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An integral method to determine the mechanical behavior of materials in metal cutting
Journal of Materials Processing Technology 142 (2003) 72–81
An integral method to determine the mechanical behavior of materials in metal cutting Y.B. Guo∗ Department of Mechanical Engineering, The University of Alabama, Tuscaloosa, AL 35487, USA Received 13 August 2002; received in revised form 13 August 2002; accepted 17 February 2003
Abstract An integral Johnson–Cook model based approach has been presented to characterize the mechanical behavior of work materials in machining. The model parameters are determined by fitting the data from both quasi-static compression and machining tests. The approach has been validated by the case studies of quasi-static compression and machining tests of 6061-T6 aluminum, which the flow stress, strain, strain rate, and the temperature are calculated by the appropriate models. It is also confirmed that the compression data can be correlated with the cutting data. The developed approach is valid for machining with continuous chips at a variety of cutting speeds. In addition, the error sources to affect the accuracy of this approach are discussed. © 2003 Elsevier Science B.V. All rights reserved. Keywords: Material properties; Metal cutting; Materials tests
1. Introduction Metal cutting is the major manufacturing operation today in view of the economic significance. A fundamental knowledge of the metal cutting process is essential to find optimum conditions and develop new cutting equipment. Analytical and numerical models can be deduced from this fundamental knowledge to predict variables of interest for increasing process efficiency and part quality. A major problem, however, in finding this knowledge is the severe deformation together with high temperatures of the workpiece material at high speed in a very small volume. The intense circumstances under which deformation takes place in metal cutting result in mechanical behavior far removed from that encountered in conventional material tests. However, information on the mechanical behavior is of utmost importance in both the analytical and numerical models, and some experimental studies. Therefore, this paper is aimed at determining the mechanical behavior of materials in metal cutting. For this purpose, a new integral approach has been proposed to solve the subproblems as follows: (1) Determination of the strain, strain rate and the temperature, under which the material is being deformed in cutting. ∗ Tel.: +1-205-348-2615; fax: +1-205-348-6419. E-mail address: [email protected] (Y.B. Guo).
(2) Development of material and cutting tests that can be used in conditions to assess and verify the mechanical behavior of a work material in cutting. (3) Development of an effective method to construct the constitutive equation of a work material in cutting.
2. Literature review Metal cutting occurs in a broad range of strains, strain rates and temperatures that are difficult to access experimentally. An overview of typical strains, strain rates and temperatures found in manufacturing processes is given in Table 1. The table reveals, that in commonly used manufacturing processes, such as machining and forming, the workpiece material are deformed under conditions far different from those encountered in conventional material tests. For example, it shows the strain rates are in the order of 103 to 106 s−1 in machining, while the strain rates in conventional material tests are in the order of 10−3 to 10−1 s−1 , that is up to a million times smaller than in machining. Hence, the predictability of any metal cutting model largely depends on the accuracy of the constitutive model for the work material properties. This paper is limited to material properties in the primary shear zone. Merchant [1] and others [2–5] assumed that the shear flow stress is equal to that obtained from the conventional tensile tests. So the stress–strain relationship obtained from the
0924-0136/$ – see front matter © 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0924-0136(03)00462-X
Y.B. Guo / Journal of Materials Processing Technology 142 (2003) 72–81 Table 1 Typical strains, strain rates, and homologous temperatures (Th = T/Tmelt ) of some manufacturing processes [31,32] Process Extrusion Forging/rolling Sheet-metal forming Machining a
Strain
Strain rate (s−1 )
Th
2–5 0.1–0.5 0.1–0.5 1–10a
10−1
0.16–0.7 0.16–0.7 0.16–0.7 0.16–0.9
102
to 100 to 103 100 to 102 103 to 106
The strain could be larger in the secondary shear zone.
conventional tensile tests can be used in metal cutting. Zorev [5] concluded that at high cutting speeds, the temperature has comparatively little influence on the shear flow stress. von Turkovich [6] found that the shear stress is a function of dislocation density and there is no influence of cutting temperature, high strain and strain rate on the shear flow stress. Spaans [7] suggested that mutual counter balancing of strain hardening and thermal softening makes the shear flow stress remain the same as in the standard material test. However, some other researchers [8–13] found the stress–strain relationships obtained from the conventional material tests, such as tensile tests, compression tests, and torsion tests, cannot be used in metal cutting. When considering combined effect of strains, strain rates and temperatures, aside from microstructural conditions, a number of constitutive models have been proposed based on machining tests [14,15], independent material tests [16–19], or combined cutting tests and independent material tests [12,13,20,21]. For each of these methods, experimental data have been provided to support the proposed models. A carefully designed cutting test may provide an effective method for estimating the average flow stress at a range of large strains and high strain rates at cutting conditions [10,22]. While the major disadvantage of this method is that the determined constitutive equation may be only valid for the range of large strains, high strain rates and temperatures. Determination of the mechanical behavior beyond this range, such as in the elastic range or low strain and strain rate range, becomes very difficult. For the independent material test method, a constitutive model relating shear stress and shear strain was proposed for torsion data [16]. The Johnson–Cook model, the same mathematical form as that for torsion data, was also proposed for the von Mises flow stress and equivalent plastic strain [23]. Johnson–Cook model or its’ revised forms has been used by several researchers [15,17,18] to characterize the flow stress in metal cutting. A more complex constitutive model was proposed to account for the effects of strains, strain rates and temperatures on the flow stress [24,25]. Among the independent material tests, split Hopkinson pressure or torsion bar (SHPB) has been widely used to perform materiel tests at strain rates up to 104 s−1 and can be tailored for tests at an elevated temperature. The SHPB method is useful to evaluate the effect of individual factor such as strain, strain rate or temperature on the flow stress. Accordingly,
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this method is suitable for the determination of the flow stress under conditions similar to those found in machining [24,26–28]. However, the following identified factors need to be considered to conduct the SHPB test and interpret the test data: (1) appreciable heating time allowing anneal softening which does not occur in such a short time deformation as in machining, (2) difficulty in measuring testing temperatures, (3) difficulty in achieving uniform temperature in a sample if the heating time is very short, (4) the compound effect of strain rate and temperature, (5) the flow stress oscillation, especially at low strain values, (6) the data in elastic range not available generally, (7) inertial effect due to high impact speeds, (8) moderate strain rates, usually below 2500 s−1 , (9) moderate testing temperatures, usually below 600 ◦ C, (10) relative low strain values, usually below 0.4, (11) strain history effect on the flow stress using incremental impact compression for achieving a large strain, (12) extrapolation required to estimate the flow stress at high strain rates and large strains, and (13) complex testing facility, especially heating the specimen by an induction coil. Errors induced by these problems are difficult to quantify, so great care have to be taken in using the SHPB test and interpreting the testing data. Compared to the SHPB test, the cutting test method is more direct and easy to perform to estimate the flow stress properties in machining with continuous chips. To overcome the disadvantages of either cutting tests or independent material tests, a number of studies [12,13,20,21] have combined the two different methods to recover the mechanical behavior in elastic and/or plastic ranges. Stevenson conducted orthogonal cutting aluminum at very low cutting speed, 0.254 m/min, so that the cutting temperature approximated room temperature. He further showed that the shear flow stress at this cutting speed and compression agrees very well when the strain hardening and strain rate hardening response of the materials was considered. Guo and Liu [20] estimated the flow stress in a tabular format based on the tensile tests at elevated temperatures and correlated the flow stress in cutting with those from tensile tests with regard to the velocity-modified temperature. However, a general method to construct a constitutive equation of the work material was not reported. Hence, this is one objective of this work. Astakhov [29] concluded that none of these factors such as strain, strain rate and temperature in cutting affects the cutting resistance (shear flow stress), and therefore, orthogonal cutting is a cold working process. It was noticed that Astakhov drew the conclusion from that the specific work done in compression and in cutting is consistent. But, even though magnitudes of the shear flow stress and shear strain vary due to the effects of strain, strain rate and temperature in cutting, their product, i.e., the specific work, may still not change and is consistent with the specific work in compression. Thus, Astakhov’s view that cutting process is a cold working process would be more convincing if the effect of individual factor on the shear flow stress can be isolated. For numerical simulations such as finite element analysis
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(FEA), in which a high accuracy can be achieved, this difference can still be relevant even if the overall forces predicted agree with the experimental data in 10%, but other cutting response factors such as deformation and temperature fields in the second shear zone and the surface integrity of a machined part are strongly affected by local phenomena in the cutting processes. Thus more detailed information of the mechanical behavior, as given in this proposed integral approach to construct the constitutive equation based on conventional material tests and cutting tests, can be considered to be fundamentally important.
3. Experimental procedure There are two parts of the experiment: quasi-static compression and cutting tests. All tests were conducted on 6061-T6 aluminum (solution heat treated at 540 ◦ C for 0.5 h and artificially aged 205 ◦ C for 3 h) with nominal compositions are given in Table 2. Each experiment was performed five times and showed good repeatability (in general the specific measurement of cutting forces and chip thickness differ less than 3% from the mean value in Tables 3–6). Compression tests. Compression tests at a series of compression speeds were performed. The samples were cut into size (5/16 in. diameter × 0.5 in. length) using the lathe at very slow speeds to minimize thermal softening of the sample. Then the top and bottom surfaces of the samples were ground gently and polished. So the influence of thermal softening and strain hardening on sample properties could
be ignored. The polished mirror-like surfaces of the samples were to minimize the friction between the compression plates and the sample’s end surfaces for achieving strains in excess of one without significant barreling. The parallelism of the two end surfaces and the perpendicularity of each end surface to the sample axis were also ensured to a tight tolerance to maintain uniaxial loading in compression. The hardened steel platens were ground, polished, and coated with easy-slide silicon release agent to minimize friction effects. The two platens were setup just close enough to touch the sample without applying pressure. Otherwise, the elastic properties of the sample would not be obtained. The frictional effects were not completely eliminated even the sample end surfaces were polished and lubricated with easy-slide silicon release agents. It was noticed that barreling is negligible after compression, which indicates the frictional effects are small. Further, the strain rates in conventional compression tests are very low (<0.1 in this study), the effect of low strain rates on flow stress of 6061-T6 aluminum is trivial. The stress–strain curves would not change significantly under the testing conditions in this study. However, if the strain rate were large, for example in the order of 1000 s−1 , the effect of strain rate on the flow stress would be obvious. But in conventional compression tests, such a large strain rate cannot be achieved. Cutting tests. Orthogonal cutting tests were performed on an outer diameter of 50.8 mm 6061-T6 aluminum tube with a wall thickness of 1.7 mm. The NARDNI-ND 1585E lathe was equipped with a Kistler force dynamometer hooked to a personal computer with DaqView software. The dynamome-
Table 2 Nominal chemical compositions of 6061-T6
wt.%
Al
Mg
Si
Cu
Cr
Fe
Mn
Ti
Zn
98
1.0
0.6
0.28
0.2
Maximum 0.7
Maximum 0.15
Maximum 0.15
Maximum 0.25
Table 3 Cutting test data in experiment group 1 (V = 0.064 m/s)a t0 (mm)
Ft (N)
Fc (N)
tc (mm)
ε¯
ε˙¯ (×103 s−1 )
Tavg (◦ C)
σ¯ (MPa)
0.053 0.061 0.066 0.074 0.076 0.079 0.084 0.086 0.094 0.097 0.107 0.117 0.122 0.127 0.132
42.4 31.0 42.6 44.9 43.3 51.0 56.2 50.5 55.7 49.8 56.4 51.2 55.2 64.0 58.1
124.6 105.9 118.2 135.1 122.6 152.9 160.2 174.9 181.5 169.7 188.0 180.0 184.1 189.5 199.1
0.168 0.194 0.188 0.196 0.200 0.216 0.236 0.258 0.250 0.253 0.262 0.284 0.342 0.330 0.294
1.89 1.91 1.74 1.64 1.62 1.68 1.72 1.81 1.64 1.62 1.54 1.53 1.71 1.61 1.43
2.20 1.90 1.96 1.88 1.85 1.71 1.56 1.43 1.48 1.46 1.41 1.30 1.08 1.12 1.25
68.4 59.2 61.7 65.7 60.6 69.3 69.3 73.5 73.8 70.1 72.7 68.5 67.0 67.0 70.2
624.4 468.7 508.3 549.6 481.8 570.8 548.4 572.8 585.7 541.2 562.5 498.5 443.7 452.6 509.5
a t : uncut chip thickness; F : thrust force; F : cutting force; ε ¯ : equivalent strain; ε˙¯ : equivalent strain rate; Tavg : average temperature; and σ: ¯ equivalent 0 t c stress.
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Table 4 Cutting test data in experiment group 2 (V = 0.064 m/s) t0 (mm)
Ft (N)
Fc (N)
tc (mm)
ε¯
ε˙¯ (×103 s−1 )
Tavg (◦ C)
σ¯ (MPa)
0.140 0.147 0.150 0.152 0.160 0.173 0.188 0.191 0.203 0.213 0.224 0.234 0.239 0.244 0.254
91.5 97.2 64.9 101.4 86.6 111.8 115.9 100.8 125.2 110.1 141.3 141.6 141.0 124.1 121.0
268.9 322.8 228.4 320.7 309.6 349.8 373.5 340.3 393.4 351.4 436.9 438.6 455.8 412.3 379.4
0.270 0.332 0.264 0.368 0.370 0.432 0.470 0.432 0.410 0.482 0.504 0.508 0.474 0.524 0.462
1.30 1.45 1.23 1.52 1.47 1.56 1.56 1.45 1.34 1.45 1.45 1.41 1.32 1.40 1.25
1.37 1.11 1.40 1.00 1.00 0.85 0.79 0.85 0.90 0.77 0.73 0.73 0.78 0.70 0.80
83.4 94.8 73.8 91.9 89.1 92.5 93.9 88.2 95.0 85.3 98.1 96.6 99.2 91.2 83.3
670.8 732.4 568.3 672.7 645.2 633.6 624.5 596.7 673.7 546.1 645.2 630.8 672.3 578.5 538.2
Table 5 Cutting test data in experiment group 3 t0 (mm)
V (m/s)
Ft (N)
Fc (N)
tc (mm)
ε¯
ε˙¯ (×104 s−1 )
Tavg (◦ C)
σ¯ (MPa)
0.053 0.053 0.053 0.053 0.076 0.076 0.076 0.076 0.097 0.097 0.097 0.097 0.117 0.117 0.117 0.117
1.285 2.057 2.571 3.214 1.285 2.057 2.571 3.214 1.285 2.057 2.571 3.214 1.285 2.057 2.571 3.214
67.2 68.8 62.8 56.5 98.1 85.6 73.4 70.4 106.7 80.1 87.5 73.0 121.3 98.3 97.7 83.7
145.6 128.1 128.0 115.4 197.5 175.0 156.0 142.3 249.1 191.0 209.5 166.0 250.0 225.2 228.0 196.3
0.256 0.292 0.352 0.294 0.498 0.394 0.404 0.396 0.536 0.462 0.536 0.420 0.636 0.524 0.488 0.502
2.79 3.16 3.80 3.18 3.76 3.00 3.07 3.01 3.21 2.78 3.21 2.54 3.15 2.61 2.44 2.51
2.88 4.04 4.19 6.28 1.48 3.00 3.65 4.66 1.38 2.56 2.75 4.39 1.16 2.25 3.03 3.68
213.3 235.2 236.4 256.8 211.7 261.9 256.0 265.7 241.5 248.9 282.6 278.1 229.1 270.7 307.7 291.1
508.6 394.6 338.3 356.8 368.3 399.9 350.3 323.3 428.2 373.4 360.9 348.8 356.9 382.1 410.2 345.5
Table 6 Cutting test data in experiment group 4 t0 (mm)
V (m/s)
Ft (N)
Fc (N)
tc (mm)
ε¯
ε˙¯ (×104 s−1 )
Tavg (◦ C)
σ¯ (MPa)
0.066 0.066 0.066 0.066 0.086 0.086 0.086 0.086 0.107 0.107 0.107 0.107 0.122 0.122 0.122 0.122
1.285 2.057 2.571 3.214 1.285 2.057 2.571 3.214 1.285 2.057 2.571 3.214 1.285 2.057 2.571 3.214
108.8 90.6 85.5 69.1 105.8 88.0 75.9 67.8 111.9 99.1 93.8 76.1 128.8 96.2 87.7 86.2
225.4 181.9 181.6 150.4 209.7 175.5 163.5 144.0 211.2 203.0 208.5 165.7 264.2 211.3 201.9 196.7
0.428 0.336 0.354 0.294 0.520 0.492 0.434 0.412 0.600 0.550 0.470 0.458 0.746 0.550 0.500 0.518
3.73 2.95 3.10 2.59 3.47 3.29 2.91 2.77 3.25 2.98 2.57 2.51 3.53 2.63 2.40 2.48
1.73 3.52 4.17 6.28 1.42 2.40 3.40 4.48 1.23 2.15 3.14 4.03 0.99 2.15 2.95 3.56
254.2 295.0 315.7 312.7 219.2 238.8 254.7 255.8 207.8 252.7 296.1 267.5 224.9 252.0 271.5 288.2
489.7 485.0 466.2 451.5 370.1 325.0 339.4 311.6 318.4 332.1 391.5 316.9 326.7 340.3 352.6 333.9
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ter was calibrated carefully by applying a known force in a given direction to the dynamometer. A sharp cutting tool (K-type carbide) with 6◦ rake angle and 8◦ clearance angle was used in every group of cutting experiments, Tables 3–6. The full width of the tubes is machined with the straight part of the tool by choosing the cutting width at least 15 times the uncut chip thickness, so the cutting operation can be modeled to be two-dimensional. In Tables 3 and 4, very slow cutting speeds with increased uncut chip thickness were used to minimize cutting temperature and achieve strain rates as low as possible. In Tables 5 and 6, orthogonal cutting tests were conducted under practical cutting speeds. Once steady-state cutting was achieved the DaqView software was started to record force data at 100 Hz. The parasitic forces were obtained and used to correct measured forces to account for plowing forces induced by cutting edge radius [30]. A chip sample was collected in each cutting condition. The thickness of the collected chips was then measured with a digital caliper to record the chip thickness. Each chip was measured in five different locations and the average of those measurements were used for the chip thickness. The resulting average chip thickness, measured cutting and thrust forces, Tables 3–6, were then used for analyzing the material behavior during machining.
4. Experimental results and analysis Engineering stress, engineering strain, their true counterparts, and temperature rise in compression tests were calculated. The temperature rise of 74 ◦ C during compression was accounted for in this study in order to correlate the mechanical behaviors accurately in compression and low speed cutting tests. The metal cutting data were analyzed in the conventional method to determine the material shear flow stress [1]. The shear strain rate was calculated using the Oxley’s
method [10]. It should be pointed out that the primary shear zone is not a regular shape assumed by Oxley. The strain rate varies in the primary shear zone. Oxley method only gives a rough average data. Average temperature in the primary shear zone was calculated by the well-regarded Loewen–Shaw model, which has been verified by a number of researchers [7,11,13]. The equivalent flow stress, strain, strain rate, and the average temperature of material in the primary shear zone were computed. In the literature, the equivalent flow stress was usually converted to the shear flow stress for comparing with the shear flow stress calculated by cutting models [11,12]. In this study, in order to correlate the mechanical behaviors in cutting and compression tests, the calculated shear strain, shear strain rate, and shear flow stress in cutting were converted to the equivalent strain, equivalent strain rate, and equivalent flow stress based on the assumption of isotropy and a von Mises yield criterion. The aluminum tube and rod 6061-T6 were solution heat treated and artificially aged, so the effect of material anisotropy on the mechanical behavior should be trivial. Before we correlate the mechanical behaviors in compression and cutting tests, it would be helpful to plot the stress vs. strain curves in compression tests together with the calculated flow stress vs. strain data in cutting in one graph so that the nature of the calculated flow stress and the strain by the cutting models would be clear. The obtained engineering stress–engineering strain and true stress–true strain curves for 6061-T6 aluminum in the compression tests are shown in Figs. 1 and 2. It is obvious that the stress vs. strain curves of 6061-T6 is not sensitive to low strain rates. The measured cutting force, the thrust force, and the chip thickness together with the calculated equivalent strain, equivalent strain rate, average temperature, and the equivalent flow stress under each cutting condition were listed in Tables 3–6. The equivalent flow stresses and strains in cutting were plotted in Figs. 1 and 2. The true strain rate for each cutting condition was labeled in Figs. 1 and 2. For the comparison
Fig. 1. Engineering compression stress–strain and cutting stress–strain (the labeled cutting strain rate is the average strain rate for each cutting group).
Y.B. Guo / Journal of Materials Processing Technology 142 (2003) 72–81
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Fig. 2. True compression stress–strain and cutting stress–strain (the labeled cutting strain rate is the average strain rate for each cutting group).
data in Figs. 1 and 2, the average temperature rise of 74 ◦ C during compression is close to the average cutting temperature, 79 ◦ C, at low cutting speeds, Tables 3 and 4. Considering cutting and compression tests are short time duration events, the low temperatures would not produce appreciable difference on the flow stress. The most significant difference is the strain rate between compression and metal cutting. Therefore the confounding effect does exist but would be small. The strains induced in cutting are indeed larger than those in conventional compression tests and SHPB tests. Strain extrapolation would be required if we correlate mechanical behaviors in compression and cutting tests. The average equivalent strain rates are still very high even at very low cutting speeds, Tables 3 and 4. The strain rates, Tables 5 and 6, are much higher than those achieved in conventional compression tests and SHPB tests if normal cutting speeds are used. The strain rate decreases at the same cutting speed if using a larger uncut chip thickness. It is not surprising because the increased chip volume reduces the average strain rate in the shear zone if a larger uncut chip thickness is used. Strain rate extrapolation would also be required to correlate flow stresses in compression and cutting tests. Average temperatures in the primary shear zone are 79 and 259 ◦ C, respectively, at very low and normal cutting speeds in Tables 3–6. It is noticed that the uncut chip thickness has very slight effect on the average shear zone temperature, while the cutting speed is a dominant factor which has been well recognized in the literature. As the average temperature of 79 ◦ C at low cutting speed (0.064 m/s) is close to the average temperature rise, 74 ◦ C, in the compression tests, we may ignore the temperature effect to correlate flow stresses. For the flow stress, the cutting temperature is the most important factor as we see that the flow stress decreases as the average temperature increases even both the strain and strain rate increase if we assume strain hardening and strain rate hardening hold. The strain rate effect can be demonstrated by extrapolating the flow stress from compression tests to the
same strain as in cutting. It is clear that strain rate hardening exist by comparing the flow stress by strain extrapolation. Strain hardening in Fig. 2 is obvious, and the strain hardening influence on the flow stress is much larger than that of the strain rate hardening in terms of the ratios of the flow stress increase per unit increment of strain and strain rate. However, the strain rate effect is larger than that of strain in terms of variation of the flow stress magnitude in the range of strains and strain rates encountered in cutting. The microstructure effect on the flow stress is beyond the scope of this study and will be investigated in the future work.
5. An integral method to construct the constitutive equation An effective method to determine the material mechanical behavior is especially important for process optimization and machine tool design. The mechanical behavior in the elastic range can be obtained by the conventional materials tests. The mechanical behavior in the plastic range can be obtained by cutting tests, independent conventional materials tests including SHPB, or combination of material tests and cutting tests. The pros and cons of these different methods are summarized in the literature review. Stevenson and Stephenson [12] showed a convincing demonstration of the correlation between conventionally compression data to the cutting data at room temperature based on a self-consistent strain rate sensitivity index. The metal cutting strains, Fig. 2, are beyond the range achieved by the conventional compression tests even though a large strain over one was achieved in the compression tests. Thus, the effects of the strain and temperature on the flow stress need to be incorporated. Further, a method to determine the flow stress by constructing a constitutive equation including the influences of strain, strain rate and temperature by using of the cutting data and conventionally determined material property data is still missing.
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Y.B. Guo / Journal of Materials Processing Technology 142 (2003) 72–81
The philosophy of determining the material flow stress in this study is to choose a relative accurate constitutive model and fit the parameters of the constitutive model using the data of conventional material tests and cutting tests. Although a universal accurate constitutive model is not currently available, the Johnson–Cook model has shown to be effective for metals and thus a good candidate [23]. In this study, we adopted the Johnson–Cook model, fitted its parameters by the data from compression and cutting tests, and then verified its effectiveness experimentally. The Johnson–Cook model [23] is cited as follows: ˙ ε¯ σ¯ = [A + B(¯ε) ] 1 + C ln (1 − Thm ) ˙ε¯ 0 n
(1)
where Th = (T − Troom )/(Tmelt − Troom ). The thermal softening term when m = 1 was used to vary in a linear form in this study since it was shown that the linear form gave reasonable results although the alternative forms for different values of the parameter m were used in the literature [16]. Obviously, the Johnson–Cook model assumes the flow stress to be independently affected by strain, strain rate and temperature represented by the terms in each set of the brackets. It should be pointed out that effects of the strain and strain rate on the temperature exist due to adiabatic deformation at high strain rates. As discussed in the literature review, metal cutting tests provide some advantages over the SHPB method and other independent material tests to estimate the material properties in cutting conditions. Considering the characteristics of material deformation in metal cutting, we propose a new integral approach to obtain the model parameters of Johnson–Cook constitutive equation using both the conventional compression and cutting tests. In this study, the compression tests were conducted under a series of low strain rates rather than the reference strain rate of 1 s−1 proposed by the Johnson–Cook model. The lowest average true strain rate of 0.01 s−1 was chosen as the reference strain rate in this study. Other parameters were determined based on the chosen reference strain rate. Actually, a small reference strain rate has negligible effect on the flow stress as it is clear that the flow stress is not sensitive to the low strain rates, Fig. 2. The parameter A is the yield stress at the reference strain rate. The parameters B and n, can be obtained from a least-squares fit of the test data to a power law equation since the strain rate effect can be neglected in the quasi-static compression tests. The parameter C can be determined by correlating the cutting data and the compression data at the same strain value. But the maximum strain achieved in the compression test is smaller than the cutting strain, strain extrapolation is necessary to find the parameter C. Let us denote σ¯ com is the extrapolated flow stress in compression and σ¯ cut the flow stress calculated by cutting models. With the calculated strain, strain rate, and the temperature in each cutting test, and the strain rate and temperature in the compression test, the parameter C can be determined as follows based on the
above determined parameters A, B, and n: σ¯ com = (A + B¯εncut )(1 − Thcom ) ˙ ε¯ cut n (1 − Thcut ) σ¯ cut = (A + B¯εcut ) 1 + C ln ε˙¯ 0
(2) (3)
Solving Eqs. (2) and (3) gives C=
[σ¯ cut (1 − Thcom )/σ¯ com (1 − Thcut )] − 1 ln(ε˙¯ cut /ε˙¯ 0 )
(4)
Each cutting condition yields a value of the parameter C. To represent the most likely equation for the cases considered, the regression method is used to determine the parameter C in the cutting experiment group at the confidence level of 95%. The obtained parameter C is a function of independent cutting parameters of the uncut chip thickness t0 and cutting velocity V for orthogonal cutting, and is regarded valid for all other cutting conditions. A relationship between the parameter C and the feed, cutting velocity, and the depth of cut in production machining can be determined similarly. The validity of the determined parameters is assessed by comparing the predicted flow stress with the experimental data in the following case study. The test conditions to determine the model parameters are different from those in validation test conditions. The proposed approach can be applied to characterize the mechanical behavior of any metals in machining with continuous chips.
6. Case study and method assessment According to the method of constructing a constitutive equation in Section 5, the model parameter A, 275 MPa, is the yield stress at the reference strain rate of 0.01 s−1 in the compression tests. The parameters B and n, 86 MPa and 0.39, were determined by fitting the compression data to a power law equation. The parameter C was determined as follows from the cutting test data using the linear regression method: C = 0.058108 − 194.091t0 − 0.00306V
(5)
With the determined model parameters, the constitutive equation of 6061-T6 aluminum in machining is as follows: ˙ ε¯ 0.39 σ¯ = [275 + 86(¯ε) ] 1 + C ln (6) (1 − Th ) 0.01 where the parameter C can be calculated by Eq. (5) for each cutting condition. With the determined constitutive equation of 6061-T6 aluminum in machining, the effectiveness of the method is verified by comparing the predicted flow stresses using Eq. (6) with the experimental data in compression and cutting tests. For different low strain rates and low temperatures in compression tests, the predicted flow stress was plotted with the test data, Fig. 3. It shows that the predicted flow stresses agree with the compression data very well. For moderate
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Fig. 3. Comparison of predicted flow stresses to the compression data at low strain rates and room temperature.
Fig. 4. Comparison of predicted flow stresses to the test data in the cutting condition (Table 3: corresponding strain, moderate strain rate, and low temperature).
strain rates and low temperatures in cutting tests, the predicted flow stresses were compared to those calculated by the cutting models, Fig. 4, which shows that the predicted flow stress agrees with the cutting data with an average difference of 8.6%. For high strain rates and relative high tem-
Fig. 6. Comparison of predicted flow stresses to the test data in the cutting condition (Table 6: corresponding strain, high strain rate, and moderate temperature)
peratures in cutting tests, the predicted flow stresses were plotted against those calculated data by the cutting models, Figs. 5 and 6. Fig. 5 shows the prediction and the experimental data agree in trend but with a difference of 11.8%. While the difference is 17.5% in Fig. 6. The flow stress comparison shows that the constitutive equation obtained by the proposed method is valid for both quasi-static compression tests of low strain rates and low temperatures, and cutting tests of large strains, high strain rates, and relative high temperatures.
7. Discussions
Fig. 5. Comparison of predicted flow stresses to the test data in the cutting condition (Table 5: corresponding strain, high strain rate, and moderate temperature).
Though the Johnson–Cook model slightly overestimates the flow stress at the relatively high temperatures, Figs. 5 and 6, it gives overall a reasonable prediction of the flow stress considering the errors associated with force measurement and other error sources. Firstly, it needs to be mentioned that the calculated flow stress also depends on strain path in metal cutting. The current compression experimental setup compressed the sample to the strain up to 1 in one
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step at room temperature. The work material in tube cutting is under sequential cutting conditions where the strain hardened material will be deformed again by subsequent cuts, so the work material in cutting tests experiences different strain path from that in the compression tests. Secondly, the strain path dependence of the flow stress is another factor to influence the slight difference between the predicted data and those of cutting data in Figs. 5 and 6. It has shown that the flow stress, determined from the tension or compression data, is generally greater than when it is determined from the torsion data [23]. It means the flow stress is also affected by the deformation modes. The major deformation mode in cutting is shearing as in torsion while it is compression in compression tests. The different deformation modes also affect the difference between the flow stresses. That how strain path and deformation mode influence the flow stress is a complex issue and beyond the scope of this study. Thirdly, although the parameters of the Johnson–Cook model are determined by fitting the experimental data, it should be realized that the mechanical behavior is far more complicated than assumed by the semi-empirical constitutive model or other models, which consider strain, strain rate and temperature only. The microstructural effects may also contribute to the difference between the model prediction and experimental data in Figs. 5 and 6 since the difference is much smaller in low cutting temperatures, Fig. 4. Lastly, the accuracy of the cutting models is limited by some oversimplifications. In addition, the Johnson–Cook model assumes the independent effect of strain, strain rate, and temperature on the flow stress although the compound effect of the strain rate and temperature exists. The errors of strain path, deformation mode, cutting models, and the Johnson–Cook model are difficult to evaluate and therefore is presently not clear. Correspondingly, the proposed method should be regarded to give only a first approximation of the actual mechanical behavior. On the other hand, for FEA of plastic flow at high strain rates such as machining, a simple analytic relationship is highly desirable. Further, very complex constitutive models are of little computational benefit if there are no data available to determine constants for these constitutive equations. So until the models have been further evolved, a phenomenological approach based on appropriate experiments such as those in this paper to assess the mechanical behavior under production conditions should prevail. It should also be pointed out that the temperature calculated by the Loewen–Shaw model is the average temperature in the primary shear zone. The accuracy of the model is demonstrated by the difference of less than 3% between the calculated temperature and the experimental data in machining AISI 1113 steel [8]. Further, it was also evident [8] that the temperatures are not sensitive to the temperature-dependent material thermal properties. Considering the small width (<0.01 mm in the case study) of the primary shear zone, the average temperature should be representative for the entire shear plane. Although the cutting speeds in the case study are relatively lower than those in
high speed machining, the proposed method are valid as long as the chip does not melt so that the method is independent of cutting speeds.
8. Conclusions A general method based on the Johnson–Cook model has been presented to characterize the flow stress of work materials in machining. The model parameters in the strain hardening term are determined by fitting the data from both conventional compression tests to a power law equation. The parameter in the strain rate term is determined by the linear regression method. Case studies of quasi-static compression tests and orthogonal cutting of 6061-T6 aluminum tubing were conducted. It shows that the model predicted flow stresses agree with the test data with reasonable accuracy from quasi-static compression tests at low strain rates and low temperatures to cutting tests with high strain rates and high temperatures. It was also confirmed that the compression data can be correlated with the cutting data by extrapolating the strain, strain rate, and the temperature. The developed method is valid for machining with continuous chips at a variety of cutting speeds. The accuracy of the proposed constitutive model is affected by the strain path, material deformation mode, microstructural effects, cutting models, and the adopted constitutive model itself.
References [1] M.E. Merchant, Mechanics of the metal cutting process, J. Appl. Phys. 16 (1945) 267–275. [2] E. Usui, Progress of predictive theories in metal cutting, JSME Int. J. Ser. III 31 (2) (1988) 363–369. [3] S. Kobayashi, E.G. Thomsen, Metal cutting analysis. I. Re-evaluation and new method of presentation of theories, ASME J. Eng. Ind. 90 (1962) 63–70. [4] J.H.L. The, R.F. Scrutton, The stress-state in the shear zone during steady state machining, ASME J. Eng. Ind. 101 (1979) 211–216. [5] N.N. Zorev, Metal Cutting Mechanics, Pergamon Press, Oxford, 1966. [6] B.F. von Turkovich, Shear stress in metal cutting, ASME J. Eng. Ind. 92 (1970) 151–157. [7] C. Spaans, Treatise of the streamlines and the stress, strain, and strain rate distributions, and on stability in the primary shear zone in metal cutting, ASME J. Eng. Ind. 94 (1972) 690–696. [8] M.C. Shaw, Metal Cutting Principles, Oxford University Press, New York, 1984. [9] D. Kececioglu, Shear zone size, compressive stress, and shear strain in metal cutting and their effects on mean shear flow stress, ASME J. Eng. Ind. 81 (1960) 79–86. [10] P.L.B. Oxley, The Mechanics of Machining, Ellis Horwood, Chichester, UK, 1989. [11] K. Ocusima, K. Hitomi, An analysis of the mechanism of orthogonal cutting and its application to discontinuous chip formation, ASME J. Eng. Ind. 82 (1961) 545–556. [12] R. Stevenson, D.A. Stephenson, The mechanical behavior of zinc during machining, ASME J. Eng. Ind. 117 (1995) 173–178.
Y.B. Guo / Journal of Materials Processing Technology 142 (2003) 72–81 [13] R. Stevenson, Study on the correlation of workpiece mechanical properties from compression and cutting tests, J. Mach. Sci. Technol. 1 (1) (1997) 67–79. [14] S. Lei, Y.C. Shin, F.P. Incropera, Material constitutive modeling under high strain rates and temperatures through orthogonal machining tests, J. Manuf. Sci. Eng. 121 (1999) 577–585. [15] M. Shatla, C. Kerk, T. Altan, Process modeling in machining. Part I. Determination of flow stress data, Int. J. Mach. Tools Manuf. 41 (2001) 1511–1534. [16] G.R. Johnson, J.M. Hoegfeldt, U.S. Lindholm, A. Nagy, Response of various metals to large torsional strains over a large range of strain rates. Part 1. Ductile metals, J. Eng. Mater. Technol. 105 (1983) 42–47. [17] Y. Zheng, J.W. Sutherland, An orthogonal cutting model based on finite deformation analysis. Part II. Constitutive equations and experimental verification, ASME MED 10 (1999) 301–309. [18] S.P.F.C. Jaspers, J.H. Dautzenberg, W.P. Vellinga, Mechanical behavior of industrial alloys at high strain rate and temperature using the split Hopkinson pressure bar, in: P.H. Osanna, D. Prostrednik, N.M. Durakbasa (Eds.), Proceedings of the Sixth ISMQC IMEKO Symposium on Metrology for Quality Control in Production, Wien, 1998, pp. 291–296. [19] G. Poulachon, A.L. Moisan, I.S. Jawahir, Evaluation of chip morphology in hard turning using constitutive models and material property data, in: Proceedings of the ASME–IMECE, vol. 12, MED, New York, 2002, pp. 179–185. [20] Y.B. Guo, C.R. Liu, Mechanical properties of hardened AISI 52100 steel in hard machining processes, J. Manuf. Sci. Eng. 124 (1999) 1–9. [21] S.A. Batzer, G. Subhash, W. Olson, J.W. Sutherland, Large strain constitutive model development for application to orthogonal machining, in: Proceedings of the Fifth CIRP International Workshop on Modeling of Machining Operations, W. Lafayette, IN, May 20–21, 2002, pp. 31–40.
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[22] M.G. Stevenson, P.L.B. Oxley, An experimental investigation of the influence of strain rate and temperature on the flow stress properties of a low carbon steel using a machining test, Proc. Inst. Mech. Eng. 185 (1970) 741–754. [23] G.R. Johnson, W.H. Cook, A constitutive model and data for metals subjected to large strains, high strain rates and high temperatures, in: Proceedings of the Seventh International Symposium on Ballistics, The Hague, The Netherlands, April 19–21, 1983, pp. 541–547. [24] T. Shirakashi, K. Maekawa, E. Usui, Flow stress of low carbon steel at high temperature and strain rate. Part 1. Property of incremental strain method in impact compression test with rapid heating and cooling systems, Bull. Jpn. Soc. Prec. Eng. 17 (3) (1983) 161–166. [25] M. Yoshino, T. Shirakashi, Flow-stress equation including effects of strain rate and temperature history, Int. J. Mech. Sci. 39 (12) (1997) 1345–1362. [26] E. Usui, T. Shirakashi, Mechanics of machining—from descriptive to predictive theory, in: On the Art of Cutting Metals—75 Years Later, vol. 7, ASME, PED, New York, 1982, pp. 13–35. [27] K. Maekawa, T. Shirakashi, E. Usui, Flow stress of low carbon steel at high temperature and strain rate. Part 2. Flow stress under variable temperature and variable strain rate, Bull. Jpn. Soc. Prec. Eng. 17 (3) (1983) 167–172. [28] W. Lee, C. Lin, High temperature deformation behavior of Ti6Al4V alloy evaluated by high strain rate compression tests, J. Mater. Process. Technol. 75 (1) (1998) 127–136. [29] A.P. Astakhov, A treatise on material characterization in the metal cutting process. Part 1. A novel approach and experimental verification, J. Mater. Process. Technol. 96 (1999) 22–33. [30] R. Stevenson, The measurement of parasitic forces in orthogonal cutting, Int. J. Mach. Tools Manuf. 38 (1997) 113–130. [31] S. Kalpakjian, Manufacturing Processes for Engineering Materials, 3rd ed., Addison-Wesley/Longman, Menlo Park, 1997. [32] J.M. Alexander, On problems of plastic flow of materials, in: A. Sawczuk, G. Bianchi (Eds.), Plasticity Today, Elsevier, London, 1985.
An integral method to determine the mechanical behavior of materials in metal cutting Y.B. Guo∗ Department of Mechanical Engineering, The University of Alabama, Tuscaloosa, AL 35487, USA Received 13 August 2002; received in revised form 13 August 2002; accepted 17 February 2003
Abstract An integral Johnson–Cook model based approach has been presented to characterize the mechanical behavior of work materials in machining. The model parameters are determined by fitting the data from both quasi-static compression and machining tests. The approach has been validated by the case studies of quasi-static compression and machining tests of 6061-T6 aluminum, which the flow stress, strain, strain rate, and the temperature are calculated by the appropriate models. It is also confirmed that the compression data can be correlated with the cutting data. The developed approach is valid for machining with continuous chips at a variety of cutting speeds. In addition, the error sources to affect the accuracy of this approach are discussed. © 2003 Elsevier Science B.V. All rights reserved. Keywords: Material properties; Metal cutting; Materials tests
1. Introduction Metal cutting is the major manufacturing operation today in view of the economic significance. A fundamental knowledge of the metal cutting process is essential to find optimum conditions and develop new cutting equipment. Analytical and numerical models can be deduced from this fundamental knowledge to predict variables of interest for increasing process efficiency and part quality. A major problem, however, in finding this knowledge is the severe deformation together with high temperatures of the workpiece material at high speed in a very small volume. The intense circumstances under which deformation takes place in metal cutting result in mechanical behavior far removed from that encountered in conventional material tests. However, information on the mechanical behavior is of utmost importance in both the analytical and numerical models, and some experimental studies. Therefore, this paper is aimed at determining the mechanical behavior of materials in metal cutting. For this purpose, a new integral approach has been proposed to solve the subproblems as follows: (1) Determination of the strain, strain rate and the temperature, under which the material is being deformed in cutting. ∗ Tel.: +1-205-348-2615; fax: +1-205-348-6419. E-mail address: [email protected] (Y.B. Guo).
(2) Development of material and cutting tests that can be used in conditions to assess and verify the mechanical behavior of a work material in cutting. (3) Development of an effective method to construct the constitutive equation of a work material in cutting.
2. Literature review Metal cutting occurs in a broad range of strains, strain rates and temperatures that are difficult to access experimentally. An overview of typical strains, strain rates and temperatures found in manufacturing processes is given in Table 1. The table reveals, that in commonly used manufacturing processes, such as machining and forming, the workpiece material are deformed under conditions far different from those encountered in conventional material tests. For example, it shows the strain rates are in the order of 103 to 106 s−1 in machining, while the strain rates in conventional material tests are in the order of 10−3 to 10−1 s−1 , that is up to a million times smaller than in machining. Hence, the predictability of any metal cutting model largely depends on the accuracy of the constitutive model for the work material properties. This paper is limited to material properties in the primary shear zone. Merchant [1] and others [2–5] assumed that the shear flow stress is equal to that obtained from the conventional tensile tests. So the stress–strain relationship obtained from the
0924-0136/$ – see front matter © 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0924-0136(03)00462-X
Y.B. Guo / Journal of Materials Processing Technology 142 (2003) 72–81 Table 1 Typical strains, strain rates, and homologous temperatures (Th = T/Tmelt ) of some manufacturing processes [31,32] Process Extrusion Forging/rolling Sheet-metal forming Machining a
Strain
Strain rate (s−1 )
Th
2–5 0.1–0.5 0.1–0.5 1–10a
10−1
0.16–0.7 0.16–0.7 0.16–0.7 0.16–0.9
102
to 100 to 103 100 to 102 103 to 106
The strain could be larger in the secondary shear zone.
conventional tensile tests can be used in metal cutting. Zorev [5] concluded that at high cutting speeds, the temperature has comparatively little influence on the shear flow stress. von Turkovich [6] found that the shear stress is a function of dislocation density and there is no influence of cutting temperature, high strain and strain rate on the shear flow stress. Spaans [7] suggested that mutual counter balancing of strain hardening and thermal softening makes the shear flow stress remain the same as in the standard material test. However, some other researchers [8–13] found the stress–strain relationships obtained from the conventional material tests, such as tensile tests, compression tests, and torsion tests, cannot be used in metal cutting. When considering combined effect of strains, strain rates and temperatures, aside from microstructural conditions, a number of constitutive models have been proposed based on machining tests [14,15], independent material tests [16–19], or combined cutting tests and independent material tests [12,13,20,21]. For each of these methods, experimental data have been provided to support the proposed models. A carefully designed cutting test may provide an effective method for estimating the average flow stress at a range of large strains and high strain rates at cutting conditions [10,22]. While the major disadvantage of this method is that the determined constitutive equation may be only valid for the range of large strains, high strain rates and temperatures. Determination of the mechanical behavior beyond this range, such as in the elastic range or low strain and strain rate range, becomes very difficult. For the independent material test method, a constitutive model relating shear stress and shear strain was proposed for torsion data [16]. The Johnson–Cook model, the same mathematical form as that for torsion data, was also proposed for the von Mises flow stress and equivalent plastic strain [23]. Johnson–Cook model or its’ revised forms has been used by several researchers [15,17,18] to characterize the flow stress in metal cutting. A more complex constitutive model was proposed to account for the effects of strains, strain rates and temperatures on the flow stress [24,25]. Among the independent material tests, split Hopkinson pressure or torsion bar (SHPB) has been widely used to perform materiel tests at strain rates up to 104 s−1 and can be tailored for tests at an elevated temperature. The SHPB method is useful to evaluate the effect of individual factor such as strain, strain rate or temperature on the flow stress. Accordingly,
73
this method is suitable for the determination of the flow stress under conditions similar to those found in machining [24,26–28]. However, the following identified factors need to be considered to conduct the SHPB test and interpret the test data: (1) appreciable heating time allowing anneal softening which does not occur in such a short time deformation as in machining, (2) difficulty in measuring testing temperatures, (3) difficulty in achieving uniform temperature in a sample if the heating time is very short, (4) the compound effect of strain rate and temperature, (5) the flow stress oscillation, especially at low strain values, (6) the data in elastic range not available generally, (7) inertial effect due to high impact speeds, (8) moderate strain rates, usually below 2500 s−1 , (9) moderate testing temperatures, usually below 600 ◦ C, (10) relative low strain values, usually below 0.4, (11) strain history effect on the flow stress using incremental impact compression for achieving a large strain, (12) extrapolation required to estimate the flow stress at high strain rates and large strains, and (13) complex testing facility, especially heating the specimen by an induction coil. Errors induced by these problems are difficult to quantify, so great care have to be taken in using the SHPB test and interpreting the testing data. Compared to the SHPB test, the cutting test method is more direct and easy to perform to estimate the flow stress properties in machining with continuous chips. To overcome the disadvantages of either cutting tests or independent material tests, a number of studies [12,13,20,21] have combined the two different methods to recover the mechanical behavior in elastic and/or plastic ranges. Stevenson conducted orthogonal cutting aluminum at very low cutting speed, 0.254 m/min, so that the cutting temperature approximated room temperature. He further showed that the shear flow stress at this cutting speed and compression agrees very well when the strain hardening and strain rate hardening response of the materials was considered. Guo and Liu [20] estimated the flow stress in a tabular format based on the tensile tests at elevated temperatures and correlated the flow stress in cutting with those from tensile tests with regard to the velocity-modified temperature. However, a general method to construct a constitutive equation of the work material was not reported. Hence, this is one objective of this work. Astakhov [29] concluded that none of these factors such as strain, strain rate and temperature in cutting affects the cutting resistance (shear flow stress), and therefore, orthogonal cutting is a cold working process. It was noticed that Astakhov drew the conclusion from that the specific work done in compression and in cutting is consistent. But, even though magnitudes of the shear flow stress and shear strain vary due to the effects of strain, strain rate and temperature in cutting, their product, i.e., the specific work, may still not change and is consistent with the specific work in compression. Thus, Astakhov’s view that cutting process is a cold working process would be more convincing if the effect of individual factor on the shear flow stress can be isolated. For numerical simulations such as finite element analysis
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(FEA), in which a high accuracy can be achieved, this difference can still be relevant even if the overall forces predicted agree with the experimental data in 10%, but other cutting response factors such as deformation and temperature fields in the second shear zone and the surface integrity of a machined part are strongly affected by local phenomena in the cutting processes. Thus more detailed information of the mechanical behavior, as given in this proposed integral approach to construct the constitutive equation based on conventional material tests and cutting tests, can be considered to be fundamentally important.
3. Experimental procedure There are two parts of the experiment: quasi-static compression and cutting tests. All tests were conducted on 6061-T6 aluminum (solution heat treated at 540 ◦ C for 0.5 h and artificially aged 205 ◦ C for 3 h) with nominal compositions are given in Table 2. Each experiment was performed five times and showed good repeatability (in general the specific measurement of cutting forces and chip thickness differ less than 3% from the mean value in Tables 3–6). Compression tests. Compression tests at a series of compression speeds were performed. The samples were cut into size (5/16 in. diameter × 0.5 in. length) using the lathe at very slow speeds to minimize thermal softening of the sample. Then the top and bottom surfaces of the samples were ground gently and polished. So the influence of thermal softening and strain hardening on sample properties could
be ignored. The polished mirror-like surfaces of the samples were to minimize the friction between the compression plates and the sample’s end surfaces for achieving strains in excess of one without significant barreling. The parallelism of the two end surfaces and the perpendicularity of each end surface to the sample axis were also ensured to a tight tolerance to maintain uniaxial loading in compression. The hardened steel platens were ground, polished, and coated with easy-slide silicon release agent to minimize friction effects. The two platens were setup just close enough to touch the sample without applying pressure. Otherwise, the elastic properties of the sample would not be obtained. The frictional effects were not completely eliminated even the sample end surfaces were polished and lubricated with easy-slide silicon release agents. It was noticed that barreling is negligible after compression, which indicates the frictional effects are small. Further, the strain rates in conventional compression tests are very low (<0.1 in this study), the effect of low strain rates on flow stress of 6061-T6 aluminum is trivial. The stress–strain curves would not change significantly under the testing conditions in this study. However, if the strain rate were large, for example in the order of 1000 s−1 , the effect of strain rate on the flow stress would be obvious. But in conventional compression tests, such a large strain rate cannot be achieved. Cutting tests. Orthogonal cutting tests were performed on an outer diameter of 50.8 mm 6061-T6 aluminum tube with a wall thickness of 1.7 mm. The NARDNI-ND 1585E lathe was equipped with a Kistler force dynamometer hooked to a personal computer with DaqView software. The dynamome-
Table 2 Nominal chemical compositions of 6061-T6
wt.%
Al
Mg
Si
Cu
Cr
Fe
Mn
Ti
Zn
98
1.0
0.6
0.28
0.2
Maximum 0.7
Maximum 0.15
Maximum 0.15
Maximum 0.25
Table 3 Cutting test data in experiment group 1 (V = 0.064 m/s)a t0 (mm)
Ft (N)
Fc (N)
tc (mm)
ε¯
ε˙¯ (×103 s−1 )
Tavg (◦ C)
σ¯ (MPa)
0.053 0.061 0.066 0.074 0.076 0.079 0.084 0.086 0.094 0.097 0.107 0.117 0.122 0.127 0.132
42.4 31.0 42.6 44.9 43.3 51.0 56.2 50.5 55.7 49.8 56.4 51.2 55.2 64.0 58.1
124.6 105.9 118.2 135.1 122.6 152.9 160.2 174.9 181.5 169.7 188.0 180.0 184.1 189.5 199.1
0.168 0.194 0.188 0.196 0.200 0.216 0.236 0.258 0.250 0.253 0.262 0.284 0.342 0.330 0.294
1.89 1.91 1.74 1.64 1.62 1.68 1.72 1.81 1.64 1.62 1.54 1.53 1.71 1.61 1.43
2.20 1.90 1.96 1.88 1.85 1.71 1.56 1.43 1.48 1.46 1.41 1.30 1.08 1.12 1.25
68.4 59.2 61.7 65.7 60.6 69.3 69.3 73.5 73.8 70.1 72.7 68.5 67.0 67.0 70.2
624.4 468.7 508.3 549.6 481.8 570.8 548.4 572.8 585.7 541.2 562.5 498.5 443.7 452.6 509.5
a t : uncut chip thickness; F : thrust force; F : cutting force; ε ¯ : equivalent strain; ε˙¯ : equivalent strain rate; Tavg : average temperature; and σ: ¯ equivalent 0 t c stress.
Y.B. Guo / Journal of Materials Processing Technology 142 (2003) 72–81
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Table 4 Cutting test data in experiment group 2 (V = 0.064 m/s) t0 (mm)
Ft (N)
Fc (N)
tc (mm)
ε¯
ε˙¯ (×103 s−1 )
Tavg (◦ C)
σ¯ (MPa)
0.140 0.147 0.150 0.152 0.160 0.173 0.188 0.191 0.203 0.213 0.224 0.234 0.239 0.244 0.254
91.5 97.2 64.9 101.4 86.6 111.8 115.9 100.8 125.2 110.1 141.3 141.6 141.0 124.1 121.0
268.9 322.8 228.4 320.7 309.6 349.8 373.5 340.3 393.4 351.4 436.9 438.6 455.8 412.3 379.4
0.270 0.332 0.264 0.368 0.370 0.432 0.470 0.432 0.410 0.482 0.504 0.508 0.474 0.524 0.462
1.30 1.45 1.23 1.52 1.47 1.56 1.56 1.45 1.34 1.45 1.45 1.41 1.32 1.40 1.25
1.37 1.11 1.40 1.00 1.00 0.85 0.79 0.85 0.90 0.77 0.73 0.73 0.78 0.70 0.80
83.4 94.8 73.8 91.9 89.1 92.5 93.9 88.2 95.0 85.3 98.1 96.6 99.2 91.2 83.3
670.8 732.4 568.3 672.7 645.2 633.6 624.5 596.7 673.7 546.1 645.2 630.8 672.3 578.5 538.2
Table 5 Cutting test data in experiment group 3 t0 (mm)
V (m/s)
Ft (N)
Fc (N)
tc (mm)
ε¯
ε˙¯ (×104 s−1 )
Tavg (◦ C)
σ¯ (MPa)
0.053 0.053 0.053 0.053 0.076 0.076 0.076 0.076 0.097 0.097 0.097 0.097 0.117 0.117 0.117 0.117
1.285 2.057 2.571 3.214 1.285 2.057 2.571 3.214 1.285 2.057 2.571 3.214 1.285 2.057 2.571 3.214
67.2 68.8 62.8 56.5 98.1 85.6 73.4 70.4 106.7 80.1 87.5 73.0 121.3 98.3 97.7 83.7
145.6 128.1 128.0 115.4 197.5 175.0 156.0 142.3 249.1 191.0 209.5 166.0 250.0 225.2 228.0 196.3
0.256 0.292 0.352 0.294 0.498 0.394 0.404 0.396 0.536 0.462 0.536 0.420 0.636 0.524 0.488 0.502
2.79 3.16 3.80 3.18 3.76 3.00 3.07 3.01 3.21 2.78 3.21 2.54 3.15 2.61 2.44 2.51
2.88 4.04 4.19 6.28 1.48 3.00 3.65 4.66 1.38 2.56 2.75 4.39 1.16 2.25 3.03 3.68
213.3 235.2 236.4 256.8 211.7 261.9 256.0 265.7 241.5 248.9 282.6 278.1 229.1 270.7 307.7 291.1
508.6 394.6 338.3 356.8 368.3 399.9 350.3 323.3 428.2 373.4 360.9 348.8 356.9 382.1 410.2 345.5
Table 6 Cutting test data in experiment group 4 t0 (mm)
V (m/s)
Ft (N)
Fc (N)
tc (mm)
ε¯
ε˙¯ (×104 s−1 )
Tavg (◦ C)
σ¯ (MPa)
0.066 0.066 0.066 0.066 0.086 0.086 0.086 0.086 0.107 0.107 0.107 0.107 0.122 0.122 0.122 0.122
1.285 2.057 2.571 3.214 1.285 2.057 2.571 3.214 1.285 2.057 2.571 3.214 1.285 2.057 2.571 3.214
108.8 90.6 85.5 69.1 105.8 88.0 75.9 67.8 111.9 99.1 93.8 76.1 128.8 96.2 87.7 86.2
225.4 181.9 181.6 150.4 209.7 175.5 163.5 144.0 211.2 203.0 208.5 165.7 264.2 211.3 201.9 196.7
0.428 0.336 0.354 0.294 0.520 0.492 0.434 0.412 0.600 0.550 0.470 0.458 0.746 0.550 0.500 0.518
3.73 2.95 3.10 2.59 3.47 3.29 2.91 2.77 3.25 2.98 2.57 2.51 3.53 2.63 2.40 2.48
1.73 3.52 4.17 6.28 1.42 2.40 3.40 4.48 1.23 2.15 3.14 4.03 0.99 2.15 2.95 3.56
254.2 295.0 315.7 312.7 219.2 238.8 254.7 255.8 207.8 252.7 296.1 267.5 224.9 252.0 271.5 288.2
489.7 485.0 466.2 451.5 370.1 325.0 339.4 311.6 318.4 332.1 391.5 316.9 326.7 340.3 352.6 333.9
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ter was calibrated carefully by applying a known force in a given direction to the dynamometer. A sharp cutting tool (K-type carbide) with 6◦ rake angle and 8◦ clearance angle was used in every group of cutting experiments, Tables 3–6. The full width of the tubes is machined with the straight part of the tool by choosing the cutting width at least 15 times the uncut chip thickness, so the cutting operation can be modeled to be two-dimensional. In Tables 3 and 4, very slow cutting speeds with increased uncut chip thickness were used to minimize cutting temperature and achieve strain rates as low as possible. In Tables 5 and 6, orthogonal cutting tests were conducted under practical cutting speeds. Once steady-state cutting was achieved the DaqView software was started to record force data at 100 Hz. The parasitic forces were obtained and used to correct measured forces to account for plowing forces induced by cutting edge radius [30]. A chip sample was collected in each cutting condition. The thickness of the collected chips was then measured with a digital caliper to record the chip thickness. Each chip was measured in five different locations and the average of those measurements were used for the chip thickness. The resulting average chip thickness, measured cutting and thrust forces, Tables 3–6, were then used for analyzing the material behavior during machining.
4. Experimental results and analysis Engineering stress, engineering strain, their true counterparts, and temperature rise in compression tests were calculated. The temperature rise of 74 ◦ C during compression was accounted for in this study in order to correlate the mechanical behaviors accurately in compression and low speed cutting tests. The metal cutting data were analyzed in the conventional method to determine the material shear flow stress [1]. The shear strain rate was calculated using the Oxley’s
method [10]. It should be pointed out that the primary shear zone is not a regular shape assumed by Oxley. The strain rate varies in the primary shear zone. Oxley method only gives a rough average data. Average temperature in the primary shear zone was calculated by the well-regarded Loewen–Shaw model, which has been verified by a number of researchers [7,11,13]. The equivalent flow stress, strain, strain rate, and the average temperature of material in the primary shear zone were computed. In the literature, the equivalent flow stress was usually converted to the shear flow stress for comparing with the shear flow stress calculated by cutting models [11,12]. In this study, in order to correlate the mechanical behaviors in cutting and compression tests, the calculated shear strain, shear strain rate, and shear flow stress in cutting were converted to the equivalent strain, equivalent strain rate, and equivalent flow stress based on the assumption of isotropy and a von Mises yield criterion. The aluminum tube and rod 6061-T6 were solution heat treated and artificially aged, so the effect of material anisotropy on the mechanical behavior should be trivial. Before we correlate the mechanical behaviors in compression and cutting tests, it would be helpful to plot the stress vs. strain curves in compression tests together with the calculated flow stress vs. strain data in cutting in one graph so that the nature of the calculated flow stress and the strain by the cutting models would be clear. The obtained engineering stress–engineering strain and true stress–true strain curves for 6061-T6 aluminum in the compression tests are shown in Figs. 1 and 2. It is obvious that the stress vs. strain curves of 6061-T6 is not sensitive to low strain rates. The measured cutting force, the thrust force, and the chip thickness together with the calculated equivalent strain, equivalent strain rate, average temperature, and the equivalent flow stress under each cutting condition were listed in Tables 3–6. The equivalent flow stresses and strains in cutting were plotted in Figs. 1 and 2. The true strain rate for each cutting condition was labeled in Figs. 1 and 2. For the comparison
Fig. 1. Engineering compression stress–strain and cutting stress–strain (the labeled cutting strain rate is the average strain rate for each cutting group).
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Fig. 2. True compression stress–strain and cutting stress–strain (the labeled cutting strain rate is the average strain rate for each cutting group).
data in Figs. 1 and 2, the average temperature rise of 74 ◦ C during compression is close to the average cutting temperature, 79 ◦ C, at low cutting speeds, Tables 3 and 4. Considering cutting and compression tests are short time duration events, the low temperatures would not produce appreciable difference on the flow stress. The most significant difference is the strain rate between compression and metal cutting. Therefore the confounding effect does exist but would be small. The strains induced in cutting are indeed larger than those in conventional compression tests and SHPB tests. Strain extrapolation would be required if we correlate mechanical behaviors in compression and cutting tests. The average equivalent strain rates are still very high even at very low cutting speeds, Tables 3 and 4. The strain rates, Tables 5 and 6, are much higher than those achieved in conventional compression tests and SHPB tests if normal cutting speeds are used. The strain rate decreases at the same cutting speed if using a larger uncut chip thickness. It is not surprising because the increased chip volume reduces the average strain rate in the shear zone if a larger uncut chip thickness is used. Strain rate extrapolation would also be required to correlate flow stresses in compression and cutting tests. Average temperatures in the primary shear zone are 79 and 259 ◦ C, respectively, at very low and normal cutting speeds in Tables 3–6. It is noticed that the uncut chip thickness has very slight effect on the average shear zone temperature, while the cutting speed is a dominant factor which has been well recognized in the literature. As the average temperature of 79 ◦ C at low cutting speed (0.064 m/s) is close to the average temperature rise, 74 ◦ C, in the compression tests, we may ignore the temperature effect to correlate flow stresses. For the flow stress, the cutting temperature is the most important factor as we see that the flow stress decreases as the average temperature increases even both the strain and strain rate increase if we assume strain hardening and strain rate hardening hold. The strain rate effect can be demonstrated by extrapolating the flow stress from compression tests to the
same strain as in cutting. It is clear that strain rate hardening exist by comparing the flow stress by strain extrapolation. Strain hardening in Fig. 2 is obvious, and the strain hardening influence on the flow stress is much larger than that of the strain rate hardening in terms of the ratios of the flow stress increase per unit increment of strain and strain rate. However, the strain rate effect is larger than that of strain in terms of variation of the flow stress magnitude in the range of strains and strain rates encountered in cutting. The microstructure effect on the flow stress is beyond the scope of this study and will be investigated in the future work.
5. An integral method to construct the constitutive equation An effective method to determine the material mechanical behavior is especially important for process optimization and machine tool design. The mechanical behavior in the elastic range can be obtained by the conventional materials tests. The mechanical behavior in the plastic range can be obtained by cutting tests, independent conventional materials tests including SHPB, or combination of material tests and cutting tests. The pros and cons of these different methods are summarized in the literature review. Stevenson and Stephenson [12] showed a convincing demonstration of the correlation between conventionally compression data to the cutting data at room temperature based on a self-consistent strain rate sensitivity index. The metal cutting strains, Fig. 2, are beyond the range achieved by the conventional compression tests even though a large strain over one was achieved in the compression tests. Thus, the effects of the strain and temperature on the flow stress need to be incorporated. Further, a method to determine the flow stress by constructing a constitutive equation including the influences of strain, strain rate and temperature by using of the cutting data and conventionally determined material property data is still missing.
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The philosophy of determining the material flow stress in this study is to choose a relative accurate constitutive model and fit the parameters of the constitutive model using the data of conventional material tests and cutting tests. Although a universal accurate constitutive model is not currently available, the Johnson–Cook model has shown to be effective for metals and thus a good candidate [23]. In this study, we adopted the Johnson–Cook model, fitted its parameters by the data from compression and cutting tests, and then verified its effectiveness experimentally. The Johnson–Cook model [23] is cited as follows: ˙ ε¯ σ¯ = [A + B(¯ε) ] 1 + C ln (1 − Thm ) ˙ε¯ 0 n
(1)
where Th = (T − Troom )/(Tmelt − Troom ). The thermal softening term when m = 1 was used to vary in a linear form in this study since it was shown that the linear form gave reasonable results although the alternative forms for different values of the parameter m were used in the literature [16]. Obviously, the Johnson–Cook model assumes the flow stress to be independently affected by strain, strain rate and temperature represented by the terms in each set of the brackets. It should be pointed out that effects of the strain and strain rate on the temperature exist due to adiabatic deformation at high strain rates. As discussed in the literature review, metal cutting tests provide some advantages over the SHPB method and other independent material tests to estimate the material properties in cutting conditions. Considering the characteristics of material deformation in metal cutting, we propose a new integral approach to obtain the model parameters of Johnson–Cook constitutive equation using both the conventional compression and cutting tests. In this study, the compression tests were conducted under a series of low strain rates rather than the reference strain rate of 1 s−1 proposed by the Johnson–Cook model. The lowest average true strain rate of 0.01 s−1 was chosen as the reference strain rate in this study. Other parameters were determined based on the chosen reference strain rate. Actually, a small reference strain rate has negligible effect on the flow stress as it is clear that the flow stress is not sensitive to the low strain rates, Fig. 2. The parameter A is the yield stress at the reference strain rate. The parameters B and n, can be obtained from a least-squares fit of the test data to a power law equation since the strain rate effect can be neglected in the quasi-static compression tests. The parameter C can be determined by correlating the cutting data and the compression data at the same strain value. But the maximum strain achieved in the compression test is smaller than the cutting strain, strain extrapolation is necessary to find the parameter C. Let us denote σ¯ com is the extrapolated flow stress in compression and σ¯ cut the flow stress calculated by cutting models. With the calculated strain, strain rate, and the temperature in each cutting test, and the strain rate and temperature in the compression test, the parameter C can be determined as follows based on the
above determined parameters A, B, and n: σ¯ com = (A + B¯εncut )(1 − Thcom ) ˙ ε¯ cut n (1 − Thcut ) σ¯ cut = (A + B¯εcut ) 1 + C ln ε˙¯ 0
(2) (3)
Solving Eqs. (2) and (3) gives C=
[σ¯ cut (1 − Thcom )/σ¯ com (1 − Thcut )] − 1 ln(ε˙¯ cut /ε˙¯ 0 )
(4)
Each cutting condition yields a value of the parameter C. To represent the most likely equation for the cases considered, the regression method is used to determine the parameter C in the cutting experiment group at the confidence level of 95%. The obtained parameter C is a function of independent cutting parameters of the uncut chip thickness t0 and cutting velocity V for orthogonal cutting, and is regarded valid for all other cutting conditions. A relationship between the parameter C and the feed, cutting velocity, and the depth of cut in production machining can be determined similarly. The validity of the determined parameters is assessed by comparing the predicted flow stress with the experimental data in the following case study. The test conditions to determine the model parameters are different from those in validation test conditions. The proposed approach can be applied to characterize the mechanical behavior of any metals in machining with continuous chips.
6. Case study and method assessment According to the method of constructing a constitutive equation in Section 5, the model parameter A, 275 MPa, is the yield stress at the reference strain rate of 0.01 s−1 in the compression tests. The parameters B and n, 86 MPa and 0.39, were determined by fitting the compression data to a power law equation. The parameter C was determined as follows from the cutting test data using the linear regression method: C = 0.058108 − 194.091t0 − 0.00306V
(5)
With the determined model parameters, the constitutive equation of 6061-T6 aluminum in machining is as follows: ˙ ε¯ 0.39 σ¯ = [275 + 86(¯ε) ] 1 + C ln (6) (1 − Th ) 0.01 where the parameter C can be calculated by Eq. (5) for each cutting condition. With the determined constitutive equation of 6061-T6 aluminum in machining, the effectiveness of the method is verified by comparing the predicted flow stresses using Eq. (6) with the experimental data in compression and cutting tests. For different low strain rates and low temperatures in compression tests, the predicted flow stress was plotted with the test data, Fig. 3. It shows that the predicted flow stresses agree with the compression data very well. For moderate
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Fig. 3. Comparison of predicted flow stresses to the compression data at low strain rates and room temperature.
Fig. 4. Comparison of predicted flow stresses to the test data in the cutting condition (Table 3: corresponding strain, moderate strain rate, and low temperature).
strain rates and low temperatures in cutting tests, the predicted flow stresses were compared to those calculated by the cutting models, Fig. 4, which shows that the predicted flow stress agrees with the cutting data with an average difference of 8.6%. For high strain rates and relative high tem-
Fig. 6. Comparison of predicted flow stresses to the test data in the cutting condition (Table 6: corresponding strain, high strain rate, and moderate temperature)
peratures in cutting tests, the predicted flow stresses were plotted against those calculated data by the cutting models, Figs. 5 and 6. Fig. 5 shows the prediction and the experimental data agree in trend but with a difference of 11.8%. While the difference is 17.5% in Fig. 6. The flow stress comparison shows that the constitutive equation obtained by the proposed method is valid for both quasi-static compression tests of low strain rates and low temperatures, and cutting tests of large strains, high strain rates, and relative high temperatures.
7. Discussions
Fig. 5. Comparison of predicted flow stresses to the test data in the cutting condition (Table 5: corresponding strain, high strain rate, and moderate temperature).
Though the Johnson–Cook model slightly overestimates the flow stress at the relatively high temperatures, Figs. 5 and 6, it gives overall a reasonable prediction of the flow stress considering the errors associated with force measurement and other error sources. Firstly, it needs to be mentioned that the calculated flow stress also depends on strain path in metal cutting. The current compression experimental setup compressed the sample to the strain up to 1 in one
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step at room temperature. The work material in tube cutting is under sequential cutting conditions where the strain hardened material will be deformed again by subsequent cuts, so the work material in cutting tests experiences different strain path from that in the compression tests. Secondly, the strain path dependence of the flow stress is another factor to influence the slight difference between the predicted data and those of cutting data in Figs. 5 and 6. It has shown that the flow stress, determined from the tension or compression data, is generally greater than when it is determined from the torsion data [23]. It means the flow stress is also affected by the deformation modes. The major deformation mode in cutting is shearing as in torsion while it is compression in compression tests. The different deformation modes also affect the difference between the flow stresses. That how strain path and deformation mode influence the flow stress is a complex issue and beyond the scope of this study. Thirdly, although the parameters of the Johnson–Cook model are determined by fitting the experimental data, it should be realized that the mechanical behavior is far more complicated than assumed by the semi-empirical constitutive model or other models, which consider strain, strain rate and temperature only. The microstructural effects may also contribute to the difference between the model prediction and experimental data in Figs. 5 and 6 since the difference is much smaller in low cutting temperatures, Fig. 4. Lastly, the accuracy of the cutting models is limited by some oversimplifications. In addition, the Johnson–Cook model assumes the independent effect of strain, strain rate, and temperature on the flow stress although the compound effect of the strain rate and temperature exists. The errors of strain path, deformation mode, cutting models, and the Johnson–Cook model are difficult to evaluate and therefore is presently not clear. Correspondingly, the proposed method should be regarded to give only a first approximation of the actual mechanical behavior. On the other hand, for FEA of plastic flow at high strain rates such as machining, a simple analytic relationship is highly desirable. Further, very complex constitutive models are of little computational benefit if there are no data available to determine constants for these constitutive equations. So until the models have been further evolved, a phenomenological approach based on appropriate experiments such as those in this paper to assess the mechanical behavior under production conditions should prevail. It should also be pointed out that the temperature calculated by the Loewen–Shaw model is the average temperature in the primary shear zone. The accuracy of the model is demonstrated by the difference of less than 3% between the calculated temperature and the experimental data in machining AISI 1113 steel [8]. Further, it was also evident [8] that the temperatures are not sensitive to the temperature-dependent material thermal properties. Considering the small width (<0.01 mm in the case study) of the primary shear zone, the average temperature should be representative for the entire shear plane. Although the cutting speeds in the case study are relatively lower than those in
high speed machining, the proposed method are valid as long as the chip does not melt so that the method is independent of cutting speeds.
8. Conclusions A general method based on the Johnson–Cook model has been presented to characterize the flow stress of work materials in machining. The model parameters in the strain hardening term are determined by fitting the data from both conventional compression tests to a power law equation. The parameter in the strain rate term is determined by the linear regression method. Case studies of quasi-static compression tests and orthogonal cutting of 6061-T6 aluminum tubing were conducted. It shows that the model predicted flow stresses agree with the test data with reasonable accuracy from quasi-static compression tests at low strain rates and low temperatures to cutting tests with high strain rates and high temperatures. It was also confirmed that the compression data can be correlated with the cutting data by extrapolating the strain, strain rate, and the temperature. The developed method is valid for machining with continuous chips at a variety of cutting speeds. The accuracy of the proposed constitutive model is affected by the strain path, material deformation mode, microstructural effects, cutting models, and the adopted constitutive model itself.
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