METALS AND MATERIALS, VoL 4, No. 3 (1998), pp. 311-314

Simulation of Springback in V-bending Process by Elasto-Plastic Finite Element Method with Consideration of Bauschinger Effect Takeshi Uemori, Tatsuo Okada and Fusahito Yoshida Department of Mechanical Engineering, Hiroshima University Kagamiyama 1-4-1, Higashi-Hiroshima 739, Japan This paper presents the prediction of springback in sheet metal forming in terms of the description of the Bauschinger effect. It addresses three different types of constitutive models: an iso-tropic hardening model (IH model), a linearly kinematic hardening model (LK model), and a model of combined linearly-nonlinearly kinematic and isotropic hardening (L-NK+IH model). The springback predicted by the L-NK+IH model is found to be more pronounced than that by the IH and the LK model. It is caused by the stressstrain response characterized by the gradual transition from fully elastic to plastic behavior during stress-reversal, which is essential to the L-NK+IH model.

Key words:springback, bauschinger effect, isotropic hardening, kinematic hardening, stress-reversal, +V-bending process

1. INTRODUCTION In the last decade the techniques of numerical simulation applicable for sheet metal forming have been dramatically developed. However, accurate prediction of the springback remains a difficult task. One of the reasons for that is that the simulations performed so far, has not paid much attention to the Baushinger effect which appears during unloading. The present paper describes a method which allows us to introduce the constitutive equation with consideration of the Bauschinger effect to the elasto-plastic finite element method. The analysis of V-bending process and subsequent springback for a two-ply clad sheet metal was conducted to investigate the influence of Bauschinger effect on the springback. 2. C O N S T I T U T I V E

MODELS

Let us consider a material is initially isotropic at the annealed state and it obeys the von Mises criterion for yielding. The material becomes anisotropic as a result of the hardening process. The Bauschinger effect appears during the stress-reversal, and certain hardening rules account for its behavior. The yield function f may be written as

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(1)

f = ~- (sii- ~ i ) - ( Y + R ) 2

where sij, oqj, and stand the deviatoric stress, the back stress, the initial yield stress and the drag stress, respectively. The anisotropy is developed with the back stress evolution, while the rotation of the yield locus and the change of its shape are ignored in our hale. In the present work, the evolution of the back stress is given by the following Eq. 1 &. = &~) + &9) lj

ij

ij

&.(.1)= 2 H" "p [linearly kinematic hardening rule ] (2)

[nonlinearly kinematic hardening rule] where ~ is the plastic strain rate, e denote the equivalent plastic strain rate, and a, C and H' represent material parameters. The evolution of the drag stress is given by the formula [2] /

R=b(Q-R)

"p

~i,

.p

~i = .~~3/ 2

~.p~p

,j ij

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Takeshi Uemori et al.

[isotropic hardening rule ]

(3)

where b and Q denote material parameters. The plastic strain rate is determined by the associated flow rule: ~= ~

,~

(4)

In our FE analysis, we used the following three types of constitutive models: the isotropic hardening model (IH model) that exclude the kinematic hardening (the back stress evolution), the linearly kinematic hardening model (LK model) in which the workhardening is described exclusively by the linear component of the back stress, &.!l) IJ ' the model of combined linearly-nonlinearly kinematic and isotropic hardening (L-NK+IH model) that includes the linear and nonlinear components of the back stress, and, as well as the isotropic hardening. -

-

3. O U T L I N E O F F I N I T E E L E M E N T

Fig. 1. V-bending of a two-ply clad sheet.

CODE

The present authors developed a FE program based on the updated Lagrangian rate formulation [3]. We determined the incremental time step for the FE analysis based on the "r-min method [4]" considering yielding in a material element, as well as the conditions of contacting/detaching of a material point (nodal point) with/from tools. The explicit time integration procedure was employed in our finite element calculation.

Fig. 2. Two-ply clad sheet metal modeled in the finite element analysis.

4. S P R I N G B A C K A N A L Y S I S The springback in V-bending operation for most of sheet metals is a process of elastic unloading (the reyielding unlikely occurs in the process). However, some specific case, such as the use of a clad sheet metal which consists of dissimilar metal components, the reyielding would occur in the weaker layer [5].,For such a case, the Bauschinger effect should be taken into account for the accurate prediction of springback. In the present research, the springback was simulated for a twoply clad sheet metal under plane strain condition. The experimental set-up and the size of the tools used in the FE analysis are shown in Fig. 1. The clad sheet metal consists of an aluminum layer and a stainless steel layer (see Fig. 2). The effect of the choice of the constitutive model for the aluminum layer on the springback behavior was examined, when the stress-strain response of the stainless steel layer was assumed to be described by the linearly kinematic hardening model. The material

Fig. 3. Cyclic stress-strain respocses of r aluminum calculated by three constitutive models.: LK=Linearly kinematic hardening, IH=Isotropic hardening, L-NK= Linearly-nonlinearly kinematic hardening

parameters in these models were determined by means of the cyclic bending tests [6,7]. The stress-strain responses calculated by the above mentioned models are shown in Figs. 3 and 4. The isoparametric element with four nodes was used for our finite element simulation.

Simulation of Sprmgback in V-bending Process by Elasto-Plastic Finite Element

Fig. 4. Cyclic stress-strain response for stainless-steel calculated by the Linearly kinematic hardening model The calculation were performed for 900 elements, while the thickness was modeled by 6 and length by 150 elements. The elements around the top of the punch were smaller than the others. The sheet was set such that the punch contacts with the aluminum layer. The movement of the punch was reversed when the clearance between the punch and the die became equal to the initial thickness of the blank. 5. R E S U L T A N D D I S C U S S I O N S The comparison of the final shapes of the sheet calculated by means of the above three types of constitutive models, LK model, IH model and L-NK+IH model, are shown in Fig. 5. The calculated angle of springback equals 2.9, 2.5, and 3.9 deg for models IH model, LK model and L-NK+IH model, respectively. Our calculation revealed that the angle of springback differs considerably depending on the types of the constitutive models. The longitudinal stress-strain history during springback for a material element which is located fight under the punch are shown in Fig. 6. The re-yielding in the aluminum layer during unloading process was predicted by

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Fig. 6. Longitudinal stress-strain history during unloading for the aluminum at a point right under the punch. all the models. We have not predicted such re-yielding in the stainless steel layer. One should notice that the difference in the calculated springback angle between the LK model and IH model is small, while the application of the model L-NK+IH model gives the largest springback angle. Since the material behavior of the gradual transition from elastic to plastic behavior during stress reversal can be described in the model L-NK+IH model, the plastic deformation generated during stress reversal by this model was larger than by the other models. From the above results, we conclude that the Bauschinger effect should be taken into account for the accurate estimation of the re-yielding stress level. 6. C O N C L U S I O N S The springback in V-bending for a clad sheet metal was analyzed by means of the FEM with consideration of Bauschinger effect. The calculated springback differs considerably depending on the type of the constitutive models. In particular, the results of numerical simulation by using a model with nonlinear kinematic hardening rule (e.g. L-NK+IH model discussed in the present research) give large springback angle, since the model can describe the material behavior of the gradual transition from fully elastic to plastic behavior during stress reversal.

REFERENCES L-NK + IH model Fig. 5. Comparison of the final shapes of sheet calculated by three typesof censtitutive models. LK model

IH model

1. P.J. Armstrong and C.O.Frederick, C.E.G.B. Report RD/ B/N., p. 731 (1966). 2. J.L. Chaboche and Rousselier, Trans ASME J.Pressure Vessel Tech., p. 105 (1983).

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3. McMeeking, R.M. & Rice, J.R, Int. J. Solids Structures. 11, 601 (1975). 4. Y. Yamada, N. Yoshimura and T.Sakurai, Int. J. Mech. Sci. 10, 343 (1968). 5. F. Yoshida, R. Yagi and M. Ohmori, Proc. 2nd Int. Conf. Technology of Plasticity (ed., K. Lange), P.185,

Elesevier (1987). 6. F. Yoshida, M. Urabe and V.V. Toropov, Int. J. Mech. Sci. 40, 237 (1998). 7. F. Yoshida, M. Urabe and V.V. Toropov, Proc. 11th Int. Conf. on Experimental Mechanics. (1998), to be published.