ESTIMATION OF WORK MATERIAL FLOW STRESS AND TOOL-CHIP

INTERFACIAL FRICTION BY INVERSE COMPUTATION OF MODIFIED

OXLEY’S MODEL OF MACHINING

Tuğrul Özel* and Erol Zeren

Department of Industrial and Systems Engineering

Rutgers, The State University of New Jersey, Piscataway, New Jersey 08854 USA

*Corresponding author. Tel: +1-732-445-1099; fax: +1-732-445-5467.

E-mail address: ozel@rci.rutgers.edu (T. Özel)

ABSTRACT

Finite Element Analysis (FEA) based techniques have become increasingly available to

simulate metal cutting process and offer several advantages including prediction of tool

forces, distribution of stresses and temperatures, estimation of tool wear, optimization of

cutting conditions, tool edge geometry and coatings, and determination of residual

stresses on machined surfaces. However, accuracy and reliability of the predictions

heavily depend on the work material flow stress at cutting regimes i.e. high deformation

rates and temperatures, and variable friction characteristics at tool-chip interface, which

are not completely understood and need to be determined. This paper utilizes an extended

metal cutting analysis originally developed by Oxley and coworkers [6] and presents an

improved methodology to expand applicability of Johnson-Cook material model to the

cutting regimes and simultaneously determine friction characteristics at tool-chip

interface from the results of orthogonal cutting tests. A number of cutting temperature

models are also assessed in characterizing the accuracy of the average temperatures at

primary and secondary deformation zones. Predictions using cutting models are

compared with the results of testing data obtained for AISI 1045 through assessment of

machining models (AMM) activity. The methodology requires experimental data for

cutting and thrust forces, chip thickness and tool-chip contact length. The methodology is

practical and estimates both the parameters of work material Johnson-Cook constitutive

flow stress model and the interfacial friction characteristics along tool rake face. The

friction model is based on estimation of normal stress distribution over the rake face.

Predicted friction characteristics include parameters of the stress distribution on the rake

face σnmax and power exponent a, the average shear flow stress at tool-chip interface, kchip,

plastic contact length, lp, at sticking region of the tool-chip interface, an average

coefficient of friction, µe, in sliding region of the tool-chip interface. The extended

Johnson-Cook work material flow stress parameters and tool-chip interfacial friction

characteristics can directly be entered into most FEA software.

KEYWORDS: Flow stress, friction, orthogonal cutting, constitutive models.

1. INTRODUCTION

There has been considerable amount of research devoted to develop analytical,

mechanistic and finite element (FE) based numerical models in order to simulate metal

cutting processes [1-6]. Especially FE based models are highly essential in predicting

chip formation, computing forces, distributions of strain, strain rate, temperatures and

stresses on the cutting edge and the chip and the machined work surface [12]. The

premise of the FE based models is to be able to lead further predictions in machinability,

tool wear, tool failure and surface integrity in the machined surfaces. Success and

reliability of FE based models are heavily dependent upon work material constitutive

flow stress models in function of strain, strain rate and temperatures, as well as friction

parameters between tool and work material interfaces [11].

In material removal processes using geometrically defined cutting edges, work

material goes severe deformations and shearing at very high deformation rates and

temperatures. Work material flow stress determined through tensile and/or compression

tests may not be valid to represent deformation behavior in the ranges of strain, strain rate

and temperature observed during especially high-speed machining, a popular operation to

increase productivity. In addition, friction between work material and cutting tool surface

plays a significant role in predicted or simulated force, stress and temperature

distributions. It has been repeatedly overstressed that a good understanding of metal

cutting mechanics and reliable work material flow stress data and friction characteristics

between the work material and the cutting tool edges must be generated for high speed

cutting conditions [12].

2. CONSTITUTIVE MODELS FOR WORK MATERIAL FLOW STRESS

The flow stress or instantaneous yield strength at which work material starts to

plastically deform or flow is mostly influenced by temperature, strain and strain rate

factors (Childs, 1998). Accurate and reliable flow stress models are considered highly

necessary to represent work material constitutive behavior under high-speed cutting

conditions for a (new) material. Unfortunately sound theoretical models based on atomic

level material behavior are far from being materialized as reported by Jaspers and

Dautzenberg [16]. Therefore, semi-empirical constitutive models are widely utilized. The

constitutive model proposed by Johnson and Cook [7] describes the flow stress of a

material with the product of strain, strain rate and temperature effects that are

individually determined as given in Eq. (1).

( ) 0

001 ln 1

mn

m

T TA B CT T

εσ εε

− = + + − − (1)

In the Johnson-Cook (JC) model, the parameter A is in fact the initial yield strength of the

material at room temperature and a strain rate of 1/s and ε represents the plastic

equivalent strain. The equivalent plastic strain rate ε is normalized with a reference

strain rate 0ε . Temperature term in JC model reduces the flow stress to zero at the

melting temperature of the work material, Tm, leaving the constitutive model with no

temperature effect. In general, the parameters A, B, C, n and m of the model are fitted to

the data obtained by several material tests conducted at low strains and strain rates and at

room temperature as well as split Hopkinson pressure bar (SHPB) tests at strain rates up

to 1000/s and at temperatures up to 600 °C [11]. JC model provides good fit for strain-

hardening behavior of metals and it is numerically robust and can easily be used in FE

simulation models [16]. Zerilli and Armstrong (ZA) derived an alternative constitutive

model for metals with a crystal structure distinction by using dislocation-mechanics

theory [8].

Many researchers used JC model as constitutive equation for high strain rate, high

temperatures deformation behavior of metals. Hamann et al. [9] investigated the effect of

metallurgical treatments after deoxidation of free machining steels on the machinability

by conducting orthogonal cutting tests, a SHPB test, and numerical simulations of the

cutting process. They used a reverse method to identify parameters for JC constitutive

model for two low carbon free cutting steels, namely S300 and S300 Si and two low-

alloyed structural free machining steels, 42CD4 U and 42CD4 Ca. Lee and Lin [10]

fitted SHPB test results in to JC model to study high temperature deformation behavior of

Ti6Al4V, a titanium alloy. Meyer and Kleponis [13] also studied high strain rate behavior

of Ti6Al4V and fitted their experiments into JC model. Both research groups also

discussed advantages and disadvantages of using JC model as constitutive equation for

Ti6A4V. Jaspers and Dautzenberg [16] have concluded that strain rate and temperature

dramatically influence the flow stress of metals, steel AISI 1045 and aluminum AA 6082-

T6, according to measurements with SHPB tests. Firstly, they used JC and ZA

constitutive models to find parameters in the equations, and then compared results to

measurements. To calculate parameters, they did not prefer to use least square estimator

method because of its difficulty. Instead the parameters are investigated by leaving only

one parameter unconstrained, whereas the others are kept constant, and consequently all

parameters are found for both models.

3. PRIOR RESEARCH IN DETERMINATION OF WORK FLOW STRESS

FROM ORTHOGONAL CUTTING TEST

Late Oxley [6] and coworkers developed a model to predict cutting forces and

average temperatures and stresses in the primary and secondary deformation zones by

using (1) flow stress data of the work material as a function of strain and velocity-

modified temperature which couples strain rate to temperature, (2) thermal properties of

the work material, (3) tool geometry and (4) cutting conditions. They also utilized slip-

line field analysis to model chip formation in metal cutting and applied this theory to

cutting of low carbon steel and obtained flow stress and friction data by empirically

fitting the results of the orthogonal cutting tests [6].

Several researchers have used Oxley’s machining analysis, in order to obtain

work flow stress data. Shatla et al. [15] modified JC flow stress model to make and used

Oxley’s parallel sided thin shear zone orthogonal cutting model and empirically

determined parameters of the modified JC model by applying orthogonal high-speed slot

milling experimentation technique developed by Özel and Altan [13]. In an attempt to

utilize orthogonal cutting tests in determining flow stress data, Shatla et al. [15] also

presented some findings for the modified JC model for tool steels AISI P20 and AISI

H13, and aluminum, Al 2007. Tounsi et al. [17] stated a re-evaluated methodology

related to analytical modeling of orthogonal metal cutting -continuous chip formation-

process to identify variables of JC constitutive model. They applied least-square

estimator method to obtain variables of the JC constitutive model. Primary shear zone are

assumed having a constant thickness, and according to experiments and data that found

from literature, its value are approximated by one-half of the uncut chip thickness. Along

with data published by Hamann et al. [9] for 42CD4U, and S300, orthogonal cutting

experiments are undertaken for stainless steel 316L and 35NCD16 to verify the

effectiveness of proposed methodology. However, they neither presented an adequate

formulation for the friction at the tool-chip interface in their model, nor attempted to

determine friction parameters. Adibi et al. [19,20] extended Oxley’s analysis and applied

to different material models including Johnson-Cook’s constitutive model.

The research focus of this paper is to determine the values of the work material

flow stress parameters represent the excessive strain rates and temperatures during

machining process. To calculate these values, results of the orthogonal cutting

experiments including measured values for machining forces, chip thickness, and tool-

chip contact length are utilized. Specifically, based on orthogonal cutting tests, the

parameters of the JC model are calculated in the primary deformation zone using Oxley’s

slip-line field analysis in reverse [6]. The same experimental data is used in determining

the frictional parameters on the tool rake face. An experimentally determined slip-line

field proposed by Tay et al. [4] of the secondary deformation zone is used to determine

the shear flow stress kchip in the sticking region and coefficient of sliding friction µe in the

sliding region along the tool-chip inteface.

4. OXLEY’S MODEL OF ORTHOGONAL CUTTING

A simplified illustration of the plastic deformation for the formation of a

continuous chip when machining a ductile material is given in Fig.1. There are two

deformation zones in this simplified model – a primary zone and a secondary zone. It is

commonly recognized that the primary plastic deformation takes place in a finite-sized

shear zone. The work material begins to deform when it enters the primary zone from

lower boundary CD, and it continues to deform as the material streamlines follow smooth

curves until it passes the upper boundary EF. Oxley and coworkers [6] assumed that the

primary zone is a parallel-sided shear zone. There is also a secondary deformation zone

adjacent to the tool-chip interface that is caused by the intense contact pressure and

frictional force. After exiting from the primary deformation zone, some material

experiences further plastic deformation in the secondary deformation zone. Using the

quick-stop method to experimentally measure the flow field, Oxley [6] proposed a slip-

line field similar to the one shown in Fig.1. Initially, Oxley assumed that the secondary

zone is a constant thickness shear zone. In this study, we assume that the secondary

deformation zone is triangular shape and the maximum thickness is proportional to the

chip thickness, i.e. δ tc .

The slip-line field and the corresponding hodograph indicated that the strain rate

in the primary deformation zone increases with cutting speed and has a maximum value

at plane AB. Based on this experimental observations, he proposed the empirical relation

in Eq. (2) for the average value of the shear strain rate along AB.

AB

SAB l

VC0=γ (2)

The velocity along the shear plane is,

)cos(cos

αφα

−=

VVS (3)

where V is the cutting speed, α is the rake angle, VS is the shear velocity and φ is the

shear angle. Given by Eq.(2), lAB is the length of AB, and C0 is a constant representing the

ratio of the thickness of the primary zone to the length of plane AB. The shear angle φ can

be estimated from Eq. (4).

−= −

α

αφ

sin1

costan 1

c

u

c

u

tt

tt

(4)

The parameters tu and tc are the undeformed, and measured chip thickness

respectively. Oxley and coworkers [6] also claimed that C0 =5.9 for the mild steel used in

their experiments. Although the strain rate is slightly higher in the vicinity of the tool

cutting edge, its value remains relatively constant over the majority of plane AB.

Therefore, Eq. (2) represents a reasonable approximation of strain rate for plane AB. The

shear strain occurring in the primary deformation zone, and the therefore corresponds to

the average shear strain in the primary deformation zone.

)cos(sin2cos

αφφαγ

−=AB (5)

In the primary zone, the shear stress stays same on plane AB, and the average

value of shear stress at AB is given by Eq. (6).

wtFk

u

SAB

φsin= (6)

The parameter w is the width of the chip, and the shear force FS is calculated from

the measured cutting force FC and feed force FT, as shown in Eq. (7).

φφ sincos TCS FFF −= (7)

4.1 Assessment of Temperature Models

Due to the difficulties associated with routinely measuring meaningful machining

temperatures, developing mathematical models for machining temperature has been

widely used as an attractive alternative. Earlier analytical work on the steady-state

temperatures generated during metal cutting was due to Trigger and Chao [21], Chao and

Trigger [22], Loewen and Shaw [23] and Leone [24]. Many other analytical models are

introduced later including the model of Boothroyd [25] as modified by Oxley and

coworkers [6]. Recently, two other models are introduced by Tounsi et al. [17] and Adibi

et al. [20]. All of the analytical temperature models consider a temperature rise due to

shearing in the primary deformation zone and a temperature rise due to additional

shearing and friction in the secondary deformation zone. Those temperature models are

assessed in this study by utilizing cutting test data obtained for AISI 1045 from the

assessment of machining models (AMM) activity [26] as given in Table 2.

Oxley and coworkers [6] utilized modified Boothroyd’s temperature model [25]

and used in their analysis. In this model, the temperature rise at the primary shear zone is

given by Eqs. (8) and (9),

0AB ABT T T= + ∆ (8)

(1 ) coscos( )

SAB

u

FTSt w

β αρ φ α

−∆ =

− (9)

where T0 is the initial work material temperature, β is the fraction of the energy generated

in the primary zone that enters in the workpiece and empirically determined [6].

)tanlog(35.05.0 φβ TR−= for 0.10tan04.0 ≤≤ φTR (10)

)tanlog(15.03.0 φβ TR−= for 0.10tan >φTR (11)

KSVtR u

Tρ

= (12)

In Eqs. (10) and (11), RT is a non-dimensional thermal number given with Eq. (12),

where K is the thermal conductivity of the work material.

The average tool-chip interface temperature is the average temperature at the

primary zone plus the maximum temperature in the chip that is given in Eq. (13).

MAB TTT ∆+= ψint (13)

where ψ is the tool-chip interface temperature factor. The maximum temperature rise in

the chip, ∆TM , can be calculated once the thickness of the tool-chip interface δ tc, and

tool-chip contact length lc are found. 1/ 2

log 0.06 0.195 0.5logM T c T c

C

T R t R tT h h

δ ∆ = − + ∆

(14)

where ∆Tc, which is the average temperature rise in chip, is given by

sin / cos( )c F uT F St wφ ρ φ α∆ = − (15)

In this study, we present an assessment of the temperature models proposed by

various researchers. The comparison of the predicted results is shown in Figs. 3, 4 and 5.

It appears to be the best results are obtained when using Oxley and coworkers’

temperature model.

4.2 Determination of Work Flow Stress

The average shear strain rate, shear strain, and shear stress in the primary

deformation zone are given by Eqs. (2), (5) and (6) respectively. Using the Von Mises

criterion, they can be related to the equivalent stress, strain, and strain rate using Eq. (16).

ABAB k3=σ , 3

ABAB

γε = , 3

ABAB

γε = (16)

In order to determine the parameters of the JC constitutive model, a set of

orthogonal cutting experiments is performed to measure the forces FC and FT in various

cutting conditions and later values of ABABAB σεε ,, , and TAB are computed.

Using inverse solution of Oxley’s model to find the desired parameters A, B, C,

m, n is in fact quite challenging due to the non-linearity in the JC constitutive model.

Other researchers proposed various techniques to address to this problem. Shatla et al.

[15] reported that their algorithm is vulnerable finding local convergence as the

unrealistic combinations of JC parameters as the solution.

The parameters of JC model are determined by using non-linear regression

algorithm based on the Gauss-Newton algorithm with Levenberg-Marquardt

modifications for global convergence [27]. The only unknown value at this point is C0,

which is given by Eq.(17).

( ) ( )0 2

nA B AB

AB

p p A BC

Bnk

ε −− += (17)

In Eq. (17), pA and pB are the hydrostatic stresses at points A and B respectively,

which can be determined from measured machining forces and stress state at point A as

shown by Oxley [6]. The parameter C0 is solved in an iterative procedure and the

parameters of the JC constitutive model are computed. Therefore, the determined JC

constitutive model as work material flow stress can be directly used in the FE simulations

of metal cutting.

5. ANALYTICAL MODEL FOR FRICTION AT TOOL-CHIP INTERFACE

Interfacial friction on the tool rake face is not continuous and is a function of the

normal and frictional stress distributions. The normal stress is greatest at the tool tip and

gradually decreases to zero at the point where the chip separates from the rake face [3] as

shown in Fig.2. The frictional shearing stress distribution is more complicated. Over the

portion of the tool-chip contact area near the cutting edge, sticking friction occurs, and

the frictional shearing stress, τint is equal to average shear flow stress at tool-chip

interface in the chip, kchip. Over the remainder of the tool-chip contact area, sliding

friction occurs, and the frictional shearing stress can be calculated using the coefficient of

friction µe. This section describes a methodology to determine the shearing stress in the

sticking region and the coefficient of friction in the sliding region, based on the measured

machining forces, FC and FT, and the measured tool-chip interface contact length lC.

The normal stress distribution on the tool rake face can be described by Eq. (18)

( )

−=

a

cNN l

xx 1max

σσ (18)

In Eq. (18), x is the distance from the cutting edge, and a is an empirical coefficient that

must be calculated. Integrating the normal stress along the entire tool-chip contact length

yields the relation in Eq. (19).

( ) ∫∫

−==

cc l a

cN

l

NN dxlxwdxxwF

00

1max

σσ (19)

FN is the normal force on the tool rake face determined from Eq. (20)

αα sincos TCN FFF −= (20)

Oxley [6] showed that maxNσ can be estimated by analyzing the hydrostatic stress along

AB, yielding Eq. (21).

+

−−+= − BABnCk n

ABABN ε

απσ 0222

1max

(21)

Combining Eqs (19), (20) and (21), it can be shown that the exponential coefficient a is

obtained as given in Eq. (22).

NnAB

ABC

N

FBA

BnCkwl

Fa−

+

−−+=

−εαπ 022

21

(22)

Normal stress distribution over the rake face is fully defined and the coefficient of

friction can be computed, once the values of the parameters maxNσ and a are found. The

shear stress distribution on the tool rake face illustrated in Fig.2 can be represented in two

distinct regions:

a) In the sticking region: int ( ) chipx kτ = and when ( ) ,0e N chip Px k x lµ σ ≥ < ≤

b) In the sliding region: int ( ) ( )e Nx xτ µ σ= and when ( ) ,e N chip P Cx k l x lµ σ < < ≤

Here kchip is the shear flow stress of the material at tool-chip interface in the chip and it is

related to the frictional force between the chip and the tool, FF, which is given in Eq.

(23).

0

( )CP

P

ll

F chip e Nl

F wk dx w x dxµ σ= +∫ ∫ (23)

The relation between the average coefficient of friction in the sliding region µe and intk is

also given in Eq. (24).

( )chip

eN P

kl

µσ

= (24)

Combining Eqs. (23) and (24) leads to the expression for chipk as shown in Eq.(25)

( )( )

C

p

Fchip l

P NN P l

Fkwwl x dx

lσ

σ

=

+ ∫ (25)

The frictional force component FF is given by Eq. (26).

αα cossin TCF FFF += (26)

As shown in Fig.1, shape of the secondary plastic zone in the sticking region can

be assumed triangular according to Tay et al. [4]. On the rake face point N is located at

the interface between the sticking and sliding regions. Line MN is a II slip-line that is

assumed to be straight. Similar approach proposed by Oxley [6] can be taken to analyze

hydrostatic stresses along MN, and to compute the length of sticking region, lP along with

using Eq. (27).

( )αφδ

−=

sin. c

Ptl (27)

The maximum thickness of the secondary zone is δ.tc as shown in Fig.1 and δ is another

parameters yet to be determined iteratively. According to Oxley [6], the average shear

strain rate and shear strain are considered constant and can be estimated from Eqs. (28)

and (29) in the secondary zone.

Cchip

c

Vt

γδ

= (28)

Pchip

c

lt

γδ

= (29)

The shear flow stress, shear strain rate, shear strain, and temperature for the

secondary zone are given by Eqs.(25), (28), (29) and (13) respectively. The only

unknown in the solution process is the value of δ which is often solved based on the

minimum work criterion proposed by Oxley [6]. As derived in Appendix, the difference

between hydrostatic stresses at points M and N (pM and pN respectively) is given again in

Eq.(28).

( )( )int int

intint

/ sin2

sin( )

np

M N n

B n lp p k

A B

ε γ φφ α

δ φ αε

− = − + −+

(28)

As shown in Fig.1, point M is on line AB, and p changes proportionally along AB as

pB<pA remains and the hydrostatic stress at point M can be given as Eq. (30).

( )cM B A B

AB

tp p p plδ

= + − (30)

The hydrostatic stress at point N is equal to the normal stress at lp as shown in Eq. (31).

)( pNN lp σ= (31)

The only unknown in Eqs. (29), (30) and (31), δ , is solved in an iterative

procedure and together with the frictional parameters; kchip, τint, σNmax, lp and a, the

coefficient of friction in the sliding region, µe, is determined as part of the entire

determination of the work material flow stress and frictional parameters at the tool-chip

interface using orthogonal cutting tests.

6. RESULTS

The goal of this work is to present a methodology in order to determine work

material flow stress and friction properties for use in FE simulations of metal cutting. For

this purpose, a work material constitutive model developed by Johnson and Cook is used

as the work flow stress model. The friction model at tool-chip interface is based on

estimation of normal and frictional stress distributions along the tool rake face. The

parameters of the JC constitutive model and frictional parameters can be computed in an

iteration scheme by utilizing an inverse solution of Oxley’s machining theory and forces,

chip thickness and tool-chip contact length measured in orthogonal cutting tests for

various cutting conditions.

The methodology presented in this work is applied to the orthogonal cutting tests

results obtained from Oxley [6] for 0.38% carbon steel. A computer program developed

using Matlab software. The parameters of the JC material model for work flow stress are

computed as A= 451.6, B= 819.5, C= 0.0000009, n= 0.1736, m= 1.0955. Those

parameters are very close to the ones obtained from the SHBT by Jasper and Dautzenberg

[16] indicating the success of the proposed methodology. The detailed results of the

process variables are given in Table 3.

The friction parameters are also computed for the use in FEA as given in Table 4.

The friction model is based on estimation of normal stress distribution over the rake face.

Predicted friction characteristics include parameters of the stress distribution on the rake

face σNmax and power exponent a, the average shear flow stress, kchip, plastic contact

length, lp, at sticking region of the tool-chip interface, an average coefficient of friction,

µe, in sliding region of the tool-chip interface.

7. CONCLUSIONS

This paper utilizes an extended metal cutting analysis originally developed by

Oxley and coworkers [6] and presents an improved methodology to expand applicability

of Johnson-Cook material model to the cutting regimes and simultaneously determine

friction characteristics at tool-chip interface from the results of orthogonal cutting tests.

A number of cutting temperature models are also used in characterizing the

accuracy of the average temperatures at primary and secondary deformation zones.

Predictions using various temperature models are compared with the results of orthogonal

cutting test data obtained for AISI 1045 through assessment of machining models

(AMM) activity [26]. The temperature model introduced by Oxley [6] is found most

closely predicting the average temperature in shear plane and chip-tool interface.

The methodology developed for determining work flow stress and friction at chip-

tool interface requires experimental data for cutting and thrust forces, chip thickness and

tool-chip contact length. The methodology is practical and estimates both the parameters

of work material Johnson-Cook constitutive flow stress model and the interfacial friction

characteristics over the rake face. The extended Johnson-Cook work flow stress

parameters and tool-chip interfacial friction characteristics can directly be entered into

most FEA software. This methodology has been effectively applied to characterize flow

stress for 0.38% carbon steel material and the friction parameters at the tool-chip

interface by using the orthogonal cutting tests.

NOMENCLATURE

A Plastic equivalent strain in Johnson-Cook constitutive model, (MPa)

AR Aspect ratio in the temperature model given by Lowen and Shaw

B Strain related parameter in Johnson-Cook constitutive model, (MPa)

B2 Fraction of heat in the secondary shear zone,

C Strain-rate sensitivity parameter in Johnson-Cook constitutive model

C0 Strain rate constant proposed by Oxley

FC , FT Cutting and thrust force components, (N)

FF , FN Frictional and normal force components at tool rake face, (N)

FS, FNS Shear force and normal to the shear force components at AB, (N)

J Mechanical equivalent of heat, (m.kg/J)

K Work material thermal conductivity, (W/m °C)

kAB Shear flow stress on AB, (N/mm2)

kchip Shear flow stress in chip at tool-chip interface, (N/mm2)

Kt Tool material thermal conductivity, (W/m °C)

L1 , L2 Temperature parameters

lc Length of tool-chip contact, (mm)

lp Length of sticking region, (mm)

m Thermal softening parameter in Johnson-Cook constitutive model

n Strain-hardening parameter in Johnson-Cook constitutive model

p Proportion of the main shear zone

pA , pB , pM , pN Hydrostatic stresses at the points A, B, M, and N, (N/mm2)

q Rate of heat liberation per unit area of shear plane

q1 Rate at which thermal energy leaves the primary interface, (J/m2 sec)

q2 Friction energy that will be dissipated at the tool-chip interface, (J/m2 sec)

R1 Proportion of the heat conducted into work material

R2 Fraction of q2 that flows into the chip

S Specific heat of work material, (J/kg °C)

SF Shape factor in the temperature model given by Lowen and Shaw

Ss Dynamic shear stress of work material, (MPa)

T0 Initial work material temperature, (°C)

TAB Average temperature along AB, (°C)

Tint Average temperature along tool-chip interface, (°C)

Tm Melting temperature of the work material, (°C)

Tr Room temperature, (°C)

tu , tc Undeformed and deformed chip thickness, (mm)

uf Friction energy per unit volume,(kg/m2)

us Shear energy per unit volume,(kg/m2)

V, VS , VC Cutting velocity, shear velocity, and chip velocity, (m/sec)

w Width of cut, (mm)

α Tool rake angle, (degree)

β Proportion of the heat conducted into work material

δ Proportion of the thickness of the secondary zone to the chip thickness

int,ABε ε Effective strain at AB and tool-chip interface, (mm/mm)

0ε Reference strain rate, (1/sec)

int,ABε ε Effective strain rate at AB and tool-chip interface, (1/sec)

φ Shear angle, (degree)

int,ABγ γ Effective shear strain at AB and tool-chip interface, (mm/mm)

int,ABγ γ Effective shear strain-rate at AB and tool-chip interface, (1/sec)

µe Coefficient of friction in the elastic contact region of tool-chip interface

ρ Density, (kg/m3)

σ0 Initial yield strength of the material at room temperature, (N/mm2)

ABσ Effective flow stress at AB, (N/mm2)

Nσ Normal stresses acting on tool-chip interface, (N/mm2)

τint Frictional shear stress at tool-chip interface, (N/mm2)

Ψ Tool-chip interface temperature factor

APPENDIX

Slip-line field analysis

The slip-line field analysis reveals the equilibrium equations as Eq.(A1) on the AB

with a I slip-line and as Eq.(A2) on the MN with a II slip-line.

1 1 22 0p kk

s s sψ∂ ∂ ∂

+ − =∂ ∂ ∂

(A1)

2 2 12 0p kk

s s sψ∂ ∂ ∂

− − =∂ ∂ ∂

(A2)

In the above equations p and k are the hydrostatic stress and shear flow stress of

the work material, ψ is the angular rotation, and s1 and s2 are distances measured along I

and II lines respectively. Both lines AB and MN are assumed straight along slip-lines I

and II respectively. On line AB the equilibrium equation becomes Eq. (A3).

2dsdk

lpp

AB

BA =− (A3)

The right side of Eq. (A3) can be written as Eq. (A4).

22 dsdt

dtd

ddk

dsdk γ

γ= (A4)

Recall that shear flow stress is a function of strain, strain-rate and temperature

constituted by Johnson-Cook model.

/3 3

n nAB AB

ABAB

dk Bnk A Bd

γ γγ γ

= +

(A5)

0 coscos( )AB

d C Vdt lγ α

φ α=

− (A6)

2

1sin

dtds V φ

= (A7)

Substituting Eqs. (A4), (A5), (A6) and (A7) into Eq. (A3) yields Eq. (A8).

02A B AB nAB

BnCp p kA Bε −− =

+ (A8)

Similarly, the equilibrium equation along line MN that is given with Eq. (A2)

becomes Eq. (A9) when it is assume to be straight, the strain-rate and temperature are

considered unchanged in the secondary zone.

( )1

2 chipM N

MN MN

kp p dkl l ds

φ α−−− = (A9)

11 dsdt

dtd

ddk

dsdk γ

γ= (A10)

int int

int/

3 3

n n

chipdk Bnk A Bd

γ γγ γ

= +

(A11)

)cos(sin

αφδφγ−

=ctV

dtd (A12)

)sin(1

1 αφ −=

Vdsdt (A13)

Substituting Eqs. (A10), (A11), (A12) and (A13) into Eq. (A9) yields Eq. (A14)

( )( )int int

intint

/ sin2

sin( )

np

M N n

B n lp p k

A B

ε γ φφ α

δ φ αε

− = − + −+

(A14)

Temperature models

Trigger and Chao’s model: Trigger and Chao [21] introduced an analytical

model for metal cutting temperatures. In their model, the proportion of the heat

conducted into work material, which was found crucial in evaluating the primary shear

zone temperature, is presented as:

ε

β35.51

1

+= , where )tan()cot( αφφε −+= (A15)

The average temperature at the primary shear zone is calculated with the following

equation:

JSS

T sAB ρ

εβ )1( −= (A16)

The average temperature at the tool-chip interface, Tint, is given by the below equation:

+=

ρπψ

SlKV

wJFB

TTc

cFAB 59

2int (A17)

where

KSVlq

Kwq

TTKwq

B

c

c

t

ABrt

ρππ

π

348

8

2

+

−+= and

JVS

q s )sin(φε= (A18)

Loewen and Shaw’s model: Loewen and Shaw [23] introduced analysis for the

cutting-tool temperatures. In their calculations, proportion of the heat conducted into

work material, R1, was found extremely important to calculate the average temperature at

the primary shear zone, TAB. On the other hand, the temperature parameter, q1, and L1 are

proposed to evaluate the TAB. The expressions for R1, L1, and TAB are provided as:

111

1 1.328 AB

u

RKS V t

γρ

= −+

(A19)

KStV

L us

4)csc(

1ρφ

= (A20)

1sin( )su VqJ

φ= (A21)

1

10 2

)csc(754.0LK

tqTT u

ABφβ

+= (A22)

The authors also considered heat source at the secondary deformation zone and derived

equations to calculate an average tool-chip interface temperature, Tint, and related

parameters, L2, q2, R2 that are given as:

0

2 0.754

f uAB

t

f u f u c

t c

u V t SFT T

J KRu V t SF u V t t S

J K J S l Kρ

ρ

− +=

+ (A23)

KSlV

L cc

42ρ

= (A24)

2f u

c

u V tq

J l= (A25)

+=

2

22int

377.0LK

lqRTT c

AB ψ (A26)

SF refers to shape factor that can be obtained from the Fig.6 in Loewen and Shaw [23]. In

order to obtain the SF value, the aspect ratio, AR, must be calculated from the following

equation:

clwAR

2= (A27)

Leone’s model: Leone [24] conducted a line-source heat analysis at the primary

shear zone in orthogonal metal cutting in order to determine an average temperature at

the primary zone. The proportion of the heat conducted into the work material is defined

as:

VKSwVc 2

13.11

1ρ

β+

= (A28)

10

2AB

q w KT TK SVw

βπ ρ

= + (A29)

Leone [24] did not consider a heat source at the secondary deformation zone. Thus, the

average tool-chip interface temperature is calculated by using the analysis provided in

Trigger and Chao [22].

Tounsi and co-workers’ model: Tounsi et al. [17] made some modifications to

Oxley’s parallel-sided shear zone model of orthogonal cutting by considering the widely

accepted Johnson-Cook material constitutive equation for work material. In their

approach, primary shear zone is examined as a shear band with constant thickness that is

approximated by one-half the uncut chip thickness (0.5tu). Proportion of the main shear

zone, p, is given by the following equation:

1 (2 )2 2 ( )

cospcos

φ αα

−= + (A30)

The most prominent modification, which has been applied to Oxley’s model, is a new

temperature equation given for the main shear plane. Additionally, different equation to

calculate the effective strain rate on the shear plane is also given in their publication.

0( ) 2 / 3

( ) ( ) 3AB

ABp cos k AT T

S sin cosα

ρ φ φ α −

= + − (A31)

2 ( )3 0.5 ( )AB

u

V cost cos

αεφ α

=−

(A32)

On the other hand, the authors did not consider any calculation for the secondary shear

zone.

Adibi and co-workers’ model: Adibi et al. [20] extended Oxley’s analysis of

machining to use different material models and introduced a new approach to calculate

the pressure variation along slip line I in the primary shear zone. In their approach, the

temperature at the middle of the shear zone is determined by integrating the plastic work

done up to the mid-plane of the primary shear zone. For the Johnson-Cook material

constitutive model the temperature of the shear plane, TAB, is given as:

0

1

0

( ) (1 ) 1 ln1

1

ABTn AB

AB ABmT r

m r

S T BdT A CnT T

T T

ρ εβ ε εε

+ = − + + + −− −

∫ (A33)

The temperature at the upper line of the shear zone, TEF, is considered by doing the same

calculation, of course, using the values for upper line. Then, TEF is calculated from the

following equation:

0

1

0

( ) (1 ) 1 ln1

1

EFTn AB

EF EFmT r

m r

S T BdT A CnT T

T T

ρ εβ ε εε

+ = − + + + −− −

∫ (A34)

where β is assumed to be constant and evaluated at TAB. They also differently calculated

the shear force along the primary shear zone and it is given as follows:

s

us V

wtVWF = (A35)

where 0

0(1 )

EF

EF

T

T

S dT

W dε

ρ

σ εβ

= =−

∫∫ (A36)

The authors also introduced a modification to the Oxley’s calculations of the equivalent

strain in the secondary shear zone.

intint 3

cEF

c

lV

γε ε= + , where EF AB= 2 ε ε (A37)

REFERENCES

[1] H. Ernst, M. E. Merchant, Chip formation, friction and high quality machined

surfaces, Transactions of American Society for Metals, (1941), 29, 299-378.

[2] E.H. Lee, B.W. Shaffer, The theory of plasticity applied to a problem of

machining, Journal of Applied Mechanics, (1951), 18, 405-413.

[3] N.N. Zorev, Inter-relationship between shear processes occurring along tool face

and shear plane in metal cutting, International Research in Production

Engineering, ASME, New York, (1963), 42-49.

[4] A.O. Tay, M.G. Stevenson, M.G., G. de Vahl Davis, Using the finite element

method to determine temperature distributions in orthogonal machining,

Proceedings of Institution for Mechanical Engineers, (1974), 188, 627-638.

[5] E. Usui, T. Shirakashi, Mechanics of machining -from descriptive to predictive

theory. In On the Art of Cutting Metals-75 years Later, ASME Publication PED 7,

(1982), 13-35.

[6] P.L.B. Oxley, Mechanics of machining, an analytical approach to assessing

machinability, Ellis Horwood Limited, (1989).

[7] G.R. Johnson, W.H. Cook, A constitutive model and data for metals subjected to

large strains, high strain rates and high temperatures, Proceedings of the 7th

International Symposium on Ballistics, The Hague, The Netherlands, (1983), 541-

547.

[8] F.J. Zerilli, R.W. Armstrong, Dislocation-mechanics-based constitutive relations

for material dynamics calculations, Journal of Applied Physics, 61, 5, (1987),

1816-1825.

[9] J.C. Hamann, V.Grolleau, F.Le Maitre, Machinability improvement of steels at

high cutting speeds – study of tool/work material interaction, Annals of the CIRP,

45, (1996), 87-92.

[10] W.S. Lee, C.F. Lin, High-temperature deformation behavior of Ti6Al4V alloy

evaluated by high strain-rate compression tests, Journal of Materials Processing

Technology, 75, (1998), 127-136.

[11] T.H.C. Childs, Material property needs in modeling metal machining,

Proceedings of the CIRP International Workshop on Modeling of Machining

Operations, Atlanta, Georgia, USA, May 19, (1998) 193-202.

[12] T. Özel, T. Altan, Determination of workpiece flow stress and friction at the chip-

tool contact for high-speed cutting, International Journal of Machine Tools and

Manufacture, (2000), 40/1, 133-152.

[13] T. Özel, T. Altan, Process simulation using finite element method- prediction of

cutting forces, tool stresses and temperatures in high-speed flat end milling

process, International Journal of Machine Tools and Manufacture, (2000), 40/5,

713-738.

[14] H.W.Meyer Jr., D. S. Kleponis, Modeling the high strain rate behavior of titanium

undergoing ballistic impact and penetration, International Journal of Impact

Engineering, 26, (2001), 509-521.

[15] M. Shatla, C. Kerk, T. Altan, Process modeling in machining. Part I:

determination of flow stress data, International Journal of Machine Tools and

Manufacture, (2001), 41, 1511-1534.

[16] S.P.F.C Jaspers, J.H. Dautzenberg, Material behavior in conditions similar to

metal cutting: flow stress in the primary shear zone, Journal of Materials

Processing Technology, 122, (2002), 322-330.

[17] N. Tounsi, J. Vincenti, A. Otho, M.A. Elbestawi, From the basic mechanics of

orthogonal metal cutting toward the identification of the constitutive equation,

International Journal of Machine Tools and Manufacture, (2002), 42, 1373-1383.

[18] Y. Huang, S. Y. Liang, Cutting forces modeling considering the effect of tool

thermal property-application to CBN hard turning, International Journal of

Machine Tools and Manufacture, (2003), 43, 307-315.

[19] A.H. Adibi-Sedeh, V. Madhavan, Effect of some modifications to Oxley’s

machining theory and the applicability of different material models, Machining

Science and Technology, (2002), 6, 3, 379-395.

[20] A.H. Adibi-Sedeh, V. Madhavan, B. Bahr, Extension of Oxley’s analysis of

machining to use different material models, ASME Journal of Manufacturing

Science and Engineering, (2003), 125, 656-666.

[21] K.J Trigger, B.T. Chao, An analytical evaluation of metal cutting temperatures,

Transactions of ASME, (1951), 73, 57-68.

[22] B.T. Chao, K.J Trigger, The significance of the thermal number in metal

machining, Transactions of ASME, (1953), 75, 109-120.

[23] E.G. Leowen, M.C. Shaw, On the analysis of cutting tool temperatures,

Transactions of ASME, (1954), 71, 217-231.

[24] W.C. Leone, Distribution of shear zone heat in metal cutting, Transactions of

ASME, (1954), 76,121-125.

[25] G. Boothroyd, Temperatures in orthogonal metal cutting, Proc. Inst. Mech. Eng.,

177, 789-802.

[26] R. W. Ivester, M. Kennedy, M. Davies, R. Stevenson, J. Thiele, R. Furness, S.

Athavale, Assessment of machining models: progress report, In CIRP 2000

Australia.

[27] D. Bates, D. Watts, Nonlinear regression analysis and its applications, John

Wiley and Sons, (1988), 271-272.

Table 1: Parameters of JC constitutive model for some metals obtained with SHPB test

Material Reference A B n C m

Ti6Al4V Lee and Lin (1998) 782.7 498.4 0.28 0.028 1.0

AISI 1045 Jaspers and Dautzenberg

(2002) 553.1 600.8 0.234 0.0134 1

AA 6082 Jaspers and Dautzenberg

(2002) 428.5 327.7 1.008 0.00747 1.31

S300 Hamann et al. (1996) 240 622 0.35 0.09 0.25

S300 Si Hamann et al. (1996) 227 722 0.40 0.123 0.20

42CD4 U Hamann et al. (1996) 598 768 0.209 0.0137

0.80

7

42CD4 Ca Hamann et al. (1996) 560 762 0.255 0.0192

0.66

0

Ti6Al4V Meyer and Kleponis

(2001) 862.5 331.2 0.34 0.0120 0.8

Table 2: Cutting conditions used for orthogonal cutting and measured process variables

for machining 0.38% carbon steel

Test V (m/min) tu(mm) tc(mm) lc(mm) φ (ο) FC(N) FT(N)

1 100 0.125 0.4 2.5 16.9 1400 1300

2 200 0.125 0.3 2.25 21.8 1300 900

3 400 0.125 0.3 1.875 21.8 1200 900

4 100 0.25 0.7 3.75 19.0 2500 1800

5 200 0.25 0.55 3.05 23.5 2500 1500

6 400 0.25 0.5 2.55 25.5 2200 1300

7 100 0.5 1.1 4.95 23.5 4500 2500

8 200 0.5 0.9 4.05 27.8 4200 2000

Table 3: Computed process variables in the deformation zones and at tool-chip interface

Test ABε ABε ΤΑΒ kAB (N/mm2) intε intε Τint kchip(N/mm2)

1 1.07 5488 428 558.5 1.55 1.50 627 223.8

2 0.87 16488 474 648.6 1.28 5.35 681 188.6

3 0.87 17766 447 579.5 1.28 10.69 701 214.9

4 0.97 5861 467 579.3 1.42 0.98 662 193.6

5 0.82 10632 494 676.1 1.21 3.18 721 206.6

6 0.78 12653 450 614.0 1.14 7.70 674 211.9

7 0.82 3399 458 624.4 1.21 0.79 649 192.9

8 0.73 4814 453 649.0 1.07 2.37 635 191.3

Table 4: Computed parameters of C0, δ and tool-chip interfacial friction

Test C0 δ σNmax(N/mm2) lp(mm) a kchip(N/mm2) µε

1 2.39 0.008 1234 0.54 0.1393 223.8 0.94

2 2.67 0.006 1396 0.33 0.1228 188.6 0.65

3 1.48 0.008 1401 0.33 0.1380 214.9 0.72

4 4.38 0.019 1024 0.86 0.2083 193.6 0.71

5 3.14 0.018 1386 0.58 0.1834 206.6 0.57

6 1.75 0.020 1450 0.49 0.1847 211.9 0.56

7 3.99 0.050 1163 1.15 0.2566 192.9 0.53

8 2.35 0.054 1447 0.83 0.2284 191.3 0.44

tool

workpiece

chip

primaryzonese

cond

ary

zone

V

M

A

N

E

C

D

F

tc

δ tc

lplc

φ B

α

tu

l /C

AB0

Fig. 1. Simplified deformation zones in orthogonal cutting

toolchip

V

A

tc

lp

lc

φ B

α

tuσ (x)NFC

FT

FF

S

NS

τ (x)f

FN

F R

R

σNmax Fig. 2. Forces acting on the shear plane and the tool with resultant stress distributions on

the tool rake face

0

200

400

600

800

1000

1200

1400

1600

1800

1 2 3 4 5 6 7 8Test No

Cut

ting

Forc

e, F

c, (N

)Experimental LBExperimental UBOxley,1989Tounsi et. al,2002Adibi et.al,2003Trigger and Chao,1951Loewen and Shaw,1954Leone,1954

Fig. 3. Comparison of predicted cutting forces with experiments for various temperature

models

0

200

400

600

800

1000

1200

1 2 3 4 5 6 7 8Test No

Thru

st F

orce

, Ft,

(N)

Experimental LBExperimental UBOxley,1989Tounsi et. Al,2002Adibi et.al,2003Trigger and Chao,1951Loewen and Shaw, 1954Leone, 1954

Fig. 4. Comparison of predicted thrust forces with experiments for various temperature

models

0

100

200

300

400

500

600

700

800

1 2 3 4 5 6 7 8Test No

Ave

rage

tem

pera

ture

at t

he p

rim

ary

shea

r zo

ne (d

eg C

) Oxley,1989Tounsi et.al,2002Adibi et.al,2003Trigger and Chao,1951Loewen and Chao,1954Leone,1954

Fig. 5. Comparison of predicted average temperature at the primary zone for various

temperature models

500

1000

1500

2000

2500

3000

1 2 3 4 5 6 7 8Test No

Ave

rage

tem

pera

ture

at t

ool-c

hip

inte

rfac

e, (d

eg C

)Experimental LBExperimental UBOxley,1989Tounsi et. al,2002Adibi et.al,2003Trigger and Chao,1951Loewen and Shaw,1954Leone,1954

Fig. 6. Comparison of predicted average temperature at tool-chip interface with

experiments for various temperature models