23

CHAPTER 2

LITERATURE REVIEW

This chapter reviews the literatures on flow stress, various methods

used to measure flow stress data, flow stress models, techniques employed to

identify and optimize material parameters of the flow stress models. It also

reviews the application of FEM to model the machining process, the

formulations used, chip separation criterion and friction models. The chapter

briefly states the limitations and advantages of the methods and models

applied so far and mentions the tools used in this thesis work.

2.1 FLOW STRESS

The flow stress is the instantaneous yield stress at which the work

piece starts to flow in the plastic state. It is defined as a function of strain,

strain rate, temperature and microstructure of the deforming material. The

flow stress is given in Equation (2.1), Ozel and Altan (2000).

= f ( , , , S) (2.1)

The flow stress of the deforming material is one of the most

important material inputs to the FE tool along with friction modelling (Childs,

1997, Shatla et al 2001 a, Boothroyd and Knight, 2006). Sartkulvanich et al

(2004) reported that the flow stress data must be obtained at high strain rates

(up to 106

s-1

), temperatures (up to 1000°C) and strains (up to 4) for FE

simulations and is generally obtained from the following methods

24

1. High speed compression tests

2. Split Hopkinson’s pressure bar tests

3. Practical machining tests

4. Reverse engineering using FE techniques.

2.1.1 High Speed Compression Tests

Oyane et al (1967) reported the behaviour of steels under dynamic

compression using compressed air and a punch to compress the specimen.

The specimen is preheated to obtain the flow stress at high temperatures. The

drawbacks of the approach was the very low strain rate (500 s-1

) and low pre -

heating rate which causes softening and age hardening of the specimen.

2.1.2 Split Hopkinson’s Pressure Bar Tests

The Split Hopkinson’s pressure bar tests (SHPB) over come the

difficulties involved in the high speed compression tests by using a faster

heating rate with induction coils and employing higher punch speeds. The

material flow stress data could be obtained for higher strain rates, further

anneal softening and age hardening experienced with the high speed

compression tests could be prevented (Shirakashi et al 1983). Jaspers (1999)

with SHPB apparatus consisting of pre - heating facility to measure the flow

stress of steel, aluminium and metal matrix composite materials and it was

reported that the pre - heated SHPB facility was capable of determining the

mechanical behaviour of materials at high strain rates and temperatures. The

drawback of this method was the slight modification in the mechanical

behaviour of the material due to the pre - heating. The SHPB method has been

extensively used by many researchers for measuring the flow stress

(Maekawa et al 1983, Obikawa and Usui 1996, Childs et al 1997, Lee and Lin

1998, Lesuer 2000, Follansbee and Gray 1989, Nemmat – Nasser et al 2001).

Khan et al (2004) used quasi static compression loading experiments at

25

various temperatures in conjunction with SHPB to measure the flow stress of

Ti6Al4V alloy.

Shatla et al (2001 a, b) summed up the limitations of the SHPB

method as follows

1. The flow stress data obtained using the test is limited to

strains less than 1 and strain rates less than 2x103 s

-1 whereas

in machining strains higher than 2 and strain rates higher than

105 s

-1 are generally encountered.

2. It is a relatively complicated and expensive technique that

requires special testing apparatus.

2.1.3 Practical Machining Tests

Oxley (1989) was one of the early researchers to use practical

machining tests to obtain flow stress data at high strains, strain rates and

temperatures. The technique was adopted earlier by Stevenson and Oxley

(1970) with limited success. Mathew and Arya (1993) obtained material

properties from machining processes. Kopac et al (2001) obtained material

flow stress data from simple compression tests in conjunction with orthogonal

machining experiments and concluded that orthogonal cutting tests were an

excellent method for determining the flow stress at high strains and strain

rates. Lei et al (1999) employed orthogonal machining tests to obtain flow

stress under high strain rates and temperatures. Childs (1998) reported that the

problems of anneal softening and age hardening which is encountered in high

speed compression tests are not observed in machining tests. Shatla et al

(2001 a) used two dimensional orthogonal slot milling experiments in

conjunction with analytical based computer code called ‘OXCUT’ to

determine flow stress data at machining conditions and suggested a

26

methodology which was less expensive than the SHPB method. Sartkulvanich

et al (2004) reported that the solution offered by Shatla et al (2001 a) required

less experimental effort but did not offer a unique solution to all

investigations.

2.1.4 Reverse Engineering using FE Techniques

The limitations of the practical machining tests and the effort

required to conduct numerous experiments to obtain flow stress prompted

researchers to use FEM techniques in conjunction with analytical and

experimental methods to obtain flow stress through inverse mapping

techniques. Kumar et al (1997) implemented the Oxley technique combining

orthogonal cutting experiments with FEM and obtained a flow stress equation

for AISI 1045 steel based on an inverse methodology. Shatla et al (2001 a)

reported that the above methodology was time consuming and suggested that

the shear friction model had its limitations. Ozel (1998) improved the

methodology for determining flow stress developed by Kumar et al (1997) by

using Zorev’s (1963) friction model at the work- tool interface and modified

the flow stress data till FEM results matched experiments. Shatla et al (2001

a) reported that the technique employed by Ozel (1998) needed more

iterations than the technique developed by Kumar et al (1997). The limitations

of this model prompted Shatla et al (2001 a) to use analytical modelling in

conjunction with orthogonal cutting experiments for obtaining flow stress

data.

Ozel and Altan (2000) obtained an expression for flow stress and

friction parameters based on error minimization between FEM and

experiments. Sartkulvanich et al (2004) suggested that the models developed

by Kumar et al (1997) and Ozel and Altan (2000) involved extensive

computations with limited success and developed their own model based on

an inverse methodology combining Oxley’s machining theory, orthogonal slot

27

milling tests and OXCUT program. The model was able to predict the cutting

forces within 12% though the thrust forces were under predicted by 40%.

2.2 FLOW STRESS MODELS

Flow stress models are semi empirical models used for computing

the flow stress for FE applications. The flow stress measured from the

techniques outlined earlier is to fit these constitutive models and compute the

material parameters from mathematical curve fitting techniques. These

parameters are used by researchers working in FE simulations of the

machining process to calculate the flow stress for a range of strain, strain rates

and temperatures and input them to the FE model.

Ozel and Zeren (2006) stated that accurate and reliable flow stress

models are highly necessary to represent work material deformation

behaviour in metal cutting. Jaspers and Dautzenberg (2002) reported that semi

empirical constitutive models are widely utilized since sound theoretical

models based on atomic level behaviour are far from being materialized.

Bariani et al (2004) presented a comprehensive list of the various types of

flow stress material models used by various researchers. Shi and Liu (2004)

reported the influence of flow stress models in FE simulations of the

machining process.

The Flow stress models widely used in numerical simulations are

1. Johnson - Cook model

2. Modified Johnson - Cook model

3. Zerilli - Armstrong model

4. Oxley model

5. Khan -Huang -Liang model

6. Maekawa model

28

7. El-Magd model

8. Litonski -Batra model

9. Power law model

10. Bodner -Partom model

11. Rhim and Oh model

12. Generic flow stress model

13. TANH model

14. Thermo -dynamical flow stress model

2.2.1 Johnson - Cook Model

Johnson and Cook (1983) developed a material constitutive model

based on a phenomenological approach to represent empirically the

deformation behaviour of materials for large strain, strain rates and

temperatures. It is widely used in FE simulations given its simplicity and

robustness in application for a wide base of materials. The Johnson – Cook

(JC) model is described in Equation (2.2)

= [A+Bn] [1+ C ln ( o)] [1-((T – Troom) / (Tmelt – T room))

m] (2.2)

2.2.2 Modified Johnson - Cook Model

Sartkulvanich et al (2004) developed two modified forms of the JC

model which accounts for the blue brittleness effect (2.3) and the absence of

blue brittleness effect (2.4) in low and medium carbon steels.

= Bn

[1+ C ln ( o)] [(Tmelt – T) / (Tmelt – T room)]

+ [a e – 0.00005(T – 700) (T – 700)

] (2.3)

= Bn

[1+ C ln ( o)] [1 - (Tmelt – T) / (Tmelt – T room)m ] (2.4)

29

The modified Johnson - Cook (MJC) model has no coupling, no

effect of strain history and the initial stress assumption factor (A) disregarded

based on Oxley theory. Sartkulvanich et al (2004) suggested that the MJC

models save computational time and the temperature factor in Equation (2.3)

are different for various materials. Figure (2.1) shows the temperature factor

accounted in Equation (2.3) while Figure (2.2) shows the temperature factor

as an exponential term similar to the JC equation.

Figure 2.1 Temperature factor versus temperature used in Equation (2.3)

(Sartkulvanich et al 2004)

Figure 2.2 Temperature factor versus temperature used in Equation (2.4)

(Sartkulvanich et al 2004)

30

2.2.3 Zerilli -Armstrong model

Zerilli and Armstrong (1987) developed a model based on the

dislocation mechanics theory to characterize the deformation behaviour of

materials. The Zerilli – Armstrong (ZA) model is numerically robust and

aptly models the strain and strain rate phenomena of Body centered cubic

(b.c.c) (2.5) and face centered cubic (f.c.c) crystals (2.6).

= C0 + C1 exp [-C3 T + C4 T ln ( ')] + C2 (n) (2.5)

= C0 + C2 (n) exp [-C3 T + C4 T ln ( ')] (2.6)

The ‘n’ is assumed to be 0.5 for all f.c.c. materials. C0 is the stress

component that accounts for the solute and the original dislocation density on

the flow stress and also for the stress related to the slip band stress

concentrations at grain boundaries needed for transmission of plastic flow

between the poly crystal grains. Zerilli and Armstrong assume the flow stress

dependence on strain to be influenced by strain rate and temperature for f.c.c

crystals unlike b.c.c crystals.

2.2.4 Oxley Model

Oxley (1989) developed a model to predict cutting forces, average

temperatures and stresses in the primary and secondary deformation zones by

using the flow stress data of the work material as a function of strain and

velocity modified temperature. Slip line field analysis was used to model chip

formation and to determine the flow stress and friction data by empirically

fitting the results of orthogonal cutting tests. The model is described in

Equation (2.7).

= 1n

(2.7)

31

1 and n are determined by using the velocity modified temperature Tmod,

which is given in Equation (2.8).

Tmod = [1- log10 o)] T (2.8)

2.2.5 Khan -Huang -Liang Model

Khan et al (2004) developed the Khan - Huang - Liang (KHL)

model given in Equation (2.9) as a modified form of the JC equation where

the material constants were identified through uni axial loading tests and a

combination of least square techniques with constrained optimization using

Matlab programs.

= [A + B (1- ln ' / ln D0)n1 n

)] [ '/ 'o] C

[Tm– T / Tm – Tr]m (2.9)

The new parameter D0 is the upper bound strain rate (106 s

-1) and n1

accounts for the decreasing work hardening effect with increasing strain rate

which apparently is not represented in the JC model. The reference strain rate

is 1s-1

.All other parameters are the same as the JC model. The Flow stress can

be computed at temperatures below the reference temperature which was not

possible in the JC model.

2.2.6 Maekawa Model

Maekawa et al (1983) proposed a unique flow stress model by

considering the effect of coupling effects of strain rate and temperature as

well as loading history effects of strain and temperature as given in Equation

(2.10).

= A (10-3

') M

ekT

(10-3

') Kt

[ T, = ' e– kT/ N

(10-3

') – m/N

] N

(2.10)

This model accounts for the blue brittleness of low carbon steels

where flow stress increases with temperature. The integral term accounts for

32

the history effects of the strain and temperature in relation to strain rate. The

model is unique due to the recovery factor. Ozel and Karpat (2007) reported

that the model could not be applied directly to FE simulations while Ozel and

Altan (2000) linearized the integral term and used it in simulations.

2.2.7 El-Magd Model

El-Magd et al (2003) developed the constitutive model given in

Equation (2.11) which describes the flow stress as a function of reference

flow stress and strain rate and the reference flow stress is given as the

function of strain, strain rate and temperature. The expression for flow stress

of CK 45 steel is also given in Equation (2.12).

f ( ), f ( , T) (2.11)

steel – CK 45 = f (T ), steel = steel – CK 45 (2.12)

2.2.8 Litonski - Batra Model

The Litonski – Batra model was initially proposed by Litonski

(1977) and generalized by Batra (1988). The model is represented in Equation

(2.13) and considers the flow stress as a combination of reference yield stress,

strain, strain rate and temperature.

f = (1+b p) m

(1+ ( p/ 0)) n (1- sT) (2.13)

2.2.9 Power Law Model

It was proposed that the dynamic stress strain curve for steel in

simple shearing is a power law relation as given in Equation (2.14) (Shi and

Liu 2004).

f = p/ 0m

p/ 0 n

T/ T0 (2.14)

33

2.2.10 Bodner – Partom Model

Bodner and Partom (1975) proposed the model given in Equation

(2.15) where the flow stress was expressed as a measure of the plastic work

done.

f = [K1 – (K1-K0) exp (- m Wp)]2/ [-2 ln ( p/ D 0

a(2.15)

2.2.11 Rhim and Oh Model

Rhim and Oh model (2006) developed a new flow stress model for

AISI 1045 steel for predicting serrated chip formation suggesting that

conventional flow stress models failed to predict thermal softening effects

such as dynamic recrystallization in the shear bands. The model is given in

Equation (2.16) and (2.17).

= h [1-exp(-k1n1

)u(T)m1

- s [1-exp (-k2 *n2

)]m2

u(T) (2.16)

h= (C0+C1p)(1+C2 ln )(C3-C4T*

q) (2.17)

where * = [( – c)/ p] u ( ) and T*= (T – Tr / Tm – Tr)

2.2.12 Generic Flow Stress Model

Baker (2006) proposed a generic isothermal flow stress law (2.18),

(2.19), (2.20) as an approximation to convention models.

( , , T) = K (T) n(T)

(1+ C ln ( / 0)) (2.18)

K (T) = K* (T), n (T) = n* (T) (2.19)

(T) = exp (- (T/TMT) µ

) (2.20)

34

2.2.13 TANH Flow Stress Model

Calamaz et al (2008) developed a material model for 2D numerical

simulation of serrated chip formation for machining titanium alloys called the

TANH model. The model given in Equation (2.21), (2.22) is a modified form

of the JC law introducing the strain softening effect. It defines the flow stress

in terms of the strain, strain rate and temperature with the hypothesis of

dynamic recovery and recrystallization mechanism.

= [A+Bn

(1/ exp (a)] [1+Cln ( / 0]

[1- (T-Tr)/ (Tm-Tr) m

][D+ (1-D) tanh (1/ ( +S) c] (2.21)

D = 1- (T/Tm) d

, S = (T/Tm) b

(2.22)

2.2.14 Thermo-dynamical Flow Stress Model

Fang et al (2009) developed a thermo - dynamical constitutive

equation based on the work by Voyiadjis and Almasri (2008) and is given in

Equation (2.23). The JC law with the flow stress parameters is fitted to the

Equation (2.23) using least square fit to convert it to a thermo - dynamical

equation.

= [B pn(1+B1T ( p)

1/m – B2Te

A (1-T/T1) ] + Ya (2.23)

2.3 COMPARATIVE STUDIES WITH FLOW STRESS MODELS

Researchers have employed various flow stress models to compare

and validate the flow stress data computed by the constitutive equations with

the flow stress measured from experiments. The flow stress models have also

been evaluated using FEM. New flow stress models have been developed

citing inadequacies in existing models. This section reviews the studies

35

related to flow stress models application and evaluation in the context of

metal cutting.

Shi and Liu (2004) compared the effectiveness of the Litonski –

Batra, Power Law , JC and Bodner – Partom material flow stress models in

finite element modeling of the orthogonal machining process of HY-100 steel

and found good consistency in cutting forces, stress and temperature patterns

for all models except the Litonski Batra model. They suggested that the

magnitude and sign of the predicted residual stresses were sensitive to the

selection of material models. Meyer and Kleponis (2001) developed new

material parameters of the JC and ZA models through ballistic simulation

tests using high strain rate data. The data was used to model a Ti6AI4V

penetrator for penetrating a semi-infinite block at impact velocities up to

2,000 m/s. They concluded that the ZA model well represented the ballistic

behavior of Titanium alloy in the velocity range of 1,200 m/s to 2,000 m/s.

Jasper's and Dautzenberg (2002) utilized the Split Hopkinson's test

to calculate the flow stress data in metal cutting and evaluated the predictive

abilities of the JC and ZA flow stress models. It was concluded that the JC

model was good in predicting the flow stress for AA 6082 (T6) aluminium

alloy while the ZA model was good for the predictions with AISI 1045 steel

material. Iqbal et al (2007) evaluated the JC, Maekawa et al, Oxley, El-Magd

et al and ZA flow stress models in the simulation of AISI 1045 steel

machining using an updated Lagrangian finite element code and concluded

that the Oxley and JC models predict the process outputs better than other

models.

Umbrello et al (2007) studied the influence of five different set of

material constants of the JC model on the machining behaviour of AISI 316 L

steel. The piezoelectric dynamometer was used for cutting forces

measurements, thermal imaging system for temperature measurements and

36

X-ray diffraction technique for residual stresses determination. It was

concluded that the residual stresses were very sensitive to the material

parameters. Umbrello (2008) evaluated three different sets of JC material

parameters in impacting conventional and high speed machining of Ti6Al4V

alloy and suggested that good prediction of both principal cutting force and

chip morphology can be achieved only if the material parameters were

identified using experimental data obtained by a methodology which permits

to cover the ranges of true strain, strain rate and temperature similar to those

reached in conventional and high speed machining.

Calamaz et al (2008) evaluated the performance of the JC and

TANH and concluded that the JC model is not accurate for machining

simulation with titanium alloys, giving rise to a continuous chip while the real

chip is segmented when machining with a cutting speed of 60 m/min and a

feed rate of 0.1 mm/rev. Fang et al (2009) evaluated a new flow stress models

based on a thermo-dynamical equation which is a modified form of the JC

model to assess the material behavior of titanium alloys. It was concluded that

a good prediction accuracy of both principal cutting temperature and tool

wear depth can be achieved by the proposed model. Lee and Lin (1998)

analyzed the high temperature deformation behavior of Ti6Al4V alloy

evaluated by high strain rate compression tests and proposed a new set of JC

parameters. It was concluded that the strength of the material and the work-

hardening coefficient decreased rapidly with an increase in temperature. The

proposed JC model has been used in FE simulations with good correlation

with machining experiments.

MacDougall and Harding (1999) conducted experiments with a

tube of titanium alloy and computed the material parameters for a ZA type

constitutive relation which was incorporated in a FE code and used to predict

the experimentally measured flow stress in the impact tests with reasonable

37

accuracy. Lesuer (2000) performed experimental investigations of material

models for Ti6Al4V titanium and 2024-T3 aluminum. The ability of the JC

material model to represent the deformation and failure responses using

DYNA 3D FE code was evaluated and a new set of material parameters were

defined for the strength component of the JC model for Ti6Al4V and 2024-T3

aluminium materials. Guo (2003) studied an integral JC model to characterize

the material behavior. The model parameters were determined by fitting the

data from both quasi-static compression and machining tests with 6061-T6

aluminum alloy. The developed approach is valid for machining with

continuous chips at various cutting speeds.

Sartkulvanich et al (2004) used orthogonal slot milling in

conjunction with quick stop tests to determine the flow stress data through a

program called OXCUT. They tested several materials like AISI 1045 steel,

P20 and H13 and obtained flow stress data from measured forces and primary

and secondary shear zone dimensions. The flow stress data was used to

predict the FE forces and temperatures with reasonable accuracy. Sun and

Guo (2009) worked on the dynamic mechanical behavior of machining

Ti6Al4V beyond the range of strains, strain rates, and temperatures in

conventional materials testing and predicted the flow stress characteristics of

strain hardening and thermal softening with the JC model. The predicted flow

stresses from the JC model at small strains agreed very well with those from

the split Hopkinson pressure bar (SHPB) tests. Fang (2005) presented a

sensitivity analysis of the flow stress of 18 materials based on the JC model

and suggested that strain hardening and thermal softening is the first

predominant factor governing the material flow stress while strain-rate

hardening is the least important factor depending on the specific material

employed and the varying range of temperatures especially when machining

aluminum alloys.

38

Childs and Rahmad (2009) assumed the flow stress to reduce non-

linearly with increasing temperature in the manner proposed by the ZA law up

to a temperature of 900° C above which rapid softening takes place and

concluded that the ZA model produced better results than the Power law

model. Shi et al(2010, a, b) reported the importance of material constitutive

laws in machining and developed a new methodology to compute flow stress

based on the distributed primary zone deformation model and validated the JC

law with SHPB, orthogonal cutting tests and FEM which gave good

predictions in simulations.

In this research work, the Johnson - Cook model, Modified

Johnson- Cook model and Oxley models were evaluated in the orthogonal

cutting simulation of AISI 1045 steel work material. The Johnson - Cook

model and Zerilli - Armstrong model was evaluated in the orthogonal cutting

simulation of AA6082 (T6) Aluminium alloy. Four different sets of the

Johnson - Cook model were evaluated in the orthogonal cutting simulation of

Ti6Al4V titanium alloy.

2.4 IDENTIFICATION OF FLOW STRESS MODEL

PARAMETERS

Bariani et al (2004) reported in a comprehensive work on various

flow stress models that whatever the approach followed in analytical

constitutive modelling, the related equations are dependent on several

material coefficients which have to be properly determined to get effective

predictions of the flow stress. The following methods are basically used to

identify the material parameters of the constitutive equations:

1. Direct calculation

2. Inverse analysis

39

2.4.1 Direct Calculation

Direct calculation is used for simple models which have very few

coefficients to determine. It is almost impossible to extend this analogy to

more complex constitutive models involving many coefficients.

2.4.2 Inverse Analysis

Inverse analysis is used in such cases where the optimum values of

the material parameters are found from comparison with experimental

measurements. The inverse approach has been used by many researchers for

computing the material coefficients (Pietrzyk 2002, Forestier et al 2002, Gelin

and Ghouati 1995, 2001, Dal negro et al 2001) in metal forming processes.

Bariani et al (2004) reported that the sensitivity analysis of the material

coefficients with the measured process parameters and the selection of the

correct process parameter are crucial to successful application of the inverse

theory.

Ozel and Zeren (2006) used the inverse methodology to identify the

flow stress parameters of the JC model. The data from orthogonal cutting tests

is combined with Oxley’s theory to predict the flow stress. Flow stress

measured from SHPB tests is input to a minimization mathematical model to

compute the flow stress parameters. Ozel and Karpat (2007) reported that the

methodology proposed by Ozel and Zeren (2006) was limited to SHPB data

only. Ozel and Karpat (2007) used evolutionary computational techniques to

identify the flow stress parameters of the JC model for AISI 1045 steel,

AA 6082 (T6) aluminium alloy, AISI 4340 steel and Ti6Al4V titanium alloy

and reported an improvement in flow stress predictions over the conventional

models which employed SHPB data.

Jaspers and Dautzenberg (2002) identified material parameters for

the JC and ZA models by fitting SHPB data to the constitutive models and

40

used the technique of not constraining one parameter and keeping the others

constant at a time and identified the coefficients. Meyer and Kleponis (2001)

identified the material parameters for the JC and ZA models for Ti6Al4V

alloy using data from SHPB and the least square fit technique for the JC

model and a computer program for the ZA model.

Lee and Lin (1998) used regression analysis to identify the

parameters for the JC model for Ti6Al4V material from SHPB data. Gray et

al (1994) used a computer program based on the optimization routine to fit

experimental data to identify material parameters of the JC model.

Sartkulvanich et al (2004) used the downhill simplex optimization technique

to identify the material parameters of the JC and MJC models for AISI 1045

steel material which is based on minimizing the root mean square error

between the predicted and measured flow stress to compute the parameters.

Khan et al (2004) reported that the Mecking and Cocks (1981) model which

was frequently used by Follansbee and Gray (1989) and in a modified form

by Nemmat – Nasser et al (2001) identified material constants by fitting

constants to uni axial stress strain curves. The Mecking and Cocks (1981)

model had many material parameters which made the process cumbersome

while the Nemmat – Nasser et al (2001) model had lesser number of

parameters. Since both models use the same method to identify the material

parameters the difference is only in the number of parameters.

Khan et al (2004) reported that the JC and KHL models with 5 and

6 material parameters respectively were easier to compute than the 23 and 8

material parameters of the Mecking and Cocks (1981) and Nemmat – Nasser

et al (2001) models. Khan et al (2004) used the least squares and constrained

optimization procedure to identify the material parameters of the KHL model

by correlating measured and experimental flow stress. Lesuer (2000) used the

least square fit to identify material constants for titanium alloys for the JC

41

model based on minimizing the error between experimental and measured

flow stress data from SHPB tests. Interestingly the five parameters were

found from stress - strain curves and stress - temperature curves using least

curve fit technique. Tounsi et al (2002) used a least square approximation

technique to minimize the error between physically measured stress, strain,

strain rate and temperature and the predicted data and identified the material

parameters for the JC model for a number of materials. It was reported that

the parameter A depends on heat treatment and hardness of the material and

found good correlation of the identified parameters with the compressive

SHPB tests.

Pujana et al (2007) reported that the number of parameters to be

identified has an effect on the number of tests to be conducted to adjust the

parameters to the least square approximation technique. It was reported that

deterministic methods such as simplex, steepest descent, Gauss – Newton etc.,

depend on the starting point and nature of constraints which creates different

set of material parameters for the same material model (JC model).They

reported the use of regularly distributed functions evaluations along with

laboratory characterization tests to identify material parameters of the JC and

ZA models.

Mulyadi et al (2006) used a hybrid optimization approach to

optimize the material parameters of titanium alloy. Genetic algorithms were

used to find an initial parameter starting point for the simplex method to

obtain a global minimum. Al Bawaneh (2007) used the central composite

design of experiments and response surface methodology to optimize the JC

material parameters for AISI 1045 steel material.

The literature suggests that the least square approximation

technique that fit to SHPB data is the most frequently used methodology to

identify and optimize the flow stress model parameters. Flow stress is

42

dependent on the nature of experiments and is sensitive to the material model

parameters. The flow stress data for machining has to accurately map the

deforming material in machining conditions for which it is necessary to

identify flow stress as a function of the machining process itself or find

methods to optimize existing parameters to fit the deformation processes

through FEM and inverse approach. Though many approaches have been used

to fine tune and optimize the material parameters, most of the models require

superior mathematical skills and are time consuming.

In this work a methodology based on the Taguchi design of

experiments is employed to identify a new set of material parameters and a

sensitivity study was carried out to analyze the effect of the JC parameters on

the FE output.

2.5 FEM IN METAL CUTTING

Klamecki (1973) developed a three dimensional FE model in metal

cutting which was restricted to the initial stages of chip formation.

Shirakashi and Usui (1974) and Usui and Shirakashi (1982) developed the

first two dimensional FE model in metal cutting and pioneered efforts in

numerical simulations. Iwata et al (1984) employed a plane strain model to

study the orthogonal metal cutting process based on a rigid plastic material

model. Strenkowski and Carroll (1985) reported the numerical simulations of

the orthogonal cutting process with a preformed chip. Childs and Maekawa

(1990) studied tool wear, chip formation and stresses in orthogonal cutting

using FE simulations. Zhang and Bagchi (1994) developed a new chip

separation criterion between chip and work piece. Marusich and Ortiz (1995)

developed a two dimensional FE model to simulate metal cutting. Shet and

Deng (2000) developed a FE model to simulate the orthogonal cutting

process. Klocke et al (2001) developed a FE model to simulate high speed

cutting using Deform 2D software. Halil et al (2004) stated that FEM is the

43

most important tool for analyzing the metal cutting process in his comparative

study of three commercial FE codes: Deform, Thirdwave and Advant Edge.

Grzesik (2006) used Advant Edge FE code for simulating metal cutting. Ozel

(2006) used Deform to analyze the effects of friction in metal cutting process.

Iqbal et al (2007) evaluated material models for AISI 1045 steel using FE

simulations. Umbrello et al (2007) studied the influence of material

coefficients in numerical simulations of AISI 316 L steel.

In recent years Umbrello (2008) used FE simulations to evaluate

three JC models in conventional and high speed machining of titanium alloys.

Calamaz et al (2008), Fang et al (2009) and Umbrello et al (2008), Shi et al

(2010 a, b), Sima and Ozel (2010) developed new material models to simulate

orthogonal cutting citing deficiencies and limitations in existing models.

Umbrello et al (2008) studied the influence of material coefficients and

material models using FE simulations. Davim and Maranhao (2009)

employed FEM to analyze the strain and strain rate effects in machining

AISI 1045 steel.

Friction modelling and methods have dominated the field in recent

times with a number of researchers developing new friction models and

theories for FE simulations in metal cutting (Arrazola et al 2008, Bonnet et al

2008, Haglund et al 2008, Arrazola and Ozel 2010, Maranhao and Davim

(2010).

Mackerle (1999, 2003) published a bibliography on FE

development in metal cutting while Ehmann et al (1997) and Vaz et al (2007)

reviewed the modelling and simulation methods used in simulating metal

cutting in well compiled works. The literature reveals that FEM has been

employed in a number of metal cutting studies over the last few decades due

to its distinct advantages over analytical, mechanistic and experimental

models in saving time and effort and improving the accuracy of predictions.

44

The availability of cutting edge technology in digital computing and the

advancements in supporting hardware resources have given further thrust to

FE studies in this direction. Many commercial FE codes are available for

numerical simulation studies with varying degrees of applicability and

advantages (Halil et al 2004).

In this work FEM has been used considering the benefits of the FE

model in predicting orthogonal metal cutting process and the same has been

employed to evaluate flow stress models and optimize a set of parameters of

the JC model for machining simulations. The FE simulations were performed

using Deform 2D© FE code which is a popular code in metal cutting

simulations and employed by many researchers (Ceretti et al 1996, 1999, Ozel

and Altan 2000, Klocke et al 2001, Halil et al 2004, Umbrello et al 2007,

Umbrello 2008).

The salient features of FEM are

1. FEM formulation

2. Chip separation criterion

3. Flow stress modelling

4. Friction modelling

The first three features are reviewed in this section while the flow

stress modelling has been reviewed in sections 2.2 to 2.4.

2.5.1 FEM formulation

In FEM three types of formulations are generally used,

1. Eulerian

2. Lagrangian

3. Arbitrary Lagrangian – Eulerian

45

2.5.1.1 Eulerian Formulation

In Eulerian formulations the mesh is fixed in space and the work

material flows through the element faces allowing large strains without

causing numerical problems. It overcomes the limitations of the Lagrangian

formulation by eliminating element distortion effects and allows simulations

of steady state machining (Vaz et al 2007). The limitations of these

formulations are that they require prior knowledge of chip geometry and chip

tool contact length restricting the application range, do not permit element

separation or chip breakage and require proper modelling of the convection

terms associated with material properties (Vaz et al 2007). Researchers have

employed iterative procedures to overcome the shortcoming by adjusting the

chip geometry and tool contact length (Iwata et al 1984, Carroll and

Strenkowski 1988, Strenkowski and Moon 1990, Tyan and Yang 1992,

Joshi et al 1994, Strenkowski and Athavale 1997, Kim et al 1999,

Raczy et al 2004).

2.5.1.2 Lagrangian formulation

The Lagrangian formulation assumes that the FE mesh is attached

to the work material during deformation (Vaz et al 2007). The chip geometry

is the direct result of the simulation and the technique provides a simple

methodology to simulate transient and discontinuous chip formation

processes. The element distortion is a matter of concern in this formulation

which limits the analysis to incipient chip formation or machining ductile

materials using large rake angle or low friction conditions (Klamecki 1973,

Lin and Lin 1992, Xie et al 1994, Hashemi et al 1994, Guo and Dornfield

2000, Lo 2000, Mamalis et al 2001, Soo et al 2004 a, b, Barge et al 2005).The

error has been minimized by using a pre distorted mesh (Shih, 1996, Huang

and Black 1996, Obikawa and Usui 1996, Lei et al 1999, Shet and Deng 2003,

Mabrouki and Rigal 2006) or by remeshing (Marusich and Ortiz 1995,

46

Madhavan and Chandrasekar 1997, Ozel and Altan 2000, Klocke et al 2001,

Mamalis et al 2002, Umbrello et al 2004, Hua and Shivpuri 2004,

Sartkulvanich et al 2005, Baker 2005, 2006, Rhim and Oh 2006, Ozel 2006).

2.5.1.3 Arbitrary Lagrangian - Eulerian formulation

The advantages of the Lagrangian and Eulerian formulations were

combined to create an Arbitrary Lagrangian - Eulerian formulation (ALE)

approach. The ALE approach used the operator split methodology

(Figure 2.3) where the Lagrangian and Eulerian steps are applied sequentially.

In the first step, the mesh follows the material flow and the displacements are

solved. In the next step, the mesh is repositioned and the velocities are solved.

The element distortion in Lagrangian formulations are avoided here but a

careful numerical treatment of the velocities is required (Vaz et al 2007). The

ALE method has been employed to good effect in FE simulations in metal

cutting (Rakotomalala et al 1993, Olovsson et al 1999, Movahhedy et al 2000,

Benson and Okazawa, 2004, Pantale et al 2004, Madhavan and Adibi-Sedeh

2005, Courbon et al 2010).

Figure 2.3 Steps in ALE formulation (Vaz et al 2007)

2.5.2 Chip Separation Criterion

The chip separation criterion is an important aspect of successful

Lagrangian formulations. Three types of strategies are adopted for modelling

this aspect in FE simulations.

47

1. Chip separation along a pre defined parting line

2. Chip separation and breakage

3. No chip separation

2.5.2.1 Chip separation along a pre defined parting line

The most common method of chip separation is using a predefined

chip separation line or plane along which a separator indicator is computed

(Vaz et al 2007).The two most common types are the geometrical and

physical separators.

Figure 2.4 shows the geometrical criterion based on the distance

between the tool tip and the nearest node along a predefined cutting direction.

As the tool advances, the distance between the node Fw,c and tool tip

decreases and at a critical distance dcr, a new node is created or a restriction in

superimposed nodes are removed to enable chip separation (Shirakashi and

Usui 1974, Usui and Shirakashi 1982).

Figure 2.4 Geometrical chip separation based on nodal distance

(Shirakashi and Usui, 1974, Usui and Shirakashi 1982)

48

Zhang and Bagchi (1994) developed a new chip separation criterion

based on ratio of separation distance and depth of cut. Huang and Black

(1996) evaluated the performance of the geometrical and physical chip

separation criterions concluding that neither criterion predicts the incipient

chip formation correctly. The geometrical chip separation criterion has been

employed by many researchers in FE simulations (Shih 1995, Obikawa et al

1997, Lei et al 1999, Mamalis et al 2001, Dae – Cheol Ko et al 2002).

Figure 2.5 shows the physical chip separation criterion developed

by Strenkowski and Carroll (1988) based on equivalent plastic strain. The

chip separates when the equivalent plastic strain calculated at the nearest node

to the cutting edge reaches a critical value (indicated as Icr). The limitations of

this method is that if the process is uncontrolled then chip separation is faster

than the cutting speed causing a large open crack ahead of the tool tip

(Vaz et al 2007).

Figure 2.5 Physical separation based on equivalent plastic strain

(Strenkowski and Carroll 1988)

Lin and Lin (1992) proposed a physical criterion based on the total

strain energy density factor suggesting that the critical value in the

49

‘Equivalent plastic strain’ criterion was found to affect the magnitude of the

residual stresses. Iwata et al (1984) introduced the ductile fracture concepts

and suggested the versions of Cockroft and Latham (1968) and Osakada et al

(1977) as the best chip separation criteria. Ko et al (2002) used the ductile

fracture criteria suggested by Cockroft and Latham (1968). A physical chip

separation criterion based on the stress index parameter where the chip

separates when the parametric measure of normal and shear failure stresses

reach a critical value has been used by many researchers (Li et al 2002 , Shi

et al 2002, Mc Clain et al 2002, Shet and Deng, 2003). Barge et al (2005) and

Mabrouki and Rigal (2006) used a critical strain to fracture criterion based on

the JC law which accounts for equivalent plastic strain rate, hydrostatic and

yield stresses and room and melting temperatures.

2.5.2.2 Chip separation and breakage

Hashemi et al (1994) employed a combination of equivalent plastic

strain to model chip separation and maximum principal stress to simulate chip

breakage. Marusich and Ortiz (1995) proposed use of either brittle or ductile

fracture criteria depending on the machining conditions for chip separation

and breakage. Obikawa and Usui (1996) and Obikawa et al (1997) obtained

general expressions for the strain to fracture, which account for the equivalent

plastic strain, equivalent plastic strain rate, hydrostatic stress and absolute

temperature. Owen and Vaz Jr. (1999) used a combined finite/ discrete

element algorithm and multi-fracturing materials, and adopted a chip

breakage criterion based on Lemaitre’s ductile damage model. Borouchaki

et al (2002) also included damage mechanics in the simulation of crack

propagation based on Lemaitre’s model. The formulation proposed by

Lemaitre (1991) postulates that damage progression is governed by void

growth. Umbrello et al (2004) studied the effect of hydrostatic stress on chip

segmentation during orthogonal cutting and adopted a chip breakage criterion

50

based on Brozzo et al (1972). Ceretti et al (1996, 1999) adopted a chip

breakage criterion based on the combination of the effective stress and

Cockroft and Latham’s (1968) maximum tensile plastic work. The latter was

also used by Hua and Shivpuri (2004) to simulate chip breakage in orthogonal

cutting of Ti6Al4V titanium alloy. Lin and Lin (2001) and Lin and Lo (2001)

extended use of the maximum strain energy density, and Ng et al (2002) and

Benson and Okazawa et al (2004) proposed use of the strain to fracture based

on the modified Johnson and Cook’s yield stress equation to simulate

discontinuous chip formation.

2.5.2.3 No chip separation

FEM model should not require chip separation criteria that highly

deteriorate the physical process simulation around the tool cutting edge

especially when there is dominant tool edge geometry such as a round edge or

a chamfered edge (Tugrul Ozel 2000). An updated Lagrangian implicit

formulation with automatic remeshing without using chip separation criteria

has also been used in simulation of continuous and segmented chip formation

in machining processes (Marusich and Ortiz 1995, Sekhon and Chenot 1992,

Ceretti et al 1996, Ozel and Altan 2000, Klocke et al 2001, Baker et al 2002).

The automatic remeshing feature creates new mesh when the old mesh is

distorted and the data from the old mesh is extrapolated to the new mesh

before the start of the new simulation step. The ALE formulation has been

used in metal machining to avoid frequent remeshing for chip separation

(Rakotomalala et al 1993, Olovsson et al 1999).

In this work the updated Lagrangian formulation with automatic

remeshing has been employed to model the chip formation process without a

chip separation criterion thereby avoiding the detrimental effects of using chip

separation as discussed above.

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2.5.3 Friction Modelling

Friction between chip and tool constitutes one of the most

important and complex aspects of machining processes. It can determine the

tool wear, quality of the machined surface, structural loads and power to

remove a certain volume of metal (Vaz et al 2007). The contact regions and

the friction parameters between the tool and the chip are influenced by factors

such as cutting speed, feed rate, rake angle, etc., mainly because of the very

high normal pressure at the surface. Friction at the tool-chip interface is

complicated and difficult to estimate. It is widely accepted that the friction at

the tool-chip interface can be represented with a relationship between the

normal and frictional stress over the tool rake face (Tugrul Ozel 2006).

Tugrul Ozel (2006) reported that the two most important factors in machining

simulations are the flow stress model and the friction model. Filice et al

(2007) analyzed the importance of various friction models in orthogonal

machining and concluded that the coefficient of friction is more important in

cutting force predictions than the frictional law. In recent years friction

modelling has been given importance in numerical simulations of the

machining process (Arrazola et al 2008, Bonnet et al 2008, Haglund et al

2008, Arrazola and Ozel 2010, Maranhao and Davim 2010).

The friction modelling approaches used by researchers are

1. Coulomb friction model

2. Zorev’s friction model

3. Shear friction model

4. Variable friction model

2.5.3.1 Coulomb friction model

The coulomb friction model assumes the frictional stresses on the

tool rake face to be proportional to the normal stress, with a coefficient of

52

friction as the proportionality factor (Equation 2.24). Tugrul Ozel (2006)

reported that the coulomb friction model works well in conventional

machining while it falls short in high speed machining ranges. The coulomb

frictional model was used by many researchers in friction modelling in

machining simulations (Strenkowski and Carroll, 1985, Komvopoulos and

Erpenbeck, 1991, Shih et al 1990, Lin and Lin, 1992).

= n (2.24)

A modified form of the coulomb law was used by some researchers

(Shet and Deng 2000, Guo and Liu 2002, Yogesh and Zehnder 2003) to

model the friction factor. The Law is given in Equation (2.25) and states that

the relative motion (slip) occurs at the contact point when the shear stress is

equal to or greater than the critical frictional stress. When the shear stress is

smaller than the critical frictional stress there is no relative motion and the

point of contact is in a state of stick.

c = min ( p, th) (2.25)

It can be noted that if the threshold value is set to infinity then the

Equation (2.25) follows the conventional coulomb law.

2.5.3.2 Zorev’s friction model

Zorev’s (1963) frictional model has been widely used for modelling

the friction as sliding and sticking types (Guo and Dornfeld, 000, Lin and Lin

1999, 2001, Shatla et al 2000, Lo 2000, Mamalis et al 2001, Guo and Liu

2002). Zorev advocated two distinct regions to be present on the tool rake

face as shown in Figure 2.6. The sticking region covers the distance between

the tip of the tool up to a point (lp) where the frictional stress (shear stress) is

equal to the average shear flow stress (shear yield strength) of the material.

53

The second region called the sliding region assumes the frictional stress to be

proportional to the normal stress and obeys the coulomb law of friction.

Figure 2.6 Stress distributions on the tool rake face (Zorev 1963)

2.5.3.3 Shear friction model

In the shear friction model (Altan 2002, Altan and Eugene 2003,

Ozel 2003) the friction in the work – tool contact region is modelled using a

shear friction factor. The model represented in Equation (2.26) states that the

frictional shear stress is proportional to the shear yield strength of the work

material.

= mKchip (2.26)

2.5.3.4 Variable friction model

In recent times researchers have attempted to model the

tool – work interface frictional characteristics using variable frictional

models. Ozel (2006) studied the influence of varying the coulomb, shear

friction and Zorev’s models over the sticking and sliding regions in the

modelling process and suggested good FE predictions with the variable

models. The variable friction modelling includes varying the shear friction

over the entire rake face, varying the coefficient of friction over the entire

54

rake face or varying the shear friction and coefficient of friction on the

sticking and sliding regions.

Filice et al (2007) studied the influence of various friction models

in orthogonal machining and concluded that cutting forces, contact length

were not sensitive to the friction models. Arrazola et al (2008) reported the

use of a variable coefficient of friction for better numerical predictions in

comparison to coulomb coefficient of friction. Haglund et al (2008) reported

that friction behaviour had little impact on the chip thickness and there were

no significant improvement in the numerical predictions with any of the

friction models investigated citing a re think on the friction modelling aspects

in machining simulations. Bonnet et al (2008) developed a new friction model

which combines the sliding velocity and friction coefficient and reports

improved predictions of the cutting process with the new model. Ozlu et al

(2009) concluded in a recent research work on analytical frictional modelling

that accurate cutting force predictions were possible only when the entire

contact area between chip and tool was modeled as sticking and sliding

regions. Arrazola and Ozel (2010) reported that the major short coming of the

stick – slip friction models is the uncertainty over the limiting shear stress

value which is dependent on local deformation conditions and temperatures.

The literature on friction models suggests that no single friction

model is fully adequate for modelling the complicate deformation process

associated with metal cutting. There appears a continuous scope for

improvement in this area with many researchers investing their time in

friction modelling to improve the simulated results. Given the constraints in

experimental measurement of friction it is natural to turn to FEM for friction

analysis and identification of the friction coefficients from inverse techniques.

In this work the shear friction model was employed for AISI 1045

steel and AA 6082 (T6) aluminium materials and the coulomb friction model

55

was used for Ti6Al4V alloy. The coefficient of friction was varied till steady

state cutting forces were obtained for the three materials in orthogonal cutting.

2.6 SUMMARY OF THE LITERATURE REVIEW

In this chapter, the techniques for measuring flow stress, the list of

various flow stress models used for representing the material constitutive

behaviour, a comparative study of the different flow stress models, methods

of identification of the material parameters and the application of FEM in

machining simulations have been reviewed. Flow stress data is the most

important input data for FE simulations. It is necessary to evaluate different

flow stress models to identify the model which suits the deformation

characteristics of the orthogonal machining process. The material parameters

of these flow stress models are sensitive to the experiments and mathematical

techniques used to compute them. The literature reports varied material

parameters for the same material. There is a need to evaluate these material

parameters and optimize them to improve the flow stress characteristics which

will suit machining conditions. This research work aims to fill the gap by

evaluating different flow stress models of three important industrial materials:

AISI 1045 steel AA 6082 (T6) aluminium alloy and Ti6Al4V titanium alloy,

and identify a new set of material parameters through an integrated

Taguchi – FE inverse methodology, which has not been used in material

parameter optimization studies before. Also a detailed material parameter

sensitivity is not reported in literature. In this research work the sensitivity of

the Johnson – Cook flow stress model parameters to the FE output has been

performed.