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A worn tool force model for three-dimensional cutting operations
International Journal of Machine Tools & Manufacture 40 (2000) 1929–1950
A worn tool force model for three-dimensional cutting operations David W. Smithey, Shiv G. Kapoor, Richard E. DeVor
*
University of Illinois at Urbana-Champaign, 361 Computer and Systems Research Laboratory, 1308 West Main Street, Urbana, IL 61801, USA Received 28 May 1999; received in revised form 25 January 2000; accepted 3 March 2000
Abstract Previous studies have shown that there is a region on the flank of a worn cutting tool where plastic flow of the workpiece material occurs. This paper presents experimental data which shows that in three-dimensional cutting operations in which the nose of the tool is engaged, the region of plastic flow grows linearly with increases in total wearland width. A piecewise linear model is developed for modeling the growth of the plastic flow region, and the model is shown to be independent of cutting conditions. A worn tool force model for three-dimensional cutting operations that uses this concept is presented. The model requires a minimal number of sharp tool tests and only one worn tool test. An integral part of the worn tool force model is a contact model that is used to obtain the magnitude of the stresses on the flank of the tool. The force model is validated through comparison to data obtained from wear tests conducted over a range of cutting conditions and workpiece materials. It is also shown that for a given tool and workpiece material combination, the incremental increases in the cutting forces due to tool flank wear are solely a function of the amount and nature of the wear and are independent of the cutting condition in which the tool wear was produced. 2000 Elsevier Science Ltd. All rights reserved.
1. Introduction It is commonly known that during the metal cutting process, the tool geometry changes as a result of tool wear and that these changes can have undesirable effects on process performance. The most significant variation from sharp tool operation is an increase in cutting forces that can lead to deterioration in process stability, part accuracy and part surface finish. A model that can predict worn tool cutting forces without the need for numerous time-consuming wear tests could * Corresponding author. Tel.: +1-217-333-3432; fax: +1-217-244-9956. E-mail address: [email protected] (R.E. DeVor).
0890-6955/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 8 9 0 - 6 9 5 5 ( 0 0 ) 0 0 0 1 7 - 1
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be used to ensure proper operation of a machining process. In order to develop such a model, it is necessary to have an understanding of the nature of the tool flank contact. Thomsen [1,2] performed some of the earliest investigations into the nature of the contact on the tool flank. From these studies, Thomsen was able to conclude that plastic flow of the workpiece material can result from the contact between the flank of a worn tool and the workpiece and that the deformation work absorbed by the workpiece during metal cutting can be significant. Later, Kobayashi and Thomsen [4] found that good force predictions can be made by assuming that the contact on the flank of the tool is plastic in nature and by using Von Mises’ stress criterion to relate the flank stresses to the material properties. Later, Liu and Barash [5] investigated the apparent strain energy, strain hardening index and residual stress distribution in the workpiece after machining and found that the flank wearland width was one of the governing factors of these values. Since these parameters are closely related to plastic deformation, this further indicated that plastic flow of the workpiece is present and that the amount of plastic flow changes with wearland width. Several authors have suggested that the size of the region on the tool flank where plastic flow of the workpiece occurs increases linearly with increases in total wearland width [1–3,8]. Thomsen et al. [1] and Kobayashi and Thomsen [4] related the linear increases in cutting forces with increases in wearland width that they observed in their experimental data to a similar growth of the plastic flow region. Waldorf [8] took the concept further and developed a model for predicting worn tool cutting forces for orthogonal cutting based on a linear growth of the plastic flow region. In this model, it is assumed that after the total wearland width, VB, (the distance from the cutting edge to the back of the wearland) increases beyond a critical value, there is plastic flow at the front of the wearland and elastic contact at the back of the wearland. The contact is assumed to be totally elastic for VB values less than this critical value. Once VB is greater than this critical value, any increase in the total wearland width results in an equal increase in the width of the plastic flow region. This results in linear increases in the cutting forces as predicted by Waldorf’s model. In this paper, it is proposed that for three-dimensional cutting operations (i.e. when the nose of the cutting tool is engaged), the plastic flow region also grows linearly as the total wearland width increases and that this growth can be modeled independently of cutting conditions for a given workpiece material. Using this concept, a three-dimensional worn tool force model is developed and validated. The organization of the paper is as follows. First, experimental data that shows how the plastic flow region grows with increases in total wearland width is presented. Based on this data, a linear model of the plastic flow region growth is developed, and it is shown that the model is independent of cutting conditions. A force model for predicting worn tool cutting forces for three-dimensional cutting operations is then presented. The model discretizes the flank of the tool into small twodimensional elements and uses a contact model to determine the stresses on the individual elements. Finally, the force model is validated by comparison to worn tool force data obtained during conventional turning operations with a range of cutting conditions and workpiece materials.
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2. Experimental observations of the width of the plastic flow region in turning Fig. 1 illustrates the situation in which both plastic flow and elastic contact are present on the flank of a worn cutting tool for an arbitrary total wearland width, VB. The width of the plastic flow region is denoted by VBp. In order to observe the growth of the plastic flow region as the total wearland width increases, six wear experiments were conducted with a Class 40 gray cast iron workpiece (242 BHN). Tests were also conducted for 4140 steel and Al 357. These tests were performed on a Mori Seiki Frontier L1 CNC turning center. For the cast iron and 4140 steel tests, Kennametal KC910 TiC coated inserts were used. For the Al 357 test, a Carboloy grade 370 TiN coated insert was used. Also, it should be noted that the inserts used in the cast iron and the steel tests had flat rake faces while the inserts used in the Al 357 test had a grooved rake face geometry. The different rake face geometries were chosen to ensure that any observed trends in the data were not specific to one type of rake face geometry. During the experiments, the inserts were subjected to periodic inspections. During these inspections, the tool was removed from the machine and a picture of the flank was taken with a Nikon 40× microscope (Fig. 2). Since the wear rates of the tools varied with the different cutting conditions used and the inspections were made at given time intervals, the total number of inspections made varied from test to test. The specifications for the cutting conditions for all eight tests are given in Table 1. The pictures of the tool flanks taken during the experiments provide a means to measure the width of the plastic flow region. In the example tool flank pictures shown in Figs 2a and b, a
Fig. 1.
Worn tool flank with both plastic flow and elastic contact.
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Fig. 2. Plastic region shown on tool flank.
Table 1 Cutting conditions for worn tool testsa Test No.
V (m/min)
f (mm/rev)
as (°)
doc (mm)
Cast Iron, 1 Cast Iron, 2 Cast Iron, 3 Cast Iron, 4 Cast Iron, 5 Cast Iron, 6 4140 Steel Al 357
132 132 215 132 132 215 215 300
0.2 0.325 0.2 0.2 0.325 0.2 0.3 0.1
⫺10 ⫺10 ⫺10 ⫺5 ⫺5 ⫺5 ⫺5 ⫺5
2.25 2.25 2.25 2.25 2.25 2.25 1.75 1.50
For all tests: ab=⫺50, gl=00, rn=0.8 mm, where: as=side rake angle; ab=back rake angle; gl=lead angle; rn=nose radius; f=feed rate; V=cutting velocity. a
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Fig. 3.
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Growth of the plastic flow region for cast iron tests 1 and 2.
distinct line is visible running horizontally through the wearland. The region above the line is the plastic flow region that is characterized by a shinier surface. This is the result of the workpiece material being deposited on the flank. In the lower region, the surface is not as shiny and the tool material is much more evident. By measuring the width of the shinier region, a measurement of the width of the plastic flow region, VBp, was obtained. This value is an average width along the wearland. By measuring the width of the total scarred region, a measurement of the total wearland width, VB, was obtained. The data from the tests are shown in Figs. 3–7. In all of the data shown in Figs. 3–7, it can be seen that initially VBp=0. However, when the total wearland width reaches about 0.04–0.08 mm for the cast iron and steel tests and about 0.12 mm for the Al 357 test, plastic flow begins to appear and continues to grow linearly as the total wearland width increases. Physically, it is likely that the initial stresses and temperatures on the flank of the tool are not sufficiently large enough to cause yielding of the asperities on the workpiece surface. However, it is known that forces and temperatures on the flank increase as the
Fig. 4.
Growth of the plastic flow region for cast iron tests 3 and 4.
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Fig. 5.
Growth of the plastic flow region for cast iron tests 5 and 6.
Fig. 6.
Growth of the plastic flow region for 4140 steel test.
wearland width increases [6,9]. Eventually, the stresses and temperature on the flank reach levels that cause the asperities to yield and plastic flow begins to occur [1]. 3. Linear model of the growth of the plastic flow region Based on the observed data for the growth of the plastic flow region, a piecewise linear function was developed to model this growth: −c0 VBp⫽c0⫹c1VB if VB⬍VBcr then VBp⫽0 where: VBcr⫽ . c1
(1)
To obtain the coefficients c0 and c1 in Eq. (1), the linear function is fit through the data in which VBp⬎0. The VBcr value is the VB–axis intercept of the linear model, which represents the
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Fig. 7. Growth of the plastic flow region for Al 357 test.
critical total wearland width at which plastic flow is initiated. The model shown in Eq. (1) was fit to each of the eight sets of data presented above by the method of least squares. The model coefficients and the sample coefficients of determination for each set are given in Table 2. 3.1. Model parameter independence of cutting conditions Examining Figs. 3–5 (the six cast iron tests), it can be seen that all of the data sets have similar slopes and VB–axis intercepts. This seems to suggest that the growth of the plastic flow region Table 2 Fitted linear models of the growth of the plastic flow region Test No.
c0
c1
R2
Cast Iron, 1 Cast Iron, 2 Cast Iron, 3 Cast Iron, 4 Cast Iron, 5 Cast Iron, 6 4140 Steel Al 357
⫺0.085 ⫺0.046 ⫺0.059 ⫺0.090 ⫺0.080 ⫺0.109 ⫺0.118 ⫺0.155
1.08 1.14 1.28 1.30 1.23 1.54 1.46 1.18
0.926 0.992 0.996 0.997 0.977 0.974 0.997 0.959
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may be independent of the cutting conditions, for a given workpiece material. To test this premise, a statistical analysis was performed on all six of the cast iron fitted models in order to investigate whether or not the fitted models across the different cutting conditions are all essentially the same. If the linear model is independent of cutting conditions, only one wear test would have to be conducted in order to obtain a model for the growth of the plastic flow region for a range of cutting conditions and a given material. First, a Bartlett’s test [13] was performed on the residual variance estimates from each of the six fitted models, and it was determined that all of these residual variances could be estimates of the same common variance. Hence, the variance estimates can be pooled. This pooled variance was then used to obtain confidence intervals for the true values of the model parameters based on the estimates of c0 and c1 for each of the six tests. In Fig. 8, the estimates of the c0 and c1 coefficients for each of the six tests are given with 95% confidence intervals based on the Student’s t-distribution. Examining Fig. 8, it can be seen that for both of the model parameters there is some sizable region of overlap of the confidence intervals for all six cutting conditions. Therefore, it is plausible
Fig. 8. Linear model parameter estimates with confidence intervals for their true values.
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that all of the parameter estimates may have the same true values and that the model for VBp is independent of cutting conditions. Therefore, by running one wear test at one cutting condition, a fitted model of the width of the plastic flow region can be obtained and applied to any cutting conditions within some range about that condition for that workpiece. As indicated in Table 2, a range that extends over 50–100% beyond the minimum value for each variable condition seems to work in the case at hand. Fig. 9 shows all of the experimental measurements of VBp taken from the cast iron tests of Table 1 and a linear curve of the form of Eq. (1) that was obtained by a fit through the data from all six tests. The model coefficients are c0=⫺0.078 and c1=1.26. The error band shown in the figures represents the 95% prediction interval for individual data realizations based on the fitted model. Examining Fig. 9, it can be seen that the number of data points within the prediction interval is well within the anticipated level.
4. Worn tool force model Now that the linear model for the prediction of VBp has been established, a force model for predicting worn tool cutting forces for three-dimensional cutting operations will be developed. For this model, the tool flank stresses and resulting forces due to tool wear, which depend on the width of the plastic flow region, will first be defined in terms of a two-dimensional flank geometry. An integral part of the determination of the flank stresses will be the development of a contact model for determining the normal stress at the tool tip. The three-dimensional cutting forces due to tool flank wear will then be determined by discretizing the rounded flank face into small twodimensional elements and summing the forces from all of the elements. The sharp tool forces will be found using the mechanistic force model [12] and will be added to the forces due to tool wear alone to calculate the total worn tool cutting forces.
Fig. 9.
Linear model of the growth of the plastic flow region with cast iron data.
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4.1. Two-dimensional stress distributions on the tool flank The cutting force and thrust force due to tool flank wear Fcw and Ftw, as shown in Fig. 10, can be found by integrating the normal and shearing tool flank stresses sw and tw, respectively, over the flank wearland width, VB, [7,8] viz.,
冕
VB
Ftw⫽w sw(x)dx
(2)
0
冕
VB
Fcw⫽w tw(x)dx 0
where w=width of cut; VB=width of wearland; x=distance from tool tip. Fig. 11 is a schematic of sw and tw on the flank of a worn tool for an arbitrary flank wearland width, VB. These are the same stress distributions used by Waldorf [8]. The variable x is the same in both Figs. 10 and 11. For the region of elastic contact at the back of the flank wearland, the stress distribution is assumed to be quadratic. For the plastic flow region at the front of the wearland (x=0→VBp), the stresses are constant and equal to the tool tip stresses, s0 and t0. Based on the stress distribution shown in Fig. 11 and applying the linear function to the growth of the plastic flow region, Eq. (3) gives the stresses sw and tw on the flank of the tool for a given total wearland width of VB. If the region of plastic flow grows to cover the entire wearland, it is assumed that it will continue to cover the entire flank wearland as VB increases further. if x⬍VBp
再
sw⫽s0 tw⫽t0
Plastic Flow
Fig. 10.
Cutting with two-dimensional worn cutting tool.
(3)
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Fig. 11. Wearland stress distribution on the tool flank.
冦
冉 冉
sw⫽s0·
if x⬎VBp
冊 冊
VB−x VB−VBp
VB−x tw⫽t0· VB−VBp
2
2
Elastic Contact
where: VBp⫽c0⫹c1VB if VB⬍VBcr then VBp⫽0 where: −c0 if VBp⬎VB then VBp⫽VB VBcr⫽ c1 The cutting forces on a two-dimensional flank element for a given total wearland width can be found by inserting Eq. (3) into Eq. (2). 4.2. Methods for obtaining tool tip stresses As seen in Eq. (3), the tool tip stresses, s0 and t0 as shown in Fig. 11, need to be determined to predict the tool flank stresses. The next two sections will discuss the contact model used to obtain the normal tool tip stress, s0, and the slip-line field equation for obtaining the shearing tool tip stress, t0.
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4.2.1. Contact model for predicting the normal tool tip stress Since plastic deformation of the workpiece occurs under the flank of the tool [1], the maximum effective stress in the workpiece must be equal to the yield strength, y, of the workpiece material. Therefore, it is assumed that the magnitude of the flank stresses is limited by the yield strength of the workpiece material. The contact model presented here calculates the normal flank stress that will cause the maximum effective stress in the workpiece to be equal to the yield strength of the workpiece material. Thomsen et al. [1] said that as the total wearland width increases, the average normal stress on the flank of the tool can become equal to the yield strength of the workpiece material such that sublayer plastic deformation occurs. Therefore, in the current contact model, it is assumed that the tangential stresses do not play a significant role in sublayer plastic deformation, and they are neglected. In the contact model, the normal stresses on the workpiece that arise from the contact between the tool flank and the workpiece are applied to an elastic half-space as a line load, as shown in Fig. 12. In order for a line load to be applied, the plane strain assumption has to be invoked. Since the bulk of the workpiece material is much larger than the loaded region, the plane strain assumption is a valid assumption. For the width of the loaded region (⫺a to a in Fig. 12), the value of VBcr that is obtained from the linear model of the plastic flow region growth is used. Although the contact on the flank of the tool is elastic in nature at this value of wearland width, any increase in VB beyond VBcr will cause plastic deformation of the workpiece material. Therefore, VBcr is used as the critical wearland width at which elastic contact is present, but the maximum effective stress in the workpiece material is equal to the yield strength of the material due to plastic yielding. The contact model
Fig. 12.
Stresses on workpiece in contact model.
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is only applied once for each workpiece material and the value of s0 calculated is assumed to remain constant for all values of VB. Kobayashi and Thomsen [4] have pointed out that although strain-hardening can cause the yield strength in the chips to vary from that obtained in uniaxial tensile tests, the yield strength in the bulk of the workpiece under the flank essentially remains constant. The stresses in the workpiece material at a point A with an arbitrary x and z coordinate due to the line load can be found by integrating a set of concentrated loads of width ds over the loaded region as described by Johnson [10], viz.,
冕 a
2z sw(s)(x−s)2ds sx⫽⫺ p {(x−s)2⫹z2}2 ⫺a
冕 a
2z3 sw(s)ds sz⫽⫺ p {(x−s)2⫹z2}2
(4)
⫺a
冕 a
2z2 sw(s)(x−s)ds txz⫽⫺ p {(x−s)2⫹z2}2 ⫺a
冉冊
2
where the normal stress distribution is sw(s)=so
s−a 2a
, and x and z are shown in Fig. 12.
The stresses in Eq. (4) are used in von Mises’ stress criterion to obtain the effective stress, seff, viz.,
冑
seff⫽ (s1−s2)2+s21+s22 where: s1,2⫽
sx+sz ⫾ 2
冪冉 冊
sx−sz 2 2 +txz. 2
(5)
To find the maximum effective stress present in the workpiece, the workpiece is divided into small square elements and the effective stress is evaluated at all four corners of each element (see Fig. 12). The maximum effective stress is the maximum of all of these values. To find the normal stress load that causes the maximum effective stress to be equal to the yield strength of the workpiece material, an iterative solution method is employed. A small value of s0 is chosen and the resulting maximum effective stress is calculated by numerically integrating Eq. (4) and inserting the results into Eq. (5). If the maximum effective stress is less than the yield strength, the value of s0 is increased and the maximum effective stress is recalculated. This procedure is repeated until the maximum effective stress exceeds the yield strength. A flow chart of this procedure is shown in Fig. 13. 4.2.2. Slip-line field for predicting the shearing tool tip stress The shearing tool tip stress, t0, is found by first assuming that the friction on the flank of the tool is adhesive in nature and the real area of contact is equal to the apparent area of contact. This has been shown to be a good assumption by [1,4,11]. Then the slip-line field developed by
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Fig. 13.
Flow chart for obtaining normal tool tip stress.
Waldorf [8] can be applied with friction factors of one as a result of full adhesion on the flank of the tool to get the shearing tool tip stress, viz., t0 ⫽
s0 1+2p
(6)
where s0 is determined from the contact model. 4.3. Three-dimensional forces through flank face discretization Once the tool tip stresses, s0 and t0, have been obtained using the contact model and Eq. (6), the tool flank stresses, sw and tw, can be found. The three-dimensional cutting forces are found by applying these flank stresses to small two-dimensional elements of the tool flank with elemental width ⌬w and elemental wearland width ⌬VB, as shown in Fig. 14. This section will discuss the geometry of the three-dimensional force model and present the equations required for the worn tool force calculations. Fig. 14 illustrates the cutting geometry for a conventional turning operation. The width of the flank wearland is assumed to increase linearly from a value of 0.5VB at the outer edge to VB at the apex of the tool nose. This shape of the wearland is based on the observations made during the wear tests described earlier. Also, the force predictions made by the model are not very sensitive to changes in the width of the wearland at the outer edge. A 50% change in the width of the wearland at the outer edge results in only a 5% change in the force predictions made by the model. Each two-dimensional element has its own value of wearland width that is substituted into Eq. (3) depending on its location on the flank edge. Fig. 15 shows the flank wearland spread out as a flat section. The letters A, B, C and D correspond to the same letters in Fig. 14. For the straight edge section (A–B), the width of each two-dimensional element, ⌬w, is found by dividing the length of the straight section by a number large enough to provide small twodimensional elements, N, viz., doc−rn . ⌬w⫽ N cos(gl)
(7)
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Fig. 14.
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Three dimensional tool geometry and flank wearland.
Fig. 15.
Flat depiction of flank wear.
The individual element’s ⌬VB values are given by: ⌬VB⫽
(u+0.5⌬w)(VB−0.5VB) p doc+rn −2 4
冉 冊
(8)
where u=distance from A. For the rounded nose section (B–D), ⌬w is found by dividing the total included angle on the
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nose of the tool into small incremental ⌬x sections. This ⌬x value is then used to find the chord length for each incremental element. The chord length is taken to be the width of each element, viz., p −g 2 l ⌬x , ⌬w⫽2rnsin . ⌬x⫽ N 2
冉 冊
(9)
For elements on the rounded nose section, if the element is between B and C (i.e. xⱕp/4⫺gl) then the elemental wearland width is (doc+rn(x−0.5⌬x)0.5VB ⌬VB⫽0.5VB⫹ p doc+rn −1−gl 4
冉 冊
(10)
If the element is between C and D (i.e. x⬎p/4⫺gl) the elemental wearland width is
冉
冊
p x− −0.5⌬x 0.5VB 4 . ⌬VB⫽VB⫺ p 4
(11)
To find the cutting forces due to tool wear, the appropriate equations from Eqs. (9)–(11) are applied to each individual element depending on its location on the flank to get elemental ⌬w and ⌬VB values. These values are then inserted into Eqs. (3) and (2) to get the cutting forces due to wear for each element, (⌬Fcw and ⌬Ftw). Since the tool tip stresses from the contact model and Eq. (6) are applied to all of the flank elements regardless of their location on the flank, the tool tip stresses are constant around the nose of the tool. This application is supported by the fact that since plastic deformation is present, the tool tip stresses are limited by the constant yield strength of the workpiece material. Also, the model is constructed such that the width of the plastic flow region is greater at the tool nose as would be expected. Therefore, the forces generated by an individual element at the nose of the tool would be greater than those at the outer edge of the wearland, as would be expected since the nose carries a concentrated load. The three-dimensional wear forces for a given element, ⌬FTanW, ⌬FLonW and ⌬FRadW, are found from ⌬Fcw and ⌬Ftw by considering the element’s orientation in the global coordinate system, viz., ⌬FTanW⫽⌬Fcw, ⌬FLonW⫽⌬Ftwcos(y), ⌬FRadW⫽⌬Ftwsin(y)
冉 冊
(12)
where: y=gl for the straight edge section (A–B); y=x+gl and x=0→ π2⫺gl for the nose (B–D). Once the elemental cutting forces are known, the total cutting forces due to tool wear can be obtained by summing all of the forces from the individual elements, viz.,
D.W. Smithey et al. / International Journal of Machine Tools & Manufacture 40 (2000) 1929–1950
FTanW⫽⌺⌬FTanW, FLonW⫽⌺⌬FLonW, FRadW⫽⌺⌬FRadW.
1945
(13)
In order to find the total cutting forces (i.e. the forces measured by a dynamometer during actual cutting), the forces due to wear from Eq. (13) have to be added to the sharp tool cutting forces, FTanS, FLonS and FRadS. These cutting forces are found using the sharp tool mechanistic force model [12]. The mechanistic force model can predict sharp tool cutting forces for a range of cutting conditions with a minimal number of calibration tests. Once the mechanistic force model is used to predict the sharp tool cutting forces, they can then be added to the forces due to tool wear to get the total cutting forces: FTan, FLon and FRad. 5. Worn tool force model validation In order to validate the worn tool force model developed above, the next section presents experimental data obtained by the authors. First, force data from worn tool cutting tests will be presented to demonstrate the predictive ability of the force model. Then an analysis will be presented that validates the assumption inherent in the worn tool force model that the cutting forces due to tool wear are independent of cutting geometry. 5.1. Comparison to worn tool force data During the wear tests conducted on the Class 40 cast iron, the 4140 steel and the Al 357 described earlier in Section 4, the cutting forces were measured with a Kistler 3-axis dynamometer, Model No. 9121. The worn tool force model along with the mechanistic force model was used to predict the worn tool cutting forces. The mechanistic model calibration ranges are given in Table 3 and the model coefficients as described in Ref. [12] are given in Table 4. Table 5 gives the yield strength values for each of the workpiece materials that were used in the contact model calculations. For the linear model of the plastic flow growth, the coefficient values in Table 2 were used for the 4140 steel and the Al 357, and the model developed in Section 3.1 was used for the cast iron tests. Examples of the experimental results and force predictions are shown in Fig. 16. In the figures, the error bands are 95% prediction intervals for an individual realization based on the mechanistic model calibration and the linear model fit of the plastic flow region growth. The figures show that the experimental data falls within the prediction intervals in all cases. Table 3 Mechanistic model calibration ranges
Uncut chip thickness, tu (mm) Cutting velocity, V (m/min) Normal rake angle, an (°)
Cast Iron Low
High
4140 Steel Low High
Al 357 Low
High
0.32 132 ⫺10
0.20 215 ⫺5
0.15 160 ⫺10
0.1 150 ⫺10
0.2 300 ⫺5
0.32 230 ⫺5
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Table 4 Mechanistic model coefficients for sharp tool prediction
a0,b0 a1,b1 a2,b2 a3,b3 a4,b4
Cast Iron Kn
Kf
4140 Steel Kn Kf
Al 357 Kn
Kf
6.42 ⫺0.214 ⫺0.052 ⫺0.121 0.111
5.63 ⫺0.275 ⫺0.111 ⫺0.040 0.029
7.08 ⫺0.117 ⫺0.022 ⫺0.001 ⫺0.036
6.15 ⫺0.164 ⫺0.079 0.014 0.015
5.50 ⫺0.277 ⫺0.088 ⫺0.004 0.067
6.02 ⫺0.305 ⫺0.038 ⫺0.048 0.026
Table 5 Yield strength values for contact model Yield strength, y (MPa) Cast Iron 4140 Steel Al 357
276 655 129
Table 6 gives the maximum and average percentage error in the model predictions of the resultant force as compared to the experimental data for all eight tests. In examining Fig. 16 and Table 6, it can be seen that the worn tool force model based on a linear growth of the plastic flow region gives good force predictions for a range of cutting conditions and workpiece materials. In Waldorf’s orthogonal model [8], the width of the plastic flow region increased with a slope of one with increases in total wearland width (c1=1). Since the slopes of the linear models presented in this paper are fairly close to one, simulations were performed in order to determine if a slope of one could give acceptable worn tool force predictions. Whenever a c1 value of one was used in the linear model instead of the values given in Table 2 and the force predictions were made, the cast iron force predictions decreased by 25%, the 4140 steel predictions decreased by 45% and the Al 357 predictions decreased by 20%. Therefore, it is concluded that the variation in the c1 values from the previous orthogonal value of one results in significant variation in the force predictions. 5.2. Validation of model independence of cutting geometry Since the linear model for the growth of the plastic flow region is independent of the cutting conditions, it is postulated that for a given tool and workpiece material combination, the incremental increases in the cutting forces due to tool flank wear are solely a function of the amount and nature of the wear and are independent of the cutting condition in which the tool wear was produced. In order to determine if this is the case, all of the experimental Fw forces (FTanW, FLonW and FRadW) from the six cast iron tests are plotted in Figs. 17–19 along with the model predictions and a 95% prediction interval based on the linear model for the growth of the plastic flow region. The experimental Fw forces were obtained by subtracting the sharp tool cutting forces for each
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Fig. 16. Worn tool force versus flank wearland width.
Table 6 Model predictions versus experimental data Test
Cast Iron, Test Cast Iron, Test Cast Iron, Test Cast Iron, Test Cast Iron, Test Cast Iron, Test 4140 Steel Al 357
Max. percentage error in resultant force 1 2 3 4 5 6
12 ⫺17 11 15 7 17 8 12
Average percentage error in resultant force 2 ⫺2 5 7 3 10 4 ⫺1
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Fig. 17. FLonW force versus total wearland width, cast iron data.
Fig. 18. FTanW force versus total wearland width, cast iron data.
cutting condition from the forces measured whenever tool wear was present. For some of the small values of VB, the experimental forces due to tool wear are less than zero. This is due to variation in the measurement of the total sharp and worn tool cutting forces. In some cases, this variation caused the measured sharp tool force to be greater than the measured worn tool forces, which resulted in experimental values of Fw being less than zero. Also, it should be noted that the errors in the total cutting forces come from measurements that were taken on a much larger scale than the scale of the Fw forces. This causes the variation in the Fw forces to be exaggerated in comparison to the same variation in the total cutting forces. Examining Figs. 17–19, it can be seen that the number of data points within the prediction intervals is well within the anticipated level. Therefore, it is concluded that the cutting forces due to tool wear are independent of cutting conditions. This also indicates that the factors introduced in the worn tool force model, such as the flank wearland geometry and the normal tool tip stress, are independent of cutting conditions.
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Fig. 19. FRadW force versus total wearland width, cast iron data.
6. Conclusions 1. Based on experimental observations from worn tool flank pictures, it has been shown that the width of the plastic flow region grows with a linear growth rate for three-dimensional cutting operations. 2. A linear model for the growth of the plastic flow region was developed and shown to be independent of cutting conditions. A new worn tool force model for three-dimensional cutting operations that uses the linear model for the growth of the plastic flow region has been developed. As part of the worn tool force model, a contact model for obtaining the normal tool tip stress has been developed. This contact model does not require any experimental tests and is based on the yield strength of the workpiece material, which is readily attainable. 3. The new worn tool force model was validated for three workpiece materials: cast iron, 4140 steel and Al 357. The worn tool force predictions for a range of cutting conditions showed excellent agreeement with the data from validation experiments (±10%). 4. It was also shown that for a given tool and workpiece material combination, the incremental increases in the cutting forces due to tool flank wear are solely a function of the amount and nature of the wear and are independent of the cutting condition in which the tool wear was produced.
Acknowledgements The authors gratefully acknowledge the continuous financial support of the University of Illinois at Urbana-Champaign NSF Industry/University Collaborative Research Center for Machine-Tool Systems Research and its member companies. The authors would also like to acknowledge the donation of workpiece materials made by Northwest Aluminum Company.
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References [1] E.G. Thomsen, A.G. MacDonald, S. Kobayashi, Flank friction studies with carbide tools reveal sublayer plastic flow, Trans. ASME, Journal of Engineering for Industry 84 (1962) 53–62. [2] E.G. Thomsen, J.T. Lapsley, R.C. Grassi, Deformation work absorbed by the workpiece during metal cutting, Trans. ASME 75 (1953) 591–603. [3] M. Okoshi, T. Sata, Friction on relief face of cutting tool, Scientific Papers of the Institute of Physical and Chemical Research 52 (1958) 216–223. [4] S. Kobayashi, E.G. Thomsen, The role of friction in metal cutting, ASME Journal of Engineering for Industry 82 (1960) 324–332. [5] C.R. Liu, M.M. Barash, The mechanical state of the sublayer of a surface generated by chip-removal process, part 2: cutting with a tool with flank wear, Journal of Engineering for Industry 98 (1976) 1202–1208. [6] E. Usui, T. Shirakashi, T. Kitagawa, Analytical prediction of cutting tool wear, Wear 100 (1984) 129–151. [7] J.T. Lapsley, R.C. Grassi, E.G. Thomsen, Correlation of plastic deformation during metal cutting with tensile properties of the work material, Trans. ASME 72 (1950) 979–986. [8] D.J. Waldorf, Shearing, ploughing and wear in orthogonal machining. Ph.D. Thesis, University of Illinois at Urbana-Champaign, 1996. [9] D.W. Smithey, Prediction of worn tool forces in metal cutting with tool flank contact analysis. Masters Thesis, University of Illinois at Urbana-Champaign, 1999. [10] K.L. Johnson, Contact Mechanics, Cambridge University Press, New York, 1985. [11] M.C. Shaw, A. Ber, P.A. Mamir, Friction characteristics of sliding surfaces undergoing subsurface plastic flow, in: MIT Research Report P-5, 1959 [12] S.G. Kapoor, R.E. DeVor, R. Zhu, R. Gajjela, G. Parakkal, D. Smithey, Development of mechanistic models for the prediction of machining performance: model building methodology, Journal of Machining Science and Technology 2 (1998) 213–238. [13] R.E. DeVor, T. Chang, J.W. Sutherland, Statistical Quality Design and Control, Macmillan Publishing Company, New York, 1992.
A worn tool force model for three-dimensional cutting operations David W. Smithey, Shiv G. Kapoor, Richard E. DeVor
*
University of Illinois at Urbana-Champaign, 361 Computer and Systems Research Laboratory, 1308 West Main Street, Urbana, IL 61801, USA Received 28 May 1999; received in revised form 25 January 2000; accepted 3 March 2000
Abstract Previous studies have shown that there is a region on the flank of a worn cutting tool where plastic flow of the workpiece material occurs. This paper presents experimental data which shows that in three-dimensional cutting operations in which the nose of the tool is engaged, the region of plastic flow grows linearly with increases in total wearland width. A piecewise linear model is developed for modeling the growth of the plastic flow region, and the model is shown to be independent of cutting conditions. A worn tool force model for three-dimensional cutting operations that uses this concept is presented. The model requires a minimal number of sharp tool tests and only one worn tool test. An integral part of the worn tool force model is a contact model that is used to obtain the magnitude of the stresses on the flank of the tool. The force model is validated through comparison to data obtained from wear tests conducted over a range of cutting conditions and workpiece materials. It is also shown that for a given tool and workpiece material combination, the incremental increases in the cutting forces due to tool flank wear are solely a function of the amount and nature of the wear and are independent of the cutting condition in which the tool wear was produced. 2000 Elsevier Science Ltd. All rights reserved.
1. Introduction It is commonly known that during the metal cutting process, the tool geometry changes as a result of tool wear and that these changes can have undesirable effects on process performance. The most significant variation from sharp tool operation is an increase in cutting forces that can lead to deterioration in process stability, part accuracy and part surface finish. A model that can predict worn tool cutting forces without the need for numerous time-consuming wear tests could * Corresponding author. Tel.: +1-217-333-3432; fax: +1-217-244-9956. E-mail address: [email protected] (R.E. DeVor).
0890-6955/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 8 9 0 - 6 9 5 5 ( 0 0 ) 0 0 0 1 7 - 1
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be used to ensure proper operation of a machining process. In order to develop such a model, it is necessary to have an understanding of the nature of the tool flank contact. Thomsen [1,2] performed some of the earliest investigations into the nature of the contact on the tool flank. From these studies, Thomsen was able to conclude that plastic flow of the workpiece material can result from the contact between the flank of a worn tool and the workpiece and that the deformation work absorbed by the workpiece during metal cutting can be significant. Later, Kobayashi and Thomsen [4] found that good force predictions can be made by assuming that the contact on the flank of the tool is plastic in nature and by using Von Mises’ stress criterion to relate the flank stresses to the material properties. Later, Liu and Barash [5] investigated the apparent strain energy, strain hardening index and residual stress distribution in the workpiece after machining and found that the flank wearland width was one of the governing factors of these values. Since these parameters are closely related to plastic deformation, this further indicated that plastic flow of the workpiece is present and that the amount of plastic flow changes with wearland width. Several authors have suggested that the size of the region on the tool flank where plastic flow of the workpiece occurs increases linearly with increases in total wearland width [1–3,8]. Thomsen et al. [1] and Kobayashi and Thomsen [4] related the linear increases in cutting forces with increases in wearland width that they observed in their experimental data to a similar growth of the plastic flow region. Waldorf [8] took the concept further and developed a model for predicting worn tool cutting forces for orthogonal cutting based on a linear growth of the plastic flow region. In this model, it is assumed that after the total wearland width, VB, (the distance from the cutting edge to the back of the wearland) increases beyond a critical value, there is plastic flow at the front of the wearland and elastic contact at the back of the wearland. The contact is assumed to be totally elastic for VB values less than this critical value. Once VB is greater than this critical value, any increase in the total wearland width results in an equal increase in the width of the plastic flow region. This results in linear increases in the cutting forces as predicted by Waldorf’s model. In this paper, it is proposed that for three-dimensional cutting operations (i.e. when the nose of the cutting tool is engaged), the plastic flow region also grows linearly as the total wearland width increases and that this growth can be modeled independently of cutting conditions for a given workpiece material. Using this concept, a three-dimensional worn tool force model is developed and validated. The organization of the paper is as follows. First, experimental data that shows how the plastic flow region grows with increases in total wearland width is presented. Based on this data, a linear model of the plastic flow region growth is developed, and it is shown that the model is independent of cutting conditions. A force model for predicting worn tool cutting forces for three-dimensional cutting operations is then presented. The model discretizes the flank of the tool into small twodimensional elements and uses a contact model to determine the stresses on the individual elements. Finally, the force model is validated by comparison to worn tool force data obtained during conventional turning operations with a range of cutting conditions and workpiece materials.
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2. Experimental observations of the width of the plastic flow region in turning Fig. 1 illustrates the situation in which both plastic flow and elastic contact are present on the flank of a worn cutting tool for an arbitrary total wearland width, VB. The width of the plastic flow region is denoted by VBp. In order to observe the growth of the plastic flow region as the total wearland width increases, six wear experiments were conducted with a Class 40 gray cast iron workpiece (242 BHN). Tests were also conducted for 4140 steel and Al 357. These tests were performed on a Mori Seiki Frontier L1 CNC turning center. For the cast iron and 4140 steel tests, Kennametal KC910 TiC coated inserts were used. For the Al 357 test, a Carboloy grade 370 TiN coated insert was used. Also, it should be noted that the inserts used in the cast iron and the steel tests had flat rake faces while the inserts used in the Al 357 test had a grooved rake face geometry. The different rake face geometries were chosen to ensure that any observed trends in the data were not specific to one type of rake face geometry. During the experiments, the inserts were subjected to periodic inspections. During these inspections, the tool was removed from the machine and a picture of the flank was taken with a Nikon 40× microscope (Fig. 2). Since the wear rates of the tools varied with the different cutting conditions used and the inspections were made at given time intervals, the total number of inspections made varied from test to test. The specifications for the cutting conditions for all eight tests are given in Table 1. The pictures of the tool flanks taken during the experiments provide a means to measure the width of the plastic flow region. In the example tool flank pictures shown in Figs 2a and b, a
Fig. 1.
Worn tool flank with both plastic flow and elastic contact.
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Fig. 2. Plastic region shown on tool flank.
Table 1 Cutting conditions for worn tool testsa Test No.
V (m/min)
f (mm/rev)
as (°)
doc (mm)
Cast Iron, 1 Cast Iron, 2 Cast Iron, 3 Cast Iron, 4 Cast Iron, 5 Cast Iron, 6 4140 Steel Al 357
132 132 215 132 132 215 215 300
0.2 0.325 0.2 0.2 0.325 0.2 0.3 0.1
⫺10 ⫺10 ⫺10 ⫺5 ⫺5 ⫺5 ⫺5 ⫺5
2.25 2.25 2.25 2.25 2.25 2.25 1.75 1.50
For all tests: ab=⫺50, gl=00, rn=0.8 mm, where: as=side rake angle; ab=back rake angle; gl=lead angle; rn=nose radius; f=feed rate; V=cutting velocity. a
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Fig. 3.
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Growth of the plastic flow region for cast iron tests 1 and 2.
distinct line is visible running horizontally through the wearland. The region above the line is the plastic flow region that is characterized by a shinier surface. This is the result of the workpiece material being deposited on the flank. In the lower region, the surface is not as shiny and the tool material is much more evident. By measuring the width of the shinier region, a measurement of the width of the plastic flow region, VBp, was obtained. This value is an average width along the wearland. By measuring the width of the total scarred region, a measurement of the total wearland width, VB, was obtained. The data from the tests are shown in Figs. 3–7. In all of the data shown in Figs. 3–7, it can be seen that initially VBp=0. However, when the total wearland width reaches about 0.04–0.08 mm for the cast iron and steel tests and about 0.12 mm for the Al 357 test, plastic flow begins to appear and continues to grow linearly as the total wearland width increases. Physically, it is likely that the initial stresses and temperatures on the flank of the tool are not sufficiently large enough to cause yielding of the asperities on the workpiece surface. However, it is known that forces and temperatures on the flank increase as the
Fig. 4.
Growth of the plastic flow region for cast iron tests 3 and 4.
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Fig. 5.
Growth of the plastic flow region for cast iron tests 5 and 6.
Fig. 6.
Growth of the plastic flow region for 4140 steel test.
wearland width increases [6,9]. Eventually, the stresses and temperature on the flank reach levels that cause the asperities to yield and plastic flow begins to occur [1]. 3. Linear model of the growth of the plastic flow region Based on the observed data for the growth of the plastic flow region, a piecewise linear function was developed to model this growth: −c0 VBp⫽c0⫹c1VB if VB⬍VBcr then VBp⫽0 where: VBcr⫽ . c1
(1)
To obtain the coefficients c0 and c1 in Eq. (1), the linear function is fit through the data in which VBp⬎0. The VBcr value is the VB–axis intercept of the linear model, which represents the
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Fig. 7. Growth of the plastic flow region for Al 357 test.
critical total wearland width at which plastic flow is initiated. The model shown in Eq. (1) was fit to each of the eight sets of data presented above by the method of least squares. The model coefficients and the sample coefficients of determination for each set are given in Table 2. 3.1. Model parameter independence of cutting conditions Examining Figs. 3–5 (the six cast iron tests), it can be seen that all of the data sets have similar slopes and VB–axis intercepts. This seems to suggest that the growth of the plastic flow region Table 2 Fitted linear models of the growth of the plastic flow region Test No.
c0
c1
R2
Cast Iron, 1 Cast Iron, 2 Cast Iron, 3 Cast Iron, 4 Cast Iron, 5 Cast Iron, 6 4140 Steel Al 357
⫺0.085 ⫺0.046 ⫺0.059 ⫺0.090 ⫺0.080 ⫺0.109 ⫺0.118 ⫺0.155
1.08 1.14 1.28 1.30 1.23 1.54 1.46 1.18
0.926 0.992 0.996 0.997 0.977 0.974 0.997 0.959
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may be independent of the cutting conditions, for a given workpiece material. To test this premise, a statistical analysis was performed on all six of the cast iron fitted models in order to investigate whether or not the fitted models across the different cutting conditions are all essentially the same. If the linear model is independent of cutting conditions, only one wear test would have to be conducted in order to obtain a model for the growth of the plastic flow region for a range of cutting conditions and a given material. First, a Bartlett’s test [13] was performed on the residual variance estimates from each of the six fitted models, and it was determined that all of these residual variances could be estimates of the same common variance. Hence, the variance estimates can be pooled. This pooled variance was then used to obtain confidence intervals for the true values of the model parameters based on the estimates of c0 and c1 for each of the six tests. In Fig. 8, the estimates of the c0 and c1 coefficients for each of the six tests are given with 95% confidence intervals based on the Student’s t-distribution. Examining Fig. 8, it can be seen that for both of the model parameters there is some sizable region of overlap of the confidence intervals for all six cutting conditions. Therefore, it is plausible
Fig. 8. Linear model parameter estimates with confidence intervals for their true values.
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that all of the parameter estimates may have the same true values and that the model for VBp is independent of cutting conditions. Therefore, by running one wear test at one cutting condition, a fitted model of the width of the plastic flow region can be obtained and applied to any cutting conditions within some range about that condition for that workpiece. As indicated in Table 2, a range that extends over 50–100% beyond the minimum value for each variable condition seems to work in the case at hand. Fig. 9 shows all of the experimental measurements of VBp taken from the cast iron tests of Table 1 and a linear curve of the form of Eq. (1) that was obtained by a fit through the data from all six tests. The model coefficients are c0=⫺0.078 and c1=1.26. The error band shown in the figures represents the 95% prediction interval for individual data realizations based on the fitted model. Examining Fig. 9, it can be seen that the number of data points within the prediction interval is well within the anticipated level.
4. Worn tool force model Now that the linear model for the prediction of VBp has been established, a force model for predicting worn tool cutting forces for three-dimensional cutting operations will be developed. For this model, the tool flank stresses and resulting forces due to tool wear, which depend on the width of the plastic flow region, will first be defined in terms of a two-dimensional flank geometry. An integral part of the determination of the flank stresses will be the development of a contact model for determining the normal stress at the tool tip. The three-dimensional cutting forces due to tool flank wear will then be determined by discretizing the rounded flank face into small twodimensional elements and summing the forces from all of the elements. The sharp tool forces will be found using the mechanistic force model [12] and will be added to the forces due to tool wear alone to calculate the total worn tool cutting forces.
Fig. 9.
Linear model of the growth of the plastic flow region with cast iron data.
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4.1. Two-dimensional stress distributions on the tool flank The cutting force and thrust force due to tool flank wear Fcw and Ftw, as shown in Fig. 10, can be found by integrating the normal and shearing tool flank stresses sw and tw, respectively, over the flank wearland width, VB, [7,8] viz.,
冕
VB
Ftw⫽w sw(x)dx
(2)
0
冕
VB
Fcw⫽w tw(x)dx 0
where w=width of cut; VB=width of wearland; x=distance from tool tip. Fig. 11 is a schematic of sw and tw on the flank of a worn tool for an arbitrary flank wearland width, VB. These are the same stress distributions used by Waldorf [8]. The variable x is the same in both Figs. 10 and 11. For the region of elastic contact at the back of the flank wearland, the stress distribution is assumed to be quadratic. For the plastic flow region at the front of the wearland (x=0→VBp), the stresses are constant and equal to the tool tip stresses, s0 and t0. Based on the stress distribution shown in Fig. 11 and applying the linear function to the growth of the plastic flow region, Eq. (3) gives the stresses sw and tw on the flank of the tool for a given total wearland width of VB. If the region of plastic flow grows to cover the entire wearland, it is assumed that it will continue to cover the entire flank wearland as VB increases further. if x⬍VBp
再
sw⫽s0 tw⫽t0
Plastic Flow
Fig. 10.
Cutting with two-dimensional worn cutting tool.
(3)
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Fig. 11. Wearland stress distribution on the tool flank.
冦
冉 冉
sw⫽s0·
if x⬎VBp
冊 冊
VB−x VB−VBp
VB−x tw⫽t0· VB−VBp
2
2
Elastic Contact
where: VBp⫽c0⫹c1VB if VB⬍VBcr then VBp⫽0 where: −c0 if VBp⬎VB then VBp⫽VB VBcr⫽ c1 The cutting forces on a two-dimensional flank element for a given total wearland width can be found by inserting Eq. (3) into Eq. (2). 4.2. Methods for obtaining tool tip stresses As seen in Eq. (3), the tool tip stresses, s0 and t0 as shown in Fig. 11, need to be determined to predict the tool flank stresses. The next two sections will discuss the contact model used to obtain the normal tool tip stress, s0, and the slip-line field equation for obtaining the shearing tool tip stress, t0.
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4.2.1. Contact model for predicting the normal tool tip stress Since plastic deformation of the workpiece occurs under the flank of the tool [1], the maximum effective stress in the workpiece must be equal to the yield strength, y, of the workpiece material. Therefore, it is assumed that the magnitude of the flank stresses is limited by the yield strength of the workpiece material. The contact model presented here calculates the normal flank stress that will cause the maximum effective stress in the workpiece to be equal to the yield strength of the workpiece material. Thomsen et al. [1] said that as the total wearland width increases, the average normal stress on the flank of the tool can become equal to the yield strength of the workpiece material such that sublayer plastic deformation occurs. Therefore, in the current contact model, it is assumed that the tangential stresses do not play a significant role in sublayer plastic deformation, and they are neglected. In the contact model, the normal stresses on the workpiece that arise from the contact between the tool flank and the workpiece are applied to an elastic half-space as a line load, as shown in Fig. 12. In order for a line load to be applied, the plane strain assumption has to be invoked. Since the bulk of the workpiece material is much larger than the loaded region, the plane strain assumption is a valid assumption. For the width of the loaded region (⫺a to a in Fig. 12), the value of VBcr that is obtained from the linear model of the plastic flow region growth is used. Although the contact on the flank of the tool is elastic in nature at this value of wearland width, any increase in VB beyond VBcr will cause plastic deformation of the workpiece material. Therefore, VBcr is used as the critical wearland width at which elastic contact is present, but the maximum effective stress in the workpiece material is equal to the yield strength of the material due to plastic yielding. The contact model
Fig. 12.
Stresses on workpiece in contact model.
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is only applied once for each workpiece material and the value of s0 calculated is assumed to remain constant for all values of VB. Kobayashi and Thomsen [4] have pointed out that although strain-hardening can cause the yield strength in the chips to vary from that obtained in uniaxial tensile tests, the yield strength in the bulk of the workpiece under the flank essentially remains constant. The stresses in the workpiece material at a point A with an arbitrary x and z coordinate due to the line load can be found by integrating a set of concentrated loads of width ds over the loaded region as described by Johnson [10], viz.,
冕 a
2z sw(s)(x−s)2ds sx⫽⫺ p {(x−s)2⫹z2}2 ⫺a
冕 a
2z3 sw(s)ds sz⫽⫺ p {(x−s)2⫹z2}2
(4)
⫺a
冕 a
2z2 sw(s)(x−s)ds txz⫽⫺ p {(x−s)2⫹z2}2 ⫺a
冉冊
2
where the normal stress distribution is sw(s)=so
s−a 2a
, and x and z are shown in Fig. 12.
The stresses in Eq. (4) are used in von Mises’ stress criterion to obtain the effective stress, seff, viz.,
冑
seff⫽ (s1−s2)2+s21+s22 where: s1,2⫽
sx+sz ⫾ 2
冪冉 冊
sx−sz 2 2 +txz. 2
(5)
To find the maximum effective stress present in the workpiece, the workpiece is divided into small square elements and the effective stress is evaluated at all four corners of each element (see Fig. 12). The maximum effective stress is the maximum of all of these values. To find the normal stress load that causes the maximum effective stress to be equal to the yield strength of the workpiece material, an iterative solution method is employed. A small value of s0 is chosen and the resulting maximum effective stress is calculated by numerically integrating Eq. (4) and inserting the results into Eq. (5). If the maximum effective stress is less than the yield strength, the value of s0 is increased and the maximum effective stress is recalculated. This procedure is repeated until the maximum effective stress exceeds the yield strength. A flow chart of this procedure is shown in Fig. 13. 4.2.2. Slip-line field for predicting the shearing tool tip stress The shearing tool tip stress, t0, is found by first assuming that the friction on the flank of the tool is adhesive in nature and the real area of contact is equal to the apparent area of contact. This has been shown to be a good assumption by [1,4,11]. Then the slip-line field developed by
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Fig. 13.
Flow chart for obtaining normal tool tip stress.
Waldorf [8] can be applied with friction factors of one as a result of full adhesion on the flank of the tool to get the shearing tool tip stress, viz., t0 ⫽
s0 1+2p
(6)
where s0 is determined from the contact model. 4.3. Three-dimensional forces through flank face discretization Once the tool tip stresses, s0 and t0, have been obtained using the contact model and Eq. (6), the tool flank stresses, sw and tw, can be found. The three-dimensional cutting forces are found by applying these flank stresses to small two-dimensional elements of the tool flank with elemental width ⌬w and elemental wearland width ⌬VB, as shown in Fig. 14. This section will discuss the geometry of the three-dimensional force model and present the equations required for the worn tool force calculations. Fig. 14 illustrates the cutting geometry for a conventional turning operation. The width of the flank wearland is assumed to increase linearly from a value of 0.5VB at the outer edge to VB at the apex of the tool nose. This shape of the wearland is based on the observations made during the wear tests described earlier. Also, the force predictions made by the model are not very sensitive to changes in the width of the wearland at the outer edge. A 50% change in the width of the wearland at the outer edge results in only a 5% change in the force predictions made by the model. Each two-dimensional element has its own value of wearland width that is substituted into Eq. (3) depending on its location on the flank edge. Fig. 15 shows the flank wearland spread out as a flat section. The letters A, B, C and D correspond to the same letters in Fig. 14. For the straight edge section (A–B), the width of each two-dimensional element, ⌬w, is found by dividing the length of the straight section by a number large enough to provide small twodimensional elements, N, viz., doc−rn . ⌬w⫽ N cos(gl)
(7)
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Fig. 14.
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Three dimensional tool geometry and flank wearland.
Fig. 15.
Flat depiction of flank wear.
The individual element’s ⌬VB values are given by: ⌬VB⫽
(u+0.5⌬w)(VB−0.5VB) p doc+rn −2 4
冉 冊
(8)
where u=distance from A. For the rounded nose section (B–D), ⌬w is found by dividing the total included angle on the
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nose of the tool into small incremental ⌬x sections. This ⌬x value is then used to find the chord length for each incremental element. The chord length is taken to be the width of each element, viz., p −g 2 l ⌬x , ⌬w⫽2rnsin . ⌬x⫽ N 2
冉 冊
(9)
For elements on the rounded nose section, if the element is between B and C (i.e. xⱕp/4⫺gl) then the elemental wearland width is (doc+rn(x−0.5⌬x)0.5VB ⌬VB⫽0.5VB⫹ p doc+rn −1−gl 4
冉 冊
(10)
If the element is between C and D (i.e. x⬎p/4⫺gl) the elemental wearland width is
冉
冊
p x− −0.5⌬x 0.5VB 4 . ⌬VB⫽VB⫺ p 4
(11)
To find the cutting forces due to tool wear, the appropriate equations from Eqs. (9)–(11) are applied to each individual element depending on its location on the flank to get elemental ⌬w and ⌬VB values. These values are then inserted into Eqs. (3) and (2) to get the cutting forces due to wear for each element, (⌬Fcw and ⌬Ftw). Since the tool tip stresses from the contact model and Eq. (6) are applied to all of the flank elements regardless of their location on the flank, the tool tip stresses are constant around the nose of the tool. This application is supported by the fact that since plastic deformation is present, the tool tip stresses are limited by the constant yield strength of the workpiece material. Also, the model is constructed such that the width of the plastic flow region is greater at the tool nose as would be expected. Therefore, the forces generated by an individual element at the nose of the tool would be greater than those at the outer edge of the wearland, as would be expected since the nose carries a concentrated load. The three-dimensional wear forces for a given element, ⌬FTanW, ⌬FLonW and ⌬FRadW, are found from ⌬Fcw and ⌬Ftw by considering the element’s orientation in the global coordinate system, viz., ⌬FTanW⫽⌬Fcw, ⌬FLonW⫽⌬Ftwcos(y), ⌬FRadW⫽⌬Ftwsin(y)
冉 冊
(12)
where: y=gl for the straight edge section (A–B); y=x+gl and x=0→ π2⫺gl for the nose (B–D). Once the elemental cutting forces are known, the total cutting forces due to tool wear can be obtained by summing all of the forces from the individual elements, viz.,
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FTanW⫽⌺⌬FTanW, FLonW⫽⌺⌬FLonW, FRadW⫽⌺⌬FRadW.
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(13)
In order to find the total cutting forces (i.e. the forces measured by a dynamometer during actual cutting), the forces due to wear from Eq. (13) have to be added to the sharp tool cutting forces, FTanS, FLonS and FRadS. These cutting forces are found using the sharp tool mechanistic force model [12]. The mechanistic force model can predict sharp tool cutting forces for a range of cutting conditions with a minimal number of calibration tests. Once the mechanistic force model is used to predict the sharp tool cutting forces, they can then be added to the forces due to tool wear to get the total cutting forces: FTan, FLon and FRad. 5. Worn tool force model validation In order to validate the worn tool force model developed above, the next section presents experimental data obtained by the authors. First, force data from worn tool cutting tests will be presented to demonstrate the predictive ability of the force model. Then an analysis will be presented that validates the assumption inherent in the worn tool force model that the cutting forces due to tool wear are independent of cutting geometry. 5.1. Comparison to worn tool force data During the wear tests conducted on the Class 40 cast iron, the 4140 steel and the Al 357 described earlier in Section 4, the cutting forces were measured with a Kistler 3-axis dynamometer, Model No. 9121. The worn tool force model along with the mechanistic force model was used to predict the worn tool cutting forces. The mechanistic model calibration ranges are given in Table 3 and the model coefficients as described in Ref. [12] are given in Table 4. Table 5 gives the yield strength values for each of the workpiece materials that were used in the contact model calculations. For the linear model of the plastic flow growth, the coefficient values in Table 2 were used for the 4140 steel and the Al 357, and the model developed in Section 3.1 was used for the cast iron tests. Examples of the experimental results and force predictions are shown in Fig. 16. In the figures, the error bands are 95% prediction intervals for an individual realization based on the mechanistic model calibration and the linear model fit of the plastic flow region growth. The figures show that the experimental data falls within the prediction intervals in all cases. Table 3 Mechanistic model calibration ranges
Uncut chip thickness, tu (mm) Cutting velocity, V (m/min) Normal rake angle, an (°)
Cast Iron Low
High
4140 Steel Low High
Al 357 Low
High
0.32 132 ⫺10
0.20 215 ⫺5
0.15 160 ⫺10
0.1 150 ⫺10
0.2 300 ⫺5
0.32 230 ⫺5
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Table 4 Mechanistic model coefficients for sharp tool prediction
a0,b0 a1,b1 a2,b2 a3,b3 a4,b4
Cast Iron Kn
Kf
4140 Steel Kn Kf
Al 357 Kn
Kf
6.42 ⫺0.214 ⫺0.052 ⫺0.121 0.111
5.63 ⫺0.275 ⫺0.111 ⫺0.040 0.029
7.08 ⫺0.117 ⫺0.022 ⫺0.001 ⫺0.036
6.15 ⫺0.164 ⫺0.079 0.014 0.015
5.50 ⫺0.277 ⫺0.088 ⫺0.004 0.067
6.02 ⫺0.305 ⫺0.038 ⫺0.048 0.026
Table 5 Yield strength values for contact model Yield strength, y (MPa) Cast Iron 4140 Steel Al 357
276 655 129
Table 6 gives the maximum and average percentage error in the model predictions of the resultant force as compared to the experimental data for all eight tests. In examining Fig. 16 and Table 6, it can be seen that the worn tool force model based on a linear growth of the plastic flow region gives good force predictions for a range of cutting conditions and workpiece materials. In Waldorf’s orthogonal model [8], the width of the plastic flow region increased with a slope of one with increases in total wearland width (c1=1). Since the slopes of the linear models presented in this paper are fairly close to one, simulations were performed in order to determine if a slope of one could give acceptable worn tool force predictions. Whenever a c1 value of one was used in the linear model instead of the values given in Table 2 and the force predictions were made, the cast iron force predictions decreased by 25%, the 4140 steel predictions decreased by 45% and the Al 357 predictions decreased by 20%. Therefore, it is concluded that the variation in the c1 values from the previous orthogonal value of one results in significant variation in the force predictions. 5.2. Validation of model independence of cutting geometry Since the linear model for the growth of the plastic flow region is independent of the cutting conditions, it is postulated that for a given tool and workpiece material combination, the incremental increases in the cutting forces due to tool flank wear are solely a function of the amount and nature of the wear and are independent of the cutting condition in which the tool wear was produced. In order to determine if this is the case, all of the experimental Fw forces (FTanW, FLonW and FRadW) from the six cast iron tests are plotted in Figs. 17–19 along with the model predictions and a 95% prediction interval based on the linear model for the growth of the plastic flow region. The experimental Fw forces were obtained by subtracting the sharp tool cutting forces for each
D.W. Smithey et al. / International Journal of Machine Tools & Manufacture 40 (2000) 1929–1950
1947
Fig. 16. Worn tool force versus flank wearland width.
Table 6 Model predictions versus experimental data Test
Cast Iron, Test Cast Iron, Test Cast Iron, Test Cast Iron, Test Cast Iron, Test Cast Iron, Test 4140 Steel Al 357
Max. percentage error in resultant force 1 2 3 4 5 6
12 ⫺17 11 15 7 17 8 12
Average percentage error in resultant force 2 ⫺2 5 7 3 10 4 ⫺1
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Fig. 17. FLonW force versus total wearland width, cast iron data.
Fig. 18. FTanW force versus total wearland width, cast iron data.
cutting condition from the forces measured whenever tool wear was present. For some of the small values of VB, the experimental forces due to tool wear are less than zero. This is due to variation in the measurement of the total sharp and worn tool cutting forces. In some cases, this variation caused the measured sharp tool force to be greater than the measured worn tool forces, which resulted in experimental values of Fw being less than zero. Also, it should be noted that the errors in the total cutting forces come from measurements that were taken on a much larger scale than the scale of the Fw forces. This causes the variation in the Fw forces to be exaggerated in comparison to the same variation in the total cutting forces. Examining Figs. 17–19, it can be seen that the number of data points within the prediction intervals is well within the anticipated level. Therefore, it is concluded that the cutting forces due to tool wear are independent of cutting conditions. This also indicates that the factors introduced in the worn tool force model, such as the flank wearland geometry and the normal tool tip stress, are independent of cutting conditions.
D.W. Smithey et al. / International Journal of Machine Tools & Manufacture 40 (2000) 1929–1950
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Fig. 19. FRadW force versus total wearland width, cast iron data.
6. Conclusions 1. Based on experimental observations from worn tool flank pictures, it has been shown that the width of the plastic flow region grows with a linear growth rate for three-dimensional cutting operations. 2. A linear model for the growth of the plastic flow region was developed and shown to be independent of cutting conditions. A new worn tool force model for three-dimensional cutting operations that uses the linear model for the growth of the plastic flow region has been developed. As part of the worn tool force model, a contact model for obtaining the normal tool tip stress has been developed. This contact model does not require any experimental tests and is based on the yield strength of the workpiece material, which is readily attainable. 3. The new worn tool force model was validated for three workpiece materials: cast iron, 4140 steel and Al 357. The worn tool force predictions for a range of cutting conditions showed excellent agreeement with the data from validation experiments (±10%). 4. It was also shown that for a given tool and workpiece material combination, the incremental increases in the cutting forces due to tool flank wear are solely a function of the amount and nature of the wear and are independent of the cutting condition in which the tool wear was produced.
Acknowledgements The authors gratefully acknowledge the continuous financial support of the University of Illinois at Urbana-Champaign NSF Industry/University Collaborative Research Center for Machine-Tool Systems Research and its member companies. The authors would also like to acknowledge the donation of workpiece materials made by Northwest Aluminum Company.
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