## 3 Numerical Examples
### Crack in a Drying Wall
If \(\alpha_{VG}=2\times 10^{-5}\frac{1}{\text{P}_{\text{a}}}\) and \(n_{VG}=2.4\) for brick and \(\alpha_{VG}=1\times 10^{-6}\frac{1}{\text{P}_{\text{a}}}\) and \(n_{VG}=1.4\) for plaster are considered (Equation (5)), a good agreement is found with recommended surfaces of [22] Figure 13. \begin{table}
\begin{tabular}{c c c c c c} \hline  & **Property** & **Dimension** & **Brick** & **Mortar** & **Plaster** \\ \hline \multirow{3}{*}{\begin{tabular}{c} Mechanical \\ parameters \\ \end{tabular} } & [PERSON]’s modulus (\(E\)) & GPa & 16.6 & 2.37 & 10 \\  & [PERSON]’s ratio (\(v\)) & & 0.15 & 0.2 & 0.15 \\  & Thermal expansion coefficient (\(\alpha\)) & \(1/\text{K}\) & \(5\times 10^{-5}\) & \(1.64\times 10^{-5}\) & \(1.04\times 10^{-5}\) \\ \hline \multirow{3}{*}{\begin{tabular}{c} Linear cohesive \\ crack \\ \end{tabular} } & Tensile Strength (\(f_{t}\)) & kPa & \(2\times 10^{6}\) & \(1.6\times 10^{6}\) & \(1\times 10^{6}\) \\  & Fracture Energy (\(G_{f}\)) & N/m & & 95 & \\ \hline \multirow{3}{*}{\begin{tabular}{c} Initial Condition \\ \end{tabular} } & Initial saturation degree (\(S_{u0}\)) & - & & 1 & \\  & Initial Temperature (\(T_{0}\)) & \({}^{\circ}\)C & & 10 & \\ \hline \multirow{6}{*}{\
\begin{tabular}{c} Hydraulic \\ parameters \\ \end{tabular} } & Residual saturation degree (\(S_{u\
u}\)) & - & & 0 & \\  & \(\alpha_{VG}\) & \(1/\text{Pa}\) & \(2\times 10^{-5}\) & \(1\times 10^{-6}\) & \(1\times 10^{-6}\) \\  & \(n_{VG}\) & - & 2.4 & 1.4 & 1.4 \\ \cline{1-1}  & \(d_{s}\) & \(1/^{\circ}\)C & & \(-1\times 10^{-5}\) & \\ \cline{1-1}  & Intrinsic permeability (\(k_{u0}\)) & m/s & & \(1\times 10^{-12}\) & \\ \cline{1-1}  & \(\alpha_{w}\) & - & & 6 & \\ \cline{1-1}  & Maximum permeability (\(k_{w}^{max}\)) & m/s & & \(1\times 10^{-15}\) & \\ \hline \multirow{6}{*}{
\begin{tabular}{c} Thermal parameters \\ \end{tabular} } & Solid heat diffusion (\(\lambda_{s}\)) & J/mS\({}^{\circ}\)C & & 3.6 & \\ \cline{1-1}  & Water heat diffusion (\(\lambda_{w}\)) & J/mS\({}^{\circ}\)C & & 0.6 & \\ \cline{1-1}  & Vapor heat Diffusion (\(\lambda_{v}\)) & J/mS\({}^{\circ}\)C & & \(2.58\times 10^{-2}\) & \\ \cline{1-1}  & Latent heat of vaporization (\(h_{fg}\)) & J/kg & & \(2.4\times 10^{6}\) & \\ \cline{1-1}  & Tortuosity factor (\(\alpha_{vnp}\)) & - & & 0.1 & \\ \hline \end{tabular}
\end{table} 
Table 2: Material properties for Problem 2 of plaster on the wall. At the first step of the simulation, the moisture reduces quickly in the surface layer of the plaster. To make up for the moisture reduction in the surface layer, moisture flows from bricks toward the plaster. Since the air entry pressure is much more in the plaster than in the bricks, as is shown in Figure 14c, the degree of saturation reduces more in the bricks compared to the plaster at their interface. As the cavity behind the plaster hinders the absorption of moisture by the plaster, the shrinkage of the plaster will not be uniform, and the plaster bulges to the cavity. The maximum stress that develops in the inner surface of plaster in place of the cavity causes nucleation of fracture in the plaster. With the continuation of drying, the crack progresses and bisects the plaster in the initial steps of simulation. The increase in temperature along the crack over time (Figure 15a) leads to the drying of plaster in the neighborhood of the crack, followed by the increase in suction around the crack (Figure 15b) that prevents excessive changes of opening along the crack due to the drying (Figure 15c). In other words, as a result of the increasing temperature, the degree of saturation decreases (Figure 14c) with increasing suction (Figure 14b) that prevents the crack from further propagation. This process limits an excessive change of opening along the crack (Figure 15c). The deformation and distribution of suction, saturation, and wall temperature two hours after the beginning of the analysis are shown in Figure 14. As the crack opens, suction increases around the crack (Figure 14b) to prevent further fracture opening.