An analysis of cutting under different rake angles

using the nite element method

Ship-Peng Lo
*
Department of Mechanical Engineering, Sze Hai Institute of Technology and Commerce, 1380LN,
Chin-yung Road, Tucheng, Taipei Hsien 236, Taiwan, ROC
Received 22 June 1999
Abstract
The elasticplastic nite element method is developed in this study to investigate the effect of the tool rake angle on the chip and the
machined workpiece during the precision cutting process. Cutting simulations were conducted under a variety of tool rake angles to explore
the effect of tool rake angle on cutting force, the geometric shapes of the chip, the equivalent stress distribution, the residual stress and the
surface of the machined workpiece. The ndings indicate that an increase in the tool rake angle leads to the following: a decrease in the
cutting force required during machining; a smoother chip contour; a smaller difference between the chip thickness and the undeformed chip
thickness; a decrease in the equivalent stress distribution; and a less pronounced curvature phenomenon at the initial cutting end of the
machined workpiece. Further, the results also show that as the tool rake angle increases from 5 to 158, the changes in the above-mentioned
physical phenomena are more pronounced. In contrast, the increase of tool rake angle from 15 to 208 hardly brought about any changes to
these physical phenomena. Therefore, to reach the goal of lowering the above requirements such as the cutting force so as to extend tool
life, it is recommended that a tool rake angle of 158 instead of 208 be adopted for cutting. #2000 Elsevier Science S.A. All rights reserved.
Keywords: Rake angle; Residual stress; Separation criterion
1. Introduction
An understanding of the material-removal process is
necessary when aiming to increase the dimension accuracy
and the surface integrity of a machined product. Factors
affecting surface integrity include the rake angle of the tool,
residual stress, metallurgical (phase) transformations and
plastic deformation of the surface generated during the
manufacturing process. In recent years, ultra-precision
machining technologies have developed very quickly to
meet the needs of numerous industrial areas, such as the
rotary polygon mirrors of scanners, magnetic discs of com-
puters and laser reective mirrors. To enhance product
dimensional accuracy and surface integrity, the effect of
the rake angle of the tool during cutting on the deformation
of the machined workpiece, the chip shape, the cutting force
and the stress distribution must be thoroughly understood.
Although there have been numerous studies on the ortho-
gonal cutting model, most are limited to certain topics, such
as the distribution of residual stress and temperature on
the machined workpiece, and the stress on the toolchip
interface. Little research effort has been made in exploring
the effects of the rake angle of the tool on the cutting
process. In this paper, the nite element method is used
to simulate the effects of the rake angle of the tool on the
machining process from the incipient cut of the tool into the
workpiece until the formation of steady state cutting.
The nite element method started to be applied in the
simulation of cutting procedures in the 1970s to improve the
error generated by the slip line solution in earlier days. Lee
and Shaffer [1] obtained the well-known slip-line solution.
The material was assumed to be rigidperfectly plastic and
the solution could not really account for the variation of
yield stress with strain, strain rate and temperature. Saito
et al. [2] and Akiyama et al. [3] used the nite element
method to simulate the thermal deformation of the tool
cutting edge and the tool thermal stress as a result of
temperature variation during cutting. Iwata et al. [4]
assumed the material to be rigidplastic, and the ow stress
as a function of strain to analyze steady-state micro-cutting
with a low cutting velocity and a low strain rate. However,
since elasticity was not accounted for, the residual stress
could not be estimated. Strenkowski and Moon [5] estimated
the approximate geometry of the chip, developed the
Journal of Materials Processing Technology 105 (2000) 143151
*
Tel.: 886-2-2273-4424; fax: 886-2-2798-8318.
E-mail address: loulin@ms17.hinet.net (S.-P. Lo).
0924-0136/00/$ see front matter # 2000 Elsevier Science S.A. All rights reserved.
PII: S0 9 2 4 - 0 1 3 6 ( 0 0 ) 0 0 6 5 0 - 6
orthogonal cutting model, and predicted the temperature
distribution of the workpiece, chip and tool, while ignoring
the effect of elasticity. Among the bounty of literature on
cutting, the cutting depth is mostly for ordinary cutting. Very
little has been done on smaller cutting depths, especially
micro-cutting with a depth of 5 mm or under being applied in
soft metals such as copper (OFHC) and aluminum. Mor-
iwaki et al. [6] used the rigidplastic nite element method
to analyze the micro-cutting of copper (OFHC), and
obtained the temperature distribution of the tool and the
workpiece. Moriwaki and Okuda [7] used experimental
apparatus consisting of a y cutting machine and a diamond
tool to conduct the high velocity cutting of copper (OFHC).
They measured the cutting force, and explored the effect of
feed rate on surface roughness. Lucca et al. [8] conducted an
experiment by energy consumption with a cutting depth
from 0.025 to 15 mm. They used diamond tools to cut copper
(OFHC), and investigated the effect of chip formation,
slipping and plowing. They compared their experimental
data with those fromthe studies of others [7]. Moriwaki et al.
[9] analyzed, by experiment and theory, the effect of cutting
heat on the thermal deformation of the cutting tool and of the
workpiece generated, and also on the machining accuracy.
However, they did not discuss the residual stress of the
machined surface. Based on the above studies mentioned, it
can be seen that most studies regarding the micro-cutting of
copper (OFHC) obtained such features as the cutting force
and the workpiece surface deformation after machining
from experiments: very few researchers used the nite
element method to simulate the conditions after cutting,
such as the phenomena of chip shape and workpiece surface
deformation after machining. Further, the impact of the tool
rake angle effect on the machined workpiece was not
investigated. In this paper, a two-dimensional elasticplastic
analytical model of metal cutting is developed based on the
large deformationlarge strain theory, the update Lagran-
gian formulation, the incremental principle and the geome-
trical separation criterion to simulate and analyze certain
characteristics, such as the residual stress and workpiece
surface deformation after machining, of the machined sur-
face in micro-cutting with diamond tools.
2. Theoretical foundation
2.1. Constitutive equation
Based on large-strain deformation theory [10], and by
means of the updated Lagrangian formulation and the
PrandtlReuss ow rule, a thermoplastic constitutive
equation for orthogonal cutting is derived. The incre-
mental stressstrain relationship in isotropically harden-
ing material for large strainlarge deformation can be
obtained as
1. In the elastic range,
ds = [D
e
[(de de
t
) (1)
2. In the plastic range,
[ds[ = [D
ep
[(de de
t
)
2s
3S
0
S
@R
@
_
e
d
_
e
@R
@T
dT

(2)
where
[D
ep
[ = [D
e
[
[D
e
[@f =@s@f =@s
T
[D
e
[
H
/
@f =@s
T
[D
e
[@f =@s
(3)
and
S = [D
e
[s
/
(4)
S
0
=
4
9
s
2
H
/
s
/

T
S (5)
in which s
/
is the stress deviator and s the effective stress, R
indicates the magnitude of the yield surface, which is
considered as a function of the effective plastic strain, the
effective strain rate and the temperature. H
/
is the strain
hardening parameter, and f is the plastic potential.
2.2. Finite-element formulation of large deformation
In this model, the Jaumann rate of Euler stress is taken as
the stress rate for the constitutive equation. The updated
Lagrangian formulation is used and a residual force term is
also incorporated into the governing equation in order to
balance the forces. By ignoring the body force, the equation
for the incremental principal of the virtual work is used:
then, the governing equation for a thermo-elasticplastic
material under large deformation can be derived as follows
[11]:
([K
A
ep
[ [K
A
G
[)
_
d
A
=
_
F
A
0

Z
vA
[B
eA
[
T

_
R
_
eT
dv

Z
vA
[B
eA
[
T
[D
ep
[_ e
t
dv (6)
Nomenclature
s
/
stress deviator
s effective stress
H
/
strain hardening parameter
f plastic potential
s
uni
flow stress
e equivalent plastic strain
e
+
plastic strain rate

_
d
A
nodal velocity
[K
A
ep
[ elasticplastic stiffness matrix
[K
A
G
[ geometric stiffness matrix

_
F
A
n
normal loading rate

_
F
A
t
tangential loading rate
144 S.-P. Lo / Journal of Materials Processing Technology 105 (2000) 143151
where
[K
A
ep
[ =
X
workpiece element
Z
vA
[B
eA
[
T
[D
ep
[[B
eA
[ dv;
[K
A
G
[ =
X
workpiece element
Z
vA
[B
e
[
T
[D
G
[[B
e
[ dv;

_
F
A
0
=
X
surface
Z
sA
[N
A
[
T

_
f
0
ds
in which
_
d
A
is the nodal velocity and
_
R
_
eT
the notation
for the rate of change of the fractional term in Eq. (6). [K
A
ep
[
and [K
A
G
[ are the elasticplastic stiffness matrix and the
geometric stiffness matrix.
For large plastic deformation, the traction rate is a func-
tion of the deformable velocity of the workpiece. Hence, the
load correction matrix [K
A
c
[, which is derived from the
tangential loading rate
_
F
A
t
component and the normal
loading rate
_
F
A
n
component, is incorporated into Eq. (6).
Frictional behavior in the cutting process is handled through
a constant friction factor and Coulomb's law, the latter being
applied in the region of the chiptool and toolworkpiece
interface. Therefore, the frictional compensation matrix
[K
A
f
[ is also introduced into the governing equation, of
Eq. (6), the latter subsequently becoming
([K
A
ep
[ [K
A
G
[ [K
A
c
[ [K
A
f
[)
_
d
A

=
_
F
A
n

Z
vA
[B
eA
[
T

_
R
_
eT
dv
Z
vA
[B
eA
[
T
[D
ep
[_ e
t
dv
(7)
where
_
F
A
n
is the normal loading rate of the contact nodes
at the chiptool interface.
3. Finite-element mesh and boundary conditions
Fig. 1 presents a schematic interpretation of the nite-
element model and the boundary conditions. ABEF on the
upper part of the workpiece is the part removed by cutting.
The left boundary ABCand the bottomboundary CDremain
xed. EB denotes the pre-dened cutting path of the tool.
The model consists of 560 CST elements and 329 nodes.
3.1. Workpiece material modeling
Oxygen-free high conductivity copper (OFHC) is used for
the orthogonal cutting simulation, the material characteris-
tics of this copper accounting for the effects of strain, strain
rate and temperature, its ow stress being expressed as
follows [12]:
s
uni
= (90 Ae
n
)(1 Bln e
+
)(1 T
+m
) (8)
where s
uni
is the owstress in MPa, A= 292 MPa (the strain-
hardening coefcient), B = 0:025 (the strain-rate coef-
cient), m = 1:09; n = 0:31 (the strain-hardening exponent),
e the equivalent plastic strain, e
+
the plastic strain rate,
T
+
= (T T
room
)=(T
melt
T
room
), where T
melt
= 1356 K.
The values of Young modulus and Poisson's ratio of OFHC
copper are 128 GPa and 0.3, respectively. The other physical
properties of the tool and the workpiece are listed in Table 1.
3.2. Chip separation criterion
The chip separation criterion must be considered in the
simulation of the chip formation process. This is usually a
continuous separation process modeled to follow along a
pre-dened path. Strenkowski and Carroll [13] used the
effective plastic strain as the chip separation criterion. When
the effective plastic strain value of the node closest to the
tool tip exceeds a pre-dened value, this node separates to
form the machined surface and the chip. Zhang and Bagchi
[14] used the conditional link element as the separation cri-
terion. These two node link elements are placed between the
chip and the workpiece along a pre-dened separation path.
In this paper, the special geometrical separation method
was used as the chip separation criterion. In Fig. 2, the chip
and the workpiece are connected by these twin nodes along a
pre-dened separation path OB. The chip will be separated
Fig. 1. The nite element model of the tool and the workpiece.
S.-P. Lo / Journal of Materials Processing Technology 105 (2000) 143151 145
when the distance D between the leading node and the tool
edge is equal to or smaller than a given value D
c
. Conse-
quently, as the tool advances, these twin nodes will be
separated one by one, resulting in the formation of the chip
and the machined surface.
Chip separation is a continuous process occurring imme-
diately ahead of the tool edge for continuous chip formation.
Therefore, it is hoped that the twin nodes connecting the chip
and the workpiece will separate immediately only when the
tool edge is very close to them. Therefore, the critical value,
D
c
, must be as small as possible, but not so small as to make
chip formation impossible. According to the trial runs
performed, D
c
was thus chosen to be
D
c
= (0:010:03)l (9)
where l is the length of element that ensures continuous chip
formation for different rake angles and cutting velocities.
4. Finite element simulation: results and discussion
In order to obtain a more complete solution for the cutting
process, it would be best to simulate the transient process
from the beginning to steady state. In this paper, ve cases of
orthogonal machining of OFHC copper at the incipient stage
were simulated with 0, 5, 10, 15 and 208 tool rake angle,
0.002 mm undeformed chip thickness and 0.1 mm unde-
formed chip width. The case of tool rake angle 08 was
chosen because it had been used in experiments by Lucca
et al. [8] and Moriwaki et al. [9], in which the cutting forces
were measured.
4.1. Cutting force
Fig. 3 shows the simulated cutting forces obtained at tool
rake angles from 0 to 208. To validate the accuracy of the
cutting model presented in this paper, the simulated cutting
force at the tool rake angle of 08 and the cutting force
reported in the experiment by Moriwaki and Okuda [7]
under the same cutting conditions are compared. As shown
in Fig. 3, the simulated cutting force at the rake angle of 08 is
3.8 N/mm [15] while the experimental value reported by
Moriwaki and Okuda [7] is 3.95 N/mm. The results indicate
that the error between the present simulated value and the
experimental value is less than 10%. Hence, the accuracy of
the simulated physical phenomena may be considered rea-
sonable and within acceptable limits. Fig. 3 also shows that
the cutting force decreases as the tool rake angle increases.
The simulated cutting forces obtained at the rake angles
of 5, 10, 15 and 208 are 3.57, 3.14, 2.49 and 2.45 N/mm,
respectively. There is a difference of 0.56 N/mm between
the cutting force at the angle of 08 and that at the angle
of 108, which is rather limited in extent. However, the
Table 1
Physical properties of the workpiece and the tool
Thermal expansion
coefficient (K
1
)
Thermal conductivity
(W/m K)
Specific heat
(J/kg K)
Density
(kg/m
3
)
Tool (diamond) 0.0000025 999.0 420.0 3520
Workpiece (OFHC) 0.0000165 393.6 385.5 8960
Fig. 2. Geometrical separation method: (a) before node separation; (b)
after node separation (twin nodes). Fig. 3. Cutting force under different rake angles.
146 S.-P. Lo / Journal of Materials Processing Technology 105 (2000) 143151
difference of cutting forces between the angles of 0 and
208 is more signicant and reaches as much as 1.35 N/mm.
The above ndings suggest that given a tool rake angle of
less than 108, the variation of cutting force is very limited.
Under these circumstances, the tool life and the smoothness
of the machined surface should be the focus of consi-
deration. If a cutting tool with a rake angle of under 208
can be employed, the tool rake angle should be adjusted at
15 or 208 from the perspective of energy consumption.
However, if tool life is the top priority, the tool rake
angle should be 158. The difference between the cutting
force obtained at the angle of 158 and that at the angle of
208 is merely 0.04 N/mm, this difference being so small
that it can be ignored. On the other hand, the tool life
under the rake angle of 158 is much longer than that under
the angle of 208.
4.2. Effect of rake angle on chip contour
Fig. 4(a)(d) shows the simulated chip contour and mesh
deformation under four different tool rake angle conditions.
Fig. 4(a) shows that at the angle of 58, the chip thickness is
greater than the undeformed chip thickness. The deforma-
tion on the chip back is smoother and there is less smooth
surface at the A1 location on the chip tip. Fig. 4(a) also
reveals an uneven surface at B1 at the starting end of the
machined workpiece and that the mesh deformation tilts
toward the right-hand side. Fig. 4(b) shows the chip contour
and mesh deformation at the tool rake angle of 108. There is
almost no difference between the chip thickness and the
undeformed chip thickness in this gure, but a less smooth
surface is also present at A2 on the chip tip. Further, mesh
deformation toward the right at B2 at the starting end of the
machined workpiece can also be found in this gure.
Fig. 4(c) and (d) shows the results at the rake angles of
15 and 208, respectively. It is obvious from these two gures
that the increase in rake angle results in a smoother surface at
A3 and A4 on the chip tip. At the same time, the deformation
at B3 and B4 at the starting end of the machined workpiece
tilts slightly toward the left.
Several physical phenomena can be identied from the
above discussion. As the tool rake angle increases from 5 to
208, the difference between chip thickness and undeformed
chip thickness decreases, i.e. the larger the rake angle, the
smaller the difference between the machined chip thickness
and the original undeformed chip thickness. Further, the chip
tip surface also becomes smoother as the rake angle
increases, and as the rake angle increases, the mesh defor-
mation at the starting end of the machined workpiece shifts
toward the left instead of the right, as for lower rake angles.
The main reason for the above phenomena is that given a
small rake angle, the cutting action often transforms into a
pushing and squeezing force. In this case, a larger cutting
force is required, which is evident from the cutting force
shown in Fig. 3. This is also why there is a less smooth
surface on the chip tip.
4.3. Effect of rake angle on equivalent stress distribution
Fig. 5(a)(d) shows the simulated equivalent stress dis-
tribution under four different tool rake angle conditions.
Fig. 5(a) shows that at the rake angle of 58, the maximum
equivalent stress is 450 MPa, which occurs at about 0.0008
0.0012 mm from the tool tip along the tool rake face. The
equivalent stress close to the upper chip edge also reaches as
high as 350 MPa. Thus, the chip region with the equivalent
Fig. 4. The effect of rake angle on chip contour and mesh deformation,
where the rake angle is: (a) 58, (b) 108, (c) 158, (d) 208.
S.-P. Lo / Journal of Materials Processing Technology 105 (2000) 143151 147
stress between 400 and 350 MPa should be the region with
greater deformation. Fig. 5(b) shows the equivalent stress
distribution at the tool rake angle of 108. In Fig. 5(b), the
maximum equivalent stress of 385 MPa is located at about
0.00080.002 mm from the tool tip along the tool rake face.
This distribution is similar to that shown in Fig. 5(a), but the
distribution range of the equivalent stress of 300 MPa is
smaller than that in Fig. 5(a). The most important feature of
this phenomenon is that an increase in the tool rake angle
(from 5 to 108) reduces the cutting force required. As a
result, the equivalent stress in the primary deformation zone
and secondary deformation zone of the chip also decreases.
Fig. 5(c) and (d) shows the results at the rake angles of 15
and 208, respectively. In Fig. 5(d), the maximum equivalent
stress is around 360 MPa and the distribution zone of the
equivalent stress of 300 MPa decreases signicantly, while
the distribution of the equivalent stress of 200 MPa can still
be found at the workpiece end in Fig. 5(a), the equivalent
stress distribution at the same spot in Fig. 5(d) being only
150 MPa. The above ndings indicate that as the tool rake
angle increases, the cutting force required decreases and so
does the equivalent stress inside the chip and the workpiece.
In other words, given a smaller tool rake angle, the cutting
action contains the function of pushing and squeezing,
which prevents the chip from deforming and results in a
higher equivalent stress inside the chip and the workpiece.
4.4. Effect of rake angle on section strain value
Fig. 6 shows the simulated equivalent strain on section A
A under four different tool rake angle conditions. Fig. 6
shows that, regardless of the rake angle, the strain reaches a
maximum value on the workpiece surface, drops abruptly as
the section depth increases and nally approaches zero at the
end of the workpiece. Fig. 6 shows that the maximum
equivalent strain values of the machined workpiece surface
are 1.92, 1.62, 1.05 and 0.98, respectively, at the rake angles
of 5, 10, 15 and 208. From the above discussion, it is quite
obvious that the maximumequivalent strain on the machined
workpiece surface decreases as the rake angle increases.
Further, it is also noted that as the rake angle increases from
5 to 108, the maximum equivalent strain on the workpiece
surface decreases from1.92 to 1.62, representing a reduction
of around 0.3. However, when the rake angle increases to
158, the maximum equivalent strain decreases from 1.62 to
Fig. 5. Effect of rake angle on equivalent stress distribution, where the
rake angle is: (a) 58, (b) 108, (c) 158, (d) 208.
Fig. 6. Effect of rake angle on section strain value.
148 S.-P. Lo / Journal of Materials Processing Technology 105 (2000) 143151
1.05, a difference of 0.57. This clearly indicates that the
decrease of the maximum equivalent strain is quite limited
when the rake angle increases from 5 to 108. However, as the
rake angle increases to 158, the decrease of maximum
equivalent strain doubles from the previous amount: this
difference can be veried from the cutting force shown in
Fig. 3. In addition, it was also found that when the rake
angle increases from 15 to 208, the maximum equivalent
strain of the workpiece surface decreases from 1.05 to
0.98, a mere difference of 0.07. This phenomenon is
identical with the physical phenomena shown in the
cutting force in Fig. 3. Therefore, to signicantly reduce
the maximum equivalent strain on the workpiece surface,
it is sufcient to increase the rake angle to 158 instead of
208 in order to prolong tool life.
4.5. Effect of rake angle on residual stress
Figs. 710 show the simulated residual stress on section
AA under four different tool rake angle conditions. In
Fig. 7, at the rake angle of 58, the tensile stress on the
workpiece surface among stress s
x
in the cutting direction is
about 1.2 MPa, and gradually transforms into a compressive
stress beneath the machined surface with a maximum value
of 120 MPa. As for the residual stress s
y
perpendicular to
the cutting direction, its maximum value on the workpiece
surface is 280 MPa, and decreases as the distance from the
workpiece surface increases. The maximum equivalent
stress appears on the workpiece surface, with a maximum
value of 280 MPa. The equivalent stress also decreases as its
distance from the workpiece surface increases. Fig. 8 shows
the residual stress on section AA of the machined work-
piece at the rake angle of 108. In Fig. 8, the maximum tensile
stress of s
x
appears not on the workpiece surface, but at
0.00052 mm beneath the workpiece surface, with a value of
around 15 MPa. Further, as the s
x
value's distance from the
workpiece surface increases, it gradually transforms into a
compressive stress, with a maximum value of about
110 MPa. Fig. 8 also shows that the maximum compres-
sion stress of s
y
is about 245 MPa and occurs on the
workpiece surface. As the distance between the s
y
value and
the workpiece surface increases, the s
y
value decreases. The
maximum equivalent stress in Fig. 8 is 250 MPa, and occurs
Fig. 7. The residual stress on section AA under 58 rake angle.
Fig. 8. The residual stress on section AA under 108 rake angle.
Fig. 9. The residual stress on AA section under 158 rake angle.
Fig. 10. The residual stress on AA section under 208 rake angle.
S.-P. Lo / Journal of Materials Processing Technology 105 (2000) 143151 149
on the workpiece surface. It also decreases as its distance
from the workpiece surface increases. The variation is
similar to the trend of equivalent stress shown in Fig. 7.
Fig. 9 shows the results at the rake angle of 158. The
maximum tensile stress of s
x
is about 60 MPa, and it occurs
on the workpiece surface. The value then decreases as its
distance from the workpiece surface increases. It transforms
into a compressive stress at about 0.0032 mm beneath the
workpiece surface. The maximum compression stress of s
y
is about 160 MPa, and it occurs on the workpiece surface,
but the maximumequivalent stress occurs at a place different
from those shown in Figs. 7 and 8. As shown in Fig. 9, the
equivalent stress on the workpiece surface is about 215 MPa,
but the value at 0.0005 mm beneath the workpiece surface is
240 MPa. In Fig. 10, at the rake angle of 208, the maximum
tensile stress of s
x
, about 80 MPa, occurs on the workpiece
surface and transforms into a compressive stress at
0.0032 mm beneath the workpiece surface. The maximum
compression stress of s
y
is 140 MPa and occurs on the
workpiece surface. The equivalent stress on the workpiece
surface is 190 MPa, while the maximum equivalent stress of
210 MPa occurs at 0.0005 mm beneath the workpiece sur-
face.
The above results indicate that the tensile stress on work-
piece surface of stress s
x
in the cutting direction increases as
the tool rake angle increases. In contrast, for s
y
, perpendi-
cular to the cutting direction, the maximum compressive
stress on the workpiece surface decreases as the tool rake
angle increases. In this study, as the rake angle increases
from 5 to 208, the s
y
value decreases from 280 to
140 MPa. In other words, as the tool rake angle increases,
the fatigue strength of the machined workpiece surface
decreases. Further, it was found that at the rake angles of
5 and 108, the maximum equivalent stress on section AA
occurs on the workpiece surface after cutting. In contrast, at
the rake angle of 15 and 208, the maximum equivalent stress
on section AA occurs at 0.0005 mm beneath the workpiece
surface after cutting instead of on the workpiece surface.
4.6. Effect of rake angle on machined workpiece surface
Fig. 11 shows the machined workpiece surface under
different tool rake angles. The initial cutting end of the
workpiece is the place where an uneven surface is present.
Under a smaller tool rake angle, the presence of the pushing
force during the cutting process often results in the phenom-
enon of curvature at the initial cutting end of the workpiece.
As shown in Fig. 11, there is curvature at the initial cutting
end of workpiece after cutting both at the angles of 0 and 58.
The curvature is especially pronounced at the angle of 08,
with a curvature value of 0.00018 mm. At the rake angle of
58, the maximum curvature value at the initial cutting end of
workpiece is about 0.00005 mm. As the tool rake angle
increases, the curvature value decreases. The curvature
values are 0.00002 0.00004 and 0.00005 mm at the
rake angles of 10, 15 and 208, respectively. It is obvious from
Fig. 6 that deformation occurs on the machined workpiece
surface. Among all of the cutting simulations, the most
smooth workpiece surface occurs in the simulation at the
tool rake angle of 158. This is because the deformation at
various points over the entire workpiece surface is more
moderate at this angle. The workpiece surface after
cutting at other angles tends to be less smooth. In addition,
it was also found that the cutting simulation at the angle
of 08 requires a proper xture to secure the initial cutting
end of the workpiece to prevent a greater curvature at this
end due to the pushing force occurring during the cutting
process.
5. Conclusions
The elasticplastic nite element method was developed
in this study to investigate the effect of tool rake angle on
cutting force, chip contour, equivalent stress distribution,
residual stress and the machined workpiece surface during
the precision cutting process. The geometrical separation
Fig. 11. Effect of rake angle on machined workpiece surface.
150 S.-P. Lo / Journal of Materials Processing Technology 105 (2000) 143151
criterion is adopted as the criterion of chip separation from
the workpiece. The entire cutting process, from the initial
cutting to the formation of steady state, under different tool
rake angles is simulated. To validate the accuracy of the
present cutting model, the simulated cutting force was
also compared with experimental values obtained under
the same cutting conditions. The comparison indicates
that the error between the simulated cutting force and
experimental values falls within acceptable limits. Based
on the previous analysis and discussion, the following
conclusions may be drawn:
1. The cutting force decreases as the tool rake angle
increases. In this paper, the extent of cutting-force
reduction is most evident as the rake angle increases
from 10 to 158. In contrast, when the rake angle
increases from 15 to 208, the cutting force experiences
only a slight reduction.
2. The maximum equivalent strain on the section decreases
as the tool rake angle increases. The extent of reduction
is also most evident when the rake angle increases from
10 to 158. The reduction of maximum equivalent strain
is very limited when the rake angle increases from 15 to
208.
3. The top of the chip contour becomes smoother as the
tool rake angle increases. The difference between the
chip thickness and the undeformed chip thickness
decreases as the tool rake angle increases.
4. The phenomenon of curvature at the initial cutting end
of the workpiece becomes more moderate as the tool
rake angle increases. As a result, the cutting simulation
at the angle of 08 requires a proper xture to secure the
initial cutting end of the workpiece to prevent a greater
curvature at this end.
5. Both the equivalent stress and the stress s
y
perpendi-
cular to the cutting direction among the residual stress
show a more obvious trend of decrease as the tool rake
angle increases. As for the stress s
x
in the cutting
direction, its maximum tensile value on the workpiece
surface increases as the tool rake angle increases.
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