# Hygro-Thermo-Mechanical Analysis of Brick Masonry Walls Subjected to Environmental Actions
## 4 Hygro-Thermo-Mechanical Model
### Coupling between Heat, Moisture, and Mechanical Fields  
The mechanical problem is linked to the thermal and moisture fields through the extended Duhamel-Neumann equation or total strain additive decomposition [65]; the total strain can be divided into different components: namely, mechanical-, thermal-, and moisture-induced strains. This is based on the assumption of infinitesimal strains and small deformations and rotations, which is generally applicable to masonry structures [24]:  
\[\varepsilon=\varepsilon_{elastic}+\varepsilon_{plastic}+\varepsilon_{thermal}+ \varepsilon_{swelling} \tag{22}\]  
The mechanical strains \(\varepsilon_{elastic}\) and \(\varepsilon_{plastic}\) account for the effects caused by forces and moments, either in the elastic or inelastic ranges. On the other hand, the thermal and hygric strains are related to deformations caused by temperature and moisture variations, respectively. The temperature-induced strain for an isotropic material is defined as:  
\[\varepsilon_{thermal}=\alpha_{T}\Delta T=\alpha_{T}\Big{(}T-T_{ref}\Big{)} \tag{23}\]  
where \(\alpha_{T}\) [1/K] is the coefficient of thermal expansion of the material, and \(T_{ref}\) [K] is the reference temperature at which the thermally induced strain is zero. An analogous relation may be written for the moisture-induced strain:  
\[\varepsilon_{swelling}=\alpha_{h}\Delta w_{g}=\alpha_{h}\Big{(}w_{g}-w_{g,ref }\Big{)} \tag{24}\]  
where \(\alpha_{h}\) [-] is the moisture swelling coefficient of the material, \(w_{g}\) [kg/kg] is the gravimetric moisture content, and \(w_{g,ref}\) [kg/kg] is the reference moisture content for which the moisture-induced strain is zero. Note that the moisture content may also appear as mass of absorbed water per volume of dry material, \(w\) [kg/m\({}^{3}\)]. In that case, if \(\alpha_{h}\) is initially given in terms of gravimetric moisture content (dimensionless), it must be divided by the density of the material, \(\rho_{bulk}\) [kg/m\({}^{3}\)], to maintain the consistency of the units. Since the moisture content is dependent on the relative humidity by means of the moisture storage function, the moisture-induced strain is also a non-linear function of the relative humidity. It is noted that the hygro-mechanical and thermo-mechanical coupling processes defined in this way involve a two-step procedure. First, the temperature and moisture profiles are obtained by means of the hygro-thermal analysis. Subsequently, the temperature and moisture results are used as input for the mechanical problem. Consequently, the proposed model consists of a one-way or unidirectional type of coupling, that is, heat and moisture fields have an impact on each other, and the hygro-thermal results are linked to the mechanical response, but the effects of the structural response on the temperature and moisture fields are not considered. It must be mentioned that the development of stresses and strains within a porous material may affect the hygric and thermal responses. For instance, [PERSON] et al. [66] studied the so-called mechano-sorptive behavior of porous materials and concluded that the amount of moisture adsorbed by the material reduces as the specimens are subjected to higher compressive stresses. Additionally, these authors observed that the stress effect on the sorption behavior is more pronounced for higher relative humidity values. However, these mechanically induced effects are expected to be of minor importance for the stress limits considered in this work [23]. Thus, a schematic diagram of the proposed HTM model is provided in Figure 9.  
A direct influence of thermal variations on the mechanical properties of masonry materials can be considered negligible for the temperature range of interest in this study. On the contrary, the mechanical performance of porous materials is generally affected by moisture, which may lead to a reduction of the mechanical properties [16].