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ing (FSW) is a solid-state welding process in which the welding is completed without melting of the base metal. A schematic diagram of the FSW system is presented in Figure 1. A rotating tool moves along the joint interface and generates heat. Movement of the solid and its deformation of a superplastic nature close to the tool forms the joint. The tool usually has a large-diameter shoulder and a smaller threaded pin. Since its development in 1991 at TWI,[1] FSW has been widely analyzed, both experimentally and using mathematical models.[2–15] However, most of these models have focused on heat transfer calculations and disregarded plastic flow near the tool. Several earlier attempts to model FSW were based on analytical solution of the heat conduction equation for a thin plate. For example, Russel and Shercliff[2] adapted analytical solutions of heat conduction equations to model FSW considering several simplified heat sources such as a point source, line source, and a distributed surface heat source. Gould and Feng[3] also used the Rosenthal’s analytical conduction model to predict a quasi-steady temperature field due to a moving point heat source at a constant velocity. They considered heat generation only at the tool shoulder. Schmidt et al.[4] proposed a more general analytical model for heat generation based on various contact conditions at the tool/matrix interface, namely sliding, sticking, and partial sticking condition. They concluded that the sticking condition prevailed at the tool/matrix interface of AA 2024-T3 alloy, based on experimental data on heat generation rate and plunge force. Several investigators used numerical heat transfer to understand the FSW process. However, the plastic flow was ignored in some of the initial studies. For example, R. NANDAN, Graduate Student, G.G. ROY, Visiting Scientist, and T. DEBROY, Professor, are with the Department of Materials Science and Engineering, The Pennsylvania State University, University Park, PA 16802, U.S.A. Contact e-mail: [email protected] Manuscript submitted August 29, 2005. METALLURGICAL AND MATERIALS TRANSACTIONS A

Frigaard et al.[5] developed a numerical three-dimensional (3-D) heat flow model for FSW based on the finite difference method. They assumed that heat was generated at the tool shoulder due to frictional heating and adjusted the coefficient of friction so that the calculated peak temperature did not exceed the melting temperature. Chao et al.[6] formulated heat transfer in the FSW process based on overall heat balance and inverse modeling into two boundary value problems (BVP)—a steady BVP for the tool and a transient BVP for the workpiece—and solved them using the finite element method (FEM). The found that only 5 pct of the heat generated was transported into the tool and about 80 pct of the mechanical work was dissipated as heat. Song and Kovacevic[7,8] presented a detailed 3D numerical model of FSW where heat input from the tool was modeled as a moving heat source. Heat transfer during tool penetration and pulling was modeled. Later, Song and K