ny

ptdia

ents an, Bay athavenrth

be well suited for deep drawing of Ti6Al4V alloy. 2014 Elsevier Ltd. All rights reserved.

1. Introduction

In recent years, Ti6Al4V alloy is inristic ance anfrommoreal form

gely depends on the input material models used and correctness ofthe input material data [7]. Particularly, in warm-forming simula-tion, selection of an appropriate yield criterion is essential becauseit provides an accurate prediction of the observed initial and sub-sequent yield behaviors of a material [5].

d locus using veability of tmparing thdata [14].

uniaxial and biaxial stress state are required. These yield criteriaconsidered symmetry in yielding between tension andcompression. However, the stress asymmetry in the case of HCPcrystal structure is considerable [16]. Considering the effect ofasymmetry in yielding further development in the yield criterionis done by Cazacu et al. [9]. However, limited study has beenreported for the implementation of anisotropic yield criteria forHCP alloys in forming applications at elevated temperatures.

Corresponding author. Tel.: +91 9010451444; fax: +91 40 66303998.E-mail address: nitink@hyderabad.bits-pilani.ac.in (N. Kotkunde).

Materials and Design 63 (2014) 336344

Contents lists availab

an

elsreduce inaccurate and expensive tryouts in sheet metal industries[6]. However, the trustworthiness of the numerical simulations lar-

et al. [8,15] developed yield criteria for metals with an increasein eight parameters. For determination of these parameters bothonal close-packed structure and high degree of springback [4].Hence, to overcome the difculty in forming of HCP crystal struc-ture alloys, the best choice would be warm forming [5].

Nowadays, nite element simulations are extensively used to

which is exible and suitable to represent a yielindependent material parameters [13]. The applicterion was tested for steel and aluminum by codicted forming limit strains with experimentalhttp://dx.doi.org/10.1016/j.matdes.2014.06.0170261-3069/ 2014 Elsevier Ltd. All rights reserved.his cri-e pre-Barlatmanufacturing processes to fabricate titanium alloy componentswhich cannot only reduce the cost due to machining but alsoenhance the performance of the products [3]. But the formabilityof Ti6Al4V alloy is very poor compared with other traditionalmetallic materials at room temperature. The main reasons for poorformability are low ductility at room temperature due to its hexag-

the plane stress condition [11]. Hill 1948 yield model is the mostpopular for nite element simulations because of limited numberof parameters which are easy to determine using uniaxial tensiletest. Barlat and Lian [12] proposed an anisotropic yield criterionwhich also required four parameters to describe a yield locus. Fur-thermore, Hill proposed a user-friendly modied yield criterion,because of their lightweight charactenation of strength, corrosion resistaimportance of this alloy is indicatedit is mostly used alloy accounting fortonnage in the world [2]. Sheet metcreasingly being usednd an excellent combi-d fabricability [1]. Thethe fact that currentlythan 50% of all titaniuming is one of the major

Since selection of a yield model is essential in nite elementsimulations, considerable effort has been made for experimentalobservations of yield loci on various types of metals [8,9]. In thelast few years, several efforts have been made for the developmentof anisotropic yield criteria which consider plastic anisotropy. Forexample, Hill 1948 [10] proposed an extension of the von-Misesisotropic criterion to cover plastic anisotropy. This model consid-ered orthotropic symmetry and four anisotropy coefcients inle, complexity in material parameter identication and computational time has shown Cazacu-Barlat toTechnical Report

Experimental and numerical investigatiofor warm deep drawing of Ti6Al4V allo

Nitin Kotkunde a,, Aditya D. Deole a, Amit Kumar GuaDepartment of Mechanical Engineering, BITS-Pilani, Hyderabad Campus, AP 500078, InbDepartment of Mechanical Engineering, GRIET, Hyderabad, AP 500072, India

a r t i c l e i n f o

Article history:Received 11 April 2014Accepted 7 June 2014Available online 25 June 2014

a b s t r a c t

Accuracy of the nite elemrectness of input propertiecriteria namely, Hill 1948mented for Ti6Al4V allomined and deformation beAlso, deep drawing experimment simulation results. Fu

Materials

journal homepage: www.of anisotropic yield criteria

a a, Swadesh Kumar Singh b

simulation of sheet metal forming is signicantly dependent on the cor-d appropriate selection of material models. In this work, anisotropic yieldrlat 1989, Barlat 1996, Barlat 2000 and Cazacu Barlat have been imple-400 C. Material constants required for the yield criteria have been deter-ior in deep drawing process has been analyzed in nite element software.ts on Ti6Al4V alloy have been performed at 400 C to validate nite ele-er, comparison of yield criteria based on thickness distribution, earing pro-

le at ScienceDirect

d Design

evier .com/locate /matdes

Therefore, applicability of these yield models in nite elementsimulations need to be validate for Ti6Al4V alloy.

Anisotropic yield function developed by Barlat and Lian [12] formodeling sheets under plane stress condition is expressed inEq. (2). The model computes the yielding of a material by consid-ering anisotropy ratio in 0, 45 and 90 orientations. The materialresponse after yielding is governed by exponential hardening rule;m is the material parameter which depends on the crystal struc-ture of a material.

U ajK1 K2jm ajK1 K2jm cj2K2jm 2rmy 2where K1 rxhry2 ;K2

rxhry

2

2 p2s2xy

r; and a, c, h are functions

of anisotropy ratio and they are expressed as

a 2 2

R00R901 R001 R90

s

h R001 R90R901 R00

sand c 2 a

where rh is anisotropy ratio measured for h orientation from rollingdirection p is found out iteratively. The following equation is usedto nd out the value of p iteratively [12].

2rm

N. Kotkunde et al. /Materials andTheobjective of the presentwork is to study applicability of differ-ent anisotropic yield criteria in nite element simulation of deepdrawing of TiAl4V alloy at 400 C and validation of results basedon thickness distribution and earingphenomenon indeepdrawncup.

2. Experimental details

In this work, Ti6Al4V alloy sheet of 0.89 mm thickness isused. The composition of the employed material is given in Table1. The experiments were carried out on the test rig which is shownin Fig. 1. Nickel based super alloy is used for manufacturing the die,blank holder and punch because of its excellent dimensional stabil-ity even at elevated temperatures.

The setup temperature was controlled and prevented fromoverheating by means of water circulation from cooling tower. Anoncontact type pyrometer was used to measure the operatingtemperature. Circular blank specimens were machined by usingwire-cut electro-discharge machining process for high accuracyand nish. Temperature higher than 400 C increases the oxygencontamination in Ti6Al4V alloy and with oxygen the materialbecomes more brittle due to formation of a-scale. Therefore, it ispreferred to perform warm forming of Ti6Al4V alloy in an inertand protective environment [17]. Considering the limitations of theexperimental facility at higher temperatures with an inert environ-ment, the experiments have been performed at 400 C. Molykotewas used as an effective lubricant for forming process at elevatedtemperatures [13]. Blanks were kept at particular temperaturefor certain duration (approximately 35 min) for uniform heatingof sheet. Deep drawing operation was performed when the blankreached the required temperature. 16 bar blank holding pressureis used for deep drawing process. The value is approximately 2%of the yield strength. Punch speed for all the deep drawing exper-iments are used as 1 mm/s. Experiments are performed three timeand average thickness distribution and cup heights are taken.

3. Anisotropic yield models

Anisotropic yield criteria namely, Hill 1948, Barlat 1989, Barlat1996, Barlat yield 2000 and Cazacu Barlat were selected for warmdeep drawing analysis of Ti6Al4V alloy. The material parametersof yield function have been determined and yield loci arecompared.

The material properties required for determination of yield cri-teria parameters were obtained from uniaxial tensile test at 0, 45and 90 orientations to rolling direction of sheet and biaxial datawas considered from previous work done by Odenberger et al.[7]. For uniaxial tensile test, the specimen dimensions are usedas per sub-size ASTM: E8/E8M-11 standard. The detail aboutexperimental setup and specimen preparation were as per Kotkun-de et al. [2,17]. Also, the anisotropic coefcient required for yieldcriteria development and nite element simulations were calcu-lated using ASTM: E517. The experiments were performed threetimes and the average value was taken for development of theyield criteria. In literature, Holloman equation is the most popularand widely used to predict stress strain in plastic region. Therefore,for present study, Holloman equation is used to calculate strainhardening exponent (n). The calculated material properties forTi6Al4V alloy at 400 C are mentioned in Table 2.

Table 1Chemical composition of as received Ti6Al4V sheet.Element Al V Fe C Ti

Comp (wt.%) 5.56 4.07 0.185 0.022 89.9973.1. Hill 1948 yield criterion

Hill 1948 [10] is one of the most popular yield criterion for niteelement simulation of sheet metal forming processes. The planestress yield function is given by Eq. (1)

Fr r211 r222 22R 1R 1 r11r22 2

2R 1R 1 r12

1=21

where R is normal anisotropy coefcient, r11, r22 are the principalstresses. The material response after yielding is governed by elas-ticplastic constitutive relation with plastic hardening modulusand yield stress of the material as an input. The yield locus for Hill1948 yield criterion is as shown in Fig. 2(a).

3.2. Barlat 1989 yield criterion

Fig. 1. Experimental test rig and enlarged view of die with induction furnace.

Design 63 (2014) 336344 337rh y@/@rx

@/@ry

rh

1

r7

andHere, h is taken as 45 to compare the value and to further iter-ate. The yield locus plotted by Barlat 1989 model is as shown inFig. 2(b).

3.3. Barlat 1996 yield criterion

Barlat et al. [18] developed a yield model for modeling aniso-tropic materials. The yield function is expressed in Eq. (3)

/ /1jS2 S3ja /2jS3 S1ja /3jS1 S2ja 2ra 3where r is equivalent stress and a is a material parameter which isconsidered 8 for FCC and 6 for BCC materials. Si corresponds to prin-cipal values of Cauchy stress deviator.

The equivalent stress deviator is dened as

S Lr

where L

c2 c3=3 c3=3 c2=3 0c3=3 c3 c1=3 c1=3 0c2=3 c1=3 c1 c2=3 00 0 0 c4

26664

37775

c1, c2, c3, c4 are material constants and /i1;2;3 depends upon the ori-entation of frame of the principal value of S and anisotropy axis [8].

/i /xP21i /yP22i /zP23iFor a plane stress condition when shear stress is zero the yield

function becomes

/ /xjSy Szja /yjSz Sxja /z0jSx Syja 2ra 4where

Sx c3 c23 rx c33ry

Sy c33 rx c11 c3

3ry

Sz c23 rx c13ry

For instance, consider /z0 = 1. When the shear stress is notequal to zero, other constants c6 and /z1 are used to introduceanisotropy. Therefore, seven tests need to be performed for deter-mining the material constants.

For uniaxial tension in rolling direction, the components of thestress deviator are

Sx c3 c23 r0

c3

Table 2Material properties for Ti6Al4V alloy.

Temperature (C) r0 (MPa) r45 (MPa) r90 (MPa)

400 681.0 591.0 691.0

338 N. Kotkunde et al. /MaterialsSy 3 r0

Sz c23 r0

By substituting these components into Eq. (4)

F /xjc2 c3ja /yj2c2 c3ja /z0jc2 2c3ja 23r=r0a 5Similarly, for uniaxial tension in 90 orientation and biaxial

stress state, the equation reduces in the form of Eq. (6)F /xjc3 2c1ja /yjc3 c1ja /z0jc1 2c3ja 2 3r=r90 aF /xj2c1 c2ja /yjc1 2c2ja /z0jc1 c2ja 2 3r=rb a

6Other three expressions are obtained by a function expressed in

Eq. (7) considering r values directionality requirement.

r0 2ra

@u@Sx

@u@Sy

r0

1

r90 2ra

@u@Sx

@u@Sy

r90

1 7

rb 2ra

@u@Sx

@u@Sy

rb

1

Therefore, six material constants are calculated by solvingabove six non-linear equations. The material constants for Barlat1996 yield criteria are determined and presented in Table 3 andits yield locus is plotted as shown in Fig. 2(c).

3.4. Barlat 2000 yield criterion

Barlat et al. [8] proposed an anisotropic yield criterion which ismade of two convex functions. The equation for the yield locus isgiven by Eq. (8)

/0 jX01 X02ja; /00 j2X002 X001ja j2X001 X002ja

/ jX 01 X02ja j2X002 X 001ja j2X001 X 002ja 2ray 8in which X01;2 and X

001;2 are the principal values of the linearly trans-

formed stress deviator matrices {s},

fX0g C 0fsg C 0Tr L0r

fX00g C 00fsg C 00Tr L00r

where T 2=3 1=3 01=3 2=3 00 0 1

264

375

When the matrices C0 and C00 are taken as identity matrix theabove criteria reduces to isotropic case. For the simplicity of calcu-lation, anisotropy parameters and the coefcients of L0 and L00 arerelated as

L011L012L0210

2666666

3777777

2=3 0 01=3 0 00 1=3 0

2666666

3777777

a1a2

264

375

C (MPa) rb (MPa) r0 r45 r90 rb

51.0 744.9 0.6 1.26 0.512 1.02

Design 63 (2014) 336344L22L066

4 5 0 2=3 00 0 1

4 5 a7

L0011L0012L0021L0022L0066

26666664

37777775

2 2 8 2 01 2 4 4 04 4 4 1 02 8 2 2 00 0 0 0 9

26666664

37777775

a3a4a5a6a8

26666664

37777775

Fig. 2. Yield loci for Ti6Al4V alloy at 400 C (a) Hill 1948, (b) Barlat 1989, (c) Barlat 1996, (d) Barlat 2000 and (e) Cazacu Barlat.

N. Kotkunde et al. /Materials and Design 63 (2014) 336344 339

Table 3Barlat 1996 model material parameters for Ti6Al4V alloy.

Temperature (C) C1 C2 C3 C4 ax ay az0 a

400 1.059 0.939 1.233 1 0.992 0.992 1 8

340 N. Kotkunde et al. /Materials andExponent a is the same parameter as in the three parameterBarlats criterion. When the value of a is two, the yield surfacebecomes Hill yield surface, i.e. von-Misses yield surface in isotropiccase. Matrices C0 and C00 are expressed in terms of eight anisotropycoefcients /i, for isotropic case /i is unity. The complete descrip-tion of the yield model is given by Barlat et al. [8].

For determination of material parameters /1 to /8, eight testsneed to be performed. In this study, yield stresses in 0, 45 and90 orientation and Lankford coefcients in the same directionare obtained from experiments. In addition to that, biaxial yieldstress and biaxial anisotropy coefcient are considered from previ-ous literature [7].

Deviatoric components are dened for loading each stress ofstate as

sx cr; sy drThe material parameters are determined by solving two equa-

tions given below for each state of stress.

F j/1c/2dja j/3c 2/4dja /z0j2/5c/6dja 2 r=r a

G qx@/@sx

qy@/@sy

0

The parameter description of c, d, qx and qy is mentioned inTable 4 [19]. The material constants for Barlat 2000 criterion arepresented in Table 5. The yield locus plotted by Barlat 2000 modelis as shown in Fig. 2(d).

3.5. Cazacu Barlat yield criterion

Cazacu et al. [9] proposed an anisotropic yield criterion whichconsists of both tension and compression asymmetry. For extend-ing this criterion, stress deviator s is linearly transformed and theprinciple values of Cauchy stress deviator in the yield functionare replaced by transformed tensor. The proposed anisotropic yieldfunction is given as Eq. (9)

jR1j kR1a jR2j kR2a jR3j kR3a F 9

R Cs

C

C11 C12 C13C12 C22 C23C13 C23 C33

C44C55

C66

2666666664

3777777775

where C is a fourth order transformation tensor with reference toorthotropic (x, y, z) axes and F is the size of yield locus, k is based

on yield stress in tension/compression and material parameter ain the yield function.

Table 4Parameter description for Barlat 2000.

Experimental test c d qx qy

0 Tension 2/3 1/3 1 r0 2 + r090 Tension 1/3 2/3 2 + r90 1 r90Balanced biaxial tension 1/3 1/3 1 + 2rb 2 + rbrTrC

2=31 ka 21=31 ka

2=31 ka 21=31 ka 1=a

Material parameter a is considered as 8 for Ti6Al4V. Thematerial constants in tensor Care determined by performing uniax-ial yield stress in tension and compression along with balancedbiaxial test and anisotropy coefcients [9].

Consider plane stress condition for thin sheet, the non-zerocomponents are the (rxx, ryy, sxy). Hence the principal values ofR are:

R1 12 Rxx Ryy Rxx Ryy2 4R2xy

q

R2 12 Rxx Ryy Rxx Ryy2 4R2xy

q

R3 Rzzwhere

Rxx 23C11 13C12 13C13

rxx 13C11

23C12 13C13

ryy

Ryy 23C12 13C22 13C23

rxx 13C12

23C22 13C23

ryy

Rzz 23C13 13C23 13C33

rxx 13C13

23C23 13C33

ryy

Rxy C66rxyLet rT0 and rC0 is yield stress in tension and compression respec-

tively along the 0 orientation. The proposed orthotropic yield cri-terion is written as

rT0 F

j/1j k/1a j/2j k/2a j/3j k/3a 1

a

rC0 F

j/1j k/1a j/2j k/2a j/3j k/3a 1

a

where

/1 23C11 13C12

13C13

/2 23C12 13C22

13C23

/3 23C13 13C23

13C33

Similarly for transverse direction,

rT90 F

jW1j kw1a jw2j kw2a jw3j kw3a 1

a

rC90 F

jW1j kw1a jw2j kw2 a jw3j kw3 a 1

a

where

w1 13C11 23C12

13C13

Design 63 (2014) 336344w2 13C12 23C22

13C23

Table 5Barlat 2000 model material parameters for Ti6Al4V alloy.

Temperature (C) a1 a2 a3 a4 a5 a6 a7 a8 a

.976

Fig. 3. Axisymmetric model of deep drawing setup.

Fig. 4. Experimental deep drawn cup at 400 C and 54 mm blank diameter.

N. Kotkunde et al. /Materials and Design 63 (2014) 336344 341w3 13C13 23C23

13C33

rTb F

jX1j kX1 a jX2j kX2 a jX3j kX3 a 1

a

where

X1 13C11 13C12 23C13

X2 13C12 13C22 23C23

X3 13C13 13C23 23C33

Let rh is Lankford coefcient under the uniaxial tension case,according to proposed orthotropic criterion

rT0 1kaUa11 W11kaUa12 W2Ua13 W3

1kaUa11 W1U11kaUa12 W2Ua13 W3Ua2Ua3

rT90 1kaWa12 U21kaWa11 U1Wa13 U3

1kaWa12 W2U21kaWa11 U1Wa13 U3Wa1Wa3

The material constants have been calculated by solving a set ofnon-linear equations using MATLAB. The material constants arepresented in Table 6. The yield locus plotted by Cazacu Barlat func-tion is as shown in Fig. 2(e).

4. Finite element analysis

The nite element analysis in the present work is done usingDynaform version 5.6.1 with LSDyna version 971 as solver. Thesimulations are carried out on a system with the conguration:Intel Xeon CPU E3-1270 V2, 3.50 GHz, 16 GB Ram and 64-bit Win-dows-7 Operating System. The deep drawing setup model usedfor the simulations is as shown in Fig. 3. Setup consists of die,punch, blank and blank holder. The model is discretized by usingBelytschko-Tsay shell elements to avoid higher computation timerequired for continuum elements [6]. In order to simplify niteelement model, only quarter geometry is modeled since thematerial properties, geometry and loading are considered to besymmetric along in-plane mutually perpendicular axes. The die,punch and blank holder are considered as rigid materials. Differ-ent material models are assigned to the blank for comparison.Selective mass scaling is done to reduce computational time,adaptive remeshing is used since it captures the deformation inblank in critical regions like punch corner and wall accurately.

400 0.982 0.853 0.787 0Further, the difculty in determining the friction coefcient isovercome by inverse approach [6,20]. In this work, a blank isdrawn to form a cup at 400 C and punch load vs. displacement

Table 6Cazacu-Balart yield model material parameters for Ti6Al4V alloy.

Temperature (C) C11 C22 C33 C12

400 1.959 1.837 1.955 0.6840.986 0.787 1.132 1.426 8is recorded. Following this, simulations are run by selecting arange of friction values. The punch load vs. displacement data

C13 C23 C44 a k

0.480 0.451 1.880 8 0.085

and342 N. Kotkunde et al. /Materialsobtained by nite element simulations is compared with that ofexperimental. Thereby, the friction coefcient for which the graphmatches with a satisfactory level of accuracy is selected for sim-ulations. The coefcient of friction value was chosen as 0.09 fornite element (FE) simulations.

Fig. 5. Simulated deep drawn cups (a) Hill 1948, (b) Barlat 19Design 63 (2014) 3363445. Results and discussion

Fig. 2(a) and (b) shows loci of Hill 1948 and Barlat 1989 yieldfunctions. Material constants evaluations for these anisotropicyield criteria are comparatively simpler than that of other yield cri-

89, (c) Barlat 1996, (d) Barlat 2000 and (e) Cazcau Barlat.

teria [16]. But, these models are not able to show yielding behaviorin uniaxial compression and biaxial tensile stress state. Inspite ofthis drawback, these criteria are frequently used in nite elementsimulations due to ease in identication of material parameters[19].

On the other hand, loci obtained by Barlat 1996 and Barlat 2000yield function shown in Fig. 2(c) and (d) nearly approximate the

Fig. 6. Thickness distribution 54 mm blank at 400 C.

Barlat1996

Relativeerror

Barlat2000

Relativeerror

CazacuBarlat

Relativeerror

0.829 0.032 0.799 0.005 0.803 0.000

Fig. 7. Earing prole of experimental and simulated cup.

Table 9Mechanical parameters required for yield criteria.

Yield model Parameters Tests required

r0 r45 r90 rb r0 r45 r90 rb rc

Hill 1948 4 Y Y Y YBarlat 1989 3 Y Y Y YBarlat 1996 8 Y Y Y Y Y Y Y YBarlat 2000 8 Y Y Y Y Y Y Y YCazacu Barlat 8 Y Y Y Y Y Y Y Y

Y = test required.

Table 10Total CPU time taken for simulation.

Yield model Time (s)

Hill 1948 7166Barlat 1989 8769Barlat 1996 82,722Barlat 2000 76,014Cazacu Barlat 37,398

N. Kotkunde et al. /Materials and Design 63 (2014) 336344 343experimental data points for Ti6Al4V alloy. But, accurate deter-mination of material response in sheet metal forming requiresmore accurate yield function which takes tensioncompressionasymmetry into account. However, Ti6Al4V alloy has HCP crys-tal structure and Bauschinger effect is more pronounced. SinceCazacu Barlat yield function considers stress asymmetry, it is verywell suited for Ti6Al4V alloy. It is also evident from Fig. 2(e) thatthe yield locus plotted by Cazacu Barlat yield function predictsyielding very accurately with all the states of stress. Similar resultswere observed by Cazacu and Barlat [9] for Mg0.5% Th alloywhich is also a HCP crystal structure.

In order to validate the yield criteria, the experiments havebeen performed at 400 C. The maximum blank diameter of54 mm was successfully drawn hence the limiting draw ratio(LDR) of 1.8 was achieved. It is seen that LDR of Ti6Al4V alloyis signicantly lower than other structural alloys such as austen-itic stainless steel. The experimental deep drawn cup at 400 Cand simulated cups using different anisotropic yield criteria areas shown in Figs. 4 and 5 respectively. Thickness distribution

Table 7Relative error for thickness distribution.

Normalizeddistance

Expt.thickness

Hill1948

Relativeerror

Barlat1989

Relativeerror

0 0.803 0.807 0.006 0.816 0.017

0.1 0.789 0.795 0.008 0.797 0.010 0.828 0.049 0.800 0.014 0.803 0.0180.2 0.788 0.808 0.026 0.817 0.036 0.829 0.053 0.809 0.026 0.806 0.0230.3 0.772 0.808 0.047 0.777 0.007 0.796 0.031 0.803 0.040 0.781 0.0120.4 0.792 0.801 0.012 0.761 0.039 0.784 0.010 0.790 0.003 0.781 0.0140.5 0.837 0.792 0.054 0.763 0.089 0.783 0.065 0.785 0.062 0.822 0.0180.6 0.834 0.897 0.076 0.799 0.042 0.799 0.042 0.818 0.019 0.835 0.0010.7 0.884 0.943 0.067 0.812 0.082 0.836 0.054 0.849 0.040 0.856 0.0320.8 0.903 0.922 0.021 0.844 0.065 0.875 0.031 0.900 0.003 0.882 0.0240.9 0.908 0.921 0.014 0.896 0.014 0.890 0.020 0.956 0.053 0.904 0.0051 0.914 0.932 0.019 0.923 0.010 0.895 0.020 0.980 0.072 0.947 0.036

Table 8Comparison of percentage relative error in thickness distribution.

Material model Hill 1948 Barlat 1989 Barlat 1996 Barlat 2000 Cazacu -Barlat

Average relative error (%) 3.1 3.7 3.6 3.0 1.6Standard deviation of error 0.0236 0.0283 0.0161 0.0232 0.0112

and earing prole are taken as quantiable measures to validatethe simulation results with experimental results. Experimentswere performed three times and the average thickness and cupheight values were taken for further analysis. Thickness distribu-tion for the 54 mm blank diameter is as shown in Fig. 6. The cal-

(2) Although similar complexity is involved in the materialparameter identication of Barlat 1996, Barlat 2000 andCazacu Barlat yield model, considering the computationaltime, Cazacu Barlat is more efcient in nite element anal-

344 N. Kotkunde et al. /Materials and Design 63 (2014) 336344culation of relative error of thickness along deep drawn cup ispresented in Table 7. Average percentage error of thickness distri-bution and its standard deviation are chosen as statistical mea-sures for comparing anisotropic yield criteria. It is clear fromTable 8 that Cazacu Barlat yield has the least error in predictioncompared to other models.

It should be noticed that earing is a pronounced phenomenon indeep drawing for anisotropic metals which is also evident fromFig. 4. This necessitates comparison of yield models based on ear-ing prediction in deep drawing. Fig. 7 shows earing prole ofexperimental and simulated cups. Earing is poorly predicted by Hill1948, Barlat 1996 and Barlat 2000 yield models. Therefore, thesemodels are less preferred for deep drawing of Ti6Al4V alloy.On the other hand, Barlat 1989 predicts the earing prole but notas precise as the earing predicted by Cazacu Barlat yield modelwhich is evident from Fig. 7. Therefore, considering these qualita-tive parameters of deep drawn cups, Cazacu Barlat yield model iswell suited for deep drawing of Ti6Al4V alloy.

For the simplicity in implementation of yield criteria, it isimportant to consider a number of required material parametersand experimental tests to be carried out for their evaluation. Table9 shows different yield models and mechanical tests required fortheir determination. Hill 1948 has only four material parameterswhich are determined by performing simple uniaxial tension tests.Also, the simulation time taken for these models is relatively lesserthan that of other models. Table 10 shows comparison of simula-tion time required for different anisotropic yield criteria. On theother hand, advanced yield models like Barlat 1996 and Barlat2000 and Cazacu Barlat yield models need different mechanicaltests to be carried out like biaxial tensile test, compression testetc. which makes them complicated for their application in niteelement simulation. Moreover, the simulation time required forthese models are very large compared to Hill 1948 and Barlat1989 yield models.

It should be noticed that the simulation time for Cazacu Barlatyield model is less as compared to Barlat 1996 and Barlat 2000with adequate accuracy along with prediction of thicknessdistribution and earing prole in the deep drawn cup. HenceCazacu Barlat yield model is most suitable for deep drawing simu-lation of Ti6Al4V alloy.

5. Conclusions

This work involves nite element simulations of deep drawingusing different yield criteria implemented for Ti6Al4V alloyand their validation with experimental results. Based on results,the important ndings are:

(1) Cazacu Barlat criterion is the most suited for Ti6Al4Valloy among other anisotropic yield criteria since anisot-ropy in yielding and stress asymmetry resulted in excellentvalidation of yield function with experimental thicknessdistribution and earing prole.ysis for deep drawing of Ti6Al4V alloy.

Future scope involves implementation of constitutive modelswith advanced anisotropic yield criteria for nite element simula-tions of warm deep drawing processes.

Acknowledgement

The nancial support received for this research work fromDepartment of Science and Technology (DST), Government of India,SERB-DST, SR/FTP/ETA-0056/2011 is gratefully acknowledged.

References

[1] Poondla N, Srivatsan TS, Patnaik a, Petraroli M. A study of the microstructureand hardness of two titanium alloys: commercially pure and Ti6Al4V. J AlloyCompd 2009;486:1627.

[2] KotkundeNitin,KrishnamurthyHansogeNitin,PuranikPavan,GuptaAmitKumar,Singh Swadesh Kumar. Microstructure study and constitutive modeling of Ti6Al4V alloy at elevated temperatures. Mater Des 2014;54:96103.

[3] Chen F, Chiu K. Stamping formability of pure titanium sheets. J Mater ProcessTechnol 2005;170:1816.

[4] Li Xiaoqiang, Guo Guiqiang, Xiao Junjie, Song Nan, Li Dongsheng. Constitutivemodeling and the effects of strain-rate and temperature on the formability ofTi6Al4V alloy sheet. Mater Des 2014;55:32534.

[5] Naka Tetsuo, Uemori Takeshi, Hino Ryutaro, Kohzu Masahide, Higashi Kenji,Yoshida Fusahito. Effects of strain rate, temperature and sheet thickness on yieldlocusofAZ31magnesiumalloy sheet. JMaterProcessTechnol2008;201:395400.

[6] Singh SK, Mahesh K, Kumar A, Swathi M. Understanding formability of extra-deep drawing steel at elevated temperature using nite element simulation. JMater Des 2010;31:447884.

[7] Odenberger E-L, Schill M, Oldenburg M. Thermo-mechanical sheet metalforming of aero engine components in Ti6Al4VPART 2: Constitutivemodelling and validation. Int J Mater Form 2013;6:40316.

[8] Barlat F, Brem JC, Yoon JW, Chung K, Dick RE, et al. Plane stress yield functionfor aluminium alloy sheet. Part 1. Theory. Int J Plast 2003;19:1297319.

[9] Cazacu O, Plunkett B, Barlat F. Orthotropic yield criterion for hexagonal closedpacked metals. Int J Plast 2006;22:117194.

[10] Hill R. The mathematical theory of plasticity. Oxford: Clarendon; 1950.[11] Jansson Mikael, Nilsson Larsgunnar, Simonsson Kjell. On constitutive modeling

of aluminum alloys for tube hydroforming applications. Int J Plast2005;21:104158.

[12] Barlat F, Lian J. Plastic behaviour and strechability of sheet metals Part I: ayield function for orthotropic sheets under plane stress condition. Int J Plast1989;5:516.

[13] Hill R. A user-friendly theory of orthotropic plasticity in sheet metals. Int JMech Sci 1993;35:1925.

[14] Xu S, Weinmann KJ. Prediction of forming limit curves of sheet metals usingHills 1993 user-friendly yield criterion of anisotropic materials. Int J Mech Sci1998;40:91325.

[15] Yoon Jeong-Whan, Barlat Frederic. Plane stress yield function for aluminumalloy sheetsPart II: FE formulation and its implementation. Int J Plast2004;20. 495-22.

[16] Hosford WF, Allen TJ. Twining and directional slip as a cause for strengthdifferential effect. Met Trans 1973;4:14245.

[17] Kotkunde Nitin, Deole Aditya D, Gupta Amit Kumar, Singh Swadesh Kumar.Comparative study of constitutive modeling for Ti6Al4V alloy at low strainrates and elevated temperatures. Mater Des 2014;55:9991005.

[18] Barlat F, Maeda Y, Hayashida Y, Murtha SJ, Hattori S, et al. Yield functiondevelopment for aluminum alloy sheets. J Mech Phys Solids 1997;45(11-12):172763.

[19] Banabic D. Sheet metal forming processes. Verlag Berlin Heidelberg: Springer;2010.

[20] Chandra Mohan Reddy G, Ravindra Reddy PVR, Janardhan Reddy TA. Finiteelement analysis of the effect of coefcient of friction on the drawability. JTribol Int 2010;43:11327.

Experimental and numerical investigation of anisotropic yield criteria for warm deep drawing of Ti6Al4V alloy1 Introduction2 Experimental details3 Anisotropic yield models3.1 Hill 1948 yield criterion3.2 Barlat 1989 yield criterion3.3 Barlat 1996 yield criterion3.4 Barlat 2000 yield criterion3.5 Cazacu Barlat yield criterion

4 Finite element analysis5 Results and discussion5 ConclusionsAcknowledgementReferences