mathematical modeling and finite element analysis of superplastic forming of ti 6al 4v alloy in a stepped rectangular die

Available online at www.sciencedirect.com ScienceDirect Procedia Engineering 64 (2013) 1209 – 1218 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 Assistant Professor, Department of Mechanical Engineering, Anna University, University College of Engineering -Ramanathapuram Campus, Ramanathapuram-623513, Tamilnadu, India Assistant Professor, Department of Mechanical Engineering, A.C College of Engg &Tech, Karaikudi- 630 004 Tamilnadu, India Associate 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) © ©2013 2013The TheAuthors Authors.Published Publishedby byElsevier Elsevier Ltd Ltd Selection Selectionand andpeer-review peer-reviewunder underresponsibility responsibilityof ofthe theorganizing organizingand andreview reviewcommittee committeeof ofIConDM IConDM2013 2013 Key Words: Finite Element method; Mathematical modeling; Stepped Rectangular; Ti-6Al-4V; Superplastic forming process _ 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 1210 M Balasubramanian et al / Procedia Engineering 64 (2013) 1209 – 1218 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 T m 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 D2 h0 k l1 Mn p R Ri+1 t Ti W1 Aluminum Half the die depth in stage two (mm) Original sheet thickness (mm) Material constant (MPa sm) Length of die in second stage (mm) Manganese Forming pressure (MPa) Radius of curvature (mm) Decrement in curvature (mm) Forming time (sec) Titanium Half width of the die (mm) D1 h hi+1 l1 m n pi+1 Ri S ti+1 V W2 Depth of die in first stage (mm) Current thickness (mm) Decrement in thickness (mm) Length of die in first stage (mm) Strain rate sensitivity index Strain hardening index Pressure increment (MPa) Radius of curvature in first stage (mm) Arc length (mm) Increment in time (sec) Vanadium Quarter the die width (mm) M Balasubramanian et al / Procedia Engineering 64 (2013) 1209 – 1218 Xi Xi+1 Yj Instantaneous in semi width in first stage (mm) 1211 Xj Instantaneous in semi width in second stage (mm) Decrement in semi width length (mm) Yi Instantaneous in depth of die in first stage (mm) Instantaneous in depth in second stage (mm) Yi+1 Decrement in depth of die (mm) Reduction in width of die (mm) Reduction in depth of die (mm) Stress (N/mm2) Stress in width direction (N/mm2) w Strain Strain rate (per sec) Strain rate in width direction (per sec) Effective strain rate (per sec) Angle suspected between radius of curvature and axis line (degree) Decrement of angle suspected (degree) Theoretical modeling 2.1 Superplastic forming process Argon Gas Before Applying Pressure (a) Vent hole (b) After Applying Pressure (c) (d) Fig (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 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 for the superplastic material can be expressed as Eq.(1) (1) 1212 M Balasubramanian et al / Procedia Engineering 64 (2013) 1209 – 1218 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 2.3 Superplastic blow forming process and geometric model R R D1 D1 S d S D1 D2 W1 W2 (b) W1 (a) R y=D2 R D1 Yi Yi+1=D2 S S D2 W2 Xi (d) (c) R1 Yi+1 W2 S1 Yj Stage R2 Xj Stage S2 (e) Fig (a), (b), (c) , (d) & (e) Illustration of different stages of blow forming D1 M Balasubramanian et al / Procedia Engineering 64 (2013) 1209 – 1218 1213 The Fig 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) R (2a) (2b) (2c) (2d) L d (2e) W1 Fig 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) (3) The forming time is calculated in each stage by Eq (4) (4) The current thickness of the sheet during blow forming in each step is obtained by the Eq (5) (5) The sheet is treated as a membrane during forming, the forming pressure is obtained by the Eq (6) (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 ( ) in width direction and positive decrement ( ) in depth direction during the lengths contacted on the bottom and sidewall respectively during each stage of processing Using Y j =Yi+1 Xj = Xi+1 and pressure increment are found from Eqs (7), (8)& (9)  Ǧ Ǧ (7) (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 1214 M Balasubramanian et al / Procedia Engineering 64 (2013) 1209 – 1218 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 Fig FEM model for rectangular sheet The finite element simulation in a sheet metal with stepped rectangular geometry, the first step depth D1 = 14 mm, width 2W1 = 28 mm and length l1 = 120 mm and second step D2 = mm, width 2W2 = 14 mm and length l2 = 106 mm with 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 and Table shows the composition and mechanical properties of Ti-6Al-4V alloy 1215 M Balasubramanian et al / Procedia Engineering 64 (2013) 1209 – 1218 Table Composition of Ti-6Al-4V Alloy Component Ti Al V _ % of weight 90 06 04 _ Table Mechanical properties of Ti-6Al-4V alloy _ S.No Mechanical properties Value _ Yield Strength 924 Mpa Ultimate Strength 993 Mpa Melting point 1500 to 1600°C Modulus of Elasticity 113.8 GPa Poiss 0.342 3.4 Blow forming components at different stages The Fig (a),(b),(c) & (d) Different stages of blow forming of sheet into the stepped rectangular die Fig (a) shows that the full rectangular sheet, before applying boundary and load conditions Fig (d) shows that after completion of blow forming process (a) (b) (c) (d) Fig (a),(b),(c) & (d) Different stages of blow forming of Ti-6Al-4V sheet into a stepped rectangular die 1216 M Balasubramanian et al / Procedia Engineering 64 (2013) 1209 – 1218 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 0.7 0.6 0.5 0.4 0.3 0.2 0.1 FEA(ABAQUS) Theoretical 20 40 60 Forming time (Min) 80 Forming pressure (MPa) Forming Pressure (MPa) Superplastic forming depends on the gas pressure and time The forming pressure with respect to forming time is shown in Fig 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 0.8 0.6 0.5 0.3 0.2 0.1 0.6 0.5 0.4 0.3 r = mm m r = mm m 0 10 20 30 40 50 60 Forming time (Min) 15 30 45 60 75 90 70 Fig Illustration of effective of optimum pressure as a function of forming time at different die entry radius initial thickness of 1.6 mm Fig Illustration of effect of pressure profile as a function of forming time at different friction coefficient Thickness distribution (mm) Optimum pressure (Mpa) 0.7 0.1 Forming time(Min) Fig Theoretically predicted forming pressure as a function of forming time, at initial thickness of 1.6 mm 0.2 μ=0.0 μ=0.3 μ=0.5 0.4 1.8 1.6 1.4 1.2 0.8 0.6 0.4 0.2 r = mm r = mm r = mm 10 20 30 40 50 60 Forming Time (Min) 70 Fig 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 off 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 M Balasubramanian et al / Procedia Engineering 64 (2013) 1209 – 1218 1217 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 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 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 of forming 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 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 f of thickness distribution from centre along the die profile 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 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 The pressure increases rapidly when the rate of change of radius is greater than the rate of change of thickness 1218 M Balasubramanian et al / Procedia Engineering 64 (2013) 1209 – 1218 The forming pressure requirement was decreased to maintain constant strain-rate deformation with increasing die entry radius At a given forming temperature, thickness distribution varies with increasing die entry radius Optimum pressure need decreases with increase in friction coefficient More thinning is found at both the bottom corner of the rectangular die 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 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 [16] Luckey.S.G, Jr,.Friedman.P.A, Weinmann.K.J, 2007 Correlation of finite element analysis to superplastic forming experiments, Journals of Materials Processing Technology, vol 194, pp 30-37  ... 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... compared to the rest of the part 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... 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