# On the Thermal Stresses Due to Weathering in Natural Stones
## 3 Results and Discussion
### Stresses Due to Daily Variation of Temperature  
The current temperature at each point inside the slab was calculated using the Fourier law, and these differences in temperature were used to evaluate the thermal stresses according to the procedure described in Section 2.4. During the heating stage, compressive stresses were acting at the boundaries, while the tensile stresses were rose in the cooling stages, as presented in Figure 4. The major absolute values of stresses were achieved between the sunrise and the sunset, as already expected. However, the peak of the thermal stresses was developed around 14:00 on 10 August, and not at 11:15, when the highest temperature was measured. The highest stresses occurred due to the rapid decrease in temperature on the external face, which corroborates the study presented by [PERSON] et al. (2020), in which the parametric analysis showed that transient heat flux can cause more damage than just considering the values of thermal variation in steady-state conditions. \begin{table}
\begin{tabular}{c l c c c} \hline \hline
**Symbol** & **Parameter** & **Value** & **Unit** & **Ref.** \\ \hline \(k\) & Thermal conductivity & 2.9 & \(\mathrm{W/m\cdot K}\) & [41] \\ \(\varepsilon\) & Emissivity & 0.95 & - & [42, 43] \\ \(c\) & Specific heat capacity & 870 & \(\mathrm{J/kg\cdot K}\) & [43] \\ \(\rho\) & Density & 2785 & \(\mathrm{kg/m^{3}}\) & [43] \\ \(\alpha_{s}\) & Solar absorptivity & 0.44 & - & [43] \\ \(\alpha\) & Thermal expansion coefficient & \(5.9\cdot 10^{-6}\) & \({}^{\circ}\mathrm{C}^{-1}\) & [40] \\ \(E\) & Young modulus & 52.4 & GPa & [40] \\ \(\
u\) & Poisson’s ratio & 0.16 & - & [40] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Properties of the Carrara marble used in this study. \begin{table}
\begin{tabular}{c l c c c} \hline \hline
**Symbol** & **Parameter** & **Value** & **Unit** & **Ref.** \\ \hline \(B\) & Building plan width & 10 & m & \\ \(L_{b}\) & Building plan length & 15 & m & \\ \(\psi\) & Surface azimuth & 194.28 & \({}^{\circ}\) & \\ \(L\) & Thickness & 0.03 & m & \\ \(DT\) & Daylight savings time & 1 & - & \\ \(UTC\) & Universal coordinated time & 1 & - & \\ \(\tau_{b}\) & Beam optical depths & 0.494 & - & [32] \\ \(\tau_{d}\) & Diffuse optical depths & 1.935 & - & [32] \\ \(h_{o}\) & Heat-transfer coefficient & 22 & \(\mathrm{W/m^{2}\cdot K}\) & \\ \(h_{i}\) & Heat-transfer coefficient & 3 & \(\mathrm{W/m^{2}\cdot K}\) & \\  & (inside) & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Properties of Pescara’s Justice Court used in this study. The physical meaning of the highest stress due to the rapid decrease in temperature can be understood by the non-gradual deformation from the external to the internal surface of the slab. Internal stresses are induced to counterpoise these unbalanced deformations in order to maintain the integrity of the body. In this case, the differential strains produced tensile stresses at the boundaries and compressive stresses in the inner part of the slab, as indicated in Figure 5.  
The parabolic distribution of stress is in line with the increase in porosity along the width, which was verified in naturally weathered marble slabs via the total optical porosity method proposed by [PERSON] et al. [44], in which one can verify that the decohesion of grains (or the increase in porosity) is greater at the boundaries than in the inner part of the slab. Moreover, the faster changes in temperature during the winter and the random orientation of the calcite grains (hypothesis not considered in this formulation) could help to explain the higher decohesion between calcite grains on the exposed surface rather than the other one as verified by those authors. Figure 4: Variation of thermal stresses along the width on 10 August 2007: (**a**) Axis orientation where the 0.000 position represents the internal (hidden) surface and the 0.030 position represents the external (exposed) surface; (**b**) distribution of thermal stress throughout the cycle.