Analytical solution of the residual temperature field in wire-arc directed energy deposition considering heat accumulation

In the process of wire-arc directed energy deposition, when the inter-layer idle time is not long enough, a heat accumulation effect occurs. That is, the residual temperature does not return to room temperature, and as the material continues to be deposited, the residual temperature of the just deposited layer significantly increases. In this paper, an analytical solution is derived for the residual temperature field during the layer-by-layer depositing process of thin-walled components. There are two main contributions. First, the Eulerian coordinate system is employed to simplify the depositing process. In comparison with the solution expressed in the Lagrangian coordinate system, this treatment avoids the updating of the solution domain, resulting from the movement of the heat source. Second, the temperature field, expressed as a continuous function of position in the Eulerian coordinate system, allows for the introduction of virtual heat sources and their mirror images to account for heat convection and radiation on the boundary. The validity and accuracy of the present analytical solution are verified through comparisons with finite element simulations and experimental results. This analytical solution of the residual temperature will be used in an inherent strain method developed by the authors, to efficiently investigate the impact of heat accumulation on residual stress and deformation, in a follow-up work.