Analytical Solution in One Dimension

We have now formulated our thermomechanical model for a Kelvin–Voigt material. The plan for the remaining three chapters of the book is as follows. In this chapter, we specialize the strong form of the model into one space dimension, and solve the constant-coefficient case analytically. In one-dimensional and two-dimensional models, the heat escaping via the surrounding air needs special consideration. For completeness, we first solve the one-dimensional model without this additional cooling term—thus corresponding to a thermally insulated domain—and then add in the cooling term, and solve it again to obtain a more realistic result. In Chap.  8 , we consider how the model changes if we make the parameters into linear functions of temperature, which matches physical observations. We will see that doing that will reduce the possibilities to obtain an analytical solution. However, we will take the opportunity to showcase some useful techniques, which allow us to obtain partial analytical solutions. After we are done with the one-dimensional models, in Chap.  9 we move on to a numerical solution in two space dimensions (including the depth direction, i.e. the previously printed layers), using the weak form of the model and the finite element method.