# Study of Materials Deformation in Nanometric Cutting by Large-scale Molecular Dynamics Simulations

- QX Pei
^{1}Email author, - C Lu
^{1}, - HP Lee
^{1}and - YW Zhang
^{1}

**Received: **22 December 2008

**Accepted: **27 January 2009

**Published: **18 February 2009

## Abstract

Nanometric cutting involves materials removal and deformation evolution in the surface at nanometer scale. At this length scale, atomistic simulation is a very useful tool to study the cutting process. In this study, large-scale molecular dynamics (MD) simulations with the model size up to 10 millions atoms have been performed to study three-dimensional nanometric cutting of copper. The EAM potential and Morse potential are used, respectively, to compute the interaction between workpiece atoms and the interactions between workpiece atoms and tool atoms. The material behavior, surface and subsurface deformation, dislocation movement, and cutting forces during the cutting processes are studied. We show that the MD simulation model of nanometric cutting has to be large enough to eliminate the boundary effect. Moreover, the cutting speed and the cutting depth have to be considered in determining a suitable model size for the MD simulations. We have observed that the nanometric cutting process is accompanied with complex material deformation, dislocation formation, and movement. We find that as the cutting depth decreases, the tangential cutting force decreases faster than the normal cutting force. The simulation results reveal that as the cutting depth decreases, the specific cutting force increases, i.e., “size effect” exists in nanometric cutting.

## Keywords

## Introduction

Nanometric cutting is a tool-based materials removal technique to remove materials at nanometer scale thickness in the surface. Nanometric cutting can be used to produce micro/nano-components with nanoscale surface finish and sub-micron level form accuracy for many applications such as micro-electro-mechanical systems (MEMS) and nano-electro-mechanical systems (NEMS) [1, 2]. Understanding the material removal mechanism and mechanics at atomistic scale in the surface, such as deformation evolution, chip formation, machined surface, cutting forces, and friction, is a critical issue in producing high precision components. However, as the nanometric cutting process involves only a few atomic layers at the surface, it is extremely difficult to observe the cutting process and to measure the process parameters through experiments. Therefore, theoretical analysis plays a major role in obtaining information on nanometric cutting. The widely used finite element method based on continuum mechanics for the analysis of conventional cutting is not appropriate to analyze the nanometric cutting process because of the discrete nature of materials at such a small length scale; therefore molecular dynamics (MD) simulation has become a very useful tool in the study of nanometric cutting.

A number of studies have used the MD simulations to analyze the nanometric cutting process [3–9]. The typical studies among them include: Maekawa et al. [3] studied the role of friction between a single-crystal copper and a diamond-like tool in nano-scale orthogonal machining. The Morse type potentials were used for the interactions between Cu–Cu, Cu–C, and C–C atoms; Zhang et al. [4] studied the wear and friction on the atomic scale and identified four distinct regimes of deformation consisting of no-wear, adherence, plowing, and cutting regimes; Komanduri et al. [5–7] carried out MD simulations of nanometric cutting of single-crystal copper and aluminum. They investigated the effects of crystal orientation, cutting direction and tool geometry on the nature of deformation, and machining anisotropy of the material; more recently, Zhang et al. [9] used MD simulations to study the subsurface deformed layers in the atomic force microscopy (AFM)-based nanometric cutting process.

All those previous studies have provided much help in understanding nanometric cutting. However, as the MD simulation of nanometric cutting is compute-intensive, small simulation models with a few thousands to tens of thousands of atoms were used in the reported studies to reduce the computing time. Although those small models have provided a lot of information on the nanometric cutting processes, a small model may induce significant boundary effects that make the results unreliable. For example, if the model is not large enough, the widely used fixed-atoms boundary in MD simulations may have strong effect on the dislocation movement and thus will affect the motion of atoms at the cutting surface. Besides, in most of the reported studies, the simulation models are two-dimensional or quasi-three-dimensional (plane strain) due to the limitation on the model size. Therefore, there is a need for large-scale MD simulations of three-dimensional (3D) nanometric cutting processes.

Another limitation of previous studies on MD simulations of nanometric cutting of metals is that the Morse potential has been 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. The pair potential does 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. The EAM potential, which has been specially developed for metals [10–12], can better describe the metallic bonding. Therefore, the EAM potential gives a more realistic description of the behavior and properties of metals than the Morse potential. Our previous study [13] showed that the two different potentials resulted in quite different simulation results and suggested that the EAM potential should be used in MD simulation of nanometric cutting.

In this article, we present large-scale 3D MD simulations of nanometric cutting of copper. In our simulations, the EAM potential is employed for the interactions between Cu atoms in the workpiece. We first studied the model size effect on the simulation results with three different model sizes of about 2, 4, and 10 million atoms. Then, we used the 4-million-atom model, which is shown to be large enough to eliminate the boundary effect, to study the detailed materials deformation, dislocation movement, and cutting forces during the cutting processes.

## Simulation Models and Conditions

*x*direction, which is taken as the [100] direction of the FCC lattice of copper. The 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.

The cutting speed used in the MD simulations ranges from 50 to 500 m/s, while the cutting depth ranges from 0.8 to 4 nm. The cutting is in the (001) plane and along the [100] direction of the workpiece. The initial temperature of the workpiece is 300 K. The three layers of atoms adjacent to the fixed-atom boundary at the workpiece bottom are set as the thermostat atoms, in which the temperatures 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.

where
is a pair potential energy function;*D* is the cohesion energy;*α* is the elastic modulus;*r*_{
ij
}and*r*_{0}are the instantaneous and equilibrium distance between atoms,*i* and*j*, respectively.

*i*and

*j*, with separation distance,

*r*

_{ ij };

*F*

_{ i }is the embedding energy of atom,

*i*; is the host electron density at site,

*i*, induced by all other atoms in the system, which is given by the following equation:

where
is the contribution to the electronic density at atom,*i*, due to atom,*j*, at distance,*r*_{
ij
}, from the atom,*i*.

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 the Morse potential for the workpiece–tool interaction with the parameters adopted from reference [4] being *D* = 0.087 eV, α = 5.14, and *r*_{0} = 2.05 Å. Since the diamond tool is much harder than the copper workpiece, it is a good approximation to take the tool as a rigid body. Therefore, the atoms in the tools are fixed relative to each other, and no potential is needed for the interaction among the tool atoms.

where *R*_{
i
} and *R*_{i+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.

The large-scale MD simulations of nanometric cutting are carried out on the IBM p575 supercomputer at the Institute of High Performance Computing (IHPC). The multi-processor parallel computing is used for the simulations. The parallel computing is realized by using message passing interface (MPI) library. The calculation time for each simulation case depends on the model size, cutting speed, cutting distance, as well as the number of CPUs used. For example, it took about 3 weeks to finish the simulation run for the 10-millino-atom model with the cutting speed of 100 m/s using 32 CPUs.

## Simulation Results

### The Simulation Model Size

For a MD simulation, the larger the model size, the less obvious the boundary effect on the simulation results. However, a very large model will take unnecessarily long computing time. Therefore, it is necessary to study the model size effect, so that we can find a suitable model size for the MD simulations of nanometric cutting. The model size should be moderate with diminished boundary effect on the simulation results.

*y*direction implies that both the workpiece and the cutting tool repeat in this direction. The repeated cutting tools may make the stresses at the periodic boundary regions higher due to stress superposition arising from the interaction of stress fields. The stress interaction is helpful for the dislocations in the cutting regions to slide to the periodic boundaries and also helpful for new dislocations to be generated at the periodic boundaries. This phenomenon was also reported by Saraev et al

*.*[21] in their study of the nanoindentation of copper. As lattice defects exist in the periodic boundaries in the simulation results, the 2-million-atom model is not large enough to eliminate the boundary effect at the periodic boundaries, though it is quite large compared with the models used in the reported works on MD simulation of nanometric cutting.

Thereafter, we performed simulations using the 4-million-atom model in Fig. 1b with the workpiece thickness (*y* direction) two times that of the 2-million-atom model. The simulation results in Fig. 3b show that the 4-million-atom model could eliminate the boundary effect of the periodic boundaries. We also carried out simulations with the 10-million-atom model in Fig. 1c. In the 10-million-atom model, the workpiece is larger than that of the 4-million-atom model in all the three dimensions with very obvious increase in both the*x* and*z* directions to test the boundary effects in these two directions. We found that the simulation results with the 10-million-atom model, shown in Fig. 3c, did not show obvious difference from those of the 4-million-atom model. Therefore, for the cutting speed of 100 m/s and cutting depth of 4 nm, the 4-million-atom-model is shown to be large enough to ignore the boundary effect in the simulations.

MD simulations were also carried out to study the effect of cutting speed and cutting depth on the boundary effect. The simulation results show that reducing cutting speed results in more obvious boundary effect, while reducing cutting depth results in less obvious boundary effect. This is because a slower cutting speed means longer cutting time, and therefore the dislocations have more time to move and are more possible to reach the boundaries, which makes the boundary effect stronger. A smaller cutting depth means less material deformation, and therefore results in a weaker boundary effect. As the cutting speed and cutting depth may make the boundary effect stronger, it is important to consider those process parameters in choosing the model size for MD simulations of nanometric cutting.

### Material Deformation, Dislocations, and Cutting Forces

*x*–

*z*plane at three different cutting distances of 8, 12, and 16 nm, respectively. In this simulation case, the cutting speed is 100 m/s and the cutting depth is 0.8 nm. 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. The materials in front of and beneath the tool are away from the perfect FCC lattice. Dislocations and other lattice defects are generated 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.

*Fx*, and the normal cutting force,

*Fz*, increase at the start of the cutting. Then the cutting forces tend to remain steady during the rest of the cutting process. The formation of dislocations results in the release of the accumulated cutting energy, which corresponds to the temporary drop of the cutting force. The fluctuation of the cutting forces in Fig. 6is due to the formation of dislocations and their complex local movement in the cutting region. It is also observed from Fig. 6that the normal cutting force,

*Fz*, shows stronger fluctuation than the tangential cutting force,

*Fx*. This is because at this very small cutting depth, the normal cutting force is higher than the tangential cutting force, and therefore the normal cutting force undergoes stronger fluctuation. With a larger cutting depth as discussed in next section, the magnitude of normal cutting force is close to that of the tangential cutting force, and so magnitude of the force fluctuations is also close. 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.

### Effect of Cutting Depth on the Cutting Process

*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 effect”. The “size effect” on the specific cutting force in nanometric cutting 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 cutting surface to cutting volume, and thus the bigger the specific cutting force.

## Conclusions

We have performed a series of large-scale 3D MD simulations using the EAM potential to study the nanometric cutting process. Three different model sizes of 2-million-atom, 4-million-atom, and 10-million-atom are used with different cutting speeds and cutting depths. It is shown that the 2-million-atom model, though quite large compared with the models used in the previously reported studies, is not large enough to eliminate the boundary effect for the simulation conditions used. It is also shown that the 4-million-atom model is large enough to eliminate the boundary effect at the cutting speed of 100 m/s and cutting depth of up to 4 nm. A detailed study on the material deformation, lattice defects, dislocation movement, and cutting forces during the cutting process is made with the 4-million-atom model. It is observed that the nanometric cutting process is accompanied by complex material deformation, chip formation, lattice defect generation, and dislocation movement. It is found that as the cutting depth decreases, both the tangential and normal cutting forces decreases; however, the tangential cutting force decreases faster than the normal cutting force. It is also found that as the cutting depth decreases, the specific cutting force increases, which reveals that the “size effect” exists in nanometric cutting.

## Declarations

### Acknowledgments

This work has been supported by the Agency for Science, Technology and Research (A*STAR), Singapore. Thanks also go to the staffs of the Computational Resource Centre at the Institute of High Performance Computing, who have provided the assistance in the large-scale computing and visualization.

## Authors’ Affiliations

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