were based on assuming constant contact conditions, led to the conclusion that the thermal loading on the coating itself is increased. As Klocke et al. [2] correctly ...
A New Approach to Cutting Temperature Prediction Considering the Thermal Constriction Phenomenon in Multi-layer Coated Tools M. H. Attia1,2, L. Kops2 (1) Aerospace Manufacturing Technology Centre Institute for Aerospace research, National Research Council Canada, 2 Department of Mechanical Engineering, McGill University Montreal, Quebec, Canada 1
Abstract A novel approach to the prediction of cutting temperature in multi-layer coated tools is presented. This approach considers the contact mechanics at asperity level and resulting thermal constriction resistance phenomenon. Micro-contact model was developed and the correlation between contact pressure and thermal constriction resistance of uncoated and multi-layer coated tools is established. The thermal interaction and redistribution of heat between the workpiece, the chip and the tool were analyzed, supported by FE model, which considers thermal characteristics of multi-layer coating. The results indicate that the tool coating may cause significant reduction in heat flowing into the tool. Keywords: Cutting, Tool coating, Thermal constriction resistance
1 INTRODUCTION The issue of improving the thermal and tribological characteristics of coated tools is at the center of efforts aiming at improving productivity and enhancing the performance of the emerging technologies of high speed machining and hard machining. Experiments simulating the machining process indicated that the coating has significant effect on temperature field in the substrate [1]. It was postulated that coatings with higher thermal diffusivity allow more heat flow from the tool-chip contact zone into the substrate. Recent FE analysis results [2] showed, however, the insignificant effect of coatings on the temperature field and maximum temperature rise in the multi-layered coating and the substrate. These observations, which were based on assuming constant contact conditions, led to the conclusion that the thermal loading on the coating itself is increased. As Klocke et al. [2] correctly pointed out, the modeling of this problem requires proper consideration of the effect of coatings on the change in the contact conditions and the conceivable decrease in the heat transfer between the chip and tool. The main objective of this paper is to present a novel approach to cutting temperature prediction in multi-layer coated tools, which is based on the contact mechanics at the asperity level and the resulting thermal constriction resistance. This approach can be used for the design of cutting tools and other related issues, with higher degree of confidence than the perfect contact approach. 2 THERMAL CONSTRICTION PHENOMENON IN THE PRESENCE OF MULTI-LAYER COATING Due to the nature of real surfaces, the physical and tribological interactions between contacting solids are limited to the highest asperities. As a result, frictional heat generated at the micro-contact areas will spread out rather than taking a straight path. This gives rise to the socalled thermal constriction (or spreading) resistance Rc. To overcome this resistance, a steep temperature gradient has to be established in the subsurface layer, giving rise to high contact temperature and high thermal
stresses. The analysis presented by Attia et al. in static and dynamic contact [3] showed that the temperature distribution in such thermally distributed layer, which is of the order of 50-100 µm, takes an exponential form. This phenomenon is quite complex when the surfaces are coated with materials whose thickness are of the order of the micro-contact areas. The thermal response behaviour of a multi-layer coated tool is controlled both by the micro-contact configuration at the tool-chip interface and the thermal constriction phenomena taking place at the tool-chip-workpiece system. These two aspects are analyzed in the following subsections. 2.1 Mechanical Contact Problem: Micro- and MacroContact Configuration The contact configuration at the tool-chip interface is defined by the apparent macro-contact area of contact Aa and the real area of micro-contacts Ar. The contact area Aa is the product of the tool-chip contact length Lc and the width of the chip b. To a first approximation, the ratio Ar/Aa is equal to the ratio pc/H between the applied pressure and the effective hardness of contacting solids. The contact length Lc depends on the undeformed chip thickness [4], the thermal conductivity of the cutting tool [5], the shear plane φ, and consequently all other process variables that affect φ, e.g., the tool rake angle α and the coefficient of friction µ [6]. The presence of surface coating affects both the macroand micro-contact configuration through its effect on the friction force [7,8], the effective surface hardness and the interface stiffness and the contact length Lc [9]. For normally distributed surface asperity heights, the attributes of the micro-contact configuration are defined [10] by relations expressed in (1), (2) and (3). In these equations ε is the constriction ratio, γ is the density of micro-contact areas, r1 is the average radius of micro-contact areas, σ is the standard deviation of the asperity heights, m is the mean absolute slope of the asperities, and M is the number of micro-contacts over the apparent area Aa.
⎛A ⎞ p 1 ε2 = ⎜ r ⎟ = c = erfc(x) ⎜A ⎟ σ 2 f ⎝ a⎠
M 1 ⎛m⎞ γ= = ⎜ ⎟ Aa 16 ⎝ σ ⎠
⎛ ⎞ 2 ⎜ exp ⎛⎜ − 2x 2 ⎞⎟ ⎟ ⎝ ⎠⎟ ⎜ ⎜ erfc(x) ⎟ ⎜ ⎟ ⎝ ⎠
8 ⎛σ⎞ r1 = ⎜ ⎟ exp⎛⎜ x 2 ⎞⎟ erfc(x) ⎝ ⎠ π ⎝m⎠
(1)
(2)
(3)
The parameter x, which appears in the complimentary error function erfc(x), is defined as:
x=
Y 2 σ
(4)
where Y is the separation between the median planes of contacting surfaces. Careful examination of equations (1) to (3) shows that the parameters ε, γ and r1 are interrelated by a single dimensionless parameter x, which is uniquely defined by the contact pressure ratio pc/H. 2.2 Thermal Constriction Model Figure 1 depicts an idealized orthogonal machining process using a sharp tool. In this idealization, there are two heat sources; the primary shear plane source Q1 and the rake face source Q2. The latter accounts for the secondary plastic deformation zone in the chip, and the tool-chip friction. The inset in Figure 1 shows that point-to-point contact is encountered in the sliding zone. The heat generated in the process enters the tool through a limited number of small contact spots, whose average radius is r1. The volume, which encompasses each of these micro-contact areas and extends some distance into the solid, is defined as the elemental heat flow channel (HFC). Since the HFCs are connected in parallel, the solution of the heat transfer process in a single channel presents the building block to model the whole process. Since no data is available regarding the thermal constriction resistance of a multi-layer coating, this problem has been investigated numerically. A finite element idealization of a typical heat flow channel is shown in Figure 2. The axi-symmetric model incorporates three layers of different coatings (c1, c2, c3). The ratio between the radius r1 of the micro-contact area and the
Figure 1: Schematic of the contact configuration and heat sources in the tool-chip-workpiece system.
radius r2 of the HFC was varied to produce constriction ratio 0.05 < ε < 0.95. The element size in the contact region was selected to be 1 µm. Each coating layer was 3 µm thick and was divided into three rows of finite elements. Two types of multi-layer coatings were considered: coating 'A' (TiN/Al2O3/TiC) and coating 'B' (TiN/TiCN/TiC), both with TiN layer on the top. All surfaces are adiabatic except the micro-contact area, where a uniform heat flux is applied. This satisfies the definition of HFC and implies that the heat transfer by conduction and convection through the interstitial fluid occupying the gap between the surface asperities at the tool-chip interface is neglected. The latter effect was dealt with separately following the approach proposed by Yovanovich [10]. The selected values of the thermal conductivity k of steel (the workpiece/chip material), tungsten carbide, TiN, TiCN, Al2O3, and TiC are 45, 60, 25, 32, 7 and 38 W/mºC, respectively. The temperature distribution in surface sub-layer along the center of a single micro-contact spot is shown in Figure 3, for coating A and ε = 0.2. From figure 3(b), one can realize that a 'pseudo' temperature drop ∆Tc exists between the extrapolated temperature values outside the so-called 'thermal disturbance zone' [3]. Knowing ∆Tc and the heat flow across the micro-contact area Q, the constriction resistance Rc is determined: Rc =
∆Tc Q
(5)
The correlation between the contact pressure pc and the constriction resistance is expressed by the dimensionless parameter ε and the constriction parameter ψ :
ψ = 4 k r1
(6)
Figure 4 shows the ψ−ε relationship for both types of coatings. In calculating ψ, the thermal conductivity of 'WC' was selected as a reference value (5). The significant increase in the constriction resistance (i.e., the constriction parameter ψ) with the reduction in the contact pressure (i.e., the constriction ratio ε) is evident. The figure also shows that coating can increase the constriction resistance (and consequently, the interface temperature rise) by a factor > 2, for small contact pressures. The curves converge, however, at high contact pressures, resulting in relatively small effects. Coatings with Al2O3 layer exhibit higher resistance for ε < 0.25. The functional relationship between constriction parameter ψ
Figure 2: Finite element model of the heat flow channel.
Constriction Parameter ψ
3 Coating A
2.5 2
Coating B
1.5 Uncoated
1 0.5 0 0
0.2
0.4
0.6
0.8
1
Constriction Ratio ε (a) Figure 4: Effect of constriction ratio ε on the thermal constriction parameter ψ.
∆ Tc 400
200
0 0 20 30 40
By combining (2), (3) and (10), the following expression is obtained
Depth z, mm
10
( )
⎛ 2π ⎞ ⎛ σ ⎞ ⎟ ⎜ ⎟ exp x 2 ⎡⎢ ψ1 + ψ 2 ⎤⎥ Rc = ⎜ ⎜ Aa ⎟ ⎝ m ⎠ ⎣ k1 k 2 ⎦ ⎝ ⎠ Substitution expression:
50
Figure 3: (a) Temperature distribution in the subsurface of a multi-layer coated micro-contact model, and (b) Temperature profile along the axis of the HFC. and the constriction ratio ε was found to be well presented by the following simple expression:
ψ = α (1- ε) β
(7)
The constriction resistance of a single ith contact spot is the sum of two resistances in series:
⎞ ⎟ ⎟ ⎠j
(8)
where the subscripts 1 and 2 refer to the two contacting solids. Over the sliding zone at the tool-chip interface, there are M contact spots that are thermally connected in parallel. Therefore, the total thermal constriction resistance Rc is determined by the following relation: 1 = Rc
M
1 = 4k1 R i =1 ci
∑
M
ri + 4k 2 ψ i =1 1,i
∑
into
(11)
yields
the
following
⎡ α (1 − ε ) β 1 β ⎛ 2π ⎞ ⎛ σ ⎞ α (1 − ε ) 2 ⎟ ⎜ ⎟ exp ⎛⎜ x 2 ⎞⎟ ⎢ 1 Rc = ⎜ + 2 ⎜ Aa ⎟ ⎝ m ⎠ ⎝ ⎠⎢ k1 k2 ⎝ ⎠ ⎣⎢
(b)
⎛ ψ ⎞ ⎟⎟ + ⎜ 2 ⎠ j ⎜⎝ 4k 2 r
(7)
60
o
Temperature T, C
⎛ ψ R cj = ⎜⎜ 1 ⎝ 4k 1 r
of
(11)
M
r
∑ ψ 2i ,i
(9)
i =1
Since the thermal process in practical problems cannot be dealt with analytically, the replacement of the continuous contact pressure and temperature distributions by their equivalent step-wise discrete representation is permissible. Over a small region of constant contact pressure and temperature, one can justifiably assume that the radius of the micro-contact area r1 and the constriction parameters are constant. Therefore, (9) is reduced to: 1 ⎛⎜ ψ1 ψ 2 ⎞⎟ Rc = (10) + 4 r M ⎜⎝ k1 k 2 ⎟⎠
⎤ ⎥ (12) ⎥ ⎦⎥
The two terms in the bracket […] in (12) represent the thermal constriction resistance of each of the two contacting bodies 1 and 2. The structure of (12), with its additive terms, allows us to develop a general expression for the contribution Rc,i of each of the contacting solid to the thermal constriction resistance:
( )
Rc , i = 2π α i exp x 2 (1 − ε ) βi , i = 1,2 (13) where the dimensionless constriction resistance is defined as: Rc,i = Rci ki Aa (m/σ). To reduce (13) into a simpler mathematical expression, the term F:
( )
F = exp x 2 (1 − ε )
βi
⎛p = C ⎜⎜ c ⎝ H
⎞ ⎟⎟ ⎠
n
(14)
is numerically evaluated. Examination of the behaviour of the function F shows that it is insensitive to the exponent βi for the range of interest. This allowed us to establish the following simple expression, in which the effect of the coating is only introduced through a scaling parameter that is embedded in the coefficient C:
( )
F = exp x 2 (1 − ε )
βi
⎛p ⎞ =C ⎜ c ⎟ ⎝H ⎠
n
(15)
The constant C = 0.136 and the exponent n = - 0.97, for 1.5 < ε < 2.0. For the practical range of contact pressures pc/H < 0.01, the error in using (15) is < + 3%. Combining (13) and (15), yields the following:
⎛p ⎞ ℜc,i = 0.34 αi ⎜ c ⎟ ⎝H⎠
0.97
(16)
Validation of the Thermal Constriction Model Equation (16) was validated by reducing it to the special case investigated by Yovanovich [10] for uncoated surfaces with pc/H < 0.02. The results were found to be identical to those reported in [10]. The results presented in this study were also validated against the experimental results and the analytical solutions presented in [11] and [12] for uncoated surfaces and for single-layered coating, respectively. The relative errors were found to be < 6%. At this point, it is instructive to comment on the significant error introduced by treating the coating as a merely thin layer of resistor connected in perfect contact with the substrate, as was presented in [9]. If one tries to express the constriction resistance in terms of equivalent additional resistance made of the 'WC' substrate, then at ε = 0.5, coating A will introduce the equivalent to 2.3 mm of WC, which is more than two orders of magnitude greater than that of the case of perfect contact. 3 MODELLING OF THE HEAT TRANSFER PROCESS IN MULTI-LAYER COATED TOOLS 3.1 Application of the Thermal Constriction Model: To apply the thermal constriction model, the distributions of the normal and shear stresses along the tool-chip interface are first estimated. The distributions of the heat flux and the corresponding thermal constriction resistance are then determined, using Muraka's model [15] and (16). This methodology is demonstrated in the following subsections. The significant effect of the multi-layer coating on the cutting temperature and the re-distribution of heat between the tool and the chip are also discussed. A portion of a two-dimensional finite element model for the heat transfer process into the tool, through its interface with the chip is presented in Figure 5. The size of the element in the first row in the contact region was selected to be 50 µm x 100 µm. The elements of this row were then treated as 'contact elements'. This selection of the element size was based on the analysis conducted by Attia et al. [3], in which the depth of the thermally disturbed zone was estimated to be of the order of 50-100 µm. Given the local contact pressure at any nodal point over the sliding region Ls, the constriction ratio was determined and then used to estimate the constriction parameter ψ (figure 4). Using (16), the thermal constriction Rc can readily be estimated. By adding Rc to the nominal thermal resistance Re of the 50 µm x 100 µm contact element, an equivalent thermal conductivity keq was calculated individually and was then used as input data to the FE solver:
k eq =
δ Rt A
cases described in Table 1 were carried out under the following operating conditions: rake angle α = 0, cutting speed V = 200 m/min, undeformed chip thickness h = 0.2 mm. Cases 1-3, which are designated as uncoated, perfect contact, represent the reference cases, in which there is no constriction resistance over the length Ls, as commonly assumed in previous investigations. In all other cases 318 (coated and uncoated), the constriction resistance along Ls was considered. The values of forces Fn and Ft, as well as the tool-chip contact length Lc were obtained experimentally using the data published in [6-9]. The distribution of the contact pressure along the tool-chip interface was assumed to follow a power law, with exponent n = 3 [14]. Knowing the local contact pressure over the sliding region Ls, the equivalent thermal conductivity of the contact element is estimated from (17). Case #
Coating type
Heat input Q (W) 50,70,90
Lc (mm)
Fn/Ft (N)
0.8
1800/800
1,2,3
Uncoated, perfect contact
4,5,6
Uncoated, imperfect contact
50,70,90
0.8
1800/800
7,8,9
Uncoated, imperfect contact
50,70,90
1.0
1800/800
10,11,12
Coating A
50,70,90
0.7
1500/530
13,14,15
Coating A
50,70,90
0.6
1500/530
16,17,18
Coating B
50,70,90
0.7
1500/530
Table 1: Conditions of the FE test cases.
(17)
where δ is the thickness of the contact element, and the total thermal resistance Rt is the sum of Re and Rc. Over the adhesion region, La, the condition of perfect contact was assumed with ψ = 0. With the exception of the tool chip interface, the convective coefficient of heat transfer h over the overhang surfaces of the tool was taken as 28 W/m2 ºC [13]. 3.2 Design of Numerical Experiments To demonstrate the effects of the constriction phenomenon and the coating, the analysis was carried out for the following groups (Table 1): (a) uncoated tool, with perfect contact along the contact length Lc, (b) uncoated tool with imperfect contact along the sliding zone Ls, and (c) coated tool, types A and B, with imperfect contact along the sliding zone Ls. The numerical test
Figure 5: Finite element idealization of the coated tool insert (top), and close-up of the cutting edge (bottom).
The heat flux boundary condition applied to the tool-chip interface Lc is based on the assumption that the flux is uniform over the adhesion zone La and drops linearly to zero over the sliding region Ls. This distribution follows that of the shear stress [15].
Average Interface Temoerature, C
4 RESULTS AND DISCUSSION 4.1 Correlation between Heat Input and the Average Temperature Rise at the Tool-Chip Interface The results of the FE analyses are compiled in Figure 6, which shows the relationship between the average temperature rise at the tool-chip interface and heat input Q for the following interface conditions: Line A (coating B, and Lc = 0.6 mm), line B (coating B, and Lc = 0.7 mm), line C (uncoated, imperfect contact and Lc = 0.8 mm), line D (uncoated, perfect contact and Lc = 0.8 mm). The average temperature-heat input line designated as E is based on the analytical model presented by Shaw [6] for uncoated, perfect contact and Lc = 0.8 mm. The slopes of the linear relationships (A to D) reflect the role of constriction resistance and tool coating. The deviation between these results and the analytical prediction (line E [6]) is quite evident. The fact that these lines converge to the origin underlines the significant effect of the interface condition at high heat input (medium and heavy cuts). It should be noted that the Q represents the resultant effect of various process variables, e.g., cutting speed, feed rate, depth of cut, rake angle. This graph is to be used to estimate the redistribution of heat among the components of the workpiece-chip-tool system and the tool temperature, as will be discussed later. 4.2 Effect of Constriction Resistance and Coating on Tool Temperature Figure 7 shows the isothermal lines in the Y-Z plane, limited for clarity to the region close to the cutting edge, corresponding to the close-up of the FE model shown in Figure 5, for Q = 90 W. Figure 7(a) represents the case of no coating and perfect contact over the full contact length. Figure 7(b) represents similar condition, but with the constriction resistance over Ls is taken into consideration. In Figure 7(c), the tool was coated with type A (TiN/Al2O3/TiC). The figure shows that for the same heat input to the tool, the presence of coating results in a significant increase in the tool-chip interface temperature rise. It also causes the maximum temperature to shift from the cutting edge to the other extreme point of the contact
1200 D
C
B
length. A change in the temperature by approximately 130 degrees should be viewed in light of the fact that the hardness of carbide and ceramic tool falls rather rapidly at temperatures above 650ºC [6]. The increase in the tool temperature from 650 to 750ºC causes a drop in hardness approximately from 57 to 52 HRC for WC. However, one should not interpret the effect of coating (which acts as a thermal barrier) as source of increase in the tool temperature. The thermal equilibrium of the workpiecechip-tool system may cause the redistribution of the heat flowing into the tool and the chip in such a way that resultant effect is reducing the tool temperature. This important issue will be clarified and discussed in section 4.3. Results obtained for Q = 50 and 70 W, which represent light and medium cutting conditions, showed similar trend. In the case of Q = 50 W, the maximum temperature changes from 575 to 585 and then 645 ºC, respectively. 4.3 Effect of Constriction Resistance and Coating on Redistribution of Heat To assess the effect of coating and thermal constriction resistance on the tool temperature and the redistribution of heat, the case presented in [6] for uncoated tool was analyzed. The selection of this case is based on the fact it was an experimental one, with well documented test conditions: Workpiece material: free machining steel, cutting tool: carbide (with rake angle α = 20º). The cutting force Fc = 3750 N, feed force, Ff = 1190 N, bulk chip speed V = 69 m/min, feed rate = 0.274 mm/rev, shear angle φ = 27.8º, chip-tool contact length Lc = 0.91 mm. Calculated maximum tool face temperature was reported to exceed 1000 ºC [6]. The problem was first solved using Shaw's analytical approach [6], and then re-examined using the tool thermal
(a)
A
1000 800 600
(b)
E
400 200 0 0
20
40
60
80
100
Heat input, W
(c) Figure 6: Effect of coating conditions and contact length on the heat input-temperature relationship.
Figure 7: Effect of thermal constriction and surface coating on temperature distribution in the tool.
characteristics produced in this study and summarized in Figure 6. The results of this analysis are given in Table 2, for the following three conditions:
7 REFERENCES
[1]
(1) Uncoated tool, assuming perfect contact over Lc, (2) Uncoated tool, with constriction resistance over Ls, (3) Coated tool (coating type A), considering the constriction resistance over Ls. The temperatures Θs and Θt represent the temperatures of the shear plane and the tool-chip interface, respectively. In case (3), the forces and contact length were scaled down to account for the effect of coating, based on the data published in [7,8]. The results show that, for these given conditions of heavy cut, the tool coating in the presence of thermal constriction resistance reduces the amount of heat flowing into the tool by a factor of two and reduces the maximum temperature by about 120 ºC.
Parameter
Uncoated tool, perfect contact (A)
Uncoated tool, with constriction resistance
Coated tool, with constriction resistance
Lc (mm)
0.91
0.91
0.75
Fc, Ff (N)
3750, 1190
3750, 1190
3125, 790
Θs (ºC)
370
370
313
Θt (ºC)
1076
1096
955
Q (tool) (W)
176
101
82
Q (chip) (W)
8,004
8,078
6,795
Q (work) (W)
1,533
1,533
1,217
[2]
[3]
[4]
[5]
[6] [7]
[8]
[9]
Table 2: Effect of constriction resistance and tool coating on heat redistribution in the workpiece-chip-tool system. [10] 5 CONCLUSIONS A novel approach to cutting temperature prediction in multi-layer coated cutting tools is developed. This approach is not based on the commonly used assumption of perfect contact at the tool-chip interface, but rather the contact mechanics at asperity level and the resulting thermal constriction resistance. A constriction model was developed, and the correlation between the contact pressure and the thermal contact resistance of uncoated and multi-layer coated tools is established. The model was validated against analytical and experimental data. It was found that coating causes reduction of the heat flow into the tool and reduction of the maximum temperature rise. These reductions can reach more than 50% and 120ºC, respectively. The importance of the present approach lies in the fact, that it can be used with a higher degree of confidence for the design of coated tools and other related issues, such as e.g. wear. 6 ACKNOWLEDGEMENTS This investigation was conducted under the support of the Natural Sciences and Engineering Research Council of Canada, which the authors greatly appreciate. The authors acknowledge the effort of Mr. A. Cameron and Mr. S. R. Sen, Research Assistants, Mech. Engrg. Dept., McGill University, in FE modeling and analysis.
[11]
[12]
[13]
[14]
[15]
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