Wear 258 (2005) 890–897
Tool-chip friction in machining with a large negative rake angle tool
N. Fang∗
Department of Mechanical and Aerospace Engineering, Utah State University, Logan, UT 84322, USA
Received 5 April 2004; received in revised form 21 September 2004; accepted 23 September 2004
Available online 6 November 2004
Abstract
A fundamental and quantitative analysis of the tool-chip friction helps to better understand the mechanism of chip formation and other
complex phenomena in machining with a large negative rake angle tool. Built upon Lee and Shaffer’s model, this paper presents an analytical
slip-line approach to investigate how the negative tool rake angle and the cutting speed affect the tool-chip friction, and how the tool-chip
friction further affects machining performances, such as the ratio of the cutting force to the thrust force, the chip thickness ratio, the geometry
of the shear zone, and the geometry of the stagnation zone of material flow adjacent to the tool rake face. Published experimental data covering
a wide range of negative tool rake angles and cutting speeds are employed to validate the analytical model. The predicted force ratio and chip
thickness ratio are in good agreement with the experimental data. Different effects resulting from the positive and negative rake angles on the
tool-chip friction are compared and analyzed.
© 2004 Elsevier B.V. All rights reserved.
Keywords: Friction; Machining; Large negative rake angle; Slip-line approach
1. Introduction
Tool-chip friction and work material properties have long
been recognized as two unsolved bottleneck problems in fundamental machining research. Previous studies on the toolchip friction have been primarily focused in machining with
a positive rake angle tool, with various theories and experimental techniques having been proposed [1–4]. The study
on the tool-chip friction in machining with a negative rake
angle tool, especially with those tools having the rake angle
greater than −40◦ , is still limited. This latter study receives
growing attention in recent years due to the need to better
understand the mechanism of chip formation in a variety of
both traditional and emerging machining techniques, such
as:
(1) grinding [5],
(2) machining with chamfered tools [6],
(3) hard turning [7],
∗
Tel.: +1 435 797 2948; fax: +1 435 797 2417.
E-mail address: [email protected].
0043-1648/$ – see front matter © 2004 Elsevier B.V. All rights reserved.
doi:10.1016/j.wear.2004.09.047
(4) micro-machining [8],
(5) nano-machining [9].
In the machining operations mentioned above, the grains
of a grinder, or the chamfered or honed tool cutting edges,
machine the surface of the work material at a very small undeformed chip thickness. These grinding and cutting processes
are actually performed within the domain of large negative
tool rake angles. Under such machining conditions, the traditional metal cutting theories proposed in [1] and [10], which
are primarily developed for machining with a positive or a
small negative rake angle tool, appear less competent or even
no longer applicable.
Slip-line theory has been recognized as one of the most
effective tools to model chip formation in machining [1].
Based on the assumptions of rigid-plastic material property
and plane-strain deformation, a slip-line model can clearly
show the material flow in the entire shear zone. It also allows for rapid computations of major machining performance
measures such as the cutting forces and the chip thickness.
A rapid computation is very important for sensitivity analysis of the tool geometry and cutting conditions. Due to these
N. Fang / Wear 258 (2005) 890–897
said advantages, various slip-line models have been developed in the past decades for different machining operations
[11–15]. For example, in their classic paper, Lee and Shaffer
[11] developed two slip-line models for orthogonal metal cutting with continuous chip formation. Their first model does
not consider built-up edge (BUE) formation and is only applicable to machining with a large positive rake angle tool
and low tool-chip friction. Their second model accounts for
BUE formation and is applicable to machining with a small
positive or a small negative rake angle tool with severe (intensive) tool-chip friction. Compared to their first model that
has been broadly cited in numerous machining literatures,
the Lee and Shaffer’s second model receives less attention
in the international machining research community. Part of
the reason, as some researchers commented, is that the BUE
size predicted from their second model is fairly small and
often not observed under the cutting conditions commonly
employed in machining tests.
In a most recent investigation of machining with a large
negative rake angle tool, Ohbuchi and Obikawa [16] showed
some photomicroscopes of material flow in the shear zone
and confirmed the existence of a triangular stagnation zone
of material flow ahead of the tool rake face, which has been
previously reported by Komanduri [5] and Kita et al. [17–19].
The stagnation zone is very similar to a BUE in appearance
except that the former has a more stable structure. Based on
the experimental observations of the geometry and dimension
of the stagnation zone, Ohbuchi and Obikawa [16] employed
the finite element method to model chip formation in machining with a large negative rake angle tool. With its rake angle
assumed to be 0◦ or +10◦ , the stagnation zone was attached
to the cutting tool as a single solid body to perform finite
element analysis.
In the present study, the Lee and Shaffer’s second model
[11] stated before is extended to model chip formation in
machining with a large negative rake angle tool. The goal is
to develop an analytical slip-line approach to investigate the
tool-chip friction under such special machining conditions.
Two major objectives of this study include understanding
how the tool rake angle and the cutting speed affect the
tool-chip friction, and how the tool-chip friction further
affects machining performances, the geometry of the shear
zone, and the geometry of the stagnation zone. Tool-chip
friction serves in this paper as an important bridge to link
the tool geometry and cutting conditions with machining
performances. The similarities and differences between
machining processes with both negative and positive rake
angle tools are compared. Published experimental data [20]
in the machining of a lead–antimony alloy are employed to
validate the analytical model. Although this type of alloy
is not extensively employed nowadays, the data [20] are
still valuable because they are very comprehensive covering
a wide range of negative tool rake angles and cutting
speeds.
It needs to be pointed out that the scope of the present
study is limited to orthogonal metal cutting with sharp edge
891
tools. Numerous studies [15,21–25] have demonstrated that
the cutting edge radius is a very important factor controlling chip formation in micro- and nano-machining. Material
flow in the tertiary shear zone near a round cutting edge is
extremely complex [15]. Much further research is needed to
model chip formation with both the large negative rake angle
and the tool edge radius taken into account.
2. Analytical modeling of chip formation
2.1. Slip-line model and hodograph
Extended from the Lee and Shaffer’s second model [11],
Fig. 1 shows the slip-line model and hodograph for machining
with a large negative rake angle tool. The slip-line field ABCF
consists of three regions as follows:
(1) a central-fan region AFC with the slip-line angle θ,
(2) a triangular region BCD with the angle η controlled by
the tool-chip friction,
(3) a triangular region ABD with the stress-free boundary
AB.
The region CFE, with the apex angle ψ at point F, represents the stagnation zone of material flow ahead of the tool
rake face. It is treated as a rigid body with a stable structure
as observed in cutting experiments [16], not a built-up edge
as defined in Lee and Shaffer’s model [11]. The top boundary
of the stagnation zone is the slip-line CF, taking the form of
a circular arc. The bottom boundary of the stagnation zone is
a straight line EF parallel to the cutting velocity Vc . The ge-
Fig. 1. Machining with a large negative rake angle tool: (a) extended Lee
and Shaffer’s model [11] and (b) hodograph.
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N. Fang / Wear 258 (2005) 890–897
ometry of the stagnation zone shown in Fig. 1 is very similar
to the triangular geometry observed in cutting experiments
[16].
In Fig. 1(a), γ 1 is the tool rake angle, t1 the undeformed
chip thickness, Vch the chip velocity, and φ is commonly
referred to as the ‘shear-plane’ angle. The hodograph in
Fig. 1(b) shows how the velocity changes from Vc to Vch
when the work material passes through the shear zone. This
is caused by the velocity discontinuity along the slip-line AF
and continuous variations in velocity in the central-fan region
AFC.
2.2. Mathematical formulation
To fully define the slip-line field shown in Fig. 1, three
input variables need to be determined first, including the toolchip friction factor τ/k, the hydrostatic pressure PA /k at point
A, and the tool rake angle γ 1 . The variable τ is the material
frictional shear stress at the tool–chip interface, and k the
material shear flow stress. τ/k varies between 0 and 1.0, with
a large value of τ/k meaning intensive tool-chip friction. The
tool-chip friction is also often quantified by the coefficient of
friction [1,10].
Once the three input variables τ/k, PA /k, and γ 1 are given,
the model can predict the geometry of the slip-line field, the
cutting forces (in a dimensionless form), the chip thickness,
and the geometry and dimensions (length and height) of the
stagnation zone. Lee and Shaffer [11] have provided major
mathematical equations for determining these output variables. The present work makes two modifications to Lee and
Shaffer’s mathematical modeling. One modification is to reexpress all output variables in terms of τ/k, PA /k, and γ 1 , as
later given in Eqs. (1–7) in order to facilitate the investigation
on the effect of tool-chip friction factor τ/k. The other modification is to develop a set of new equations, as later shown
in Eqs. (8–14) for calculating the forces acting on the bottom
boundary EF of the stagnation zone. The method suggested
by Lee and Shaffer [11] for calculating the forces acting on
EF does not apply to machining with a large negative rake
angle tool, due to the stagnation zone being usually of a large
size in this machining situation.
2.2.1. The chip thickness and the geometry of the
stagnation zone
In Fig. 1(a), the angle η caused by the tool-chip friction
factor τ/k is given by
η=
1
τ
cos−1
2
k
(1)
The friction angle λ, which is commonly defined as the
angle between the resultant force and the normal force acting
on the tool rake face BC, is expressed as
λ=
π
−η
4
(2)
The slip-line angle θ of the central-fan region AFC is determined by
θ=
(PA /k) − 1
2
(3)
From the geometrical relationship shown in Fig. 1(a), the
apex angle ψ of the stagnation zone is expressed as
ψ=
π
+ λ − γ1 − θ
4
(4)
Note that γ 1 in Eq. (4) takes a negative value, or −γ 1 in Eq.
(4) takes a positive value.
The chip thickness ratio t2 /t1 is calculated as
t2
cos((π/4) − λ)
=
t1
cos ψ
(5)
The length l (i.e., CE) and the height h (i.e., EF) of the
stagnation zone are determined by
l
sin((π/4) + λ) − sin(ψ + γ1 )
=
t1
cos ψ cos γ1
(6)
h
l
sin((θ/2) + ψ)
=
t1
t1 cos ((θ/2) + ψ + γ1 )
(7)
2.2.2. The cutting forces
The forces acting on the entire tool rake face BE consist
of (1) the forces transmitted through the slip-line AF and (2)
the forces acting on the bottom boundary EF of the stagnation zone. If the friction on EF is τ ef /k, which takes 1.0 in
the present study to best fit the experimental results, the frictional force Fef and the normal force Nef acting on EF can be
determined by using the plastic slip-line theory as
Fef = klw cos(2ηef )
(8)
Nef = klw [1 + 2θ + π − 2ψ + 2ηef + sin(2ηef )]
(9)
where w is the width of cut, and by analogy to Eq. (1), the
angle ηef caused by τ ef /k is expressed as
ηef =
1
τef
cos−1
2
k
(10)
Therefore, the cutting force Fc and the thrust force Ft are
determined by
Fc = kt1 w (1 + 2θ + tan ψ) + Fef
(11)
Ft = kt1 w [(1 + 2θ) tan ψ − 1] + Nef
(12)
The first terms on the right-hand side of Eqs. (11) and (12)
are two force components transmitted through the slip-line
AF. Note that Eqs. (11) and (12) contain the material shear
flow stress k, which is an unknown variable. Therefore, the
N. Fang / Wear 258 (2005) 890–897
893
dimensionless forces Fc /kt1 w and Ft /kt1 w are employed in
the present study and determined by substituting Eqs. (6),
(8), and (9) into Eqs. (11) and (12) as
Fc
= 1 + 2θ + tan ψ + cos(2ηef )
kt1 w
×
sin((π/4) + λ) − sin(ψ + γ1 )
cos ψ cos γ1
(13)
Ft
= (1 + 2θ) tan ψ − 1 + [1 + 2θ + π − 2ψ + 2ηef
kt1 w
+ sin(2ηef )]
sin((π/4) + λ) − sin(ψ + γ1 )
cos ψ cos γ1
(14)
The force ratio Fc /Ft can then be easily determined from
Eqs. (13) and (14) without the need to determine the unknown
material shear flow stress k.
Fig. 2. Admissible combinations of the tool-chip friction factor τ/k and the
hydrostatic pressure PA /k for different negative tool rake angles.
is suggested:
2
Fc
Fc
−
Ft expe
Ft pred
D = 2
+ tt21
− tt21
pred
2.3. Determination of the tool-chip friction and the
hydrostatic pressure
In theory, multiple combinations of the tool-chip friction
factor τ/k and the hydrostatic pressure PA /k exist for a tool
rake angle γ 1 . The combinations that result in a negative slipline angle θ, a negative ‘shear-plane’ angle φ, or a negative
apex angle ψ of the stagnation zone are apparently inadmissible. Fig. 2 shows admissible combinations of τ/k and PA /k
for the tool rake angles of −60◦ and −30◦ . As seen from
Fig. 2, the admissible value of PA /k increases with increasing absolute value of the negative tool rake angle, implying
that the stress state in the shear zone varies with the tool rake
angle.
In order to determine a particular combination of τ/k and
PA /k for a specific cutting operation, the following equation
(15)
expe
where D stands for the overall prediction error for the force
ratio Fc /Ft and the chip thickness ratio t2 /t1 , and the subscripts
‘pred’ and ‘expe’ represent the predicted and experimental
values, respectively. The combination of τ/k and PA /k that
makes D minimum is chosen as the best-fitted combination
for the specific cutting operation investigated.
Fig. 3 shows how the method stated above is employed.
The input and output variables of the developed analytical
model are also shown in Fig. 3.
3. Experimental validation
Findley and Reed [20] have performed extensive orthogonal cutting tests covering a very broad range of cutting speeds
and negative tool rake angles up to −60◦ . The tool material
Fig. 3. Computational flow chart of the analytical model.
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N. Fang / Wear 258 (2005) 890–897
Fig. 5. Predicted and experimental force ratio and chip thickness ratio at the
tool rake angle of −40◦ for varying cutting speeds.
Fig. 4. Predicted and experimental force ratio and chip thickness ratio at the
cutting speeds of (a) Vc = 120 m/min and (b) Vc = 170 m/min for varying tool
rake angles.
employed was primarily tungsten carbide. The work material employed was a casting of lead–antimony alloy with the
following chemical composition: copper, none; tin, 0.06%;
lead, 88.85%; antimony, 10.9%; arsenic, 0.13%. Their experimental data of the cutting forces and the chip thickness, which were carefully measured in the cutting tests,
are employed in the present study to validate the analytical
model.
Note that Findley and Reed [20] also reported that discontinuous and segmented chips were generated for some
particular combinations of the negative tool rake angle
and the cutting speed. Only those combinations that generate continuous chip formation are selected for experimentally validating the analytical model presented in this
paper.
3.1. Varying tool rake angles
Fig. 4 shows the comparison of the predicted and experimental force ratio Fc /Ft and chip thickness ratio t2 /t1 for
the tool rake angles ranging from 0◦ to −60◦ . The predicted
results are generated using the method described in Section
2.3, with the best-fitted combination of τ/k and PA /k for each
cutting operation given later in Section 4.
As seen in Fig. 4, both the predicted Fc /Ft and t2 /t1 are
in good agreement with the experimental results for the tool
rake angles γ 1 from −20◦ to −60◦ . A relatively large error
of prediction exists for γ 1 of 0◦ , possibly due to the predicted
size of the stagnation zone being larger than its actual size
at this specific rake angle. Comparing the prediction error
for γ 1 from −60◦ to −20◦ and the prediction error for γ 1
from −20◦ to 0◦ , it can be said that the developed analytical
model does work, as expected for large negative rake angles.
For small negative rake angles, the models developed by other
researchers, i.e., by Kopalinsky and Oxley [26], may generate
more accurate predictions.
A more detailed analysis of the results shown in Fig. 4
demonstrates that a negative rake angle and a positive rake
angle can affect machining processes quite differently. For
example:
(1) When using a negative γ 1 , the force ratio Fc /Ft is
typically less than 1.0, which means the cutting force
Fc is less than the thrust force Ft . Other researchers
[5,16,19,26] also have confirmed this unique phenomenon. However, as well known, Fc is generally larger
than Ft in machining with a positive γ 1 .
(2) The force ratio Fc /Ft decreases with increase in the absolute value of negative γ 1 . A decrease in positive γ 1
decreases Fc /Ft .
(3) The chip thickness ratio t2 /t1 decreases with decrease in
the absolute value of negative γ 1 . An increase in positive
γ 1 decreases t2 /t1 .
3.2. Varying cutting speeds
Fig. 5 shows the comparison of the predicted and experimental force ratio Fc /Ft and chip thickness ratio t2 /t1
at the cutting speeds ranging from 120 to 1120 m/min.
The tool rake angle γ 1 kept constant at −40◦ . As seen
from Fig. 5, good agreement between the predicted and experimental Fc /Ft and t2 /t1 is also achieved. The predicted
chip thickness ratio t2 /t1 agrees very well with the experimental results, with the average error of predictions less
than 1%.
Fig. 5 also shows that as the cutting speed increases, the
force ratio Fc /Ft increases, while the chip thickness ratio t2 /t1
decreases. The effect of the cutting speed appears similar for
both negative and positive tool rake angles.
N. Fang / Wear 258 (2005) 890–897
895
Fig. 8. Effect of tool-chip friction factor τ/k on the force and chip thickness
ratios.
Fig. 6. Tool-chip friction factor τ/k and hydrostatic pressure PA /k for the
same cutting conditions given in Fig. 4.
4. Effect of tool rake angle and cutting speed on the
tool-chip friction
Figs. 6 and 7 show the combination of the tool-chip friction factor τ/k and the hydrostatic pressure PA /k for the same
cutting conditions given in Figs. 4 and 5, respectively. As seen
from Figs. 6 and 7, the tool-chip friction factor τ/k decreases
with increasing absolute value of negative tool rake angle γ 1
and with increasing cutting speed Vc . The hydrostatic pressure PA /k increases with increasing absolute value of γ 1 and
keeps nearly constant with Vc . Both τ/k and PA /k follow certain varying trends that make physical sense, which further
demonstrates that the method described in Section 2.3 is not
an ‘all over the map’ approach to make the basic analytical
model work.
As known, the tool-chip friction generally decreases with
increasing γ 1 in machining with a positive rake angle tool [1].
However, Fig. 6 demonstrates that this common knowledge
does not apply to machining with a negative rake angle tool.
Supposing the tool rake angle γ 1 changes from −60◦ to 0◦ ,
and then to +30◦ , the tool-chip friction factor τ/k will first
increase from a low value at γ 1 of −60◦ to a high value at γ 1
of 0◦ , and then drops to another low value at γ 1 of +30◦ .
The effect of cutting speed on the tool-chip friction appears
similar for both negative and positive tool rake angles, as
illustrated in Fig. 7.
Fig. 7. Tool-chip friction factor τ/k and hydrostatic pressure PA /k for the
same cutting conditions given in Fig. 5.
5. Effect of tool-chip friction on the force ratio, the
chip thickness ratio, the shear-zone geometry, and the
stagnation-zone geometry
To separately study the effect of tool-chip friction factor
τ/k, it is reasonable to change τ/k while keeping two other
input variables (i.e., the hydrostatic pressure PA /k and the
tool rake angle γ 1 ) of the analytical model constant. For the
cutting tests shown in Fig. 5, γ 1 remains unchanged at −40◦ ,
PA /k keeps nearly constant as shown in Fig. 7 and only τ/k
varies slightly with the cutting speed. Therefore, the predicted
results for the cutting tests shown in Fig. 5 are employed in
this section.
Figs. 8–10 show the effect of τ/k on the force and chip
thickness ratios (Fc /Ft and t2 /t1 ), major angles (the friction
angle λ, the ‘shear-plane’ angle φ, the slip-line angle θ, and
the apex angle ψ of the stagnation zone), the coefficient of
friction µ (i.e., tan λ), and the length and the height of the
stagnation zone (l/t1 and h/t1 ), respectively. Fig. 8 was re-
Fig. 9. Effects of tool-chip friction factor τ/k on (a) major angles and (b) the
coefficient of friction.
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N. Fang / Wear 258 (2005) 890–897
observed in cutting experiments (see Fig. 4 in [16]). This
further implies that if the analytical model presented in this
study is integrated into a finite element model, the need for
the latter model to assume the stagnation-zone geometry before conducting detailed stress and thermal analysis can be
eliminated.
6. Conclusions
Fig. 10. Effect of tool-chip friction factor τ/k on the length and height of the
stagnation zone.
produced using the same set of data given in Figs. 5 and 7. It
shows that τ/k has nearly no effect on the force ratio Fc /Ft , but
significantly affects the chip thickness ratio t2 /t1 . The higher
τ/k, the larger t2 /t1 is.
Fig. 9 shows that as τ/k increases, the friction angle λ,
the slip-line angle θ, the apex angle ψ of the stagnation zone,
and the coefficient of friction µ all increases, while the ‘shearplane’ angle φ decreases. Fig. 10 shows that both the length
l/t1 and the height h/t1 of the stagnation zone increase with
increasing τ/k, and τ/k has more effect on l/t1 than on h/t1 .
All analytically modeled results shown in this study have
real physical sense. As an additional example, Fig. 11 shows
the predicted stagnation-zone geometry at three different cutting speeds to further demonstrate the effect of τ/k. As seen
from Fig. 11, the size of the stagnation zone decreases with
increase in the cutting speed. To the best knowledge of the
author of this paper, to date, no other analytical approach
has been developed to predict the stagnation-zone geometry.
The predicted geometry shown in Fig. 11 is similar to that
A fundamental and quantitative analysis of the tool-chip
friction helps to better understand the mechanism of chip
formation and various phenomena in machining with a large
negative rake angle tool. Compared to the amount of experimental and finite element modeling work that has been conducted on this topic, analytical modeling work still seems
limited. This paper has extended Lee and Shaffer’s physical
slip-line model [11], with two mathematical modifications,
to study the tool-chip friction in machining with a large negative rake angle tool. Published experimental data [20] have
been employed to validate the analytical model. Encouraging good agreement has been reached between the predicted
and experimental force ratio and chip thickness ratio over a
broad range of cutting speeds and negative tool rake angles.
The following paragraphs summarize major findings made
from the present study.
(1) The negative tool rake angle affects the force ratio Fc /Ft ,
the chip thickness ratio t2 /t1 , and the tool-chip friction
factor τ/k in a manner contrary to a positive tool rake
angle does. This has been confirmed by both the predicted
and experimental results.
(2) The effect of cutting speed on Fc /Ft , t2 /t1 , and τ/k appears
similar for both negative and positive rake angle tools.
(3) The tool-chip friction τ/k decreases with increasing absolute value of negative tool rake angle γ 1 and with increasing cutting speed Vc .
(4) In machining with a negative rake angle tool, an increase
in the tool-chip friction factor τ/k increases the chip thickness ratio t2 /t1 , the friction angle λ, the slip-line angle θ,
the apex angle ψ of the stagnation zone, and the coefficient of friction, but decreases the ‘shear-plane’ angle φ.
τ/k has nearly no effect on the force ratio Fc /Ft .
(5) The length and the height of the stagnation zone can be
predicted by using the analytical approach presented in
this paper. It is shown that the size of the stagnation zone
decreases with increase in the cutting speed.
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Fig. 11. Predicted stagnation-zone geometry for the tool rake angle of −40◦
at different cutting speeds (to scale).
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