## 2 Materials and Methods  
If the Earth is to be considered as a sphere in the calculations, the solution will be challenging, so it is appropriate to introduce a simpler planar model in the initial approximation that can be extended in the future. However, if the dimensions of the heat source (the Sun) and its distance from the Earth are taken into account, it can be concluded that the heat rays incident on the Earth are almost parallel. This approach is common in mechanics. An acceptable simplification for the calculations can be achieved by replacing the Earth's crust by an infinite hollow cylinder (see Figure 2), the so-called 'Earthcylinder', and defining the heating or cooling of the Earth's outer surface as a time-varying heat function \(\mathrm{T}=\mathrm{f}(\varphi,t)\), where \(\varphi\) is the angle /deg/ and t is time /s/ [9]. The Earth's crust is composed of lithospheric plates and is mainly made up of rocks and minerals, where various faults, fractures, and other geological formations occur. In addition to rocks and minerals, there are also gases and water. It is, therefore, an inherently highly anisotropic, heterogeneous and inhomogeneous material [2; 32; 33; 34; 35; 36; 37; 38; 39]. The creation of such a material model would be very complicated, and so some appropriate simplifications are adopted in the calculations. From the point of view of material properties, our computational model is a composite [4; 5; 9]. The model of the Earth's crust is divided into 24 sections with different material properties; it is a piecewise isotropic homogeneous material model, which appears to be anisotropic from the outside. On the surface of each material section, there are temperature functions \(\mathrm{T}_{1},\mathrm{T}_{2}\ldots,\mathrm{T}_{24}\)\(=\mathrm{f}(\varphi,t)\), which correspond to time-varying temperature values over a two-year period, according to [6; 7], as shown in Figure 3.  
Figure 2: Model of the Earth as an infinite hollow cylinder, the so-called ‘Earthcylinder’ with heating from the Sun (not to scale). For the FEM calculation, the problem is treated as a planar problem with an assumed plane strain. In Figures 3 and 4, the crustal thickness is highly enlarged (for clarity), and the planar model is shown as being divided into 24 sections, with assigned material properties and loading indicated from the temperature. The angular perimeter is divided by 15 degrees, with a different material in each of the 15 degrees, as shown in Figures 3 and 4. The 'Earthcylinder' is a model of a cylinder replacing the earth's crust with constant inner and outer radii, \(R_{1}=6348\) km and \(R_{2}=6378\) km, as shown in Figures 1-3. Below the crust, i.e., below the radius \(R_{1}\), is the upper mantle, whose assumed constant temperature is \(T_{\rm p}\). This temperature is mostly transferred to the inner radius of the crust by thermal conduction, as shown in Figures 3 and 4. In the model, the upper mantle is replaced by a Winkler elastic foundation with the Modulus of the Foundation K = 10\({}^{7}\) Nm\({}^{-3}\), as shown in Figures 1 and 3 [9; 13; 14; 15; 19]. The elastic foundation is also applied in many engineering problems [24; 25; 26; 27]. The outer radius of the model \(R_{2}\) is affected by the temperature \(T=f(\varphi,t)\). This temperature depends on the angle \(\varphi\) and the time t. Such a dependence respects the fact that the temperature varies at different locations on the Earth's crust at different times, i.e., there are temperature differences between day and night or, possibly, seasons [9]. Figure 3: ‘Earthcylinder’ with materials and temperature loads (not to scale).