0
TECHNICAL PAPERS

# Prediction of Necking in Tubular Hydroforming Using an Extended Stress-Based Forming Limit Curve

[+] Author and Article Information
C. Hari Manoj Simha, Alexander Bardelcik

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

Institute for Aerospace Research, National Research Council, Aerospace Manufacturing Technology Center, 5145 Decelles Avenue, Campus of the University of Montreal, Montreal, PQ, Canada H3T 2B2

Michael J. Worswick1

Department of Mechanical Engineering, University of Waterloo, Waterloo, ON, Canada, N2L 3G1worswick@lagavulin.uwaterloo.ca

1

Corresponding author.

J. Eng. Mater. Technol 129(1), 36-47 (Aug 09, 2006) (12 pages) doi:10.1115/1.2400269 History: Received September 17, 2005; Revised August 09, 2006

## Abstract

This paper presents an extended stress-based forming limit curve (XSFLC) that can be used to predict the onset of necking in sheet metal loaded under non-proportional load paths, as well as under three-dimensional stress states. The conventional strain-based $ϵ$FLC is transformed into the stress-based FLC advanced by Stoughton (1999, Int. J. Mech. Sci., 42, pp. 1–27). This, in turn, is converted into the XSFLC, which is characterized by the two invariants, mean stress and equivalent stress. Assuming that the stress states at the onset of necking under plane stress loading are equivalent to those under three-dimensional loading, the XSFLC is used in conjunction with finite element computations to predict the onset of necking during tubular hydroforming. Hydroforming of straight and pre-bent tubes of EN-AW 5018 aluminum alloy and DP 600 steel are considered. Experiments carried out with these geometries and alloys are described and modeled using finite element computations. These computations, in conjunction with the XSFLC, allow quantitative predictions of necking pressures; and these predictions are found to agree to within 10% of the experimentally obtained necking pressures. The computations also provide a prediction of final failure location with remarkable accuracy. In some cases, the predictions using the XSFLC show some discrepancies when compared with the experimental results, and this paper addresses potential causes for these discrepancies. Potential improvements to the framework of the XSFLC are also discussed.

<>

## Figures

Figure 1

Cross section of dies used in the hydroforming experiments

Figure 2

Schematic of conditions under which a neck originates in tubular hydroforming

Figure 3

(a) Strain-based FLCs for EN-AW 5018 aluminum alloy and DP600 steel, (b) stress strain curves for the alloys, (c) stress-based FLCs for the alloys obtained assuming isotropic hardening, and (d) XSFLCs for the two alloys. The mean stress is assumed to be positive in tension.

Figure 4

Contour plots of straight tube hydroforming (EN-AW 5018 aluminum) using solid elements. Formability variable γ (Eq. 2) indicates whether the load path (Σhyd,Σeq) in the element has crossed the XSFLC. (a) Shows one-quarter of the tube when the internal pressure is 32.2MPa. Several elements on the inside of the tube have crossed the XSFLC. (b) Close up of the tube center where the locations under a three-dimensional state of stress are labeled as 3D. At these locations, the load paths in all of the elements through the thickness of the tube have crossed the XSFLC (γ=1). The regions designated as free expansion are under plane stress loading (approximately). The regions labeled die contact stay in contact with the die during the entire process.

Figure 5

Load paths from solid elements computations of straight tube hydroforming computations (EN-AW 5018 aluminum). These results were obtained using explicit time integration. Load path from an element at location 3D. This load path crosses the XSFLC and while doing so the path changes slope point when the mesh comes into contact with the die. (c) Load paths from elements in the die contact and at the free expansion regions. These paths do not cross the XSFLC. The load paths are from elements that are located in the middle layer. (b),(d) Results obtained using implicit time integration.

Figure 6

Load paths from Beytschko-Lin-Tsay plane stress shell element computations of straight tube hydroforming (EN-AW 5018 aluminum alloy). (a) Load path (Σhyd,Σeq) from element at location 3D plotted with the XSFLC. (b) Load path (Σhyd,Σeq) from an element in the free expansion region plotted with the XSFLC. The load path in the free expansion region crosses the XSFLC, whereas, the path in the 3D region does not.

Figure 7

Load paths from Beytschko-Lin-Tsay plane stress shell element computations of straight tube hydroforming (EN-AW 5018 aluminum alloy). (a) Load path (σ2,σ1) from element at location 3D plotted with the σFLC. (b) Load path (σ2,σ1) from an element in the free expansion region plotted with the σFLC. The load path in the free expansion region crosses the XSFLC, whereas, the path in the 3D region does not.

Figure 8

Schematic illustrating Assumption 3. (a) Material response under monotonic single loading (solid line). Dashed lines illustrate Type 1 and Type 2 responses. These curves are shifted by the prestrain values. (b) Effect of Assumption 3. Load paths oa and ob have been drawn as straight lines for the purpose of illustration, in reality they are not.

Figure 9

Strain paths and stress paths for bending and hydroforming of pre-bent OCF EN-AW 5018 aluminum alloy tube. Plots in each column correspond to element locations at the inside, outside, and the failure location, respectively (see mesh outline plot at bottom left). Plots in rows correspond to strain, stress and XSFLC space. The linear paths shown are simplifications to those computed. For the purpose of discussion, each plot is treated as an element in a matrix. Note that the equivalent stress axes in the XSFLC plots (Outside-c and Fail-c) do not start at zero. Also note that the augmented formability, as per Assumptions 3 and 4, is shown as a dashed horizontal line in plots Outside-c and Inside-c, respectively.

Figure 10

Comparison of contour plots predicting failure locations (left) and photographs (right) from the experiments. The white arrows indicate the failure location. The gray arrows indicate additional locations in the DP600 tubes where the XSFLC approach indicates failure. Though the variable γ=1 in the location indicated by the black arrow (DP600 SCF), all the elements throught the thickness of the mesh have not crossed the XSFLC. The binary definition of the formability variable, γ, is used to plot these contours.

## Discussions

Some tools below are only available to our subscribers or users with an online account.

### Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related Proceedings Articles
Related eBook Content
Topic Collections