fe-model for titanium alloy v cutting based

8/8/2019 FE-Model for Titanium Alloy V Cutting Based

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ORIGINAL RESEARCH

FE-model for Titanium alloy (Ti-6Al-4V) cutting based

on the identification of limiting shear stress

at tool-chip interface

Yancheng Zhang & Tarek Mabrouki & Daniel Nelias &

Yadong Gong

Received: 9 December 2009 /Accepted: 5 May 2010# Springer-Verlag France 2010

Abstract Modeling of metal cutting has proved to be

particularly complex, especially for tool-chip interface. The present work is mainly aimed to investigate the limiting

shear stress at this interface in the case of Titanium alloy

(Ti-6Al-4V) dry cutting based on a FE-model. It is first

shown that the surface limiting shear stress was linked to

the contact pressure and the coefficient of friction (CoF). A

relationship between CoF and the limiting shear stress was

given, and the effect of the temperature on the limiting

shear stress was also considered. After that, an orthogonal

cutting model was developed with an improved friction

model through the user subroutine VFRIC in Abaqus/

Explicit software. The numerical results obtained were

compared with experimental data gathered from literature

and a good overall agreement was found. Finally, the

effects of cutting speed, CoF and tool-rake angle on chip

morphologies were analyzed.

Keywords FE cutting model . Limiting shear stress .

Fracture energy. Titanium alloy Ti-6Al-4V

Abbreviations

FE Finite element

J-C Johnson-Cook

SDEG Scalar stiffness degradation

CoF Coefficient of friction

Nomenclature

A initial yield stress (MPa)a half contact width (mm)

ap cutting depth (mm)

B hardening modulus (MPa)

C strain rate dependency coefficient (MPa)

Cp specific heat (J/kg1C

1)

D overall damage variable

D1...D5 coefficients of Johnson-Cook material shear

failure initiation criterion

E1 tool insert Youngs modulus (MPa)

E2 machined material Youngs modulus (MPa)

f feed rate (mm/rev)

fc chip segmentation frequency (kHz)

Fc cutting force (N)

Ff feed force (N)

L characteristic length (mm)

Lc chip segmentation wavelength (mm)

m thermal softening coefficient

m1 contact coefficient

n work-hardening exponent

n1 contact coefficient

p hydrostatic pressure (MPa)

R radius of a cylinder (mm)

Pc contact force per unit length (N/mm)

p0 maximum contact pressure (N/mm2)

qo tangential traction at x = 0 (N/mm2)

tf limiting shear stress (MPa)

tY shear stress calculated by yield stress (MPa)

g shear strain in the primary shear zone

Rn cutting edge radius (m)

T temperature at a given calculation instant (C)

tc cutting time (s)

Tm melting temperature (C)

Tr room temperature (C)

Y. Zhang : T. Mabrouki (*) :D. NeliasUniversit de Lyon, CNRS, INSA-Lyon, LaMCoS, UMR5259,

69621 Lyon, France

e-mail: [email protected]

Y. Gong

School of Mechanical Engineering & Automation,

Northeastern University,

Shenyang 10004, China

Int J Mater Form

DOI 10.1007/s12289-010-0986-7


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T* homologous temperature T

T Tr = Tm Tr uf equivalent plastic displacement at failure (mm)

u equivalent plastic displacement (mm)

VC cutting speed (m/min)

Vchip local chip sliding speed (m/min)e plastic strain rate (s1)

e0 reference strain rate (s1)e

normalized effective strain ratee

e= e0e0i equivalent strain at the onset of damage

ef equivalent plastic strain at failure

e equivalent plastic strain increment

e equivalent plastic strain

a0 flank angle (deg)

ad expansion coefficient (m/m/C)

g0 rake angle (deg)

l thermal conductivity (W/m/C)

Poissons ratio

KC fracture toughness (MPa ffiffiffiffimp

)

Gf fracture energy (N/m) CoF

density (kg/m3)

I principal stresses (i=1...3) (MPa)

s von Mises plastic equivalent stress (MPa)

Y yield stress (MPa)

* stress triaxiality s p=sw damage initiation criterion

Introduction

For the high strength-to-weight ratio, combined with an

excellent corrosion resistance at high temperature, the

cutting of titanium alloys has recently received considerable

interest due to their wide range of application in aerospace,

automotive, chemical, and medical industry [1].

However, titanium alloys are classified as hard machin-

ing materials because of their high chemical reactivity and

low thermal conductivity [2]. The high chemical reactivity

increases with temperature and produces an early damage

of the cutting tool affecting the surface quality and

increasing the production costs [3]. Their low thermal

conductivity hinders the evacuation of the heat generated

during the cutting process resulting in a temperature rise of the

workpiece [3]. This leads to the characteristic segmented

chip feature even at very low cutting speed, as it was

mentioned by Hou and Komanduri [4]. These authors

predicted the critical cutting speed value corresponding to

the onset of shear localization of Ti-6Al-4V to be approx-

imately equals to 9 m/min whereas this value is equal to

130 m/min when cutting AISI4340 steel. In order to increase

productivity and tool-life in machining of titanium alloys, it

is necessary to study the mechanics of chip segmentation and

the effects of the working parameter variation on material

cutting performance. For that, it is more judicious to build a

reliable FE-model allowing more physical comprehension in

relationship with this type of material cutting.

Nevertheless, the robustness of a given numerical model

is strongly dependent on the work material contact nature at

cutting tool-chip interface, material fracture criterion, and

both tool and workpiece material thermal parameters [58].In the present study, Material fracture energy was put

forward to reduce the mesh dependency and achieve

material degradation during cutting process. Moreover, the

mesh sensitivity was also analyzed in order to obtain the

appropriate mesh size. The limiting shear stress in the friction

model was refined with the maximum shear stress at the tool-

chip contact surface.

Comparison of predicted results (in terms of chip

morphologies and cutting forces) with those obtained from

experimental studies was carried out to validate the

numerical model for positive and negative tool-rake angle.

Finally, effects of cutting speed, coefficient of friction(CoF), and rake angle on the chip morphology were studied

with the numerical cutting model.

Numerical approach

Modeling data and geometrical model

To improve physical comprehension of segmented chip

formation, friction properties during Titanium alloy Ti-6Al-

4V cutting, the commercial software Abaqus 6.8-2 with its

explicit approach was employed. A 2D orthogonal cutting

model was developed as shown in Fig. 1. Linear quadrilat-

eral continuum plane strain element CPE4RT with reduced

integration was utilized for a coupled temperaturedisplace-

ment analysis.

Machining parameters were taken similar to those

adopted by Jiang and Shivpuri [9] and Umbrello [5]. For

the workpiece, the uncut chip thickness was 0.127 mm with

a cutting depth ap=2.54 mm. The WC ISO-P20 cutting tool

considered has normal rake and flank angles of 15 and 6,

respectively. The tool entering and inclination edge angles

were 90 and 0, respectively. The tool-cutting edge radius

was of 0.030 mm. The tool-workpiece interaction was

considered under dry machining conditions.

To optimize the contact management during simulation,

a multi-part model (Fig. 1) was typically developed with

four geometrical parts: (1) Part1the insert active part, (2)

Part2the uncut chip thickness, (3) Part3the tool-tip

passage zone, and (4) Part4the workpiece support.

It should be noticed that the thickness of tool-tip passage

zone (Part3), which is the sacrificial layer, was usually

recommended larger than that of the cutting edge radius

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[10] to avoid mesh distortion problem. In the proposed model,

the critical layer size above which the element can climb up

rake face without mesh distortion was found equal to0.026 mm. The sacrificial zone was meshed with five

elements to study the contact effect around the cutting edge.

Also, the passage zone (Part3) was divided into three small

parts: Part3-1, Part3-2, and Part3-3. The CoF in Part3-1should

be very small to reduce the influence of the sacrificial zone for

the new formed chip surface, or the chip surface can be torn

out by the sacrificial zone before it enters into contact with the

tool surface. As Part3-2 and Part3-3 are usually the stick zones

[11], the CoF should be set very high. However, there is no

more contact problem for them when they are finally deleted.

For boundary conditions used in the present model, the

cutting tool is assumed to be fixed on its top and right sides,and the workpiece is allowed to move horizontally from the

left to the right while restrained vertically.

Material behaviour and chip formation criterion

Material constitutive model

The material constitutive material model of Ti-6AL-4V

follows the Johnson-Cook (J-C) model [12]. It provides a

satisfactory description of the behaviour of metals and alloys

since it considers large strains, high strain rates and

temperature dependent visco-plasticity. This model isexpressed by the following expression of the equivalent stress.

s A B"n 1 Cln e 1 Tmh ih

1

The J-C material parameters of the workpiece made in

Ti-6Al-4V can be found in Table 1, whereas the physical

parameters of both the workpiece and the tool-insert aregiven in Table 2.

Chip separation criterion

For different software and material constitutive models, the

approach to deal with element damage is different. Indeed,

Umbrello [5] and Jiang and Shivpuri [9] employed the

Cockroft and Lathams criterion to predict the effect of

tensile stress on chip segmentation, but the elements are

deleted at the onset of damage initiation. This induces

instability in numerical simulation. The J-C model is widely

adopted for its satisfactory description of material flowstress, while different approaches are used to achieve the

degradation of material to get segmented chip. Calamaz

et al. [15] introduced the strain softening through

improving the J-C model by adding the strain softening

effects. While the fracture energy as a failure evolution

criterion after damage initiation can well achieve material

degradation, and this approach was adopted for the present

study.

Damage initiation The initiation of damage in the J-C

material model is derived from the following strain

cumulative damage law:

w Xe

e0i2

Vc X

Y

No displacement

in Y-direction

Fixed boundaries

Part2Part1

Rn=30m

25

0

0Part3

Part4

Damagezones

Part2+Part3

26m

Part3-1

Part3-2

Part3-3

:

Fig. 1 Model mesh and bound-

ary conditions

Materials A (MPa) B (MPa) n C m D1 D2 D3 D4 D5

Ti-6Al-4V 1098 1092 0.93 0.014 1.1 0.09 0.25 0.5 0.014 3.87

Table 1 Johnson-Cook Materi-

al Model [13]

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Where $e is the increment of equivalent plastic strain

during an increment of loading, and e0i the equivalent strain

at the onset of damage and is expressed as following:

"0i D1 D2 exp D3s h i

1 D4 ln e 1 D5Th ih

3The parameters D1, D2, D3, D4, and D5 are experimental

data (Table 1). Damage is assumed to be initiated when the

parameter w=1.

Damage evolution The Hillerborgs fracture energy [16]

was introduced in this model mainly for two reasons:

Firstly to control the material degradation after the damage

initiates (Fig. 2), which makes the failure process more

stable, secondly, to capture high strain localization

during chip segmentation even for relatively large size

element.

When material damage occurs, the stress-strain relation-

ship does no-longer accurately represent the materials

behavior. Continuing to use the stress-strain relation

introduces a strong mesh dependency based on strain

location, such that the energy dissipated as the mesh is

refined. The mesh size in the Calamazs [15] model is

around 2 m near the cutting edge and along the primary

shear zone. The Hillerborgs fracture energy, Gf, was

adopted to reduce the mesh dependency by creating a

stress-displacement response after damage initiation.

Gf Z"pl

f

"pl

0

LsYd"pl

Zuplf

0

sYdupl 4

Where, the characteristic length L is the square root of

the integration point element area based on a plane strain

element CPE4RT.

The scalar stiffness degradation for the linear damage

process used for part 3 is given by:

D Leuf

uuf

5

Where the equivalent displacement is uf 2Gf=sY.

Whereas an exponential damage parameter used for part 2,

evolves according to:

D 1 exp Zu

0

s

Gfdu

6

The formulation of the model ensures that the energydissipated during the damage evolution process is equal to

Gf, and the scalar stiffness degradation approaches to one

asymptotically at an infinite equivalent plastic displace-

ment. In the case of plane strain condition Gf can be

obtained by Gf K2C 1 n2 =E where KC is the fracturetoughness [14].

Mesh sensitivity During machining process, large defor-

mation is a common phenomenon, especially for the

segmented chip. According to Eq. 4, to get this large

deformation with constant element characteristic length L,

the value of plastic strain must be very high, which alsodepends greatly on the local mesh. The characteristic

length L could also be increased to reduce the mesh

dependency. Consequently, the question that can be asked

in the present formulation is: which size is appropriate for

the characteristic length L?

Two constraints can be retained for the evaluation of L:

First, the size should be relative large (to save computing

Physical parameters Workpiece (Ti-6AL-4V) Tool(WC ISO-P20)

Density, (kg/.m3) 4430 15700

Elastic modulus, E (GPa) 110 705

Poissons ratio, 0.33 0.23

Specific heat, Cp (J/kgC) 670 178

Thermal conductivity, l (W/mC) 6.6 24

Expansion coef., d (m/m/C) 9 5

Tmelt (C) 1630

Troom (C) 25 25

Table 2 Workpiece and tool

physical parameters [14]

Damage initiation

( =1, D=0)

Damage

evolution

Material stiffness

is fully degraded

(D=1)E

E

0if

~D

(1-D)E

y

c

d

b

ad

Fig. 2 Typical uniaxial stress-strain response of a metal specimen

[17]

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time), second the result should be similar or close to

experimental values (in terms of cutting force and chipmorphology). For that, four mesh mean values are

discussed with characteristic length of 6, 8, 12, and

14 m, where the simulated cutting forces are compared

with the experimental ones. The dynamic cutting force

versus the characteristic length is first investigated.

Furthermore, the fracture energy was refined to get the

appropriated material degradation evolution and conse-

quently the real segmented chip morphology.

Figure 3 presents the cutting force sensitivity for

different mesh sizes. For both characteristic length of 12

and 14 m, the computed average cutting forces are 481 N

and 458.5N, respectively, which are far below the value of

550 N measured experimentally under the same working

conditions [5, 9]. Whereas for characteristic lengths of

8 m and 6 m, the cutting force reaches 541 N and 549 N,

respectively, which is close to the cutting force obtained

especially. To contain the computing costs, the mesh size of

8 m will be now chosen.

Limiting shear stress from the aspect of contact

mechanics

The Zorevs temperature independent stick-slip friction

model [11], see Eq. 7, was widely adopted by many

authors [10, 15, 18, 19] to define the friction properties at

the toolchip interface. Zorev advocated the existence of

two distinct toolchip contact regions: In the stick zone

near the tool tip the shear stress tfis assumed to be equal

to the yield shear stress of the material being machined,tY,, whereas, in the sliding region, the frictional stress is

lower than the yield shear stress. Note that a constant

Coulomb friction coefficient is assumed here.

if tf < tY & tf msn ! Sliding regionif tf tY & tf msn ! Stick region

&7

The yield shear stress tY, in Eq. 7 is usually related to the

conventional yield stress Y of the workpiece material

adjacent to the surface [18, 20, 21]. A reasonable upper

X

Z

aaVchip

P Friction Q

Friction Q

Cutting tool

Chip contact zone

VcRn

X

ZY

(a) PcVchip

Y

(b)

X

Z

Yp0

+a

-aap

Pc

For=0

Cylinder 2

Cylinder 1

Cylinder 2-chip

Cylinder 1-cutting tool

(c)

c

Workpiece

Fig. 4 Simplified contact model

at the tool-chip interface

400

450

500

550

600

650

0. 0 46 0. 0 914 0. 1 368 0. 1 822 0. 2 276 0. 2 73 0. 3 184 0. 3 638 0. 4 092 0. 4546 0.5

Cutting time

Cutting

force(N)

mesh 6mesh 8

mesh 12mesh 14

Average experimental force

(ms)

mmmm

Fig. 3 Cutting force sensitivity versus cutting time with different

mesh sizes

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bound estimation [17] of the yield shear stress can be

calculated by the von Mises criterion:

1

6s1 s2 2 s2 s3 2 s3 s1 2

n o t2Y s2Y

3

8

In which 1, 2 and 3 are the principal stresses, and tYand Y denote the yield stress values of the material in both

simple shear stress and tension, respectively.

However, when the CoF is relatively small, the maxi-

mum shear stress is found beneath the contact surface. If

this maximum value is adopted to define the limiting shear

0.35

0.375

0.4

0.425

0.45

0.475

0.5

0.525

0.55

0.575

0.6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Maximum shear stress in contact zone

Maximum shear stress at contact surface

/

Y

CoF -

Fig. 6 Ratio of the limiting shear stress and Yversus CoF at a given

temperature

0.028704

0.057407

0.086111

0.11481

0.14

35

2

0.1

7222

0.20

093

0.20093

0.2296

3

0.22963

0.25833

0.28704

0 0.5 1 1.5 2

-2

-1.8

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.05

0.1

0.15

0.2

0.25

z/a

x/a

p0

(at X = 0.0, Z = -0.72)

max /po = 0.31574

( /po)contact surface

contact zone

(a)

0.06010.091556

0.12301

0.15447

0.1

8592

0.21738

0.21738

0.24883

0.28029

0.28

029

0.31174

0.343

2

0 0.5 1 1.5 2

-2

-1.8

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.1

0.15

0.2

0.25

0.3

z/a

x/a

p0

(at X = 0.6, Z = -0.45)

max /po = 0.37465


contact zone

(b)

0.11267

0.149

13

0.18559

0.22205

0.22205

0.25851

0.29497

0.33143

0.36

789

0.40435

0.44081

0.44081

0 0.5 1 1.5 2

-2

-1.8

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.15

0.2

0.25

0.3

0.35

0.4

z/a

x/a

p0

(at X = 0.69, Z = -0.27)

max /po = 0.47727


contact zone

(c)

0.135

51

0.17466

0.2

1382

0.25298

0.25298

0.29213

0.29213

0.33129

0.37044

0.4096

0.44876

0.48791

0 0.5 1 1.5 2

-2

-1.8

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.15

0.2

0.25

0.3

0.35

0.4

0.45

z/a

x/a

p0

(at X = 0.42, Z = 0.0)

max /po =0.5271


contact zone

(d)

Fig. 5 Contour plot of shear stress in chip contact zone for different CoFs: a =0.0; b =0.2; c =0.4; d =0.48

0

100

200

300

400

500

600

0 100 200 300 400 500 600 700 800 900 1000

CoF= 0

CoF= 0.1

CoF= 0.15

CoF= 0.2

CoF= 0.3

CoF= 0.4

CoF= 0.48

Temperature (C)

f(MPa)

Fig. 7 Limiting shear stress versus temperature for different CoFs

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stress, the material beneath the contact surface will yield

before the surface material reaches the critical value. So, it

is necessary to redefine the critical limiting shear stress.

The latter is introduced in detail from the point of view of

contact mechanics by considering the relationship between

the CoF and temperature.

Figure 4(a) schematizes the tool and chip contact. The

tool is fixed and the workpiece moves at the cutting speed

Vc from left to right. To analyze the contact condition, a

small region at the interface was considered with a sliding

rate Vchip and a normal force Pc per length unit. More

precisely the contact model was simplified according toFig. 4(b) which consists of a flat surface slider moving from

left to right over a curved profile and with a steady velocity

Vchip. For frictionless elastic contact and when the contact

dimensions are small compared to the size of the contacting

bodies (Hertz assumptions), the solution is well described

by the Hertz theory [22]. The plane strain problem is

equivalent to the contact between two cylinders, see

Fig. 4(c), with a is the half contact width and p0 is the

maximum pressure at the contact center. It is important to

note that the contact pressure is distributed along a semi-

elliptical shape between [-a, a] along the x-axis.

The load at which material yield begins is related to theyield threshold of the softer material (here the workpiece) in

a simple tension or shear test through an appropriate yield

criterion [22].

Frictionless contact between two cylinders

According to Johnson [22], under a normal force Pc (per

length), the stress field distribution in the chip (here

schematized by cylinder2) is given by Eq. 9 below:

sx p p0a m1 1 z2n2

1

m21n2

1

2z

n osz p p0a m1 1

z2n21

m21n2

1

sy

p v sx p sz p

txz p p0a n1

m21z2

m21n2

1

m2

1

p 1

2

a2

x2

z2

4x2z2 0:5 a

2

x2

z2

h in21

p 1

2a2 x2 z2 4x2z2 0:5 a2 x2 z2 h i

8>>>>>>>>>>>>>>>>>>>:9

In the case of plane strain, the principal stresses can be

calculated by Eq. 10.

s1;2 sx p sz p2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

sx p sx p2

2 txz p

r

s3 sy

p v sx p sz p

:

8>: 10

Combining Eqs. 810, the shear stress distribution in the

(X, Z) plane shown in Fig. 5 (a) with a maximum shearstress of 0.31574 p0 at a depth z=0.72a. Note that for

frictionless contact, the maximum shear stress is found at a

location far below the surface whereas it reaches the surface

Chip valley

Considering crack

Chip peak

Free surface

New formed

free surface

Segmentation

wavelength

Lc=

h2=

(a) (b)

F ig . 8 C h i p m o r p h ol o g y

obtained at VC=120 m/min,

f=0.127 mm/rev: a computation

considering the scalar stiffness

degradation (SDEG = 0.74), and

b Experimental comparison with

[9]

VC=120 (m/min) Lc (m) Peak (m) Valley (m) Fc [5] (N)

Experiment 140 165 46.5 559

Simulation 133 161.5 48 541.3

Table 3 Comparison between

experimental and numerical

chip geometry

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when CoF is equal or higher than 0.48. Consequently, for

frictionless contact or when the CoF is lower than 0.48, it is

necessary to consider the shear stress at the contact surfaceas the limiting shear stress, not the maximum one which is

located at the hertzian depth.

Frictional contact between two cylinders

In presence of friction the contribution of the tangential force

Q, acting on each contact surface along the direction opposed

to the motion (see Fig. 4(b)) should be also considered:

sx q q0a n1 2 z2m2

m21n2

1

2x

n osz q q0p0 txz psy

q v sx q sz q

txz q q0p0 sx p:

8>>>>>>>>>:

11

Where q0=p0 is the tangential traction at x =0, and the

suffixes p and q refer to the stress components due to

normal pressure and tangential traction, respectively. Also,it is assumed that the tangential traction has no effect upon

the normal pressure distribution. When superimposed to the

effect of the contact pressure (normal effect) it yields:

sx sx q sx psy sy

q sy

p

sz sz q sz ptxz txz q txz p

8>>>:

12

Figure 5(b) presents the counter plots of the shear stress

in the contact zone between the chip and the tool for a

relatively low CoF, here 0.2. It is noticed that the maximumshear stress is still below the contact interface. This

maximum shear stress moves towards the interface when

Fig. 9 Comparaison between

the present simulations and

experiments [15] of chip mor-

phologies for f=0.1 mm/rev,

(Vc=60 m/min case of a and b)

and (Vc= 180 m/min case

of c and d)

Vc (m/min) Methods Lc (m) Peak h1 (m) Valley h2 (m) Fc (N)

60 Experimentation [15] 100 N/A N/A 557

Simulation 96 136 94 600

180 Experimentation [15] 100 131 62 548

Simulation 96 132 77 580

Table 4 Comparison between

experimental and numerical

chip geometry

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increasing the CoF as shown in Fig. 5(c) and (d). It reaches

the surface when 0.48. Whereas only one point is

concerned when =0.48 (see Fig. 5(d)).

Figure 6 presents the maximum shear stress conditions at

the surface and below the contact versus the CoF. Results

are normalized by the material yield stress Y. It can be seen

that the maximum shear stresses are totally different when

the CoF is less than 0.48. When the CoF is higher than

0.48, the maximum shear stress is located at the contact

interface and is equal to 1= ffiffiffi3p

sY.

In practical condition, the yield stress Y is updated

according to variations with temperature, strain rate, etc.

So, it is necessary to identify accurately the limiting shear

stress at the contact interface. According to Komanduri

[23], the yield stress of Ti-6Al-4V evolving with tempera-

ture, by taking into account the results given in Fig. 6, it is

possible to draw the limiting shear stresses versus temper-

ature for different CoFs as shown in Fig. 7. It can be

noticed that the higher the CoF, the higher the limiting

shear stress, whereas the latter decreases monotonously

when the temperature increases. Therefore, both CoF and

temperature influence the limiting shear stress. The as-

sumption of constant threshold shear stresses regardless the

frictional properties and temperature is no more valid.

It is important to underline that this figure is drawn only

for CoFs varying between 0 and 0.48. Indeed, for CoF

higher than 0.48, the limiting shear is found at the surface

and is equal to 1=ffiffiffi

3p

(see Fig. 6), and the limiting shear

stress is applied in the cutting model by user subroutine

VFRIC.

Results and discussions

Numerical model validation with experimental works

In order to validate the model developed in the present work,

the numerical results presented are compared with experi-

mental data mainly gathered from literatures [5, 9, 15]. In the

present model, the failure of material is based on Pirondiswork [24], as described in experiment and simulation, crack

starts to take place when the scalar stiffness degradation

(SDEG) is larger than 0.74, so this value can be taken as the

critical value for determining the material failure occurrence.

The simulation carried-out following Jiang and Shivpu-

ris [9] data gives the result shown in Fig. 8(a). It can be

seen that the periodic cracks initiate at the free surface of

the cutting material ahead of the tool and is propagated

towards the tool tip. This result agrees with the works of

Vyas and Shaw [25]. The chip valley is measured from the

boundary of failure zone (SDEG=0.74) to the new formed

free surface for considering crack.A comparison between results obtained by experimenta-

tion and simulation is shown in Table 3. It can be remarked

a good agreement between them. The average cutting force

(Fig. 3 with mesh size of 8 m) is nearly equal to

Umbrellos result [5] with the same cutting parameters.

Another comparison was made with Calamazs [15]

experiment (Fig. 9). In the case of a cutting speed of 60 m/

min, the chip segment morphologies as a result of the

periodic cracks initiation take place both at tool tip and the

free surface ahead of tool tip, where the failure zone at tool

tip is presented in the form of secondary shear zone [26]

(Fig. 9(a)), while the propagation of the failure zone at the

free surface only arrives at half of chip thickness arising

from the lower shear strain rates in the upper region of the

primary shear zone.

When the cutting speed increases to 180 m/min, the

failure zone (with the stiffness degradation larger than 0.74)

goes through the primary shear zone from two sides in and

out, as shown in Fig. 9(c), which is different form

Table 5 Simulation parameters used in the study

Feed rate, f(mm) 0.06, 0.1, 0.127

Cutting speed, Vc (m/min) 60, 120, 180

CoF, 0.3, 0.7, 1

Rake angle, 0=15, 6, 4 Clearance angle, 0=6, 12

Cutting width ap=2.54 mm Cutting edge radius Rn=30 m

Fig. 10 SDEG distribution in

chip according to VC variation

for=0.7 and f=0.127 mm/rev: a

VC=60 m/min; b VC=120 m/min;and c VC=180 m/min

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Luttervelt [27], who discovered the occurrence of a crack at

the tool tip and its propagation towards the free surface,

which is also different from Vyas and Shaw [25] as

mentioned above for rake angle is positive.

When the cutting speed increases, the stiffness degrada-

tion in the primary shear zone accelerates chip deformationin further step i.e. the chip valley will become smaller

(Table 4), while the chip wavelength and peak nearly dont

change. The cutting forces in these two conditions are

larger than the experimental value for the negative rake

angle, while the errors are less than 10%.

As a summary, it can be said that the new developed model

is in good corroboration with Shivpuris and Calamazs

experiments. It permits to predict the chip morphology and

the cutting force with a good approximation. Moreover, it

seems that the appropriate values of the limiting shear stress

are well expressed in this new proposed model.

It is also found that the chip morphology is different with

positive and negative rake angle cutting tool. Indeed, for

positive rake angle tool, the period cracks initiate at the free

surface of the machined material ahead of the tool and

propagate partway toward the tool tip. For negative rake

angle tool, the period cracks initiate at two free faces in and

out, go through the primary shear zone, and join in together

forming the shear band zone with high cutting speed.

Parametric study of Ti-6Al-4V cutting

With the proposed new model, a numerical parametric

study concerning chip morphology in the in orthogonal

cutting of Ti-6Al-4V was performed. The process simula-tion parameters are listed in Table 5.

Cutting speed effect on chip morphology

The SDEG distribution in the shear zone is shown in

Fig. 10 with increased cutting speed, hereinto the value

above 0.74 directly indicates the crack possibility of the

chip as mentioned in Numberical model validation with

experimental works, which is also a reason for segmented

chip, as described in [8], among other phenomena. When

cutting speed is equal to 60 m/min, the chip presents a

continuous part and a segmented. Nonetheless, the region withthe value of SDEG above 0.74 is just a small zone beneath the

free surface of the cutting material ahead of the tool tip. When

the cutting speed increases to 120 m/min, the segmentation

takes place on the entire chip, and part of the region with the

value of SDEG above 0.74 passes through half of the chip.

When the cutting speed increases to 180 m/min, the regions

with the value of SDEG above 0.74 nearly arrive at the free

surface around tool tip for all segmentations. So, the higher the

cutting speed the higher chip segmentation.

The segmentation frequency fc can be approximated by

the ratio VcLc, where Lc is chip segmentation wavelength as

it is indicated in Fig. 8(b) [28, 29]

fc VcLc

13

The computed chip segmentation wavelengths, for

cutting speeds 60, 120, and 180 m/min (as shown in

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Cutting speed(m/min)

L

c(mm)

0

5

10

15

20

25

30

35

fc(kHz)

50 75 100 125 150 175 200

For Lc

For fc

Segmentation wavelength

Segmentation frequency

Fig. 11 Effect of VC on Lc and fc for f=0.127 mm, and =0.7

Fig. 12 Temperature distribution with increased for f=0.127 mm/rev, VC=120 m/min: a =0.3; b =0.7; and c =1

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Fig. 11), are constant, whereas the segmentation frequency

is directly proportional to the cutting speed according to

Eq. 13. This result agrees with the experiment study carried

out by Sun [28].

CoF effect on chip morphology

To study the influence of CoF for the cutting of Ti-6Al-4V,

friction values were taken 0.3, 0.7 and 1 for tool-chip

interfaces. The CoF for sacrificial passage zone (see

(Fig. 1), Part 3) is taken constant. As shown in Fig. 12,

the temperature distribution on chip and tool is greatly

influenced by the CoF variation. It is remarked that the

insert-zone where the temperature is of 581720C

migrates from tool-tip to the rake face-chip contact end,

as CoF varies from 0.3 to 1.

Moreover, it is noted the absence of friction effect oncutting force which has an average value around 540N, as it

is illustrated in Fig. 13. This result can be expected because

the global chip morphology is remained the same with the

CoF variation. Whereas, the feed force does not rise with

the increased CoF, by contrary the values decline when CoF

evolves from 0.3 to 1. This phenomenon mainly attributes

to the thermal softening effect of the high temperature

ahead of tool tip and rake face, and the limiting shear

stresses directly decrease as discussed in Fig. 7.

Rake angle effect on chip morphology

Two feed rates 0.06 and 0.1 mm/rev were checked to study

rake angle effect on chip morphology. According to Fig. 14

it can be underlined that when passing from positive to

0

100

200

300

400

500

600

0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95 1.05

Averageforce(N)

Cutting force

Feed force

CoF -

Fig. 13 Evolution of the average cutting and feed force versus for

f=0.127 mm/rev, and VC=120 m/min

Fig. 14 Chip SDEG distribu-

tion at VC=120 m/min and =

0.7: a g0=15, f=0.06 mm/rev;

b g0=6, f=0.06 mm/rev;

c g0=15, f=0.1 mm/rev; and

d g0=4, f=0.1 mm/rev

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negative angle the chip segmentation phenomenon becomes

more pronounced.

For the same tool-rake angle, the increase in feed rate

from 0.06 to 0.1 mm/rev implies more chip segmentation.

This can be observed when comparing Fig. 14(a) and (c).

So, the higher the feed rate the higher the material SDEG as

it is also shown by Fig. 14(b) and (d) (see the propagation

of SDEG between 0.74 and 1). It can be also underlinedthat more negative rake angle, higher is the tool-workpiece

contact pressure and so the chip segmentation.

Concluding remarks

The aim of this contribution concerns the a numerical finite

element cutting model for an aeronautic Titanium alloy

referenced as Ti-6Al-4V. The major findings are summa-

rized as follows:

The present work based on an energetic concept andrefined limiting shear stress to build a multi-part cutting

model under Abaqus\Explicit was presented. The

adopted approach can well predict the cutting force

and chip morphology, with the corresponding mis-

match less than 10 % compared with experiments. And

the cutting model can be used to simulate high

coefficient of friction (CoF) and low feed rate with

large cutting edge radius tool without convergence

problem. In order to obtain the appropriate model mesh

size for the present study, a mesh sensitivity analysis

was performed.

The limiting shear stress at the tool-chip contact surfaceis used as a friction model from the aspect of contact

mechanics. It is first shown that the surface shear stress

is linked to the contact pressure and the CoF. A

relationship between the CoF and the limiting shear

stress is given. The effect of the temperature on the

limiting shear stress is also considered.

The presentation of a parametric study is carried out

with the new cutting model. This allowed bring some

physical comprehension accompanying chip formation

according to the variation of cutting speed, CoF, and

rake angle. The simulation result show that the higher

the cutting speed, the more marked chip segmentation phenomenon, the distribution of material scalar stiff-

ness degradation on the chip seems to be located

closely to the tool surface.

A refined measure of the computed chip segmentation

wavelengths show that they are constant regarding to

cutting speed variation, whereas chip segmentation

frequency is directly proportional to it. Moreover, the chip

segmentation is also more pronounced when the insert

geometry passes from positive to negative rake angle.

As CoF varies among 0.3 and 1, the highest temper-

ature located in the insert zone migrates from tool-tip to

the rake face-chip contact end. This temperature

migration has implied a decrease in feed force whereas

the variation of CoF is inactive regarding cutting force

variation.

In the meaning time, the FE model is being tested forother materials in the objective to propose to the scientific

community a robust cutting model.

Acknowledgments The authors would like to acknowledge Prof.

Shivpuri and Prof. Calamaz for providing the experimental data in this

work, and especially for the financial support of China Scholarship

Council (CSC).

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