Analysis of the influence of thermal loading on the behaviour of the Earth’s crust

Citation: Wandrol, I.; Frydrýšek, K.; ˇ Cepica, D. Analysis of the Influence of Thermal Loading on the Behaviour of the Earth’s Crust. Appl. Sci. 2023, 13, 4367. https://doi.org/10.3390/ app13074367 Academic Editor: Daniel Dias Received: 19 July 2022 Revised: 20 March 2023 Accepted: 21 March 2023 Published: 29 March 2023 Copyright: © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). applied sciences Article Analysis of the Influence of Thermal Loading on the Behaviour of the Earth’s Crust Ivo Wandrol 1,* , Karel Frydrýšek 2,3 and Daniel ˇ Cepica 2,3 1Institute of Physics in Opava, Silesian University in Opava, Bezruˇcovo Námˇestí1150/13, 746 01 Opava, Czech Republic 2 Department of Applied Mechanics, Faculty of Mechanical Engineering, VSB—Technical University of Ostrava, 17. Listopadu 2172/15, 708 00 Ostrava, Czech Republic; [email protected] (K.F.); [email protected] (D. ˇ C.) 3Institute of Emergency Medicine, Faculty of Medicine, University of Ostrava, Syllabova 19, 703 00 Ostrava-Vítkovice, Czech Republic *Correspondence: ivo.wandr[email protected] Abstract: The article focuses on the deformation and strain-stress analysis of the Earth’s crust under external thermal loading. More specifically, the influence of cyclic changes in the surface temperature field on the stress and displacement inside the crust over a two-year time span is investigated. The finite element program MSC.Marc Mentat was used to calculate the stresses and displacements. For practical analysis reasons, the Earth’s crust is simplified as a planar, piecewise homogeneous, isotropic model (plane strain), and time-varying temperature functions of illumination (thermal radiation) from the Sun are considered in the local isotropy sections of the model. Interaction between the Earth’s crust and mantle is defined by the Winkler elastic foundation. By applying a probabilistic approach (Monte Carlo Method), a new stochastic model of displacements and stresses and new information on crustal displacements relative to the Earth’s mantle were obtained. The results proved the heating influence of the Sun on the Earth’s crust and plate tectonics. Keywords: geomechanics; Earth’s crust; Finite Element Method; Sun heating (radiation); stress; displacement; elastic foundation; stochastic approach; Earth’s crust; tectonics 1. Introduction The geological structure of planet Earth is complicated and full of ‘unknowns’. Therefore, it is suitable to make a discretization for numerical modelling; see Figure 1[1–3]. The Earth’s crust is affected by many external influences, such as tidal forces, cyclic changes in crustal surface temperatures caused by the Sun, recurrent changes in atmospheric air pressure, the transmission of ocean and sea water mass waves to the Earth’s crust, and geological processes within the core and mantle of the planet [2–9]. This paper focuses on the effect of cyclic surface temperature changes on the stress and strain of the Earth’s crust. The crust (comprising lithospheric plates) is periodically heated (thermally deformed and stressed) by radiation induced by stellar physical processes inside the Sun. These are mainly diurnal periods (alternation of day and night—24 h cycle) or annual periods (alternation of seasons in about 365.4 days). Both of these periods have been applied to the input of the surface temperature of the Earth [9–12]. Due to the complexity of the whole process, the Finite Element Method (FEM) is used and the problem is solved as a planar one [ 2 – 5 ]. The Winkler elastic foundation model, which is commonly used in mechanics, is used for the interaction of the Earth’s crust and mantle, as shown in Figure 1[ 9 , 13 – 17 ]. However, application of the Winkler elastic foundation in geomechanics is a new way of solving the problem. Appl. Sci. 2023,13, 4367. https://doi.org/10.3390/app13074367 https://www.mdpi.com/journal/applsci Appl. Sci. 2023,13, 4367 2 of 24 Appl. Sci. 2023, 13, x FOR PEER REVIEW 2 of 25 Figure 1. Structure of the Earth and the possible discretization applied (not to scale). The Earth’s crust is an anisotropic, heterogeneous upper part of the geoid and is simplified in the analysis as a planar, piecewise, homogeneous, isotropic model with timevarying temperature functions of illumination, i.e., thermal radiation, from the Sun. The latter are considered in the local isotropy sections of the model. The generally anisotropic crust is partitioned into piecewise isotropic regions and the whole problem is solved as a 2D (plane strain) problem. The theory of large deformations, i.e., true (logarithmic) strains and true stresses, was used because it gives more accurate results, but leads to the solution of a non-linear problem. This is a non-stationary coupled (thermal + structural) problem and more details are given in [9,13–15]. A new and modern probabilistic (stochastic) approach is applied, which respects the real variability of the obtained records of thermal radiation from the Sun. There are not enough global models of the Earth’s crust, in connection with the FEM and stochastic approaches. This article fills the information gap. However, 2D models are commonly used in geomechanics [9,12]. In the first step, the influence of gravitational and tidal effects from the Sun and Moon, and the effect of rotational and centrifugal forces on the Earth’s crust, are neglected in the model. The reason for this is the thermal heating of the crust, which is the focus and research of this article. We focused on thermal heating of the crust because it is usually neglected or not considered, but it has a significant influence. Nevertheless, a somewhat simple probabilistic model has been developed which respects the real variability of input and output quantities [9,15–25]. Other possible approaches connected with groundwater investigations and uncertainty quantification are presented in [12,23]. In reference [23], there is a state-of-the-art in approach for uncertainty quantification in geomechanics. However, this approach cannot yet be applied to the complicated problem of crust-mantle interaction because of the lack of global information about the crust. It has recently been proved that the Sun’s heating of the Earth’s surface can have a partial and important influence on the creation and propagation of plate tectonics. Figure 1. Structure of the Earth and the possible discretization applied (not to scale). The Earth’s crust is an anisotropic, heterogeneous upper part of the geoid and is simplified in the analysis as a planar, piecewise, homogeneous, isotropic model with timevarying temperature functions of illumination, i.e., thermal radiation, from the Sun. The latter are considered in the local isotropy sections of the model. The generally anisotropic crust is partitioned into piecewise isotropic regions and the whole problem is solved as a 2D (plane strain) problem. The theory of large deformations, i.e., true (logarithmic) strains and true stresses, was used because it gives more accurate results, but leads to the solution of a non-linear problem. This is a non-stationary coupled (thermal + structural) problem and more details are given in [9,13–15]. A new and modern probabilistic (stochastic) approach is applied, which respects the real variability of the obtained records of thermal radiation from the Sun. There are not enough global models of the Earth’s crust, in connection with the FEM and stochastic approaches. This article fills the information gap. However, 2D models are commonly used in geomechanics [9,12]. In the first step, the influence of gravitational and tidal effects from the Sun and Moon, and the effect of rotational and centrifugal forces on the Earth’s crust, are neglected in the model. The reason for this is the thermal heating of the crust, which is the focus and research of this article. We focused on thermal heating of the crust because it is usually neglected or not considered, but it has a significant influence. Nevertheless, a somewhat simple probabilistic model has been developed which respects the real variability of input and output quantities [9,15–25]. Other possible approaches connected with groundwater investigations and uncertainty quantification are presented in [ 12 , 23 ]. In reference [ 23 ], there is a state-of-the-art in approach for uncertainty quantification in geomechanics. However, this approach cannot yet be applied to the complicated problem of crust-mantle interaction because of the lack of global information about the crust. It has recently been proved that the Sun’s heating of the Earth’s surface can have a partial and important influence on the creation and propagation of plate tectonics. So far, there have been no simpler nonlinear geomechanical stochastic/probabilistic models incorporating crust-mantle interaction, with a focus on deformation and stress Appl. Sci. 2023,13, 4367 3 of 24 states in the Earth’s crust, induced by solar radiation. This is the main contribution of this paper. Our work builds on simpler, previously developed, models [9,18,19,26–31]. Alternatively, the findings or procedures presented in [ 32 ], which were carried out for an area in Sweden, can also be used, but their application to the entire surface of the Earth is too challenging and complicated. The nomenclature for all of the variables and abbreviations used is presented at the end of this article. 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 T=f(ϕ,t), where ϕis the angle /deg/ and t is time /s/ [9]. Appl. Sci. 2023, 13, x FOR PEER REVIEW 3 of 25 So far, there have been no simpler nonlinear geomechanical stochastic/probabilistic models incorporating crust-mantle interaction, with a focus on deformation and stress states in the Earth’s crust, induced by solar radiation. This is the main contribution of this paper. Our work builds on simpler, previously developed, models [9,18,19,26–31]. Alternatively, the findings or procedures presented in [32], which were carried out for an area in Sweden, can also be used, but their application to the entire surface of the Earth is too challenging and complicated. The nomenclature for all of the variables and abbreviations used is presented at the end of this article. 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 T=f(φ,t), where φ is the angle /deg/ and t is time /s/ [9]. Figure 2. Model of the Earth as an infinite hollow cylinder, the so-called ‘Earthcylinder’ with heating from the Sun (not to scale). 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 [2,32–39]. 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 T,T…,T = f(φ,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). 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 [2,32–39]. 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 T1 , T2. . . , T24 =f(ϕ,t) , which correspond to time-varying temperature values over a two-year period, according to [6,7], as shown in Figure 3. Appl. Sci. 2023,13, 4367 4 of 24 Appl. Sci. 2023, 13, x FOR PEER REVIEW 4 of 25 Figure 3. ‘Earthcylinder’ with materials and temperature loads (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. Figure 3. ‘Earthcylinder’ with materials and temperature loads (not to scale). For the FEM calculation, the problem is treated as a planar problem with an assumed plane strain. In Figures 3and 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 3and 4 . The ‘Earthcylinder’ is a model of a cylinder replacing the earth’s crust with constant inner and outer radii, R1=6348 km and R2=6378 km, as shown in Figures 1–3. Below the crust, i.e., below the radius R1 , is the upper mantle, whose assumed constant temperature is Tp . This temperature is mostly transferred to the inner radius of the crust by thermal conduction, as shown in Figures 3and 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 1and 3[ 9 , 13 – 15 , 19 ]. The elastic foundation is also applied in many engineering problems [24–27]. The outer radius of the model R2 is affected by the temperature T=f(ϕ,t) . This temperature depends on the angle ϕ 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]. Appl. Sci. 2023,13, 4367 5 of 24 Appl. Sci. 2023, 13, x FOR PEER REVIEW 5 of 25 Figure 4. ‘Earthcylinder’ with temperature loads (not to scale). The ‘Earthcylinder’ is a model of a cylinder replacing the earth’s crust with constant inner and outer radii, R= 6348 km and R= 6378 km, as shown in Figures 1–3. Below the crust, i.e., below the radius R, is the upper mantle, whose assumed constant temperature is T. 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 = 107 Nm−3, as shown in Figures 1 and 3 [9,13–15,19]. The elastic foundation is also applied in many engineering problems [24–27]. The outer radius of the model R is affected by the temperature T=f(φ,t) . This temperature depends on the angle φ 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]. The material properties of individual sections of the Earth’s crust, as shown in Figure 3, vary within given ranges and are determined by random combinations: • Young’s modulus E ∈〈3×10; 4.5 × 10〉 Pa; • Poisson number µ ∈ 〈0.30; 0.35〉; • Density (mean value) ρ = 2760 kgm−3; • Conductivity (thermal conductivity coefficient) λ = 3 Wm−1K−1; • Specific heat c = 1100 Jkg−1K−1; • Thermal expansion coefficient α ∈ 〈3.5×10; 4.5 × 10〉 K −1. Figure 4. ‘Earthcylinder’ with temperature loads (not to scale). The material properties of individual sections of the Earth’s crust, as shown in Figure 3, vary within given ranges and are determined by random combinations: •Young’s modulus E ∈3×1010; 4.5 ×1010Pa; •Poisson number µ∈h0.30;0.35i; •Density (mean value) ρ= 2760 kgm−3; •Conductivity (thermal conductivity coefficient) λ= 3 Wm−1K−1; •Specific heat c = 1100 Jkg−1K−1; •Thermal expansion coefficient α∈3.5 ×10−5; 4.5 ×10−5K−1. For the individual sections 1–24 in Figure 3, the material properties are determined according to the ranges mentioned above. The material properties are given in Table 1. The real material properties of the materials contained in the earth’s crust are given in [ 9 , 37 – 41 ]. 2.1. Temperature Loading A constant (time independent) mean temperature Tp = 648 K is considered to be on the inner surface of the cylinder, i.e., the Earthcylinder, at a depth of 30 km in the Earth’s crust, see [9,41]. Appl. Sci. 2023,13, 4367 6 of 24 Table 1. Material properties of individual sections of the planar crustal model of the Earth’s crust. Section Angle φ/deg/ Young’s Modulus E/Pa/ Poisson Number /1/ Density /kg·m−3/ Conductivity /W·m−1·K−1/ Specific Heat /J·K−1·kg−1/ Thermal Expansion /K−1/ 1 345–360 3.00 ×1010 0.30 2760 3 1100 3.50 ×10−5 2 0–15 3.50 ×1010 0.35 2760 3 1100 4.00 ×10−5 3 15–30 4.00 ×1010 0.30 2760 3 1100 4.50 ×10−5 4 30–45 4.50 ×1010 0.35 2760 3 1100 3.50 ×10−5 5 45–60 3.00 ×1010 0.30 2760 3 1100 4.00 ×10−5 6 60–75 3.50 ×1010 0.35 2760 3 1100 4.50 ×10−5 7 75–90 4.00 ×1010 0.30 2760 3 1100 3.50 ×10−5 8 90–105 4.50 ×1010 0.35 2760 3 1100 4.00 ×10−5 9 105–120 3.00 ×1010 0.30 2760 3 1100 4.50 ×10−5 10 120–135 3.50 ×1010 0.35 2760 3 1100 3.50 ×10−5 11 135–150 4.00 ×1010 0.30 2760 3 1100 4.00 ×10−5 12 150–165 4.50 ×1010 0.35 2760 3 1100 4.50 ×10−5 13 165–180 3.00 ×1010 0.30 2760 3 1100 3.50 ×10−5 14 180–195 3.50 ×1010 0.35 2760 3 1100 4.00 ×10−5 15 195–210 4.00 ×1010 0.30 2760 3 1100 4.50 ×10−5 16 210–225 4.50 ×1010 0.35 2760 3 1100 3.50 ×10−5 17 225–240 3.00 ×1010 0.30 2760 3 1100 4.00 ×10−5 18 240–255 3.50 ×1010 0.35 2760 3 1100 4.50 ×10−5 19 255–270 4.00 ×1010 0.30 2760 3 1100 3.50 ×10−5 20 270–285 4.50 ×1010 0.35 2760 3 1100 4.00 ×10−5 21 285–300 3.00 ×1010 0.30 2760 3 1100 4.50 ×10−5 22 300–315 3.50 ×1010 0.35 2760 3 1100 3.50 ×10−5 23 315–330 4.00 ×1010 0.30 2760 3 1100 4.00 ×10−5 24 330–345 4.50 ×1010 0.35 2760 3 1100 4.50 ×10−5 2.1.1. Initial Conditions The initial preheating condition of the model determines the temperature at the beginning of the analysis in MSC.Marc Mentat 2006 software. On the inner surface, the temperature Tp is prescribed, and on the outer surface of the crust, the temperature Tc = 287 K is prescribed. From the mentioned initial temperatures Tp and Tc , the initial temperature distribution T initial ∈Tc;Tp is acquired. The initial temperature is independent of the angle ϕbut dependent on the radius R, as shown in Figure 5. 2.1.2. Boundary Conditions The temperature boundary condition T=f(ϕ,t) (i.e., T1 , T2. . . , T24 =f(ϕ,t) , as shown in Figures 2–4), depends on time t and the angular dimension of the Earth and acts on the outer surface of the crust. This boundary condition simulates the cyclic temperature changes based on the daily cycle of the Earth’s rotation and the annual cycle of the planet’s orbit around the Sun. An example of T2 dependence loading for 2 years and 5 days is shown in Figure 6. Appl. Sci. 2023,13, 4367 7 of 24 Appl. Sci. 2023, 13, x FOR PEER REVIEW 7 of 25 Figure 5. Earthcylinder and its initial temperature distribution Tinitial. 2.1.2. Boundary Conditions The temperature boundary condition T=f(φ,t) (i.e., T,T…,T =f(φ,t) , as shown in Figures 2–4), depends on time t and the angular dimension of the Earth and acts on the outer surface of the crust. This boundary condition simulates the cyclic temperature changes based on the daily cycle of the Earth’s rotation and the annual cycle of the planet’s orbit around the Sun. An example of T dependence loading for 2 years and 5 days is shown in Figure 6. Figure 5. Earthcylinder and its initial temperature distribution Tinitial. For the calculation, surface temperatures from the ALA monitoring system [ 10 , 11 ] (in addition to surface temperature and air temperature) were taken from stations located between latitudes of approximately 40 and 50 degrees north for the period October 2007 to August 2009, at 15 min increments. The basic characteristic values of the statistical data of temperature measurements T1,T2. . . ,T24 =f(ϕ,t)are: •Minimum temperature = 263.65 K, •Mean temperature = 282.20 K, •Median temperature = 282.15 K, •Standard deviation of temperature = 7.68 K, •Maximum temperature = 303.35 K. The data were then processed for importing into the MSC.Marc Mentat 2006 software [ 42 ]. The resulting time-dependent temperature series contained 35,424 values, with 110 real temperature anomalies where the temperature continuity check (large variability between two measurements) was not met. These were mainly large fluctuations in air temperature, which could have been caused by the outbreak of a severe storm, etc. The surface temperature followed the air temperature pattern. In this case, the above-mentioned anomalies occurred only in the spring and summer periods, from 3 April 2008 to 30 August 2008. Of course, these anomalies, which are real in nature, were also used in the solution. Appl. Sci. 2023,13, 4367 8 of 24 Appl. Sci. 2023, 13, x FOR PEER REVIEW 8 of 25 (a) (b) Figure 6. Temperature T distribution (a) Boundary condition—for whole 2 years at node 1; (b) Boundary condition at node 1 and calculated temperature at node 54 for 99th to 105th day of solution. For the calculation, surface temperatures from the ALA monitoring system [10,11] (in addition to surface temperature and air temperature) were taken from stations located between latitudes of approximately 40 and 50 degrees north for the period October 2007 to August 2009, at 15 min increments. The basic characteristic values of the statistical data of temperature measurements T,T…,T =f(φ,t) are: • Minimum temperature = 263.65 K, • Mean temperature = 282.20 K, • Median temperature = 282.15 K, Figure 6. Temperature T2 distribution ( a ) Boundary condition—for whole 2 years at node 1; ( b ) Boundary condition at node 1 and calculated temperature at node 54 for 99th to 105th day of solution. It can, therefore, be concluded that the temperature variability is significantly higher in the rising temperature phase than in the falling temperature phase. A similar phenomenon could be observed in the diurnal time cycle. For this reason, the idea of modelling the temperature function using goniometric functions was abandoned as it would introduce errors (the application of Fourier series will not bring accuracy and is not necessary). The processed data were used for the section ϕ∈h345◦;360i . For all subsequent sections, the entire time series was successively shifted by one hour. For Section 2of Figure 3( ϕ∈h0◦;15i ), the date 1 February 2008 0:00:00 corresponded to the original date Appl. Sci. 2023,13, 4367 9 of 24 of 1 February 2008 01:00:00 and, for each subsequent section, the time series is shifted by another hour, as already mentioned. The different temperature boundary conditions in each section represent a virtual rotation of the model, i.e., the change in temperature simulates the rotation of the Earth around the Sun. The temperature time series, however, had to be reduced for importing into the MSC.Marc Mentat software due to the large amount of data, so that the original time step of 15 min had to be increased to a time step of 3519 s (i.e., 58 min 39 s). This leads to the faster solution by FEM. Since the initial surface temperature of the model is constant around the perimeter and the boundary temperature conditions are angle dependent, it is necessary to let the model settle at the beginning of the analysis. According to the results obtained, the model is considered to be steady after three days of loading, i.e., the values from 4 February 2008 00:48:37 are taken for processing the results. More information regarding the boundary conditions is given in [9]. 2.2. Finite Element Mesh Figure 7shows the finite element mesh used for the calculation. At the transition points between the two sections, i.e., different material properties and surface temperatures, bias subdivision is used with refinement of the mesh towards the outer surface, since larger temperature variations are expected near the outer surface than at greater depths. The FE mesh contains 8973 quadratic elements and 10,116 nodes. Appl. Sci. 2023, 13, x FOR PEER REVIEW 10 of 25 Figure 7. Detail of biased mesh (refinement towards the surface of the Earth). 3. Results From the FE calculation performed on an Earthcylinder, the mechanical stress and displacement distributions were evaluated. The equivalent von Mises mechanical stress (HMH) σ /Pa/ is calculated according to Equation (1) σ =σ σ σ −(σσσσσσ), (1) where σ,, /Pa/ are the principal stresses, see [43]. The total displacement 𝑢/m/ can be expressed by 𝑢=𝑢 𝑢 , (2) where 𝑢 /m/ is the displacement in the X-axis direction, and 𝑢/m/ is the displacement in the Y-axis direction, see Figure 2 or Figure 3. Equations (1) and (2) were applied in the finite element analysis (FEA). 3.1. Results from FEA As for the finite element model, the accuracy was checked by comparison. First, a coarser finite element mesh with a longer time step was solved. Then a finer mesh with a shorter time step was solved, e.g., this finite element id described in Figure 7. There was no significant difference between these solutions and the presented solution can then be declared sufficiently accurate. Because of problems where the analytical solution is not Figure 7. Detail of biased mesh (refinement towards the surface of the Earth). Appl. Sci. 2023,13, 4367 16 of 24 Table 2. Statistical parameters of tangential displacements. Node Angle φ/deg/ Min /m/ Mean /m/ Median /m/ Max /m/ 1 0.0 −22.405145 −4.016917 −4.155867 19.706385 7646 7.5 −13.167980 1.980264 2.203522 12.992962 7094 15.0 −29.331777 7.971764 8.205548 40.070023 51 22.5 −6.667464 1.772484 1.822645 8.890464 3 30.0 −8.410809 0.015862 0.108814 8.880879 53 37.5 −2.733061 −0.369284 −0.444654 2.724747 4 45.0 −12.265832 −2.029589 −2.114735 11.494079 55 52.5 −1.619803 0.019862 0.053834 1.584367 5 60.0 −11.991985 2.225961 2.311249 13.202844 57 67.5 −1.703642 0.148204 0.202752 1.321986 6 75.0 −8.316151 −1.156488 −1.263755 9.357056 59 82.5 −2.202902 0.168749 0.243539 1.710141 7 90.0 −10.964751 1.914396 2.002252 11.641080 61 97.5 −1.717808 −0.088355 −0.117302 1.903794 11491 105.0 −14.292981 −2.386571 −2.480774 13.402903 63 112.5 −2.136792 −0.321966 −0.361097 2.297582 11585 120.0 −6.676715 −0.051041 −0.041540 6.316006 65 127.5 −2.620005 0.196727 0.304310 1.918440 11782 135.0 −10.374590 1.212906 1.450878 8.992166 67 142.5 −3.431526 0.537606 0.601411 3.128304 11925 150.0 −12.217613 2.033221 2.138069 12.652762 69 157.5 −3.368985 −0.578049 −0.604307 3.254372 12044 165.0 −23.284528 −4.365703 −4.469298 18.689545 71 172.5 −2.412931 −0.367422 −0.407727 2.594586 13 180.0 −10.843711 2.072278 2.139710 12.114409 73 187.5 −2.344523 0.396470 0.427631 2.438536 12234 195.0 −9.007515 0.961226 1.080069 7.542674 9225 202.5 −2.045828 0.180401 0.245550 1.597627 12424 210.0 −6.245059 −0.379791 −0.502522 7.471727 9372 217.5 −3.101930 −0.514486 −0.545275 3.085332 12500 225.0 −12.566509 −2.093027 −2.173286 11.838935 9471 232.5 −2.061168 0.018279 0.060519 1.922392 12599 240.0 −13.427477 2.207517 2.396578 13.803798 9594 247.5 −2.078801 0.147412 0.237499 1.551424 12674 255.0 −8.193241 −1.127393 −1.232754 8.960823 9736 262.5 −3.634378 0.168002 0.216868 2.734815 19 270.0 −18.844958 1.926130 1.995067 16.550423 9910 277.5 −2.515986 −0.117068 −0.148072 2.376055 10004 285.0 −14.114205 −2.368588 −2.561178 14.466366 12796 292.5 −2.534931 −0.399108 −0.459958 2.579560 12919 300.0 −6.537996 −0.083045 −0.093447 6.221886 Appl. Sci. 2023,13, 4367 17 of 24 Table 2. Cont. Node Angle φ/deg/ Min /m/ Mean /m/ Median /m/ Max /m/ 13017 307.5 −2.130202 0.200543 0.266355 1.685076 13116 315.0 −9.207633 1.157171 1.258941 8.282640 13259 322.5 −3.271788 0.624200 0.663308 3.667678 13384 330.0 −12.221221 2.082060 2.173141 12.821112 13569 337.5 −2.657303 −0.430821 −0.459640 2.697734 10258 345.0 −23.800367 −4.473265 −4.585868 18.998453 10135 352.5 −8.195234 −1.552937 −1.599643 6.756131 TOTAL −29.331777 0.063933 0.055363 40.070023 Table 3. Statistical parameters of radial displacements. Node Angle φ/deg/ Min /m/ Mean /m/ Median /m/ Max /m/ 1 0.0 −0.247709 0.045662 0.048070 0.264486 7646 7.5 −1.069984 −0.220166 −0.225239 0.744611 7094 15.0 −0.225554 0.050253 0.052106 0.273853 51 22.5 −2.949429 −0.597693 −0.609791 2.074209 3 30.0 −2.840615 −0.575473 −0.589606 1.996382 53 37.5 −2.451921 −0.498885 −0.509240 1.719541 4 45.0 −2.948489 −0.593957 −0.604933 2.117883 55 52.5 −3.122978 −0.632979 −0.646084 2.204687 5 60.0 −3.688951 −0.746602 −0.762182 2.596254 57 67.5 −3.284413 −0.665639 −0.679903 2.317962 6 75.0 −3.600448 −0.728494 −0.744516 2.538881 59 82.5 −2.910104 −0.590158 −0.600756 2.054802 7 90.0 −3.126893 −0.632771 −0.646967 2.203372 61 97.5 −3.260287 −0.660954 −0.672389 2.301363 11491 105.0 −3.269007 −0.658896 −0.672805 2.344291 63 112.5 −3.078366 −0.623752 −0.635934 2.171800 11585 120.0 −3.613587 −0.732255 −0.748583 2.538011 65 127.5 −3.173468 −0.643556 −0.659318 2.232398 11782 135.0 −3.171930 −0.642370 −0.657984 2.232703 67 142.5 −3.036209 −0.616722 −0.629598 2.138323 11925 150.0 −3.896294 −0.789001 −0.806090 2.740027 69 157.5 −3.991671 −0.808510 −0.825623 2.816905 12044 165.0 −3.199840 −0.645226 −0.658717 2.294331 71 172.5 −2.829153 −0.573808 −0.584828 1.996196 13 180.0 −2.850070 −0.577022 −0.589402 2.007131 73 187.5 −3.290911 −0.667605 −0.679138 2.323781 12234 195.0 −3.451112 −0.699088 −0.712300 2.430043 Appl. Sci. 2023,13, 4367 18 of 24 Table 3. Cont. Node Angle φ/deg/ Min /m/ Mean /m/ Median /m/ Max /m/ 9225 202.5 −3.446525 −0.698456 −0.712473 2.433441 12424 210.0 −3.481797 −0.705072 −0.719768 2.452611 9372 217.5 −3.153902 −0.639660 −0.651908 2.224569 12500 225.0 −2.889573 −0.583077 −0.595777 2.063464 9471 232.5 −3.225099 −0.653850 −0.670153 2.270460 12599 240.0 −3.643746 −0.739617 −0.755094 2.553612 9594 247.5 −3.900572 −0.792134 −0.808253 2.746832 12674 255.0 −3.421864 −0.694008 −0.705438 2.417951 9736 262.5 −2.969488 −0.607632 −0.618616 2.114656 19 270.0 −3.264306 −0.662024 −0.676559 2.271688 9910 277.5 −3.891178 −0.791065 −0.807631 2.689200 10004 285.0 −3.323049 −0.665437 −0.681295 2.371287 12796 292.5 −3.731120 −0.756405 −0.773357 2.623065 12919 300.0 −3.443254 −0.697380 −0.711933 2.418230 13017 307.5 −3.195928 −0.648019 −0.661250 2.255990 13116 315.0 −3.107899 −0.628968 2.190442 2.190442 13259 322.5 −3.210917 −0.650901 −0.664147 2.267816 13384 330.0 −3.473598 −0.702698 −0.716477 2.446307 13569 337.5 −3.841410 −0.777634 −0.794236 2.709838 10258 345.0 −1.664193 −0.333632 −0.341247 1.236957 10135 352.5 −1.383713 −0.281419 −0.288036 0.969561 TOTAL −3.991671 −0.613224 −0.565425 2.816905 The findings in Table 3show that the displacement at the Earth’s surface can be as large as h−3.992;2.817i m for radial displacement over two years. This is interesting and, again, is not inconsistent with crustal motion. The above-mentioned displacements are usually quasi-static but, in extreme situations associated with earthquakes, they can also occur in a dynamic manner. The histogram of the resulting radial and tangential displacements (stochastic evaluation of tangential and radial displacement) is shown in Figure 17. A large number of different relationships can be used to estimate the error of the Mote Carlo method. For example, consistent with the literature [ 45 , 46 ], the error of the Monte Carlo method can be approximately estimated by the relation: errorMonteCarlo =StDeviation pSteps , (3) where ‘StDeviation’ is the standard deviation and ‘Steps’ is number of Monte Carlo simulations. Appl. Sci. 2023,13, 4367 19 of 24 1 Figure 16. Histograms of the tangential component of the displacement at nodes on the Earth’s outer surface induced by thermal radiation from the Sun. From the data in Figure 17a, Equation (3) gives 3.90418181 √2000000 = 0.00276 and, from the data in Figure 17b, Equation (3) gives 1.05741154 √2000000 = 0.00075. The error is too small, and the stochastic approach is reliable. Appl. Sci. 2023,13, 4367 20 of 24 Appl. Sci. 2023, 13, x FOR PEER REVIEW 20 of 25 Radial displacements, which are not presented in this article, can also be evaluated in a similar way [9]. The findings of Table 2 show that the thermally induced displacements on the Earth’s surface can be as large as the interval 〈−29.332; 40.070〉 m for tangential displacement over two years, an interesting finding that is not generally inconsistent with crustal motion. Some places on Earth move less and some move more, e.g., in areas of tectonic plate contact or faulting. The findings in Table 3 show that the displacement at the Earth’s surface can be as large as 〈−3.992;2.817〉 m for radial displacement over two years. This is interesting and, again, is not inconsistent with crustal motion. The above-mentioned displacements are usually quasi-static but, in extreme situations associated with earthquakes, they can also occur in a dynamic manner. The histogram of the resulting radial and tangential displacements (stochastic evaluation of tangential and radial displacement) is shown in Figure 17. (a) (b) Figure 17. Histograms and distribution functions of (a) tangential component of total crustal surface displacement, (b) radial component of total crustal surface displacement (sw MSC.Marc Mentat and Anthill). A large number of different relationships can be used to estimate the error of the Mote Carlo method. For example, consistent with the literature [45,46], the error of the Monte Carlo method can be approximately estimated by the relation: error  =StDeviation  Steps , (3) where ‘StDeviation’ is the standard deviation and ‘Steps’ is number of Monte Carlo simulations. From the data in Figure 17a, Equation (3) gives . √ = 0.00276 and, from the data in Figure 17b, Equation (3) gives . √ = 0.00075. The error is too small, and the stochastic approach is reliable. Figure 17. Histograms and distribution functions of ( a ) tangential component of total crustal surface displacement, ( b ) radial component of total crustal surface displacement (sw MSC.Marc Mentat and Anthill). 4. Discussion An initial new model of the Earth’s heat loading has been developed. Its advantage is the simplicity of the plane strain problem and the original combination of the finite element method and the probabilistic Monte Carlo Method. FEA is performed for the non-stationary and non-linear coupled geomechanical problem (i.e., temperature task and structural task). A simple measure of the material Inhomogeneity of the Earth’s crust (partitioned into 24 materials) was respected in a simple way. The interaction between the crust and the upper mantle is replaced by the bilateral Winkler elastic foundation boundary condition. This results in a simplified model, which respects many random (pseudo-random) inputs, i.e., 2 × 10 6 Monte Carlo simulations. Nonlinear geomechanics tasks, in the case of large deformation caused by Sun heating, were solved. From the stochastic modelling, the tangential and radial displacements of the Earth’s surface and crustal stresses are obtained, and their values can initiate the possible fractures of the Earth’s crust connected with its tectonics. The model is considered to be continuous and, therefore, does not allow for the formation of cracks in the Earth’s crust. However, the obtained stresses and displacements clearly confirm sufficient capacity to induce tectonic changes in the Earth’s surface. The periodic and inhomogeneous thermal radiation from the Sun acting on the Earth’s surface is sufficient to induce these tectonic changes. This is the important finding for understanding the complexity of the Earth’s crust behavior. It can be said that tectonic changes in the Earth’s crust are also caused by thermal radiation from the Sun (external heating of the Earth), which is still a new idea. Our results are not inconsistent with earlier foreign works [26,28,29] or our works [9,15,18,19,30,31]. As is well known, tectonic changes in the Earth’s crust are also due to tidal effects from the gravitational pull of the Sun and Moon, centrifugal forces from the Earth’s rotation, Appl. Sci. 2023,13, 4367 21 of 24 and internal magmatic processes within the Earth. The influence of tidal, centrifugal, and internal magmatic effects is not addressed in this paper. In the future, the dynamic and cyclic loading of the Earth’s crust from tidal effects and magmatic processes can also be considered, or a more complex spatial (3D) model can be developed that could also account for the other geological and geomechanical phenomena mentioned above. In the future, by applying mechanical contacts between the Earth’s plates, it will be possible to deal with tectonic phenomena on our own or foreign planets in a similar but more complex and complicated way. TheimportantinfluenceofthermalradiationontheEarth’stectonicsarenewfindings,proved by our work and by the different models, but consistent with the literature [9,15,18,19,26,28–31]. 5. Conclusions Using a simple model for the heating of the Earth from a heat source, i.e., the Sun, over a two-year period, the displacement fields and equivalent mechanical stresses (von Mises) within the Earth’s crust were obtained. The Sun causes relatively large displacements in the Earth’s crust during annual periods (and, to a lesser extent, over a daily period), reaching up to 40.07 m in the tangential direction and 3.99 m in the radial direction at the crustal surface. However, it is a slow process that is almost imperceptible to humans, compared to earthquakes, where the crustal displacement is visible to the naked eye. The numerical analysis further revealed stresses that reached up to 52 MPa. These stresses, from temperature alone, are normal for the Earth and are not very damaging to it but can cause tectonic movements. However, in combination with other geomechanical phenomena, e.g., tidal effects, the Earth’s rotation, or convection currents within the Earth, etc., it can cause faults in the Earth’s crust. Our findings, with respect to the important influence of solar thermal heating on the Earth’s tectonics, are new as well as proven by different models, but consistent with the chosen references. We have proved that the Sun can ‘move’ the Earth’s crust. Author Contributions: Methodology, K.F.; Formal analysis, I.W.; Writing – original draft, D. ˇ C., K.F. and I.W. All authors have read and agreed to the published version of the manuscript. Funding: This article was supported by Specific Reasearch SP2023/027 “Application of Modern Computational and Experimental Approaches in Applied Mechanics”; Specific Research SP2022/26 “Computational and experimental modeling in the tasks of applied mechanics and biomechanics” and by international projects CZ.02.1.01/0.0/0.0/17_049/0008441 “Innovative Therapeutic Methods of Musculoskeletal System in Accident Surgery” and CZ.02.1.01/0.0/0.0/17_049/0008407 “Innovative and additive manufacturing technology—new technological solutions for 3D printing of metals and composite materials” within the Operational Programme Research, Development and Education financed by the European Union and from the state budget of the Czech Republic. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Conflicts of Interest: The authors declare no conflict of interest. Nomenclature Name Physical Unit Explanation ALA Monitoring system for measuring c Jkg−1K−1Specific heat of Earth’s crust E Pa Young’s modulus of Earth’s crust Earthcylinder Simplified model of geoid of Earth errorMonteCarlo 1 Possible error of Monte Carlo Method Appl. Sci. 2023,13, 4367 22 of 24 Name Physical Unit Explanation f Function FE Finite element FEA Finite Element Analysis FEM Finite Element Method K Nm−3Modulus of the foundation material1, material2, . . . Name of materials in Earth’s crust material24 Max Maximum value Mean Mean value Median Median value Min Minimum MSC.Marc Mentat Finite element software Node Node of finite element mesh R m Radius of Earth’s crust R1m Inner radius of Earth’s crust R2m Outer radius of Earth’s crust StDeviation Standard deviation Steps Number of Monte Carlo simulations Stress HMH Equivalent von Mises stress T K Temperature T1, T2, . . . T24 K Temperature loading on section 1, 2, . . . , 24 of Earth’s crust, see Figure 3 TcK Initial temperature on inner radius of Earth’s crust, see Figure 5 Tinitial K Initial temperature of Earth0s crust from interval Tc;Tp TpK Temperature on inner radius of Earth’s crust t s Time um Total displacement in point of Earth’s crust ut1 m Tangential component of crustal surface displacement for node 1 ut7094 m Tangential component of crustal surface displacement for node 7094 ut12044 m Tangential component of crustal surface displacement for node 12044 utTot m Tangential component of total crustal surface displacement ur1 m Radial component of crustal surface displacement for node 1 ur7094 m Radial component of crustal surface displacement for node 7094 ur12044 m Radial component of crustal surface displacement for node 12044 urTot m Radial component of total crustal surface displacement uxm Displacement in X axis direction uym Displacement in Y axis direction X m Axis of coordinate system Y m Axis of coordinate system Z m Axis of coordinate system αK−1Thermal expansion coefficient of Earth’s crust ϕdeg Angle of Earth’s crust λWm−1K−1Conductivity of Earth’s crust µ1 Poisson number of Earth’s crust ρKgm−3Density of Earth’s crust σHMH Pa Equivalent von Mises Stress σ1,σ2,σ3Pa Principal stress Appl. 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