Temperature-Dependent 1D Models, with Analytical Techniques and Their Limitations

In the previous chapter, we solved the 1D model where we the material parameters were taken to be independent of temperature. However, in 3D printing of metals, the range of temperatures encountered is large—from the solidus temperature of the metal down to room temperature—and the material parameters typically vary significantly in this range. So the natural question to ask next is, if we consider the material parameters as functions of temperature, is it still possible to get an analytical solution for some simplified case? It turns out that this is tricky, and a complete analytical solution is either not possible, or at least too unwieldy for practical use.