Additive manufacturing (3D printing) of metals has risen in popularity in industrial applications. A key aspect is that there are very few or no geometric restrictions. Particularly, the 3D printing of metals allows one to make complex, durable, custom parts. For the last few years, our group has investigated the thermomechanical behavior of 3D printing of metals in the laser-based powder bed fusion (L-PBF) process, also known as selective laser melting (SLM). Here we highlight the main results of the research. We construct a thermoviscoelastic continuum model for the case where a thin fin is being printed at a constant velocity. We use a coordinate frame that moves with the printing laser, and apply an Eulerian perspective to the moving solid. We consider a steady state similar to those used in the analysis of production processes in the process industry, in the field of research known as axially moving materials. We demonstrate the model with material parameters for 316L steel. As our main result, we obtain the steady-state deformation shape in the wake of the focus spot of the laser in a two-dimensional setting. Another important result are nondimensional parameters derived from a one-dimensional model. Similarly to how the Reynolds number governs flows, these parameters govern the behavior of the printing process. Finally, we extend traditional linear rheology by introducing a kinetic inductor element, inspired by electrical RLC circuits, to complement the classical spring and dashpot elements. This adds the inertial response of the material to the constitutive law. This modification becomes important in settings with quickly varying loading. Although the effect is negligible in most cases for steel, for soft materials under acoustic or ultrasound loading, the magnitude of the inertial term becomes important.