Scholarly article on topic 'Mathematical Modeling and Finite Element Analysis of Superplastic Forming of Ti-6Al-4V Alloy in a Stepped Rectangular Die'

Mathematical Modeling and Finite Element Analysis of Superplastic Forming of Ti-6Al-4V Alloy in a Stepped Rectangular Die Academic research paper on "Materials engineering"

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Abstract of research paper on Materials engineering, author of scientific article — M. Balasubramanian, K. Ramanathan, V.S. Senthil kumar

Abstract Superplastic forming has become a viable process in manufacturing of aircraft and automobile parts such as compressor blades, window frames and seat structures, turbine disc etc., which require relatively low tooling and assembly cost. In this paper, the attempt was made to analyze the Ti-6Al-4V alloy sheet using a stepped rectangular die by superplastic blow forming technique. This alloy is most suitable material for producing complex shapes using superplastic forming methods. The forming characteristics of thickness distribution, bulge forming time and optimum pressure with and without die entry radius and friction coefficient in a two step rectangular die have been analyzed by the theoretical model and numerical simulation using Finite Element Method (Abaqus).

Academic research paper on topic "Mathematical Modeling and Finite Element Analysis of Superplastic Forming of Ti-6Al-4V Alloy in a Stepped Rectangular Die"

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Procedía Engineering 64 (2013) 1209 - 1218

Procedía Engineering

www.elsevier.com/locate/procedia

..International Conference On DESIGN AND MANUFACTURING, IConDM 2013

Mathematical Modeling and Finite Element Analysis of Superplastic Forming of Ti-6Al-4V Alloy in a Stepped

Rectangular Die

M.Balasubramanian1*, K.Ramanathan2, V.S.Senthil kumar3

1Assistant Professor, Department of Mechanical Engineering, Anna University, University College of Engineering -Ramanathapuram Campus,

Ramanathapuram-623513, Tamilnadu, India. 2Assistant Professor, Department of Mechanical Engineering, A.C. College of Engg &Tech, Karaikudi- 630 004 Tamilnadu, India. 3Associate Professor, Department of Mechanical Engineering, College of Engineering, Guindy Campus, Anna University, Chennai-600 025,

Tamilnadu, India.

Abstract

Superplastic forming has become a viable process in manufacturing of aircraft and automobile parts such as compressor blades, window frames and seat structures, turbine disc etc., which require relatively low tooling and assembly cost. In this paper, the attempt was made to analyze the Ti-6Al-4V alloy sheet using a stepped rectangular die by superplastic blow forming technique. This alloy is most suitable material for producing complex shapes using superplastic forming methods. The forming characteristics of thickness distribution, bulge forming time and optimum pressure with and without die entry radius and friction coefficient in a two step rectangular die have been analyzed by the theoretical model and numerical simulation using Finite Element Method (Abaqus).

© 2013TheAuthors.PublishedbyElsevierLtd.

Selectionandpeer-review underresponsibilityoftheorganizingandreview committeeofIConDM 2013 Key Words: Finite Element method; Mathematical modeling; Stepped Rectangular; Ti-6Al-4V; Superplastic forming process.

1. Introduction

The superplastic forming is a valuable tool for fabrication of complex parts used in the aircraft and automobile industries. Superplastic forming of the sheet metal has been used to produce complex shapes and integrated structure that are often light weight and stronger than the assembled components. Superplasticity is a property of certain metallic materials, which enable them to achieve very high elongation of 1000% without necking in hot

Corresponding Authors: Tel: +91-04567-291599, fax: +91-04567-291699. Email : *annaunivbala76@gmail.com, Email address : 2kalirams@yahoo.com, 3vsskumar@annauniv.edu.

1877-7058 © 2013 The Authors. Published by Elsevier Ltd.

Selection and peer-review under responsibility of the organizing and review committee of IConDM 2013 doi: 10. 1016/j .proeng.2013.09.200

tensile test under certain condition and also this material undergo extreme elongation at the proper temperature and strain rate. Superplastic deformation is carried out very close to isothermal conditions under controlled strain rate. The optimum strain rate varies with the superplastic material, which is usually in the range of 0.001 s 1 to 0.00001s 1. This is attributed to the viscous material behaviour exhibited by some metals and alloys with very fine and stable grain structure at temperature above 0.5 Tm (Tm-melting point of materials). Few materials like Ti-6Al-4V alloy undergo extensive tensile plastic deformation prior to failure under a specific temperature and particular strain-rate.

Ghosh and Hamilton (1980) used the plane strain condition to explore the shape of the die on the optimized pressurization profile during blow forming into a rectangular die. Jovane (1968) used a uniform-deformation method to analyze the relationship between the optimum pressurization profile and the strain-rate sensitivity during the blow forming of a circular diaphragm. Hwang et al. (1997) developed a generalized mathematical model considering uniform and non-uniform thinning in the free bulge region to examine the optimized pressurization profile and thickness distribution of the product in blow forming into a circular die. Padmanabhan and Davies (1980) achieved long elongation at slow strain rate at temperature 0.5 Tm. Viswanathan et al. (1980) investigated the theoretical and experimental models of the thermo pressure forming process of the Ti-6Al-4V alloy into a hemispherical shape. Viswanathan et al. (1990) analysed macro, micro and re-entrant shape in a Ti-6Al-4V alloy to optimize the forming pressure, time and thickness distribution.

Yogesha and Bhattacharya (2011) studied superplastic deformation capability of the Ti-Al-Mn alloy by thermoforming route. SenthilKumar et al. (2006) analyzed the finite element modeling of superplastic forming of AA7475 aluminium alloy in a hemispherical die. Chandra and Chandy (1991) used a finite element analysis model choosing the membrane element model for the superplastic forming process in a box with a complex shape. Bonet et al. (1994) developed a finite element analysis model using incremental flow formulation in thick and thin sheet components. Xing et al. (2004) developed a rigid- viscoplastic finite element program, to predict the microstructure variation to improve the uniformity of wall thickness. Mimaroglu and Yenihaya (2003) analysed the superplastic forming process under constant strain rate by the ANSYS finite element analysis code, parametric design language and ANSYS-visco108 element. Giuliano (2008) considered four-node, isoparametric and arbitrary quadrilateral elements for Finite element analysis in a Ti-6Al-4V alloy. Chen et al. (2001) used the continuum element for finite element analysis in Ti-6Al-4V alloy.

Balasubramanian et al. (2004) has developed a theoretical model and C++ coding in a long rectangular die and analysed superplastic parameters like radius of curvature, bulge forming time, thickness distribution and pressure profile for 8090 Al-Li alloy. Many work have been carried out in the related field. but only less work has been reported on Ti-6Al-4V alloy. To best of our knowledge there is no literature focused on two and more than two stages in a rectangular die under plane strain condition using titanium alloy. Hence in this paper an attempt has been made for two stages in a long rectangular die with plane strain condition using Ti-6Al-4V alloy. Superplastic forming process have been done by a simple theoretical model and by numerical analysis using finite element method (FEM- Abaqus) simulation with accurate prediction of the deformation characteristics.

Nomenclature

Al Aluminum Di Depth of die in first stage (mm)

D2 Half the die depth in stage two (mm) h Current thickness (mm)

ho Original sheet thickness (mm) hi+i Decrement in thickness (mm)

k Material constant (MPa sm) li Length of die in first stage (mm)

li Length of die in second stage (mm) m Strain rate sensitivity index

Mn Manganese n Strain hardening index

P Forming pressure (MPa) Pi+i Pressure increment (MPa)

R Radius of curvature (mm) Ri Radius of curvature in first stage (mm)

Ri+i Decrement in curvature (mm) S Arc length (mm)

t Forming time (sec) ti+i Increment in time (sec)

Tl Titanium V Vanadium

Wi Half width of the die (mm) W2 Quarter the die width (mm)

X, Instantaneous in semi width in first stage (mm) Xj Instantaneous in semi width in

second stage (mm)

Xi+i Decrement in semi width length (mm) Yi Instantaneous in depth of die in

first stage (mm)

Yj Instantaneous in depth in second stage (mm) Yi+1 Decrement in depth of die (mm)

AX Reduction in width of die (mm) AY Reduction in depth of die (mm)

o Stress (N/mm2) "w Stress in width direction (N/mm2)

e Strain £ Strain rate (per sec)

£w Strain rate in width direction (per sec) £ Effective strain rate (per sec)

0i Angle suspected between radius of curvature and axis line (degree)

0i+i Decrement of angle suspected (degree)

2. Theoretical modeling

2.1 Superplastic forming process Argon Gas

(c) (d)

Flg. i. (a), (b), (c) & (d) Different stages of pressure blow forming technique in a stepped rectangular die.

Many number of metal forming process such as pressure forming, vacuum forming, thermo forming, deep drawing, etc have been developed in recent years. Pressure forming is the most widely used method for forming of superplastic metal into desired components shown in Fig. i. In superplastic forming process a material is heated to the superplastic temperature within a closed sealed die, and inert gas pressure was applied, sheet to take the shape of the pattern. The flow stress of the material during deformation, increases rapidly with increase in pressure.

In order to simulate mathematically, the pressure profile, thickness distribution and forming time in the superplastic forming process, the numerous constitutive equations have been proposed to characterize the material flow stress response. The flow stress (o) for the superplastic material can be expressed as Eq.(i)

a = kém

2.2 Basic assumptions

The following basic assumptions have been made during the theoretical modeling of the superplastic forming process, (i) The material is isotropic and incompressible, (ii) The diaphragm is rigidly clamped at the periphery of the die, (iii) Process is assumed to be plane strain condition, (iv) The specimen thickness is very small when compared with the die radius, so that bending and shearing effects are negligible.

r Stage 1

Stage 2

Fig. 2. (a), (b), (c) , (d) & (e) Illustration of different stages of blow forming.

The Fig. 2 clearly evident that, geometric relationship established to predict the thickness variation, radius of curvature, arc length, time required to form the curvature and forming pressure during both the step of bulge forming.

The mathematical relationship [1] is obtained from above geometric blow forming process diagram. In this theoretical analysis , it is assumed that the depth (D1) of the die is equal to half of the width (W1) of the die in step one (D1=W1) and step two (D2=W2).

R2 = Wi2 + L2

R2 = W2 + (R - d )2

R2 = w±2 + R2 + d2 - 2 X R X d

2xRxd = d2 + W2

R = (d2+^!2) 2xd

(2a) (2b) (2c) (2d)

Fig. 3 Geometric configuration of radius of curvature.

From Fig. 3, the radius of curvature is obtained by the Eq. (2e) Arc length of bulge is described by the Eq. (3) S = 2xRx sin"1^)

The forming time is calculated in each stage by Eq. (4)

«-UsM®--*©}

The current thickness of the sheet during blow forming in each step is obtained by the Eq. (5) h — h0exp (— ¿wt)

The sheet is treated as a membrane during forming, the forming pressure is obtained by the Eq. (6)

Using the above equations, the various superplastic forming parameters are analyzed at every stage of forming until the profile reaches the bottom of the die.

Subsequently, the forming takes towards the edge direction in both the steps. Assume positive decrement (A X) in width direction and positive decrement (AY) in depth direction during the lengths contacted on the bottom and sidewall respectively during each stage of processing. Using Yj =Yi+1 - AY and Xj = Xi+1 - AX, for each process assign a small positive value of AX and AY. Simultaneously in both the stages, the time increment, thickness drop and pressure increment are found from Eqs. (7), (8)& (9).

. , ( 2 ^ [(Ri+10i+1+f+-

ti+l-ti + ^jln ^AYAX)

L (■

hi+1 = h0exp(- ewti+1~)

(8) (9)

The time, thickness and pressure computation are carried out in this manner until profile reaches the edge of the die. Same equation is used to find all parameter in the second step of rectangular shape.

3. Finite Element Modeling

3.1 FEM model

Superplastic blow forming is a complicated process involving large strain, large deformation and material nonlinearity. Usually deformation is dependent on boundary conditions. Consequently, the numerical analysis of a highly nonlinear system presents formidable computational problems. Fortunately, the superplastic behaviours of materials are characterized by the dependency of the flow stress upon the strain rate, which allows the material to be described as rigid visco-plastic. Therefore, the simulation of superplastic blow forming can be performed using the creep strain rate control scheme within FEM (Abaqus). The die and sheet model of quarter stepped rectangular is shown in Fig. 4.

SIMUtIA

Fig. 4. FEM model for rectangular sheet.

The finite element simulation in a sheet metal with stepped rectangular geometry, the first step depth Di = 14 mm, width 2Wi = 28 mm and length h = 120 mm and second step D2 = 7 mm, width 2W2 = 14 mm and length l2 = 106 mm with 3 mm flange all around it.

The initial dimension of the blank is 126 mm x 34 mm x 1.6 mm; the blank was rigidly clamped on all its edges. The finite element mesh was generated using brick element in a rectangular sheet. The modified Newton Raphson method adopted for solving non-linear equation in Abaqus. The material constants of k = 250 MPasm, T = 927°C and m = 0.58 chosen for Ti-6Al-4V alloy in a numerical simulation analysis.

The nodes of element have three degrees of freedom i.e. X, Y and Z direction, the finite element model and boundary condition nodes on the blank outer edge had all their degree of freedom constrained. All nodes of the die surface were totally restricted for any movement in any direction. Pressure has applied to the blank surface in the Y direction as a distributed load, now several load steps corresponding to each operational procedure are carefully modeled to obtain an accurate simulation of a superplastic blow forming process in FEM (Abaqus).

3.2 Material selection

Titanium alloys can be used in the fabrication of airframe control surface and small scale structural elements where low weight and high stiffness are required. Ti-6Al-4V alloy is used for the theoretical modeling and finite element simulation of the superplastic forming process. Table 1 and Table 2 shows the composition and mechanical properties of Ti-6Al-4V alloy.

Table 1. Composition of Ti-6Al-4V Alloy

Component Ti Al V

% of weight 90 06 04

Table 2. Mechanical properties of Ti-6Al-4V alloy

S.No Mechanical properties Value

Yield Strength Ultimate Strength

Melting point

Modulus of Elasticity Poisson's Ratio

924 Mpa 993 Mpa 1500 to 1600°C 113.8 GPa 0.342

3.4 Blow forming components at different stages

The Fig. 5. (a),(b),(c) & (d) Different stages of blow forming of sheet into the stepped rectangular die. Fig. 5 (a) shows that the full rectangular sheet, before applying boundary and load conditions. Fig. 5 (d) shows that after completion of blow forming process.

Fig. 5. (a),(b),(c) & (d) Different stages of blow forming of Ti-6Al-4V sheet into a stepped rectangular die.

4. Results and discussion

A simple mathematical modeling of superplastic forming of two stepped rectangular box has been developed and the finite element package is used to predict the superplastic forming parameters such as the thickness distribution, forming time and optimization of pressure profile.

4.1 Variation of forming pressure as a function of forming time

Superplastic forming depends on the gas pressure and time. The forming pressure with respect to forming time is shown in Fig. 6.

From Fig. 6, observed that, the rate of change in pressure initially increases then slightly decreases and further rapidly increases. This observation is due to the rate of change of the thickness which is less than rate of change of the radius. The forming of the sheet continues, the rate of change of thickness increases while that of the radius decreases, and pressure reduced to continue the constant flow stress. Once the sheet contacts die surface, the rate of change of the radius again dominates in both the stages, and a rapid pressure increase obtained.

0.7 0.6 0.5

S ■o ta

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

-FEA(ABAQUS) -Theoretical

20 40 60

Forming time (Min)

Fig. 6. Theoretically predicted forming pressure as a function of forming time, at initial thickness of 1.6 mm.

0.4 0.3 0.2 0.1 0

15 30 45 60 Forming time(Min)

Fig. 7. Illustration of effect of pressure profile as a function of forming time at different friction coefficient.

0 10 20 30 40 50 60 Forming time (Min)

1.8 1.6 1.4 1.2 1 0.8 ■3 0.6

n io ti

r = 1 mm r = 2 mm r = 3 mm

0 10 20 30 40 50 60 Forming Time (Min)

Fig. 8. Illustration of effective of optimum pressure as a function of forming time at different die entry radius. initial thickness of 1.6 mm.

Fig. 9. Illustration of effective of thickness distribution as a function of forming time at different die entry radius

4.2 Effect of pressure profile as a function of forming time at different friction coefficient

The friction is widely recognized as an important factor for affecting the thinning of superplastic forming components. For an initial study, the friction coefficient is assumed uniformly along the contact surface. In addition, it is reasonable to assume that the friction coefficient ranges from 0.0 to 0.5. The effect of the frictional coefficient between the die and sheet during superplastic forming is shown in Fig. 7.

In the free bulged region, the forming pressure initially rises and remains constant at different friction coefficients. After the bulge profile touches side or bottom wall the friction coefficient has take place. The Fig .7 shows that the forming pressure decreases while the friction coefficient increases. The forming time needed to complete the blow forming process and also increases with increasing friction coefficient.

4.3 Effects of optimum pressure as a function of forming time at different die entry radius by FEA

The Fig. 8 shows the optimum pressure as a function of time with respect to different die entry radius. This profile indicates that, the forming time is decreasing to maintain constant strain-rate deformation with increasing die entry radius. During the flow formation to maintain constant strain-rate deformation with increasing die entry radius, the need of forming pressure was slightly decreased. The optimum pressure is obtained when the die entry radius and radius of the corner increases.

4.4 Effects of thickness distribution as a function offorming time at different die entry radius by FEA

The thickness distribution and forming time were changes with respect to die entry radius as shown in Fig. 9. This picture clearly conclude that, the thickness distribution is increasing to maintain constant strain-rate deformation with increasing die entry radius. The forming time rapidly decrease and thickness distribution values increase when the die entry radius increases. This abrupt thinning is due to the large tension exerted upon the sheet with free bulged region. As the free bulged region begins to make contact with the wall in both the steps, this rapid thinning become more profound when the die entry radius increases.

4.5 Effective of thickness distribution from centre along the die profile

Distance from I In center ulong the die profile (mm)

Fig. 10. Illustration of thickness distribution along the die surface profile at initial thickness of 1.6 mm.

The Thickness distributions are measured along with die surface and it is represented in Fig. 10. The thinning is measured from bottom centre point to top flange along die surface. The degree of thinning over the die profile can be calculated to a good accuracy at different die entry radius. From the Fig. 10, more thinning at second step bottom corner and first step bottom corner was observed. It indicates that more thinning occurs in the bottom corner compared to the rest of the part.

5. Conclusion

The mathematical modeling and Finite Element analysis of superplastic forming of Ti-6Al-4V alloy in a stepped rectangular die leads to the following conclusions.

1. The pressure increases rapidly when the rate of change of radius is greater than the rate of change of thickness.

2. The forming pressure requirement was decreased to maintain constant strain-rate deformation with increasing die entry radius.

3. At a given forming temperature, thickness distribution varies with increasing die entry radius.

4. Optimum pressure need decreases with increase in friction coefficient.

5. More thinning is found at both the bottom corner of the rectangular die.

6. References

[1]. Ghosh. A.K, Hamilton.C.H, 1980. Superplastic forming of a long rectangular box section analysis and experiment, modeling fundamentals and applications to metals, ASM, Metals Park, OH, pp 303-329.

[2]. Jovane.F, 1968. An approximate analysis of the superplastic forming of a thin circular diagram theory and experiments, International Journals of Mechanical Science, vol 10, pp 403-427.

[3]. Hwang.Y.M, Yang.J.S, Chen.T.R, Huang J.C, 1997. Analysis of superplastic sheet metal forming in circular closed die considering non uniform thinning, Journals of Materials Processing technology, vol 65, pp 215-227.

[4]. Padmanabhan.K.A and Davies G.J, 1980. Superplastisity, Materials Research and Engineering, Springer- Verlag, Berlin, Vol 2, pp 1-6.

[5]. Viswanathan.D, Venkatasamy.S, Padmanabhan.K.A, 1980. Theoretical and experimental studies on the pressure thermoforming of hemispheres of Ti-6Al-4V, International Conference on Superplasticity and superplastic forming, The Minerals, Metals and Materials Society, Warrendale, U.S.A; pp 321-326.

[6]. Viswanathan.D, Venkatasamy.S, Padmanabhan.K.A, 1990. Macro, Micro and re-entrant shape forming of sheets of alloy Ti-6Al-4V, Journal of Materials Processing Technology, vol 24, pp 213223.

[7]. Yogesha.B, Bhattacharya.S.S, 2011.Superplastic hemispherical Bulge forming of a Ti-Al-Mn alloy, International Journals of Scientific & Engineering Research, Vol 2. pp 1-4.

[8]. Senthil Kumar.V.S. Viswanathan.D, Natarajan.S, 2006. Theoretical prediction and FEM analysis of superplastic forming of AA7475 aluminum alloy in a hemispherical die, Journal of Materials Processing Technology, vol 173, pp 247-251.

[9]. Chandra.N, Chandy.K, 1991. Superplastic process modeling of plane strain components with

complex shapes, Journals of Materials Processing technology, vol 9, pp 27-37.

[10]. Bonet,J, Bhargava.P, Wood.R.D,1994. The incremental flow formulation for the finite element analysis of 3-dimensional superplastic forming processes, Journals of Materials Processing technology, vol 45, pp 243-248.

[11]. Xing.H.L, Zhang.K.F, Wang.Z.R, 2004. A preform design method for sheet superplastic bulging with finite element modeling, Journals of Materials processing technology, vol 151, pp 284-288.

[12]. Mimaroglu.A, Yenihayat.O.F, 2003. Modelling the superplastic deformation process of 2024 aluminium alloys under constant strain rate : use of finite element technique, International Journals of Material Design, vol 24, pp 189-195.

[13]. Giuliano.G, 2008. Constitutive equation for superplastic Ti-6Al-4V alloy, International Journals of Material Design, vol 29, pp 1330-1333.

[14]. Chen.Y, Kibble.K, Hall.R, Huang.X, 2001. Numerical analysis of superplastic blow forming of Ti-6Al-4V alloys, International Journals of material Design, vol 22, pp 679-685.

[15]. Balasubramanian.M, Senthil Kumar.V.S, Natarajan.S, Viswanathan.D, 2004. Analysis and numerical simulation of superplastic forming in a long rectangular die, National conference on Advances in Materials and Manufacturing Technology (CAMMT), IIT Madras, Chennai, pp 97-98.

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