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
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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)
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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)
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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.