly sharp tool were also reported in previously published works [21,40,44].
The proposed method predicted temperatures using the J-C model and the chip formation model.The availability of J-C constants for metal materials commonly used in machining is increasing; however, it is still a major limitation.For materials such as AISI 316L steel and the Ti6Al4V titanium alloy, the J-C constants must be chosen from a large number of J-C constants available for the same material [45,46].The error bars of documented temperatures in machining AISI 1045 steel were found to be generally larger than the error bars of documented temperatures in machining Al 6082-T6 aluminum because of the large input force errors in predicting temperatures for AISI 1045 steel.Therefore, the accurate measurements of cutting force and chip thickness were required for the temperature prediction.

Conclusions
In this work, an original methodology was presented for the prediction of temperature distribution in orthogonal cutting.Currently, the temperature distribution in machining is investigated with either arduous experiments using IR cameras, tool-chip interfaces, inserted thermocouples, etc., or inefficient numerical simulations, or complex analytical models, in which the determination of intermediate parameters remains difficult.This method was developed based on the J-C model and the chip formation model.The cutting conditions, the J-C model constants, the experimental cutting force, and the experimental chip thickness were used as inputs.The reliable and easily measurable cutting force and chip thickness were utilized in predicting the temperatures at the PSZ and at the SSZ.The temperatures were determined when the differences between the calculated shear stresses using the chip formation model and those using the J-C model reached acceptably low values.To validate the proposed methodology, machining data for AISI 1045 steel and Al 6082-T6 aluminum under various cutting conditions were adopted from the literature to predict the corresponding temperatures.The predicted temperatures were then compared to the documented results.Good agreement was observed for the tests with AISI 1045 steel and Al 6082-T6 aluminum.In light of the fact that cutting force and chip thickness are reliable and easily measurable, the calculations in the J-C model and the chip formation model were simple and easy to use.The proposed methodology has the advantages of less experimental complexity, less mathematical complexity, and high computational efficiency.In the future, this methodology can be employed in the analytical modeling of machining force and tool wear.
Author Contributions: J.N. conceived and developed the proposed analytical model, extracted and analyzed the data, and wrote the paper.S.Y.L. provided general guidance, and proofread the manuscript writing.
Funding: This research was funded by US National Science Foundation grant number CMMI-1404827.The author would like to acknowledge the funding support from the US National Science Foundation.

Conflicts of Interest:
The authors declare no conflict of interest.The founding sponsors had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, and in the decision to publish the results.ε 0 reference strain rate k AB material flow stress on shear plane AB (calculated using the J-C model) τ int shear stress at the tool-chip interface (calculated using the chip formation model) k int shear stress at the tool-chip interface (calculated using the J-C model)

Nomenclature
σ N normal stress at the tool-chip interface (calculated using the chip formation model) σ N normal stress at the tool-chip interface (calculated using the J-C model) η heat partition ratio in calculated temperature in the PSZ

Figure 3 .
Figure 3. Schematic drawing of the temperature measurement with an infrared camera on the chip's free side in orthogonal cutting[7].

Figure 4 .
Figure 4. Schematic drawing of the temperature measurement using a tool-work thermocouple technique[38].To enforce orthogonal cutting, a tubular workpiece with uniform wall thickness was used[39].The cutting force was in the tangential direction of the tubular workpiece, and perpendicular to the tool's cutting edge, as shown in the side-view drawing (blue box).

Figure 3 .
Figure 3. Schematic drawing of the temperature measurement with an infrared camera on the chip's free side in orthogonal cutting[7].

Figure 3 .
Figure 3. Schematic drawing of the temperature measurement with an infrared camera on the chip's free side in orthogonal cutting[7].

Figure 4 .
Figure 4. Schematic drawing of the temperature measurement using a tool-work thermocouple technique[38].To enforce orthogonal cutting, a tubular workpiece with uniform wall thickness was used[39].The cutting force was in the tangential direction of the tubular workpiece, and perpendicular to the tool's cutting edge, as shown in the side-view drawing (blue box).

Figure 4 .
Figure 4. Schematic drawing of the temperature measurement using a tool-work thermocouple technique[38].To enforce orthogonal cutting, a tubular workpiece with uniform wall thickness was used[39].The cutting force was in the tangential direction of the tubular workpiece, and perpendicular to the tool's cutting edge, as shown in the side-view drawing (blue box).

Figure 5 .Figure 5 .
Figure 5. Flowchart for the algorithm of temperature prediction based on the Johnson-Cook (J-C) model and the chip formation model.

Figure 6 .
Figure 6.Temperature predictions for machining AISI 1045 steel.The average temperatures in the primary shear zone are shown in (a); The average temperatures in the secondary shear zone are shown in (b).The documented values are predicted values adopted from the literature [42].

Figure 6 .Figure 6 .
Figure 6.Temperature predictions for machining AISI 1045 steel.The average temperatures in the primary shear zone are shown in (a); The average temperatures in the secondary shear zone are shown in (b).The documented values are predicted values adopted from the literature [42].

Figure 7 .
Figure 7. Temperature predictions for machining Al 6082-T6 aluminum.The average temperatures in the primary shear zone are shown in (a); The average temperatures in the secondary shear zone are shown in (b).The documented values are predicted values adopted from the literature [41].

Table 2 .
Cutting conditions and experimental measurements for machining AISI 1045 steel and Al 6082-T6 aluminum.

Table 3 .
Predicted temperature distribution, and other process variables.
the angle between the resultant force R and the primary shear plane AB C 0 Oxley constants (the ratio of the shear plane length to the thickness of the PSZ) δ strain rate constant (and the ratio of the thickness of the SSZ to the chip thickness)