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Large scale molecular dynamics study of nanometric machining of copper
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Computational Materials Science 41 (2007) 177–185 www.elsevier.com/locate/commatsci
Large scale molecular dynamics study of nanometric machining of copper Q.X. Pei b
a,*
, C. Lu a, H.P. Lee
a,b
a Institute of High Performance Computing, 1 Science Park Road, Singapore 117528, Singapore Department of Mechanical Engineering, National University of Singapore, 9 Engineering Drive 1, Singapore 117576, Singapore
Received 16 October 2006; accepted 3 April 2007 Available online 1 June 2007
Abstract Nanometric machining involves removal of materials at the order of a few nanometers or less. At such a small length scale, molecular dynamics (MD) simulation is an important tool in studying the nanometric machining mechanism and process. In this study, a series of large scale MD simulations with the model size of more than four-million atoms have been performed to study the nanometric machining of copper. The dislocations at finite temperature during the cutting processes are identified and their nucleation and movement are studied. The effects of cutting depth, cutting speed, crystal orientation and cutting direction on the material deformation, lattice defects and cutting forces are investigated. The simulation results show that a smaller cutting depth results in less plastic deformation and fewer dislocations in the workpiece and thus result in a smoother machined surface. It is found that as the cutting depth decreases, the specific cutting force increases rapidly, which shows that the ‘‘size effect’’ exists in nanometric machining. It is observed that a higher cutting speed results in more lattice defects at the cutting region and higher cutting forces. It is revealed that the crystal orientation and cutting direction have a strong effect on material deformation, dislocation movement and cutting forces. 2007 Elsevier B.V. All rights reserved. PACS: 02.70.Ns; 07.05.Tp; 81.05.Bx; 81.20.Wk; 85.40.Hp Keywords: Molecular dynamics; Nanometric machining; Copper
1. Introduction Nanometric machining is a tool-based materials removal technique to fabricate components with nanometric surface finish and sub-micron level form accuracy. Studying the nanometric cutting mechanism and process, such as the material deformation, chip formation, dislocations and cutting forces, is very important in producing well made components. However, as nanometric machining involves changes in only a few atomic layers at the surface region, it is extremely difficult to observe the machining process and to measure the process parameters through experiments. Moreover, it may not be appropriate *
Corresponding author. Tel.: +65 64191225; fax +65 64191280. E-mail address: [email protected] (Q.X. Pei).
0927-0256/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2007.04.008
to use the conventional theory based on ‘‘continuum mechanics’’, such as FEM, to analyze the nanometric machining due to the discrete nature of materials at such a small length scale. Therefore, molecular dynamics (MD) simulation has become an important tool in the study of nanometric machining. MD simulations of nanometric machining have attracted some research interests [1–9] in the past years. The typical works among them are those by Komanduri et al. [6–8], who performed MD simulations of the nanometric machining of single crystal copper and aluminium to study the chip formation, material deformation, cutting forces, atomic scale friction and exit failure during nanometric machining process. The previous works are helpful in understanding nanometric machining. However, as the MD simulation of nanometric machining process is
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compute intensive, small simulation models with a few thousands to tens of thousands of atoms were used in all the reported works to reduce the computing time. Although those small models have provided a lot of information on the nanometric machining, a small model may generate significant boundary effects, which make the results unreliable. For example, if the model is not big enough, the fixed atoms boundary that is commonly used would have strong effect on the dislocation movement and thus affects the motion of atoms at the machining zone. Besides, in most of the reported simulation works, the cutting is two-dimensional or quasi-three-dimensional (plane strain) due to the limitation of the small models. Therefore, there is a need for large scale MD simulation of three-dimensional nanoscale machining processes. Another limitation of the reported works on MD simulations of nanometric machining of metals is that the Morse potential was widely adopted to model the interatomic force between metal atoms. Morse potential is a pair potential which considers only two-body interactions, thus it provides a rather poor description of the metallic bonding. The strength of the individual bond in metals has a strong dependence on the local environment. It decreases as the local environment becomes too crowded due to the Pauli’s ‘‘exclusion principle’’ and increases near surfaces and in small clusters due to the localization of the electron density. Pair potentials do not depend on the environment and, as a result, cannot reproduce some of the characteristic properties of metals, such as the much stronger bonding of atoms near surfaces or in small clusters. Another shortcoming of a pair potential is that it predicts the ratio between the elastic constants of C12 and C44 being 1 for a cubic crystal, which is a so-called Cauthy relation. However, experiments show that the ratio of C12/C44 for FCC metals is generally well above 1 due to many-atoms effect. For example, the ratio of C12/C44 for copper is about 1.5. Therefore, the manybody EAM potential [10–12], which can give a more realistic description of the behavior and properties of metals, has to be used in the MD simulation of nanometric machining process. Our previous work [13] showed that the two different potentials resulted in quite different simulation results and suggested that the EAM potential be used in MD simulation of nanomachining. In this work, we have performed a series of large scale MD simulations of nanometric cutting of copper using EAM potential. The effects of the cutting depth, cutting speed, crystal orientation and cutting direction on materials deformation, dislocation movement and cutting forces during the cutting processes are studied.
Fig. 1. The MD simulation model of nanometric machining.
boundary conditions of the cutting simulations include: (1) three layers of atoms at the bottom of the workpiece materials (lower z plane) are kept fixed; (2) periodic boundary conditions are maintained along the y direction. In nanometric machining, as the cutting depth can be as small as a few nanometers, the edge of the cutting tool is not sharp compared with this very small cutting depth. The edge radius of the cutting tool is usually much bigger than the cutting depth. Therefore, in our large scale MD simulations, we used a round edge cutting tool with the edge radius being 6 nm instead of a sharp cutting tool. The geometry of the cutting tool is shown in Fig. 2. The
2. Simulation methodology Fig. 1 shows the large scale MD simulation model. The workpiece size is 40 nm · 40 nm · 30 nm containing 4,098,686 atoms. The diamond tool contains 8446 carbon atoms. The cutting is along the x direction, which is taken as the [1 0 0] direction of the FCC lattice of copper. The
Fig. 2. The geometry of the cutting tool. The tool edge radius r = 6 nm. The rake angle a = 12 and clearance angle b = 12.
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tool rake angle a = 12 and the tool clearance angle b = 12. The thickness of the tool is 3 nm. The cutting speed used in the MD simulations ranges from 50 to 500 m/s. The initial temperature of the workpiece is 300 K. The three layers of atoms adjacent to the fixed atoms boundary at the workpiece bottom are set as the thermostat atoms, in which the temperature are maintained at 300 K by rescaling the velocities of the atoms. The velocity Verlet algorithm with a time step of 2 fs is used for the time integration of Newton’s equations of motion. In most of the simulations, the depth of cut is 4 nm and the cutting distance is 16 nm. The force acting on an individual atom is obtained by summing the forces contributed by the surrounding atoms. The forces are calculated from the interatomic potentials. The Morse potential is relatively simple and computationally inexpensive compared to the EAM potential. The Morse potential is as follows: /ðrij Þ ¼ Dfexp½2aðrij r0 Þ 2 exp½aðrij r0 Þg
ð1Þ
where /(rij) is a pair potential energy function; D is the cohesion energy; a is the elastic modulus; rij and r0 are the instantaneous and equilibrium distance between atoms i and j, respectively. The EAM method, which has been evolved from the density-function theory, is based upon the recognition that the cohesive energy of a metal is governed not only by the pair-wise potential of the nearest-neighbor atoms, but also by embedding energy related to the ‘‘electron sea’’ in which the atoms are embedded. For EAM potential, the total atomic potential energy of a system is expressed as X 1X Etot ¼ Uij ðrij Þ þ F i ð qi Þ ð2Þ 2 i;j i where Uij is the pair-interaction energy between atoms i and i is the host j and Fi is the embedding energy of atom i. q electron density at site i induced by all other atoms in the system, which is given by X i ¼ qj ðrij Þ ð3Þ q
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each other. In this situation, the interaction force between the tool atoms will have no effect on the tool atoms and thus the potential for the tool atoms is actually a dummy one. Dislocations play a crucial role in the plastic deformation of materials. However, accurately identifying dislocations at room temperature in MD simulations is very difficult due to the thermal vibration of atoms. This might be the reason why almost all the previous MD studies of dislocations were carried out at extremely low temperature of 0 K or 1 K [15–19]. The widely used methods to identify dislocations and other lattice defects in MD simulations are the atomic coordinate number [15], the slip vector [16], and the centro-symmetry parameter [17]. We have compared the different methods and found that the methods of atomic coordinate number and the slip vector become less effective in identifying the lattice defects at finite temperature due to thermal fluctuations of atoms. Therefore we have chosen to use the centro-symmetry parameter, which is less sensitive to the temperature increase. In a centrosymmetric material (such as copper and other FCC metals), each atom has pairs of equal and opposite bonds among its nearest-neighbors. As the material is distorted, these bonds will change direction and/or length, but they will remain equal and opposite under homogeneous elastic deformation. If there is a defect nearby, however, this equal and opposite relation no longer holds. In a perfect bulk FCC lattice, each atom has 12 nearest-neighbor bonds or vectors. The centro-symmetry parameter for each atom is defined as follows: X ! ! 2 CSP ¼ j Ri þ R j ð4Þ i¼1;6
iþ6
where Ri and Ri+6 are the vectors corresponding to the six pairs of opposite nearest-neighbors in the FCC lattice. By definition, the centro-symmetry parameter is zero for an atom in a perfect FCC material under any homogeneous elastic deformation and non-zero for an atom which is near a defect such as a cavity, a dislocation or a free surface.
j6¼i
There are three different atomic interactions in the MD simulations of nanometric cutting processes: (1). The interaction in the workpiece; (2). The interaction between the workpiece and the tool; and (3). The interaction in the tool. For the interaction between the copper atoms in the workpiece, we used the EAM potential for copper constructed by Johnson [14]. For the interaction between the copper workpiece and the diamond tool, as there is no available EAM potential between Cu and C atoms, we still use Morse potential for the workpiece-tool interaction with the parameters adopted from reference [4] as D = ˚ . The Morse potential 0.087 eV, a = 5.14 and r0 = 2.05 A is also used for the interaction between the tool atoms. As the copper workpiece is much softer than the diamond tool, it is a good approximation to take the tool as a rigid body, and so the atoms in the tools are fixed relative to
3. Simulation results 3.1. The nanometric machining process Figs. 3a–d show the cross-sectional views of the x–z plane at four different cutting distances of 4, 8, 12 and 16 nm respectively. The cutting depth is 4 nm with a cutting speed of 100 m/s. The cutting is in the (0 0 1) plane and along the [1 0 0] direction of the workpiece. It can be seen from the figures colored by CSP that the workpiece materials deform during cutting and the material removal takes place via the chip formation as in conventional cutting. As the cutting proceeds, more chips are formed in front of the tool. The CSP values show that the materials in front of and beneath the tool are away from the perfect FCC lattice. Dislocations and other lattice defects are generated
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Fig. 3. Showing the cross-sectional views of the cutting process at the cutting distances of (a) 4 nm, (b) 8 nm, (c) 12 nm and (d) 16 nm.
250
200
Cutting force (nN)
in these regions. It can be clearly observed that the dislocations emit from the cutting region and some of them glide deep into the workpiece. The cutting forces in the MD simulations are obtained by summing the atomic forces of workpiece atoms on the tool atoms. The variations of the cutting forces with the cutting distance during the cutting process for this simulation case are shown in Fig. 4. It can be seen that both the tangential cutting force Fx and the normal cutting force Fz increase at the start of the cutting. When the cutting distance is bigger than the cutting depth of 4 nm, the values of the cutting forces tend to be steady. The simulated cutting force in the thickness direction of the workpiece (y direction) is not shown here as it is very small with its time-averaged value over the whole cutting process being zero. Figs. 5a–d present the side views of the three-dimensional lattice defects at the cutting distances of 4, 8, 12 and 16 nm respectively. Note that the inside atoms with CSP smaller than three are removed in the visualization, as these atoms are assumed to be in perfect FCC configuration. Also note that the isolated atoms distributed inside the model are not lattice defects. Those atoms having CSP above three are due to the thermal vibration of atoms at finite temperature. Although the CSP method is not perfect in identifying the lattice defects at finite temperature, it is more accurate than other methods such as the atomic coor-
150
100 Tangential force Fx Normal force Fz
50
0 0
2
4
6
8
10
12
14
16
Cutting distance (nm)
Fig. 4. Variations of the cutting forces with the cutting distance. The cutting depth is 4 nm.
dinate number and the slip vector. It can be seen from Figs. 5a–d that lattice defects are formed in the workpiece during the cutting process. These lattice defects include Shockley partial dislocations and stacking fault. In FCC crystals, a perfect dislocation dissociates into Shockley partial dislocations since the reaction accompanies an elastic energy reduction [20,21]. The elastic energy reduction, however, is balanced by the energy of the stacking fault created in between the partial dislocations.
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Fig. 5. Side views of the lattice defects during the cutting process at the cutting distance of (a) 4 nm, (b) 8 nm, (c) 12 nm and (d) 16 nm.
At the cutting distance of 4 nm, see Figs. 3a and Fig. 5a, a small dislocation loop was formed. As the cutting proceeds, more dislocation loops are formed and those dislocation loops keep competing to grow until one of the candidates is finally triggered to evolve into an isolated dislocation loop as seen in Fig. 5b. In Fig. 5c, one more isolated dislocation loop is formed and the two isolated dislocation exist at the same time. The isolated dislocation loops glide in the ½1 0 1 direction, which corresponding to h1 1 0i slip direction in FCC lattice. In Fig. 5d, the first isolated dislocation loop has disappeared as it travels to the workpiece bottom and only the second isolated dislocation loop remains. The formation of dislocations results in the release of the accumulated cutting energy, which corresponds to the temporary drop of the cutting force. The oscillatory cutting force curves in Fig. 4 are due to the complex local motions of the dislocations beneath the cutting tool. 3.2. Effect of cutting depth In addition to the cutting depth of 4 nm, MD simulations were also conducted for another two cutting depths of 2 nm and 0.8 nm. Figs. 6a–c show the partial cross-sectional views of the x–z plane for the three different cutting depths of 0.8, 2.0 and 4.0 nm respectively. A comparison of
the different figures shows that with the increase of the cutting depth, more materials are deformed ahead of the cutting tool. Correspondingly, more dislocations are observed in the cutting region with a larger cutting depth. It can also be observed that as the cutting depth increases, the machined surface becomes rougher. This may be due to the higher cutting forces required for the larger cutting depth, which will be shown in the follows. The variations of the tangential cutting force Fx with cutting distance for the different cutting depths are shown in Fig. 7. It can be seen that the smaller the cutting depth, the earlier the cutting force reaches the steady state. Besides, a smaller cutting depth results in a lower cutting force. In order to give a more quantitative comparison of the cutting force, the time-averaged tangential and normal cutting forces for the cutting distance of 6–16 nm are calculated for the different cutting depths as shown in Fig. 8. Although both the tangential and normal cutting forces decrease as the cutting depth decreases, the normal cutting force decreases less rapidly than the tangential cutting force. As a result, the ratio of normal force to tangential force changes from smaller than 1.0 for the cutting depth 4.0 nm to greater than 1.0 for the cutting depths of 2.0 and 0.8 nm. This shows that for nanometric cutting with small cutting depth, as the tool edge radius is quite large compared to the cutting depth, the nanometric cutting is
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Fig. 6. The partial cross-sectional views of the cutting process for different cutting depths of (a) 0.8 nm, (b) 2.0 nm and (c) 4.0 nm. 280
300 Cut depth 0.8 nm
Resultant force Fr
Cut depth 2 nm
250
Average cutting force (nN)
Cutting force (nN)
240
Cut depth 4 nm
200 160 120 80 40 0 0
2
4
6
8
10
12
14
16
Cutting distance (nm)
Specific force Fs
200 150 100 50 0 0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Cutting depth (nm)
Fig. 7. Variations of the tangential cutting force Fx with cutting distance for the different cutting depths of 0.8, 2.0 and 4.0 nm.
Fig. 9. The resultant cutting force and the specific cutting force for the different cutting depths of 0.8, 2.0 and 4.0 nm.
250
Average cutting force (nN)
Tangential force Fx
200
198.7 176.7
Normal force Fz
150
133.3 108.2 86.9
100
Fx Fz
59.3
Fx Fz
50
effect’’. This ‘‘size effect’’ on the specific cutting force in nanometer machining can be explained by the metallic bonding. The special feature of metallic bonding is that the strength of the individual bond has a strong dependence on the local environment. The bonding becomes stronger at the surface due to the localization of the electron density. The smaller the cutting depth, the larger the ratio of surface to cutting depth, and thus the bigger the specific cutting force.
Fx Fz 0
3.3. Effect of cutting speed 0.8 nm
2.0 nm
4.0 nm
Cutting depth
Fig. 8. The time-averaged cutting forces for the different cutting depths of 0.8, 2.0 and 4.0 nm.
more similar to the conventional grinding with a large negative tool rake angle. Fig. 9 shows the changes of the resultant cutting force and the specific cutting force with cutting depth. Here the resultant cutting force Fr is the vector sum of the tangential force Fx and normal force Fz. Note that the average cutting force along the thickness direction Fy is zero. The specific cutting force Fs is the resultant cutting force divided by the cutting depth. It can be seen that with the decrease of cutting depth the resultant cutting force decreases. However, the specific cutting force increases rapidly with the decrease of cutting depth, which shows a very obvious ‘‘size
To study the effects of the cutting speed on the nanometric machining process, simulations with different cutting speeds have been conducted. Figs. 10a–d show the partial cross-sectional views of the x–z plane for the nanocutting process at the cutting speeds of 50, 100, 200 and 500 m/s respectively. It can be seen that the higher the cutting speed, the more the lattice defects in the cutting region. However, the cutting speed seems to have no obvious effect on the machined surface quality for the four different cutting speeds studied. The effect of the cutting speed on the cutting forces is shown in Fig 11. It can be seen that both the tangential and the normal cutting forces increase with the increase of the cutting speed. This is because a higher cutting speed results in shorter time for the dislocations to move away from the cutting region and thus the dislocations pile up
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Fig. 10. The partial cross-sectional views of the cutting process for the cutting speeds of (a) 50 m/s, (b) 100 m/s, (c) 200 m/s and (d) 500 m/s.
Average cutting force (nN)
300 250 200 150 100
Tangential force Fx Normal force Fz
50 0 0
50
100
150
200
250
300
350
400
450
500
550
Cutting speed (m/s)
Fig. 11. Variations of cutting forces with cutting speed.
in the cutting region and strengthen the materials there, which results in higher deformation force. However, the increase of the tangential cutting force is more obvious than the normal cutting force. The more rapid increase of the tangential cutting force with the cutting speed may be due to the fact that for a higher cutting speed an additional cutting force is needed to accelerate the chips in front of the tool to the higher cutting speed. 3.4. Effect of crystal orientation and cutting direction In the above simulations, the cutting is in the (0 0 1) plane and along the [1 0 0] direction of the workpiece. We also performed MD simulations for two more crystal ori-
entations of (0 0 1)½1 1 0 and (1 1 1)½2 1 1. The cutting speed is 100 m/s with the cutting depth of 4 nm. Figs. 12a–c show the simulation results for the three different lattice setups. The upper figures are the cross-sectional views along x–z plane, while the lower figures are the top views. The figures are colored by the CSP. It can be seen that nanometric machining with different crystal orientations result in quite different deformation behavior of the workpiece materials. In the case of (0 0 1)[1 0 0] the dislocations seem to propagate at 45 to the cutting direction as shown in Fig. 12a. In the case of (0 0 1)½1 1 0 the dislocations are in parallel to the cutting direction as shown in Fig. 12b. While in the case of (1 1 1)½2 1 1 shown in Fig. 12c, the dislocations propagate at 30 to the cutting direction at the top surface and at 60 to the cutting direction inside the workpiece. From Figs. 12a–c, it can also be seen that the propagation directions of the dislocations belong to the h1 1 0i direction, which is the slip direction of FCC crystals. Besides, the material deformation at the cutting region is quite different for the three crystal orientations. The crystal orientation of (0 0 1)½1 1 0 results in the largest chips ahead of the tool, while the (1 1 1)½2 1 1 results in the smallest ones. The simulated cutting forces for the three lattice setups are shown in Fig. 13. It can be seen that the lowest resultant force is with the case of (0 0 1)½1 1 0, while the highest one is with the case of (1 1 1)½2 1 1. For the case of (0 0 1)[1 0 0] the resultant force is in between. As the dislocations in the workpiece propagate at the angles of 0, 45 and 60 to the cutting direction for the cases of (0 0 1)½1 1 0, (0 0 1)[1 0 0] and (1 1 1)½2 1 1, respectively, the simulation results show that the larger the dislocation
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Fig. 12. Showing cross-sectional views (upper figures) of the x–z plane and top views (lower figures) for three different crystal orientations of (a) (0 0 1)[1 0 0], (b) (0 0 1)[1 1 0] and (c) (1 1 1)[2 1 1].
400 Tangential force Fx
Average cutting force (nN)
350
Normal force Fz Resultant force Fr
300
265.9
275.2
241.7
250 200 150 100
Fx Fz Fr
Fx Fz Fr
Fx Fz Fr
(001)[110]
(111)[211]
50 0 (001)[100]
Crystal orientation
Fig. 13. The cutting forces for the different crystal orientations of (0 0 1)[1 0 0], (0 0 1)[1 1 0] and (1 1 1)[2 1 1].
propagation angle, the higher the resultant cutting force. This can be explained by the projection of the cutting forces on the dislocation slip directions. The larger the propagation angle, the smaller the cutting force projected to the dislocation slip direction, and thus large cutting force is required to move the dislocations to generate plastic deformation during cutting. 4. Conclusions Large scale MD simulations using the EAM potential have been carried out to study the nanometric machining
of copper. The material deformation, lattice defects, dislocation movement, and cutting forces during the cutting process are studied. The effects of cutting depth, cutting speed, crystal orientation and cutting direction on the nanoscale cutting process are investigated. From the simulation results, the following conclusions are obtained: (1) The nanometric machining process is accompanied with complex dislocation formation and movement, which have strong effect on the material deformation and cutting forces. (2) With the decrease of the cutting depth, less plastic deformation and fewer dislocations are generated in the workpiece, and correspondingly a smoother machined surface is obtained. As the cutting depth decreases, the cutting forces decreases, however the specific cutting force increases rapidly, which shows that the ‘‘size effect’’ exists in nanometric machining. (3) A higher cutting speed results in more lattice defects at the cutting region and higher cutting forces. However, no obvious difference in the machined surface quality was observed for the different cutting speeds studied. (4). The crystal orientation and cutting direction have a strong effect on the material deformation, dislocation movement and cutting forces. For the three crystal setups of (0 0 1)½1 1 0, (0 0 1)[1 0 0] and (1 1 1)½ 2 1 1, the dislocations in the workpiece have been found to propagate at 0, 45 and 60 to the cutting direc-
Q.X. Pei et al. / Computational Materials Science 41 (2007) 177–185
tion respectively. Due to the difference in the dislocation propagation angles, the lowest cutting forces are with the case of (0 0 1)½1 1 0, while the highest cutting forces are with the case of (1 1 1)½ 2 1 1. Acknowledgement This work was supported by the Agency for Science, Technology and Research (A*STAR), Singapore. References [1] S. Shimada, N. Ikawa, H. Tanaka, J. Uchikoshi, Ann. CIRP 43 (1994) 51–54. [2] T. Inamura, N. Takezawa, Y. Kumaki, T. Sata, Ann. CIRP 43 (1994) 47–50. [3] K. Maekawa, A. Itoh, Wear 188 (1995) 115–122. [4] L. Zhang, H. Tanaka, Wear 211 (1997) 44–53. [5] T.H. Fang, C.I. Weng, Nanotechnology 11 (2000) 148–153. [6] R. Komanduri, N. Chandrasekaran, L.M. Raff, Wear 242 (2000) 60–88.
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[7] R. Komanduri, N. Chandrasekaran, L.M. Raff, Phys. Rev. B 61 (2000) 14007–14019. [8] R. Komanduri, N. Chandrasekaran, L.M. Raff, Mater. Sci. Eng. A 311 (2001) 1–12. [9] Y.Y. Ye, R. Biswas, J.R. Morris, A. Bastawros, A. Chandra, Nanotechnology 14 (2003) 390–396. [10] M.S. Daw, M.I. Baskes, Phys. Rev. B 29 (1984) 6443–6453. [11] M.W. Finnis, J.E. Sinclair, Philos. Mag. A 50 (1984) 45–55. [12] M.S. Daw, S.M. Foiles, M.I. Baskes, Mater. Sci. Rep. 9 (1993) 251– 310. [13] Q.X. Pei, C. Lu, F.Z. Fang, H. Wu, Comput. Mater. Sci. 37 (2006) 434–441. [14] R.A. Johnson, Phys. Rev. B 37 (1988) 3924–3931. [15] J. Li, K.J. Van Vliet, T. Zhu, S. Yip, S. Suresh, Nature 418 (2002) 307–310. [16] J.A. Zimmerman, C.L. Kelchner, P.A. Klein, J.C. Hamilton, S.M. Foiles, Phys. Rev. Lett. 87 (2001) 165507. [17] C.L. Kelchner, S.J. Plimpton, J.C. Hamilton, Phys. Rev. B 58 (1998) 11085–11088. [18] O. Kum, Mol. Simulat. 31 (2005) 115–121. [19] D. Saraev, R.E. Miller, Acta Mater. 54 (2006) 33–45. [20] G. Lasko, D. Saraev, S. Schmauder, P. Kizler, Comput. Mater. Sci. 32 (2005) 418–425. [21] J.P. Hirth, J. Lothe, Theory of Dislocations, Willey, New York, 1982.
Computational Materials Science 41 (2007) 177–185 www.elsevier.com/locate/commatsci
Large scale molecular dynamics study of nanometric machining of copper Q.X. Pei b
a,*
, C. Lu a, H.P. Lee
a,b
a Institute of High Performance Computing, 1 Science Park Road, Singapore 117528, Singapore Department of Mechanical Engineering, National University of Singapore, 9 Engineering Drive 1, Singapore 117576, Singapore
Received 16 October 2006; accepted 3 April 2007 Available online 1 June 2007
Abstract Nanometric machining involves removal of materials at the order of a few nanometers or less. At such a small length scale, molecular dynamics (MD) simulation is an important tool in studying the nanometric machining mechanism and process. In this study, a series of large scale MD simulations with the model size of more than four-million atoms have been performed to study the nanometric machining of copper. The dislocations at finite temperature during the cutting processes are identified and their nucleation and movement are studied. The effects of cutting depth, cutting speed, crystal orientation and cutting direction on the material deformation, lattice defects and cutting forces are investigated. The simulation results show that a smaller cutting depth results in less plastic deformation and fewer dislocations in the workpiece and thus result in a smoother machined surface. It is found that as the cutting depth decreases, the specific cutting force increases rapidly, which shows that the ‘‘size effect’’ exists in nanometric machining. It is observed that a higher cutting speed results in more lattice defects at the cutting region and higher cutting forces. It is revealed that the crystal orientation and cutting direction have a strong effect on material deformation, dislocation movement and cutting forces. 2007 Elsevier B.V. All rights reserved. PACS: 02.70.Ns; 07.05.Tp; 81.05.Bx; 81.20.Wk; 85.40.Hp Keywords: Molecular dynamics; Nanometric machining; Copper
1. Introduction Nanometric machining is a tool-based materials removal technique to fabricate components with nanometric surface finish and sub-micron level form accuracy. Studying the nanometric cutting mechanism and process, such as the material deformation, chip formation, dislocations and cutting forces, is very important in producing well made components. However, as nanometric machining involves changes in only a few atomic layers at the surface region, it is extremely difficult to observe the machining process and to measure the process parameters through experiments. Moreover, it may not be appropriate *
Corresponding author. Tel.: +65 64191225; fax +65 64191280. E-mail address: [email protected] (Q.X. Pei).
0927-0256/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2007.04.008
to use the conventional theory based on ‘‘continuum mechanics’’, such as FEM, to analyze the nanometric machining due to the discrete nature of materials at such a small length scale. Therefore, molecular dynamics (MD) simulation has become an important tool in the study of nanometric machining. MD simulations of nanometric machining have attracted some research interests [1–9] in the past years. The typical works among them are those by Komanduri et al. [6–8], who performed MD simulations of the nanometric machining of single crystal copper and aluminium to study the chip formation, material deformation, cutting forces, atomic scale friction and exit failure during nanometric machining process. The previous works are helpful in understanding nanometric machining. However, as the MD simulation of nanometric machining process is
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compute intensive, small simulation models with a few thousands to tens of thousands of atoms were used in all the reported works to reduce the computing time. Although those small models have provided a lot of information on the nanometric machining, a small model may generate significant boundary effects, which make the results unreliable. For example, if the model is not big enough, the fixed atoms boundary that is commonly used would have strong effect on the dislocation movement and thus affects the motion of atoms at the machining zone. Besides, in most of the reported simulation works, the cutting is two-dimensional or quasi-three-dimensional (plane strain) due to the limitation of the small models. Therefore, there is a need for large scale MD simulation of three-dimensional nanoscale machining processes. Another limitation of the reported works on MD simulations of nanometric machining of metals is that the Morse potential was widely adopted to model the interatomic force between metal atoms. Morse potential is a pair potential which considers only two-body interactions, thus it provides a rather poor description of the metallic bonding. The strength of the individual bond in metals has a strong dependence on the local environment. It decreases as the local environment becomes too crowded due to the Pauli’s ‘‘exclusion principle’’ and increases near surfaces and in small clusters due to the localization of the electron density. Pair potentials do not depend on the environment and, as a result, cannot reproduce some of the characteristic properties of metals, such as the much stronger bonding of atoms near surfaces or in small clusters. Another shortcoming of a pair potential is that it predicts the ratio between the elastic constants of C12 and C44 being 1 for a cubic crystal, which is a so-called Cauthy relation. However, experiments show that the ratio of C12/C44 for FCC metals is generally well above 1 due to many-atoms effect. For example, the ratio of C12/C44 for copper is about 1.5. Therefore, the manybody EAM potential [10–12], which can give a more realistic description of the behavior and properties of metals, has to be used in the MD simulation of nanometric machining process. Our previous work [13] showed that the two different potentials resulted in quite different simulation results and suggested that the EAM potential be used in MD simulation of nanomachining. In this work, we have performed a series of large scale MD simulations of nanometric cutting of copper using EAM potential. The effects of the cutting depth, cutting speed, crystal orientation and cutting direction on materials deformation, dislocation movement and cutting forces during the cutting processes are studied.
Fig. 1. The MD simulation model of nanometric machining.
boundary conditions of the cutting simulations include: (1) three layers of atoms at the bottom of the workpiece materials (lower z plane) are kept fixed; (2) periodic boundary conditions are maintained along the y direction. In nanometric machining, as the cutting depth can be as small as a few nanometers, the edge of the cutting tool is not sharp compared with this very small cutting depth. The edge radius of the cutting tool is usually much bigger than the cutting depth. Therefore, in our large scale MD simulations, we used a round edge cutting tool with the edge radius being 6 nm instead of a sharp cutting tool. The geometry of the cutting tool is shown in Fig. 2. The
2. Simulation methodology Fig. 1 shows the large scale MD simulation model. The workpiece size is 40 nm · 40 nm · 30 nm containing 4,098,686 atoms. The diamond tool contains 8446 carbon atoms. The cutting is along the x direction, which is taken as the [1 0 0] direction of the FCC lattice of copper. The
Fig. 2. The geometry of the cutting tool. The tool edge radius r = 6 nm. The rake angle a = 12 and clearance angle b = 12.
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tool rake angle a = 12 and the tool clearance angle b = 12. The thickness of the tool is 3 nm. The cutting speed used in the MD simulations ranges from 50 to 500 m/s. The initial temperature of the workpiece is 300 K. The three layers of atoms adjacent to the fixed atoms boundary at the workpiece bottom are set as the thermostat atoms, in which the temperature are maintained at 300 K by rescaling the velocities of the atoms. The velocity Verlet algorithm with a time step of 2 fs is used for the time integration of Newton’s equations of motion. In most of the simulations, the depth of cut is 4 nm and the cutting distance is 16 nm. The force acting on an individual atom is obtained by summing the forces contributed by the surrounding atoms. The forces are calculated from the interatomic potentials. The Morse potential is relatively simple and computationally inexpensive compared to the EAM potential. The Morse potential is as follows: /ðrij Þ ¼ Dfexp½2aðrij r0 Þ 2 exp½aðrij r0 Þg
ð1Þ
where /(rij) is a pair potential energy function; D is the cohesion energy; a is the elastic modulus; rij and r0 are the instantaneous and equilibrium distance between atoms i and j, respectively. The EAM method, which has been evolved from the density-function theory, is based upon the recognition that the cohesive energy of a metal is governed not only by the pair-wise potential of the nearest-neighbor atoms, but also by embedding energy related to the ‘‘electron sea’’ in which the atoms are embedded. For EAM potential, the total atomic potential energy of a system is expressed as X 1X Etot ¼ Uij ðrij Þ þ F i ð qi Þ ð2Þ 2 i;j i where Uij is the pair-interaction energy between atoms i and i is the host j and Fi is the embedding energy of atom i. q electron density at site i induced by all other atoms in the system, which is given by X i ¼ qj ðrij Þ ð3Þ q
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each other. In this situation, the interaction force between the tool atoms will have no effect on the tool atoms and thus the potential for the tool atoms is actually a dummy one. Dislocations play a crucial role in the plastic deformation of materials. However, accurately identifying dislocations at room temperature in MD simulations is very difficult due to the thermal vibration of atoms. This might be the reason why almost all the previous MD studies of dislocations were carried out at extremely low temperature of 0 K or 1 K [15–19]. The widely used methods to identify dislocations and other lattice defects in MD simulations are the atomic coordinate number [15], the slip vector [16], and the centro-symmetry parameter [17]. We have compared the different methods and found that the methods of atomic coordinate number and the slip vector become less effective in identifying the lattice defects at finite temperature due to thermal fluctuations of atoms. Therefore we have chosen to use the centro-symmetry parameter, which is less sensitive to the temperature increase. In a centrosymmetric material (such as copper and other FCC metals), each atom has pairs of equal and opposite bonds among its nearest-neighbors. As the material is distorted, these bonds will change direction and/or length, but they will remain equal and opposite under homogeneous elastic deformation. If there is a defect nearby, however, this equal and opposite relation no longer holds. In a perfect bulk FCC lattice, each atom has 12 nearest-neighbor bonds or vectors. The centro-symmetry parameter for each atom is defined as follows: X ! ! 2 CSP ¼ j Ri þ R j ð4Þ i¼1;6
iþ6
where Ri and Ri+6 are the vectors corresponding to the six pairs of opposite nearest-neighbors in the FCC lattice. By definition, the centro-symmetry parameter is zero for an atom in a perfect FCC material under any homogeneous elastic deformation and non-zero for an atom which is near a defect such as a cavity, a dislocation or a free surface.
j6¼i
There are three different atomic interactions in the MD simulations of nanometric cutting processes: (1). The interaction in the workpiece; (2). The interaction between the workpiece and the tool; and (3). The interaction in the tool. For the interaction between the copper atoms in the workpiece, we used the EAM potential for copper constructed by Johnson [14]. For the interaction between the copper workpiece and the diamond tool, as there is no available EAM potential between Cu and C atoms, we still use Morse potential for the workpiece-tool interaction with the parameters adopted from reference [4] as D = ˚ . The Morse potential 0.087 eV, a = 5.14 and r0 = 2.05 A is also used for the interaction between the tool atoms. As the copper workpiece is much softer than the diamond tool, it is a good approximation to take the tool as a rigid body, and so the atoms in the tools are fixed relative to
3. Simulation results 3.1. The nanometric machining process Figs. 3a–d show the cross-sectional views of the x–z plane at four different cutting distances of 4, 8, 12 and 16 nm respectively. The cutting depth is 4 nm with a cutting speed of 100 m/s. The cutting is in the (0 0 1) plane and along the [1 0 0] direction of the workpiece. It can be seen from the figures colored by CSP that the workpiece materials deform during cutting and the material removal takes place via the chip formation as in conventional cutting. As the cutting proceeds, more chips are formed in front of the tool. The CSP values show that the materials in front of and beneath the tool are away from the perfect FCC lattice. Dislocations and other lattice defects are generated
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Fig. 3. Showing the cross-sectional views of the cutting process at the cutting distances of (a) 4 nm, (b) 8 nm, (c) 12 nm and (d) 16 nm.
250
200
Cutting force (nN)
in these regions. It can be clearly observed that the dislocations emit from the cutting region and some of them glide deep into the workpiece. The cutting forces in the MD simulations are obtained by summing the atomic forces of workpiece atoms on the tool atoms. The variations of the cutting forces with the cutting distance during the cutting process for this simulation case are shown in Fig. 4. It can be seen that both the tangential cutting force Fx and the normal cutting force Fz increase at the start of the cutting. When the cutting distance is bigger than the cutting depth of 4 nm, the values of the cutting forces tend to be steady. The simulated cutting force in the thickness direction of the workpiece (y direction) is not shown here as it is very small with its time-averaged value over the whole cutting process being zero. Figs. 5a–d present the side views of the three-dimensional lattice defects at the cutting distances of 4, 8, 12 and 16 nm respectively. Note that the inside atoms with CSP smaller than three are removed in the visualization, as these atoms are assumed to be in perfect FCC configuration. Also note that the isolated atoms distributed inside the model are not lattice defects. Those atoms having CSP above three are due to the thermal vibration of atoms at finite temperature. Although the CSP method is not perfect in identifying the lattice defects at finite temperature, it is more accurate than other methods such as the atomic coor-
150
100 Tangential force Fx Normal force Fz
50
0 0
2
4
6
8
10
12
14
16
Cutting distance (nm)
Fig. 4. Variations of the cutting forces with the cutting distance. The cutting depth is 4 nm.
dinate number and the slip vector. It can be seen from Figs. 5a–d that lattice defects are formed in the workpiece during the cutting process. These lattice defects include Shockley partial dislocations and stacking fault. In FCC crystals, a perfect dislocation dissociates into Shockley partial dislocations since the reaction accompanies an elastic energy reduction [20,21]. The elastic energy reduction, however, is balanced by the energy of the stacking fault created in between the partial dislocations.
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Fig. 5. Side views of the lattice defects during the cutting process at the cutting distance of (a) 4 nm, (b) 8 nm, (c) 12 nm and (d) 16 nm.
At the cutting distance of 4 nm, see Figs. 3a and Fig. 5a, a small dislocation loop was formed. As the cutting proceeds, more dislocation loops are formed and those dislocation loops keep competing to grow until one of the candidates is finally triggered to evolve into an isolated dislocation loop as seen in Fig. 5b. In Fig. 5c, one more isolated dislocation loop is formed and the two isolated dislocation exist at the same time. The isolated dislocation loops glide in the ½1 0 1 direction, which corresponding to h1 1 0i slip direction in FCC lattice. In Fig. 5d, the first isolated dislocation loop has disappeared as it travels to the workpiece bottom and only the second isolated dislocation loop remains. The formation of dislocations results in the release of the accumulated cutting energy, which corresponds to the temporary drop of the cutting force. The oscillatory cutting force curves in Fig. 4 are due to the complex local motions of the dislocations beneath the cutting tool. 3.2. Effect of cutting depth In addition to the cutting depth of 4 nm, MD simulations were also conducted for another two cutting depths of 2 nm and 0.8 nm. Figs. 6a–c show the partial cross-sectional views of the x–z plane for the three different cutting depths of 0.8, 2.0 and 4.0 nm respectively. A comparison of
the different figures shows that with the increase of the cutting depth, more materials are deformed ahead of the cutting tool. Correspondingly, more dislocations are observed in the cutting region with a larger cutting depth. It can also be observed that as the cutting depth increases, the machined surface becomes rougher. This may be due to the higher cutting forces required for the larger cutting depth, which will be shown in the follows. The variations of the tangential cutting force Fx with cutting distance for the different cutting depths are shown in Fig. 7. It can be seen that the smaller the cutting depth, the earlier the cutting force reaches the steady state. Besides, a smaller cutting depth results in a lower cutting force. In order to give a more quantitative comparison of the cutting force, the time-averaged tangential and normal cutting forces for the cutting distance of 6–16 nm are calculated for the different cutting depths as shown in Fig. 8. Although both the tangential and normal cutting forces decrease as the cutting depth decreases, the normal cutting force decreases less rapidly than the tangential cutting force. As a result, the ratio of normal force to tangential force changes from smaller than 1.0 for the cutting depth 4.0 nm to greater than 1.0 for the cutting depths of 2.0 and 0.8 nm. This shows that for nanometric cutting with small cutting depth, as the tool edge radius is quite large compared to the cutting depth, the nanometric cutting is
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Fig. 6. The partial cross-sectional views of the cutting process for different cutting depths of (a) 0.8 nm, (b) 2.0 nm and (c) 4.0 nm. 280
300 Cut depth 0.8 nm
Resultant force Fr
Cut depth 2 nm
250
Average cutting force (nN)
Cutting force (nN)
240
Cut depth 4 nm
200 160 120 80 40 0 0
2
4
6
8
10
12
14
16
Cutting distance (nm)
Specific force Fs
200 150 100 50 0 0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Cutting depth (nm)
Fig. 7. Variations of the tangential cutting force Fx with cutting distance for the different cutting depths of 0.8, 2.0 and 4.0 nm.
Fig. 9. The resultant cutting force and the specific cutting force for the different cutting depths of 0.8, 2.0 and 4.0 nm.
250
Average cutting force (nN)
Tangential force Fx
200
198.7 176.7
Normal force Fz
150
133.3 108.2 86.9
100
Fx Fz
59.3
Fx Fz
50
effect’’. This ‘‘size effect’’ on the specific cutting force in nanometer machining can be explained by the metallic bonding. The special feature of metallic bonding is that the strength of the individual bond has a strong dependence on the local environment. The bonding becomes stronger at the surface due to the localization of the electron density. The smaller the cutting depth, the larger the ratio of surface to cutting depth, and thus the bigger the specific cutting force.
Fx Fz 0
3.3. Effect of cutting speed 0.8 nm
2.0 nm
4.0 nm
Cutting depth
Fig. 8. The time-averaged cutting forces for the different cutting depths of 0.8, 2.0 and 4.0 nm.
more similar to the conventional grinding with a large negative tool rake angle. Fig. 9 shows the changes of the resultant cutting force and the specific cutting force with cutting depth. Here the resultant cutting force Fr is the vector sum of the tangential force Fx and normal force Fz. Note that the average cutting force along the thickness direction Fy is zero. The specific cutting force Fs is the resultant cutting force divided by the cutting depth. It can be seen that with the decrease of cutting depth the resultant cutting force decreases. However, the specific cutting force increases rapidly with the decrease of cutting depth, which shows a very obvious ‘‘size
To study the effects of the cutting speed on the nanometric machining process, simulations with different cutting speeds have been conducted. Figs. 10a–d show the partial cross-sectional views of the x–z plane for the nanocutting process at the cutting speeds of 50, 100, 200 and 500 m/s respectively. It can be seen that the higher the cutting speed, the more the lattice defects in the cutting region. However, the cutting speed seems to have no obvious effect on the machined surface quality for the four different cutting speeds studied. The effect of the cutting speed on the cutting forces is shown in Fig 11. It can be seen that both the tangential and the normal cutting forces increase with the increase of the cutting speed. This is because a higher cutting speed results in shorter time for the dislocations to move away from the cutting region and thus the dislocations pile up
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Fig. 10. The partial cross-sectional views of the cutting process for the cutting speeds of (a) 50 m/s, (b) 100 m/s, (c) 200 m/s and (d) 500 m/s.
Average cutting force (nN)
300 250 200 150 100
Tangential force Fx Normal force Fz
50 0 0
50
100
150
200
250
300
350
400
450
500
550
Cutting speed (m/s)
Fig. 11. Variations of cutting forces with cutting speed.
in the cutting region and strengthen the materials there, which results in higher deformation force. However, the increase of the tangential cutting force is more obvious than the normal cutting force. The more rapid increase of the tangential cutting force with the cutting speed may be due to the fact that for a higher cutting speed an additional cutting force is needed to accelerate the chips in front of the tool to the higher cutting speed. 3.4. Effect of crystal orientation and cutting direction In the above simulations, the cutting is in the (0 0 1) plane and along the [1 0 0] direction of the workpiece. We also performed MD simulations for two more crystal ori-
entations of (0 0 1)½1 1 0 and (1 1 1)½2 1 1. The cutting speed is 100 m/s with the cutting depth of 4 nm. Figs. 12a–c show the simulation results for the three different lattice setups. The upper figures are the cross-sectional views along x–z plane, while the lower figures are the top views. The figures are colored by the CSP. It can be seen that nanometric machining with different crystal orientations result in quite different deformation behavior of the workpiece materials. In the case of (0 0 1)[1 0 0] the dislocations seem to propagate at 45 to the cutting direction as shown in Fig. 12a. In the case of (0 0 1)½1 1 0 the dislocations are in parallel to the cutting direction as shown in Fig. 12b. While in the case of (1 1 1)½2 1 1 shown in Fig. 12c, the dislocations propagate at 30 to the cutting direction at the top surface and at 60 to the cutting direction inside the workpiece. From Figs. 12a–c, it can also be seen that the propagation directions of the dislocations belong to the h1 1 0i direction, which is the slip direction of FCC crystals. Besides, the material deformation at the cutting region is quite different for the three crystal orientations. The crystal orientation of (0 0 1)½1 1 0 results in the largest chips ahead of the tool, while the (1 1 1)½2 1 1 results in the smallest ones. The simulated cutting forces for the three lattice setups are shown in Fig. 13. It can be seen that the lowest resultant force is with the case of (0 0 1)½1 1 0, while the highest one is with the case of (1 1 1)½2 1 1. For the case of (0 0 1)[1 0 0] the resultant force is in between. As the dislocations in the workpiece propagate at the angles of 0, 45 and 60 to the cutting direction for the cases of (0 0 1)½1 1 0, (0 0 1)[1 0 0] and (1 1 1)½2 1 1, respectively, the simulation results show that the larger the dislocation
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Fig. 12. Showing cross-sectional views (upper figures) of the x–z plane and top views (lower figures) for three different crystal orientations of (a) (0 0 1)[1 0 0], (b) (0 0 1)[1 1 0] and (c) (1 1 1)[2 1 1].
400 Tangential force Fx
Average cutting force (nN)
350
Normal force Fz Resultant force Fr
300
265.9
275.2
241.7
250 200 150 100
Fx Fz Fr
Fx Fz Fr
Fx Fz Fr
(001)[110]
(111)[211]
50 0 (001)[100]
Crystal orientation
Fig. 13. The cutting forces for the different crystal orientations of (0 0 1)[1 0 0], (0 0 1)[1 1 0] and (1 1 1)[2 1 1].
propagation angle, the higher the resultant cutting force. This can be explained by the projection of the cutting forces on the dislocation slip directions. The larger the propagation angle, the smaller the cutting force projected to the dislocation slip direction, and thus large cutting force is required to move the dislocations to generate plastic deformation during cutting. 4. Conclusions Large scale MD simulations using the EAM potential have been carried out to study the nanometric machining
of copper. The material deformation, lattice defects, dislocation movement, and cutting forces during the cutting process are studied. The effects of cutting depth, cutting speed, crystal orientation and cutting direction on the nanoscale cutting process are investigated. From the simulation results, the following conclusions are obtained: (1) The nanometric machining process is accompanied with complex dislocation formation and movement, which have strong effect on the material deformation and cutting forces. (2) With the decrease of the cutting depth, less plastic deformation and fewer dislocations are generated in the workpiece, and correspondingly a smoother machined surface is obtained. As the cutting depth decreases, the cutting forces decreases, however the specific cutting force increases rapidly, which shows that the ‘‘size effect’’ exists in nanometric machining. (3) A higher cutting speed results in more lattice defects at the cutting region and higher cutting forces. However, no obvious difference in the machined surface quality was observed for the different cutting speeds studied. (4). The crystal orientation and cutting direction have a strong effect on the material deformation, dislocation movement and cutting forces. For the three crystal setups of (0 0 1)½1 1 0, (0 0 1)[1 0 0] and (1 1 1)½ 2 1 1, the dislocations in the workpiece have been found to propagate at 0, 45 and 60 to the cutting direc-
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tion respectively. Due to the difference in the dislocation propagation angles, the lowest cutting forces are with the case of (0 0 1)½1 1 0, while the highest cutting forces are with the case of (1 1 1)½ 2 1 1. Acknowledgement This work was supported by the Agency for Science, Technology and Research (A*STAR), Singapore. References [1] S. Shimada, N. Ikawa, H. Tanaka, J. Uchikoshi, Ann. CIRP 43 (1994) 51–54. [2] T. Inamura, N. Takezawa, Y. Kumaki, T. Sata, Ann. CIRP 43 (1994) 47–50. [3] K. Maekawa, A. Itoh, Wear 188 (1995) 115–122. [4] L. Zhang, H. Tanaka, Wear 211 (1997) 44–53. [5] T.H. Fang, C.I. Weng, Nanotechnology 11 (2000) 148–153. [6] R. Komanduri, N. Chandrasekaran, L.M. Raff, Wear 242 (2000) 60–88.
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