A mathematical model has been developed in describing the temperature distribution, the flow of the molten fluid and the stress field in the solid during welding. In modeling the properties of the material during welding, the solid phase is assumed to behave as a thermoviscoplastic solid obeying Bodner-Partom/Walker type constitutive equation, whereas the fluid phase as a thermoviscous incompressible fluid. Three regions exist: pure solid, pure fluid, and the transition (solid-fluid mixture). In the formulation of the boundary value problem, the energy equation is coupled to the equation of motion through the terms of mechanical work and the latent heat of the phases, whereas the equations of motion of the solid and the fluid are decoupled. Appropriate thermal and traction boundary conditions are detailed in the text. Phase transformation activities during cooling are monitored by CCT diagram and Avrami equation. An arbitrary Lagrangian and Eulerian method is used to accommodate the kinematic description of both the solid and the fluid phases. A representative plane perpendicular to the moving heat source is analyzed. Results of sample calculations are presented to show the temperature and the stress evolution in time. Residual stress and microstructure patterns are presented.

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