MONTE CARLO (MC) SIMULATION
OF NANOMETRIC CUTTD\fG
By
RUTUPARNA NARULKAR
Bachelor of Engineering
Pt. RaviShankar University
Raipur, India 1999
Submitted to the Faculty of the
Graduate College of the
Oklahoma State University
in partial fulfillment of
the requirements for
the degree of
MASTER OF SCIENCE
May, 2003
MONTE CARLO (MC) SIMULATION
OF NANOMETRIC CUTTING
Thesis Approved:
Thesis Advisor
Dean of the Graduate College
11
ACKNOWLEDGEMENTS
I would like to sincerely thank my parents for their faith and confidence in me and
for their help, support, sacrifice, love and understanding in the time of need. Without
them, I would have never come so far. Special thanks are to my sister who always
supported me.
I would like to pay my sincere appreciation to my adviser, Dr. R. Komanduri, for
his inspiration, technical guidance, motivation, financial SUppOlt and intelligent
supervision throughout this project. I would also like to express my sincere appreciation
to Dr. L. M. Raff for introducing me to Monte Carlo and Molecular Dynamic simulation
and all his technical guidance and encouragement throughout this study, without which
this project would not have been possible. I would also like to thank Dr. Z. B. Hou for
helping me in understanding the thermal aspects of manufacturing.
This project has been funded by the grant (No. DMI0200327) from the National
Science Foundation (NSF). The author thanks Dr. W. DeVries, Dr. G. A. Hazelrigg, and
Dr. 1. Chen for their interest and suppOli of this work
1 would also like to thank Dr. N. Chandrasekaran for helping me initially in
understanding the basics of molecular dynamic simulation. 1would also like to extend my
sincere thanks to Mr. M. Lee, for helping me with Unix system, and Mr. J. Hershberger
and Mr. M. Malshe for their valuable discussions.
Finally, I would like to thank the Department of Mechanical and Aerospace
Engineering of Oklahoma State University for providing me the opportunity to pursue my
graduate studies.
III
TABLE OF CONTENTS
Chapter Page
1 Introduction 1
1.1 UltraPrecision Machining 1
1.1.1 Advantages ofUPM 1
1.1.2 Disadvantages ofUPM 2
1.2 Simulation 2
1.3 Molecular Dynamic (MD) Simulation 3
1.3.1 Advantages ofMD simulation 4
1.3.2 Disadvantages ofMD simulation 5
1.4 Monte Carlo (MC) Simulation 5
1.5 Thermal analysis of Metal Cutting Process 9
2 Literature Review 12
2.1 Molecular Dynamic (MD) Simulation 12
2.2 Methods to enhance the Computational Speed 16
2.2.1 BookKeeping Technique 16
2.2.2 Linked Li st Method 16
2.3 Interatomic Potential 17
2.3.1 Morse Potential 17
2.4 Monte Carlo (MC) Simulation 20
2.5 AcceptanceRej ection method 21
2.5.1 Single Variate Case 22
2.5 Thennal Analysis in the Metal Cutting Process 22
3 Problem Statement 26
4 Monte Carlo Simulation of Nanometric Cutting 29
4.1 Introduction 29
4.2 .:vIC simulation of nanometric cutting 30
5 Application of Thermal Model of Metal Cutting to MD 37
and MC Simulation
5.1 Introduction 37
5.1.1 Shear Band Heat Source 37
5.1.2 Frictional Heat Source at Toolchip Interface 40
5.1.3 Combined Effect of Shear Plane Heat Source and Toolchip 42
Interface Frictional Heat Source
5.2 Application to MC Simulation of Nanometric Cutting 45
5.3 Cutting Velocity in MC Simulation 50
6 Algorithm for MC sjmulation of Nanometric Cutting 54
7 Results and Discussions 57
7.1 Comparison between MD and MC Simulation of Nanometric 57
Cutting at a Cutting Velocity of 500 ms·]
7.1.1 MD Simulation of Nanometric Cutting 57
7.1.2 MC Simulation of Nanometric Cutting 61
7.2 MC simulation of Nanometric Cutting of Aluminum at a 65
Cutting Speed of 5 msI
7.2.1 Rake Angle 10° 65
IV
8
7.2.2
7.2.3
7.2.4
7.2.5
7.2.6
7.2.7
7.2.8
7.2.9
Rake Angle 30°
Rake Angle 45°
Effect of Rake Angle on Cutting and Thrust Forces per Unit
Width
Effect ofRake Angle on Force Ratio
Effect of Rake Angle on Specific Energy
Effect of rake angle on frictional force and shear force per
unit width
Effect of rake angle on shear angle
Temperature in Cutting Zone
Conclusions
Monte Carlo Simulation of Nanometric Cutting
Thermal Model
Future Work
References
Appendix A
v
67
70
70
72
73
74
74
75
77
77
79
79
82
89
7.4 Results of MC simulation of nanometric cutting of aluminum at 5 ms']
for various rake angles, namely 10°, 30°, and 45° (See Table 7.3)
Table
2.1
7.1
7.2
7.3
7.5
LIST OF TABLES
Morse Parameters for some single crystal metals
Workpiece dimension and cutting parameters used in MD and MC
simulation ofnanometric cutting of single crystal aluminum at a cutting
speed of 500 ms· 1
Comparision between MD and Me simulation for nanometric cutting of
aluminum at 500 ms· 1 and 10° rake angle (wokpiece dimensions are
given in Table 7.1)
Computational parameters used in MC simulation of nanometric cutting
of single crystal aluminum at cutting velocity of 5 ms· 1
Results of MC simulation of nanometric cutting of aluminum at 5 ms· 1
for various rake angle using workpiece dimesions [Tom Table 7.3
vi
Page
18
58
64
65
70
72
LIST OF FIGURES
60
43
53
61
19
50
31
34
51
52
59
Page
7
18
45
46
41
47
48
Monte Carlo system for a general physical system
Variation of Morse potential with bond distance for Aluminum
Variation of Morse potential with interatomic distance for
different values ofD and a
Schematic of MC simulation of nanometric cutting showing
various regions of interest
Schematic showing two zones in MC simulation
Schematic of the analytical model of the cutting process for the
determination of the temperature rise in the chip and
workmaterial caused by the shear plane heat source, after
Komanduri and Hou 37
Schematic of Hahn's model of a band heat source moving 39
obliquely in an infinite medium
Schematic showing the heat transfer model for the frictional
heat source at the toolchip interface on the chip side
considering as a movingband heat source problem. (b)
Schematic showing the heat transfer model of the frictional heat
source at the tool}chip interface on the tool side considering as
a stationary rectangular heat source problem
Schematic of the heat transfer model with a common coordinate
system combined effect of two principal heat sources
Schematic showing the region of observation in the cutting
zone for temperature estimation
Comparison of Morse potential and Harmonic oscillation
Variation of (a) cutting and (b)thrust force with lime in the
during MD simulation of nanometric cutting of single crystal
aluminum
Merchant's force circle for Orthogonal Cutting
Comparison of the temperature in the cutting region near the
tool tip between the thermal model after Komanduri and Hou
and the MD simulation
Side spread due to absence of momentum in the chip in the MC
simulation of nanometric cutting
Effect of reflecting boundaries on MC simulation
Effect of switching off the interaction between the chip atoms
and the boundary atoms
MD simulation at various stages of nanometric cutting of
aluminum with 10° rake tool and a cutting velocity of500 ms·1
Variation of cutting force and thrust force, respectively with
cutting time in MD simulation of nanometric cutting aluminum
at 500 ms· 1
Variation of temperature in the cutting zone in MD simulation
of aluminum at a cutting velocity of 500 ms· 1
5.4
5.5
5.6
5.7 (a)
(b)
5.8
5.9
5.10 (a)
5.10 (b)
5.10 (c)
7.1 (a)
(d)
7.2 (a)
(b)
7.3
5.3
5.2
4.1
Figure
1.1
2.1
2.3 (a)
(b)
4.2
5.1
VIl
7.4 (a) MC simulation at varIOUS stages of nanometric cutting 62
(d) aluminum with 10° rake tool at a cutting velocity of 500 ms· l
7.5 (a) Variation of cutting and thrust forces, respectively with cutting
(b) time in MC simulation of aluminum at a cutting velocity of 500
ms·1 63
7.6 Variation of temperature in cutting zone with cutting time in
MC simulation of aluminum at a cutting velocity of 500 ms· 1 64
7.7 (a) MC simulation at vanous stages of nanometlic cutting
(d) alwl1inum with 10° rake tool at a cutting velocity of 5 ms·1 66
7.8 (a) MC simulation at varlOUS stages of nanometric cutting
(d) aluminum with 30° rake tool at a cutting velocity of 5 ms· 1 68
7.9 (a) MC simulation at vanous stages of nanometric cutting
(d) alwnimUll with 45° rake tool at a cutting velocity of 5 ms· 1 69
7.10 (a) Variation of cutting and thrust forces with cutting time in Me
(b) simulation of nanometric cutting for different rake angles at a
cutting speed of5 ms· l 71
7.11 Variation of cutting force per unit width and thrust forces per
unit width with rake angle in MC simulation of aluminum at a
cutting speed of 5 illS· I 72
7.12 Variation of forces ratio with rake angle in MC simulation of
nanometric cutting at a cutting speed of 5 ms· l 73
7.13 Variation of specific energy with rake angle in MC simulation
ofnanometric at a cutting speed of5 I11S·
1 73
7.14 Variation of frictional force per unit width and shear force per
unit width rake angle in Me simulation at a cLltting speed of 5
ms·1 74
7.15 Variation of shear angle in rake angle in MC simulation of
nanometric cutting at a cutting speed 0[5 ms· 1 75
7.16 Variation of temperature during the MC simulation of
nanometric cutting at a cutting speed of 5 ms·1 with 10° rake
angle 75
VIII
A
a
air
AB
c
D
F
Fe
F,
Fs
Ffr
J
K
Ko(u)
111a
Npe
Nwp
NT
P(qO)
qo
qpl
qpls
qpli
R
rij
r e
R
T
t
LJV
v
NOMENCLATURE
Area
Thermal diffusivity, (Jlem2
)
Acceptance to rejection ratio
Length ofshear plane heat source, (Aj
Specific heat, (Jig °C)
Equilibrium dissociation energy, (e V)
Force, (nN)
Cuttingforce, (nN)
Thrust force, (nN)
Shearforce, (nN)
Frictional force, (nN)
Joule's mechanical equivalent ofheat
Boltzmann constant
BesselJunction
Mass ofan atom
Peclet number
Number ojatoms in the workpiece
Number ofatoms in the tool
Boltzmann distribution Junction
Configuration ojN particles in 3D space
Heat liberation intensity ofa moving line heat source, (Jlern::,)
Heat liberation intensity ofa moving plane heat source. (Jlem s)
Heat liberation intensity ofa moving  induced plane heat source, (Jlcm s)
Distance betvveen the movingline heat source and point M, where temperature is
concerned, (em)
Inter particle distance, (A)
Equilibrium distance, (A)
Distance between the moving line heat source and the point M, (A)
Temperature, (K)
Depth ofcut, (A)
Change in Morse potential, (e V)
Minimum potential, (e V)
Velocity ofmoving plane heat source, (cmls)
Velocity ofi'li atom
Cutting velocity, (cmls)
Chip velocity, (cmls)
Width ofcut, (A)
Coordinates ofthe point where the temperature rise is concerned in moving
coordinate system, (em)
ix
Greek letters
ex range ofinteraction, (AI)
e temperature rise, ( °C)
eM Temperature rise at point M,(oC)
A Thermal conductivity, (J/cm s °C)
~ Shear angle, (deg)
q; Oblique angle, (deg)
p Density, (g/cm J
)
?: Random number between aand 1
B Fraction ofthe heat conducted into he workmaterial
(IB) Fraction ofthe heat conducted into the chip
x
Chapter 1
Introduction
1.1 UltraPrecision Machining
Nanometric cutting involves material removal at extremely small depths of cut on
the order of a few nanometers. Processes involved in executing such small depths of cut
are ultra precision machining (UPM), ultra precision grinding (UPG), and polishing.
UPM requires a single point cutting tool, generaHy a single crystal diamolld, to finish
parts to a high degree of accuracy using a highprecision, extremely ligid machine tool.
UPG and UPM are used in semiconductor and optical industries in the machining of
mirrors and lenses of nonferrous metals, such as copper, aluminum. Applications of
UPGIUPM include manufacturing of certain components of microprocessors, lenses,
plisms, optical mirrors, computer hard drives. Unlike UPM, UPG uses a multi pointgrinding
wheel, generally diamond, to finish materials, such as ceramics and glass
components. UPM has several advantages and some limitations over conventional
machining process [1].
1.1.1 Advantages of UPM
1. High [onn accuracy and surface finish
2. Moderate material removal rate
3. N.C controlled machine tools provide reliability and repeatability
4. Superior local surface roughness and global flatness as well as parallelism
5. No possibility of imbedding foreign materials in the machined surface
6. Relatively large machinable workmaterial surface
1.1.2 Disadvantages of UPM
1. Use of expensive diamond cutting tools and high capital costs of ultra
precision, high rigidity machine tools.
2. High cost of experimentation
3. Accuracy dependent on the mechanical movement resulting from the control
system and physical condition of the cutting tool edge
An alternate approach to investigate the mechanics of nanometric cutting IS
computer simulation using molecular dynamics and/or Monte Carlo methods followed by
a few careful experiments to verify the findings.
1.2 Simulation
Simulation has been an important tool for engineers a for long time, whether it is
simulation of the metal clltting process, a supersonic jet flight, a maintenance operation
or a largescale military battle. Naylor ef al. [2] defined simulation as a numerical
technique for conducting experiments on a digital computer, involving certain types of
mathematical and logical models that describe the behavior of a physical system over
extended periods of real time. According to them, analysis by simulation can be useful
for the following reasons:
1. Simulation makes it possible to study and experiment complex internal interactions of
a given system whether it be a firm, an industry, an economy, or some subsystem of
these.
2
2. Detailed observations of a system being simulated can lead to a better understanding
of the system and suggestions for improvementsp, namely, suggestions that otherwise
would not be apparent.
3. Simulation can be used as a pedagogical teaching device to students and practitioners
alike in basic skills in theoretical analysis, statistical analysis, and decision making.
4. Simulation of complex systems can yield valuable insights as to which variables are
more important than others in a system and how they interact.
5. Simulation can be used to experiment new situations about which very little or no
information is available so as to prepare for such situations.
6. Simulation can serve as a "preservice test" to tryout new policies and decision rutes
for operating a system before running the risk of experimenting on the real system.
7. Simulations are sometimes valuable in that they afford a convenient way of breaking
down a complex system into subsystems, each of which may be modeled by an expert in
that particular area.
1.3 Molecular Dynamic (MD) Simulation
MD simulations generate infonnation at the nanometric level, including atomic
positions, velocities, and forces. It was first applied some 50 years ago by Alder and
Wainwright at Lawrence Radiation Laboratory (LRL) [3,4] to study the interactions of
hard spheres. The simulations conducted were in the field of equilibrium and nonequilibrium
statistical mechanics to calculate the response of several hundred interacting
classical particles.
3
MD simulations are widely applied in such diverse fields as chemistry, physics,
biology, material science, and engineering. MD simulation of metal cutting involves
calculation of the trajectory of workmaterial atoms cut by a cutting tool by solving the
Newtonian equations of the motion using an empirical interatomic potential. Thus, the
workmaterial subjected to an external force is studied. The mechanism of chip formation
and the effects of cutting parameters on the nature and deformation of the workmaterial
can be studied without using costly singlecrystal diamond tool and/or expensive submicron
diamond turning machine. The results of MD simulation can be validated by
experimental studies of ultTa precision machining.
1.3.1 Advantages of MD simulation
MD simulation has many advantages over FEM and continuum analysis [5].
Unlike in FEM, the nodes and the distance between the nodes are based on more
fundamental units of matelials, namely, lattice constant rather than arbitrary values as in
FEM. MD simulations give higher temporal and spatial resolution than by continuum
analysis. In addition, MD simulation does not require complex and expensive diamond
cutting tools, such as single crystal diamond tool and expensive ultraprecision machine
tools to obtain experimental data. Also, the effect of various process parameters in
machining, such as tool geometry and cutting conditions can be effectively studied
without expensive setup costs. It is simpler to vary tool geometry and cutting conditions
than by actual cutting. Also, the cutting process can be investigated in situ in MD
simulation while experiments using UPM involve post mortem analysis.
4
1.3.2 Disadvantages of MD simulation
As powerful as MD simulations are, there are, however, some difficulties
associated with their use [5]. In fact, MD simulations in many cases provide too much
information, which may not be of interest but is obtained at the expense of premium
computational time. For example, when we execute a simulation of nanometric cutting,
the NID simulation would provide information that gives precisely how the atoms move
to each of the new positions, their corresponding potential energy, and kinetic energy. If
we are interested only in the steady state conditions of the chip formation process or the
subsurface deformation, MD simulation provides too much information. The major direct
cost associated with the MD simulation is the computational time. The indirect effects are
high memory usage, high cutting speeds (l00500 mls), and fewer numbers of atoms (a
few thousand) to be considered in the simulation. Due to the nonavailability of
appropriate potentials for metallic alloys, MD simulation is restricted to a limited
member of pure elements and simple systems. When applied to nanometric cutting, the
depth of cut of necessity is somewhat limited and is much lower than that used in UPM.
In most of the MD simulations, the tool is considered as infinitely hard and thus toolwear,
and tool deformations are not considered. To circumvent some of the problems
stated, a Monte Carlo method is used in the present investigation.
1.4 Monte Carlo (Me) Simulation
Monte Carlo technique, used some 50 years ago to model atombomb explosions,
has become today's hot simulation tool. In the late 1940s (the generally accepted birth
date of MC is 1949) scientists at Los Alamos National Laboratory (LANL) programmed
5
their early computers to create random combinations of known variables to simulate the
range ofpossible nuclear reactions [6]. Metropolis and Ulam [6] nicknamed the program,
Monte Carlo, after that city's famous roulette wheels and used it to find patterns that
would enable them to plot the probability of different outcomes. MC methods are
stochastic techniques based on random numbers and probability statistics used in the
investigation of complex problems.
One advantageous feature of the MC method is the simple structure of the
computational algorithn1 used. Statistical simulation methods may be contrasted to
conventional numerical methods which typically apply ordinary or partial differential
equations that describe some underlying physical or mathematical system. In many
applications of Monte Carlo, the physical process is simulated directly and there is no
need to write down the differential equations that describe the behavior of the system.
Instead, the physical or mathematical system is described by probability density functions
(PDFs). Once the PDFs are known, the Monte Carlo simulation can proceed by random
sampling from the PDFs.
Simulations are performed with multiple trials, and the desired result is taken as
an average over the number of observations, which can be a single observation or
millions of observations. In many practical applications, one can predict the statistical
error in the average result, and hence an estimate of the number of Monte Carlo trials that
are needed to achieve the desired results within a given limit of error.
Figure 1.1 illustrates the basic idea behind Monte Carlo, or statistical simulation
as applied to an arbitrary physical system. Assuming that PDFs can describe the physical
system, the Monte Carlo simulation then proceeds by sampling from these PDFs. The
6
outcomes of these random samplings must be accumulated in an appropriate manner to
produce the desired result. But the essential characteristic of Monte Carlo is the use of
random sampling techniques and perhaps other algebra, to manipulate the outcomes and
arrive at a solution of the physical problem. In contrast, a conventional numerical
solution approach would start with the mathematical model of the physical system,
implementing the differential equations and then solving a set of algebraic equations for
the unknown state of the system.
Arbitrary
Physical system
.....
Probability
density function
(PDFs)
0.'
Random numbers [0,1]
St> S2> S3".
Results of the simulation
Figure 1.1 Monte Carlo system for a general physical system [7]
This general description of the Monte Carlo method may not be directly
applicable to all systems. It is natural to assume that Monte Carlo methods are used to
simulate random or stochastic processes, since PDFs can describe those processes.
7
However, this coupling is actually too restrictive because many Monte Carlo applications
have no apparent stochastic content, for example, the evaluation of a definite integral or
the inversion of a system of linear equations. However, one can pose the desired solution
in terms of PDFs, and while this transfonnation may seem artificial, this step allows the
system to be treated as a stochastic process for the purpose of simulation and hence
Monte Carlo methods can be applied to simulate the system. It is, therefore necessary to
take a broader view of the definition of Monte Carlo methods and include in the Monte
Carlo rubric all methods that involve statistical simulation of some underlying system,
whether or not the system represents a real physical process.
MC simulation method has been used extensively m such diverse fields as
mathematics, physical sciences, engineering, and medicine, for many years. Although,
Monte Carlo methods are used in diverse ways, in the context of molecular computations
there are five types most commonly encountered [8]. They are given in the following:
1. Classical Monte Carlo (CMC)  Samples are drawn from a probability distribution,
often the classical Boltzmann distribution, to obtain thermodynamic properties,
minimumenergy structures, and rate coefficients;
2. Quantum Monte Carlo (QMC)  Random walks are used to compute quantummechanical
energies and wave functions, often to solve electronic structure problems,
using Schroedinger's equation as a formal starting point;
3. Pathintegral quantum Monte Carlo (PMC)  Quantum statistical mechanical integrals
are computed to obtain them10dynamic properties, or even rate coefficients, using
Feynman's path integral as a formal starting point;
8
4. Volumetric Monte Carlo (VMC)  Random and quasirandom number generators are
used to generate molecular volumes and sample molecular phasespace surfaces.
5. Simulation Monte Carlo (SMC)  Stochastic algorithms are used to generate initial
conditions for quasiclassical trajectory simulations, or to actually simulate processes
using scaling arguments to establish time scales or by introducing stochastic effects into
molecular dynamics. Kinetic Monte Carlo is an example or an SMC method and so is the
thennalization of a molecular dynamics trajectory.
1.5 Thermal analysis of Metal Cutting Process
One of the earliest papers on the thennal aspects of machining is on "an inquiry
into the source of heat which is excited by friction" by Benjamin Thompson (Count
Rumford) in 1798 [9]. He demonstrated the interdisciplinary nature of tribology, namely
friction and wear especially in machining, and breaking the notions about the nature of
heat. Joule, who established the mechanical equivalent of heat same 50 years later, paid
tribute to the pioneering work of Count Rumford in developing the concept of
mechanical equivalent of heat [10).
Jaeger in his 1942 classical paper [1 I] pointed out that problems involving
moving sources of heat arise frequently in practice in the calculation of temperature at
sliding or cutting contacts, but despite their importance they have not been studied
systematically. He developed the heat source method, an aJtemative to the classical
partial differential equations (PDEs) approach, and applied to a range of moving heat
source problems. The plastic deformation energy and/or the frictional energies involved
in some of the manufacturing processes are very high. Most, if not all, of this energy is
9
converted into heat, which manifests as high temperatures In vanous regIOns of the
process. These processes also produce high temperature gradients that can have
detrimental effect on the finished product including subsurface deformation and residual
stresses. Thermal analysis of these processes is, therefore, very important as it can shed
light on the physics of these processes. It can also enable optimization of the process
parameters, selection of appropriate tools and lubricants, reliability in service, and
improve the economics of the operation.
The commonly used approaches for thermal modeling of various processes are:
1. analytical or heat source methods,
2. simulation fmite element method (FEM), or finite difference method (FDE),
and
3. experimental.
Due to various assumptions and inherent limitations In each method, the
validity of one method has to be checked with the other method.
The thesis is divided into 8 chapters. Chapter 1 gives an introduction to MD/MC
simulation of nanometric cutting. In Chapter 2, a review of literature on various topics of
interest to the present investigation, namely MD simulation, MC simulation, interatomic
potential, Morse potential, methods of enhancing computational speed, acceptancerejection
method, and thermal aspect of machining are presented. Chapter 3 gives the
problem statement of this investigation. In Chapter 4, details of the MC simulation 0 f
nOl1ometric cutting are given. It includes the description of the principle and the
formulation of MC simulation 0 f nanometric cutting, description of Markov chain,
MaxwellBoltzmann probability density function, and a description of Markov moves in
10
the workpiece. In MC simulation of nanometric cutting, speed does not enter directly.
However, it is well known that cutting speed is related to the temperature generated in the
cutting process. If the temperature can be estimated for a given cutting speed, then it can
be used in the Maxwell Boltzmann PDF and the acceptancerejection criterion. This
way, cutting speed effects can be introduced indirectly.
In this investigation, an attempt has been made to determine the temperature in
the cutting zone in MD and MC simulation using the thennal model of the metal cutting
process developed by Komanduri and HOll [1214]. Chapter 5 presents some details on
the thennal model of the metal cutting process developed by Komanduri and Hou [1214].
The two principle heat sources of heat in metal cutting, namely, the shear plane heat
source in the primary shear zone and the frictional heat source at the chiptool interface
are combined in this analysis to obtain the temperature distribution in the workpiece, the
chip, and the tool. Chapter 6 describes the algorithm used for MC simulation of
nanometric cutting. Chapter 7 presents results and discussion of MC simulation of
nanometric cutting of single crystal aluminum for three different tool rake angles, namely
10°, 30°, and 45°. The workpiece was oriented in (001) plane and in [100] direction.
These results are compared with conventional MD simulations of nanometric cutting.
Chapter 8 gives conclusions arising out of the present investigation and offers some
suggestions for future work.
11
Chapter 2
Literature Review
2.1 Molecular Dynamic (MD) Simulation
In 1950s, Alder and Wainwright [3,4] initiated MD simulation studies at the
Lawrence Radiation Laboratories in the fields of equilibrium and nonequilibrium
statistical mechanics. Since then, MD simulation has been applied in such diverse fields
as biology, chemistry, physics, material science, and engineering. Belak et al. [15,16]
investigated both 2D and 3D cutting of copper llsing a LennardJones potential at a
cutting speed of 100 ms· 1 with different edge radii tools and different depths of cut. In
this paper, as in most other papers that followed, the cutting tool was considered to be
infinitely hard. They reported disorder in the shear zone as well as in the chip. They
observed that large radii tools require larger force to achieve the same depth of cut as
sharp tools. Belak et al. [17] also investigated machining of si licon using a diamond tool
at a cutting speed of 540 ms· l
. Howevcr the tool was considered infinitely hard.
lkawa et al. [18] and Shimada et al. [1921] and Shimada [22] of Japan conducted
2D MD simulation of nanometric cutting of copper using a diamond tool. They studied
the effect of edge radius and depth of cut on the chip formation process, subsurface
deformation, and specific cutting energy. Most of the tcsls were simulated at 200 I11s· 1
although a few tests were conducted at 5 ms· l
. Shimada et ai. [22] also investigated the
size effect in nanometric cutting of copper with a diamond tool. The results of MD
simulation were compared with the experimental results. They observed an increase in
12
force with increase in the cutting edge radius. This, they attributed to the plowing force at
the cutting edge radius. They found the specific energy to exceed the heat of fonnation of
copper when the chip thickness was half the tool edge radius.
Shimada et al. [20] also investigated the potential application ofMD simulation to
detennine the attainable accuracy i.n nanometric cutting. They observed dislocations in
both copper and aluminum workpieces. They also studied the generation and movement
of dislocations in the workpiece. The recovery of atoms underneath the 1001 once the tool
passed the machined surface was observed. They attributed this behavior to the elastic
recovery of the material. From the MD simulation results, they concluded that 0.5 nm is
the ultimate attainable surface roughness in UPM of copper. Shimada [22] studied ductile
and brittle phenomenon in microindentation and micro machining. Silicon substrate was
indented with an octahedral diamond tool. Threebody Tersoff potential was used to
evaluate the forces. Indentation was carried out at 100 ms I to a depth of cut of 1.5 nm
after that, speed was increased to 200 msI
. No generation and movement of dislocations
were observed. However, the results showed that atoms were densely packed at the outer
areas and relatively sparse near the tool. Inamura et al. [2326] investigated MD
simulation of nanometric cutting under quasi static conditions on a copper workpiece and
an infinitely hard djamond tool. Inamura et al. [21] investigated the effect of interatomic
potential used to model the interaction of the tool with the workpiece.
Maekawa et al. [27] introduced the idea of area restricted molecular dynamics
(ARMD). In this method, instead of running the simulation for the entire workpiece,
simulations are carried out in a region near the tool nose with a radius of 7.3 nm. This
restIicted region moves along with the tool as the simulation proceeds. Thjs process
13

reduces the overall computational time but makes the process dependent on the cutting
geometry as well as the size of the area chosen.
At Oklahoma State University, Professors Komanduri and Raff have conducted a
wide range ofMO simulation studies in nanometric cutting and tribology [5,2835]. For
example, Chandrasekaran et al. [36] introduced the lengthrestricted molecular dynamics
(LRMD) simulation. The LRMD is based on the fact that once the tool has advanced into
the workmaterial through some distance, the workrnaterial atoms in the machined region
will exert minimum influence on future simulation results since their interaction with the
tool atoms will be negligible. Atoms from the machined part of the workmaterial that are
not going to be affected by the simulation results significantly are discarded but their
memory positions are retained. These memory positions are used to add new atoms to the
workmaterial. With this method, a smaller workpiece can be used to simulate cutting to
practically unIlmited length.
Komanduri et al. [31] conducted MD simulations of indentation and scratching on
a single crystal aluminum in various crystal orientations and directions of scratching to
investigate the anisotropy in hardness and friction. They also conducted MD simulation
of nanometric cutting on single crystal aluminum in specific combinations of crystal
orientation {(l10), (110), and (OOl)} and cutting directions <lIIO], [211], and
[lOO]>with tools of different rake angles, namely 10°, 30°, and 45° to investigate the
nature of deformation and the extent of anisotropy in aluminum [32]. When the
aluminum crystal was oriented in the plane (111) and cut in the [110] direction, they
observed that plastic deformation ahead of the tool was accomplished predominantly by
compression along with shear in the clltting direction. They also found the defonnation in
14
the workmaterial underneath the depth of cut region to be along the cutting direction. In
the (001)[110] combination, they found the dislocations to be generated parallel to the

cutting direction. In the case of (110) orientation and [It OJ cutting direction, the
dislocations were found to be parallel as well as perpendicular to the cutting direction. In
contrast, for (001)[100] combination, extensive dislocation motion at 450 to the cutting
direction were observed. Similarly, for the (111) [211] combination, they observed the
dislocation motion to be at 600 to the cutting direction. In both cases, the material in the
shear zone was observed to deform at an angle equivalent to the shear angle, which is the
mirror image of the dislocations generated in the workmaterial. They also investigated
the vaIiation of the cutting forces, the ratio of thrust to cutting force (or the force ratio),
the specific energy (i.e. energy required for removal of unit volume of workmaterial), and
the nature of deformation ahead of the tool as well as the subsurface deforrnation of the
machined surface with crystal orientation and direction of cutting.
The review of the literature on MD simulation of nanometric cutting presented
here is restricted to a limited number of papers directly related to the present
investigation. For a more complete review of the literature in MD simulation of
nanometric cutting in general and the results of Professors Komanduri and Raff, the
readers are referred to an extensive review article they authored [5].
15
2.2 Metbods to Enhance the Computational Speed
Several methods have been developed to enhance the computational speed in MD
simulations. Two of the techniques commonly used are the bookkeeping technique and
the linked list method which will be briefly described in the following.
2.2.1 BookKeeping Technique
In the bookkeeping technique [37), a 11st of neighboring atoms based on the
cutoff radius for each atom in the system is created. Instead of taking an infinite cutoff
radius, which means interaction potential between two atoms is zero at infinite distance, a
specific value is chosen for the cutoff radius. This limits the interaction of the atoms to
only those neighboring atoms that are within the cutoff radius. This fonnulation
decreases the computational time significantly. On the other hand, some amount of
potential is lost. When the neighbor list has to be updated, the entire system is checked
for neighboring atoms for each atom. This is the actual drawback of this method. Since in
a cutting operation the position of the atoms changes with each time step, the
corresponding changes have to be reflected in its neighbor list.
2.2.2 Linked List Method
In order to overcome the computational time involved in refreshing the neighbor
list in the bookkeeping technique, another technique, called linkedlist method, was
introduced. Rentsch et al [37] used this method to optimize the book keeping technique
and overcome the problem of long computational time involved in updating the neighbor
list for each atom. Allen and Tudesley [38] divided the simulation space into small
16
volumes and neighboring volumes for each atom and tabulated them. So, when an update
for neighbor list is needed instead of going through the entire system only the volumes
associated with a particular atom have to be checked and updated for each time step.
This method is applied in the current investigation of MC simulation of nanometric
cutting of single crystal aluminum.
2.3 Interatomic Potential
The accuracy of any simulation depends on the appropriateness of the potential
used in the simulation. There are various potentials developed to address MD simulation
of a wide range of workmaterials. They are Morse potential, LennardJones potential,
BornMeyer potential, Tersoffpotential, MEAM potential. Inamura et al. [24] used two
different potentials, namely, a pairwise Morse potentia! and a BomMeyer potential, to
calculate the force between a copper workmaterial and a diamond tool (infinitely hard).
They concluded that the nature of cutting differs somewhat depending on the potential
used. The pairwise stands for the interaction only between two atoms ignoring the rest of
the surrounding atoms present in the system. The potential llSed in present investigation
ofMC simulation ofnanometric cutting of aluminum is a pairwise Morse potential.
2.3.J Morse Potential
The Morse potential is a pairwise potential. It can be represented as:
V(r) = D{l exp[ a(r  rc )]2 or V(r) = D{exp[ 2a(r  rJ] 2exp[ a(r  rJh
where r is the distance between any two atoms and re is the equilibrium distance, D is
the equilibrium dissociation energy, and a denotes the range of interactions. The three
17
constants re , D, and a are material properties and are based on bulk modulus, rigidity
modulus, lattice spacing, etc.
Table 2.1 Morse parameters for some single crystal metals
Symbol Metal Crystal a (A" ) re(A) D (cV)
Structure
Pb Lead FCC 1.18360 3.733 0.23480
Ag Silver FCC 1.36900 3.115 0.33230
Ni Nickel FCC 1.41990 2.780 0.42050
Cu Copper FCC 1.35880 2.866 0.34290
Al Aluminum FCC 1.16460 3.253 0.27030
Mo Molybdenum BCC 1.50790 2.976 0.80320
W Tungsten BCC 1.41160 3.032 0.99060
Cr Chromium BCC 1.57210 2.754 0.44140
Fe Iron BCC 1.38850 2.845 0.41740
10 r,
8 ..•......: ~ ~ : ~ ~ .
I I • • I •
I • I I I •
• • I • I I
6 :.....•... ~ : : ~ : .
·I .• • • • • , . , .
• , • I • I
> 4 ....•..•.:..••..•.. ~. ••• • .•• : .•..••..•:..•....•. ; ..•.•••.. ~ .
• • I I , •
~ : : : : : :
]i 2 ....•..••:..•...•.. ~. . . .. . •. : .••.•....:.....•... ;  ..   . ~ .
c .. ",
Q) :: ::: &0 ,. ."
..  r ..
,, ',
, ,
·2  : ~ .
, . , ,
4 : ~  .. .. .. .. .. .. .. .. .. . ~ ..  ~ ..  ..
• I • • • ,
• • I I , •
• I • I I ,
·6.1......:·.:..·:..':..'''
Bond distance(A)
Figure 2.1 Variation of Morse potential with bond distance for aluminum
18
12
10  ...... ~ ..  ~ .. ... · .. . . .. ~ .. ... :. ..  ...:...  .. ~ .  ... , .
0=3 ()
8 .   .•
6· _••••
>' ~ 4· 
ro 2 2
o a...
o 
2
.._._,·• _.•,. •  .. _,I . , ,;::
. ..   ..  .. '" ~......•......
,~_. . . . • • D=4.0 •
• • I , •
• • .. • ",  •••• "J r   ,  ..  .
" .
.6 L....._..:_~_ __:..._ ____:__.:..._....:......_ _..:..__:.._.::....__I
Bond distance(A)
(a)
14
12
10
• • I I.
  •    •• _ "0 • _ ••••• _ _ • _ .. 0.' _ •
0.=1 1 :
.. ."
""'"' '•:"•" '" •••••••••• ,J •••••• j •••••• I,. ••••••••••• __
. . . . . . , .
• .0 4 __ __ • _._ _. _ _1 _
. , .
2 ••.•• ~ ··'·:····~a:;f.D:·.:·:·····.:'·····:···.:.. 
_____11~.~.~.:::.~.,~.~..~.~.~.~;5~ ..~_~.~_~.~~.~_~_.~_~:~__ . __ , , , ,
o
2
4
r +'': 11=1.3:
8    : ~ ~  : ~   .. : .  ~ ": .
• • • I • •
>" 6 ..
..!.. :g 4
OJ o
ll.
6 L.._........_""__""'_"__.L.. '__......_'_l
Bond distance(A)
(b)
Figures 2.3 (a) and (b) Variation of Morse potential with interatomic djstance
for different values ofD and a
19
2.4 Monte Carlo (Me) Simulation
The first use of Monte Carlo method as a research tool stems from the work on
the simulation of atom bomb during World War II [6). The success of MC method
subsequently in nuclear engineering application has paved the way for other applications
of this method. One nuclear application involved a direct simulation of the probabilistic
problems concerned with random neutron diffusion in fissile material.
By nature, any physical system tends to move to a minimum potential. Ford et al.
[39] computed the configuration for an equilibrium state using the damped trajectory
(DT) method. In this method, kinetic energy of each lattice atom is set to zero and the
Hamiltonian equations of motion for the lattice atoms are integrated until the total
potential energy reaches a minimum. This entire cycle is designated as one DT cycle.
This cycle is repeated on the system until the minimum potential Vlllin is reached, i.e. Vmin
is conserved to at least four significant digits from one cycle to the next. Alternatively,
the steepest descent method can be used to locate the minim um potenti al [40J. The
steepest descent method simply moves the function parameters to decrease the function
value. It will move toward the minimum when the starting point is anywhere within the
ridge ofhins surrounding the minimum.
Both DT and the steepest descent method always converge to a local minimum
rather than the global minimum, which we seek, called canonical Markov. Kelmedy et at.
[41J showed the use ofMC method to evaluate any type of infinitedimensional integral.
20
The Markov process is used to generate a sequence of configurations. The
concept of Markov process is based on the fact that the new configuration has to be based
on its predecessor.
Spath and Raff [42] used Monte Carlo method for modeling diffusioncontrolled
molecular reactions in matrices. Komanduri et al. [40] conducted MC simulation of
uniaxial tension of FCC metals combined with damped trajectory. They conducted tests
on four FCC metals, namely, AI, Cu, Ag, and Ni using combined MCDT simulations
and the results were compared with conventional MD simulation employing the same
potentialenergy surface. They combined DTs or steepestdescent method with MCMarkov
chains to converge the lattice atom coordinates rapidly to equilibrium. They
found the simulation time to be significantly reduced compared to MD simulations. They
also found the ultimate strengths and the corresponding strains, Young's modulus, and
strain at fracture to nearly follow the same ranking order as the intrinsic strength and
ductility of the materials and these values agreed reasonably well with the theoretical
strength calculation as well as with pure MD simulations.
2.5 AcceptanceRejection Method
von Neumann introduced the acceptancerejection method [43]. It consists of a
sampling a random variate from an appropriate distribution and subjecting it to a test to
determine whether or not it will be acceptable for LIse.
21
2.5.1 Single Variate Case [44}
Let X to be generated from fy (x), x E I .To carry out the method, let us assume
fAx) =Ch(x)g(x),
where C 2: 1, hex) is a PDF and 0 ~ g(x) .$ I . Then generate two random variables U and
Y from 9(0,1) and h(y), respectively and test to see whether or not the inequality
U ~ g(Y) holds. If the inequality holds, then accept Yas a variate generated from fl' (x).
If the inequality is violated, reject the pair U, Y and repeat the process.
2.5 Thermal Analysis of the Metal Cutting Process
The heat generated in machining was one of the first topics investigated.
Benjamin Thompson (Count Rumford) in 1798 pioneered the work in this area by
relating the heat generated during the boring of a carulOn to the mechanical work [9].
Joule who established the mechanical equivalent of heat some 50 years, \ater,
acknowledged the seminal contribution of Count Rumford that lead to his work [10].
Rosenthal [45] applied the theory of heat flow due to a moving point and a
moving line heat source using partial differential equations (PDE) and determined
analytically the temperature distribution in welding. He introduced the moving coordinate
system and for mathematical simplicity considered it as a quasistatic state for the
analysis. He .plotted the temperature distribution contours using the analysis and
compared it experimentally.
Jaeger [11] introduced the heat source method instead of solving the PDE of heat
conduction (Fouriers Law) directly for addressing a wide range of moving heat source
problems. He developed solutions for the temperature rise for plane heat sources of
22
different shapes (band, square and rectangular) starting from the solution of an
instantaneous line heat source. Carslaw and Jaeger [46] contributed significantly to the
moving heat source problems which became the basis for much of the work that followed
in machining and tribology. Jaeger [11] presented the solutions for uniform stationary
heat source using the heat source method. For mathematical simplicity, he considered the
heating time, t =00 in the very early stages of the derivation thus limiting the analysis to
quasisteady state conditions. Jaeger not only introduced the exact solutions for uniform
moving band and moving rectangular heat sources but also gave a series of approximate
equations for very high and very low values ofPeclet numbers.
Trigger and Chao [47] presented an analytical method for the evaluation of the
metal cutting temperature in machining. They considered the heat liberation intensity to
be uniformly distributed and determined the average chip temperature as it leaves the
shear zone by considering the total mechanical energy input as well as the shear energy at
the shear plane. Based on the work of Schmidt and Roubik [48], they assumed the
partition of heat into the chip and the workpeice as 90% and 10%, respectively.
The average temperature rise in the chip as it leaves the shear plane due to the
shear plane heat source is gjven by
(2.1 )
Here, es cpVctw is the increment of internal heat energy in the material passing through
the shear plane heat source per unit time, A[FcVc 0 B)  FVch ] is the sensible heat, and
J is the Joule's mechanical equivalent of heat.
23
Trigger and Chao [47] applied Blok's ingenious principal of heat partition [50] in
the investigation of temperature rise in metal cutting. They considered the frictional heat
source between the chip and the tool as a heat source that is moving in relation to the
chip, and at the same time, stationary in relation to the tool. They calculated the average
heat partition fraction for the chip and the tool from the resulting average temperature at
the tool chip interface.
Hahn [51] presented a radically different analysis of the temperatures in the shear
plane due to the shear plane heat source without the need for heat partition. He llsed an
oblique moving band heat source model based on the fact that the depth of the layer
removed from the workmaterial passes continuously through the shear plane thereby
undergoing extensive plastic defonnation to [onn the chip.
Komanduri and Hou [1214] presented a thermal model based on Halm's moving
oblique band heat source solution [51] with an appropriate image heat source and
boundary conditions for the shear plane heat source. They extended Chao and Trigger's
[49] work on the frictional heat source at the chiptool interface using modified Jaeger's
moving band theory (for the chip) and stationary rectangular heat source (for the tool)
solutions with nonunifonn distribution of heat intensity.
Komanduri and HOll [1214] considered the two principal heat sources in metal
cutting, namely, the shear plane heat source and the toolchip interface frictional heat
source operating simultaneously on a common coordinate system. They considered the
heat flow from the shear plane through the chip and the chiptool interface into the tool to
be continuous. They also considered the effect of the shear plane heat source on the
temperature rise at the toolchip interface on the chip side as well as on the tool side to be
24
the same. Thus, the contribution of the shear plane heat source on the temperature rise at
the toolchip interface was included in the total temperature rise on both sides. They
considered only the upper surface of the chip to be an adiabatic boundary. Consequently,
through the lower surface of the chip, which is in contact with the tool rake face, part of
the heat from the shear plane heat source flows into the tool. Therefore, the shear plane
heat source will contribute towards the temperature rise at the toolchip interface not only
on the chip side but also on the tool side.
Komanduri and HOll [1214] applied the model to two cases of metal cutting,
namely, conventional machining of steel with a carbide tool at high Peclet number
(NPe "" 5  20) [52] and ultraprecision machining of aluminum with a single crystal
diamond at low Peelet number (Nl'e "" 0.5) [52]. They found the analytical results to
agree reasonably well with the experimental results.
25
Chapter 3
Problem Statement
In MD simulations of nanometric cutting (both 2D and 3D), the computational
time is significant even at very high cutting speeds (100500 IDSI). The problem becomes
even more serious if simulations were conducted at conventional cutting speeds at 25
ms I
. The use of such high cutting speeds has always been a concern to researchers but
there was no choice. Even the fastest computers available (including massive parallel
processors) calIDot reduce this significantly. Apart form this, there are other issues that
need to be addressed, such as reduction in the computational time and increase in the total
number of atoms considered in the simulation. Adaptive integration can be used to reduce
the simulation time for MD simulation but this is not very significant. Due to inherent
complexity of the simulation process, the operational cutting speeds in the simulation are
still orders of magnitude higher than the conventional cutting speed (500 rns I instead of
25 IDSI).
Monte Carlo methods facilitate to circumvent some of the problems associated
with computational complexity. Since speed does not enter into the analysis directly, its
effect becomes of secondary importance. However, speed effect can be introduced in MC
simulation via the temperature generated in the cutting zone, since cutiing speed is related
to cutting temperatures. In this investigation, the cutting temperature in the cutting zone
for a given cutting speed is estimated using the thennal analysis of the metal cutting
process developed by Komanduri and Hou [1214]. This temperature, in tum, is
26
substituted in the MaxwellBoltzmann acceptance rejection criterion. This way speed
effect is introduced into MC simulation of nanomelTic cutting and it is feasible to run MC
simulations of nanometric cutting at practical cutting speeds. Since the computational
time in MC simulation should be significantly less than that in MD, the number of atoms
in the simulation can be increased considerably and/or the cutting speed can be reduced
to conventional cutting speeds.
The objectives of this investigation are the following:
1. To develop Monte Carlo method of simulation of nanometric cutting process using
2D or orthogonal machining model. TIle method is intended to reduce the
computational time significantly as well as reduce the cutting speeds to practical
values. This simulation is based on the hypothesis that at every stage of a physical
process, the system goes through local minimum potential. Consequently, the
simulation is designed to take the entire toolworkpiece system to a minimum
potential
2. To demonstrate MC simulation of nanometlic cLltting at practical cutting speeds of 25
ms1
3. To validate the results of MC simulations of nanometric cutting with conventional
MD simulations at a cutting speed of 500 ms1
4. To estimate the temperature in the cutting region for MD and MC simulations using
the thermal model of the metal cutting process developed by Komanduri and Hou
[1214].
27
5. To conduct MC simulation of nanometric cutting of single crystal aluminum in the
(DOl) crystal orientation and [100] cutting direction using three rake angle tools,
namely, 10°, 30°, and 45° at a cutting speed of 500 ms I
.
6. To investigate the effect of tool rake angle on the cutting force, thrust force, force
ratio, and specific energy, in MC simulation of nanometric cutting of aluminum at
500msl
.
28
Chapter 4
Monte Carlo Simulation of Nanometric Cutting
4.1 Introduction
Monte Carlo method has been applied extensively to numerous complex
problems. Initially, it was used in the game of chances in gambling but currently it is llsed
routinely in such diverse application as engineering, medical, chemical, and physical
systems. The basis for Me method is that the physical system should be described as a
probability density function. Monte Carlo method has proved to be an effective way in
reducing the total computational time from N 2 for MD simulation to the order of
N log(N) for Me, where N is the number of atoms used in the model. For example, for
a system of 10,000 atoms the total computational time in Me simulation is reduced by a
factor of 2,500 times then MD simulation. Komanduri et al [40] have shown that
combined MCDT method used for nanomctric uniaxial testing of FCC metals takes 6.3
times less time than MD simulation for a system of 2,500 atoms. This difference
increases significantly as the number of atoms considered increases.
Most MD simulations of nanometric cutting are conducted at very high cutting
speeds (l00500 rnsI) to reduce the computational time. The major part of the
computational time is associated with the numerical integration of the classical
Newtonian equations of motion for the interacting particles to obtain new configuration
in space over a certain period of time. If some of the calculations are not needed than it
should result in significant savings in the computational time.
29
In MC method, the computational time is reduced because the velocity parameter
is not involved directly. The position of the atom is not computed using Newtonian
equations but is based on random Markov moves as described in the following section
(Sec. 4.2) of this Chapter.
4.2 Me simulation of nanometric cutting
Figure 4.1 is a schematic of the MC simulation of nanometric cutting. The
workpiece atoms are divided into movmg atoms and boundary atoms. Unlike in MD
simulation, there is no need for the peripheral zone in MC simulation. The use of
peripheral atoms in MD simulation is to dissipate the heat generated during the cutting
process. In other words, to reset the high value of velocity of the moving atoms
corresponding to high temperature or high energy. The velocity reset function that
operates upon the atoms in the peripheral zone serve to represent the bulk effects upon
the energy that would present for an extended lattice model. In Me, velocity term in not
directly involved. Hence, there is no need for peripheral atoms in the simulation.
The purpose of boundary atoms in the workpiece is to simulate the bulle The
boundary atoms of the workpiece are not allowed to move during any time in the
simulation. Thus, the interaction between any two boundary atoms is not calculated. All
tool atoms are designated as boundary atoms, or in other words, the tool is considered
infinitely hard. Also, the boundary atoms of the tool do not interact with the boundary
atoms of the workpiece.
30
Rake angle
Cutting direction
,...
Work rnateri.al
I
Clearance angle
• Boundary atoms o Moving atoms
Figure 4.1 Schematic ofMe simulation of nanometric cutting showing various regions of
interest
Since nature always drives any system to its most stable configuration, in MC
simulation, we select randomly a configuration in 3D for the minimum potential. A
common method to handle this type of problem involves the generation of random
positions of the workpiece atoms by Markov chain using the Metropolis sampling
procedure instead of totally random selection.
Consider an initial configuration of qo for an N particle system. The Boltzmann
distribution function, p(qa) for this system is given by
(4.1)
where V and T are the potential and the temperature, respectively for corresponding
location qo in space, K is the Boltzmann constant, and C is a proportionality constant. A
subset of m particles from a total of N particles is now selected and the coordinates of
these particles are randomly varied. For the i 'h atom, the new configuration is obtained
using the following equations:
31
(Xi) /leW = (X; ) old + (0.5  ~;I),ix
(y;) new =(y; )Old + (0.5  ~i2 )~y
(Z;)lIew =(ZJold + (0.5  ~;JLiz
(4.2)
(4.3)
(4.4)
where, (Xi) lIew' (Y, )new ,and (z,. )lleU.,' are the new configurations in space and (x; )old' (y; ) old
and (z; ) old are the old configurations for t" atom. fu, ~y, and /j,z are the maximum
step sizes along X, Y, and Z directions, respectively and are chosen randomly. Their
values are chosen by the user by an iteration technique.
Generally, the maximum step size, ~x = .6.y = /j,z lS taken the same in all
directions. An example of such a process is the computation of the diffusion rate of a
hydrogen atom through a perfect crystalline solid in which the three primitive translations
are equivalent [55]. For such a system, there is no reason to choose different maximum
step sizes for different directions. On the other hand, if the Monte Carlo simulation is
executed to simulate uniaxial tension, we might reasonably expect that the maximum step
size pennitted in the loading direction, /j,z to be di fferenl from the cOITesponding
maxima in the transverse directions, .6.y and Liz, which most likely be taken to be the
same [55].
Now, 111 a system of N particles, a subset of m particles is sampled and the
coordinates of these particles are changed randomly. This is called sampling. Here, m IS
very small compared to N. In MC simulations, the value of m is typically in the range of
1 to 5 atoms in a system comprising of several thousand atoms. If we choose In and the
maximum step size to be very large, we have the possibility of rapidly sampling an
regions of the configuration space for all particles in the system. As a result, the change
in the potential ~v will be very high and eV(qo)/KT('!o) very small. Consequently, the
32
acceptance of the given Metropolis move will be very low and thus a great deal of
computational time is wasted because of the rejected moves. On the other hand, if we
take the maximum step size to be very small, ilV wi)] be very small and the probability
of acceptance will be very high and we may be accepting unnecessarily most of the
moves. These two extremes are, therefore, not beneficial. The values chosen should be
such that the acceptance to rejection ratio (a / r) is in the range of 0.7 ~ a/ r ~ 1.3 . In this
investigaiton, the value of m is taken as 1 and the value of maximum step size as 0.2 A.
~il '(2 and c;i3 are a set of random numbers generated between 0 and 1.
Depending on the ratio of ~'eljPold , this new configuration point will be either
accepted or rejected which is equal to eM(qo)/KT(qo). The move is accepted, if
(4.5)
where, ~j is the random number generated between 0 and 1 in the i fh Morkov move and
Consideration of the MaxwellBoltzmann criterion eM(l/u)/KT(l/o) ~ ~i will show
that all moves leading to lower potential are accepted. In other words, if tlV < 0, the
move is accepted. Thus, the system eventually moves to the minimum potent:al. On the
other hand, if the change in the potential is positive, i.e. the potential rises. Then the value
of el>V(qo}/KT(qo) should be greater than the random number generated, c;i for the particular
Markov move to get accepted. The move is accepted with a probability that decays
exponentially with increasing tlV. If the move is rejected old position of the atom is
retained. This entire cycle is called a Markov move. The Markov moves are repeated
until a global minimum is reached.
33
Two methods are generally employed in selecting a particular subset of In
particles to be moved in a given set. In the first method, the m particles in the subset are
selected randomly in each move from the N palticles in the system. In the second
method, some systematic method of selection is used that ensures that all particles are
moved before any particle is moved again. For an infmite number, both methods become
equivalent. But for a fInite number, one method may produce rapid convergence than the
other. In the present Me simulation of nanometric cutting, one atom is chosen randomly
at any time and is given a Markov move. This process is repeated until the local potential
is conserved.
Cutting direction
Zone 1;,E
Zone 2
Rake angle
Clearance angle
Figure 4.2 Schematic showing two zones in Me simulation
In the cutting process there is much more disturbance or displacements of the
atoms in workpiece at or near the tool than the atoms further down in the workpiece. In
other words, the change in potential is more at or near the tool tip in the workpiece than
farther down in the workpiece. The workpiece atoms near the surface are removed from
the bulk in the fonn of chip and are moved to an elevated potential. Hence, more number
34
of Markov moves are required for the atoms in the cutting region. The Monte Carlo step
is divided into two zones. Firstly, an atom is cllosen from the atoms from the cutting
region (zone 1) as shown in Figure 4.2 and is given a Markov move. Tills process is
repeated until local potential is conserved. Secondly, an atom is chosen randomly from
the workpiece, including those interacting with the tool and is given a Markov move.
This entire process is continued until the overall system potential is minimized.
The empirical potential used for the MC simulation is a pairwise Morse Potential
as it is found to be more suitable for the FCC materials [40]. So, the potential of the entire
toolworkpiece system is the sum of pairwise summation of potential between the lattice
atoms of the workpiece and a second summation of pairwise Morse potentials between
the atoms of the tool and the atoms of the workpiece.
(4.6)
where rij is the i  j inter particle distance
(4.7)
(4.8)
(4.9)
vt (rij) =0 for rij > r; , and NWp and NT are the total number of atoms in the workpiece
and the tool, respectively, V::' and vt are the pairwise Morse potential for the
workpiece atoms and the workpiece  tool (interface) atoms, respectively; re is the
35
equilibrium distance between two atoms; D is equilibrium dissociation energy, and a lS
the range ofinterac6on.
36
Chapter 5
Application of the Thermal Model of Metal Cutting to MD and MC Simulations
5.1 Introduction
In metal cutting with a sharp tool there are two principal heat sources namely, the
shear plane heat source at the primary shear zone and the frictional heat source at the
toolchip interface [1214J. In the following, thermal modeling of these two heat sources
developed by Komanduri and Hou [1214] will be briefly reviewed as this model IS
adopted in present investigation to estimate the temperature III the cutting region m
nanometric cutting.
5.1.1 Shear Band Heat Source
M~lcriaJ 110\\
(a) 1TlIIJd for lhermal anulysi, of wor~
"JiJh~lie
Jnlllgc heill source
_"'_ I
J /t WI, f:
~/: , "/Milter'liI,l'low
Il1laglllary part ..J
(h; ITJO(JcJ LJrtht'l'fllal allilly,i, 01 ..hip
(Kolll<lllduri <lila HOll's modd)
Figure 5.1 Schematic of the analytical model of the cutting process for the determination
of the temperature rise in the chip and workmaterial caused by the shear plane
heat source, after Komanduri and Hou' s [12]
37
Kornanduri and Hou [12] presented a thennal model for the determination of the
temperature distribution in the chip and the workmaterial caused by the shear plane heat
source in machining. Hahn [51] originally derived the general equation for a band heat
source moving obliquely in an infmite medium. Komanduri and Hou [12] modified this
equation for application to a semiinfinite body for the oblique moving shear plane heat
source. An advantage of this method is that it is not necessary to make an explicit a priori
assumption regarding partitioning of heat between the workrnaterial and the chip, as was
common in most of the work reported in the literature. Instead this information is
provided as part of the solution.
Komanduri and Hou [12] introduced a new approach to thermal model of the
metal cutting process. They divided the analysis into two parts, namely, the workmaterial
side and the chip side of the shear plane and then combined to determine the temperature
distribution in the workmaterial and the chip. The workmaterial (or the chip) is extended
beyond the shear plane (as an imaginary region for continuity) to detemline the
temperature distribution in the workmaterial (or the chip) near the shear plane as shown
in Figure 5.1. The imaginary regions are the regions either of the workmaterial that was
cut by the cutting tool prior to this instance and became the chip or will be cut by the
cutting tool prior to becoming the chip. An appropriate image heat source with the same
intensity as the shear plane heat source is considered for each case. The temperature
distributions in the chip and the workmaterial were determined separately by this method
and combined to obtain isotherms of the total temperature distribution in the
workmaterial and the chip.
38
For continuous chip formation under orthogonal machining conditions, the shear
plane heat source is moving in a semiinfinite medium with the work surface and the chip
surface being the boundaries of a semiinfinite media. Thus, Hahn's oblique moving heat
source solution (see Figure 5.2) was modified with consideration for the effect of the
boundaries and use of appropriate image sources. The following equation was derived to
calculate the temperature distribution.
x
Ii
semiinfinite medium
"primary (uhlique ham.!) he'lll soun.:c
I.
Figure 5.2 Schematic of Hahn's model of a band heat source moving obliquely in an
infinite medium [12].
(5.1)
Please refer to the nomenclature for the definition of various parameters used 111 this
investigation.
39
The thennal model developed by Komanduri and Hou [12] was compared with
experimental data reported in the literature (Loewen and Shaw [53], Trigger and Chao
[52], Leone [54], Nakayama [56]) and found to be in good agreement.
5.1.2 Frictional Heat Sonrce at Toolchip Interface
Chao and Trigger [47] conducted pioneering studies on the thermal aspects of
metal cutting. For the case of frictional heat SOUTce at the chiptool interface they showed
that heat partition and consequent temperature rise at the chiptool interface on the chip
side and tool side cannot be matched by assuming unifonn distribution of heat intensity.
This is because the temperature rise distribution caused by a moving band heat SaUTee for
the chip is quite different from the unifonn stationary rectangular heat source tor the tool.
Therefore, nonuniform heat partition along the chiptool interface should be considered.
Chao and Trigger [47] developed an analytical model that incorporated classical solution
of Jaeger's moving band heat source (for the chip) and stationary rectangular heat source
(for the tool).
Chao and Trigger [47] proposed two approaches to address the variable heat
partition problem, namely, functional analysis approach and an iterative method. They,
however, preferred the later method as the former method is based on cutdry approach.
Komanduri and Hou [13], on the other hand, determined the heat partition and
temperature distribution in the moving chip and stationary tool due to frictional heat
source at the chip tool interface in metal cutting analytically using the functional analysis.
It takes into account appropriate boundary conditions and considers nonurriform
distribution of the heat partition fraction along the toolchip contact for the purpose of
matching the temperature distribution in the chip side and tool side.
40
Figure 5.3 is a schematic of the heat transfer model for the frictional heat source
at the toolchip interface on the chip side. The interface frictional heat source relative to
the chip is a band heat source moving with a velocity, of Veil' Considering the heat
partition fraction for the chip to be B, the heat liberation rate Bq of the movingband heat
source is considered totally transferred into the chip. Thus, the interface boundary is
considered as adiabatic and the solution used is for a semiinfinite medium.
.y;
o
L ..dxi xi
x ~
...''T~+...,..::=~.f_;;,,~~.'/
/
.,
.,J"'='h
Ich
)" )
dii Ii
z ...
Chip IChtQ?:'1
dli Ii
Ri'
o
M(X,z)
x
Figure 5.3. Schematic showing the heat transfer model for the frictional heat source at the
toolchip interface on the chip side considering it as a movingband heat
source problem (b) Schematic showing the heat transfer model of the
frictional heat source at the toolchip interface 011 the tool side considering it
as a stationary rectangular heat source problem [13]
Komanduri and Hou [13] considered nonunifoml distribution of heat partition
fraction along the toolchip interface for the purpose of matching the temperature
distribution on the chip side and the tool side. The total temperature rise at any point in
the chip caused by the entire moving interface frictional heat source, including its image
source, is given by
L
eM = q~: { fe(Xli)1'120[Ko(Rjv/2a) + Ko(R;v/2a)]dli
}
lfA 1,=0
(5.2)
41
Kornanduri and Hou [13] applied the thennal model to two cases of metal cutting
at macro level, namely, conventional machining of steel with a carbide tool at high PecIet
number (NPe ~ 5  20) [52] and ultraprecision machining of aluminum with a single
crystal diamond at low Peclet number (NPe >:::; 0.5) [52] and reported good agreement
between analytical and expelimental results.
5.1.3 Combined Effect of Shear Plane Heat Source and Toolchip Interface
Frictional Heat Source
Figure 5.4 is a schematic of the heat transfer model showing the two principal
heat sources, namely, the shear plane heat source AB and the chiptool interface frictional
heat source OA operating simultaneously on a common coordinate system. By
considering that part of the heat flow from the shear plane through the chip and the toolchip
interface into the tool to be continuous, Komanduri and Hou [14] found the effect of
the shear plane heat source on the temperature rise at the toolchip interface on the chip
side as well as on the tool side to be the same. Thus, the contribution of the shear plane
heat source on the temperature rise at the toolchip interface was included in the total
temperature rise for both sides.
Referring to Figure 5.4, both the shear plane heat source and the toolchip
interface frictional heat source moves relative to the chip at the chip velocity of Veil but in
an opposite direction to the chip flow. As the entire shear plane heat source is under the
upper boundary surface of the chip except for point B (which is at the boundary surface
all the time), an image heat source A'S was considered, Komanduri and HOll [14] used an
oblique moving band heat source for an infinite medium for both plimary shear plane
42
heat source and its image heat source. Similarly, the total temperature rise at any point in
the tool caused by the two principal heat sources also consist of two paIts. One is the
frictional heat source at the to01 chip interface from the tool side and the other is due to
shear plane heat source. However, a different heat transfer model is needed to address
this problem. They considered that pmt of the heat coming from the shear plane heat
source through the chip and the toolchip interface into the tool, acting as a stationary
heat source located at the tool chip interface. Thus heat source is considered as induced
stationary rectangular heat source caused by the shear plane beat source.
\I.:h
,/
0'
z
::4l"i:;;;...=:.=;=::=;l,./
'
R.i' /// dli Li
/~ f Upper surface
,oflhe chip
IImage h~atsourcc~
A' / 7 " '   ~v")J7
';(cjwi
,,S,.hcarp,.Ianc....., ~/
heal sounx
+ \
Chip [CiiipJ1OW] i. \1
...~ ~_\___+l!~_._......:;.;,l.l...dl~Ji__;lkhJ
X A ilt'>......".L () /r
I
TOOIChiP interhu.:c / Lheal source , ./'
'1'001 ///'
,/
~_.......
Fi gure 5.4 Schematic of the heat transfer model with a common coordi nate system for the
combined effect of two principal heat sources [14]
The analysis developed by Komaduri and HOll [14] is eXlensive and somewhat
involved hence, here only the final results of that investigation are given. For details the
reader is referred to References 1214.
43
The temperature rise due to frictional heat source at the toolchip interface is
given by
L
eM = qpl {(BchiP 1:18) Je(XI,)VI20[Ko(Riv/2a) + K o(R;v/2a)]dl; +
JrA I,~O
21:18 J(!J...)lIIe{XI,)VI2a[Ko(Riv/2a) + K o(R;V/2a)]dli +
1,=0 L
eM f(1:Ye(X,,)VI2°[Ko(R;VI2a)+Ko(R;vI2a)]dl;} (5.3)
1,=0
The temperature rise at any point in the chip including all points along the toolchip
interface caused by the shear plane heat source is given by
K v ~ 2 2 o[ (XX;) +(2trh zz;) ]}dwi 2a
The total rise in the temperature at any point in the workpiece is given by,
q L e =~{(B. 1:18) fe(XI,)VI2"[K (Rv/2a)+K (R:v/2a)]dl + M JrA ,/1Ip 0 , 0, ,
1,=0
21:18 J(i)'''e(XI;)vJ2°[Ko(R;v/2a) + Ko(R;v /2a)]dl; +
1,=0
L I
Clill J(1/e (XI,)VI2"[Ko(R;vI2u) + Ko(R;V/2a)]dl;} +
I, =0
'</, 1 co~(¢Ct) :;; J e(XX,)vI 2a {Ko[;a ~(X _X;)2 +(ZZ;)2]+
Wi =0
K v ~ 2 2 o[ (X Xi) + (2t ch ZZi) ]}dw;
2a
44
(5.4)
(5.5)
Komanduri and Hou [14] applied the thennal model to two cases of metal cutting
at macro level, namely, conventional machining of steel with a carbide tool at high Peelet
number (NPe ~ 5  20) [52] and ultraprecision machining of aluminum with a single
crystal diamond at low Peclet number (NPe ~ 0.5) [52]. TIley found good agreement
between the analytical and the experimental results.
5.2 Application to MC Simulation of Nanometric Cutting
As pointed out earlier, the thermal model developed by Komanduri and Hou [14]
was verified with the experimental results reported in the literature at macro level. In this
investigation, this model is applied for the first time to nanometric cutting. To check the
viability of this model, MD simulation of nanometric cutting of single crystal aluminum
at a cutting speed of 500 ms1was conducted and results were compared with the thermal
analysis of the metal cutting process at nanoscale.
Region of
observation
Cutting Direction
...
Depth of cut
Clearance angle
Figure 5.5 Schematic showing the regIon of observation in the cutting zone for
temperature estimation
45
For this analysis, a small region ahead of the tool in the cutting zone (see Figure
5.5) was selected and observed continuously during the entire simulation. The width of
this region is the same as the width of the workpiece. The temperature in this region was
calculated at every tool movement using the following equation [59]
p 3 m vl T=La
,
;=02 K
(5.6)
where P is the number of aU the atoms falling in the region considered, ma is the mass
ofthe atom, v, is the velocity of the i,h atom in the same region.
The size of the region is such that there is enough number of atoms in the region
to obtain a good statistical average value of temperature.
1.2
.0
.Il
0.0
 Morse potential
Harmolil.c approirimanon
1 o 2
Bond length
5
Figure 5.6 Comparison of Morse potential and Harmonic Oscillation
Usually, some 50 to I00 atoms are considered to obtain a good statistical average.
For simplicity Eqn. 5.6 assumes the harmonic relation for the oscillation in the workpiece
46
atoms instead of the Morse potential (see Figure 5.6). The cutting and thrust forces were
obtained from the MD simulation of nanometric cutting [Figures 5.7 (a) and (b)].
90 , .
, , .
80   .  ;.   :   : ..     ~ .   .:  . ·· ., ..
70 ~ : ~ ; ~ ..
• I I •
I • I •
600 800 1000
Time (t.u.)
200 400
I I I , .................... ~ : : ~ : .
, . . . .
. ,
......... __ .; __ .: : ; : ..
I. • I
,. I.
• , I • .. ,' ~ · .. · ., .. . ..
....................... .. ." .. ··, ... .., o I__.L...:... ____+:.... ~ ____:._J
o
10
20
Z 60
Eo
~50 ;;;
LL
g' 40
E
6 30
(a)
600 800 1000
Time (l.u)
 ~ . . . .. ~  ..   .. . ~. . . .   . . .
200 400
· , _ •• __ .... ~ __ •• ~ •••• _ ••'. _ • _ •• _ ~ •••• II. __ • __ • _ ...... .. _ • _ •• ~~ _ •• __
• • • • I
·• .• .• .• .I   .       . • • I • • ~ .... ..... . ~ . .   ..  .. ~ ·· .. . .. ..,.   ...   ..... ,.... ...      ......  ..  . • , I I •  ..    ~  : .   .  .   ; .  .     .. ~ ..  ..     .: . · . . . .
• I • • • · .
· . . . . ................... ~ ." , i ..      ~  .     . ·· .,..
5 ....  ...
., ., • .. , •••   •••   ,_  •••   • ~ ~ •  ••••••••• 'I     •    ••
.., I.. t.. '.. o !.__...J~ .......;.:.... ~. ;._.1
o
45
40
35
Z 30
.s
III 25
~
.E
v; 20
2
.c f 15
10
(b)
Figure 5.7 Variation of (a) cut6ng force and (b) thmst force (moving average values)
with time in MD simulation of nanometric cutting of single crystal
aluminum. Cutting speed: 500 ms· l
, rake angle: 10°, depth of cut: 5.IA, and
width of the workpiece: 43 A
47
Figure 5.8 Merchant's Force Circle for Orthogonal Cutting
These forces were resolved along the shear plane and the tool face using Merchant's
force circle (see Figure 5.8 for details) to detemline the intensities of the heat sources at
the shear plane and the toolchip frictional interface, respectively. The resolved forces
were then used to calculate the heat generated at the shear plane and the toolchip
interface. The heat generated at the chiptool interface is given by,
(5.7)
The heat generated at tbe shear plane is given by,
(5.8)
where Fir is the frictional force acting along the chiptool surface, Fe is the cutting force,
L is the length of frictional heat source, Vch is the chip velocity, w is the width of cut,
and AB is the length of the shear heat source( t / sin ¢ ).
48
In MD simulation, the tool is considered infinitely hard. So, the atoms do not
vibrate at their respective equilibrium positions. In other words, there is no molecular
kinetic energy in the tool atoms which is reflected by the fact that temperature rise in the
tool is zero in the entire simulation. Or, in other words when we apply the lhen11al model,
the tool does not share the heat generated at the chiptool interface and as a first
approximation the chip carries away all the heat. So, when Eqn. 5.5 is applied, M =0
and Beh;p =1.0. Thus, Eqn. 5.5 reduces to
8M = qpl {fe(XI;)VI2G[Ko(RjvI2a)+Ko(R;VI2a)]df;}+
ll'A. 1,=0
'"/cos¢a) ;; f e(XX,)v/2a {Ko[~ ~(X  XJ2 + (z  Zj)2] +
w,=o
(5.7)
The temperature rise in the cutting region was calculated usmg Eqn. 5.7 and
compared with the MD simulation results. It can be seen from Figure 5.9, the agreement
is reasonably good. It is well known that temperature in the cutting zone increases with
cutting speed. For example, Shaw [60] developed a simple relationship between cutting
temperature and cutting velocity,
(5.8)
Once the cutting temperature in the cutting region is known, it can be substituted
in the MaxwellBoltzmann equation that detcnnines the acceptancerejection criterion.
This way cutting velocity effect is indirectly introduced into the Me simulation of
nanometric cutting. Thus Me simulation of nanometric cutting can be conducted at
conventional cutting speeds, say at 5 ms· l
, as adopted in this investigation. To conduct a
49
similar simulation by MD, it will take a very, very long time even with the fastest
computer available. Appendix A. gives an example calculation of the temperature
generated in the cutting zone using the thennal model developed by Komanduri and Hou
.......  _..~ •...• " .. . ~  " .. ·t~Brmal·niodel·.;" ; _ .
600 800 1000 1200
Time (t.u)
200 400
· . ,
.. 6 .. _:_ _ _:_ ~ _: _ ; _. _ .. __ .. ·· .. . ·· ..
[14].
1400
1200
1000
~
~
(l) ~ 800
::J
~
(l)
0. 600
E
(l)
I
400
200
0
0
Figure 5.9 Comparison of the temperature generated in the clltting region near the tool
tip between the thennaJ model after Komanduri and Hou [14] and the MD
simulation
5.3 Cutting Velocity in MC Simulation
Since there is no implied velocity in the MC simulation, the momentum in the
chip is zero. Consequently, the chip tends to spread sideways. Alternately, the boundary
atoms attempt to pull the chip atoms down towards them. Hence, the chip atoms find
minimum potential on the sides near to the workpiece atoms rather than the atoms
moving up in the direction of the chip velocity as in MD simulation. Figure 5.1 0 shows
the side spread of the chip in the absence of momentum in the chip in Me simulation of
nanometric cutting.
50
Two approaches were considered to circumvent this problem. The first one is to
set reflecting boundaries along the Y2 plane, so that the chip atoms find their minimum
potential in the direction of the chip velocity rather than near the workpiece.
Figure 5.10 (a) Side spread due to absence of momentum In the chip In the Me
simulation of nanometric cutting
Figure 5.10 (b) shows the effect of reflecting boundaries on the side spread of the
chip. We can see that the side spread has been considerably restricted with the use of
reflecting boundaries. However, the disadvantage of this method is loss of acceptance to
rejection ratio (air), which means
5l
Reflec~ng boundaries
'I
Figure 5.10 (b) Effect ofreflecting boundaries on Me simulation
the system is not reaching its minimum potential, thus not simulating the actual physical
process.
The second method is to switchoff the interaction of the chip atoms with the
boundary atoms once chip starts fonning. This way, the boundary atoms do not pull the
moving chip atoms towards them, thus avoiding the side spread. It can be seen from
Figure 5.10 (c) that very little or no side spread ofthe chip occurs in this case.
52
Fig. 5.10 (c) Effect of switchingoff of the interaction between the chip atoms and the
boundary atoms
An additional, advantage of this method is that there is no loss of acceptance to
rejection ratio, which means the system will reach its global minimum position. In this
investigation, the second method was chosen, for the Me simulation of nanometric
cutting, for this reason.
53
Chapter 6
Algorithm for :vIC simulation of Nanometric Cutting
The following is the step by step substantiation of MC simulation of nanometric
cutting using the Morse potential:
1. The workmaterial and the tool atoms are set In space, with units In angstroms,
according to the crystal structure defined by the user (in this case single crystal
aluminum workpiece, FCC crystalline structure in (001) plane and cutting in [100]
direction. The workpiece is divided into two parts, namely, the moving atoms and the
boundary atoms. The boundary atoms of the workpiece are not allowed to move any
time in the simulation, i.e. the boundary atoms do not subjected to random moves.
The tool is set up at a distance where is out of reach of cutoff distance of the
workpiece atoms. Since, the tool is considered infinitely hard, all the atoms in the
tool are designated as boundary atoms.
2. We know that atoms down in the workpiece are out of the cutoff radius of the tool.
So, to reduce the computational time, a list of workpiece atoms is created that could
possibly interact with the tool atoms during the entire simulation.
3. The next step is to calculate the bond list for individual atoms based on the cutoff
radius. The atoms which are father than cutoff and less than three times the cutoff
radius are considered as probable atoms which may interact with a given atom. All
atoms farther than that cause negligible potential and hence are neglected. The atoms
of the tool are not allowed to interact with the boundary atoms of the workpiece. The
54
bond list of each workpiece atom contains a count of all the workpiece atoms as well
as the tool atoms that are within its cutoffradius.
4. Room temperature is allocated to all the moving atoms at the start of nanometric
cutting and the potential of the system is calculated.
5. Monte Carlo steps are executed until the system reaches its minimum potential. In
each Monte Carlo step, one atom is selected for the Marokov move and is given a
new random position. Change in potential is calculated due to a change in the position
of the atom and according to the criteria fulfilled by MaxwellBoltzmann inequality,
the move is either accepted or rej ected.
6. The bond list is updated during the Monte Carlo step according to the need. For
example, when an atom is moved during the Monte Carlo step to a new position, new
atoms may fall in its cutoff radius and/or old atoms in the existing bond list may go
out of its cutoff radius. If the move is accepted as per MaxwellBoltzmann criterion
the bond list has to be updated for all the atoms. If the move is not accepted, the
original position of the atom is regained and no change in the bond list is needed.
7. Forces are now calculated along the X, Y, and Z directions. From this, the cutting
force and the thrust forces are determined. Using Merchant's force circle diagram, the
forces along the shear plane and the toolchip interface were resolved. Using these
forces, the heat intensities at the shear plane and the chiptool interface were
calculated by using the thermal model of conventional cutting by Komanduri and Hou
[14]. The temperature rise in the cutting zone at a given cutting speed is estimated.
This temperature is substituted in the MaxwellBoltzmann inequality during the
Markov moves to determine, if the move has to be accepted or rejected. Thickness of
55
the chip and length of contact of the chip with the tool are calculated with the use
from the simulations.
8. The reallocation of temperature is done for the atoms in the cutting region.
9. Steps 3 to 8 are repeated until the specified final position of the tool is reached.
56
Chapter 7
Results and Discussion
The mam objective of this investigation is to conduct Monte Carlo (MC)
simulations of nanometric cutting and obtain the cutting and thrust forces, force ratio,
frictional force on the tool face, shear force on the shear plane and specific energy, as
well as subsurface deformation in the workmaterial and compare the results with MD
simulation. MC simulation of nanometric cutting was carried out for a single crystal
aluminum in (001) crystal orientation and [100] clltting direction with different rake
angle tools, namely 10°, 30°, and 45° at a cUlting speed of 5 ms· l
.
The simulation generates the two components of the force, namely, the cutting
force (Fe) and the thrust force (FI
). The forces are calculated in terms of electron volts
per unit angstrom. The conversion factor used to obtain the forces in nano Newton is 1
eV / A = 1.6021 nN. The force ratio is given by the ratio of the thmst force to the cutting
force. The specific energy is the energy required for removal of unit volume of
workmaterial and is given by cutting force / (width of the workpiece) x (depth of cut)]
7.1 Comparison Between MD and Me Simulation of Nanometric Cutting at a
Cutting Velocity of 500 ms·1
7.1.1 MD Simulation of Nanometric ClItting
MD simulation of nanometric cutting was perfOf:l1ed on a single crystal aluminum
in the (100) plane and [001] cutting direction at a cutting velocity of 500 ms·l
. Table 7.1
57
gives the workpiece and tool dimensions used in the simulation. Morse potential is used
for the simulation (see Table 2.1 for details of the Morse parameters used).
Table 7.1 Workpiece dimensions and cutting parameters used in MD and MC simulation
ofnanometric cutting of single crystal aluminum at a cutting speed of 500 ms I
Configuration ,2D orthogonal cutting
Work material Dimension 11a x 50a x lla, along X,Y, and Z directions a is
the lattice constant of the work piece
Tool Dimension 22a x lOa x lOa, along X,Y, and Z directions ais
the lattice constant of the tool
Depth of cut 4.05 A5.l A
At the beginning (see Figures 7.1 (a)(d» very little deformation ahead of the tool
is observed. As cutting continues, plastic defonnation in the shear zone can be seen in the
workpiece. As tool proceeds further, dislocation generation and propagation in the work
piece [Figure 7.1 (d)] can be seen. Figures 7.2 (a) and (b) show the variation of cutting
and thrust forces with cutting time. The temperature in the cutting zone is calculated
using Eqn. 5.6, for the case of hannonic oscillation.
Figure 7.3 shows the variation of the temperature rise in the cutting region with
cutting time. From this, the mean temperature rise is estimated to be around ~900 uK or
the cutting temperature of 500°C during the cutting.
58
(d)
(c)
. f .' f alun>iIlUlfi with a \0' lake tool at a
fig
UleS
1.I (a){d) MD s\roulatiO~ at wnous ~~es 0 nan
oroetnC
cut\l!lg 0 .
cutting'le\octty of 5()() !us
V'la
160 r:.....
140
120
z
c
 100
(1)
2~
80
OJ c
·B 60
:::J
U
40
     ,   :  : :. :  ..  f.   ...
I I , • • _  _ _ . _. __  _ _._
I • , • •
• •• I , ................. .. : ~ ; : ~ ; ..
• • I I I . . • I I I ................ .. : ~ ~ ~ : .. ··· . ... . , . .. .. .. .. : ~.. .. .. .. .. .. .. ~ .. .. .. .. .. .. .. .. .. , ..
20 I I. I I
.... .. 0 I ..
200 400 600 800 1000 1200 1400
Ol.L........:.....:":........:......:.__.....:. ...J
o
Time (Lu)
(a)
(b)
200 400 600 800 1000 1200 1400
. . ..........  ..   .. , ..   ..,     _.. ..   
. .
............................... _ " ' "J> ..
, I I I
. .
____ _ \.. _ _ 1_ _' ~ ~ .. J .J _ _ .. .. .. L _ _ _ _ ..
, I , I I I
.    : ..    .. :    : :   .. ~ .. ~    ;. ..    .  ..
120
100
Z 80 .
EQ)
()
~ ..0.... 60
iii
2
.!: 40
I
20
0
0
Time (Lu)
Figures 7.2 (a) and (b) Variation of the cutting force and thrust force, respectively with
cutting time in MD simulation of nanometric cutting of
aluminum.
60
200 400 600 800 1000 1200 1400
. .
.... .. : ~ : : ; : ..
·• • .·• ..• I •
I • I • .... .. :" ~     ..  .. ~ ~  ..
............ .. ," " ,•. I ,• "',I'" .
, ··I ..• ..I .I · . .
, • I I I • .............. ... ' ~ "'" ..  ' "" ..  ..
• I • • • •
··, .I ..I I
1400
1200
:2
1000 Q) I 800
.3
~
Q) Q. 600
E
Q)
I 400
200
0
0
Time (t.U)
Figure 7.3 Variation of the temperature rise in the cutting zone in MD simulation of
aluminum at a cutting velocity of 500 ms· 1
7.1.2 Me Simulation of Nanometric Cutting
MC simulation was performed 011 a single crystal aluminum workmatelial at the
same cutting speed of 500 ms· 1 with a 10° rake tool. Figures 7.4 (a)(d) show MC
simulation at various stages of nanomctric cutting of aluminum at 500 ms· 1
• It can be seen
that the nature of deformation ahead of the tool and subsurface defonnations, in both Me
and MD simulation, are very similar. Figures 7.5 (a) and (b) show the variation of cutting
and thrust force, respectively, with cutting time. The temperature in the cutting region
was calculated after each tool advancement using the thermal model of Komanduri and
Hou [14]. Figure 7.6. shows the variation of the temperature in the cutting zone with
cutting time. A mean temperature of 1000 OK was estimated during the cutting process.
61
~
cJ . 0 ak \ t a . ...."0 of aluminu'" ""tn a lOt e \00 a
. . t",a Sof nanornetrtC cU\..u>"::;l
figure, 1 A (a)(d) Me ,unulatlon at vaIt
ou
' ~ ~e
cutt1
U
g ve\OC\t'j of 50
0
tIl!:> .
On comparing Figures 7.5 (a) and (b) and Figures 7.2 (a) and (b) it can be seen
that both the cutting force and the thrust force are higher in Me simulation than in MD
simulation. Consequently, the specific energy and temperature generated at the cutting
.......... • p·· ..  . .. .. · . . . , . .... ·.. ,".".."..
. . ,. ""
........................... ......................... ........ 4 ~ ' ~ ~ ..
• .• ..• .• ,• I .• · .. ..,.. .. ............. ~ _> ~ J.... .. ~ ~ _:_ ~ _ · " . I'.. .•.••.•.
" '.. . .. _ ... :....  ..:.... ··:·······i·······; .. ···:·:···· .. "':':···
• • , , fl. · "" • ,.. I
• ,.. I •
, • • • I •
............. t'...... .. . .., " r"""""" ~ ..
" .,... , .".'".. ,. ..
120
100
Z 80 S
Q)
(,)
u0.. 60
Ol cE
::J 40
()
20
0
0 200 400 600 800 1000 1200 1400 1600 1800
Time (l.u)
Figure 7.5 (a)
800 1000 1200 1400 lGOO 1800
Time (l.u.)
200 400 600
.. __ .... _ .·... _ _ _ • _.. • _ ..... e ..... __ .• .............. _ ...... _,_ • ... e • _ .. _ .... _ .... e ..... _ .. ., ., , ., .,
• I •• • I
" .. ....",
..... _ ..'_ •••.e _ • "" __ ... _ • I ~ ..... " •• _' • ... ... _ • _ • __ • _ ....... e ...,.. .""""... .. ",. ,. ,, "..
· . . • _ • _ •• J _ • __ .. .. _' ••• __ • _ ... .. _ ... ~ · . . e e .... ... ' .... • _ ... L • _ • .. , . , , ·· ., " ., , .' , , , .' · , , .,.'    . ~ : . .: ~ .. .;. : .  : .  . ;   : .
, I • • .. , • • · . .. . ... .. . .
120
100
Z 80 E.
Ql
U
u0.. 60
(;j
::J .c 40
~
20
0
0
Figure 7.5 (b)
Figures 7.5 (a) and (b) Variation of cutting and thrust forces, respectively with cutting
time in MC simulation of aluminum at cutting velocity of 500
msI
in MC simulation will be higher than in MD simulation (see Figure 7.6). While the exact
reason for this is known at the present time this may be attributed to the absence of
63
momentum in the chip in MC simulation. Similarly, we observe more fluctuations in the
MD data than MC data. In the MD data, all the chip atoms are repelled simultaneously
by the tool atoms as represented by its fluctuation in forces. But in MC simulation, the
atoms in the chip are closer to the tool atoms in simulation and the chip atoms do not
experience repulsion simultaneollsly. Consequently, they experience greater force.
1400 . __.
1200
.. ..
600 ' •..•.• ; •.•.•.•:••••.••:.•.•••• ; •.•.•• ~ •...•.•:.••.•.•;•••...• ; •..•••
• • I • 1 "
·• .I . ..,. .•
400 ~~.  _. .: _ : _. ., _.. ~ _ : _ :._ .. _ ; ..
• • • I I • • I
• • I • I • I 1
• • • I • • , I
200 ~ : : ; ~_ : : ;. ..
• I • I I • I •
• • I • • • • •
I •• I".
~ 1000
:::.:::
~ 800 :::l ro....
Q) a.
E
Q)
f
800 1000 1200 1400 1600 1800
Time (t.u)
200 400 600
OLI"__........_ ........__l.. ......._....J
o
Figure 7.6 Variation of temperature rise in the cutting zone with cutting time in MC
simulation with 10° rake angle tool and a cutting velocity of 500 ms'\
Table 7.2. Comparison between MD and MC simulation for nanometric cutting of
aluminum at 500 ms·1 and 10° rake angle (workpiece dimensions are given in Table 7.1)
MD MC
Cutting force per unit width (nN/A) 1.255 2.53
Thrust force per unit width (nN/A) 0.9583 1.17
Force ratio 0.7635 0.4624
Width of cut (A) 40.5 43.13
Depth of cut (A) 4.1 5.l
Specific energy (N/mmL
) * I OJ 0.306 0.4956
Computational Time (hrs) 810 1416
We observe the computational time taken by MC simulation is more than MD
simulation. But, the cutting velocity in the simulation used was 500 ms· l
. The point to be
64
noted is the Me calculation is independent of cutting velocity of the tool. But the same is
not true for MD simulation. If we reduce the cutting velocity of the simulation from 500
msI to 5 ms·) the total computational time for MC simulation would remain the same.
But for MD simulation it would take at least 1015 times more for this configuration [40J.
Ifwe want to simulate nanometric cutting at 500 ms· 1 we probably would not want to use
MC simulation as there is no gain in the computational time and the speeds are way too
high anyway.
Table 7.3
Computational parameters used in the MC simulation of nanometric cutting of single
crystal aluminum at cutting velocity 5 ms· 1
Configuration 3D cutting
Work material Dimension Sa x 25a x ISa , a lattice constant of work piece
Tool Dimension 20a x Iia x lIa, a latti ce constant of tool
Depth of cut 5.1 A
7.2 MC Simulation of Nanometric Cutting of Aluminum at a Cutting Speed of 5 IllS·1
MC simulation of nanometric cutting on a single crystal of aluminum for crystal
oriented in (001) plane and cutting of[100] direction. Three rake angles, namely 10°, 30°,
and 45° were used. Figures 7.7 (a)(d), Figures 7.8 (a)(d) and Figures 7.9 (a)(d) show
various stages of the nanometric cutting of aluminum with 10°, 30°, and 45° rake tools,
respecti vely.
7.2.1 Rake Angle of 10°
Figures 7.7 (a)(d) shows Me simulation at various stages of nanomctric cutting
of a single crystal aluminum work material with 10° rake angle at cutting velocity of 5
·1
illS .
65
Figures 7.7 (a)(d) Me si%ulation at various stages of nanornetric cutting of aluminuJdJ.,;th a 10° rake tool and a
cutting velocity of 5 msI
0\
0\
As the tool proceeds into the workpiece we observe fonnation of primary shear
zone and some subsurface defonnation in the machined surface. We also observe elastic
recovery of the atoms in the machined surface as the tool moves away fonn the machined
surface just generated [Figure 7.7 (d)]. The high elastic recovery can be attributed to the
basic principle of Monte Carlo, namely, the system attempts to reach minimum potential.
The basic principle of minimum potential, also explains the crystalline structure observed
in the chip during nanometric cutting instead of highly deformed structure in the chip
either in conventional cutting and/or MD simulations.
7.2.2 Rake Angle 30°
Figures 7.8 (a)(d) show vanous stages of nanometric clltting of aluminum
workmaterial with a 30° rake. We observe similar features, such as plastic defol1natiol1 in
the shear zone and subsurface deformation in the machined surface. The thickness 0 f the
chip is less and the extent of deformation is also less than that with a 10°
67
0\
CXl
(b)
(c)
Figures 7.8 (a)(d) Me simulation at various stages of nanometric cutting of aluminum. with a 30° rake tool at a
cutting velocity of 5 msI
Cd)
(a)
(c)
Figures 7.9 (a)(d) Me simulation at various stages of nanometric cutting of aluminum with a 45° rake tool at a
cutting velocity of 5 msI
0\
\0
7.2.3 Rake Angle 45°
Figures 7.9 (a)(d) show various stages of nanometric cutting of aluminum with a
45° rake tool. We again observe similar characteristics, such as plastic defoffilation in the
shear zone and subsurface deformation in the machined surface. The high elastic
recovery can be attributed to the basic principle of Me, namely, the system attempts to
reach minimum potential.
Table 7.4 Results of MC simulation of nanomelric cutting of aluminum at 5 ms 1 for
various rake angles, namely 100
, 30°, ar:.d 45° (see Table 7.3 for workpiece
dimensions)
rake angle Cutting Force Thrust Force Force ratio Specific Energy
(degrees) per unit width per unit width xl05 N/mm2
(N/mm) (N/mm)
10 3.0253 1.854 0.613 0.593
30 2.0885 1.128 0.54 0.384
45 1.5556 0.796 0.512 0.305
7.2.4 Effect of Rake Angle on Cutting and Thrust Forces per Unit Width
Figures 7.10 (a) and (b) show variation of the cutting force and the thrust force,
respectively with cutting time for three rake angles, namely, 10°, 30°, and 45°. [t can be
seen that both cutting force and thrust forces decrease as the rake angle increases. The
70
1,5 2
~
. ... ' . _.. , .(a"Ke '(ingre '4"5' . ~ , "'" . ··· ..· ..
0.5
.. : _ .. _: .. __._ : __ . __ .. _ . ··· .. .. · .. .. , . .
120
140 r.
rake;angle10 : ·•· •.·••·•••. ,,:l:"9::30~~·· ••• ··· ...
Q) 80 u
L a
u..
0> 60
c
:8
::J
0 40
20
0
0
~ 100
Z
c
Time (t.u.)*10e5
(a)
70...,
45
60 _.•...•   •. , , •..........•.•..'  •  ....•  . L •••••••••••••••
..... ,.;~'0~ .... , _ 50
Z
c
10
Q) 40
~
a
1.L
Vi 30
:..J..
.e
f 20
0.5 1.5 2
O+......r:.~+__++___1
o
Time (t.u)*10e5
(b)
Figures 7.10 (a) and (b) Variation of cutting and thrust forces with cutting time in Me
simulation ofnanometric cutting for different rake angles at a
cutti.ng speed of 5 ms· 1
71
Table 7.5 Results of MC simulation of nanometric cutting of aluminum at 5 ms I for
various rake angles (see Table 7.3 for workpiece dimensions)
rake angle Shear force per Shear angle Frictional Force
(degrees) unit width (degrees) per unit width
(N/rom) (N/mm)
10 2.46 14.54 3.30
30 1.48 23.84 2.25
45 0.844 34.24 1.67
3.5,,
. . .. . ....  ,  .. 1 .. .. .. .. .. .. .. .. .. .. .. .. .. __'0__ _ ..
• I • I • I • .,.. I,
.• . • I .,,. . I
....  :  ~ ~  ' ~  ~ ; ..  ~      
• • • • • I • 3      .'   ..  ..      L      w',. ~ ........ .I .. _ .. _ .. _ L ... .'.... _  _ .I ...... L .. _ ..
·: : : : : : Cutti~g For~ : , . . . ~ . . .
_._ ...:..  .... ~_ .. _~, ~.:.. _._._~_._._.~_ ..
• I • • • • • • •
I •• ..
I •• ~
Ql
u~
1
:? 25
Zc
'.;; 2
0
~ 1.5
5 10 15 20 25 30 35 40 45 50
• • • I I • 0.5 " .•.••:.  " ••. ~ •••..• : ••..••:. _•••• ~ ..•••  :    •  :      ~   .• " " :   •. " "
·• .• .• .• .• •, •. I
• • • • • I I o++'';+'+;.....+'+.++1
o
Rake angle (degrees)
Figure 7.11 Variation of cutting force per unit width and thrust Force per unit width with
rake angle in MC simulation at a cutting speed of 5 ms I
7.2.5 Effect of Rake Angle on Force Ratio
Figure 7.11 shows the variation of force ratio (ratio of the thrust force to the
cutting force) with rake angle. The force ratio decreases from 0.6638 to 0.496 as rake
angle increases from 10° to 45° as shown in Figure 7.11 and Table 7.4
72
0.62, ...,
0.6  ~   .....    .. ~ ..... 
, .   ....   ..    ...     .. ...  ..  , .  ....     .~      .  ..  
. , . o58 .   ..     .•• ~ .   . . . . . .. . ~  .    .•.  •  . ~   .      ..  :  ....  .     
.Q
~
Q) 056
() oLL
054
0.52
. . ..........   _...   ..     .. .'" .. _ ".' ..           ..
...... .. ~ ..   ~  ..   a:.  .. _  .
, .
_ _~_ __.:_ .. _ .. __ :.. _.~_.__ .
20 30 40 50
Rake angle (degrees)
10
0.5 1;........+;.\
o
Figure 7.12 Variation of the force ratio with rake angle in Me simulation of nanometlic
cutting at a cutting speed of 5 ms I
7.2.6 Effect of Rake Angle on Specific Energy
OJ ,,
• _ • • _ • • J __ • • oJ .... ••• '. _ • _ • • " •• _ • • _
,• ,I ,, •
, . ,
........................ J .J __ .. _ .. , .
, , , . ........................ , .. .. .. .. .. .. .. .. .. .. ""  • ", I""    · ,.
........... _,• _ I _ ,I • .
06 .......•...............  .:      ..    . :.... .  ....  : _.        .
N
E 0.5
E
Z
';: 0.4
~
Q)
~ 03
!E
u~
0.2
(/)
, .
0.1 ~ ~ ~ ~ .. ~ ~ ~. ~ ~ ~ ~ ~ ~ ~. ~ ~. ~ ~ .. ~ ~ ~ ~ ..... ~ •• ~ .... ~:~ ~ .•...      .:•.. ~ .•   ~ _. _.. · . . .
20 30 40 50
Rake angle (Degrees)
10
0+._.....1
o
Figure 7.13 Variation of specific energy wi lh rake angle in Me simulation of nanometric
cutting at a cutting of aluminum speed of 5 ms· j
73
_.~~~_._
Figure 7.13 shows the variation of specific energy with rake angle. It can be seen that the
specific energy decreases from 0.593 to 0.309 x 105 N/mrn2 as the rake angle increases
from 10° to 45°. (Table 7.4 details)
7.2.7 Effect of Rake Angle on Shear Force per Unit Width and Frictional Force per
Unit Width
Figure 7.14 shows the variation o[ frictional per unit width and shear forces per
unit width with rake angle. It can be seen that the frictional force per unit width and the
shear force per unit width decreases as the rake angle increases from 10° to 45° (See
Table 7.5 for details).
3.5..,....,
20 25 30 35 40 45 50
Rake angle (degrees)
5 10 15
3  . _.  [      ~ .   , .  :  _.  ~·Frlc~iorlaITorce· 1' _. 
_ 2.5  .. _; ••.  ..; .••. ;_ •• : ·_··f; ; .. .;
~ : :::':::
.zs , ""'" i 2  ' ., " '. '~;~~~f~~~~'7::] 1 .. :
"a; 1.5 ,.  _.: _.... .  .   :     :  •. _.. :  _.   •  ..  .:    _. : ..    : ,   
~ I •• r • • , • o . .'"".
LL 1 _ •• ~ •••••• ~~ •••• ~. •• : ••• ~ •.•.•• ~ •• .; ••  •• pi •••• : ••••••
• • • • I •
• • I • " I I
I • • • " "
0.5 ; ~   i      ; . _ f· .. · f· · .. ~ .. _ ;  ~  
I • • I I • I I •
• • • • I , • • •
• • • I • , • I •
0+"';'i';"";""';'"T.i'....;..i._.;..I
o
Figure 7.14 Variation of the frictional force per unit width and the shear force per unit
width with rake angle in Me simulation of nanometric cutting at cutting
speed of 5 rns1
7.2.8 Effect of Rake Angle on Shear Angle
Figure 7.15 shows the variation of shear angle with rake angle. It can be seen that
shear angle increases with increase in rake angle. This is because the chip thickness
74
decreases with increase in rake angle as can be seen from the Me simulations of
nanometric cutting with different rake tool.
40 T""",
 ..  ·  , ,   .. · , ·· , .. . . . ., ... .... , ." , ..  r . _  
• I ~ • ..
" . ,    .. ~ ;. : ~ ;. :  ..  ..:  ..
• I I ..
, ' .
.. .. .. .. _ · . _ . . ,or. __ _
• I I •
I • I I
· . . " . . ...... _ .. _, _ _ .. _ _ L. _ 1_ J I. _ .. _" ..
• • ( • I • • I • · .. "
" . 35 ..    :...    ; •.     i·· _. _. r" .:.  _., .; ... _. i····· :.....  ,'. _.,.,
• I I • • · "" • • • • • , • I
.......... _' _ .' __ _ _". _ J 1.. .. .. _coO"" _ .. _
• , • I • I •
• I • I • • •
• • I • • · "
r;; 30
Q)
~
as> 25
82
20
OJ c
C1l ro 15
Q)
..c
(J) 10
20 25 30 35 40 45 50
Rake angle (degrees)
5 10 15
.. .
5· ...    ~ .     ~    .  . :   ..  . ~ ...  ..:.  ..  . ~   .  .  :  ..    ~ ..  .. :.     
, . ....
• .., I
• ••• I oliiii'i...;.'...;.'...;.'j.'l
o
Figure 7.15 Variation of the shear angle with rake angle in MC simulation of nanometric
cutting at a cutting speed of 5 ms I
7.2.9 Temperature Rise in the Cutting Zone
450 T"""...,
1.2 14 16 18 2
, "
o.J . _ .' . ,,.' o.J _ .. • I. • I
02 04 0.6 0.8
• I • I I • • • ,
...... : ~ : ~ :  "" .;_ .; :  ~ ..   _! · . . ., .,
• I • " .,
, • I • • • I • • ......... "," 1  , ," ..  r  , i , ~ ..  
I • , • • • I • •
I I • • I I , ,
, • I , • , I , •
.......... ·. , ,     ,. ..   ..  ., T  ••• "I.   .. , . . . . , .. .... .,
· ,
............• .............I.......... _• I  ..
• ·I .• • , ,
350
 ,• • I • • ~  .,  , .  ..
• I • , I • , •
• I • I , • • •
• I • • • • I • 100 ....  :  _.   :   _.  .:    .. ~ .. _.  :.   .  :.    _.:    .  :.  .. , ~.  •.  
I · '" I. SO : ~  : ~ ~ : ~ :" ~  .. . I...
• " I I o +_+_1'i.~_i___+_+'+__+_';'1
o
400
~ 300
Q) !:i 250
'§
~ 200
E
~ 150
Time (LU) *e5
Figure 7.16 Variation ofternperature rise with cutting time MC simulation ofnanometric
cutting at a speed of 5 illS·
1 with 10° rake angle
75
Figure 7.16 shows the variation of temperature rise In the cutting zone with
cutting time. It can be seen that temperature in the cutting zone at 5ms· l is 400 K or the
cutting temperature of 100°C and the graph looks more stable. This is in contrast with
the average temperature rise in the cutting zone 1000 K or the temperature of 700°C at
500ms·) .
76
Chapter 8
Conclusions
In this investigation Monte Carlo (MC) simulation of nanornetric cutting of
aluminum single crystal using Morse Potential was conducted at a cutting velocity of 5
ms 1 to Shldy ultraprecision machining (UPM). The temperature in the cutting zone in the
MD or MC simulation of nanometric cutting at a given clltting speed was estimated using
the thennal model of the cutting process developed by Komanduri and HOll [14]. This
temperature is substituted in the MaxwellBoltzmann acceptancerejection criterion.
Thus, cutting speed is indirectly used in the MC simulation via the temperature in the
cutting zone. This chapter summarizes the conclusions and offers some suggestion for
future work.
Monte Carlo Simulation of Nanornetric cutting
Monte Carlo method of nanometric cutting developed in this investigation
incorporates Markov random walk, metropolis acceptancerejection criterion, and the
thermal model of the cutting process developed by Komanduri and Hou [14] to estimate
the temperature in the cutting region. The model incorporates cutting velocity effect in
MC simulation via the temperature in the cutting region. A code in C programming
language was developed to implement this method.
77
Monte Carlo simulation of nanometric cutting Was conducted on single crystal
aluminum workmaterial in the (00l) plane and [100] cutting direction at a cutting
velocity of 500 ms· 1 as well as at a practical cutting speed of 5 ms·! .
At a cutting speed of 500 ms· 1
? the nature of deformation ahead of the tool, the
dislocation generation and propagation, and the temperature in the cutting region in Me
simulation were fund to be in reasonable agreement with MD simulation conducted under
the same cutting conditions.
MC simulations of nanometric cutting were conducted at 5 ms· 1 using three rake
angle tools, namely, 10°, 30°, and 45°. The variation of cutting force, thrust force, force
ratio, frictional force on the tool face, shear force on the shear plane, shear angle, and
specific energy with tool geometry was investigated.
The clltting and thrust forces were found to decrease with increase in rake angle.
The cutting force per unit width decreases from 3.0223 x 102 N/mm at a rake angle of
10° to 1.5556 X 10'2 N/mm. at rake a angle of 45°. Similarly, the thrust force per unit
width decreases foml 1.854 x 10.2 N/mm at rake angle of 10° to 0.796 x 10.2 N/mm for a
rake angle of 45°. The rate of decrease for cutting force was found to be 1.389 times than
that of thrust force. The force ratio was found to decrease from 0.6638 to 0.496 as the
rake angle increase from 10° to 45°.
Similarly, the specific energy was found to decreases from 0.593 N/mm2 for a 100
rake angle tool to 0.305 N/mm2 for a 45° rake angle tool.
Frictional force per unit width also decreases form 3.0223 x 10.2 N/mm for 100
rake tool to 1.67 x 10.2 N/mm rake 45° tool. The shear angle increases from 14.5412 to
34.245034 as rake angle increases [rom 10° to 45° . Shear force decrease from 2.46 x 10'2
78
N/mm to 0.844 x 102 N/mm as rake angle increases fonn 10° to 45° . All the trends
described above were found to be in accordance with the experimental as well as MD
simulation results, reported in the literature
Thermal Model
The thennal model of conventional machining developed by Komanduri and Hou
[14] for conventional machining was incorporated in MC and MD sUnulations of
nanometric cutting of aluminum to estimate the temperature generated in the cutting zone
ahead of the tool. Results of the thennal model of cutting at nanoscale were compared
with the MD simulation ofnanometric cutting and found to be in reasonable agreement.
The temperature rise in the cutting zone was found to be 1000 K when cutting
aluminum at a cutting velocity of 500 ms 1 and ~400 K when cutting at a cutting velocity
of5msl
.
Future Work
1. There is ample scope for optimization of the Monte Carlo simulation of nanometric
cutting. The computational time for Monte Carlo simulation of nanometTic cutting
can be further reduced by using the damped trajectory (DT) or the steepest descent
technique.
2. Cutting velocity is incorporated in the MC simulation is indirectly as a function of
temperature. In the future, the relationship between temperature and cutting speed has
to be firmly established over a wide range of cutting speeds and for different
79
workmaterial. This way, simulation ofnanometric cutting can be conducted routinely
for different metals and over a range of cutting speeds.
3. One Disadvantage of Me simulation is absence of momentum in the chip. In the
present investigation some approaches are given to address this problem. More
alternatives have to be considered to overcome the problem or addressing absence of
momentum in the chip. For example, a mathematical formulation involving forward
and inverse integration of the Hamiltonian equation proposed by Kennedy et al [41 j
may be used at conventional speeds for temperature characterization. The resulting
functional fonn can be incorporated as a criterion for acceptancerejection in the Me
method. The second choice would be to use Newtonian equations of motion only on
the chip, instead of applying it on the entire workpiece, as we do in MD simulation, to
produce the effect of momentum in the chip.
4. To further reduce the computational time, use of parallel processmg of Me
simulation of nanometeric cutting should be considered using massive parallel
processors. We have built a 26node parallel processor at OSU and this system should
be used using appropriate parallel processing program or software.
5. In this investigation, MC simulations were conducted on a perfect single crystal
workmaterial with no defects. However, real workmaterials deviate from this
assumption. It is, therefore, necessary to incorporate various types of defects in the
wokmaterial for a better understanding of the nanometric cutting process.
6. The MC method can be extended to other cutting and tribological applications, such
as indentation, grinding, nanometric cutting with negative rake angle tool, oblique
80
cutting, extrusion, laser machining. Also Me method can be extended to microscale
cutting, instead of nanoscale cutting, as in UPM.
7. The thermal model applied can be extended to negative rake angle tool to simulate
cUitting with negative rake angle tools or to oblique machining to simulate cutting
with tools of different inclination angles.
81
References
1. Komanduri, R., Lucca, D. A., and Y. Tani, "Tec1mological Advances In Fine
Abrasive Processes," Annals ofCIRP, 41/2 (1997) 545596.
2. Naylor, T. J., Balintfy, J. L., Burdick, D. S., and K. Chu, "Computer Simulation
Techniques," Wiley, New York, 1966.
3. Alder, R, and T. E. Wainwright, "Studies 111 Molecular Dynamics.1. General
Method," J of Chemical Physics, 31 (1959) 459.
4. Alder, 8., and T. E. Wainwright, "Studies in Molecular Dynamics.2.Behavior of a
Small Number of Elastic Spheres," J of Chemical Physics, 33, (1960) 1439.
5. Komanduri, R., and L. M. Raff, "A Review on the Molecular Dynamics Simulation
of Machining at the Atomic Scale," Proc. Inst. Mech. Engrs. Vol. 215 Part B.
6. Metropolis, N., and S. M. Ulam, ''The Monte Carlo Method," J Am. Stat. Assoc., 44
(1949) 335341.
7. Chandrasekaran A., "Monte Carlo Simulation of Uniaxial Tension," M.S. Thesis,
Mechanical and Aerospace Engineering, Oklahoma State University, Stillwater OK
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Appendix A
Example Calculation of the Temperature in the Cutting Zone
In the following, an example calculation of the temperature generated m the
cutting zone using the thennal model developed by Komanduri and Hau [14] is given.
MD or Me simulation of nanon1etric cutting generates data on cutting force and thrust
force. First, the chip thickness is either measured from simulation or calculated using
geometric relation of the cutting process. Once the chip thickness, t,,,, is known, the
shear angle is calculated using equation
tan ¢ = rcosa 1(1  r sin a)
where, r is the ratio of depth of cut, t" to chip thickness, teh , and ex is rake angle
Then length of the shear plane is calculated by the equation
AB = t eh /cos(¢  a) or AB = te Isin¢
(A 1)
(A2)
Then shear force and frictional force are calcu lated by the use of Merchant's
Circle as follows
Fs =Fc cos CAfI.  F f sin tdp.
Chip velocity is calculated by the equation
v =V *r eh e
(A3)
(A4)
(AS)
The resolved forces were then used to calculate the heat generated at the shear
plane and the toolchip interface.
The heat generated at the chiptool interface is given by,
q pi  Ffr *Vell /L *w
89
(A6)
and the heat generated at the shear plane is given by,
(A7)
L is the length of contact at the toolchip interface and its maximum length is defined by
the user based on experimental data correlating depth of cut to length of contact between
the tool and the chip during the simula60n.
Using the following equation developed by Komanduri and Hall [14], the
temperature calculations are done.
L
eM = q~; {fe(XI/)VI2