Prediction of forming limit diagrams based on shear failure criterion Jun Lin, L.C. Chan * , Lin Wang Department of Industrial and Systems Engineering, Hong Kong Polytechnic University, Hung Hom, Hong Kong, SAR, China article info Article history: Received 22 December 2009 Received in revised form 12 May 2010 Available online 20 June 2010 Keywords: Shear failure criterion Forming limit diagram Sheet metal Constitutive model abstract Maximum shear stress theory, also called the ‘Third Strength Theory’, is a classical theory used to predict the failure of common metal, but it cannot be used directly to predict sheet metal failure due to anisot- ropy and the loading path. Therefore, this paper proposes a maximum shear stress calculating method, which has been named ‘‘shear failure criterion” for the purpose of this paper. In order to validate the shear failure criterion, a general program was developed, and two typical materials, steel, and aluminum alloy, were used to study the new shear failure criterion in this study. The two materials were modeled by advanced constitutive models, including Hill1948 and Yld2000-2d yield functions and several types of isotropic hardening models. Experimental validation has indicated the accuracy of predicted FLD using shear failure criterion, which is able to provide a new alternative method to numerically predict FLD. Ó 2010 Elsevier Ltd. All rights reserved. 1. Introduction The demand for energy conservation has led the automotive industry to look for new materials and plastic forming methods to reduce the weight of vehicle body structures. In order to explore the new forming process of the metal sheets, a fundamental under- standing of the forming limit of the sheet metal is necessary. One of the important methods for doing this is to use the Keeler–Good- win approach. The Keeler–Goodwin approach (Keeler and Backho- fen, 1963; Goodwin, 1968), which can also be called the forming limit diagram (FLD), has been the predominant method of estimat- ing the forming limit of sheet metal for many years in the sheet metal forming industry. The FLD can be achieved before the onset of local necking. The commonly used necking models are the Swift diffuse criterion (Swift, 1952) and the Hill local criterion (Hill, 1952). The Swift dif- fuse criterion was derived by assuming there is a maximum loading force and the Hill localization criterion is obtained by assuming there is a maximum principal stress. Another well-known methodol- ogy is the geometric imperfection model proposed by Marciniak and Kuczynski (1967, 1973), which is referred to as the M–K model. These models are used extensively in sheet metal FLD prediction. An alternative to the above-mentioned predicting methods is proposed by the assumption that there is a maximum shear stress in one special plane. A classical framework for the shear failure cri- terion of plastic deformation was presented by Rice (1976). Based on this, the following theory was engaged in this paper. If the shear stress is greater than a critical one, s P s cr ; ð1Þ the material will have necking failure. There is a classical calcula- tion formula for the maximum shear stress, s ¼ 1 2 ðr 1 r 3 Þ; ð2Þ where r 1 and r 3 are the first and third principal stress, respectively. This theory is also called ‘‘Maximum Shear Stress Theory” or ‘‘Third Strength Theory”. But Eq. (2) is not reasonable under sheet metal forming conditions due to anisotropy and the strain ratio, which is defined in the Section 2.2. The angle of maximum shear plane to the direction of maximum principal strain direction will not always be 30° due to anisotropy and the strain ratio. In order to use the theory to predict the sheet forming limit, an alternative calculating method should be explored. In 1983, an alternative shear failure theory was proposed and a novel formula, used to cal- culate the maximum shear stress, was derived by Bressan and Wil- liams (1983). However, the Bressan–Williams model can only be used to calculate the right region of FLD, as they did in their origi- nal paper. The analysis of the maximum shear stress will be discussed in this paper, and the paper is organized as follows. In Section 2,a new calculating method for maximum shear stress is proposed, which can be used to calculate both right and left regions of FLD. This has been named as ‘shear failure criterion’ for the purpose of the present work. In order to validate the shear failure criterion, two yield functions and several isotropic hardening models were adopted, and these are presented in the 3rd section. In the 4th sec- tion, a general program was written with stress updating algo- rithm; one steel material and two aluminum alloy sheets were engaged to validate the failure model, and finally the FLD analysis, 0020-7683/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijsolstr.2010.06.004 * Corresponding author. Tel.: +852 27666634. E-mail address: mfl[email protected] (L.C. Chan). International Journal of Solids and Structures 47 (2010) 2855–2865 Contents lists available at ScienceDirect International Journal of Solids and Structures journal homepage: www.elsevier.com/locate/ijsolstr
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