ELSEVIER Journal of Materials Processing Technology 68 ¢1997j 215- 220

Journal of

Materials Processing Technology

A finite-element model for the superp]astic bulging deformation of Ti°aHoy pipe

Nihat Akkus *, Ken-ichi Manabe, Masanori Kawahara, Hisashi Nishimura Tokyo Metropolitan UniversiO', Department ol Mechanical Engineering, i-! Minamiosawa. Hachioji-shi, Tokyo 192-03. Japan

Abstract

The superplastic bulging deformation process of Ti-alloy pipe under constant strain rate at the apex has been analyzed using a finite-element model (FEM). For the simulation of the process and the prediction of the final thickness distribution of the deformed part, MARC-MENTAT was used as a computational tool. An incremental approach based on a rigid-plastic flow formulation was employed. Special sub-routines were developed and linked to the main program for the superplastic material properties and the results of pressure calculations of constant apex-strain-rate deformation for each step. By FE modelling it is confirmed that the constant apex-strain-rate deformation process is superior to the constant pressure deformation process with respect to wall-thickness bulging. The results of calculation are compared to experimental results. '~ 1997 Elsevier Science S.A.

Keyword~: Finite element model; Bulging deformation: Ti-ailoy pipe

1. Introduction

Using superplastic forming processes (SPF), very complex structural components can be produced easily with particular materials such as titanium and alu- minium alloys, ceramics, metal-matrix composites etc., because of their extremely large deformation ability. The application of SPF has expanded very quickly in the aerospace industry and more recently it has been used to produce some consumer goods such as golf clubs and cookery-ware [1]. This expanding application also has brought some unsolved problems and lead many researchers to put effort into the optimization of SPF processes.

To obtain the optimum conditions for the SPF pro- cess, the primary requirement is to define suitable SPF parameters, especially the deformation temperature, the strain rate and the strain-rate-sensitivity index. As re- ported earlier by several researchers [2-4], maximum formability is possible with maximization of the m value, which latter depends strongly on the strain rate of the deformation process. For the optimization of the m value, the pressure path of the deformation process is very important, and moreover, it becomes crucial in the

*Corresponding author. Tel./fax: +81 426 772941; e-mail: ni- hat@mech.metro-u.ac.jp

0924-0136/97/$17.00 © 1997 Elsevier Science S.A. All rights reserved.

PII S0924-0136(96)00104-5

pipe-bulging process, where the strains in the merid- ional and circumferential directions are quite different. From this aspect, FE modelling shows great ability to simulate the deformation process. FEM application in superplasticity has two advantages: first, any step of the whole process can be observed easily because of the graphic capabilities of FEM; and second, unnecessary costs and time spent for experimental set-up and de- signing of the part can be reduced. If the desired final thickness is defined, the initial thickness of the pipe can be estimated by several try-and-see efforts using FEM.

The SPF pipe-bulging process, which has potential for producing spherical shells as one deformation pro- cess, has been studied by only a few researchers. AI- naib et al. [5] presented the first idea of producing spherical shapes using tin-lead pipes under SPF sub- jected to hydrostatic pressure. Bingkin et al. [6] showed only a picture of a Ti-alloy sphere formed by combining spinning technology and the SPF process. Kawahara et al. [7] proposed producing a balloon which is very thin, but sufficiently strong for explo- ration in Venus' atmosphere, by using an extra-thin titanium liner and filament-winding technology.

The main interest of this paper is to simulate the deformation process of Ti-alloy pipe into a spherical shape by FEM using a commercial computational code. A 500% enlargement in the diameter under rigid-per-

216 N. Akkus et al./ Journal of Materials Processing Tecknology 68 (1997) 215-220

fectly plastic flow was aimed at for a pipe with a 30-mm initial diameter. A decreasing pressure path with an increasing bulging height was also presented, to keep the strain rate within a particular range.

2. Basic equations in SPF analysis

Large deformation problems need non-linear solu- tions, since force-displacement relationships depend on the particular material characteristics in the current state of the particular deformation step. In every defor- mation step, the shape, thickness, applied pressure and the stress-strain relationship of the deformed part are different than that of the previous steps. A generalized approximation can be written as in Eq. (1):

P ~- K(P, u) u (1)

where u is generalized displacement vector, P is a generalized force vector and K(P, u) is the stiffness matrix. In the SPF pipe-bulging process, it is necessary to take into account large deformations and large strains, which means that incremental numerical proce- dures are necessary. An accurate description of the material properties and the numerical calculations in each step will ensure good results for the next step in the calculations. Although there are several constitutive equations describing superplastic metal flow, that most commonly used is:

a = K~" (2)

where tr is flow stress of the material used, K is th~ superplastic strength coefficient, ~ is the strain rate and m is the strain-rate-sensitivity parameter. The above equation is valid only under constant temperature. As mentioned by several researchers [8-10], work harden- ing in the superplastic deformation process occurs mostly in two ways: strain-rate-induced grain growth; and temperature-dependent coarsening. No work hard- ening was assumed in the present work, since the deformation temperature was chosen to be sufficiently below the transition temperature of the material and grain growth was kept below an acceptable I~,vel [11]. According to the Von Mises criteria, material flow starts when the equivalent stress equals the yield stress and can be calculated from:

1 e = SijSij , Sij = O'ij - - ~ij ~ O'kk (3)

where # is the equivalent stress tensor, Sij is the devia- toric tensor, and for a plane-stress system O'3a = 0.

For the definition of the material flow, the strain rate has to be defined once yielding has begun. Elastic strains are very small in SPF compared to plastic strains. Assuming that the material is incompressible and that the effect of elasticity can be ignored in a

rigid-perfectly plastic material, the equivalent strain can be calculated from:

= e~j~j (4)

where g is the equivalent strain. For a plane-stress system, the strain in the thickness direction is:

c~3 = - (el, + 92) (5)

In this study, rigid-perfectly plastic flow, which ig- nores the effect of elasticity, is chosen for the material flow, and the flow analysis is based on the Eulerian (or current) reference system. In the Eulerian system, the velocity field, calculated at each step, depends on the current geometry and load conditions instead of on history-dependent calculations. For an incompressible and rigid-perfectly plastic material, the normal flow condition may be written in the form:

ai) = -~ ~i =/1(~) 4, (6)

where:

J3 A = (7)

To obtain better accuracy for the whole deformation process, a time step which results in less than 5% strain increment for any given increment, was chosen. For the accuracy of any given increment, convergence testing is done to check whether the reaction forces are in equi- librium with the external forces and the residual loads are several orders of magnitude smaller than the reac- tion forces.

For the solution procedure, an incremental approach is required, since the SPF process is a non-linear prob- lem and the deformation parameters are changing whilst the deformation progresses. Direct substitution switched to the full Newton-Raphson method was employed to solve the above necessary equations. In the direct-substitution method the last solution is used as a trial for the next iteration, where if the computational accuracy or convergence is not satisfied the program switches to the full Newton-Raphson method. The full Newton-Raphson method requires recalculation and refactorization of the stiffness matrix for each iteration, which takes a major portion of the calculation time, but provides better results for non-linear problem than do any other methods.

3. Pressure-loading algorithm

Several models have been proposed for pressure pre- diction [12,13] to keep the strain rate constant whilst the m value remains at maximum. These studies suggest that to define a target or ot3timum strain rate, then all

N. Akkus et al./ Journal (~/ Materials Processing Technology 68 (1997~ 215 220 2~ 7

of the elements must be checked to find the domain where the strain rate is optimum. [n the present study. a special algorithm for the constant apex-strain-rate pipe bulging process was used to define the deformation pressure at the beginning of each h~crement. The pres- sure algorithm, linked to the main program, can be explained briefly as follows.

First, the strength coefficient K, the m value, the flow stress try, and the strain rate k~ are specified and input to the program. Then the equivalent stress is calculated by use of Eq. (3). The calculated equivalent stress is put into Eq. (8) to obtain the first-step pressure:

p = ~ho/Ro ! 8)

where p is the initial pressure, ho is the initial thickness, and Ro is the initial radius of the pipe. The results of the above calculation were then input to the program to calculate the plastic strains. With the deformation at the first step accomplished and the strains obtained, the radius in the circumferential direction is calculated from:

R~ = Ro + H (10)

In the above equation, R~ is the radius in the circumfer- ential direction and H is the bulging height. Assuming that the radius of curvature in the meridional direction is a part of a circular arc, then it can be obtained from:

R~ = ( L 2 + H 2 ) / 2 H ( 11

where L is the half length of the pipe, and R~ is the radius of curvature in the meridional direction. Finally, the pressure for next step is calculated from:

p = 2 6 h / { ( q 2 - q + l ) ~ 2 R ~ } q = 2 - R d R ~ (12)

where h is the current thickness. Once the initial pres- sure required for obtainin:3 the initial strain rate ~'~o has been calculated using Eq. (8), then the deformation pressure for next and further steps under the constant strain-rate ~o condition are calculated by the use of Eq. (12). Here the equivalent yield stress is assumed to be constant since it is a function of K, m and ~ (Eq. (2)). Thus the deformation pressure p as a variable for the next step can be calculated easily if R~, R~ and h are known. The above approach gives good results if the strain rate is acceptable within a particular range. If the pressure results of the approach are plotted as a func- tion of bulging height, a pressure path, shown in Fig. 1, is obtained under constant apex-strain-rate.

4. Finite-element modelling

Using a pre-processor program called MENTAT, a mesh, as shown in Fig. 2, was generated for Ti-alloy pipe. The initial dimensions of the pipe were 30 O x 1.8 t x 240 L (mm). Due to the double symmetry of the deformed part only a quarter of the pipe was modelled.

2.0

1.5

- ~ 1 . o

0.5

0 0 l0 29 30 40 50 60 70

Bulging Height

Fig. 1. Pressure path for pipe bulging under constant apex strain rate condition.

The mesh used to model of a quarter of the pipe consists of 400 axisymetric elements of 4-nodes in two layers, which corresponds to 1.8 mm thickness of the pipe with a 603 x 2 degrees of freedom. The six nodes, three on the left end and three on the right end, were fixed in the X direction and left free in the Y direction in order to simulate test-piece clamping. In order to avoid severe bending effects at the pol tion of die where the sphere starts, a 2.5 mm radius was given to the die. The pressure was applied to the elements faces as distributed loads. When any deformable elements of the pipe touches to the rigid-die, the program contact al- gorithm prevents the touching element penetrating the rigid-die. Since boron nitrite lubricant was used be- tween the die and the pipe in the real experimental work no friction was taken into account in the present program input.

The material properties of the pipe were input to the program by a special sub-routine. The test-pieces used for the experimental were made of a new kind of Ti-alloy piece, called SP-700 commercially, containing

] - .. I , j ~ -- ~ Rigid Die I / /

] ""N , \ I I \, I '

A Quarter of pipe x\ I \,, x ~ 2"5 . . . . . . . .

Fig. 2. Original mesh and rigid die Idimensions in ram).

218 N. Akkus et aL / Journal of Materials Processing Te, chnology 68 (1997) 215-220

Table 1 Experimental conditions for constant apex-strain-rate deformation of pipe bulging

Initial dimensions of pipe Deformation temperature Strength coefficient (K, Mpa Strain r a t e Strain-rate-sensitivity index (m) (mm) (T, °C) s-t) (L s -I )

30 EIx 1.8 tx240 L 825 5220 4.8 × 10 - 4 0.78

4.5%A1, 3%V, 2%Fe and 2%Mo, and showing super- plasticity in the temperature range of about 700-850°C. The m value varies between 0.4 ~ 0.8 depending on the deformation temperature and the strain rate [14]. The experimental conditions in this study were modelled as shown in Table 1.

5. Results and discussion

Two ways of pressure control, constant pressure and decreasing pressure (constant apex-strain-rate control), were employed in the simulation program. An internal pressure (0:3 MPa) was chosen to check how the defor- mation progresses under constant pressure. As a result, when the pipe deformed to a diameter of 95 mm, the program stopped because of an abnormal increase in the strain rate, which results in unsatisfied convergence. The abnormal increase in the strain rate (especially in the later stage of deformation) is shown in Fig. 3. It may be possible to deform a pipe up to a very large diameter under constant pressure if a very low defor- mation pressure is chosen. Under the present FEM simulation two pipes were deformed to a diameter of 95 mm, one under constant pressure and the other under constant apex-strain-rate. As shown in Fig. 4, the pipe deformed under constant strain rate has a better thick- ness dJstfibuti0n than that deformed under constant pressure, This shows dearly that the non-uniformity in thickness distribution increases under constant pressure deformation. In the present modelling, the deformation pressure for constant apex-strain-rate was required to

keep the strain-rate constant at the apex. However, the real strain rate at the apex is not completely constant and varies within a small range. Fig. 5 shows the variation of strain rate at the apex in the case of the present pressure algorithm. A change of strain rate within about 3 x 10- ~ s - t under constant temperature does not affect the m value chosen for this simulation [14].

To see the effect of the m value on the thickness distribution, two pipes under different strain rates, which results in different m values, were simulated. Fig. 6 shows that a pipe deformed under a strain rate of 4.8 x 10 - 4 S - t with an m value of 0.78 has a better thickness distribution than that of a pipe deformed under a strain rate of 5 x 10 - s s - ] with an m value of 0.56. This confirms that a greater m value gives a better thickness distribution, as stated by Ghosh and Hamil- ton [15] and later by Guo et al. [16]. Comparing the experimental results to the calculated results in Fig. 6, very good agreement is seen for the thickness distribu- tion. This shows also the validity of the model ex- plained in this paper. A quarter of the deformed mesh and a fully deformed pipe are shown in Figs. 7 and 8, as the result of simulation and experimental test, re- spectively.

6. Conclusions

The superplastic deformation process of Ti-alloy pipe into a spherical shape was analyzed using the FEM. The following conclusions were obtained.

. 1 I l i i i i ! i l i i [ i i i i i I i i = .,,... '= 5 ............... , .................... . . p . . - . ° - ~ k ~ ........... l ....... --:

~:~': 4 i i m = O ' F 8 i / ::

• ............... i ................. i ................. i ................. i ............. -

2 ............... i i J i , . . . . . . . . . .

° ! : • ; ; : : : . . . . . . . . . . . . . . . . . . . . . . . .

5 Number of Increments

Fig. 3. Abrupt increase in strain rate under constant pressure (in ram).

! 'Constant piessdre ii . -t 0.9 ~- - - --Consi~n(a~x~rain !rate ~ 1 o.8 .- ........ i .......... ! ........... i ........... i ........... ! .......... i ......... ~,i,--

'-" : i i i ! :: i r l :

=° ~ ! i . . . i ~ N 0.6 0.7 = ........ ~ ......... ~ .......... ~ ~ '>- ~" ~ ! t .......... ~ ........ i r d ....... -

¢ 0.5

0.4 0 20 40 60 80 100120 140 160

Distance from Apex

Fig. 4. Thickness distributions of two pipes, deformed under different pressure-control systems.

N. A k k u s et al. / Journal o ] Mater ia ls Processing Technology 68 (1997) 2 t 5--220 219

10

8

;< 7

" ~ 6

.t=

input+valud+ + =4i8xi0+ s-li . . . . . . . . . + . . . . . . . . . . .¢ . . . . . . . . . . . + . . . . . . . . . . . ' . . . . . . . . . . + . . . . . . . . . . ; . . . . . . . . . . . ~. . . . . . . . . i

i " 2 I: . . . . . . . . i . . . . i . . . . I . . . . o . . . . . . . . . . .

0 50 100 150 200 250 300 350 400

Number o f increments

Fig. 5. Strain rate changes during d,zformation.

Fig. 7. A quarter of the predicted final shape,

(1) FEM simulation shows that the strain rate ef- fect is a crucial point in obtaining large circumferen- tial strains, such as 500'7o, especially at the domain where the thickness is minimum. Thus, the deforma- tion pressure should be well controlled for superplas- tic deformat ion of pipes.

(2) The proposed pressure algorithm can be used to control the strain rate at the apex to a particular limit. This algorithm is very simple and can be linked to program by a user's sub-routine in M A R C code

[171. (3) The effect of the m value and the strain rate

on the thickness distribution can be investigated easily by F EM simulation. A more uniform thickness distri- but ion would be obtained with the maximization of the m value and the optimization of the strain-rate

control .

1.2 " ' 1 ' " 1 " ' 1 ' ' ' 1 " ' 1 ' " 1 . . . . . . -4 ,--, i - - ~" --d=5x10"5 s-li (Sit~.) m=0.56 q

1 ......... " ......... : .......... " ........ : ......... £ .......... :" ........... ;""f""l " i ~ . 8 x 1 0 : 4 ~ (Sire.'.) m=0.78 ! • : .. : ~ : , .~..~..

0 . 8 . . . . . . . . . . . . . . . . i . . . . . . . , , , - -

-~ 0.6 ......... " .......... :- .......... .: .......... " .......... ": ....... ~ " ~ " - : ........ + ........ a -

. . . . . . . . . 0.4 ........................

0 F° - ° " i ' " l . . . . , . . . . . . . . , I ' " :

0 2 0 4 0 6 0 8 0 100 120 140 1 6 0

D i s t a n c e f r o m A p e x

Fig. 6. Final thicknesses of two different pipes simulated under two differenl strain rates.

Fig. 8. The final bulged shape obtained experimentally.

A c k n o w l e d g e m e n t s

The authors wish to thank Mr Yutaka Yamaki of Nippon-Marc for helpful discussions.

R e f e r e n c e s

[1] K. Osada, H. Yoshida, Recent application of superplastic mate- rials in Japan, Mater. Sci. Forum 170/172 (1994) 715-723.

[2] D. Lee, W.A. Backofen, Trans. Met. Soc. AIME 239 (1967)

1034- 1040. [3] J. Bonet, R.D. Wood, R. Collins, Pressure-control algorithm for

the numerical simulation of superplastic forming, Int. J. Mech. Sci. 36 (4) (1994) 297-309.

[4] N. Chandra, S.C. Rama, Application of finite element method to the design of superplastic forming process, Trans. ASME J. Eng. Ind. 114 (1992) 452-458.

[5] T.Y.M. AI-Naib, J.L. Duncan, Int. J. Mech. Sci. 12 (1970)

463 477. [6] C. Bingkin, H. Jintao, in: C.H. Hamilton, N.E. Paton (Eds.h

Superplasticity and Superplastic Forming, 1988, pp. 315-319. [71 M. Kawahara, T. Takatsu, N. Akkus, H. Hata, Application of

CRFP spheres as models of balloon to explore in Venus atmo- sphere, in: I. Kimpara, H. Miyaira, N. Takeda IEds.), Com- posites '95, Japan Society for Composite Materials, 1995, pp.

467-472. [8] R.S. Sadeghi, Z.S. Pursell, Finite element modelling of super-

plastic forming, Mater. Sci. Forum 170/172 (1994) 571-576.

220 N. Akkus et al./Journal of Materials Processing Technology 68 (t997) 215-220

[9] O,A. Kaibyshev, Superplasticity of Alloys, lntermetallides and Ceramics, Springer-Verlag, Berlin, 1992.

[10] E. Sato, K. Kuribayashi, R. Horiuchi, A mechanism of super- plastic deformation and deformation induced grain growth based on grain switching, in: M.J. Mayo, M. Kobayashi, J. Wadsworth (Eds.), Symposium Proceedings of MRS, Vol. 196, 1990, pp. 27-32.

[! 1] N. Akkus, M. Kawahara, H. Nishimura, A technological analy- sis of superplastic deformation of titanium alloy pipes into spherical shape, Mater. Sci. Forum 170 (1994) 633-638.

[12] S.C. Rama, N. Chandra, Int. J. Non-Linear Mech. 25 (5) (1991) 711.

[13] N. Rebelo, T.B. Wertheimer, Finite simulation of superplastic forming, Trans. NAMRI SME 16 (1988) 00.

[14] A. Ogawa, H. Fukai, K. Minakawa, C. Ouehi, The effects of alloying elements on superplastieity of T-6A1-4V, in: N. lgata, I. Kimpara, T. Kishi, 1. Nakata, A. Okura, T. Uryu (Eds.), Proc. First Japan International SAMPE, 1989, pp. 75~-81.

[15] A.K. Ghosh, C.H. Hamilton, Influences of material properties and microstructure on superplastic forming, Metali. Trans. A (1982) 733-743.

[16] Z.X. Guo, J. Pilling, N. Ridley, Bulge-forming of domes; A comparison of theoretical prediction and experiment, in: C.H. Hamilton, N.E. Paton (Eds.}, Superplasticity and Superplastic Forming, TMS, 1988, pp. 303-308.

[17] MARC-MENTAT Manuals, Marc Analysis Research Corpora- tion, USA, 1991.