A Buckingham-Pi dimensionless analysis for melt pool stability and defect prediction in additive manufacturing

Abstract

Melt pool instabilities limit the reliability of additive manufacturing. Here, we demonstrate that a minimal Buckingham-$\pi$ framework, supplemented by a normalized enthalpy (NE) metric, consolidates process outcomes across heat source settings (power, speed, spot) and material properties. IN738LC was processed on an EOS M290; single-track and bulk responses, melt pool geometric features, part relative density (${\rho }^{\ast }$), and areal roughness parameters $S_a$ and $S_z$, were quantified and subsequently mapped onto compact NE-dimensionless number spaces after the normalized-enthalpy metric had been calibrated using an effective absorptivity inferred from the measured melt pool depth. The recoil number cleanly delineates modes: $\mathrm{Recoil}\lesssim 2$ (conduction$\to$stable keyhole) maintains ${\rho }^{\ast }\gtrsim 99\text{\%}$ with low $S_a$, whereas $\mathrm{Recoil}\gtrsim 4$–5 marks an unstable keyhole with spatter and porosity. Within this map, favorable transport balances are $\mathrm{Re}\lesssim 100$, $\mathrm{We}<1$, small $\mathrm{Ca}$ and not-too-small $\mathrm{Oh}$, and $\mathrm{Fo}>0.1$; external convection remains negligible ($\mathrm{Nu}\ll 1$). Rather than VED, we advocate working directly in $\Pi$-space $(\mathrm{NE},\mathrm{Recoil},\mathrm{Re},\mathrm{We},\mathrm{Ca},\mathrm{Oh},\mathrm{Fo},\mathrm{Nu})$—to define, compare, and transfer qualifiable process windows across machines and alloys.