Development of a heterogeneous microstructurally based finite element model for the prediction of forming limit diagram
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ING the limits of formability is important for full utilization of any material’s capabilities. The forming limit diagram (FLD) is perhaps the most accepted means of characterizing formability. The FLD defines the extent to which a sheet material can be deformed over a wide monotonic range of strain paths. This capability is limited by the occurrence of localized thinning in the form of an intense shear band, as shown in Figure 1 for both aluminum alloys and steel under uniaxial tension, and in Figure 2 under biaxial loading. However, the measurement of FLDs is tedious and time-consuming. Thus, many attempts have been made to predict FLDs.[1–6] One challenge lies in relating the way in which the FLD is determined experimentally to the methodology used in the numerical predictions. For example, the deterministic parameters used to get the predicted FLD lead to a single curve in major-minor strain space, while experimentally measured FLDs often show a large scatter.[7] Such a pronounced difference adds to the discrepancy between the experimental and numerical FLDs. One of the major sources of the scatter in the determination of an FLD comes from the selection of measurement points near the fracture plane. There are two generally accepted rules: (1) the grids (square or circle) must still be intact and (2) the grids must be imminently adjacent to the fracture surface. A combination of these two rules implies that, in Figure 2, points A, B, and C are all valid measurement points while point D is invalid. XINJIAN DUAN, Senior Stress Analyst, is with the Atomic Energy of Canada Limited, Mississauga, ON Canada. MUKESH JAIN, Associate Professor, Department of Mechanical Engineering, and DAVID S. WILKINSON, Professor, Department of Materials Science and Engineering, Director, McMaster Manufacturing Research Institute, and Director, McMaster Centre for Automotive Materials, are with McMaster University, Hamilton, ON, Canada L8S 4L7. Manuscript submitted July 5, 2005. METALLURGICAL AND MATERIALS TRANSACTIONS A
In sheet materials, an intense shear band is frequently observed ahead of fracture. Once the band forms, the strain value within the band increases dramatically. If a complete grid is partially located within the intense shear band (point E in Figure 2), the validity of the grid strains becomes somewhat questionable. The relative position of the FLD will then change depending on whether such grids are included. There is no generally accepted evaluation strategy. From the experimental method, it is clear that the FLD is associated with the localized necking. Specifically, the forming limit corresponds to the onset of localized necking. It is often difficult to experimentally stop the test just at this point due to the test control and signal measurement limitations of formability test machines. Hence, highly necked or fractured specimens are generally used to obtain the forming limit strains. Numerical simulation is not subjected to the same restrictions, and a suitable localization criterion (Hill,[8] for example) can be ap
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