CUTTING TEMPERATURE MODELING BASED ON NON-UNIFORM HEAT INTENSITY AND PARTITION RATIO

Transcription

1 Machining Science and Technology, 9: Copyright # 2005 Taylor & Francis Inc. ISSN: print/ online DOI: / CUTTING TEMPERATURE MODELING BASED ON NON-UNIFORM HEAT INTENSITY AND PARTITION RATIO Yong Huang & Department of Mechanical Engineering, Clemson University, Clemson, South Carolina, USA Steven Y. Liang & George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, Georgia, USA & The understanding of temperature distribution along the tool-chip interface is important for machining process planning and tool design. Among many temperature modeling studies, uniform heat partition ratio and=or uniform heat intensity along the interface are frequently assumed. This assumption is not true in actual machining and can lead to ill-estimated results at the presence of sticking and sliding. This paper presents a new analytical cutting temperature modeling approach that considers the combined effect of the primary and the secondary heat sources and solves the temperature rise along the tool-chip interface based on the non-uniform heat partition ratio and nonuniform heat intensity along the interface. For the chip side, the effect of the primary shear zone is modeled as a uniform moving oblique band heat source, while that of the secondary shear zone is modeled as a non-uniform moving band heat source within a semi-infinite medium. For the tool side, the effect of the secondary heat source is modeled as a non-uniform static rectangular heat source within a semi-infinite medium; and the primary heat source affects the temperature distribution on the tool side indirectly by affecting the heat partition ratio along the interface. Imaginary heat sources are considered as a result of the adiabatic boundary condition involved along the toolchip interface and of the insulated boundary conditions along both the chip back side and the tool flank face. The temperature matching condition along the tool-chip interface leads to the solution of distributed heat partition ratio by solving a set of linear equations. The proposed model is verified based on the published experimental data of the conventional turning process and it shows both satisfactory accuracy and improved match. Keywords Metal Cutting, Tool-chip Interface, Heat Intensity, Heat Partition, Thermal Modeling INTRODUCTION The understanding of temperature in machining is important for process planning and tool design inasmuch as cutting temperature is the dominating Address correspondence to Yong Huang, Department of Mechanical Engineering, Clemson University, Clemson, SC, , USA. yongh@clemson.edu

2 302 Y. Huang et al. factor for tool life, part quality, chip morphology, etc. The role that cutting temperature plays has been studied in detail as early as 1907 by Taylor (1). Following Taylor s pioneering work, numerous experimental, numerical, and analytical works have been documented to model the temperature along the tool-chip and=or tool-workpiece interfaces. In the early 20th century, much of the study has been directed toward the determination of average cutting temperature based on experimental measurement of thermo-emf (thermocouples), radiation (pyrometry, infra-red photography, etc.), and thermo-chemical reactions (thermo-colors) (2). Recently, other experimental methods have been pioneered by some researchers, such as a metallographic method (3) and a PVD film method (4), to name a few. Alternatively, a reverse estimation scheme was attempted to solve the cutting temperature profile based on indirectly measured temperature information (5). Apparently, experimental measurements are laborious and costly to carry out. As alternatives, numerical and analytical investigations have been receiving more and more attention. Numerical methods (FEM or other computational methods) were also applied to determine the temperature distribution with some important results documented by Tay et al. (6) and Dawson et al. (7) More recent progress has been reviewed by da Silva et al. (8) and Ng et al. (9). Although significant advances in computing technology have been witnessed recently, numerical methods are still time consuming and their accuracy depends on formulation methods, workpiece material constitutive models, the toolchip condition, and boundary conditions. Most importantly, at present, they are not efficient when used in process optimization, especially when using an exhaustive searching method. As a much less computationally intensive approach, analytical modeling attracts a lot of interest in efforts to optimize the machining processes. Further, analytical modeling approach can provide better insight to the underlying physical mechanisms. At the forefront of analytical modeling, based on the moving heat source method (10, 11), the analytical modeling of steady-state temperature in metal cutting has been presented by Hahn (12), Trigger and Chao (13 16), Loewen and Shaw (17), and more recently by Komanduri and Hou (18 20). The commonalties of these models are the consideration of heat sources at the primary and the secondary shear zones with related boundary conditions and the assumption that the bulk of the deformation energy is converted into heat while a negligible amount is stored as latent energy in the deformed metal. However, there are differences in these models in the way the heat source properties and boundary conditions were treated. In treating the primary heat source, the models are different in considering the nature, the moving direction, and velocity of the heat source, as well as in estimating the heat partition ratio and the boundary conditions. For example, when modeling the temperature rise on the chip

3 Cutting Temperature Modeling 303 side, the velocity of the moving heat source was treated differently as the chip velocity in work (12, 18), as the cutting velocity in work (14), and as the shear velocity in work (17). Furthermore, except for those of work (12, 14, 18), most models considered the primary heat source as the result of two bodies in sliding contact and used Blok s heat partition approach (21) for the evaluation of the average temperature within the primary shear zone. Realizing the fact that there is actually only one body involved in the primary shear zone, Komanduri et al. (18) proposed that the temperature rise on chip due to the primary heat source should be the effect of an oblique band heat source moving at chip velocity within a semi-infinite medium. They also argued that the temperature rise on the workpiece is the result of an oblique band heat source moving at the cutting velocity within a semi-infinite medium. No boundary along the tool-chip interface was considered when modeling the effect of the primary heat source, though an insulated boundary condition along the tool-chip interface was used when modeling the effect of the secondary heat source. In evaluating the combined effects of two heat sources, Komanduri et al. (20) considered the effect of the primary heat source on the final temperature rise within the tool by introducing an induced stationary rectangular heat source caused by the primary heat source. When modeling the effect of the secondary heat source, the heat partition approach is commonly applied by considering a contact pair of the tool and the chip. Chao et al. (15) used Blok s partition principle (21) by assuming a uniform heat partition ratio along the tool-chip contact length, but failed to achieve a temperature rise on the chip in agreement with that on the tool along the interface. To resolve this issue, Chao et al. suggested there was a non-uniform distribution of the heat partition ratio along the contact length and tried the functional analysis method, the discrete numerical iterative method (15), and a method in which the linear algebraic equations (16) are simultaneously solved. On the other hand, Chao et al. (16) simply assumed that the heat intensities along the tool-chip and tool-workpiece interfaces were uniform, which actually should have been the combination of plastic and elastic zones with different heat intensities, respectively. Recently, Komanduri et al. (19) furthered the functional analysis approach based on the idea of Chao et al. (15), but no perfect function has been found. Although those studies (16, 19) have adopted the concept of a non-uniform distribution of the heat partition ratio, they all treated heat intensity along the tool-chip interface as uniform. Wright et al. (22) modeled the effect of the secondary heat source by applying non-uniform heat intensity, but applied a uniform heat partition ratio which was determined empirically. Until now, there is no documented model that considers both the non-uniform heat partition ratio and non-uniform heat intensity along the tool-chip interface.

4 304 Y. Huang et al. This paper presents an analytical modeling approach to describe the temperature distribution along the tool-chip contact length in metal cutting. It addresses related boundary conditions, considers the combined effect of the primary and the secondary heat sources, and solves the temperature rise along the tool-chip interface based on the non-uniform heat partition ratio and non-uniform heat intensity along the interface. The following sections give the basic assumptions, theoretical formulation, and experimental evaluation of the model. PROPOSED ANALYTICAL MODEL Introduction and Basic Assumptions The temperature distribution along the tool-chip interface at a location mid-way across the width of cut is of key interest in this study since the temperature takes on its highest level at that location. The temperature at other locations can also be calculated by applying the approach described in this section. The heat source method introduced by Jaeger (10) and Carslaw et al. (11) is applied as the basis for the proposed analytical model. As a well-adopted assumption, the temperature rise on the chip side and on the tool side along the interface should be the same, thus the tool-chip interface boundary is considered to be adiabatic for the tool and the chip. The temperature rise on the chip side is attributed to the primary and the secondary heat sources, and that on the tool side is attributed to the secondary heat source. Considering the primary heat source on the chip as a uniform moving oblique band heat source, and the secondary heat source as a non-uniform moving band heat source within a semi-infinite medium, the temperature rise on the chip side can be expressed as: h chip shear þ h chip friction. Considering the secondary heat source acting on the tool to be a non-uniform static rectangular heat source, within a semi-infinite medium, the temperature rise on the tool side at a position midway across width of cut can be expressed as: h tool friction. The heat partition ratio of the secondary heat source going to the chip is specified as a function BðxÞ (or Bðx 0 Þ) along the contact length, so the remaining (1 BðxÞ) (or 1 Bðx 0 Þ) of heat is going to the tool as shown in Figure 1. The primary heat source affects the temperature distribution on the tool side indirectly by affecting the heat partition ratio along the interface. The basic assumptions involved in the study are: 1. The generated heat flow and the temperature distribution are in steady state; 2. All of the deformation energy within the deformation zones is converted into heat; a negligible amount is stored as latent energy in the deformed

5 Cutting Temperature Modeling 305 FIGURE 1 Heat sources and heat partition along the tool-chip interface. metal; and heat loss along the interface and at all surfaces of the tool and the chip is insignificant; 3. The dimensions of the tool are so large compared to the chip cross section that the tool can be considered as infinite; 4. The primary and the secondary heat sources are plane heat sources and the nature of the secondary heat source is not affected by the possible crater wear; 5. There is no redistribution of thermal shear energy going into the chip during the very short time when the chip is in contact with the tool; this assumption appears to be well founded for the normal cutting operation involving continuous chip formation without built-up-edge (17). Thermal Modeling of the Oblique Moving Band Heat Source The moving heat source method was first addressed by Jaeger (10) and applied in many engineering applications. Jaeger s general problem of a moving band heat source within an infinite medium with moving velocity coinciding with the direction of heat source length was extended to handle the oblique moving band heat source often encountered in metal cutting. The primary heat source is commonly modeled as an oblique moving band heat source. As shown in Figure 2, the temperature field under the effect of the oblique moving band uniform heat source with velocity V within an infinite medium can be expressed as (11, 18): hðx ; ZÞ ¼ q 2pk Z L 0 V 2a K 0 2a R l cos uþv ðx e dl ð1þ

6 306 Y. Huang et al. FIGURE 2 Schematic of the oblique moving band heat source in an infinite medium. Chip Side Temperature Modeling along the Tool-Chip Interface Modeling the Effect of the Primary Heat Source The effect of the primary heat source on the chip side is modeled as an oblique moving band heat source as shown in Figure 3. Recently, Komanduri et al. (18) proposed that the temperature rise on the chip side is a problem of a band heat source moving within the chip under the surface OA, which is considered as the insulated boundary for a semi-infinite medium (chip). They considered the chip as a semi-infinite body, treated the part below the shear plane (OB) as imaginary, and there no boundary condition specified along BC. What is considered herein, however, is that there is an adiabatic boundary condition along the tool-chip contact length (BC) in light of the fact that the temperature rise on both the chip and tool sides of the interface is identical. This paper also considers that the oblique heat source moves at the chip velocity into the workpiece below the shear plane which is considered as an infinite body as shown in Figure 3. As the insulated condition extends from AO to the workpiece while the adiabatic condition extends from CB to the workpiece, it can be assumed that there FIGURE 3 Schematic of the oblique moving band heat source relative to the chip side (the primary heat source moves at the chip velocity).

7 Cutting Temperature Modeling 307 FIGURE 4 Heat transfer model of the primary heat source relative to the chip side. are two main imaginary heat sources AA and BB, as shown in Figure 4, with intensity equivalent to that of the primary heat source (11). In reality, there are no boundaries extending from AO and CB on the workpiece side, therefore the imaginary heat intensity cannot be the same as that of the primary heat source. To simplify the problem, the intensities of two imaginary heat sources are treated to be half of that of the primary heat source. By considering two main imaginary heat sources, AA and BB, the boundaries in Figure 4 can be disregarded based on the equivalent thermal effect. The temperature rise due to the primary heat source and other two imaginary heat sources can be calculated individually based on the solution of the oblique moving band heat source within the infinite medium. When applying Equation (1), as shown in Figure 4, u equals 90 ð/ aþ for the heat source AA and 90 þð/ aþ for both the primary heat source and the heat source BB. The temperature rise on the chip side due to the primary heat source and those imaginary heat sources can be expressed as: h chip shear ðx ; ZÞ ¼ q shear 2pk chip Z L 0 XiÞVchip e ðx 2 a chip qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V chip K 0 ðx X i Þ 2 þðz Z i Þ 2 2 a chip þ 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 K V chip 0 ðx X i Þ 2 þð2t ch Z Z i Þ 2 2 a chip þ 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 K V chip 0 ðx X i Þ 2 þðz þ Z i Þ 2 dl i 2 a chip ð2þ where X i ¼ l li sinð/ aþ, Z i ¼ li cosð/ aþ, and L ¼ t ch = cosð/ aþ.

8 308 Y. Huang et al. Modeling the Effect of the Secondary Heat Source The heat generation rate of the secondary heat source along the contact length cannot simply be treated as uniform due to the presence of sticking and sliding zones. In the sticking zone the shear stress is s s and in the sliding zone it linearly decreases from s s to zero. Assuming the plane strain condition, the shear stress of the chip along the interface can be computed as: s s ¼ F wðl s þ 0:5l f Þ ð3þ Although Wright et al. (22) and Tay et al. (6) treated the heat generation mechanisms differently, they reached the same heat generation rate model as shown in Figure 5. It was considered that the heat intensity q frictional ðxþ along the sticking zone as s s V chip, and it linearly decreases from s s V chip to zero along the sliding zone. This heat generation model is used in this study to evaluate the required heat intensity for thermal modeling along the interface. A schematic of the heat transfer model from the non-uniform secondary heat source to the chip side is shown in Figure 6. For very small chip thicknesses, the boundary effect of the upper surface of the chip cannot be ignored (19). Here, the interface is considered as adiabatic, and the upper surface of the chip is considered as insulated. To relax those boundary conditions, the three main imaginary heat sources CC, DD, and EE are considered in this case based on the method of images (11). Heat source CC has the same intensity as the secondary heat source, while DD and EE have double that intensity. The temperature rise on the chip side due to the secondary heat source can be modeled as the effect of non-uniform band heat sources moving at the chip velocity: FIGURE 5 Heat generation rate model per unit area.

9 Cutting Temperature Modeling 309 FIGURE 6 Heat transfer model of the secondary heat source relative to the chip side. h chip friction ðx ; ZÞ ¼ 1 2pk chip 2K 0 ¼ 1 pk chip K 0 Z l 0 Z l 0 xþvchip BðxÞq frictional ðxþe ðx 2a chip R i V chip 2a chip þ 2K 0 Ri 0V chip R þ 00 2K 0 2a chip BðxÞq frictional ðxþe xþv ðx chip 2a R i V chip 2a chip þ K 0 Ri 0V chip R þ 00 K 0 2a chip i V chip 2a chip i V chip 2a chip dx dx qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where R i ¼ ðx xþ 2 þ Z 2 ; Ri 0 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx xþ 2 þð2t ch ZÞ 2, and Ri 00 ¼ ðx xþ 2 þð2t ch þ ZÞ 2. ð4þ Tool Side Temperature Modeling along the Tool-Chip Interface As discussed before, 1 BðxÞ ðor 1 Bðx 0 ÞÞ of the secondary heat source is transferred to the tool as a non-uniform static rectangular heat source. The interface boundary is considered as adiabatic and the tool clearance face as an insulated boundary. Here, the rake angle effect is ignored, and a cutting wedge angle of 90 is considered to simplify the problem, which is a good approximation for most cutting tools. Thus, there are two main imaginary heat sources FF and GG. The heat intensity of

10 310 Y. Huang et al. FIGURE 7 Heat transfer of the secondary heat source relative to the tool side. the imaginary heat source FF is the same as that of the secondary heat source and the heat intensity of the imaginary heat source GG is double that of the secondary heat source. The related heat transfer model is shown in Figure 7. The total temperature rise at any point M(X 0,Y 0,Z 0 ) on the tool side is then: h tool friction ðx 0 ; Y 0 ; Z 0 Þ¼ 1 Z l Z w ½1 Bðx 0 ÞŠq frictional ðx 0 Þdx þ 2 4pk tool 0 w R 2 i Ri 0 ¼ 1 Z l Z w ½1 Bðx 0 ÞŠq frictional ðx 0 Þdx þ 1 2pk tool 0 R i Ri 0 w 2 dy 0 dy 0 ð5þ where R i ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx 0 x 0 Þ 2 þðy 0 y 0 Þ 2 þz 02 ; Ri 0 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx 0 2l þx 0 Þ 2 þðy 0 y 0 Þ 2 þz 02. Solution for Temperature Rise along the Tool-Chip Interface It is assumed that along the tool-chip interface the temperature rise on the chip side (Z ¼ 0) and on the tool side at a mid way location across the width of cut (Y 0 ¼ 0 and Z 0 ¼ 0) should be equal, that is, h chip shear ðx ; 0Þþh chip friction ðx ; 0Þþh 0 ¼ h tool friction ðx 0 ; 0; 0Þþh 0 ð6þ

11 Cutting Temperature Modeling 311 FIGURE 8 Schematic for numerical computation of the temperature rise in thermal modeling. To numerically solve the heat partition ratio BðxÞ (or Bðx 0 Þ) along the interface, the contact length along the tool-chip interface is divided into n segmental sections and the heat partition ratio within each section is represented as constant values B 1 ;...; B n as shown in Figure 8. By considering these segmental sections along the interface, Equation (6) can be rewritten based on Equations (2), (4), and (5) as: ½A 1 Š nn ½BŠ n1 ¼ ½A 2 Š n1 ð7þ where along the overlapped X and X 0 axes, the elements of matrices ½A 1 and ½A 2 Š n1 ð0 i; j nþ are calculated as: ½A 1 Š nn ði; jþ ¼ 1 2pk tool q frictional ðx 0 j Þ Z w 2 w 2 þ 1 pk chip q frictional ðx j Þe 0 Š nn B 1 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiady 0 ðxi 0 x0 j Þ2 þ y 02 ðxi 0 2l þ x0 j Þ2 þ y qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 < ðx i x j Þ 2 V chip 2a K A : ðx i x j ÞV chip 2a chip 0qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx i x j Þ 2 þ 4tch 2 þ2k 2a chip V chip 19 = A ; 1

12 312 Y. Huang et al. ½A 2 Š n1 ðiþ ¼ 1 2pk tool Z w 2 w 2 0 Z l 0 q frictional ðx 0 Þdx 0 B 1 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiady 0 ðxi 0 x0 Þ 2 þ y 0 2 ðxi 0 2l þ x0 Þ 2 þ y 0 2 q 2a chip V chip K 0 q Z L Xi ÞVchip shear e ðxi 4pk chip 0 þ3k 0 2a chip V chip 2a chip 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx i X i Þ 2 þð2t ch Z i Þ 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx i X i Þ 2 þðz i Þ 2 dl i The temperatures at every section are under not only the effect of the heat source of this section, but also of the heat sources of all other sections. q frictional ðxþ is calculated based on Equation (3) and Figure 5, and q shear is computed based on Equation (9) in the following section. The partition ratio B is predicted in terms of discrete B i s by solving Equation (7). There is no direct effect from the primary heat source on the tool side as seen from Equation (6). As a matter of fact, the contribution from the primary heat source on the tool side has not been ignored, and the solved heat partition ratio implies this effect indirectly. The primary heat source changes the temperature distribution on the chip side as well as on the tool side indirectly by affecting the heat partition ratio. With the estimated BðxÞ, the temperature distribution hðxþ along the tool-chip interface at the position mid way across width of cut for a fresh tool can be predicted as follows by including the room temperature h 0. hðxþ ¼h chip shear ðx ; 0Þþh chip friction ðx ; 0Þþh 0 ð8þ MODEL VALIDATION Comparison with Data of Loewen et al. To verify the proposed temperature model, experimental data from Loewen and Shaw (17) is considered here. The cutting conditions for this orthogonal machining process were: workpiece material was SAE B1113 steel; tool was K2 S carbide with 20 rake angle; the cut was orthogonal with cutting speed of 139 m=min (232 cm=s), undeformed chip thickness (feed) of cm, width of cut (depth of cut) of cm, chip contact length of cm, cutting force of 356 N, thrust force of 125 N, chip thickness ratio of 0.51, shear angle of 30, at room temperature of 24 C.

13 Cutting Temperature Modeling 313 The measured sticking and sliding lengths are indispensable in implementing this model. Since the sticking and sliding lengths were not documented for this case, it is assumed that sticking and sliding lengths are equal in this case and this assumption was justified (6, 23). The input variables required by the proposed model, including the chip velocity V chip, the frictional force on the rake face F, and the primary heat intensity q shear can be calculated as: V chip ¼ rv cutting F ¼ F c sinðaþþf t cosðaþ q shear ¼ ðf c cosð/þ F t sinð/þþðv cutting cosðaþ=cosð/ aþþ =tch rw cscð/þ ð9þ Further, q frictional ðxþ is determined based on Equation (3) and Figure 5. Ideally, in applying the proposed temperature model, the thermal properties of the workpiece and the tool should be evaluated at every temperature point because they are expected to be temperature dependent. As suggested (16), a temperature value intermediate between the bulk workpiece material and the average interface temperature is considered to evaluate thermal properties of the workpiece. In this case (17), thermal properties of workpiece material are evaluated at 260 C resulting in a thermal diffusivity of cm 2 =s and a thermal conductivity of J= cm(s) C. Carbide tool thermal conductivity is considered to be temperature independent in cutting (16), having a value of 0.57 J=cm(s) C (17). Although it has been argued recently that the thermal conductivity of carbide changes over a cutting temperature range, it is taken as temperature independent here. The predicted temperature distribution along the contact length starting from the tool tip is shown in Figure 9. The dotted line is the temperature distribution on the chip side, and the solid line is that on the tool side. Those two lines match perfectly along the tool-chip interface, which confirms the adiabatic condition. The experimental average temperature along the tool-chip interface was 391 C, and the predicted average temperature is 408 C. No temperature distribution information was provided by Loewen et al. (17) From Figure 9, the maximum temperature is located around the middle of the contact length, which is reasonable in metal cutting. As a comparison, the predicted result is shown in Figure 10 for the same experimental conditions of Loewen et al. (17), but by applying the nonuniform heat partition ratio and uniform heat intensity. The average temperature is C. Although the average temperature is little closer

14 314 Y. Huang et al. FIGURE 9 The predicted temperature distribution along the tool-chip interface for Loewen et al. s case. (The dotted line is temperature distribution on the chip side, and the solid line is on the tool side.) to the experimental result, the maximum temperature point is located near the end of contact length, which contradicts general observations. Comparison with Data of Wright et al. For further evaluation of the model, the experimental data presented by Wright et al. (22) are examined. The cutting conditions for this semiorthogonal machining process were: workpiece material of annealed low carbon iron (0.07% C); tool of M34 high speed steel with 6 rake angle; FIGURE 10 The predicted temperature distribution along the tool-chip interface for Loewen et al. s case by applying uniform heat intensity. (The dotted line is temperature distribution on the chip side, and the solid line is on the tool side.)

15 Cutting Temperature Modeling 315 FIGURE 11 The experimental and analytical temperature distribution for the case of Wright et al. semi-orthogonal cut with cutting speed of 100 m=min (167 cm=s), undeformed chip thickness (feed) of 0.02 cm, width of cut (depth of cut) of 0.15 cm, chip contact length of 0.3 cm, sticking region length of cm, sliding region length of cm, cutting force of N, thrust force of N, chip thickness ratio of 0.29, shear angle of 16.5, and at room temperature of 20 C. Work material thermal diffusivity is evaluated as cm 2 =s and the thermal conductivity as J=cm(s) C at 350 C based on provided material properties (22). Thermal conductivity for HSS tool is taken as J=cm(s) C (24) and considered unaffected by temperature changes. Assuming the plane strain condition still holds for this semi-orthogonal cutting case, the predicted temperature profile along the rake face is shown in Figure 11. The predicted average temperature of the proposed model is 716 C, but the experimental average temperature is not provided there. The experimental temperature distribution based on the metallographic method and the analytical results of Wright et al. (22) are also shown in Figure 11 for comparison. It can be seen that the result of the proposed model agrees with the experimental curve. An improved accuracy can also be found by comparing to the analytical result of Wright et al., who applied the non-uniform heat intensity but not the uniform heat partition ratio. DISCUSSION By simply connecting the solved B i s, the determined heat partition ratios for the aforementioned validation cases are also shown in

16 316 Y. Huang et al. FIGURE 12 Heat partition ratios along the contact length from the tool tip for the cases of Loewen et al. (left) and Wright et al. (right). Figure 12. For comparison, Figure 13 presents several typical heat partition curves developed based on the functional analysis approach by Komanduri et al. (19) Although the heat partition ratio curves of Figure 12 and 13 resemble each other, there are several fundamental differences in finding these curves. First, instead of finding such functional relationships for every machining case by a trial and error approach, no prior knowledge on functional relationships is required to solve the heat partition ratio based on the proposed approach previously. Second, the approach proposed here gives the exact temperature matches along the interface as shown in Figures 9 and 10, whereas the functional analysis approach can only provide close temperature matches. Finally, the curves in Figure 12 are predicted by considering the combined effect of the primary and the secondary heat sources FIGURE 13 Heat partition ratios along the contact length from the tool tip based on Komanduri s approach (DB ¼ 0.3 and B ¼ 0.8).

17 Cutting Temperature Modeling 317 at the same time, but the curves in Figure 13 are developed to match the temperature rises only based on the effect of the secondary heat source. The deviation between the experimental data and the model prediction may be attributed to a number of factors: 1. The insulating boundary condition imposed on the back of the chip tends to overestimate temperature. The assumption that all the deformation energy is converted into heat may also overestimate temperature. 2. The temperature distribution at the position mid-way across the width of cut is the interest of the presented model. This temperature distribution should be higher than that of average temperature across the whole toolchip contact area, as compared with the experimental results (17). The underestimated temperature distribution in Figure 11 is considered to be due to applying the plane strain condition for the semi-orthogonal cutting process. 3. A 90 cutting wedge angle is used in modeling the secondary heat source by ignoring the actual rake angle and clearance angle values for simplicity, but Loewen et al. (17) used a 20 rake angle tool and Wright et al. (22) used a 6 rake angle one. 4. Generally speaking, there is still a lack of a precise experimental method that can be used to benchmark the analytical results (8). 5. The material properties are treated as constant with respect to the temperature gradient. Some factors discussed above may lead to the overestimation, and some may lead to the underestimation. The estimated temperature profiles may be higher or lower than the measurements depending on how strong these factors are. The combination of these factors determines the final error. DISCUSSION ON THE ADIABATIC BOUNDARY CONDITION ALONG THE TOOL-CHIP INTERFACE In this paper, the tool-chip interface boundary is considered adiabatic, and the combined effect from the primary and the secondary heat sources is calculated within the same coordinate system at the same time. The existence of the secondary heat source does not affect the thermal shear energy distribution along the primary shear plane as in Assumption 5, but it presents an adiabatic boundary to the primary heat source. In modeling the thermal contribution from the primary heat source, the existence of this adiabatic boundary condition should be considered as previously. Based on this modeling approach, there is no direct effect of the primary heat source on the tool side temperature rise, and the primary heat source indirectly contributes to the temperature distribution on the tool side by

18 318 Y. Huang et al. affecting the solved heat partition ratio. Some researchers shared the same approach in treating this boundary as adiabatic (13, 17), although there are differences in modeling the effect of individual heat source. Differing from the modeling approach in this paper, there is an alternative approach (20) in considering the tool-chip interface boundary condition for the primary heat source. In that approach, part of the heat from the primary heat source is considered to be continuously flowing through the tool-chip interface and contributing to the temperature rise on the tool side directly. This direct contribution from the primary heat source to the tool side is calculated by introducing an induced heat source by the primary heat source. The heat partition ratio along the tool-chip interface is solved only based on the secondary heat source. Under that modeling scheme, the overall temperature rise for the system is the simple superposition of the effect of the primary heat source and that of the secondary heat source. Loewen s case (17) is investigated again to appreciate the alternative modeling approach (20) in treating the tool-chip interface boundary condition; that is, no adiabatic boundary condition is considered along the interface. Different from that seen previously, the imaginary heat source BB of Figure 4 is not included in modeling the temperature distributions. In this case, the temperature rise on the chip side and along the tool-chip interface due to the primary heat source and the imaginary heat source AA can be calculated as: Z L XiÞVchip e ðx 2 a chip h 0 chip shear ðx ; ZÞ ¼ q shear 2pk chip 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V chip K 0 ðx X i Þ 2 þðz Z i Þ 2 2 a chip þ 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 K V chip 0 ðx X i Þ 2 þð2t ch Z Z i Þ 2 dl i 2 a chip ð10þ where X i ¼ l li sinð/ aþ, Z i ¼ li cosð/ aþ, and L ¼ t ch = cosð/ aþ as shown in Figure 4. When solely considering the effect of the secondary heat source, the temperature rise on the chip side (Z ¼ 0) and on the tool side (Y 0 ¼ 0 and Z 0 ¼ 0) along the tool-chip interface still should be equal, then this relationship can be expressed as: h chip friction ðx ; 0Þ ¼h tool friction ðx 0 ; 0; 0Þ ð11þ Without requiring any functional relationship information for the heat partition ratio BðxÞ, BðxÞ can be easily solved from Equation (11) using a

19 Cutting Temperature Modeling 319 FIGURE 14 Heat partition ratio and temperature rise along the contact length from the tool tip for Loewen et al. s case without adiabatic boundary along the interface. (The dotted line is temperature rise on the chip side, and the solid line is on the tool side.) similar approach described previously. With the solved BðxÞ, temperature distribution hðxþ along the tool-chip interface at the position mid way across width of cut for a fresh tool can be further predicted as Equation (12). The primary heat source has the same contribution to the temperature rise of both the tool and chip sides along the interface. FIGURE 15 The predicted temperature distribution along the tool-chip interface for Loewen et al. s case without adiabatic boundary along the interface. (The dotted line is temperature distribution on the chip side, and the solid line is on the tool side.)

20 320 Y. Huang et al. hðxþ ¼h 0 chip shear ðx ; 0Þþh chip frictionðx ; 0Þþh 0 ð12þ Some results are presented in Figures 14 and 15 using the aforementioned alternative approach (Equations (10), (11), and (12)) based on the non-uniform heat intensity and partition ratio scheme. The predicted temperature distribution is shown in Figure 15 and the average temperature along the tool-chip interface is 356 C. The experimental average temperature was 391 C (17). Compared with the previously predicted average temperatures which is 408 C, the temperature value from this alternative approach has a relatively larger absolute error and also underestimates the thermal effect. CONCLUSION Based on the extended Jaeger s heat source method, the temperature distribution along the interface is modeled by considering the combined effect of the primary and the secondary heat sources with associated boundary conditions. The effect of the non-uniform heat partition ratio and nonuniform heat intensity due to the presence of sticking and sliding zones are incorporated in the model. In modeling the temperature rise on the chip side, the primary shear zone is modeled as a uniform moving oblique band heat source and the secondary shear zone is described as a non-uniform moving band heat source within a semi-infinite medium. In modeling the temperature rise on the tool side, the secondary heat source is modeled as a non-uniform static rectangular heat source within a semi-infinite medium. With the application of imaginary heat sources the adiabatic and insulated boundary conditions are accounted for with the introduction of imaginary heat sources. The distribution of the heat partition ratio is found by matching the temperature rise on the tool to that on the chip along the interface. The estimation of cutting temperature based on the developed model is compared to the published experimental data of conventional turning processes. The comparison in terms of average cutting temperature as well as the distribution of temperature suggests reasonable accuracy of the model. ACKNOWLEDGMENTS The authors wish to express their gratitude to the reviewers for their penetrating inputs and to Dr. Richard E. Teets of the Delphi Research Lab, Michigan for his constructive comments.

21 Cutting Temperature Modeling 321 NOMENCLATURE a Thermal diffusivity a chip Thermal diffusivity of the chip BðxÞ Fraction of the secondary heat source transferred into the chip F Frictional force along the rake face F c ; F t Cutting force and thrust force k Thermal conductivity k chip ; k tool Thermal conductivity of the chip and the tool K 0 Modified Bessel function of the second kind of order zero l Tool-chip contact length l f Sliding length of the tool-chip contact zone l s Sticking length of the tool-chip contact zone L Length of the band heat source or shear plane M Point in the #medium to be measured about the temperature rise q Heat intensity of the heat source q frictional ðxþ Heat intensity of the secondary heat source along the interface q shear Heat intensity of the primary heat source r Chip thickness ratio R Distance between point M to be measured and the segment dl R i ; Ri 0; R i 00 Distance between the point M and heat source segments t ch Deformed chip thickness V Velocity V chip Chip velocity V cutting Cutting velocity X, Y, Z, X 0,Y 0,Z 0 The right handed Cartesian coordinates used in related figures w Width of cut a Tool rake angle / Shear angle u Angle between the band heat source and its moving direction hðxþ Temperature distribution along the tool-chip interface h 0 Room temperature h chip friction Temperature rise on chip side due to the secondary heat source

22 322 Y. Huang et al. h chip shear h 0 chip shear h tool friction s s Temperature rise on chip side due to the primary heat source Temperature rise due to the primary heat source without considering the adiabatic boundary condition along the interface Temperature rise on tool side due to the secondary heat source Shear stress REFERENCES [1] Taylor, F.W. (1907). On the Art of Cutting Metals. Transactions of the ASME, 29: [2] Barrow, G.A. (1973). Review of Experimental and Theoretical Techniques for Assessing Cutting Temperatures. Annals of the College International pour la Recherche en Productique (CIRP), 22(2): [3] Wright, P.K. and Trent, E.M. (1973). Metallographic Methods of Determining Temperature Gradients in Cutting Tools. Journal of The Iron and Steel Institute, 211(5): [4] Kato, T. and Fuji, H. (1996). PVD Film Method for Measuring the Temperature Distribution in Cutting Tools. ASME Journal of Engineering for Industry, 118: [5] Yen, D.W. and Wright, P.K. (1986). Remote Temperature Sensing Technique for Estimating the Cutting Interface Temperature Distribution. ASME Journal of Engineering for Industry, 108(4): [6] Tay, A.O., Stevenson, M.G., and de Vahl Davis, G. (1974). Using the Finite Element Method to Determine Temperature Distributions in Orthogonal Machining. Proceedings of the Institution of Mechanical Engineers, 188: [7] Dawson, P.R. and Malkin, S. (1984). Inclined Moving Heat Source Model for Calculating Metal Cutting Temperatures. Journal of Engineering for Industry, 106(3): [8] da Silva, M.B. and Wallbank, J. (1999). Cutting Temperature: Prediction and Measurement Methods A Review. Journal of Materials Processing Technology, 88: [9] Ng, E.G., Aspinwall, D.K., Brazil, D., and Monaghan, J. (1999). Modeling of Temperature and Forces when Orthogonally Machining Hardened Steel. International Journal of Machine Tools and Manufacture, 39: [10] Jaeger, J.C. (1942). Moving Sources of Heat and the Temperatures at Sliding Contacts. Proceedings Royal Society of NSW, 76: [11] Carslaw, H.S. and Jaeger, J.C. (1959). Conduction of Heat in Solids. Oxford, UK: Oxford University Press. [12] Hahn, R.S. (1951). On the Temperature Developed at the Shear Plane in the Metal Cutting Process. Proc. of First U.S. National Congress of Applied Mechanics, [13] Trigger, K.J. and Chao, B.T. (1951). An Analytical Evaluation of Metal Cutting Temperatures. Transactions of the ASME, 73: [14] Chao, B.T. and Trigger, K.J. (1953). The Significance of the Thermal Number in Metal Cutting. Transactions of the ASME, 75: [15] Chao, B.T. and Trigger, K.J. (1955). Temperature Distribution at the Tool-chip Interface in Metal Cutting. Transactions of the ASME, 77(2): [16] Chao, B.T. and Trigger, K.J. (1958). Temperature Distribution at Tool-chip and Tool-work Interface in Metal Cutting. Transactions of the ASME, 80(1): [17] Loewen, E.G. and Shaw, M.C. (1954). On the Analysis of Cutting Tool Temperatures. Transactions of the ASME, 76: [18] Komanduri, R. and Hou, Z.B. (2000). Thermal Modeling of the Metal Cutting Process, Part 1: Temperature Rise Distribution Due to Shear Plane Heat Source. International Journal of Mechanical Sciences, 42:

23 Cutting Temperature Modeling 323 [19] Komanduri, R. and Hou, Z.B. (2001). Thermal Modeling of the Metal Cutting Process, Part 2: Temperature Rise Distribution Due to Frictional Heat Source at the Tool-chip Interface. International Journal of Mechanical Sciences, 43: [20] Komanduri, R. and Hou, Z.B. (2001). Thermal Modeling of the Metal Cutting Process, Part 3: Temperature Rise Distribution Due to the Combined Effects of Shear Plane Heat Source and the Tool-chip Interface Frictional Heat Source. International Journal of Mechanical Sciences, 43: [21] Blok, H. (1938). Theoretical Study of Temperature Rise at Surface of Actual Contact under Oiliness Lubricating Conditions. Proceedings of General Discussion on Lubrication and Lubricants. Institution of Mechanical Engineers, [22] Wright, P.K., McCormick, S.P., and Miller, T. R. (1980). Effect of Rake Face Design on Cutting Tool Temperature Distributions. ASME Journal of Engineering for Industry, 102(2): [23] Zorev, N.N. (1966). Metal cutting mechanics. Oxford: Pergamon Press. [24] Childs, T.H.C., Maekawa, K., Obikawa, T., and Yamane, Y. (2000). Metal Machining: Theory and Applications. New York: John Wiley & Sons Inc.