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Modelling of the cutting temperature distribution underthe tool ank wear effect
Y Huang1 an d S Y Liang2*1
Department of Mechanical Engineering, Clemson University, Clemson, South Carolina, USA2
George W Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, Georgia, USA
Abstract: The understanding of cutting temperature distribution at the presence of tool wear can aid
in addressing important metal cutting issues such as part surface integrity, tool life and dimensional
tolerance under practical operating conditions. The effect of tool wear on the cutting temperature
distribution was rst modelled by Chao and Trigger and there have been very few followers since. In
Chao’s model, the primary heat source was assumed to have no effect on the workpiece temperature
rise and the chip temperatur e rise was treated a s a bu lk quan tity. This paper a nalytically quan ties the
tool wear effect by taking into account the contributions of the primary heat source and consideringthe distribution of chip temperature rise. On the chip side, the primary shear zone is modelled as a
uniform moving oblique band heat source and the secondary shear zone as a non-uniform moving
band heat source within a semi-innite medium. On the tool side, the effects of both the secondary
and the rubbing heat sources are modelled as non-uniform static rectangular heat sources within a
semi-innite medium. For the workpiece side, the study models the primary shear zone as a uniform
moving oblique band heat source and the rubbing heat source as a non-uniform moving band heat
source within a semi-innite medium. The proposed model is veried based on the published
experimental data in the orthogonal cutting of Armco iron. Furthermore, a comparison case is
presented on the temperature variation with respect to cutting speed, feed rate and ank wear length.
Keywords: cutting temperature, ank wear, heat source method
NOTATION
achip thermal diffusivity of the chip
aworkpiece thermal diffusivity of the workpiece
B1… x † [or B1… x 0†]
fraction of the secondary heat source
transferred into the chip
B2… x 00† [or B2… x 000†]
fraction of the rubbing heat source
transferred into the workpiece
F frictional force along the rake face
F c, F t cutting force and thrust forceF cw rubbing force along the tool–workpiece
interface
k chip, k tool thermal conductivities of the chip and tool
K 0 modied Bessel function of the second
kind of order zero
l tool–chip contact length
L length of the shear plane
M point in the medium to be measured about
the temperature rise
qfrictional… x† [or qfrictional … x 0†]
heat intensity of the secondary heat
source
qrubbing… x 00† [or qrubbing… x 000†]
heat intensity of the rubbing heat source
qshear heat intensity of the primary heat source
r chip thickness ratio
R i, R 0i, R
00i distance between the point M and heat
source segments
t undeformed chip thickness or feed rat e inthe ort hogonal cutting
t ch deformed chip thickness
V chip chip velocity
V cutting cutt ing velocity
VB ank wear length
w width of the cut or depth of cut in
orthogonal cutting
X , Y , Z , X 0, Y 0 , Z 0 , X 00, Y 00 , Z 00 , X 000, Y 000 , Z 000
right-handed Cartesian coordinates used
in related gures
a tool rake angle
The M S was received on 25 September 2002 and was accepted after
revision for publication on 26 August 2003.* Corresponding author: George W . W oodruff S chool of M echanical
Engin eering, Georgia Inst itu te of T echnology, A tlant a, GA 30332–0405,
U S A .
1195
C12802 # IM echE 2003 Proc. Instn Mech. Engrs Vol. 217 Part C: J. M echanical Engineering Science
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y0 room temperature
ychip¡friction temperature rise on the chip side due to
the secondary heat source
ychip¡shear temperature rise on the chip side due to
the primary heat source
ytool¡friction temperature rise on the tool side due to
the secondary heat source
ytool¡rubbing temperature rise on the tool side due tothe rubbing heat source
ytoolflank temperature rise along the tool ank face
ytoolrake temperature rise along the tool rake face
yworkpiece¡rubbing
temperature rise on the workpiece side
due to the rubbing heat source
yworkpiece¡shear
temperature rise on the workpiece side
due to the primary heat source
f shear angle
1 INTRODUCTION
The role of the cutting temperature in metal cutting has
been studied in great detail, beginning as early as 1907
by Taylor [1]. Since the early twentieth century, much of
the work on the thermal aspects of metal cutting has
been directly experimental, pr oviding mostly tempera-
ture in an average sense. T hese works can be categorized
as thermo-e.m.f (thermocouples), radiation (pyrometry,
infrared photography, etc.) and thermochemical reac-
tions (thermo-colours) [2]. Other experimental methods
have included the metallographic method [3] an d th e
physical vapour deposition (PVD) lm method [4], to
name just a few. Alternatively, the reverse estimation
scheme has been tried to solve the cutting temperature
prole based on the indirectly measured temperature
information [5]. N umerical metho ds were also a pplied t o
determine the temperature distribution with some
important results documented by Tay et al. [6] an d
Dawson and M alkin [7].
On analytical modelling, the steady state temperature
in metal cutting has been estimated by Hahn [8], Trigger
an d Ch ao [9–12], Loewen and Shaw [13], Komanduri
and Hou [14–16] and most recently by Huang and Liang
[17] based on the premise of a moving heat source
[18, 19]. This better understanding of the temperature
distribution a long the tool–workpiece interface at the
presence of tool wear helps to provide insight into
several important issues in metal cutting, such as tool
wear progression, dimensional tolerance and workpiece
surface integrity, etc. Unfortunately, most of the
analytical studies documented thus far focus on thermal
modelling only for a fresh tool, except that of Chao and
Trigger [12]. In the work of Chao and Trigger, the
primary heat source was tak en as having no effect on t he
workpiece temperature rise, and the temperature rise on
the chip side was modelled as an average bulk value.
However, the temperature rise due to the primary heat
source on the workpiece surface underneath the tool
ank face can be as high as 200 8C depending on the
thermal number …tV cutting=aworkpiece† in conventional
cutting [14]. Furthermore, the temperature rise due tot he p rim a ry h ea t so ur ce h as b een sh own t o b e
distributed, rather than constant, along the chip side
[14].
The objective of this study is to model the tempera-
ture distributions analytically, especially along the t ool–
workpiece contact length in orthogonal cutting for a
worn t ool. The study ut ilizes the heat source method [18,
19] to treat the effects of heat sources. On the chip side,
the effect of t he primary shear zone is modelled as a
uniform moving oblique band heat source and that of
the secondary shear zone as a n on-uniform moving band
heat source within a semi-innite medium. On the tool
side, the effects of both the secondary and the rubbing
heat sources are considered as non-uniform static
rectangular heat sources within a semi-innite medium.
On the workpiece side, the primary shear zone is
modelled as a uniform moving oblique band heat source
and the rubbing heat source as a non-uniform moving
rectangular heat source within a semi-innite medium.
The proposed model is veried ba sed on the published
experimental data of orthogonal cutting Armco iron. In
addition, a case is presented to analyse t he effects of
cutting speed, feed rate and ank wear length on the
temperature distributions.
2 ANALYTICAL MODELLING
2.1 Introduction and basic assumptions
As shown in Fig. 1, there are three main heat sources in
metal cutting with a tool worn on the ank face, namely
primary, secondary and rubbing heat sources. Tempera-
ture distribution along the tool–workpiece interface at a
location midway across the width of cut is of interest in
this study since temperature assumes its highest level at
that location. Temperatures a t other locations along the
tool–workpiece interface can also be calculated by
applying the appro ach d escribed h erein. T he heat source
method introduced by Jaeger [18] and Carslaw and
Jaeger [19] is applied in this study. Th e temperatu re rises
on the chip side and also on the tool side along the
interface; thus the tool–chip interface boundary is
ad iabatic fo r th e too l an d th e chip r esp ectively.
Similarly, the tool–workpiece interface boundary is
considered to be adiabatic for the tool and the work-
piece respectively. R egarding the adiabatic boundary
conditions along the interfaces, the primary heat source
Y HUANG AND S Y LIANG1196
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is considered to have no direct effect on the temperature
rise on the tool side, but the determined heat partition
ratios along the interfaces will imply the contribution
from the primary heat source indirectly. To simplify the
problem further, th e effect of t he rubbing heat source on
the chip side and the effect of the secondary heat source
on the workpiece side are taken to be negligible,
considering the relative distance between them and their
partitioned heat intensities [12, 15, 17].
It is also considered that the primary heat source to
the chip is the uniform moving oblique band h eat source
with chip velocity and proper boundary conditions and
the secondary heat source is the non-uniform moving
rectan gular heat source within the semi-innite medium.
Thus the temperature rise on the chip side can be
expressed as ychip¡shear ‡ ychip¡friction [17]. Additionally,
the secondary heat source to the tool can be considered
as the non-uniform static rectangular heat source within
the semi-innite medium and the rubbing heat source
due to ank wear the non-uniform static rectangular
heat source within the semi-innite medium. Therefore,
the temperature rise on the tool side along the tool–chip
interface can be expressed as ytoolrake, wh ile t he
temper atu re r ise o n th e too l sid e alo ng th e to ol–
workpiece interface can be expressed as ytoolflank . Also
considered are the primary heat source to the workpiece
as the uniform moving oblique band heat source with
cutting velocity within the semi-innite medium and the
rubbing heat source due to ank wear as the non-
uniform moving band heat source within the semi-
innite medium. It follows that the temperature rise on
the workpiece side can be expressed as yworkpiece¡shear ‡yworkpiece¡rubbing.
In calculating the above temperature rises along the
interfaces, the heat partition ratio of the secondary heat
source going to the chip is specied as a function B1… x †[or B1… x 0†] along the rake face contact length. Thus t he
remaining heat size 1 ¡ B 1… x † [or 1 ¡ B 1… x 0†] goes to the
tool. The heat pa rtition ra tio of the rubbing heat source
going to the workpiece is specied as a function B2… x 00†[or B2… x 000†] along the ank wear land, so the remaining
heat size 1 ¡ B 2… x 00† [or 1 ¡ B2… x 000†] goes to the tool.
These are shown in Fig. 1 as well.
The basic assumptions involved in the study are:
1. The generated heat ow and temperatur e distribution
are in steady states.
2. All of the deformat ion energy within the deformation
zones is converted into heat. A negligible amount is
stored as latent energy in the deformed metal. Heat
loss along the interfaces and at all surfaces of the
tool, chip and workpiece is insignicant.
3. The dimensions of the t ool are so large compared to
the chip cross-section that the tool size can be
considered a s innite.
4. Primary, secondary and rubbing heat sources are
plane heat sources and the nature of the secondary
heat source is not affected by the possible crater
wear.
5. It is assumed that there is no redistribution of
thermal shear energy going into the chip during the
very short time when the chip is in contact with the
tool. This assumption appears to be well founded for
the normal cutting operation involving continuous
chip formation without a built-up edge [13].
6. The effect of the rubb ing heat source on th e chip and
th e effect o f th e seco nd ary h eat so ur ce o n th e
workpiece a re considered to be negligible.
Fig. 1 Heat sources and h eat part itions along the tool–chip and t ool–workpiece interfaces
MODELLING OF THE CUTTING TEMPERATURE DISTRIBUTION UNDER THE TOOL FLANK WEAR EFFECT 1197
C12802 # IM echE 2003 Proc. Instn Mech. Engrs Vol. 217 Part C: J. M echanical Engineering Science
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7. For simplicity, both heat partition ratios B1… x 0† an d
B2… x 00† are assumed to be unchanged along the Y 0 or
Y 00 directions respectively, if applicable [12].
8. A cutting wedge angle of 908 is considered to simplify
the problem, which is reasonable for most cutting
tools.
The coordinates X Y Z in Fig. 2 are attached to the
chip to model the chip t emperature while the coordi-nates X 000Y 000 Z 000 are attached to the workpiece while
modelling the temperature rise in the workpiece. To
model the heat partition along the tool–chip and tool–
workpiece interfaces, two coordinates X 0Y 0 Z 0 an d
X 00Y 00 Z 00 are adopted for the tool as
X 0 ˆ l ¡ Z 00
Y 0 ˆ ¡Y 00
Z 0 ˆ VB ¡ X 00
…1†
For the coordinates used here, X overlaps with X 0 an d
X 00 overlaps with X 000, so B1… x † a nd B1… x 0† are the same
an d B2… x 00† an d B2… x 000† are the same in this study.
2.2 Chip side temperature modelling
2.2.1 Eff ect of the primary heat source
As proposed by Huang and Liang [17], by considering
the primary heat source as the obliquely moving band
heat source with a velocity V chip , in the coordinates X Y Z
the temperature rise on the chip side due to the primary
heat source can be expressed as
ychip¡shear… X , Z †
ˆ qshear
2pk chip
… L0
e¡… X ¡ X i †V chip=…2achip †
6 K 0V chip
2achip … X ¡ X i†
2 ‡ … Z ¡ Z i†2
q µ ¶‡
1
2K 0
V chip
2achip
… X ¡ X i†
2 ‡ …2t ch ¡ Z ¡ Z i†2
q µ ¶
‡1
2K 0
V chip
2achip
… X ¡ X i†
2 ‡ … Z ‡ Z i†2
q µ ¶¼dli
…2†
where
X i ˆ l ¡ li sin…f ¡ a†
Z i ˆ li cos…f ¡ a†
L ˆ t ch
cos…f ¡ a†
2.2.2 Eff ect of the secondary heat source
By considering the secondary heat source as the non-
uniform moving band heat source with a velocity V chip,
in the coordinates X Y Z the temperature rise on the chip
side due to the secondary heat source can be expressed
Fig. 2 The used right-handed Cartesian coordinates and associated heat sources in thermal modelling
Y HUANG AND S Y LIANG1198
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as [17]
ychip¡friction … X , Z †
ˆ 1
2pk chip
… l0
B… x †qfrictional … x†e¡… X ¡ x †V chip=…2achip †
6
µ2K 0
R iV chip
2achip
´‡ 2K 0
R 0iV chip
2achip
´
‡ 2K 0 R 00
i V chip
2achip
´¶d x
ˆ 1
pk chip
… l0
B… x †qfrictional … x †e¡… X ¡ x †V chip=…2a†
6
µK 0
R iV chip
2achip
´‡ K 0
R 0iV chip
2achip
´
‡ K 0 R 00
i V chip
2achip
´¶d x …3†
where
R i ˆ
… X ¡ x †2 ‡ Z 2
q
R 0i ˆ
… X ¡ x †2 ‡ …2t ch ¡ Z †2
q
R
00
i ˆ
… X ¡ x †
2
‡ …2t ch ‡ Z †
2q
2.2.3 Chip side temperature
Th e temper atu re r ise o n the ch ip side, wh ich is
ychip¡shear … X , Z † ‡ ychip¡friction … X , Z †, is attributed to the
primary and the secondary heat sources. Along the tool–
chip interface, the temperature rise on the chip side can
be given as ychip¡shear… X , 0† ‡ ychip¡friction … X , 0† in th e
coordinates X Y Z .
2.3 Workpiece side temperature modelling
2.3.1 Eff ect of the primary heat source
Komanduri and Hou [14] considered that the primary
shear heat source was a band heat source obliquely
moving under the workpiece surface of a semi-innite
body with a velocity V cutting. The right part next to the
shear zone is imaginary and is extended for continuity in
modelling. The boundary condition for the workpiece
surface is considered to be insulated in this study. An
imaginary heat source HH with the same heat intensity
as that of the primary heat source is considered in this
model [19]. As shown in Fig. 3, the temperature rise due
to the primary heat source on the workpiece, which
contacts the tool worn ank face, can thus be shown as
yworkpiece¡shear … X 000, Z 000†
ˆ qshear
2pk workpiece
… L0
e¡… X 000 ¡li sin y¡VB†V cutting=…2aworkpiece †
6
K 0
µ V cutting
2aworkpiece
6
…VB ‡ li co s f ¡ X 000†2 ‡ … Z 000 ‡ li sin f†2
q ¶
‡ K 0µ V
cutting2aworkpiece
6
…VB ‡ li co s f ¡ X 000†2 ‡ …2t ‡ Z 000 ¡ li sin f†2
q ¶¼dli
…4†
2.3.2 Eff ect of the rubbing heat source
The rubbing heat source is considered as the band heat
source moving along the workpiece surface within a
semi-innite body with a velocity V cutting. The boundary
condition for the workpiece surface is considered to be
insulated in this study. It is a classical Jager’s moving
Fig. 3 Heat transfer model of the primary heat source relative to the workpiece side
MODELLING OF THE CUTTING TEMPERATURE DISTRIBUTION UNDER THE TOOL FLANK WEAR EFFECT 1199
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heat source conguration within a semi-innite medium.
An imaginary heat source II with t he same heat intensity
as that of the rubbing heat source is considered in thismodel [19]. As shown in Fig. 4, the temperature rise due
to the rubbing heat source on the workpiece, which
contacts the tool worn ank face, can be expressed as
yworkpiece¡rubbing… X 000, Z 000†
ˆ 1
pk workpiece
… VB
0
B2… x 000†qrubbing … x 000†
6e¡… X 000 ¡ x 000 †V cutting =…2aworkpiece †
6 K 0V cutting
2aworkpiece
… X 000 ¡ x 000†2 ‡ … Z 000†2
q µ ¶ ¼d x 000
…5†
2.3.3 W orkpiece side temperature
The temperature rise o n the workpiece, yworkpiece¡shear
… X 000, Z 000† ‡ yworkpiece¡rubbing… X 000, Z 000†, is m ainly co n-
tributed by primary and rubbing heat sources. Along
the tool–workpiece interface, the workpiece temper-
atur e r ise can b e wr itten as yworkpiece¡shear… X 000, 0† ‡yworkpiece¡rubbing… X 000, 0† in the coordinates X 000Y 000 Z 000 .
2.4 Tool side temperature modelling
2.4.1 Eff ect of t he secondary heat source
For a fresh tool, the boundary condition on the ank
can be considered to be insulated [17]. For a worn tool,
it is considered as adiabat ic because the t emperature rise
on both sides of the tool–workpiece interface should be
equal. The heat transfer model for the effect of the
secondary heat source on the to ol side is the same as tha t
for the fresh tool, except for the adiabatic boundary
condition along the tool–workpiece interface. The
temperature rise on the tool side due to friction can
thus be expressed as [17]
ytool¡friction … X 0, Y 0 , Z 0†
ˆ 1
4pk tool
… l0
‰1 ¡ B1… x 0†Šqfrictional … x0† d x 0
6
… w=2
¡w=2
2
R i
‡ 2
R 0i
´d y 0
ˆ 1
2pk tool
… l0
‰1 ¡ B1… x 0†Šqfrictional … x0† d x 0
6
… w=2
¡w=2
1
R i
‡ 1
R 0i
´d y 0 …6†
where
R i ˆ
… X 0 ¡ x 0†2 ‡ …Y 0 ¡ y 0†2 ‡ Z 02
q
R 0i ˆ
… X 0 ¡ 2l ‡ x 0†2 ‡ …Y 0 ¡ y 0†2 ‡ Z 02
q
2.4.2 Eff ect of the rubbing heat source
I t fo llows fr om sect io n 2.1 t ha t 1 ¡ B 2… x 00† [or
1 ¡ B 2… x 000†] of the rubbing heat source is transferred
to the tool as the non-uniform static rectangular heat
source. Both interface boundaries are considered as
adiabatic, considering the assumptions that temperature
rises are equal along both the tool–chip and tool–
workpiece interfaces. Thus, there are two main imagin-
ary heat sources JJ and KK [12, 19]. The heat intensity
of the imaginary heat source JJ is equivalent to that of
the rubbing heat source, and the heat intensity of the
imaginary heat source KK is twice the rubbing heat
source. The related heat transfer model is shown in
Fig. 5.
Fig. 4 H eat tran sfer model of th e rubbing heat source relative to the workpiece side
Y HUANG AND S Y LIANG1200
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The temperature rise at any point M … X 000, Y 00 , Z 00† on
the tool side due to rubbing can thus be shown to be
ytool¡rubbing… X 00, Y 00 , Z 00†
ˆ 1
4pk tool
… VB
0
‰1 ¡ B2… x 00†Šqrubbing … x 00† d x 00
6
… w=2
¡w=2
2
R i
‡ 2
R 0i
´d y 00
ˆ 1
2pk tool
… VB
0
‰1 ¡ B2… x 00†Šqrubbing … x 00† d x 00
6
… w=2
¡w=2
1
R i
‡ 1
R 0i
´d y 00 …7†
where
R i ˆ … X 00 ¡ x 00†2 ‡ …Y 00 ¡ y 00†2 ‡ … Z 00†2
q R 0
i ˆ
…2VB ¡ X 00 ¡ x 00†2 ‡ …Y 00 ¡ y 00†2 ‡ … Z 00†2
q
2.4.3 T ool rake and ank temperature rises
The temperature rise on the tool is mainly attributed
t o t he seco nd ar y a nd t he r ub bin g h ea t so ur ces
…ytool¡friction ‡ ytool¡rubbing†. When considering the effect
of the rubbing heat source on the tool temperature rise
in the coordinates X 0 Y 0 Z 0 , the temperature rise due
t o t he ru bb in g h ea t so ur ce ca n b e r ewr it t en a s
ytool¡rubbing …VB ¡ Z 0 , ¡ Y 0 , l ¡ X 0† based on the coordi-
nate transformation given in equation (1). Therefore the
temperature rise in the middle of the t ool rake face along
the tool–chip interface …Y 0 ˆ 0 a nd Z 0 ˆ 0) ca n b e
estimated by
ytoolrake … X 0, 0 , 0† ˆ y tool¡friction … X 0, 0, 0†
‡ ytool¡rubbing …VB, 0, l ¡ X 0† …8†
The effect of the secondary heat source on the tool
temperature rise can be given as ytool¡friction …l ¡ Z 00 ,
¡Y 00 , VB ¡ X 00† followed from the coordinate transfor-
mation. Thus, in the middle of the tool ank face side,
the temperature rise along the tool–workpiece interface
…Y 00 ˆ 0 and Z 00 ˆ 0) can be expressed as
ytoolflank … X 00 , 0 , 0† ˆ ytool¡friction …l,0 ,V B ¡ X 00†
‡ ytool¡rubbing… X 00, 0 , 0† …9†
2.5 Solution method for temperature distributions
It is assumed that the temperature rise on the chip side
and on t he tool side along the to ol–chip interface should
be equal as follows:
ychip¡shear … X , 0† ‡ ychip ¡friction … X , 0†
ˆ ytoolrake … X 0, 0 , 0† …10†
Similarly, on the workpiece side and on the t ool side
along the tool–workpiece interface the temperature rise
Fig. 5 Heat transfer model of the rubbing heat source relative to the tool side
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should be equal, i.e.
yworkpiece¡shear … X 000, 0† ‡ yworkpiece¡rubbing … X 000, 0†
ˆ y toolflank … X 00, 0 , 0† …11†
To solve the heat pa rtition ratios B 1 an d B2 along the
interfaces numerically, the contact length along the
tool–chip interface is divided into n1 sections and the
heat partition ratio in each section is represented as
B1, 1, . . . , B1, n1, as shown in Fig. 6. Similarly, the contact
length along the tool–workpiece interface is divided inton2 sections and the heat partition ratio in each section is
also represented as B2, 1, . . . , B2, n2, as shown in Fig. 6.
Equations (10) and (11) can be described by a set of
linear equations as
A 1 A 2
A 3 A 4
µ ¶…n1 ‡n2†6…n1 ‡n2†
B1
B2
µ ¶…n1‡n2 †61
ˆ A 5
A 6
µ ¶…n1 ‡n2†61
…12†
where
B1 ˆ ‰ B1, 1, . . . , B1, n1Šn161, B2 ˆ ‰ B2, 1, . . . , B2, n2
Šn261
an d A 1, . . . , A 6 are dened by ychip¡shear, ychip¡friction ,
ytoolrake, yworkpiece¡shear, yworkpiece¡rubbing, ytoolflank , a n d
positions along the X … X 0† or X 00… X 000† axis accordingly.
The temperatures at every section are inuenced not
only by the effect of the heat source of this section but
also of the heat sources of all the other sections. The
partition ratios B1 an d B2 can be estimated by solving
equation (12). Subsequently, the temperature rises
ytoolrake a nd ytoolflank can be predicted based on equations
(8) and (9) respectively. Finally, the temperature
distributions can be determined by including the room
temperature y0.
3 MODEL VALIDATION
3.1 Process parameter estimation for the chip formation
process
Given the cutting conditions in the orthogonal cutting,
namely, the cut ting speed V cutting, width of cut (as of thedepth of cut) w, undeformed chip thickness (as of the
feed ra te) t and material properties of the workpiece and
tool for a fresh tool, th e process information, such as the
cutting forces, shear angle and shear ow stress, can be
estimated with acceptable accuracy by applying Oxley’s
predictive ma chining th eory [20] or its modication [21].
Then the input variables required by the proposed
thermal model, including the chip velocity V chip,
frictional force on the rake face F and primary heat
intensity q shear can be calculated as follows:
V chip ˆ rV cutting
F ˆ F c sin…a† ‡ F t co s…a†
qshear ˆ ‰F c cos…f† ¡ F t sin …f†Š‰V cutting cos…a†= cos…f ¡ a†Š
t ch rw csc…f†
…13†
The heat intensity of the secondary heat source can be
determined based on the frictional force F , as discussed
in reference [17].
As the tool wears, neither the shear angle nor the chip
thickness cha nges noticeably [22]. The rubbing force F cw
due to ank wear can be modelled based on the process
information of chip formation and ank wear length VB
Fig. 6 Schematic for numerical computation o f the temperature rise in thermal modelling
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[22, 23]. If there is only an elastic contact under the tool
ank face, the heat source is uniform along the tool–
workpiece interface. Then the rubbing heat source
density a long the tool–workpiece interface can be simply
expressed as
qrubbing ˆ F cwV cutting
w VB…14†
Figure 7 depicts the modelling process of the tool–
workpiece temperature distribution.
3.2 Results and validation
3.2.1 M odel validation
To acquire the temperature distribution a long the tool–
workpiece interface, Boothroyd [24] applied an infrared
photographic technique in orthogonal cutting of a
tubular workpiece. The cutting conditions were an
Armco iron workpiece, cemented carbide t ool with a
208 rake angle, 0.60mm/rev feed, 6.35mm depth of cut
and 0.17 m/s cutting velocity. T he tools had articial
wear lengths (VB) of 0.381, 0.762 and 1.143mm.
Thermal conductivity of both the workpiece and the
chip is taken as 0.762J/cm(s) 8C and thermal diffusivity
is 0.220 cm2 /s, th e sam e as th at of a pu re iro n. Cemen -
ted carbide t ool thermal conductivity is considered to
be temperature independent in cutting [12] as 0.57
J/cm(s) 8C (K series carbide) [13].
Oxley’s predictive machining theory [20] is used to
estimate the required process information for the chip
formation process of cutting 0.03 per cent carbon
Armco iron. Waldorf’s worn tool force model [22, 23]
is applied to estimate the rubbing force at the tool–
workpiece interface. In the cutting steel workpiece, if the
ank wear length is greater than a particular value,
elastic contact and plastic contact coexist along the
tool–ank interface; otherwise, there is only elastic
contact [25]. As no information exists regarding this
particular value for Armco iron, it is assumed that there
was only uniform elastic contact along the interface for
these selected ank wear lengths. The predicted process
information is summarized in Table 1.
The estimated temperature distributions and the
measured temperature for the above conditions are
shown in Figs 8 to 10. The comparisons are within 10
per cent of error and the distribution predictions
resemble tho se of the measurements. Since the contribu-
tions of both the secondary heat source on the work-piece side and the rubbing heat source on the chip side
are ignored in this study, this simplication leads to the
temperature underestimation and it needs to include
these contributions for a more accurate prediction. The
observed error may also come from the estimated
process information, which is indispensable in predicting
the tool–workpiece temperature distribution. The aver-
age temperature information along the rake face and
ank face is listed in Table 2. The ratio between the
average rake face temperature and average ank face
temperature in kelvin is also shown in Table 2, which
ranges from 78 to 81 per cent under t he investigatedcutting conditions. This range is reasonable when
Fig. 7 Approach o f thermal modelling of the tool–workpiece interface
Table 1 Process information under the conditions of Booth-
royd [24]
VB (mm) F c…N† F t…N † f l (mm)Shear owstress (MPa) F cw…N †
0.381 18 616 20 196 5.5 16.5 414.3 563.80.762 18 616 20 196 5.5 16.5 414.3 1127.51.143 18 616 20 196 5.5 16.5 414.3 1691.2
Table 2 Temperature information under the conditions of Boothroyd et al. [24]
VB(mm)
Average
rake facetemperature( 8C)
Average
ank facetemperature( 8C)
Ratio between average
rake face temperatureand average ank facetemperature in kelvin (%)
0.381 888.9 638.3 78.40.762 890.8 652.8 79.6
1.143 893.5 664.9 80.4
MODELLING OF THE CUTTING TEMPERATURE DISTRIBUTION UNDER THE TOOL FLANK WEAR EFFECT 1203
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compared to the ratio of 82–95 per cent determined
from measurements [20, 24, 26].
3.2.2 Case study
To stu dy th e ap plicab ility o f th e mod el, ano ther
orthogonal machining case is researched here. The
cutting conditions were an AISI 1018 steel workpiece,
K3H carbide tool with a 78 rake angle and 5.16mm
depth of cut. As suggested in reference [12], this paper
considers a temperature value intermediate between the
bulk workpiece material and the average interface
temperature for evaluation of the workpiece thermal
properties. For simplicity, thermal conductivity of both
the workpiece and chip is taken as 0.489J/cm(s) 8C and
thermal diffusivity as 0.121cm2 /s at 200 8C [27]. C arbide
tool thermal conductivity is considered to be tempera-
ture independent in cutting [12] as 0.57J/cm(s) 8C (K
series carbide) [13].
Oxley’s predictive machining theory [20] is used to
estimate the required process information for the chip
formation process of cutting AISI 1018. Waldorf’s worn
tool force model [22, 23] is applied to estimate the
rubbing force a t the tool–workpiece interface. T hree
scenarios are investigated herein: (a) varying t he feed
Fig. 8 Temperature comparison along the tool–workpiece interface with a 0.381mm ank wear length
Fig. 9 Temperature comparison along the tool–workpiece interface with a 0.762mm ank wear length
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rate when VB ˆ 0.1 mm and cutting speed ˆ 1.32m/s;
(b ) v ar yin g th e an k wear len gth VB wh en feed
rate ˆ 0.28mm/rev and cutting speed ˆ 1.32m/s; (c)
varying the cutting speed when VB ˆ 0.2mm and feed
rate ˆ 0.28mm/rev. Under the selected ank wear
length, only t he un iform elastic contact along the tool–
workpiece interface is expected. The predicted process
information is summarized in Tables 3 to 5.
The estimated temperature distributions for the above
conditions are shown in Figs 11 to 13 and the average
temperature information along the rake face and ank
face is l isted in Tables 6 to 8. Th e temper atu re
distributions resemble those of Boot hroyd [24]. Th e rat io
between the average rake face temperature and average
ank face temperature in kelvin is investigated here for
the purpose of model validation. Tables 6 to 8 list the
Fig. 10 Temperature comparison along the tool–workpiece interface with a 1.143mm ank wear length
Table 3 Process information under the conditions VB ˆ 0.1mm and cutting speed ˆ 1.32m/s
Feedrate t (mm/rev) F c…N† F t …N† f l (mm) Shear ow stress (MPa) F cw…N†
0.17 2365.9 2050.5 12.6 1.1 469.6 136.30.19 2515.1 2095.7 13.5 1.1 465.8 135.20.28 3112.1 2262.5 16.2 1.3 455.2 132.0
Table 4 Process information under t he conditions feedrate ˆ 0.28mm/rev and cutting speed ˆ 1.32m/s
VB (mm) F c…N† F t …N † f l (mm) S hear o w st ress (M P a) F cw…N†
0.1 3112.1 2262.5 16.2 1.3 455.2 132.00.2 3112.1 2262.5 16.2 1.3 455.2 264.10.35 3112.1 2262.5 16.2 1.3 455.2 396.1
Table 5 Process information under the conditions VB ˆ 0.2mm/rev and feedrate ˆ 0.28mm
Cutting speed (m/s) F c…N† F t …N † f l (mm) Shear ow stress (MPa) F cw…N†
1.32 3112.1 2262.5 16.2 1.3 455.2 264.1
2.03 2812.5 1828.1 18.4 1.1 457.0 265.33.05 2561.1 1472.3 20.7 0.9 458.8 266.4
MODELLING OF THE CUTTING TEMPERATURE DISTRIBUTION UNDER THE TOOL FLANK WEAR EFFECT 1205
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comparisons. The ratio ranges from 75 to 82 per cent
under the investigated cutting conditions. Again t his
range is reasona ble when compared to the ra tio of 82–95
per cent acquired from measurements [20, 24, 26].
Based on the presented results, several conclusions
can be drawn:
1. The progression of ank wear changes the average
rake face temperature only slightly, as seen from
Tables 2 and 7.
2. Both the average ank temperature and average rake
temperature increase with ank wear length and
cutting speed.
Fig. 12 Temperature pro le under the different ank wear lengths along the tool–workpiece interface (feedrate ˆ 0.28mm/rev and cutting speed ˆ 1.32m/s)
Fig. 11 Temperature prole under th e different feed rat es along the tool–workpiece interface (VB ˆ 0.1mmand cutting speed ˆ 1.32m/s)
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3. It is found that both average temperatures decrease
with feed rate. Since shear ow stress decreases with
feed rate, as estimated here and by Oxley [20], the
rubbing force, which is a function of shear ow
stress, decreases with feed rate as well. Then heat
intensities of both the secondary a nd the rubbing
heat sources are expected to decrease with feed rate
accordingly. However, the conicted phenomena
were observed in orthogonal cutting of the AISI
1018 steel by Chao et al. [28].
4 CONCLUSIONS
This study investigates the temperature distributions by
considering the effect of the tool wear land. On the chip
side, the effect of the primary shear zone is modelled as
the uniform moving oblique band heat source and that of
the secondary shear zone as the non -uniform moving
band heat source within the semi-innite medium. For
the tool side, the effects of both the secondary and the
rubbing heat sources are mo delled as no n-uniform static
rectangular heat sources within the semi-innite medium.
On the workpiece side, the effect of the primary shear
zone is modelled as the uniform moving oblique band
heat source a nd that of the rubbing heat source as the
Fig. 13 Temperature prole under the different cutting speeds along the tool–workpiece interface(VB ˆ 0.2mm and feed rate ˆ 0.28mm/rev)
Table 6 Temperature information under the conditions
VB ˆ 0.1mm and cutting speed ˆ 1.32m/s
Feedrate t
(mm/rev)
Averagerake facetemperature
( 8C)
Averageank facetemperature
( 8C)
Ratio betweenaverage rake facetemperature andaverage ank facetemperature in kelvin
(%)
0.17 820.1 608.7 80.70.19 813.0 593.1 79.80.28 791.4 547.5 77.1
Table 7 Temperature information under the conditionsfeedrate ˆ 0.28mm/rev and cutting speed ˆ 1.32m/s
VB (mm)
Averagerake facetemperature
( 8C)
Averageank facetemperature
( 8C)
Ratio betweenaverage rake face
temperature andaverage ank facetemperature in kelvin
(%)
0.1 791.4 547.5 77.10.2 791.6 586.9 80.80.35 793.5 596.1 81.5
Table 8 Temperature information under the conditionsVB ˆ 0.2mm a nd feedrate ˆ 0.28mm/rev
Cutting
speed (m/s)
Averagerake facetemperature
( 8C)
Averageank facetemperature
( 8C)
Ratio between averagerake face temperatureand average ank facetemperature in kelvin
(%)
1.32 791.6 586.9 80.82.03 851.4 603.4 77.93.05 931.4 630.4 75.0
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