# Hygro-Thermo-Mechanical Analysis of Brick Masonry Walls Subjected to Environmental Actions
## 2 Hygro-Thermal Model
### Hygro-Thermal Coupling
#### 2.3.2 Temperature Dependency of Hygric Parameters  
The coefficient of thermal expansion of most building materials lies in the order of \(10^{-5}\) to \(10^{-6}\)\({}^{\circ}\)C\({}^{-1}\) for normal temperatures. As a result, any volumetric change induced by temperature variations within the normal temperature range falls outside ordinary measurement capabilities [44]. Thus, it is generally accepted that open porosity, \(\phi_{0}\) [-], and bulk density, \(\rho_{bulk}\) [kg/m\({}^{3}\)], can be considered as constant for the range of temperatures of interest in this study [45]. Moreover, once the volumetric variation of the porous solid matrix is disregarded, the influence of temperature on the saturation moisture content, \(w_{sat}\) [kg/m\({}^{3}\)], stems solely from the water density, which has a weak dependency on temperature with less than a 1.20% decline between 0 \({}^{\circ}\)C and 50 \({}^{\circ}\)C [46]. As a result, \(w_{sat}\) [kg/m\({}^{3}\)] can be considered as a constant independent of temperature. Here, an analogous logic is followed for the definition of the capillary moisture content, \(w_{cap}\) [kg/m\({}^{3}\)], which is considered constant. However, it must be noted that a widely accepted theory to describe the relationship between \(w_{cap}\) and temperature is still lacking in the literature. Apart from the saturation state boundaries, \(w_{cap}\) and \(w_{sat}\), the equilibrium moisture content of unsaturated porous materials is dependent on the temperature. The thermal dependency of moisture storage implies a shift in the moisture storage function (sorption isotherm or water retention curve). This behavior is expected since higher temperatures are associated with higher energy levels, which cause faster transport and release of water molecules [47]. For this reason, both adsorption and desorption curves corresponding to higher temperatures lie below the 'colder' isotherms [48]. The thermal dependency of moisture storage can be expressed analytically through temperature-dependent fitting parameters. As an example, [PERSON] et al. [49] proposed the following expression:  
\[w(\varphi)=w_{cap},\frac{\psi(T)-1}{\psi(T)-\varphi}\cdot\varphi, \tag{9}\]  
where \(\psi(T)\) [-] is a fitting parameter dependent on temperature. It is noted that \(\psi\) is inversely proportional to the temperature. The correlation between \(\psi\) and \(T\) must be established from experimental data. However, the influence of temperature on moisture storage has not been sufficiently studied and experimental data are scarce. The available information suggests a negligible temperature dependency for barely hygroscopic materials, such as fired-clay brick [44]. For the sake of simplicity, the thermal effects on moisture storage are not considered in the simulations presented in this paper. The influence of temperature on vapor diffusion is partly considered through the water vapor permeability of still air, \(\delta_{a}\) [kg/(m\(\cdot\)s\(\cdot\)Pa)], defined empirically as [50]:  
\[\delta_{a}=\frac{2.31\cdot 10^{-5}}{R_{v}\cdot T}\frac{p_{0}}{p}\bigg{(}\frac{T}{273.15 }\bigg{)}^{1.81}, \tag{10}\]  
where \(p_{0}=101\),325 Pa is the standard atmospheric pressure, and \(p\) [Pa] is the ambient barometric pressure. Additionally, the saturation vapor pressure, \(p_{v,sat}\) [Pa], is a non-linear function of temperature defined empirically as [51; 52]:  
\[p_{v,sat}=610.7[\text{Pa}]\cdot 10^{7.5(\frac{T-273.15}{T-30.85})}. \tag{11}\]  
The impact of temperature on the liquid transport term has been studied by different authors: in particular, its effect on the water absorption coefficient, \(A_{w}\) [kg/(m\({}^{2}\cdot\)s\({}^{0.5}\))], or sorptivity, \(S=A_{w}/\rho_{w}\) [m/s\({}^{0.5}\)], e.g., [53; 54; 55; 56; 54].