An analytical computation of temperature field evolved in directed energy deposition

This paper presents an analytical computation of temperature field evolved in a directed energy deposition process, using single-bead walls as illustrating examples. Essentially, the temperature field evolution during the deposition of a wall is computed by superposition of the temperature field generated by the laser source depositing the current bead and that induced from each of the past beads (layers). First, the transient solution to a point heat source in a semi-infinite body is applied to describe each individual temperature field. Then, to better describe temperature contribution from a past bead, a pair of virtual heat sources with positive and negative powers is assigned for each past bead to compute the temperature field under cooling. In addition, mirrored heat sources through a reflexion technique are introduced to define adiabatic boundaries of the part and to account for substrate thickness. In the end, three depositions of Ti-6AL-4V walls with different geometries and interlayer dwell times on an Optomec^VR laser engineering net shaping (LENS) system are used to validate the proposed analytical computation, where predicted temperatures at several locations of the substrate show reasonable agreement with the in situ temperature measurements with prediction error rate ranging from 12% to 27%. Furthermore, temperature distributions predicted by the proposed model are compared to finite element simulations. The proposed analytical computation for temperature field could be potentially used in model-based feedback control for thermal history in the deposition, which could affect microstructure evolution and other properties of the final part.