Modeling of the tool edge radius effect on the mechanics of micromachining

MODELING OF THE TOOL EDGE RADIUS EFFECT ON
THE MECHANICS OF MICROMACHINING


















WOON KENG SOON

NATIONAL UNIVERSITY OF SINGAPORE

2009

MODELING OF THE TOOL EDGE RADIUS EFFECT ON
THE MECHANICS OF MICROMACHINING













WOON KENG SOON
(B.Eng. Hons.)

A THESIS SUBMITTED

FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

DEPARTMENT OF MECHANICAL ENGINEERING

NATIONAL UNIVERSITY OF SINGAPORE

2009



i

Acknowledgements


My PhD journey has never been easy. Despite the long and exhausting

working hours during term time, there was also constant mental stress during
semester breaks. It was undoubtedly difficult at the beginning. But as time goes by
the stress became less burdening, as it transformed from worries into a constant urge
of seeking for truths and answers.
I then realized the PhD journey is not just a
scientific quest, but also a training course to prepare oneself for a much greater
journey ahead - a lifelong self-learning, self-discovery and self-challenge expedition
of his or her own.

In this journey of mine, I wish to express my heartfelt gratitude to my teacher,
Professor Mustafizur Rahman for his unconditional support and guidance and for
being the source of inspiration. His remarkable influences on me will be timeless
and without boundaries.

For university staffs that helped in experiments and FE-analysis, I hope they
know how much I value their generosity. Specifically, KS Neo, CH Tan, Nelson
Yeo, SC Lim, YC Ho, CL Wong and Simon Tan of AML; Alvin Goh and Joe Low
of IML; and EH Yeo from SVU. My thanks are also due to Drs. F.Z. Fang, K. Liu,
J H. Ko and Narrisara of SIMTech for their valuable advice. Special thanks to my
colleagues for their help in different ways and more importantly for their friendships.
They are Subra, Masheed, Sharon, Angshuman, Indraneel, Pervej, Li Tao, Poh
Ching, Shaun, Rajenish, Lingling, Hesham, Haiyan, Liu Yuan, Liqing, Anwar,
Biddut, Ibrahim, Sonti, Chin Seong, Masahiro, Michael and Zhigang.

Moreover, I would like to thank my parents for their precious love and
encouragement that have kept me strong to face numerous challenges. I am also
deeply indebted to my elder sister and brother for their understanding of my long
absence from home throughout these years.

ii

Table of Contents


Acknowledgements

i
Table of Contents

ii
Summary

vi
List of Tables

viii
List of Figures

ix
List of Symbols

xviii
Chapter 1 Introduction

1
1.1 Background

1
1.2 Scope and Objectives

8
1.3 Thesis Outline

11


Chapter 2 The Review of Literature

14
2.1 Specific Cutting Force and Energy

14
2.2 Chip Formation Mechanism

17
2.3 Surface Finish

19
2.4 Minimum Undeformed Chip Thickness

23
2.5 Contact Phenomenon

25
2.6 Finite Element Method in Machining Simulation

28

2.7 Summary

31


Chapter 3 Finite Element Modeling: An Arbitrary Lagrangian-
Eulerian Method

33
3.1 Limitations of Pure Lagrangian and Eulerian Methods

33
3.2 The Advantages of Arbitrary Lagrangian-Eulerian Method

35


iii
3.3 Governing Equations

36
3.3.1 Description of Kinematics

38
3.3.2 Conservation Equations

41
3.4 Working Principles

42

3.4.1 Mesh Layout

42
3.4.2 Mesh Smoothing

43
3.4.3 Solutions Advection

44
3.5 Deformation Behavior

47
3.6 Contact Characteristics

48
3.7 Heat Generation and Conduction

49
3.8 Model Configuration and Boundary Conditions

50


Chapter 4 Experimental Verification

56
4.1 Technical Challenges

56
4.1.1 Mechanical Aspects

56
4.1.2 Optical Aspects

57
4.2 Machine Tool

58
4.3 Cutting Tools

58
4.4 Workpiece Material

60
4.6 Experimental Setup

61
4.7 Experimental Design

63
4.8 Model Validation

64
4.8.1 Tool-Chip Contact Length

65
4.8.2 Deformed Chip Thickness

67
4.8.3 Machining Force 69

iv

4.9 Summary

72


Chapter 5 Contact Phenomenon

74
5.1 Material Separation and Frictional Shear Contact

74
5.2 Contact Stress and Stick-Slide Regions

80
5.3 Stick-Slide Characteristics

85
5.4 Tool-Chip Contact Length: A Linear Regression Analysis

92
5.5 Tool Wear Phenomenon of Edge-Radiused Cutters

96
5.6 Summary

99

Chapter 6 Chip Formation Behavior

101
6.1 Transitional Chip Formation Behavior

101
6.2 Deformation Intensity of the Primary Deformation Zone

107
6.2.1 Width of PDZ

109
6.2.2 Thickness of PDZ

113
6.2.3 Depth of PDZ

117
6.3 Deformed Chip Thickness: A Linear Regression Analysis

122
6.4 Summary


127
Chapter 7 Tool Edge Radius Effect: A Non-Dimensional Analysis

129
7.1 Tool-Chip Contact Length

129
7.2 Deformed Chip Thickness

133
7.3 Machining Force

137
7.4 Summary

143
Chapter 8 Extrusion-like Chip Formation Mechanism: A Sensitivity
and Behavioral Study
145


v

8.1 A Mesh-Refined Arbitrary Lagrangian-Eulerian Model

145
8.2 Sensitivity Analysis: Nodal Displacement Vectors

147
8.2.1 Initial Chip Formation

148
8.2.2 Chip Growth 151

8.3 Behavioral Analysis: Stress Distributions

162
8.3.1 Principal Stress

164
8.3.2 Hydrostatic Pressure and Shear Stress

167
8.3.3 In-Plane Principal Stress

171
8.4 Discussions

179
8.4.1 The Stress Model

181
8.4.2 Shear Stress and Le-Chatelier’s Principle

183
8.4.3 The Role of Compressive Stress

185
8.4.4 Surface Roughness and Extrusion-like Mechanism

189
8.4.5 Summary

196

Chapter 9 Conclusions

198
9.1 Contact Phenomenon

199
9.2 Chip Formation Behavior

200
9.3 Tool Edge Radius Effect

200
9.4 Extrusion-like Chip Formation Mechanism 201

9.5 Future Work

202
Bibliography
205


Appendix
216


Publication List
242

vi

Summary


The mechanics of tool-based micromachining is significantly different from
that of conventional-macro machining due to the tool edge radius effect. Such effect
arises in micromachining as the micron-scale undeformed chip thickness approaches
the size of the tool edge radius of several microns. A predictive cutting model that
assumes a perfectly sharp cutting edge yields reasonable agreement with
experimental results in conventional-macro machining. But such an assumption is
not valid in micromachining under the governance of the tool edge radius effect.
This phenomenon is especially profound in metallic materials, but its operating
mechanism is not fully understood at present.

This thesis presents a fundamental modeling study of the tool edge radius
effect in tool-based micromachining. Through an advanced finite element modeling
technique based on the arbitrary Lagrangian-Eulerian method, such effect is
quantified as the relative tool sharpness a/r, the ratio of undeformed chip thickness a
to tool edge radius r. The analysis was performed concurrently on both tool-chip
tribology and plastic deformation, as the main physical phenomena in metal
machining.

Findings of this study have revealed that the flow stagnation phenomenon
during material separation could be driven by the counterbalance of frictional shear
contact components on the tool edge radius, with a constant stagnation point angle
under different machining conditions. A contact model for micromachining was
proposed following the identification of three distinctive sticking and sliding regions
along the tool edge radius. With this contact model, the tool wear phenomenon of
‘edge-radiused’ cutting tools was reasonably elucidated.

Furthermore through the decrease of a/r from the above to below unity,
transitions in chip formation behavior were encountered. Material deformation
became increasingly more localized ahead of the rounded tool edge due to the effect
of tool edge radius on the contact length, which resulted in an increase of


vii
deformation intensity. At a critical a/r threshold below unity, the chip formation
mechanism transformed from concentrated shear deformation to a thrust-oriented,
extrusion-like behavior alongside the formation of an effective negative rake angle.
The extrusion-like mechanism was found to operate under severe deviatoric stress
alongside intense compressive stress and hydrostatic pressure as governed by the Le-
Chatelier's Principle. Thus the formations of continuous chip through such a
mechanism would prevent micro-void nucleation in the primary deformation zone
and produces high quality surface finish comparable to that in surface grinding.



viii

List of Tables


Table 3.1 Machining conditions, tribological parameters, tool and
workpiece properties.

55
Table 4.1 Mechanical properties of the WC-Co cutting tool.

59
Table 4.2 Machining conditions for experimental verifications.

64
Table 5.1 Summary of the stick-slide characteristic in tool-based
micromachining.

87
Table 5.2 Linear relationships between contact length, L
c
and undeformed
chip thickness, a for tool edge radius of 10 µm, at different
cutting speeds and tool rake angles.

94
Table 5.3 Linear relationships between contact length, L
c
and undeformed
chip thickness, a for tool edge radius of 5 µm, at different
cutting speeds and tool rake angles.

94
Table 8.1 Surface roughness at different a/r and cutting speeds (r =10 μm;
γ

= +10° ).

194

ix

List of Figures


Figure 1.1 Hydrostatic stress distributions along the shear plane in
conventional machining of metallic materials. (after Roth and
Oxley, 1972)

9
Figure 2.1 (a) Indentation force, R
n
and resultant force, R
c
induced by
induced by tool radius and tool face respectively. (b)

Differences in power, F
P
and feed components, F
Q
with
undeformed chip thickness in cutting force. (after Masuko,
1953)

15
Figure 2.2 Exponential increment in total specific energy with fine
reductions in undeformed chip thickness. (after Lucca, 1991)

16
Figure 2.3 SEM micrographs of chip root region produced with a
spring-operated quick-stop device on Al2024-T3, a = 37 μm
and r = 38 μm. (after Subbiah and Melkote, 2008)

19
Figure 2.4 Influences of feedrates and tool edge radii effect on ferrous
materials. (after Volger et al., 2004)

20
Figure 2.5 Varying machined qualities on silicon wafers at different
combinations of undeformed chip thicknesses and tool edge
radii. (after Arefin et al., 2004)

22
Figure 2.6 Relationships between cutting forces and chip loads on
pearlite.
(after Liu et al., 2004)

25
Figure 3.1 Referential domain Rχ, material domain RX, and spatial
domain Rx in the ALE framework associated with reference
coordinates χ, material particles X, and spatial points x
respectively. The domains are connected through single
mapping φ, Ф and Ψ. (after Stein et al, 2004)

39
Figure 3.2 Computational mesh layout. (a) One-dimensional; and (b)
two-dimensional.

42
Figure 3.3 (a) Deformations of computational mesh during chip
formation (b) undeformed state; and (c) deformed state.

43
Figure 3.4 Illustration of the Half-Index Shift (HIS) method.

46
Figure 3.5 Illustration of the building elements and work flow of the
arbitrary Lagrangian-Eulerian finite element model for tool-
based micromachining.
51


x

Figure 3.6 Coupled temperature-displacement elements. (a) 4-node
element for the workpiece; and (b) 3-node element for the

cutting tool.

52
Figure 3.7 Configuration of computationally meshed tool-workpiece
entities with predefined boundary conditions in the ALE
domain. γ = +10°, 0°, -10°; α = 6° and r = 5 µm, 10 µm are
determined from actual WC-Co cutting tools used in the
experiment.

53
Figure 3.8 Mean friction coefficient, µ
mean
of tool-workpiece under dry
condition.
54

Figure 4.1 Ultraprecision machine tool for micromachining
experiments.

58
Figure 4.2 Tool geometries of the micro-grain WC-Co cutting tool. It is
made up of carbide grains with an average size of 0.7 μm.

59
Figure 4.3 Schematic diagram of the complete experimental setup which
includes the orthogonal machining configuration, the
imaging, optical and lighting systems.

61
Figure 4.4 High speed images of tool-based micromachining for

different magnitudes of undeformed chip thickness, a
attempted.

63
Figure 4.5 Illustration of the complete experimental setup for tool-based
micromachining of medium carbon steel with cemented
tungsten carbide cutters.

63
Figure 4.6 Distributions of contact length, Lc for varying undeformed
chip thickness, a from 2 µm to 20 µm (with an interval of 2
µm), at cutting speed, V of 100 m/min.

66
Figure 4.7 Distributions of contact length, Lc for varying undeformed
chip thickness, a from 2 µm to 20 µm (with an interval of 2
µm), at cutting speed, V of 250 m/min.

66
Figure 4.8 Distributions of contact length, Lc for varying undeformed
chip thickness, a from 2 µm to 20 µm (with an interval of 2
µm), at cutting speed, V of 500 m/min.

67
Figure 4.9 Distributions of deformed chip thickness, tc for varying
undeformed chip thickness, a from 2 µm to 20 µm (with an
interval of 2 µm), at cutting speed, V of 100 m/min.

68

xi
Figure 4.10 Distributions of deformed chip thickness, tc for varying
undeformed chip thickness, a from 2 µm to 20 µm (with an
interval of 2 µm), at cutting speed, V of 250 m/min.

68
Figure 4.11 Distributions of deformed chip thickness, tc for varying
undeformed chip thickness, a from 2 µm to 20 µm (with an
interval of 2 µm), at cutting speed, V of 500 m/min.

69
70 Figure 4.12 Distributions of cutting force, Fc and thrust force, Ft for
varying undeformed chip thickness, a from 2 µm to 20 µm
(with an interval of 2 µm), at cutting speed, V of 100 m/min.


Figure 4.13 Distributions of cutting force, Fc and thrust force, Ft for
varying undeformed chip thickness, a from 2 µm to 20 µm
(with an interval of 2 µm), at cutting speed, V of 250 m/min.

70
Figure 4.14 Distributions of cutting force, Fc and thrust force, Ft for
varying undeformed chip thickness, a from 2 µm to 20 µm
(with an interval of 2 µm), at cutting speed, V of 500 m/min.

71
Figure 5.1 Flow stagnation point on the rounded curvature of tool edge
radius, r. (a) r = 33 µm; and (b) r = 2 µm. Stagnation point
angle, δ

s
is approximately 60°. (after Jaspers and
Dautzenberg, 2002)

75
Figure 5.2 Spatial velocities on the stagnation point and flow retardation
zone are zero and close to zero respectively. For a = 2-20
µm, stagnation point angles, δ
s
= δ
sv
. Examples are shown for
(a) γ = +10°, a = 10 µm, V = 100 m/min; (b) γ = 0°, a = 14
µm, V = 250 m/min; and (c) γ = -10°, a = 18 µm, V = 500
m/min.

76
Figure 5.3 Spatial velocities on the stagnation point and flow retardation
zone are zero and close to zero respectively. For a = 2-20
µm, stagnation point angles, δ
s
= δ
sv
. Examples are shown for
(a) γ = +10°, a = 10 µm, V = 100 m/min; (b) γ = 0°, a = 14
µm, V = 250 m/min; and (c) γ = -10°, a = 18 µm, V = 500
m/min.

78
Figure 5.4 Constant stagnation point angle, δ

sf
= δ
s
= 58.5°±0.5° for
undeformed chip thickness, a = (a) 20 µm; (b) 18 µm; (c) 16
µm; (d) 14 µm; (e) 12 µm; (f) 10 µm; (g) 8 µm; (h) 6 µm; (i)
4 µm; and (j) 2 µm at tool rake angle, γ

= +10°.

79
Figure 5.5 Constant stagnation point angle, δ
sf
= δ
s
= 58.5°±0.5° for
undeformed chip thickness, a = (a) 20 µm; (b) 18 µm; (c) 16
µm; (d) 14 µm; (e) 12 µm; (f) 10 µm; (g) 8 µm; (h) 6 µm; (i)
4 µm; and (j) 2 µm at tool rake angle, γ

= 0°.
79


xii

Figure 5.6 Constant stagnation point angle, δ
sf
= δ
s

= 58.5°±0.5° for
undeformed chip thickness, a = (a) 20 µm; (b) 18 µm; (c) 16
µm; (d) 14 µm; (e) 12 µm; (f) 10 µm; (g) 8 µm; (h) 6 µm; (i)
4 µm; and (j) 2 µm at tool rake angle, γ

= -10°.

80
Figure 5.7 Stick-slide interactions on 3 different regions over the tool-
chip and tool-work contact interfaces at different time steps, t
= (a) 30 µs; (b) 40 µs; and (c) 50 µs. Sticking is mainly
concentrated on the rounded tool edge while sliding happens
above and below the sticking region on the rake and
clearance faces. (Example is shown for γ

= +10°, a = 8 µm, V
= 100 m/min)

81
Figure 5.8
Distributions of (a) Sticking Region (b-x), Sliding Region I
(b-d) and Sliding Region II (x-z) on the cutting tool and; (b)
Corresponding frictional shear contact stress on (b-x), (b-d)
and (x-z). Stagnation Region (a-w) is contained within the
Sticking Region with the Stagnation Point (o) as the centre of
the regions while both Sliding Regions I and II are consisted
of primary and secondary regions

83
Figure 5.9 Impositions of high normal pressure perceived as indenting

actions by the rounded curvature of tool edge radius. (a) r

=
10 µm, γ

= +10°; (b) r

= 10 µm, γ

= 0°; (c) r

= 10 µm, γ

= -
10°; (d) r

= 5 µm, γ

= +10°; (e) r

= 5 µm, γ

= 10°; and (f) r

= 5
µm, γ

= +10°. Note that a larger tool edge radius imposes
greater density of normal pressure than that of a smaller tool
edge radius.

84
Figure 5.10 Development of tool-chip contact length, L
c
in two distinct
stages, Stage-I and Stage-II associated with the stick-slide
characteristics: (1) Sliding-I; (2) Sticking and; (3) Sliding-II.

86
Figure 5.11 Examples of time-history plot of contact length evolutions
for tool edge radius, r of 5 µm at cutting speed, V of 100
m/min: (a) γ = +10°; (b) γ = 0° and; (c) γ = -10°.

89
Figure 5.12 Examples of time-history plot of contact length evolutions
for tool edge radius, r of 10 µm at cutting speed, V of 100
m/min: (a) γ = +10°; (b) γ = 0° and; (c) γ = -10°.

90
Figure 5.13 Variations in the vector of normal contact pressure, σ
n
on tool
rake of different rake angles. (a) Negative rake angle, γ = -
10°; and (a) Positive rake angle, γ = +10°.

91
Figure 5.14 Linear evolutions of contact length with undeformed chip
thickness for tool edge radius of 10 µm.
93

xiii

Figure 5.15 Linear evolutions of contact length with undeformed chip
thickness for tool edge radius of 5 µm.

93
Figure 5.16 Crater and flank wear on an ‘edge-radiused’ cemented
carbide cutting tool. (after Kountanya and Endres, 2004)

97
Figure 5.17 Relationships between the tool wear phenomenon of ‘edge-
radiused’ tools and the contact phenomenon in tool-based
micromachining. (a) Experimental findings of Kountanya
and Endres (2004); and (b) Contact model for tool-based
micromachining as proposed in Section 5.2.

98
Figure 6.1 Transitions in plastic deformation behavior with decreasing
a/r, using different tool rake angles, γ
tool
of +10°, 0° and -10°
at the cutting speed, V of 100 m/min: (a-b) a/r > 1.0; (c) a/r
= 1.0; (d-e) a/r < 1.0.

103
Figure 6.2 Transitions in plastic deformation behavior with decreasing
a/r, using different tool rake angles, γ
tool
of +10°, 0° and -10°

at the cutting speed, V of 250 m/min: (a-b) a/r > 1.0; (c) a/r
= 1.0; (d-e) a/r < 1.0.

103
Figure 6.3 Transitions in plastic deformation behavior with decreasing
a/r, using different tool rake angles, γ
tool
of +10°, 0° and -10°
at the cutting speed, V of 500 m/min: (a-b) a/r > 1.0; (c) a/r
= 1.0; (d-e) a/r < 1.0.

104
Figure 6.4 Contour plots of von Mises stress distributions during the
transitional formations of effective rake angles, γ
eff
at
decreasing a/r. (a) Effective positive rake angle, +γ
eff
at a/r =
0.4; and (b) Effective negative rake angle, -γ
eff
at a/r = 0.2. A
nominal tool rake angle γ
tool
of +10° is used in both (a) and
(b).

106
Figure 6.5 Highly localized von Mises stress distributions and thus
plastic deformation ahead of the tool edge radius with the

formation of effective negative rake angles, -γ
eff
. Such
phenomenon is encountered for different combinations of
machining conditions at a/r = 0.2. (a-b-c) represents V of
500 m/min, 250 m/min, 100 m/min and (1-2-3) denotes γ
tool

of +10°, 0°, -10°.

107
Figure 6.6 Characterization of the primary deformation zone (PDZ)
from its width, thickness and depth.

109
Figure 6.7
Linear evolutions of WPDZ with undeformed chip thickness
using different tool rake angles and tool edge radius, at the
110


xiv
cutting speed, V of 100 m/min.

Figure 6.8
Linear evolutions of WPDZ with undeformed chip thickness
using different tool rake angles and tool edge radius, at the
cutting speed, V of 250 m/min.

111

Figure 6.9
Linear evolutions of WPDZ with undeformed chip thickness
using different tool rake angles and tool edge radius, at the
cutting speed, V of 500 m/min.

111
Figure 6.10
Linear evolutions of TPDZ with undeformed chip thickness
using different tool rake angles and tool edge radius, at the
cutting speed, V of 100 m/min.

114
Figure 6.11
Linear evolutions of TPDZ with undeformed chip thickness
using different tool rake angles and tool edge radius, at the
cutting speed, V of 250 m/min.

115
Figure 6.12
Linear evolutions of TPDZ with undeformed chip thickness
using different tool rake angles and tool edge radius, at the
cutting speed, V of 500 m/min.

115
Figure 6.13
Linear evolutions of DPDZ with undeformed chip thickness
using different tool rake angles and tool edge radius, at the
cutting speed, V of 100 m/min.

118

Figure 6.14
Linear evolutions of DPDZ with undeformed chip thickness
using different tool rake angles and tool edge radius, at the
cutting speed, V of 250 m/min.

119
Figure 6.15
Linear evolutions of DPDZ with undeformed chip thickness
using different tool rake angles and tool edge radius, at the
cutting speed, V of 500 m/min.

119
Figure 6.16 Illustration of the deformed chip thickness t
c
, as a collective
indicator of plastic deformation induced during chip
formation.

122
Figure 6.17 Linear evolutions of deformed chip thickness with
undeformed chip thickness for tool edge radius of 10 µm.

124
Figure 6.18 Linear evolutions of deformed chip thickness with
undeformed chip thickness for tool edge radius of 5 µm.

124
Figure 7.1 The response of normalized contact length, Lc/a to the ratio
of undeformed chip thickness to tool edge radius, a/r at
different tool rake angles, γ and cutting speeds, V for r of 10

µm.
130


xv

Figure 7.2 The response of normalized contact length, Lc/a to the ratio
of undeformed chip thickness to tool edge radius, a/r at
different tool rake angles, γ and cutting speeds, V for r of 5
µm.

132
Figure 7.3 Evolutions of normalized undeformed chip thickness, tc/a to
the ratio of undeformed chip thickness to tool edge radius,
a/r at different tool rake angles, γ and cutting speeds, V for r
of 10 µm.

134
Figure 7.4 Evolutions of normalized undeformed chip thickness, tc/a to
the ratio of undeformed chip thickness to tool edge radius,
a/r at different tool rake angles, γ and cutting speeds, V for r
of 5 µm.

136
Figure 7.5 Evolutions of normalized machining force, Ft/Fc to the ratio
of undeformed chip thickness to tool edge radius, a/r at
different tool rake angles, γ and cutting speeds, V for r of 5
µm.

140

Figure 7.6 Evolutions of normalized machining force, Ft/Fc to the ratio
of undeformed chip thickness to tool edge radius, a/r at
different tool rake angles, γ and cutting speeds, V for r of 10
µm.

143
Figure 8.1 Configuration of mesh-refined computational tool-workpiece
entities with predefined boundary conditions in the ALE
domain for the behavioral and sensitivity study of extrusion-
like chip formation mechanism. γ = +10°, α = 6° and r = 10
µm.

147
Figure 8.2 Nodal displacement plot of initial chip formation at a/r ≥
0.275.

148
Figure 8.3 Evolution of nodal displacement vectors during initial chip
formation at a/r ≥ 0.275.

149
Figure 8.4 Unique nodal displacement plot of the extrusion-like chip
formation mechanism in the preliminary stage at a/r ≤ 0.250.

150
Figure 8.5 Nodal displacements evolution of the extrusion-like
mechanism during initial chip formation at a/r ≤ 0.250.

151
Figure 8.6 Illustration of the formations of effective positive rake angles

for CAT-I chip formation mechanism at a/r of 0.8, 0.6, and
0.4.

152


xvi
Figure 8.7 Evolutions of nodal displacement vectors in the PDZ for
CAT-I chip formation mechanism with +γ
eff
.

154
Figure 8.8 Illustration of the formations of effective negative rake
angles for
155
CAT-I chip formation mechanism at a/r of 0.300 and 0.275.


Figure 8.9 Evolutions of nodal displacement vectors in the PDZ for
CAT-I chip formation mechanism with -γ
eff
.

157
Figure 8.10 Illustration of the formations of effective negative rake
angles for CAT-II extrusion-like chip formation mechanism
at a/r of 0.250, 0.225 and 0.200.

158

Figure 8.11 Evolutions of nodal displacement vectors in the PDZ for
CAT-II chip formation mechanism with -γ
eff
.

160
Figure 8.12 Effective negative rake angle formation in CAT-II extrusion-
like mechanism.

162
Figure 8.13 Application of Cauchy’s stress system for the description of
stress states in the primary deformation zone (PDZ) during
chip formation in tool-based micromachining.

163
Figure 8.14 Main characteristics of horizontal principal stress
distributions in CAT-II chip formation mechanism.

165
Figure 8.15 Main characteristics of horizontal principal stress
distributions in CAT-I chip formation mechanism.

165
Figure 8.16 Main characteristics of vertical principal stress distributions
in CAT-II chip formation mechanism.

167
Figure 8.17 Main characteristics of vertical principal stress distributions
in CAT-I chip formation mechanism.

167
Figure 8.18 Main characteristics of hydrostatic pressure distributions in
CAT-II chip formation mechanism.

169
Figure 8.19 Main characteristics of hydrostatic pressure distributions in
CAT-I chip formation mechanism.

169
Figure 8.20 Main characteristics of shear stress distributions in CAT-II
chip formation mechanism.

170
Figure 8.21 Main characteristics of shear stress distributions in CAT-I
chip formation mechanism.

170


xvii
Figure 8.22 Convention in stress definitions. (a) Principal stress; and (b)
In-plane principal stress.

172
Figure 8.23 Vector distributions of maximum and minimum in-plane
principal stress in CAT-II chip formation mechanism.

174
Figure 8.24 Main characteristics of maximum in-plane principal stress
distributions in CAT-II chip formation mechanism.

175
Figure 8.25 Main characteristics of minimum in-plane principal stress
distributions in CAT-II chip formation mechanism.

175
Figure 8.26 Vector distributions of maximum and minimum in-plane
principal stress in CAT-I chip formation mechanism.

178
Figure 8.27 Main characteristics of maximum in-plane principal stress
distributions in CAT-I chip formation mechanism.

179
Figure 8.28 Main characteristics of minimum in-plane principal stress
distributions in CAT-I chip formation mechanism.

179
Figure 8.29 Illustration of combined stress states during CAT-II
extrusion-like mechanism.

181
Figure 8.30 Redistribution of the elastic-plastic deformation boundary
with chip growth through its increasing inclinations as
reflected from the evolution of shear stress distribution,
under the governance of the Le-Chatelier’s principle.

184
Figure 8.31 Impositions of increasingly intense compressive stress
around the primary deformation zone during chip growth

which could suppress micro-void nucleation.

187
Figure 8.32 Distributions of surface roughness Ra (µm) with the
variations of a/r at different cutting speeds V of (a) 100
m/min; (b) 250 m/min; and (c) 500 m/min.

190
Figure 8.33 Characteristics of material plowing at low a/r of 0.05 with an
enormously large effective negative rake angle. (a) Nodal
displacement vectors; (b) hydrostatic pressure distribution;
and (c) shear stress distribution.

193
Figure 8.34 Grinding chip produced by an abrasive grain (after M.E.
Merchant and Kalpakjian, 2005) which resembles the
characteristics of CAT-II extrusion-like mechanism in tool-
based micromachining at a similar level of a/r.

196



xviii

List of Symbols


a
Undeformed chip thickness

r
Tool edge radius


V
Cutting speed


N
x

Nodal displacement


NJ
M

Mass matrix


J
P
Applied load vector


J
I

Internal force vector


ˆ
v

Nodal velocity


t
Time


tΔ
Time increment


N
T
Nodal temperature



N
T

Rate of change in temperatures


NJ

C
Lumped capacitance matrix


J
F
Internal flux vector


Rχ
Referential domain


RX
Material domain


Rx
Spatial domain


χ
Reference coordinates


X
Material coordinates


x

Spatial coordinates


φ, Ф, Ψ Numerical mappings between domains


0
T

Null row-vector



xix

D
Gradient of deformation


c
Relative velocity between material and mesh


λ

Elongation ratio


L

Elongation tensors


R
Rotation tensors


ε

Strain


t
D

Gradient of deformation


el
t
D
Gradient of elastic deformation



p
l
t
D
Gradient of plastic deformation

S

Total gradient of deformation velocity


el
S
Gradient of elastic deformation velocity


p
l
S
Gradient of plastic deformation velocity


ε


Strain rate


el
ε


Elastic strain rate

p
l
ε


Plastic strain rate


Y
Symmetric part of deformation velocity gradient


el
Y
Symmetric part of elastic deformation velocity gradient


pl
Y
Symmetric part of plastic deformation velocity gradient


Z
Antisymmetric part of deformation velocity gradient


el
Z
Antisymmetric part of elastic deformation velocity gradient

Z
p
l

Antisymmetric part of plastic deformation velocity
gradient


E
Total specific energy


ρ
Mass density




xx
v
Material velocity vector


σ
Cauchy stress tensor / flow stress tensor


b

Specific body force vector


e
Specific internal energy


S
v∇

Strain rate tensor


l
min
Minimum characteristic length of the computational
element


k
Thermal diffusivity


c
Specific heat


ξ

Solution variable

ϑ

Momentum variables


A
Linear transformation of momentum variables


M
Lump mass matrix


ϖ

Transport mass


0
σ

Null yield stress


eq
ε

Equivalent plastic strain

A
Yield strength coefficient of Johnson-Cook model


B
Hardening modulus of Johnson-Cook model


m
Thermal softening coefficient of Johnson-Cook model


n
Hardening coefficient of Johnson-Cook model


T Working temperature


T
r

Room temperature


T
m

Melting temperature


C
Strength coefficient of Johnson-Cook model




xxi
ε

eq

Equivalent plastic strain rate


0
ε


Reference strain rate


σ
n

Normal contact pressure

h
Contact overclosure


τ
f

Frictional shear stress


µ
Mean coefficient of friction (kinetic)


k
f

Material flow stress


V
s

Relative sliding velocity


pl
q
Heat flux density generated through plastic straining

p
l
η

Fraction of plastic strain heat


r
q
Heat flux density generated through frictional sliding


fr
η

Fraction of frictional heat


t
q
Flow of heat flux densities towards the contacting surfaces
of cutting tools


q
w

Flow of heat flux densities towards the contacting surfaces
of workpiece/chip

t
η

Heat fraction of cutting tool


w
η

Heat fraction of workpiece


c
q
Heat conduction across contact interfaces


t
θ

Surface temperatures on cutting tool


w
θ

Surface temperatures on workpiece

tw
θ
−
Δ
Surface temperature differences between tool and
workpiece


k
t-w

Interfacial conductance


γ / γ
tool
Tool rake angle


α Tool clearance angle



xxii

σ
y
Yield strength

K1c
Fracture toughness


W
Spindle rotational speed


D
w

Workpiece diameter


f
In-feed rate


Lc
Tool-chip contact length


tc
Deformed chip thickness


Fc
Cutting force

Ft
Thrust force


δ
s

Stagnation point angle


δ
sv

Stagnation point angle (spatial velocity)


δ
sf

Stagnation point angle (frictional shear contact)


R
2

Coefficient of determination


γ

eff

Effective rake angle


WPDZ
Width of primary deformation zone


TPDZ
Thickness of primary deformation zone


DPDZ
Depth of primary deformation zone


Lc/a Normalized contact length


tc/a
Normalized deformed chip thickness


Ft/Fc
Normalized machining force


θ
R

Inclinations of resultant force

a/r
Relative tool sharpness


sp
n
Initial engagement point on the tool edge radius


ep
n

End point of tool-chip contact length




xxiii
ij
σ

Stress tensor


i
F
Active components of the force vector

j
A

Normal vector components of the surface area.


n
(e )
T
Normal stress vectors to the primary axes


1
I
First normal stress invariant


2
I

Second normal stress invariant


3
I

Third normal stress invariant

ij
s

Deviatoric stress tensor


ij
p
δ

Mean hydrostatic stress tensor


p

Mean principal stress


1
J

First deviatoric stress invariant


2
J

Second deviatoric stress invariant

3
J

Third deviatoric stress invariant


M
ises
σ

Von Mises equivalent stress


1
σ

Horizontal principal stress


2
σ

Vertical principal stress


P
σ

Hydrostatic pressure

12
τ

Shear stress


'
1
σ

Maximum in-plane principal stress


'
2
σ

Minimum in-plane principal stress


Ra
Surface roughness