Post-print of “A comparative study of the extended forming limit diagrams considering strain path, through- thickness normal and shear stress”, by Mozhdeh Erfanian, and Ramin Hashemi, Published in International Journal of Mechanical Sciences, Volume 148, November 2018, Pages 316-326. https://doi.org/10.1016/j.ijmecsci.2018.09.005 1 A comparative study of the extended forming limit diagrams considering strain path, through-thickness normal and shear stress Mozhdeh Erfanian, Ramin Hashemi* School of Mechanical Engineering, Iran University of Science and Technology, Tehran, Iran *Corresponding author: e-mail: rhashemi@iust.ac.ir (R. Hashemi) Tel.: +98-21-77240068, Fax: +98-21-77240540 Abstract The substitute presentations of the conventional forming limit diagram (FLD) are stress-based forming limit diagram (FLSD), extended stress forming limit diagram (XSFLD), and polar effective plastic strain forming limit diagram (PEPS-FLD). These diagrams have already been proposed as alternative criteria to the conventional FLD for predicting forming limits in processes that sheet does not experience proportional or in-plane loading conditions. The present study provides a complete comparison of different forms of forming limit diagrams. For this purpose, FLD and extended FLDs are predicted based on the modified Marciniak and Kuczynski (M-K) model with Yld2011 anisotropic yield function while various factors that cause a change in the FLD of a specific alloy including initial imperfection coefficient (f0), non-linear strain path, through-thickness normal and shear stresses are considered. The results indicate that all forms of extended diagrams are independent of strain path and the diagrams in stress space (i.e., FLSD and XSFLD) are less sensitive to the strain path than PEPS-FLD. In this regard, the main weakness of XSFLD, especially on the right-hand side, is that the safety margin cannot be visualized easily. By increasing the initial imperfection coefficient, the level of all forms of diagram increases. However, the effect of through thickness stresses on different diagrams is not the same. Increase in normal and shear stresses results in a downward shift in FLSD and an upward shift in other types of forming limit diagrams. Keywords Extended forming limit diagram; Prestraining; Through-thickness normal stress; Through-thickness shear stress; M-K model . Introduction In the process of sheet metal forming, knowing the maximum allowable strains before necking is a critical requirement. For this purpose, forming limit diagram has been introduced. For many years, this diagram has been obtained both experimentally and theoretically with assumptions of plane stress state and linear strain path. But soon it became clear that due to these simplifying assumptions, the forming limit diagram was not accurate enough and no longer could predict forming limits in manufacturing processes that impose a non-linear and 3D state of stress on the sheet metal. The experimental evidence and theoretical results proved that the conventional FLD was highly sensitive to the strain path. Graf and Hosford investigated the effect of bi-linear strain paths on the aluminium alloys AA2008-T4 and AA6111 [1,2] through careful experiments. Kohara [3]