mathematical modeling of friction in the cutting zone

Proceedings of the International Conference on Industrial Engineering and Operations Management Riyadh, Saudi Arabia, November 26-28, 2019

© IEOM Society International

Mathematical Modeling of Friction In The Cutting Zone During Orthogonal Machining

Sunday Joshua Ojolo, Gbeminiyi Musbau Sobamowo and Mercy Adedayo Department of Mechanical Engineering

University of Lagos, Akoka-Yaba Lagos, 100213, Nigeria

[email protected], [email protected], [email protected]

Abstract

In this work, effects of friction on the machining process during orthogonal cutting are theoretically investigated. Mathematical models were developed to predict the effects of friction and cutting force on the machining process. The results showed that the coefficient of friction increased with increased cutting forces, cutting stress and temperature. The frictional stress decreased as the exponent (P) increased. However, the frictional stress increased as the coefficient of friction increased. Also, study revealed that as the temperature increased, the flow stress decreased. Furthermore, it is shown that as the feed speed increased, there is an increase in the flow stress of the material used. The cutting force increased with increasing the feed rate, depths of cut but decreased with increasing cutting speed. This work will enhance the influence of friction on the cutting process and also assist in the development of good products with good surface finish.

Keywords: Friction, Temperature, Force, Machining

1. IntroductionTool-chip friction and work material properties have long been recognized as two unsolved problems in

fundamental machining research. Previous studies on the tool chip friction have been primarily focused in machining with a positive rake angle tool, with various theories and experimental techniques have been proposed. In order to improve metal cutting processes, it is necessary to model metal cutting processes at a system level. A necessary requirement of such is the ability to model interactions at the tool chip interface and thus, predict cutter performance. Many approaches such as empirical, mechanistic, analytical and numerical have been proposed. Some level of testing for model development, either material, machining, or both is required for all. However, the ability to model cutting tool performance with a minimum amount of testing is of great value, reducing costly process and tooling iterations. In the cutting process, friction is mainly present in the rake face and the flank of the tool. The friction that acts on the rake face has a major influence, the other one can become also important and could take part in the stability of the system. Therefore, it is very necessary to set mathematical models for the determination of the friction on the flank.

Molinari et al (2011) observed that the heating of the chip is essential in the cutting zone during orthogonal cutting. The heats from the chip have a lot of influence in the sticking contact of the material as it leads to delay of the flow stress. They also focused on the roles of cutting speed and feed. As the cutting speed increases, the feed rate also increases thereby influencing the sticking contact.

Shi et al. (2002) studied the effect of friction on thermo-mechanical quantities in a metal cutting process. They observed that the shear strain is in the primary shear zone of the cutting process while the material near the tool tip undergoes the largest plastic strain rate. They observed that the maximum temperature rises in the cutting process occurs along the tool-chip interface and not in the primary zone. In orthogonal cutting the maximum temperature, the contact length, the shear angle and the cutting force is influenced strongly by the friction at the cutting zone.

Marusich (2001) compared cutting forces and chip morphologies for the AI6061-T6 where he made use of Finite element simulations. It was observed that in the orthogonal cutting process, the increase in speed leads to

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decrease in chip thickness and significant increase in temperature at the tool-chip interface in the cutting zone, while the temperature in the primary zone rise only modestly and have little effect in the change in cutting force.

Ojolo et al (2013) observed that during orthogonal cutting, the temperature of the cutting tool used in cutting zone plays an important role in thermal distortion and the machined part’s dimensional accuracy, as well as in tool life in machining. They used the Finite Element Analysis (FEA) to simulate the thermal behaviour of a carbide cutting tool in three-dimensional dry machining. Also a model was developed to determine the temperature rise at the shear plane and this was used to determine the effect of various parameters on temperature rise.

Salem et al (2012) studied the mathematical models for the cutting force which facilitates the choice of cutting conditions in machining of the tool steel. They studied how the cutting parameters influence the type of chip that is produced during cutting at the tool-chip interface. It was observed that the removal of material from tool increases the plastic deformation and also led to more heat generation

The aim of this study is to theoretically investigate the effects of friction at the cutting zone during orthogonal cutting. The objective is to develop and simulate mathematical model for friction at the cutting zone during orthogonal cutting. 2. Merchant’s Force Model The development of force model is based on Merchant force circle in Fig. 1.

Figure 1. Force equilibrium in orthogonal cutting (a) Forces acting on the chip, and (b) forces on the tool

According to Merchant’s force model, the cutting (Fc) and thrust (Ft) force components can be transformed to normal (N) and friction (F) force components applied on the tool face as follows:

sin cosc tF F Fα α= + (1)

cos sinc tN F Fα α= − (2) Where, α is the normal rake angle. According to Coulomb’s law, the apparent coefficient of friction on tool-chip interface becomes:

sin coscos sin

c t

c t

F FFN F F

α ατµσ α α

+= = =

− (3)

Based on the Fig. 1, the forces cannot be measured but expressed as: cos sins c tF F Fφ φ= − (4)

sin cosn c tF F Fφ φ= + (5) The total tangential, radial and axial forces can be obtained by summing the force elements in the same direction. The coefficient of friction is thus written as

tantan

t c

c t

F FF F

αµ

α+

=−

(6)

The shear stress in the sticking and sliding regions can be model as

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0 c

t c

k l l

l l lτ

µσ

< <= < ≤

(7)

Also, from Fig. 1, the following can be written ( )coscF R λ α= − (8)

( )sintF R λ α= − (9)

s ABF K lw= (10) cosN R λ= (11) sinF R λ= (12)

where

cos sin coss ABF K tw

Rθ φ θ

= = (13)

Following Eqns. (9) - (13), an expression for the friction coefficient is ( ) ( )( ) ( )

sin cos tancos sin tan

t

r

R RFF R R

λ α λ α αµ

λ α λ α α− + −

= =− − −

(14)

which gives ( ) ( )( ) ( )

sin cos tancos sin tan

λ α λ α αµ

λ α λ α α− + −

=− − −

(15)

In this work, an improved model of Johnson-Cook flow stress is developed, the shear force can be written as

( )00

1 log 1m

n AB rAB m AB

m r

T TB C

T Tε

σ σ σ εε

− = + + + − −

(16)

Recall that the frictional force is given as AB s ABF F µσ= = (17)

Therefore, from Eqns. (15), (16) and (17), the frictional force is

( ) ( )( ) ( ) ( )0

0

sin cos tan1 log 1

cos sin tan

mn AB r

AB m ABm r

T TF B C

T Tλ α λ α α ε

σ σ ελ α λ α α ε

− + − − = + + + − − − − −

(18)

Where, T determined from the temperature rise along the shear zone (X, Z) can be calculated as;

( )( )

( ) ( )

( ) ( )

( ) ( )

2 20

2220

0

2 20

2

1, 22 2 2

12 2

i chipi

chip

chipi i

chipx x Vl

chipashearchip shear i chip i

chip chip

chipi i

chip

Vk X X Z Z

a

VqT X Z e k X X t Z Z

k a

Vk X X Z Z

a

π

−−

−

− + − +

= − + − − +

− + −

∫ idl

(19)

where

( )sini iX l l φ α= − − ( )cosi iZ l φ α= − ( )cos

chiptl

φ α=

−

For the frictional force at tool-chip interface, the temperature rise along the tool-chip interface is

( ) ( )0

1 1 1, , 12

h b

tool chip frictiontool ib i

T X Y Z B x q dx dyk R Rπ−

−

′ = − + ′

∫ ∫ (20)

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where

( ) ( )2 2 2i i iR X x Y y Z ′= − + − + ( ) ( )2 2 22i i iR X h x Y y Z′ ′= − + + − +

s sshear

F Vq

lw= chip

friction

FVq

hw=

Based on the temperature rise profiles along the primary shear zone and tool-chip interface, the average temperatures are categorized as

( )0

0

,l

chip shear i

shear average

T X Z dlT T

l

−

− = +∫

(21)

and

( )0

0

,0,0nh

chip shear

tool chip

T X dXT T

h

−

− = + +∫

(22)

The calculated average temperature is used to compute the flow stress along the primary shear zone and at the tool-chip interface. It should be noted that the normal stress acting on the tool-chip interface is

21 2

4At

N ABAB

c IKγπσ τ α

= + − −

(23)

where

AB

TIT

τ τ ε τγ ε γ γ

∂ ∂ ∂ ∂ ∂= = + ∂ ∂ ∂ ∂ ∂

(24)

where

( ) ( )( ) ( ) ( )0

0

sin cos tan1 log 1

cos sin tan

mn AB r

m ABm r

T TB C

T Tλ α λ α α ε

τ σ σ ελ α λ α α ε

− + − − = + + + − − − − −

(25)

From Eqns. (24) and (25), I become

( ) ( )( ) ( )

( )

( )

1

0

1

00

1 log 1sin cos tancos sin tan

1 log

mn AB rAB

m r

mn AB r

m ABm r

T TnB C

T TI

T T dTm B CT T d

εε

ελ α λ α αλ α λ α α ε

σ σ εε γ

−

−

− + − − − + − = − − − − − + + + −

(26)

Maximum shear and shear strain rate at the tool-chip interface are given respectively as

( )1 cos2 sin cos

αγφ φ α

= −

(27)

intchip

in

Vt

γδ

= (28)

It should be noted that s

ABV

Cl

γ = (29)

where

( )cos

sin cosc

sV

Vα

φ φ α=

− (30)

Therefore,

900



( )cos

cosc

ABVC

lα

γφ α

=−

(31)

Where, ϕ can be numerically determined using Oxley’s model (1961) ( ) ( ) ( )

cos 2tan 1 2 sin 2

2 tana

φ γπφ φ φ γ ρ γρ−

= + − + − − − −

(32)

And the friction angle as

tan 1 24

At

AB

c IKγπθ φ = + − −

(33)

3. Results and Discussion Figures 2a-c show the comparison of the experimental results of Xu et al. (2010) with the present developed models. The model results cutting and tangential forces are compared with the experimental results as shown in Fig. 2. Fig. 2b presents the comparison of the experimental results of cutting stress with the results of the cutting stress in the present developed models while Fig. 1c displays the comparison of the experimental results of cutting temperature with the results of the cutting temperature in the present developed models. Also, the effects of coefficient of friction on the forces, cutting stress and temperature are presented. It is depicted that the coefficient of friction has direct relationship on the forces, cutting stress and temperature in a way that as coefficient of friction increases, the forces, cutting stress and temperature increase. This is because as the coefficient of friction increase, the frictional force that enhances the cutting process increases which consequently increase the cutting stress and more heat are generated in the material and the cutting zone thereby the cutting temperature is increased.

Figure 2. Model results of (a) cutting and tangential forces; (b) cutting stress; (c) cutting temperature

Figure 3 shows the effect of temperature on the flow stress. It is shown that as the temperature increases, the flow stress decreases. It is shown that increase in temperature results in decrease in flow stress due to softening effects on the material. Also, it is established that the cutting action and related friction at cutting surfaces increase the temperature of the tool material, which further accelerates the physical and chemical processes associated with tool wear. In order to remove the unwanted material as chips, these forces and motions are necessary; therefore, cutting tool wear is an economic penalty that must be accounted for in order to machine the part. The magnitude of this economic penalty can be minimized if the cutting process is planned and controlled based on sound knowledge of the cutting engagement, wear process and its dependency on the selection of cutting conditions. The cutting conditions normally controlled are the engagement of the cutting edge with the workpiece, the relative velocity of the cutting edge with the workpiece, and the feed velocity used to keep the tool engaged into uncut material. During the process planning stage, an assessment must be made concerning how difficult the material is to machine, the correct tool for the surface to be created must be selected, the appropriate tool material must be selected, and the type of cutting fluid needed must be determined. To make such choices, the wear environment in metal cutting must be understood and the related friction must be analyzed as carried in the present study.

0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4300

350

400

450

500

550

600

650

700

750

800

850

Coefficient of Friction,µ

For

ce

Expereimental, Ft, Xu et al. (2010)

Present Model, Ft

Expereimental, Fc, Xu et al. (2010)

Present Model, Fc

0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.41300

1350

1400

1450

1500

1550

1600

1650

1700

1750


Cut

ting

Str

ess

Expereimental, σc, Xu et al. (2010)

Present Model, σc

0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4600

650

700

750

800

850

900

950


Cutt

ing T

em

pera

ture

, oC

Expereimental, Tc, Xu et al. (2010)

Present Model, Tc

901



Figure 3. Effects of Temperature on the flow stress

Fig. 4 illustrates the variation of the tool rake angle with the flow stress. It shows that as the rake angle moves from the negative angle to zero, the flow stress reduces. It is clearly seen that the increase of cutting speed, feed rate and axial depth causes the friction stress increase dramatically. Meanwhile, the friction coefficient, angle and force increases with decreases of feed rate and increases of cutting speed and axial depth. It is established in literature that that the feed rate was the most dominant cutting condition on the cutting force, followed by the axial depth, radial depth of cut and then by the cutting speed. The cutting force increases with increasing the feed rate, depths of cut but decreases with increasing cutting speed.

Figure 4. Effects of tool rake angle on the flow stress for the two types on aluminum

4. Conclusion In this work, effects of friction on the machining process during orthogonal cutting has been theoretically investigated. Mathematical models were developed and parametric studies of the effects of friction and cutting force parameters on the machining process were carried out. The developed models were validated. It was established that as coefficient of friction increases, the forces, cutting stress and temperature increase. The frictional shear stress directly proportional to the normal stress increases. However, the frictional stress increases as the coefficient of friction increases. Also, the present work revealed that as the temperature increases, the flow stress decreases. Furthermore, it is shown that as the feed speed increases, there is an increase in the flow stress of the material used. The cutting force increases with increasing the feed rate, depths of cut but decreases with increasing cutting speed. This work will enhance the

100 200 300 400 500 600 7000

50

100

150

200

250

300

350

400

450

500

Temperature (oC)

Flow

stre

ss (N

/mm

2 )

AL 6061-T6AL 6082-T6

1 1.5 2 2.5 3 3.5 4 4.5 588

88.5

89

89.5

Feed speed (m/s)

Flow

stre

ss (N

/mm

2 )

-6 -4 -2 0 2 4 650

60

70

80

90

100

110

120

130

140

150

Tool rake angle (o)

Flow

stre

ss (N

/mm

2 )

AL 6061-T6AL 6082-T6

902



understanding of the influences of friction coefficient on the cutting process and also assist in the development of good products with good surface finish. References Abukhshim, N.A, Mativenga, P.T and Sheikh, M.A., Heat generation and temperature prediction in metal cutting: A

review and implications for high speed machining, International Journal of Machine Tools & Manufacture, vol. 46, pp. 782–800, 2006.

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McClain, B., Stephen, A., Batzer, G. Ivan Maldonado, A numeric investigation of the rake face stress distribution in orthogonal machining, Journal of Material Processing Technology, vol. 123, pp. 114-119, 2002.

Brocail, J., Laurent, D., Watremez, M., Identification of a friction model for modeling of orthogonal cutting, International Journal. Mach. Tool Manuf, vol. 50, pp. 807–814, 2010.

El Hakim, M.A. Shalaby, M.A, Veldhuis, S.C., Dosbaeva, G.K., Effect of secondary hardening on cutting forces, cutting temperature, and tool wear in hard turning of high alloy tool steels, Measurement, vol. 65, pp. 233–238, 2015.

Fang, N., Machining with tool–chip contact on the tool secondary rake face - Part I: A slip-line model, International Journal of Mechanical Sciences, vol. 44 pp. 2337–2354, 2002.

Fang, N., Jawahir, I.S., A new methodology for determining the stress state of the plastic region in machining with restricted contact tools, International Journal Mechanical Science, vol. 43, pp. 1747-1770, 2001.

Guoqin, S., Xiaomin, D., Chandrakanth, S. A., Finite element study of the effect of friction in orthogonal metal cutting, Finite Elements in Analysis and Design, vol. 38, no. 9, pp. 863–883, 2002.

Jaspers, P. F. C and Dautzenberg, J.H., Material behaviour in metal cutting: strain rates and temperatures in chip formation, Journal of Materials Processing Technology, vol. 121, no. 1, pp. 123-135, 2002.

Karpat, Y., Ozel, T., Predictive Analytical and thermal Modeling of Orthogonal Cutting Process Part 1: Predictions of Tool Forces, stresses, and Temperature Distributions, J. Manuf. Sci. Eng., vol. 128, pp. 435-444, 2006.

Lin, Z.C., Pan, W.C., Lo., A study of orthogonal cutting with tool flank wear and sticking behavior on the chip-tool interface, J. Mat. Proc. Tech., vol. 52, pp. 524-538, 1995.

Maity, K.P., Das, N.S., A Class of slip line field solutions for metal machining with sticking-slipping zone including elastic contact, Mater. Design, doi:10.1016/j.matdes, 2006.

Merchant, M. E., Mechanics of the metal cutting process I: Orthogonal cutting and a type 2 chip, Journal of Applied Physics, vol. 16, no. 5, pp. 267-275, 1945.

Moufki, A., Alain Molinari, Dudzinski, D., Modelling of Orthogonal Cutting with a Temperature Dependent Friction Law, Journal of the Mechanics and Physics of Solids, vol. 46, no. 10, pp. 2103-2138, 1998.

Mustafa, G., Ihsan, K., Ersan, A., Ulvi, S., Experimental investigation of the effect of cutting tool rake angle on main cutting force, Journal of Materials Processing Technology, vol. 166, pp. 44-49, 2005.

Oxley, P., Mechanics of metal cutting, International Journal of Machine Tool Design and Research, vol. 1, nos. 1-2, 89-97, 1961.

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Zhenhua, T., Yang, J.C., Lovell, M.R., Evaluation of interfacial friction in material removal processes: the role of workpiece properties and contact geometry, Wear, vol. 256, no. 7, 664-670, 2004.

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Biographies Sunday Joshua Ojolo is an Associate Professor of Machine Design and Manufacturing Engineering in University of Lagos since 2016.His area of specialisation is in machinery development, manufacturing engineering and energy systems. He has published over 70 research papers in reputable journals. He is a member of America Society of Mechanical Engineers, Nigerian Society of Engineers; a Fellow at Nigerian Institution of Mechanical Engineering; COREN Registered Engineer. He is on the editorial board of five reputable international journals. He is a consultant to many industries.

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Gbeminiyi Musbau Sobamowo is a Senior Lecturer in Themo-fluids at the University of Lagos. His area of specialisation is in thermal sciences, energy studies and modelling. He is a member of the Nigeria Society of Engineers and registered Engineer with COREN. He has published over 40 reasearch articles majorly in international journals. Mercy Adedayo is a Principal Engineer in the Department of Works and Services at the University of Lagos. She holds a masters degree in Mechanical Engineering. Her area of specialisation is Manufacturing Engineering.

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mathematical modeling of friction in the cutting zone

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