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Research Papers

# Application of an Extended Stress-Based Forming Limit Curve to Predict Necking in Stretch Flange Forming

[+] Author and Article Information
C. Hari Simha

Department of Mechanical and Mechatronics Engineering, University of Waterloo, Waterloo, ON, N2L 3G1, Canadasimha@lagavulin.uwaterloo.ca

Rassin Grantab

Department of Mechanical and Mechatronics Engineering, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

Michael J. Worswick

Department of Mechanical and Mechatronics Engineering, University of Waterloo, Waterloo, ON, N2L 3G1, Canadaworswick@lagavulin.uwaterloo.ca

J. Manuf. Sci. Eng 130(5), 051007 (Aug 14, 2008) (11 pages) doi:10.1115/1.2844593 History: Received November 24, 2006; Revised October 16, 2007; Published August 14, 2008

## Abstract

An extension of the stress-based forming limit curve (FLC) advanced by Stoughton (2000, “A General Forming Limit Criterion for Sheet Metal Forming  ,” Int. J. Mech. Sci., 42, pp. 1–27) is presented in this work. With the as-received strain-based FLCs and stress-strain curves for $1.6-mm$-thick AA5754 and $1-mm$-thick AA5182 aluminum alloy, stress-based FLCs are obtained. These curves are then transformed into extended stress-based forming limit curves (XSFLCs), which consist of the invariants, effective stress, and mean stress. By way of application, stretch flange forming of these aluminum alloy sheets is considered. The AA5754 stretch flange displays a circumferential crack during failure, whereas the AA5182 stretch flange fails through a radial crack at the edge of the cutout. It is shown that the necking predictions obtained using the strain- and stress-based FLCs in conjunction with shell element computations are inconsistent when compared with the experimental results. By comparing the results of the shell element computations with those in which the mesh comprises eight-noded solid elements, it is demonstrated that the plane stress approximation is not valid. The XSFLC is then used with results from the solid-element computations to predict the punch depths at the onset of necking. Furthermore, it is shown that the predictions of failure location and failure mode obtained using the XSFLC are in accord with the differences observed between the two alloys/gauges.

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## Figures

Figure 1

(a) Schematic of the stretch flange forming experiment. (b) Tooling dimensions. Both (a) and (b) are excerpted and modified from the dissertation by (21). (c) Photograph of the AA5754 stretch flange with circumferential crack. (d) Photograph of AA5182 with radial crack at the cutout edge. Experiments after (21). Note that the orientation of the photographs is upside down relative to the schematic.

Figure 2

(a) Strain-based FLCs of an AA5754 1.6-mm-thick aluminum alloy and are AA5182 1-mm-thick sheet. (b) Stress-strain curves from uniaxial tensile test. Extrapolations of the data and Voce law fits. (c) Stress-based FLC derived using the equations of (10) assuming isotropy. (d) XSFLCs for the aluminum alloys. The dotted line has a slope of 1∕3 and represents Σhyd=Σ∕3. Dotted lines are also used to extrapolate the XSFLCs to the line of slope 1∕3.

Figure 3

Mesh used in the shell element computations. In the case of solid-element computations, five such layers were used.

Figure 4

Comparison of computed main-punch force and displacement curves with the experimentally measured curves

Figure 7

(a) Formability analysis for the AA5754 stretch flange using the σFLC. The white arrow indicates the failure location observed in the experiment. (b) Formability analysis for the AA5182 stretch flange using the σFLC. The white arrow indicates the failure location observed in the experiment, and the gray arrow indicates the predicted failure location. The predicted failure location is not in agreement with the experimental observation.

Figure 8

(a) AA5754: Comparison of mean stress versus main punch depth from the shell element computation and the solid-element computation. The elements are from the location indicated by the white arrow in Fig. 7, and the results are from integration points that span the thickness of the sheet. (b) AA5182: Comparison of mean stress versus main punch depth from the shell element computation and the solid-element computation. The elements are from the location indicated by the white arrow in Fig. 7. There are five curves since there are five layers modeling the sheet. The orientation of the lower surface can be found in Fig. 5.

Figure 9

(a) AA5754: Contour plots of formability variable obtained using the XSFLC. Gray arrow indicates the predicted failure location, and the white arrow indicates the experimentally observed location. (b) AA5182: Contour plots of formability variable obtained using the XSFLC. The gray arrow indicates the predicted failure location, and the white arrow indicates the experimentally observed location. An additional failure location is predicted at A.

Figure 10

(a) AA5182: load paths (Σhyd,Σ) for location A and the experimentally observed failure location

Figure 11

AA5182: (Top) Cut-section orthogonal views of the mesh outline at various punch depths when the indicated element is subjected to compressive mean stress. (Middle and bottom) Plots of effective strain effective stress and mean stress plotted with respect to the backup and main punch displacement. The plots are from the element indicated by the arrow in the cut-section views. The dotted horizontal line in the graph at the bottom left corresponds to the quasistatic yield strength.

Figure 12

(Top) AA5754: Effect of changing the COF in the contact algorithm. The failure location relative to the bottom of the flange is not affected, but the punch depth at the onset of necking is. (Bottom) AA5754: Effect of changing the XSFLC on the necking prediction. For the −3% case, the failure location shifts relative to the top of the flange.

Figure 5

(a) Formability analysis for the AA5754 stretch flange using the ϵFLC. Contour plots are plotted at a depth of 50.8mm and for three integration layers through the thickness of the sheet (lower, upper, and last). The white arrow indicates the failure location observed in the experiment. The layer in which the strain paths intersect the ϵFLC at a punch depth of 50.8mm is designated as last. The orientation of the shell’s lower surface is also shown. No failure is indicated. (b) Formability analysis for the AA5182 stretch flange using the ϵFLC. The white arrow indicates the failure location observed in the experiment, and the gray arrow indicates the predicted failure location. The predicted failure location is in agreement with the experimental observation.

Figure 6

(a) AA5754: strain paths from the location indicated by the white arrow in Fig. 5 plotted with the ϵFLC. (b) AA5182: strain paths from the location indicated by the white arrow in Fig. 5 plotted with the ϵFLC.

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