Experimental verification of superplastic sheet-metal ...koasas.kaist.ac.kr/bitstream/10203/3486/1/ji06.pdf · Experimental verification of superplastic sheet-metal forming analysis

Journal of Materials Processing Technology

ELSEVIER Journal of Materials Processing Technology 49 (1995) 355-369

Experimental verification of superplastic sheet-metal forming analysis by the finite-element method

H. H u h a,,, S.S. H a n a, J.S. Lee b, S.S. H o n g b

aDepartment of Precision Engineering and Mechatronics, Korea Advanced Institute of Science and Technology, Taejon, South Korea

bDivision of Materials Processing, Korea Agency for Defense Development, Taejon, South Korea

Received 7 November 1993

Industrial Summary

The numerical result for superplastic sheet-forming simulation is compared with the experimental result to verify the validity of the finite-element code developed. The code calculates the optimum pressure cycle, the deformed shapes, and the distributions of the strain and strain-rate in the blow-forming processes. The calculation deals with the maximization of the strain-rate sensitivity, the protection of the localized deformation, the consistency of the desired strain-rate, and the control of the hour-glass mode and locking phenomenon. The comparison demonstrates the validity of the present algorithm and its results, especially in respect of the optimum pressure cycle, where there is only a slight difference between the two results.

Notation

c(v) D

f, K m

Av Ap

~kk

suitable function weighting the strain-rate grain size body force material constant strain-rate sensitivity virtual velocity increment velocity increment pressure increment equivalent strain-rate volumetric strain-rate object equivalent strain-rate

*Corresponding author.

0924-0136/95/$9.50 © 1995 Elsevier Science S.A. All rights reserved SSDI 0 9 2 4 - 0 1 3 6 ( 9 4 ) 0 1 5 8 1 - K

356 H. Huh et al./ Journal of Materials Processing Technology 49 (1995) 355-369

f f i j a' u

penalty constant viscosity equivalent stress Cauchy stress tensor deviatoric stress tensor

1. Introduction

The superplastic blow-forming process is one of the typical forming processes to produce very complex, but light and strong, thin-walled components [1-3]. In the process, it is necessary to control the strain-rate for large elongation with a high value of strain-rate sensitivity. Since the amount of elongation is also sensitive to the temperature, a certain range of temperature and strain-rate affords excellent stability of plastic deformation and large elongation that is less susceptible to localized necking. In such a superplastic sheet-forming process, the prediction of the pressure cycle is indispensable for optimization of the process, which is evaluated by the formability of the material and the forming time [4-6]. It is, however, not easy to predict the optimum pressure cycle by experimental exploration of real processes. An alternative method to predict the optimum pressure cycle will be an analytical method or a finite-element method which is easily applicable to a complex shape. In finite- element analysis, the deformation has to be calculated accurately, since the corresponding optimum pressure is determined using the calculated strain-rate. Once the process is analyzed and the optimum pressure cycle is obtained, there arise questions on the quality and validity of the numerical results obtained. Therefore, it is necessary to verify the numerical results with experimental results in addition to the accurate numerical calculation for the direct application of the present algorithm to forming processes of complex shapes.

In this paper, the analysis of superplastic sheet-forming processes is studied by the use of the finite-element method. In the formulation, the large inelastic behavior of the superplastic material is described as incompressible, non-linear, viscous flow. In the calculation procedure, the directional reduced integration scheme for hour-glass control is adopted to reduce the locking phenomenon [7-9]. In addition, a pressure-cycle control algorithm is combined in the analysis for optimization of the forming time, which is indispensable to minimize the relatively long forming time. The algorithm deals with the maximization of the strain-rate sensitivity, the protection of thickness reduction due to localized deformation, and the consistency of the desired strain-rate.

2. Constitutive model

The behavior of superplastic materials can be described mathematically as a function of state variables such as the flow stress ~, the strain-rate ~, the strain e and the

H. Huh et aL /Journal of Materials Processing Technology 49 (1995) 355-369 357

grain size D. One of the simplest constitutive relationships is given by

a = Ko~menD p, (1)

where the material is assumed to be purely inelastic and incompressible. Eq. (1) can be extended to a multi-axial constitutive equation by replacing the uniaxial variable a, ~, and e with the equivalent variables 6, ~ and ~. The uniaxial flow stress a ofa superplastic material is a strong function of the inelastic strain-rate ~ and a weak function of the strain c and the grain size D. For simplification, the equivalent flow stress may be regarded as a function of the strain-rate only, as follows;

6- = K~-', (2)

K = KognD t', (3)

where the exponent m is the strain-rate sensitivity and the strength coefficient K can be regarded as a constant or a step-wise constant modified considering the effect of the grain size D and the accumulated total strain c during the deformation. The equation can also be cast into a non-linear viscous model [10-12] in terms of the equivalent stress 6- and the equivalent strain-rate ~, as

6- = 3#~ (4)

which is expressed in a component form as

a'ij = 2#~, (5)

where the viscosity/~ can be obtained from the relationship

= ½ K ~ m - I (6)

3. Finite element formulation

The differential form of the equilibrium equations can be transformed into the integral form by applying the principle of virtual work. After applying integration by parts and the divergence theorem, the equation can be expressed as

f a i j 3 ~ i j d V - f t i S v l d S - f f i Sv idV =O, VSvi. (7)

D t~Dr D

Although the above integrals are related to the instantaneous configuration, they could be referred in the updated configuration with a small incremental step. To restate the internal energy in Eq. (7) in terms of a deviatoric stress tensor a'ij, the stress can be rewritten as

, Okk~__ , _ alj = aij + 3 'J aij p6i~, (8)

358 11. Huh et al./ Journal of Materials Processing Technology 49 (1995) 355-369

where p is the mean hydrostatic pressure. Since the behavior of the superplastic material is described as incompressible, non-linear, viscous flow, with Eq. (5) and Eq. (6), it can be shown that

atij ~ i j = 3#k ~ . (9)

Thus, the weak form for the equilibrium equations with boundary conditions and an incompressibility constraint can be written in a penalty form of

f3 ,- ZdV + A f kkO k dV-- f hSvidA = 0 , (10)

D' D ~ ~Dt+

where/1 is a penalty parameter. The formulation can be combined with a contact algorithm to deal with contact problems in forming processes. The contact condition in the normal direction is

trn(Au n -- Agn) = 0 on t3D~. (11)

and the contact condition in the tangential direction can be expressed as

O'T,(VT, - - UT, ) - - #a,(IVxl -- lUTI) ~> 0, V / ) T . (12)

After some manipulation with the aid of real analysis using the penalty method, the finite-element formulation can be expressed as

f 3gkfkdV + A f ekk6ekkdV + f k,(u,-gn,)6v, dA D' D' 6D~

f UT ~ f + Ilan(a'e(IUTI)I-~+I~VT, dA -- ~i6vidA = 0, (13)

where 4~, is a smoothing function in order to avoid the abrupt change of the friction force. The friction boundary is taken by the following vector form in terms of the nodal point force using the modified friction rule and replacing the smoothing function with the hyperbolic tangent function

UT f UT \ F T = ~-~/~FNtanh~T-U-~) on ODe, (14)

where ~ is an augmented constant and FN is the normal force on the contact surface. Then, the fourth term in Eq. (13) can be expressed as

f t UT- ~_ #a,¢~(lUxl)I-~TI SVT, dA f FTSVT, dA. (15)

The finite-element formulation is approximated into finite-dimensional space by discretizing the domain into finite elements, which results in a set of linear algebraic

H. Huh et al. / Journal of Materials Processing.Technology 49 (1995) 355-369 359

equations for each incremental step. In the approximation procedure, the 4-node quadrilateral isoparameteric bilinear element is adopted and the numerical integration is carried out with the directional reduced integration scheme to control the hour-glass mode and to prevent the locking phenomenon due to bending and large rotation.

4. Pressure cycle control

To predict the optimum forming-pressure, a pressure-cycle control algorithm is adopted in the analysis, which was proposed originally by Bonet et al. [4]. The difference of the present algorithm from the proposed algorithm is that I(v, x) and F(v,x) are changed at each step because of the contact formulation. The governing equations of Eq. (13) for superplastic forming can be written as

I(v, x) = p(t) F(v, x), (16)

where I(v, x) is the internal strain energy term, F(v, x) in the applied force term and p is a scale factor for the pressure at each step. The solution of these equations is a set of nodal velocities v, geometry x, and pressure p which additionally satisfy the scalar strain-rate constraint equation

c ( v , x ) = {o, (17)

where C is a proper function weighting the strain-rates, and ~o is the target equivalent strain-rate. Solutions of simultaneous equations after first-order Taylor's expansion of Eqs. (16) and (17) for Av and Ap become

Av = Kf~I {R + A p F } (18)

which gives the pressure increment as

s - - L T K ~ I R A p - L T K F 1 F , (19)

where Kt is the tangent stiffness matrix, R is the residual force vector, L is the constraint gradient vector and s = ~o - C(v) is the strain-rate constraint error. The velocity vector and pressure scaling factor can be updated as

U k+l = U k "~ AU, p k + l = pk q_ Ap (20)

for each incremental step with respect to time.

5. Directional reduced integration for hour-glass control

Deformation dominated by bending often causes the numerical locking phenomenon called the hour-glass mode in the finite-element method. There are various

360 H. Huh et aL /Journal of Materials Processing Technology 49 (1995) 355-369

remedies with integration schemes and artificial strains to control the hour-glass mode [13,14]. In the selective reduced integration scheme, the terms related to the volumetric change and the shear deformation are under-integrated to avoid locking due to incompressibility and shear deformation. Whilst the selective reduced integration under-integrates particular terms equally in the coordinate directions, the directional reduced integration (DRI) performs under-integration for all terms in the special direction, which excludes difficulties for the selective reduced integration to avoid shear locking even with the transformation of the finite elements. [7-9] The strain vector can be derived and decomposed by differentiating the displacements with respect to the coordinates, as follows.

s t = 8o + fes + fie,, (21)

where J is the Jacobian matrix of the normalized coordinates with respect to the Cartesian coordinates. In Eq. (21), eo is a constant strain vector and the last two terms denote the directional variation of e in an element with respect to st or (s, t). In bending-dominated cases, the variational part of the shear strain can be eliminated to prevent the shear locking phenomenon. Thus, the directional reduced integration adopts directional separation instead of strain separation by separating directional control parameters fls and fit such as

s t = ~o +/~q(~D, + ~v~v) +/~,2(~o, + ~vev,), (22)

where eDs = gNs "[- gSs and ~Dt = /~Nt + ~St

01 for incompressible material, ~v = otherwise,

{~ under-evaluation in s-direction, fls = otherwise,

01 under-evaluation in t-direction, /3, = otherwise.

Here subscripts N, S and V denote the deviatoric normal, the deviatoric shear and the volumetric strains, respectively and subscript D denotes the deviatoric part. By setting ~v = 0, DRI can eliminate the locking for incompressible material, and by properly setting fls or fit = 0, DRI can also eliminate shear locking. In addition, DRI can avoid the difficulty of the selective reduced integration due to the element rotation. In fact, this difficulty of the selective reduced integration is caused by employing different integration rules to strain components in the numerical sense; thus, this can be excluded by employing the same integration to all components, which is the way DRI follows exactly.

H. Huh et al./ Journal of Materials Processing Technology 49 (1995) 355-369 361

6. Experiments

Experiments into superplastic blow-forming processes were carried out to verify the validity of the finite-element code developed. The experimental apparatus was especially designed and manufactured to control the forming pressure and the temperature of the furnace in which the forming dies were situated. The superplastic sheet metal used in the experiments was SUPRAL 150. The material properties of SUPRAL 150 were obtained from the tensile test under various conditions of temperature and strain-rate. The selected material properties for the optimum condition are as follows:

Strength coefficient: K = 105.94 (s)" MPa Maximum strain-rate sensitivity: m = 0.40 Corresponding equivalent strain-rate: ~o = 6.93 × 10- 3 (s)- 1 at the forming temperature: T = 470°C

The thickness of the sheet material and the diameter of the die blank used in the experiments are as follows:

Cylindrical bulging: 2.54 mm thickness and 100 mm diameter Conical bulging : 0.88 mm thickness and 80 mm diameter Cap forming: 0.88 mm thickness and 100 mm diameter

The experiments were carried out at a temperature of 470°C, at which the strain-rate sensitivity becomes maximum. The bulging processes were carried out under the condition that the material flow is constrained by a sharp bead at a constant distance from the die profile. In the experiments, the forming pressure was set according to the numerical results to enable comparison of the numerical and experimental results for polar height, deformed shape, and thickness strain at the several-selected steps.

7. Results and discussion

The algorithm described has been implemented in the finite-element code SUPERB to simulate the superplastic sheet-forming process. The numerical simulations are performed with the material data in the previous section for the cylindrical-bulging process, the conical-forming process, and the cap-forming process. The simulation provides the optimum forming-pressure versus the forming time as well as the deformed shape, the thickness strain, the elongation, the equivalent strain, and the equivalent strain-rate at each time step. Since the numerical results are diversified with the use of a different integration scheme [7-9], investigation is needed to substatiate the validity of the finite-element code developed by experimental verification.

The experiments of the forming processes were carried out with the optimum forming-pressure calculated from the code at the given temperature. The experiments were well executed, without the localized necking or irregular forming found often in real forming processes, when carried out in accordance with the calculated optimum pressure-cycle: this is one of the verifications of the validity of the present algorithm for the optimum pressure-cycle. The cylindrical bulging was carried out in accordance


with the calculated optimum pressure-cycle as shown in Fig. 1. The polar heights of the deformed specimens were measured at several steps and plotted along with the calculated heights, fairly good coincidence being displayed.

The small difference between the two results is thought to be experimental error rather than numerical error, as the sequence of the polar heights of the specimens show a slight fluctuation. The deformed specimens at several experimental steps are shown in Fig. 2 compared with the calculated deformed mesh configuration, the deformed configurations of the two cases showing good agreement in their shapes. The thickness distributions measured from the experiments show a little difference from the numerical result, as shown in Fig. 3, the difference in the thickness between the two results, however, being within the moderate error bound of less than 5% for the maximum and the distribution patterns being in good agreement with each other. The difference between the two results is almost uniform throughout the domain and through all the steps, the reason of the difference possibly being because a small amount of draw-in of the flange is allowed during the real forming process in the experiments whilst this is not considered in the numerical analysis.

The conical bulging was also carried out according to the calculated optimum pressure-cycle shown in Fig. 4. The polar heights measured from the experiments virtually coincide with the numerical results, the difference between the two results being much smaller than that in the cylindrical-bulging process: the difference must be due to the draw-in of the flange since the thickness of the specimens is small in comparison with the die radius. The deformed specimens at several experimental steps

1.0

"2.0

Pressure 0.8

'1.5

~ o . 6

. L o ~2

0 4

• 0.5 Z 0.2

0.0 , , , 0 .0 0 100 200 300 400

Time (see)

Fig. 1. Optimum pressure-cycle and normalized polar height with respect to the forming time in the cylindrical-bulging process.

H. Huh et al./ Journ6 ~f Materials Processing Technology 49 (1995) 355-369 363

(a)

(b) Fig. 2. Deformed configurations at several steps in the cylindrical-bulging process; (a) finite element method; (b) experiments.

are shown in Fig. 5 along with the calculated deformed mesh configurations. The deformed configurations of the two cases show good agreement in their shapes, especially in their contact regions with the die. The thickness distributions measured from the experiments also shows a little difference from the numerical result as shown

364 H. Huh et aL / Journal of Materials Processing Technology 49 (1995) 355-369

1.0

0.8

M 0.4

~ j w , , B w ' J • Exp. (h = 32.7) • Exp. (h = 38.6)

0.2 " I - n Exp. (h = 46,2)

• Exp. (h = 54.8) • Exp. (h = 65.6)

0.0 . . . . 0 10 20 30 40 50

Initial Distance from Center (mm)

Fig. 3. Comparison of the calculated normalized thickness with the experimental values in the cylindrical-bulging process.

0.20 2.0

0.15 Pressure Pole Height 1.5 .d

~o.~o ~.o~

°-°'t °,'~ °

0.00 T • , - , - , 0.0 0 100 200 300

Time (sec)

Fig. 4. Optimum pressure-cycle and normalized polar height with respect to the forming time in the conical-forming process.

H. Huh et al. / Journal of Materials Processing Technology 49 (1995) 355-369 365

(a)

(b)

Fig. 5. Deformed configurations at several steps in the conical-bulging process: (a) finite-element method; (b) experiments.


1.0 •

0.8 ..'.: o 0 . 6 • •

i n o

N 0.4 ' - - FEM .3)

Z • • / • Exp . (h=32 .1) J n Exp. (h = 40.0)

0.2 • Exp. (h = 49.4)

0.0 . . . . 0 10 20 30 40 50

Initial Distance from Center (mm)

Fig. 6. Compar ison of the calculated normalized thickness with the experimental values in the conical-forming process.

4.0

3.0

~ 2.0

1.0 ¸

0.0 f

! | |

0 50 100 150

Time (sec)

Fig. 7. Op t imum pressure-cycle with respect to the forming time in the cap-forming process.


in Fig. 6, the reason of the difference possibly being because of the draw-in of the flange and deviation of the friction coefficient.

The cap forming was carried out according to the calculated optimum pressure cycle as shown in Fig. 7. The pressure cycle is almost same as that for the cylindrical-bulging process until incipient contact when it increases thereafter. The cap forming was also well executed and the deformed specimens, including the final product, are shown in Fig. 8. The thickness distributions measured from the experiments are plotted and compared with the numerical results in Fig. 9. The thickness distribution of the final shape shows a greater difference from the numerical results

f

(a)

iiiiiiii!iiiiiiiii~i

(b) Fig. 8. Deformed configurations at several steps in the cap-forming process: (a) finite-element method; (b) experiments.


1.0

0 . 8 ¸

~ 0.6 ~ ,

~. 0.4 '

0.2

0.0

FEM m Exp. (incipient contact)

Exp. (after fill up)

I i I i

0 10 20 30 40 50

Initial Distance from Center (ram)

Fig. 9. Comparison of the calculated normalized thickness with the experimental values in the cap-forming process.

than those in cylindrical bulging or conical bulging, the main reason for which in this case may be because of the friction coefficient: the sliding mechanism on the upper fiat die would be different from that on the cylindrical die wall, which is very difficult to take into account in calculations. The experimental results show that there was less sliding on the upper fiat die and more sliding on the cylindrical die wall than do the numerical results, and there was also some amount of the draw-in as in the other two bulging processes.

The comparison demonstrates fully that the calculated forming pressure can be regarded as the optimum. In the real forming process, a slight deviation of the forming pressure resulted in almost same result, so that the forming pressure can be considered as less sensitive to some extent.

8. Conclusions

The comparison of the numerical results with the experimental results promotes confidence in the validity of the present finite-element code. In the code, an approximated, updated Lagrangian finite-element formulation is used with the directional reduced integration scheme. The formulation is associated with a contact algorithm and a pressure-control algorithm to predict the optimum pressure-cycle and the corresponding deformed shapes in the superplastic blow-forming processes. The numerical results are in good agreement with the experimental results, especially for


the polar height, with respect to the optimum pressure cycle. The experiments were well executed, without the localized necking, when carried out in accordance with the calculated optimum pressure-cycle. The difference in the thickness distribution between the numerical results and the experimental results is within a moderate error bound and would be much less without draw-in of the materials.

Acknowledgement

The development work reported herein was partially funded by the Korean Science and Engineering Foundation. The authors wish to gratefully acknowledge their support during this work.

References

[1] G.C. Cornfield and R.H. Johnson, The forming of superplastic sheet metal, Int. J. Mech. Sci., 12 (1970) 479.

[2] A.K. Ghosh and C.H. Hamilton, Mechanical behavior and hardening characteristics of a superplastic Ti-6A1-4V alloy, Met. Trans. A, 10A (1979) 699.

[3] G. Kuperfarb, Y. Germain and M. Abouaf, A mechanical study of superplastic forming of Ti-6A I-4V sheet, J. Mech. Work. Technol. 14 (1987) 159.

[4] J. Bonet, A.H.S. Wargadipura and R.D. Wood, A pressure cycle control algorithm for superplastic forming, Comm. Appl. Numer. Methods, 5 (1989) 121.

[5] M. Bellet and J.L. Chenot, Numerical modelling of thin sheet superplastic forming, in: Proc. Numiform 89, 1989, 401.

[6] J. Bonet, P. Bhargava and R.D. Wood, Finite element simulation of 3-dimensional superplastic formig with diffusion bonding, in: Proc: Numiform 92, 1992, 843.

[7] B.C. Koh and N. Kikuchi, New improved hourglass control for bilinear and trilinear elements in anisotropic linear elasticity, Comput. Methods Appl. Mech. Eng. , 65 (1987) 1.

[8] H. Huh, S.S. Han and D.Y. Yang, Elasto-plastic finite element analysis with directional reduced integration in sheet metal forming process, in: Advanced Technology of Plasticity 1990, Proc. 3rd ICTP, Vol. 3, 1990, 1375.

[9] H. Huh and S.S. Han, Superplastic sheet forming analysis by the finite element method with directional reduced integration, in: Proc. Numiform 92, (1992), 851.

[10] J.F. Brandon, H. Lecoanet and C. Oytana, A new formulation for the bulging of viscous sheet metal, Int. J. Mech. Sci., 21 (1979) 379.

[11] J.H. Argyris and J.S.L. Doltsinis, A primer on superplasticity in natural formulation, Comput. Methods Appl. Mech. Eng., 45 (1984) 83.

[12] N. Chandra, Analysis of superplastic metal forming by a finite element method, Int. J. Numer. Methods Eng., 26 (1988) 1925.

[13] D.P. Flanagan and T. Belytchko, A uniform strain hexahedron and quadrilateral with orthogonal hourglass control, Int. J. Numer. Methods Eng., 17 (1981) 679.

[14] J.S.T. Ong and T. Belytchko, Hourglass control in linear and nonlinear problems, Comput. Methods Appl. Mech. Eng., 43 (1984) 251.

Documents

Experimental verification of superplastic sheet-metal ...koasas.kaist.ac.kr/bitstream/10203/3486/1/ji06.pdf · Experimental verification of superplastic sheet-metal forming analysis