Jiang Hua Post Doctoral Researcher e-mail: [email protected]

Rajiv Shivpuri Professor e-mail: [email protected] 1971 Neil Avenue, Room 210, Industrial, Welding and Systems Engineering, The Ohio State University, Columbus, Ohio 43210

A Cobalt Diffusion Based Model for Predicting Crater Wear of Carbide Tools in Machining Titanium Alloys In machining titanium alloys with cemented carbide cutting tools, crater wear is the predominant wear mechanism influencing tool life and productivity. An analytical wear model that relates crater wear rate to thermally driven cobalt diffusion from cutting tool into the titanium chip is proposed in this paper. This cobalt diffusion is a function of cobalt mole fraction, diffusion coeficient, interface temperature and chip velocity. The wear analysis includes theoretical modeling of the transport-diffusion process, and obtaining tool–chip interface conditions by a nonisothermal visco-plastic finite element method (FEM) model of the cutting process. Comparison of predicted crater wear rate with experimental results from published literature and from high speed turning with WC/Co inserts shows good agreement for different cutting speeds and feed rate. It is seen that wear rates are independent of cutting time. 关DOI: 10.1115/1.1839192兴 Keywords: Tool Wear, Diffusion, Crater, Machining, Ti-6Al-4V

1

Introduction

The major factors controlling the wear of cutting tools during the machining process include the mechanical and physicochemical properties of the workpiece and that of the tool, the tool edge geometry, and the cutting conditions, such as cutting speed and feed 关1兴. Tool wear mechanisms are often abrasion, adhesion, and diffusion 关2兴, with diffusion being most significant for cemented carbide tools in cutting hard-to-machine materials, such as titanium alloys and nickel alloys, where the temperature along the tool–chip interface can reach as high as 1000°C 关3,4兴. Diffusion in cutting is a process in which atoms of diffusing constituent transfer from the tool to the flowing chip 共and vice versa兲, and are carried away by the flow of chip material along the tool–chip interface 关5,6兴. This material transport results in a ‘‘crater’’ that forms on the rake surface of the cutting tool at a short distance from the cutting tip. This crater gradually becomes deeper and deeper with cutting time, resulting in a larger positive rake angle, weakening of the tool tip, and ultimately catastrophic failure of the cutting edge. In the past decades a large number of investigations have been done on tool wear in cutting operation by adopting Taylor’s tool life model and its extended equations. These equations describe the relationship between tool life and process parameters 共cutting speed, feed rate, etc.兲 and involve a few constants and exponents that need to be experimentally determined for a specific combination of tool–workpiece system. In addition to the tool life models, tool wear rate models have also been developed to evaluate wear resistance of cutting tool. For example, Takeyama proposed a fundamental wear rate model for abrasive wear that take into account cutting distance and diffusive wear 关7兴. Usui et al. derived a wear rate equation on the basis of adhesive wear which is a function of cutting temperature, normal stress, and sliding velocity at the tool–chip interface 关8兴. In both Takeyama’s and Usui’s models, there are constants that need to be determined experimentally for a given combination of tool and workpiece materials. Cook et al. 关5兴 developed a wear rate model for diffusion wear based on thermo-mechanics theory, and recently, Molinari et al. 关1兴 proManuscript received September 10, 2003; revision received July 14, 2004. Review conducted by: S. Mall.

136 Õ Vol. 127, JANUARY 2005

posed a diffusion wear model which considers the temperature on the tool–chip interface to be the decisive factor controlling the diffusion wear in the normal direction to the tool rake face. In the Cook model, only the diffusion in the normal direction to the tool–chip interface is taken into account. Molinari’s model does account for the effect of material convection due to the sliding of the chip along tool rake face 共mass transport along tool–chip interface兲. However, it does not predict the crater profile over the tool rake face but assumes it based on experimental observations. None of these models have specifically been developed for machining titanium alloys. The issues fundamental to the modeling of crater wear in the machining of titanium alloys are summarized in the following: • Average crater wear rates of straight WC/Co grades such C2 are far lower in the machining of titanium alloys than wear rates of ceramics, cermets, Cubic Boron Nitride 共CBN兲 or coated grades. Table 1 from Ref. 关6兴 includes results of machining tests with many different tool material combinations. Similar conclu-

Table 1 Average crater wear rates of various tool materials in the turning of Ti-6Al-4V at 200 sfpm „61 mÕmin… †6‡ Tool material Al2 O3 共Carboloy 030兲 HfO2 -coated Al2 O3 ZrC-coated Kennametal K7H 共C8兲 HfC-coated Kennatmetal K68 共C2兲 Cemented TiC 共XLO XL88兲 TiB2 -coated K7H Cast TiCN 共Teledyne SD-3兲 CBN (BorazonR) TiC-coated WC 共Carboloy 523兲 HfC-coated WC 共Surftech H-25X兲 HfN-coated WC 共Teledyne HN⫹兲 TiN-coated WC 共Teledyne TC⫹兲 Kennametal K7H 共C8兲 Surftech H-25X 共C3兲 Surftech H2S 共C2兲 Kennametal K68 共C2兲 Carboloy 820 共C2兲 Diamond (SynditeR) Diamond (CompaxR)

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Crater wear rate 共␮m/min兲 790. 560. 56. 52. 43. 39. 33. 30. 30. 22. 11. 11. 8.5 8.0 4.5 2.5 2.2 1.4 1.3

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Table 2 Diffusion coefficients of different tool materials in Ti6Al-4V †6‡ Tool Material Diamond NbC TiC VC WC ZrC TiN ZrO2 TiB2

Diffusion Coefficient 共cm2/sec兲 2.28 ⫻ 3.66 ⫻ 4.82 ⫻ 2.92 ⫻ 1.05 ⫻ 1.40 ⫻ 4.82 ⫻ 1.40 ⫻ 4.82 ⫻

10⫺6 10⫺9 10⫺9 10⫺9 10⫺10 10⫺8 10⫺9 10⫺8 10⫺9

sions are reached in Ref. 关9兴 which states that ‘‘Straight grade 共WC-Co兲 cemented carbides are the most suitable tool materials commercially available for machining titanium alloys. The best results are obtained with cobalt content ⬃6 wt % and WC grain size of not less than 0.8 mm. Cobalt contents ⬎6 wt % promote plastic deformation of the cutting edge.’’ • Diffusion pre-dominates crater wear on the rake face in the uncoated cemented carbide and ceramic cutting tools. Attrition wear and flank wear is more common in the other grades and CBN 关6,9兴. • The diffusion coefficient of cobalt D (D⫽D 0 e ⫺Q/R ( 273⫹ ␪ ) , where D 0 ⫽1.9 mm2 /s, the D max in this study is 8 ⫻10⫺4 mm2 /s) is much higher than that of tungsten carbide (1.05⫻10⫺8 mm2 /s, see Table 2兲 in titanium at 1270 K. Consequently, the diffusion of cobalt dominates the crater wear mechanism. This is contrary to the approach taken in Ref. 关1兴 where crater wear is modeled as diffusion of tungsten into the 1018 steel chip. • In the proposed model, as shown in Fig. 1, cobalt diffusion leads to the dislodging and breakaway of the WC grains and the formation of crater on the rake face. Larger WC grain size will lead to lower specific surface area and consequently, lower wear. The lower diffusion rate needs to be balanced against lower toughness associated with larger grain size. In this study, a crater wear model is presented for machining titanium alloys that takes into account loses in tool strength due to diffusion of Cobalt species into the titanium alloy chip resulting in decohesion of the WC grains, Fig. 1. The interface parameters affecting diffusion rate such as chip sliding velocity and temperature are determined using a nonisothermal visco-plastic FEM model of the orthogonal turning process. Experimental results obtained by Hartung et al. 关6兴 and Dearnley et al. 关9兴 are used to validate the model at lower cutting speeds. Finally, a nonorthogonal turning test is conducted to validate the model for higher cutting speeds.

2

Governing Equations

Accurate prediction of tool wear during machining is highly dependent on both the numerical model of cutting process, which provides reliable process parameters such as temperature and chip velocity along the rake face, and the wear model, which uses the input parameters obtained by numerical model to predict tool wear. 2.1 Diffusion Model of Crater Wear. An important step in the establishment of a predictive tool wear model is to relate the rate of tool wear to the machining conditions. Since crater wear is mainly caused by the high temperature generated at the tool surface and the convection transport of diffusing species to the freshly cut chip, the wear rate should be a function of temperature and cutting speed. Assuming that the tool wear in machining of titanium alloys occurs mainly due to diffusion and that the cutting process can be represented as ‘‘orthogonal,’’ the problem can be treated as twodimensional determination of wear magnitude at the rake face of the tool. Consequently, it can be assumed that the boundary surface between the tool rake surface and the work chip is a plane. The formulation in this section assumes dilute solution, i.e., the concentration of diffusing species is very low in the solvent 共titanium chip兲. Therefore, there is a concentration gradient in the chip. Diffusion of atoms due to concentration gradient is governed by Fick’s first and second laws: Law I

Law II

J⫽⫺D

⳵c ⳵x

(1a)

冉 冊

⳵c ⳵ ⳵c D ⫽ ⳵t ⳵x ⳵x

(1b)

Equation 共1a兲 relates the flux to special change in concentration gradient while Eq. 共1b兲 relates concentration rate to the change in flux. In the case of three dimensions and if a vector notation is used, the general statement of Eq. 共1b兲 becomes:

⳵c ⫽⫺ⵜJ ⳵t

(1c)

This is also called a continuity equation and originates from the conservation of matter. In the above equations ‘‘c’’ is the concentration of the diffusion constituent, and D is the diffusion coefficient. These diffusion equations were originally derived for systems in which there is no relative sliding motion between the two surfaces across which atoms are diffusing. In titanium machining, a thin titanium layer adheres to the face of the cutting tool. Consequently, the ‘‘no relative sliding motion’’ condition is approximately satisfied 关10兴. Therefore, the diffusion problem can be treated as follows: 1. Two-dimensional diffusion occurs at tool rake face; 2. transport of diffusion species due to bulk motion of the chip perpendicular to the rake face. 2.1.1 The Model of Transport-Diffusion in Orthogonal Cutting The geometric configuration used to calculate transport diffusion is shown in Fig. 2. The diffusing constituent of the tool at the rake face is dissolved in the flowing chip at a wear rate W. A chip element ‘‘dx’’ by ‘‘dy’’ with a unit thickness in Z direction which is moving with the chip at cutting velocity, V, is analyzed. The respective coordinates and concentrations at the four sides of this element are: Plane 1-2

Fig. 1 Diffusion mechanism in cutting titanium alloys with tungsten carbide tool

Journal of Engineering Materials and Technology

Plane 3-4

共x兲

共 x⫹⌬x 兲

‘‘c’’ ‘‘c⫹

⳵c dy’’ ⳵x

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ⵜJ⫽ 共 J 1x ⫺J 2x 兲 ⫹ 共 J 1y ⫺J 2y 兲 ⫽ ⫹

冉 冊

冉 冊

⳵ ⳵c D dxdy ⳵x ⳵x

⳵ ⳵c ⳵c dxdy D dxdy⫺V ⳵y ⳵y ⳵x

The concentration variation rate 共atoms per unit volume per unit time兲 in the small element 1234 is then obtained through dividing both sides of the above equation by the elemental volume 1•dxdy and using Eq. 1共c兲:

⳵c ⳵ 2c ⳵ 2c ⳵c ⫽D 2 ⫹D 2 ⫺V ⳵t ⳵ x ⳵x ⳵y

Fig. 2 Diffusion and transport element in metal cutting process

Plane 2-3 Plane 4-1

共y兲

‘‘c’’

共 y⫹⌬y 兲

‘‘c⫹

⳵c dx’’ ⳵y

where c is the concentration of the diffusing constituent at point (x,y). Due to the variation of diffusion and transport rate, there is a buildup 共or loss兲 of the diffusion constituent inside the element. Since there is no bulk motion of mass across planes 4-1 and 2-3, the flux of atoms across 2-3 is only by diffusion and is given as J 1y ⫽J 1yD⫽⫺D

⳵c dx ⳵y

(2)

And that across 4-1 as J 2y ⫽J 2yD ⫽J 1y ⫹

冉

冊

⳵ J 1y ⳵c ⳵ ⳵c D dy⫽⫺D dx⫺ dx dy ⳵y ⳵y ⳵y ⳵y (3)

Since there is transport across planes 1-2 and 3-4 due to the motion of the chip, flux across these planes include both diffusion and transport. For plane 1-2, we have J 1xD ⫽⫺D

⳵c dy ⳵x

J 1xT ⫽Vcdy

due to diffusion

In this the chip velocity is assumed constant for the elemental distance dx. 2.1.2 Modeling of Diffusion Wear Rate. A close form solution of Eqs. 共10兲 will produce a complete representation of the state of the diffusing constituent in the chip and workpiece 共with appropriate modification of the velocity term and axes rotation兲. An analytical solution would be fairly difficult to obtain due to the complex thermo-mechanical boundary conditions that pertain to metal cutting. Therefore, as a first step, a number of simplifying assumptions are employed to obtain a closed form solution, as follows: 1. The concentration of the diffusing constituent is constant at the chip–tool interface and can be denoted as C 0 . This assumption implies fresh tool material is offerred to the flowing chip as the crater develops; 2. the concentration gradient ⳵ c/ ⳵ y within the chip is independent of time. This assumption implies that the chip velocity is much higher than the velocity of the diffusing species; 3. diffusion coefficient D does not depend on position and time and is only a function of temperature throughout the chip; 4. the diffusion in the x-direction is negligible compared with that in the y-direction. This is supported by the fact that the temperature gradient is much larger in the y-direction compared to x-direction. Since there is almost no sliding between the chip being formed and the tool rake face near the cutting edge in cutting titanium alloys 关10兴, the process of cutting titanium alloys can be considered as a settled or stationary process. Therefore,

(4)

due to transport

(10)

⳵c ⫽0. ⳵t

(5)

(11a)

Based on the above assumptions, Eq. 共10兲 reduces to

Therefore, J 1x ⫽J 1xD ⫹J 1xT ⫽⫺D

⳵c dy⫹Vcdy ⳵x

And across 3-4, J 2xD ⫽J 1xD ⫹

冉

冉 冊

⳵ J 1xT ⳵c dx⫽Vcdy⫹V dxdy ⳵x ⳵x

Therefore, J 2x ⫽J 2xD ⫹J 2xT ⫽⫺D

冉 冊

冊

⳵ J 1xD ⳵c ⳵ ⳵c dx⫽⫺D dy⫺ dy dx (7) D ⳵x ⳵x ⳵x ⳵x

J 2xT ⫽J 1xT ⫹

⫹V

(6)

冉

冊

(9)

The net flux into the element is obtained from Eqs. 共2兲 to 共9兲 with neglecting the third-order terms as: 138 Õ Vol. 127, JANUARY 2005

or

⳵c ⳵ 2c ⫽a 2 ⳵x ⳵y

(11b)

where a⫽D/V. Equation 共11b兲 can be solved for the following boundary conditions: c 共 0,y 兲 ⫽0;

(8)

⳵c ⳵ ⳵c dy⫺ dy dx⫹Vcdy D ⳵x ⳵x ⳵x

⳵c dxdy ⳵x

⳵ c D ⳵ 2c ⫽ ⳵x V ⳵y2

c 共 x,⬁ 兲 ⫽0;

c 共 x,0兲 ⫽C 0 ,

The boundary condition c(0,y)⫽0 can be explained as: In the workpiece material close to the tip of cutting tool along the tool– chip interface, the concentration of diffusing constituent is zero since the tool tip keeps penetrating into a fresh material where the concentration of diffusing constituent is zero. The general solution of it for the two-dimensional case with a stationary interface is obtained as 关11兴:

冋 冉 冊册

c 共 x,y 兲 ⫽C 0 1⫺erf

y

2 冑ax

(11c)

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This result shows that the concentration of the diffusing constituent is a function of x and y in the combination of y/2冑ax. The function ‘‘erf’’ in 共11c兲 is known as the Gauss error integral or Gauss error function and is expressed as: erf共 ␩ 兲 ⫽erf

冉 冊 冕 y

⫽

2 冑ax

2

冑␲

␩ ⫽y/2冑ax

␩ ⫽0

exp共 ⫺ ␩ 2 兲 d ␩ (11d)

where a⫽D/V. The coefficient of diffusion D is assumed only a function of temperature but may change with time, as temperature gradually varies with time due to increasing wear. Furthermore, the diffusion rate of the constituent from the cutting tool into the chip is mainly determined by its concentration at the interface. To obtain a closed form expression for the wear rate, the problem is simplified by considering only diffusion into the chip 共y-direction兲 and transport within the chip 共x-direction兲. For the simplification of our derivation, it is further assumed that the concentration gradient ⳵ c/ ⳵ y 兩 y⫽0 at any point over the interface does not change with time. It varies from infinity at x ⫽0 关3兴 to a finite value at x⫽L 共the contact length兲. Thus, ⳵ c/ ⳵ y 兩 y⫽0 is a function of x, i.e.,

⳵c ⳵y

冏

⫽ f 共x兲

(12a)

y⫽0

From Eq. 共11c兲, we get:

⳵c ⳵y

冏

⫽⫺ y⫽0

2C 0

⫽⫺2C 0

冑␲ ax

冑

V ␲ Dx

(12b)

Then, from Eq. 1共a兲, the flux rate over the tool–chip interface along the y-direction at any point in the tool–chip interface is given as: J y⫽0 ⫽⫺D

⳵c ⳵y

冏

⫽2C 0 y⫽0

冑

VD ␲x

(13)

The wear rate of the tool material at position x on the tool–chip interface by diffusion is then given as: W⫽

J y⫽0 ␳

(14a)

where ␳ is the density of the diffusing constituent. The diffusion coefficient D varies very strongly with the temperature ␪ 关12兴: D⫽D 0 e ⫺Q/RT ⫽D 0 e ⫺Q/R 共 273⫹ ␪ 兲 Therefore, W⫽

冉 冊

2C 0 VD 0 ␳ ␲x

1/2

e ⫺Q/2R 共 273⫹ ␪ 兲

(14b)

(15)

where Q is the activation energy of the diffusion process, containing both entropy and enthalpy terms; R is the gas constant; and D 0 is the frequency factor, related to the frequency of atomic oscillations and the availability of sites for a diffusing atom to move into, and ␪ is the temperature in centigrade degree. In the present study, it is assumed that the process is controlled by diffusion of the cobalt binder from the tool into the work chip, and that the friable and brittle WC crystals fracture away as soon as exposed. That is, a weak bond between WC grain and Co matrix is assumed. The corresponding data used in Eq. 共15兲 are as follows: ␳ ⫽14.9⫻10⫺3 g/mm3 , C 0 ⫽0.0253 mole/mm3 , D 0 ⫽1.9 mm2 /s, Q⫽114.1 kJ/mole, R⫽8.315⫻10⫺3 kJ/mole/°K. We can see from Eq. 共15兲 that wear is infinite at x⫽0. According to Cook 关3兴, if very rapid wear occurs at the tool edge (x ⫽0), the contact stress and temperature will decrease there, reducing diffusion coefficient D. The result will be a quasi-stable temperature-stress-wear distribution that will permit the relatively Journal of Engineering Materials and Technology

uniform wear that can be observed in practice production and experiment 关3兴. Therefore, this equation can be used next to the edge area (x⬎0). In the present study, the temperature and the chip velocity, on which the diffusion wear depends, are obtained from a FEM simulation of the cutting process. With this approach, the temperature and chip velocity in Eq. 共15兲 for each point along the tool–chip interface can be predicted with greater accuracy. The tool material is usually a composite mixture and consists of many elements; for example, a cemented tungsten carbide material of type K20 consists of atoms of carbon, tungsten, and cobalt. When diffusion takes place, atoms of different elements present in the tool material are simultaneously transported into the work material. Consequently, the wear rate will depend on the diffusion of the element most favored by material thermodynamics at the interface. Diffusion coefficients for selected constituents of tool material are included in Table 2. On the other hand, if the pure tungsten carbide is used, it can be assumed that the wear process is controlled by diffusion of tungsten into the workpiece chip, which results in lowering the strength of WC. For model validation, orthogonal cutting of Ti-6Al-4V was modeled with an uncoated cemented tungsten carbide cutting tool of straight grade. The weight percentage of Cobalt of 6.0 and 10.0 were investigated. 2.2 FEM Model of Orthogonal Cutting. The machining of titanium alloys is investigated by a nonisothermal numerical model of the orthogonal cutting process. Workpiece is modeled as a rigid visco-plastic material, and the cutting tool as a rigid. With the FEM simulation results, the temperature and velocity in Eq. 共15兲 for each point along the tool–chip interface can be predicted with greater accuracy. The FEM model used to obtain the temperature distribution along tool rake face is shown in Fig. 3共a兲. During the FEM simulation, the Cockcroft–Latham fracture criterion is used to model the chip segmentation due to the ductile nature of chip formation during cutting process. Material model employed in this research takes into account the strain, strain rate, temperature and, phase 共␣-␤兲 evolution in titanium. Details of model development are included in Refs. 关13,14兴. The following is a summary of governing equations 关15兴. A rigid-visco-plastic material model is assumed with Von Mises yield criterion and associated flow rule. In the deformation zone, ␧˙ i j ⫽

冑

3 ␧ថ ␴⬘ 2 ¯␴ i j

(16)

冑

with ¯␴ ⫽ 32 兵 ␴ i⬘j ␴ i⬘j 其 1/2 and ␧ថ ⫽ 23 兵 ␧˙ i j ␧˙ i j 其 1/2. In Eq. 共1兲, the effective stress ¯␴ depends on the strain-ratedependent function, which is to be determined by the properties of the material being analyzed. ¯␴ ⫽ ¯␴ 共 ¯␧ ,␧ថ ,T,microstructure兲

(17)

The rigid-viscoplastic finite element formulation is based on the variational approach. According this approach, the actual velocities 共i.e., the actual solution兲 among all admissible velocity field u i that satisfy the conditions of compatibility and incompressibility, as well as the velocity boundary conditions, gives the following functional a stationary value:

␲⫽

冕

V

E 共 ␧˙ i j 兲 dV ⬘ ⫺

冕

F i u i dS

(18)

Sv

where E denotes the work function which gives: ␴ i⬘j ⫽ ⳵ E/ ⳵ ␧˙ i j . The incompressibility constraint on admissible velocity fields is removed by introducing a penalized form of the incompressibility in the variation of the functional. Therefore, the actual velocity field is determined from the stationary value of the variation as follows: JANUARY 2005, Vol. 127 Õ 139

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density, grain boundaries, phases, and so on. This energy is usually recoverable by annealing. The energy balance is written in the variation form

冕

k 1 T ,i ␦ TdV ⬘ ⫹

V

冕

冕

␳ c ⬘ T˙ ␦ T⫺

V

␬ ␴ i j ␧ i j ␦ TdV ⬘ ⫺

V

⫽0

冕

q n ␦ TdS

Sq

(22)

where q n is the heat flux across the boundary surface S q , and n denotes the unit normal to the boundary surface, and q n ⫽k q T ,n

(23)

To solve problems of this nature, it is required that the temperature field satisfies the prescribed boundary conditions and Eq. 共22兲 for arbitrary perturbation ␦ T. The finite element formulation for temperature analysis can be expressed as 关 C ⬘ 兴 兵 T˙ 其 ⫹ 关 K c 兴 兵 T 其 ⫽ 兵 Q 其

(24)

The theory necessary to integrate equation 共24兲 can be found in various numerical analysis textbooks. The convergence of the time marching scheme depends on the choice of the ␤ parameter 关15兴. T t⫹⌬t ⫽T t ⫹⌬t 关共 1⫺ ␤ 兲 T˙ t ⫹ ␤ T˙ t⫹⌬t 兴

(25)

It is usually considered that ␤ should be larger than 0.5 to ensure an unconditional stability and a value of 0.75 is usually selected. 2.4 Chip Segmentation. The fracture criterion during machining process must be considered in the simulation of the chip formation process. In metal cutting processes, cracks initiate within regions of deformation fields that are highly strained by localized flow produced by the external stress raising effect of sharp tooling. In this study, whether the crack occurs or not inside the primary deformation zone is determined according to the Cockcraft–Latham criterion for ductile fracture C⫽

冕

␧f

¯ ␴ * d␧

(26)

0

Fig. 3 FEM model and Chip formation: „a… FEM mesh; „b… Chip morphology from FEM simulation; „c… Chip morphology from FEM simulation; Chip morphology from experiment

␦␲⫽

冕

V

¯␴ ␦ ␧ថ dV ⬘ ⫹K

冕

␧˙ v ␦ ␧˙ v dV ⬘ ⫺

V

冕

F i ␦ u i dS⫽0 (19)

SF

¯ ,␧ថ ), and ␧˙ v⫽␧˙ ii is the volumetric strain-rate. Inwhere ¯␴ ⫽ ¯␴ (␧ compressibility is imposed by chosing penalty constant K to be very large positive constant. 2.3 Thermal Model. The temperatures generated in high speed machining process can be quite high and have a considerable influence on the mechanical response. This heat generation and transfer is expressed in the form of energy balance as follows: k 1 T ,ii ⫹r˙ ⫺ ␳ c ⬘ T˙ ⫽0

(20)

where k 1 T ,ii presents the heat transfer rate, ␳ c ⬘ T˙ is the internal energy-rate. It is assumed that the heat generation in the deformation zone is due only to plastic deformation and frictional conditions at the tool–work piece interface. Therefore, r˙ ⫽ ␬ ␴ i j ␧˙ i j

(21)

The fraction of mechanical energy transformed into heat ␬ and is usually assumed to be 0.9. The fraction of the rest part of the plastic deformation energy 共1⫺␬兲 causes changes in dislocation 140 Õ Vol. 127, JANUARY 2005

where ␴* is the maximum principal stress and C the critical damage value. The criterion predicts the occurrence of the crack when the critical value C is reached. The C value is a workpiece material constant that does not depend on the working operation or on the tool material 关16兴. In general, the C value is a function of temperature and material state 共m兲 at a point of interest and can be expressed as: C⫽ f 共 T,m 兲

(27)

In titanium alloys, when the temperature is below the ␤ transformation temperature, the material state is presented by ␣⫹␤ microstructure, whereas, at higher temperatures the microstructure is ductile ␤. The critical damage value by the program DEFORM-2D™ is calculated for each element under deformation at each time step. Once the damage value in an element reaches the critical one, a crack is initiated in two steps: 共1兲 This element is deleted along with all parameters related to this element including element connectivity definition, the strain value and damage value, and the stress value; 共2兲 the rough boundary produced by element deletion is smoothed by cutting out the considered rough angle and adding some new points. Consequently, there is some material lose associated with the crack initiation and propagation. The determination of C value is an iterative process. The following procedure was used in this study: Step 1: Initially, the C value, which was obtained by integrating a published flow stress curve 关17兴 from beginning of deformation to fracture, was used as the first guess for preliminary simulation. Transactions of the ASME

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Table 3 Composition, average grain size of WC, and hardness of cemented carbides †9‡

1 2 3 4 5 6 7 8 9

Straight grade

steel cuting grade

Grain size 共␮m兲

HV30

Co 共wt %兲

WC 共wt %兲

1.4 0.8 0.8 0.3 0.3 0.8 0.3 0.3

1590 1750 1643 1796 1698 1900 1509 1405

6 6 10 8 10 3 5.5 9.5

94 93.9 89.5 91.6 89.5 96.9 85.9 68.6

2.6 11.9

3.6 6

2.4 4

0.3

1533

9.5

55.5

19

12.2

3.8

It should be noted that those flow stress data are usually obtained at a small strain rate of 0.01–1 s, while strain rates in cutting are of the order of 104 s. Step 2: FEM simulations of orthogonal cutting were initially performed with the C value calculated in step 1. Step 3: The predicted chip dimension was then compared with the measured chip geometry 共see Fig. 2兲. The C value was then modified until the difference between the predicted and measured chips is minimized. Step 4: The corrected C value from step 3 was then held constant in the subsequent simulations that involve different cutting parameters, such as tool speeds, feed, rake angles, and so on.

3

Application to Titanium Machining

TiC 共wt %兲

TaC 共wt %兲

NbC 共wt %兲

VC 共wt %兲

Cr3C2 共wt %兲

0.1 0.5 0.4 0.5 0.1

Turning experiments were carried out with Ti-6Al-4V rod of 152.4 mm diameter and uncoated 共cemented carbide CNMG 433MR4 883兲 cutting inserts at The Ohio State University. Machine used in the turning experiments was a Cincinnati Milacron Cinturn CT, with a maximum speed rating of 8000 rpm. A constant depth of cut of 1.27 mm was maintained for all experiments. Cimperial 7% coolant was used during the testing process. The cutting conditions were as follows: feed rates 0.127 and 0.35 mm/ rev, rake angle 5°, Relief angle 5°, and cutting speed⫽60, 120, and 240 m/min. Crater depth was measured using a Z-scope with a continuously variable focal length. The change in focal length is calibrated for the object distance from the lens. By focusing it on different surfaces, this device was used to find the difference in height for the two surfaces or and used to find the depth of a crater on any surface.

3.1 Experimental Results From Literature and OSU. Hartung et al. 关6兴 conducted a turning test on Ti-6Al-4V with a conventional C2 grade 共Carbology 820兲 and C3 grade 共Kennametal K68兲 of carbide tool. SNG432 geometry was used with a lead angle of 15°. Cutting speeds from 61 to 610 m/min were employed in their experiment. Some of the results are listed in Table 1. Continuous turning tests were made using a 15 kW Dean Smith and Grace center lathe, with a continuously variable spindle speed by Dearnley et al. 关9兴. Cutting speeds used were from 35 to 250 m/min, and feed rate were from 0.12 to 0.35 mm/rev. Different grades of carbide tool material were employed. Table 3 shows the tool information about uncoated carbide tool. Detail information about tool material and tool geometry and style can be found in Ref. 关9兴.

3.2 Comparisons of Experiment and Prediction. Figure 4 shows the predicted chip velocity and temperature at the tool– chip interface along the tool rake face under the cutting speed of 240 m/min. and feed rate of 0.127 mm/rev. It can be seen that the chip velocity near the cutting edge is relatively low and this area can be regarded as sticking zone. About 0.165 mm away from the cutting edge, there is a sliding zone in which the chip velocity increases almost linearly. Between the sticking zone and sliding zone, there exists a constant velocity zone, which can be considered as the transient zone. The maximum temperature on the rake face appears in the transient zone. Figure 5 shows the predicted crater wear rate profile over the tool rake face at a feed rate of 0.35 mm/rev under different cutting speeds. The maximum wear rate occurs very close to the cutting edge and it corresponds to the highest temperature on the rake face, Fig. 4. The above result is in qualitative agreement with the observation of Dearnley 关9兴 and

Fig. 4 Chip velocity along the tool rake face and the temperature over the tool rake face

Fig. 5 Predicted wear rate over the tool–chip interface „feed rateÄ0.127 mmÕrev…

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Fig. 6 Measured crater depth in cutting Ti-6Al-4V

Komanduri 关10兴. Figure 5 also indicates that the location of maximum wear rate on the rake face moves to the tool cutting edge as the cutting speed increases. Figure 6 presents the measured crater depth under different cutting speeds and feed rates. From this figure it is seen that the crater depth increases greatly with feed rate. For example, when cutting speed is 60 m/min, the wear at a feed rate of 0.35 mm/rev is more than twice that of feed rate 0.127 mm/rev. On the other hand, when cutting at high feed rate 共for example, 0.35 mm/rev兲, the tool wear at cutting speed of 120 m/min is about twice that at cutting speed of 60 m/min. This indicates that the wear is more sensitive to the feed rate. Note that the crater depth increases linearly with the cutting time for all feeds and speeds. This validates Eq. 共15兲 which indicates that the wear rate is independent of time.

4

Discussion of Results

Figures 7 and 8 include comparisons of the predicted crater wear rate from the present analysis with three sets of experimental results: 共1兲 From present turning test, 共2兲 from Ref. 关6兴, and 共3兲 from Ref. 关9兴. It can be seen from these two figures that the values and trends of the predicted result are in good agreement with the experimental data. Lower cobalt mole fraction 共6%兲 cemented carbide inserts were used in the experiments done at the Ohio State University, while higher Co content 共10%兲 cemented carbide inserts were used in the experiments of Hartung 关6兴 and Dearney 关9兴, Table 3. The predicted results are always higher than the experimental data and the maximum difference is between 10% and 15%. The possible reasons for these differences can be the following: • The difference between the orthogonality assumption in FEM model and the nonorthogonal cutting condition in the titanium turning tests 共a diamond shaped insert with 80° nose angle兲. If an adjustment factor is assumed 共␩⫽0.85—eta in Figs. 7 and 8兲 to compensate for the nonorthogonality condition. This is based on the ratio of the actual chip width to predicted chip width.

Fig. 7 Comparison of predicted crater wear rate with the experimental results „Feed rateÄ0.127 mmÕrev…

142 Õ Vol. 127, JANUARY 2005

Fig. 8 Comparison of predicted crater wear rate with the experimental results „Feed rateÄ0.35 mmÕrev…

• The temperature obtained from orthogonal FEM simulation may be higher than that in turning test. Although the flow stress and the cutting conditions were verified by comparing with measured cutting forces and chip morphology 关14兴, the friction factor and the interface heat transfer coefficient was estimated from published results. The latter govern the heat transfer into the cutting tool. • The frequency factor D 0 of diffusion is selected based on experiments reported in Ref. 关12兴 that were conducted in the temperature range 1000°C–1200°C. However, the tool temperature in machining of Ti-6Al-4V is below that range 共about 600–700°C兲 providing for lower D 0 . • The proposed model does not take into account the barrier effect of the grain boundary between WC grains and the cobalt matrix. This grain boundary can retard the rate of cobalt diffusion into the cut chip. • The proposed model calculates the wear rates based on the unworn flank face of the tool insert. In actual experiments the tool edge and rake face rapidly deteriorate which increases the positive rake angle thereby reducing the temperature increase and consequently, the slope of both the wear depth with time plot 共Fig. 6兲. A

Fig. 9 Crater wear at cutting speed of 120 mÕmin. „a… After 30 s, „b… After 60 s.

Fig. 10 Temperature distribution along rake face at various cutting speeds

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software DEFORMគ2DTM, from Dr. Anil Srivastava and James David Layne at TechSolve for machining experiments, and from Peeyush Mittal at the Ohio State University for testing support. This work was funded by a grant from the Department of Energy’s SBIR program to UES Inc. 共Dr. Rabi Bhattacharya and Dr. Satish Dixit兲.

Nomenclature

Fig. 11 Chip velocity along rake face at various cutting speeds

definite change in slope is seen at the 20 s time marker. A photograph of the worn tool edge included in Fig. 9 shows that the entire nose of the cutting tool has worn away. This results in an increase in contact area between the chip and the tool rake face 共along the crater profile兲. • The plots of temperature and velocity distribution along the cutting edge 共rake face兲 as a function of cutting velocities 共Figs. 10 and 11兲 show that as the cutting speeds increase both the temperature and cutting velocities increase while the chip contact length decreases. The above changes result in an increase in wear rate with cutting speeds 共Figs. 7 and 8兲. Since the cutting velocity along the rake face is constant the crater depth profile is governed by the temperature distribution with the peak in temperature coinciding with the maximum crater depth. It should be noted that this peak moves closer to the cutting insert nose as the cutting speed increases. This observation is supported by the photograph in Fig. 9.

5

Conclusions

A thermal diffusion based wear model for predicting crater wear rate in cutting Titanium alloys is developed in this research. The temperature and chip flow speed over the tool rake face are obtained by FEM simulation. Crater wear development during cutting Ti-6Al-4V at cutting speeds up to 240 m/min are investigated both numerically and experimentally. Based on the analysis of present research results, the following conclusions are reached: • The proposed cobalt diffusion model which assumes weak bonding between cobalt and WC grains wear rate model can predict both the crater depth and wear rate with reasonable accuracy across different cobalt concentrations and for different cutting speeds and depth of cut. • Crater wear increases with the cutting speed with the slope of the wear rate remaining constant after and initial change. For example, the crater depth doubles when the feed rate is doubled. • Wear rate, slope of the wear depth curve, seems to be independent of time. A small slope change is observed after 20 s. This cannot be explained completely by the diffusion model. Possibly this may be due to the change in calibration or fixture during measurement of the wear profile. • The temperature and velocity increase while the chip contact length decreases with the cutting velocity. Consequently, the wear rate increases. The maximum depth of crater profile coincides with the location of maximum temperature along the rake face. • For more accurate determination of both the wear rate and wear profile, the evolving state of the cutting edge has to be included in the wear model.

Acknowledgments The authors wish to acknowledge the support received from the Scientific Forming Technologies Corporation for providing FEA Journal of Engineering Materials and Technology

C c c⬘ D0 D J Fi K k1 Q q R r˙ S T T˙ t V V⬘ W y x ui ␤ ␧˙ i j ␧ថ ¯␧ ␴m ␧f ␬ ␪

␩ ␳ ␴* ␴ij ¯␴ ␴ i⬘j ␴1 ⵜ

⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽

Critical damage value Concentration Heat capacity Frequency factor Diffusion coefficient Flux of diffusing constituent Surface traction Penalty constant Thermal conductivity Activation energy of the diffusion Heat flux Gas constant Heat generation rate Boundary surface Absolute temperature Temperature variation Time Chip velocity along rake face Volume Wear rate Coordinate axis into the tool Coordinate axis along the tool rake Velocity field components Parameter varying between 0-1 Strain rate components Equivalent strain rate Effective strain Hydrostatic stress Strain at the fracture Heat generation efficiency Temperature in centigrade degree Adjustment factor density Maximum stress Stress components Equivalent stress Deviatoric stress Principal stress Gradient

References 关1兴 Molinari, A., and Nouari, M., 2002, ‘‘Modeling of Tool Wear by Diffusion in Metal Cutting,’’ Wear, 252, pp. 135–149. 关2兴 Trent, E. M., and Wright, P. K., 1991, Metal Cutting, 3rd ed., ButterworthHeinemann, Washington, DC. 关3兴 Ezugwu, E. O., and Wang, Z. M., 1997, ‘‘Titanium Alloy and Their Machinability—A Review,’’ J. Mater. Process. Technol., 68, pp. 262–274. 关4兴 Hua, J., 2002, Chip Mechanics and Its Influence on Chip Segmentation and Tool Wear, Ph.D. Dissertation, The Ohio State University. 关5兴 Cook, N. H., and Nayak, P. N., 1966, ‘‘The Thermal Mechanics of Tool Wear,’’ ASME J. Eng. Ind., 88共1兲, pp. 93–100. 关6兴 Hartung, P. D., and Kramer, B. M., 1982, ‘‘Tool Wear in Titanium Machining,’’ CIRP Ann., 31共1兲, pp. 75– 80. 关7兴 Takeyama, H., and Murata, R., 1963, ‘‘Basic Investigation of Tool Wear,’’ ASME J. Eng. Ind., 85, pp. 33–38. 关8兴 Usui, T., Hirota, A., and Masuko, M., 1978, ‘‘Analytical Prediction of ThreeDimensional Cutting Process: Part 1-Basic Cutting Model and Energy Approach,’’ ASME J. Eng. Ind., 100, pp. 236 –243. 关9兴 Dearnley, P. A., and Grearson, A. N., 1986, ‘‘Evaluation of Princinpal Wear Mechanisms of Cemented Carbides and Ceramics Used for Machining Titanium Alloy IMI 318,’’ Mater. Sci. Technol., 2, pp. 47–58.

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关10兴 Komanduri, R., 1982, ‘‘Some Clarifications on The Mechanics of Chip Formation When Machining Titanium Alloys,’’ Wear, 76, pp. 15–34. 关11兴 Gebhart, B., 1961, Heat Transfer, McGraw-Hall, New York. 关12兴 Diffusion Data, 1972, Diffusion Information Center, Cleveland, Ohio 44107, Vol. 6 共2–3兲. 关13兴 Hua, J., and Shivpuri, R., 2002, ‘‘Influence of Crack Mechanics on the Chip Segmentation in the Machining of Titanium Alloys,’’ Proceedings of 9th ISPE International Conference on Concurrent Engineering Canfield, United Kingtom, pp. 27–31.

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关14兴 Shivpuri, R., Hua, J., Mittal, P., and Srivastava, A. K., 2002, ‘‘MicrostructureMechanics Interactions in Modeling Chip Segmentation During Titanium Machining,’’ CIRP Ann., 51共1兲, pp. 71–74. 关15兴 Kobayashi, S., Oh, S. K., and Altan, T., 1989, Metal Forming and The FiniteElement Method, Oxford University Press, New York. 关16兴 Oh, S. I., Chen, C. C., and Kobayashi, S., 1989, ‘‘Ductile Fracture in Axisymmetric Extrusion and Drawing,’’ ASME J. Eng. Ind., 101, pp. 36 – 44. 关17兴 Kattus, J. R., 1976, Nonferrous Alloy, Aerospace Structural materials Handbook, BELFOUR STULEN, Inc., Traverse City, Michigan.

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