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Research Papers

Bifurcation Analysis of Forming Limits for an Orthotropic Sheet Metal

[+] Author and Article Information
Shuhui Li

State Key Laboratory of Mechanical
System and Vibration,
Shanghai 200240, China
Shanghai Key Laboratory of Digital
Manufacture for Thin-Walled Structures,
Shanghai Jiao Tong University,
Shanghai 200240, China
e-mail: lishuhui@sjtu.edu.cn

Ji He

State Key Laboratory of Mechanical
System and Vibration,
Shanghai 200240, China
Shanghai Key Laboratory of Digital
Manufacture for Thin-Walled Structures,
Shanghai Jiao Tong University,
Shanghai 200240, China

Z. Cedric Xia, Danielle Zeng

Research and Innovation Center,
Ford Motor Company,
Dearborn, MI 48121

Bo Hou

Department of Engineering Technology
Non-Linear Solid Mechanics,
University of Twente,
Enschede 7500 AE, The Netherlands

1Corresponding author.

Contributed by the Manufacturing Engineering Division of ASME for publication in the JOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING. Manuscript received July 17, 2013; final manuscript received May 27, 2014; published online August 6, 2014. Assoc. Editor: Brad L. Kinsey.

J. Manuf. Sci. Eng 136(5), 051005 (Aug 06, 2014) (10 pages) Paper No: MANU-13-1281; doi: 10.1115/1.4027757 History: Received July 17, 2013; Revised May 27, 2014

A bifurcation analysis of forming limits for an orthotropic sheet metal is presented in this paper. The approach extends Stören and Rice's (S–R) bifurcation analysis for isotropic materials, with materials following a vertex theory of plasticity at the onset of localized necking. The sheet orthotropy is represented by the Hill’48 yield criterion with three r-values in the rolling (r0), the transverse (r90) and the diagonal direction (r45). The emphasis of the study is on the examination of r-value effect on the sheet metal forming limit, expressed as a combination of the average r-value raverage and the planar anisotropy (Δr). Forming limits under both zero extension assumption and minimum extension assumption as well as necking band orientation evolution are investigated in detail. The comparison between the experimental result and predicted forming limit diagram (FLD) is presented to validate the extended bifurcation analysis. The r-value effect is observed under uniaxial and equal-biaxial loadings. However, no difference is found under plane strain condition in strain-based FLD which is consistent with Hill's theory. The force maximum criterion is also used to analyze FLD for verification.

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Figures

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Fig. 1

The illustration of a uniform thin sheet metal with applied in-plane stress

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Fig. 4

The comparison between the experimental results of Al6111-T4 from Graf and Hosford and predicted results

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Fig. 3

The comparison between the three r-values introduced bifurcation analysis and Stören–Rice bifurcation analysis with the minimum energy principal searching algorithm on left hand side

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Fig. 2

The comparison between the three r-values introduced bifurcation analysis and Stören–Rice bifurcation analysis with Hill's zero extension direction on left hand side

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Fig. 6

The predicted forming limit results with different r-value combinations

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Fig. 5

The comparison of yield surfaces for material scenario 1 to 4

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Fig. 7

The necking angle distributions with different r-value combinations

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Fig. 8

The necking band angle evolution under equibiaxial (β = 1) stretching mode with three different r-values

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Fig. 9

The necking band angle evolution under equibiaxial (β = 1) stretching mode with an average r-value

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Fig. 10

The comparison between the yield surface with or without r0 effect

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Fig. 11

Illustration of the flow direction determined by the changed curvature of the yield surface

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