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Research Papers

A Phenomenological Model for the Hysteresis Behavior of Metal Sheets Subjected to Unloading/Reloading Cycles

[+] Author and Article Information
P.-A. Eggertsen

Division of Material and Computational Mechanics, Department of Applied Mechanics,  Chalmers University of Technology, SE-412 96 Göteborg, Swedeneggepera@chalmers.se

K. Mattiasson

Division of Material and Computational Mechanics, Department of Applied Mechanics,  Chalmers University of Technology, SE-412 96 Göteborg, Sweden, kjellm@chalmers.se Department 91430, PV22,Volvo Cars Safety Center, SE-405 31 Göteborg, Swedenkmattias@volvocars.com

J. Hertzman

 Industrial Development Centre, Västra Storgatan 20, SE-393 38 Olofström, SwedenJoergen.Hertzman@iuc-olofstrom.se

J. Manuf. Sci. Eng 133(6), 061021 (Dec 21, 2011) (16 pages) doi:10.1115/1.4004590 History: Received October 06, 2010; Revised June 29, 2011; Published December 21, 2011; Online December 21, 2011

The springback phenomenon is defined as elastic recovery of the stresses produced during the forming of a material. An accurate prediction of the springback puts high demands on the material modeling during the forming simulation, as well as during the unloading simulation. In classic plasticity theory, the unloading of a material after plastic deformation is assumed to be linearly elastic with the stiffness equal to Young’s modulus. However, several experimental investigations have revealed that this is an incorrect assumption. The unloading and reloading stress–strain curves are in fact not even linear, but slightly curved, and the secant modulus of this nonlinear curve deviates from the initial Young’s modulus. More precisely, the secant modulus is degraded with increased plastic straining of the material. The main purpose of the present work has been to formulate a constitutive model that can accurately predict the unloading of a material. The new model is based on the classic elastic-plastic framework, and works together with any yield criterion and hardening evolution law. To determine the parameters of the new model, two different tests have been performed: unloading/reloading tests of uniaxially stretched specimens, and vibrometric tests of prestrained sheet strips. The performance of the model has been evaluated in simulations of the springback of simple U-bends and a drawbead example. Four different steel grades have been studied in the present investigation.

Copyright © 2011 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

A schematic illustration of the elastic stiffness degradation at various levels of prestrain

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Figure 2

A schematic illustration of the hysteresis behavior during an unloading/reloading cycle

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Figure 3

Evolution of the initial tangent modulus Et as a function of accumulated effective plastic strain according to Kubli [26]

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Figure 4

(a) Picture of the vibrometric test equipment. (b) Illustration of the induced vibration mode.

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Figure 5

The tangent stiffness modulus Et measured with the RFDA-system as a function of effective plastic prestrain for three directions. Material: TKS-DP600.

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Figure 6

The tangent stiffness modulus Et measured with the RFDA-system as a function of effective plastic prestrain for three directions. Material: SSAB-DP600.

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Figure 7

The tangent stiffness modulus Et measured with the RFDA-system as a function of effective plastic prestrain for three directions. Material: TKS-220IF.

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Figure 8

The tangent stiffness modulus Et measured with the RFDA-system as a function of effective plastic prestrain for three directions. Material: Voest-DX56.

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Figure 9

Evolution of the tangent stiffness modulus Et with time. The materials have been pretensioned to 10% plastic strain in the rolling direction. The asteriks and the open circles represent measured and calculated points, respectively.

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Figure 10

Experimental, uniaxial loading-unloading-reloading curve for the TKS-DP600 material

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Figure 11

A detailed illustration of the unloading-reloading relationship. Material: TKS- DP600.

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Figure 12

Experimental values of the unloading modulus variation with plastic strain for the four materials

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Figure 13

Illustration of the yield surface and the inner surface

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Figure 14

Non-linear stress curve at reverse loading

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Figure 15

Evolution of the microplastic strain p

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Figure 16

The location of the inner surface during plastic loading

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Figure 17

Illustration of experimental values needed for the parameter identification

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Figure 18

Variation of the parameter a with effective plastic strain. The “stars” indicate experimental values, calculated according to Eq. 18. Material: TKS-DP600.

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Figure 19

Predicted and experimental unloading-reloading curves Material: TKS-DP600

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Figure 20

Detailed unloading-reloading curves at 2% strain. Material: TKS-DP600.

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Figure 21

Detailed unloading-reloading curves at 4% strain. Material: TKS-DP600.

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Figure 22

Detailed unloading-reloading curves at 8% strain. Material: TKS-DP600.

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Figure 23

Comparison of the new method and some other frequently used methods. Material: TKS-DP600.

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Figure 24

The NUMISHEET’93 benchmark problem. (a) Experimental set-up; (b) Definition of the tip deflection, used for the evaluation of the springback.

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Figure 25

Comparison between the new, proposed method, the unloading modulus method and usage of a constant elastic modulus in application to the U-bend springback problem. Material: TKS-DP600.

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Figure 26

Comparison between the new, proposed method, the unloading modulus method and usage of a constant elastic modulus in application to the U-bend springback problem. Material: TKS-220IF.

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Figure 27

Stress distribution through the thickness before and after springback for the U-bend problem for a point in the middle of the sidewall (the circle in Fig. 2). Left: Constant E-modulus. Middle: unloading modulus according to Eq. 7. Right: New method.

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Figure 28

Basic set-up of the draw-bead problem

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Figure 29

Comparison between the new, proposed method, the unloading modulus method and usage of a constant elastic modulus in application to the draw-bead springback problem. Material: TKS-DP600.

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Figure 30

Stress distribution through the thickness before and after springback for the draw-bead problem for a point illustrated by the circle in Fig. 2. Left: Constant E-modulus. Middle: unloading modulus according to Eq. 7. Right: New method.

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Figure 31

Definition of the parameters in the Yoshida-Uemori hardening model

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