Thermal analysis of the arc welding process: Part I. General solutions
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I. INTRODUCTION
ARC welding is one of the most common manufacturing operations for the joining of structural elements for a myriad of applications, including bridges, building structures, cars, trains, farm equipment, and nuclear reactors, to name a few. In these applications, it is desirable, and oftentimes critical, to determine the temperature-rise distribution in relation to the location, time, and welding conditions, for it affects the metallurgical conditions at and near the weld and the consequent strength and reliability of the joint. In this investigation, an analytical solution for the temperature rise distribution in arc welding of short workpieces is presented based on the moving heat source theory of Jaeger[1] and Carslaw and Jaeger[2] to predict the transient thermal response. Rosenthal laid the foundation for the analytical treatment of the heat distribution in welding during the late 1930s and in the mid-1940s.[3,4,5] Using the Fourier partial differential equation (PDE) of heat conduction, he introduced the moving coordinate system to develop solutions for the point and line heat sources and applied this successfully to address a wide range of welding problems. His analytical solutions of the heat flow made possible for the first time the analysis of the process from a consideration of the welding parameters, namely, the current, voltage, welding speed, and weld geometry. To facilitate solution of the PDE, Rosenthal assumed R. KOMANDURI, Professor and MOST Chair in Intelligent Manufacturing, and Z.B. HOU, Visiting Professor, are with the School of Mechanical and Aerospace Engineering, Oklahoma State University, Stillwater, OK 74078. Manuscript submitted September 13, 1999. METALLURGICAL AND MATERIALS TRANSACTIONS B
quasi-steady-state conditions that can be justified experimentally when the length of the weld is long compared to the extent of heat. This means that an observer stationed at the point heat source fails to notice any change in the temperature around him as the source moves on. Rosenthal also gave an alternate analogy for this, wherein the temperature distribution around the heat source is represented by a hill that moves as a rigid body on the surface of the plane without undergoing any modification either in size or shape. Starting from the following PDE of heat conduction, Rosenthal applied it for welding (by assuming the heat source to be a moving point or a moving, infinitely long line heat source) by considering a moving coordinate system. 2u 2u 2u 1 u 1 1 25 x2 y2 z a t
[1]
When the origin of the moving coordinate system coincides with the moving heat source and moves along with it at the same speed (with its X-axis coinciding with the x-axis of the original absolute coordinate system), the relationship between the coordinates of the point where the temperature rise is concerned along the X- (or x-) axis at any time t is given by X 5 x 2 vt. Substituting this in Eq. [1], the general PDE of heat conduction in a moving coordinate system can be obtained as 2u 2u 2u v u
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