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Research Papers

# Orientation and Path Dependence of Formability in the Stress- and the Extended Stress-Based Forming Limit Curves

[+] Author and Article Information
C. Hari Simha

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

Kaan Inal

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. Eng. Mater. Technol 130(4), 041009 (Sep 17, 2008) (14 pages) doi:10.1115/1.2931152 History: Received August 10, 2007; Revised February 22, 2008; Published September 17, 2008

## Abstract

This article analyzes the formability data sets for aluminum killed steel (Laukonis, J. V., and Ghosh, A. K., 1978, “Effects of Strain Path Changes on the Formability of Sheet Metals  ,” Metall. Trans. A., 9, pp. 1849–1856), for Al 2008-T4 (Graf, A., and Hosford, W., 1993, “Effect of Changing Strain Paths on Forming Limit Diagrams of Al 2008-T4  ,” Metall. Trans. A, 24A, pp. 2503–2512) and for Al 6111-T4 (Graf, A., and Hosford, W., 1994, “The Influence of Strain-Path Changes on Forming Limit Diagrams of Al 6111 T4  ,” Int. J. Mech. Sci., 36, pp. 897–910). These articles present strain-based forming limit curves $(ϵFLCs)$ for both as-received and prestrained sheets. Using phenomenological yield functions, and assuming isotropic hardening, the $ϵFLCs$ are transformed into principal stress space to obtain stress-based forming limit curves $(σFLCs)$ and the principal stresses are transformed into effective stress and mean stress space to obtain the extended stress-based forming limit curves (XSFLCs). A definition of path dependence for the $σFLC$ and XSFLC is proposed and used to classify the obtained limit curves as path dependent or independent. The path dependence of forming limit stresses is observed for some of the prestrain paths. Based on the results, a novel criterion that, with a knowledge of the forming limit stresses of the as-received material, can be used to predict whether the limit stresses are path dependent or independent for a given prestrain path is proposed. The results also suggest that kinematic hardening and transient hardening effects may explain the path dependence observed in some of the prestrain paths.

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Topics: Stress , Steel , Hardening

## Figures

Figure 1

Schematic illustrating Assumption 3. Primary loading path may be along uniaxial path ou or equibiaxial path ob. Initial yielding is at Σ0, and the hardening at the terminal point of the primary path is Σp. No necking occurs when the secondary load path, os, intersects the as-received XSFLC, but necking occurs when yielding occurs at point s.

Figure 2

Variation of Lankford coefficients and yield strength to the RD and yield loci at 5%, 10%, and 20%. (a) Aluminum killed steel, (b) Al 2008, and (c) Al 6111.

Figure 3

Schematic illustrating definition of path dependence. Path dependence is assumed if Δ is less than ≈3%. PI—path independent, PD—path dependent, and AR—as-received.

Figure 4

(a) ϵFLCs from Laukonis and Ghosh (3). (b) σFLCs obtained using the KB model. The dashed line denotes a path independent FLC, and the dotted lines denote path dependent ones. (c) XSFLCs obtained using the KB model. The open symbols are the effective stress and mean stress at the terminal point of the primary loading path. Here, effective stress is given by (ϕ∕2)1∕6, where ϕ is given by Eq. 2. The dashed-dotted line is the plane strain path.

Figure 5

(a) σFLCs obtained using the BL model. (b) XSFLCs obtained using the BL model. The effective stress is given by (ϕ∕2)1∕2, where ϕ is given by Eq. 1.

Figure 6

Stress-based FLCs for Al 2008-T4 obtained using the formability data of Graf and Hosford (2) and the KB model. (a) σFLCs in the RD and (b) path independent σFLCs in the TD. (c) Path dependent σFLCs in the TD. (d) Path dependent XSFLCs in the TD. Open symbols in (d) are the values of the effective stress and mean stress at the terminus of the prestrain path. The dashed-dotted line represents the plane-strain path.

Figure 7

Stress-based FLCs for Al 2008-T4 obtained using the formability data of Graf and Hosford (2) and the BL model. (a) σFLCs in the RD. UTD18/RD is path dependent in contrast to the KB model. (b) Path independent σFLCs and (c) path dependent σFLCs in the TD. (d) Path dependent XSFLCs in the TD. The solid lines depict as-received forming limits. The open symbols in (d) are the values of the effective stress and mean stress at the terminus of the prestrain path. The dashed-dotted line is the plane-strain path.

Figure 8

Stress-based FLCs for Al 26111-T4 obtained using the data of Graf and Hosford (4) and the KB model ((a) and (b)) Path independent and path dependent σFLCs in the TD. (c) Path dependent XSFLCs in the TD. The solid lines depict as-received forming limits. The open symbols in (c) are the values of the effective stress and mean stress at the terminus of the prestrain path.

Figure 9

Stress-based FLCs for Al 26111-T4 obtained using the data of Graf and Hosford (4) and the BL model. ((a) and (b)) Path independent and path dependent σFLCs in the TD. (c) Path dependent XSFLCs in the TD. The solid lines depict as-received forming limits. The open symbols in (c) are the values of the effective stress and mean stress at the terminus of the prestrain path.

Figure 10

Criterion for path dependency. (a) KB model. The upper edge of the gray band is the as-received forming limit obtained using the KB model, and the ordinates of lower edge are 90% of that of the as-received curve. Effective strains and yield contours at the terminus of the primary paths are also shown. Contours that overlap with the gray band are shown as dashed lines, and this overlap indicates possible path dependence. (b) BL model.

Figure 11

Criterion for path dependency. (1) KB model. The upper edge of the gray region is the as-received forming limit in the RD obtained using the KB model, and the ordinates of lower edge are 90% of that of the as-received curve. The right edge of the gray region is the as-received forming limit in the TD, and the abscissas of the inner edge are 90% of the right edge. Effective strains and yield contours at the terminus of the primary paths are also shown. Contours that overlap with the gray band are shown as dashed lines, and this overlap indicates possible path dependence. (2) BL model. The filled symbols indicate instances where the criterion has a discrepancy with the results of Figs.  67.

Figure 12

Criterion for path dependency. (1) KB model. The tray region for Al6111-T4 is obtained as before. (2) BL Model. The filled symbols denote instances where the criterion has a discrepancy with the results of Figs.  89.

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