# Hygro-Thermo-Mechanical Analysis of Brick Masonry Walls Subjected to Environmental Actions
## 3 Hygro-Thermal Simulations and Discussion
### Hygro-Thermal Behavior of a Brick Masonry Wall
#### 3.2.3 Time-Dependent Analysis with Variable External Boundary Conditions (TD2)  
For the second group of time-dependent analyses, labeled TD2, the steady-state equilibrium (resulting from the SS analysis or the final stage of TD1) was taken as the initial condition of the wall. Moreover, the sinusoidal expressions for variable temperature and relative humidity described in Equations (20) and (21) were adopted so that the effect of time-dependent boundary conditions could be studied. The results of the time-dependent analysis with variable boundary conditions, TD2, are shown in Figures 7 and 8, in terms of temperature and relative humidity variations, respectively. Note that the corresponding graphs show the imposed boundary conditions together with the response obtained for different points on the external and internal faces of the wall. When considering the thermal response (see Figure 7), there are some discrepancies between the wall with lime mortar (LM) and the one with cement mortar (CM). In particular, the thermal properties of bricks and lime mortar are rather similar; thus, the obtained trends show comparable temperature values for the selected points on each side of the wall. Conversely, for the CM case, the results show different temperatures for the points located on mortar joints and brick surfaces. Moreover, the thermal excursion on the external side of the wall is similar in both cases. However, the temperature variation range for the CM case is slightly greater on the internal face. This can be explained by the higher conductivity of cement mortar. It is noted that the temperature on the outer surface shows a phase shift with respect to the external input. There is an additional shift between the external and the internal faces of the wall. In particular, the maximum and minimum temperature values are detected on the internal surface around 9-10 h later with respect to the peak values on the outer surface. The external/internal delay is shorter for the CM case, which can be related to the higher thermal conductivity of the cement mortar mentioned above. Concerning the moisture response (see Figure 8), there are considerable differences between the two studied cases. In particular, the relative humidity values obtained for the LM case show only small disparities between the external points on mortar joints and brick surfaces. On the contrary, for the CM case, the results show significant differences between the relative humidity values obtained for the different points depending on their location. Furthermore, the relative humidity variation on the external side of the wall is greater for the cement mortar joints. In this context, the slower moisture response of the cement mortar (lower liquid diffusivity and water vapor permeability) can explain the greater and more shifted relative humidity excursion obtained for the mortar joints on the exterior surface. On the internal face, the CM case also shows a wider relative humidity range. It must be noted that for any of the cases, the moisture response obtained for the internal face is directly related to the temperature variation, which affects the relative humidity through the saturation vapor pressure, such as \(\varphi=p_{v}/p_{v,sat}(T)\). There is a significant delay between the external boundary condition and the moisture evolution on the external surface. It can be observed that the shift between the imposed conditions and the response of the external face of the wall is greater for moisture than for temperature. This was expected since mass transport is a slower process. Finally, there is an additional shift between the external and the internal faces of the wall. The external/internal shift is shorter for the CM case due to the shorter temperature delay mentioned before. Additionally, a parametric study was performed for the time-dependent analysis TD2 in order to evaluate the role of the brick-mortar interface on the moisture response. Consequently, the same analysis was performed using higher and lower values of the hydraulic resistance.