“…In this study, the SPH method is used to model the thermo-mechanical behavior of an Al-Si alloy consumable rod that has been treated with a solid solution during the FS process. The equations of conservation of momentum, energy, and mass in their SPH forms, based on previous studies, 9,24 can be defined as follows: The SPH formulation of equation of material momentum, mass conservation, and energy conservation are as below: where t, ρ, Vfalse→, P, μ, .1emf→ext, e, εv, K, T, ϑi, q, and qH is time, velocity, pressure, viscosity, external force, internal energy, energy dissipation rate, thermal conductivity, temperature, kinematic viscosity of the particle i, heat flux, and heat generation rate, respectively. The deformation behavior of the consumable rod was imported into the model through the Johnson-Cook equation as follows: 9 The constant used in the Johnson-Cook equation is reported in Table 1.…”