Hygro-thermo-mechanical analysis of brick masonry walls subjected to environmental actions

Citation: Ramirez, R.; Ghiassi, B.; Pineda, P.; Lourenço, P.B. Hygro-Thermo-Mechanical Analysis of Brick Masonry Walls Subjected to Environmental Actions. Appl. Sci. 2023,13, 4514. https://doi.org/ 10.3390/app13074514 Academic Editors: Angelo Luongo and Simona Di Nino Received: 5 March 2023 Revised: 29 March 2023 Accepted: 29 March 2023 Published: 2 April 2023 Copyright: © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). applied sciences Article Hygro-Thermo-Mechanical Analysis of Brick Masonry Walls Subjected to Environmental Actions Rafael Ramirez 1, Bahman Ghiassi 2, Paloma Pineda 3and Paulo B. Lourenço 1,* 1Department of Civil Engineering, ISISE, ARISE, University of Minho, Campus de Azurem s/n, 4800-085 Guimaraes, Portugal 2 Department of Civil Engineering, School of Engineering, University of Birmingham, Birmingham B15 2TT, UK 3Department of Building Structures and Geotechnical Engineering, University of Seville, Avenida Reina Mercedes 2, 41012 Seville, Spain *Correspondence: [email protected] Featured Application: This paper presents a numerical methodology for the hygro-thermo- mechanical analysis of masonry structures. The proposed approach can be used to evaluate the hygro-thermal performance of masonry walls as well as to complement the structural assessment considering other static or dynamic loads. The methodology can be further extended to include durability and aging aspects, and to study the compatibility between masonry units and mortars. Abstract: Masonry walls comprise an important part of the building envelope and, thus, are exposed to environmental effects such as temperature and moisture variations. However, structural assessment usually neglects the influence of these hygro-thermal loads and assumes ideal conditions. This paper presents a hygro-thermo-mechanical model and its application to simulate the impact of temperature- and moisture-related phenomena on the structural behavior of masonry walls. A fully coupled heat and mass transfer model is presented and a 2D finite element model is prepared to simulate the behavior of a brick masonry wall under various hygro-thermal scenarios. Two different mortars are considered: namely, cement mortar and natural hydraulic lime mortar. The results are evaluated in terms of temperature and moisture content distribution across the wall thickness. The hygrothermal model is further extended to incorporate mechanical effects through the total strain additive decomposition principle. It is shown that the hygro-thermo-mechanical response of the brick masonry wall is a complex 2D phenomenon. Moreover, the environmental loads change the natural stress distribution caused by gravitational loads alone. Finally, the wall with cement mortar develops higher levels of stress when compared to the one with lime mortar, due to the dissimilar hygro-thermal behavior between the constituent materials. Keywords: masonry; fired-clay brick; cement mortar; natural hydraulic lime mortar; multi-layered material; heat and mass transfer; multi-physics model; environmental loads; hygro-thermal stress; hygro-thermo-mechanical analysis 1. Introduction Ancient and contemporary buildings use masonry as one of the most common construction materials for both structural and non-structural applications. As an integral part of the building envelope, masonry walls are exposed to environmental actions, such as direct solar radiation, wind-driven rain, thermal and hygric variations, moisture condensation, and rising damp. The continuous exposure to environmental loads affects the mechanical response of the material and leads to long-term weathering and degradation, e.g., strength loss, damage (cracks) due to induced dimensional changes and internal stresses, and spalling caused by salt crystallization and freeze-thaw [ 1 , 2 ]. The mechanical behavior of masonry structures has been well-documented in the literature under both Appl. Sci. 2023,13, 4514. https://doi.org/10.3390/app13074514 https://www.mdpi.com/journal/applsci Appl. Sci. 2023,13, 4514 2 of 35 static and dynamic loading scenarios [ 3 – 5 ]. However, the structural assessment usually neglects the influence of hygro-thermal loads and assumes ideal conditions. As a result, there is still a limited understanding of the hygro-thermo-mechanical (HTM) behavior of masonry. For the analysis of coupled hygro-thermal phenomena in masonry structures, it is necessary to first define a multi-physics model to describe heat and mass transfer in porous materials and multi-layered components. Several hygro-thermal models for porous building materials are available in the literature, e.g., [ 6 – 8 ]. Their application to multilayered cases, however, demands specific requirements: namely, the need to use a driving potential continuous across the interfaces or the consideration of interfacial effects on the heat and mass transfer mechanisms. Perfect contact is usually assumed for heat conduction through the interface. Consequently, the hygro-thermal models commonly disregard interfacial effects on heat transfer. On the other hand, the simulation of moisture transport in multi-layered cases still presents some challenges. Only a few investigations have focused on interface modeling for mass transfer problems in masonry, e.g., [ 9 – 12 ], and a unified methodology to characterize the effect of brick–mortar interfaces has not been achieved yet. Vereecken and Roels [ 13 ] proved that masonry can be modeled as a homogeneous continuum with brick properties to simulate certain water absorption scenarios. Other authors have employed a similar approach for one-dimensional (1D) simulations, e.g., [ 14 , 15 ]. Conversely, for two-dimensional (2D) analyses, the most common approach is still the differentiation of brick units and mortar joints, although the interfacial effects are usually neglected [ 14 , 16 , 17 ]. Nonetheless, several experimental studies have proven the impact of interfaces on water absorption in masonry [ 9 – 12 , 18 ]. Thus, further research on the modeling of hydraulic interfaces is required. Most of the scientific advances in heat and mass transfer in porous building materials have come from the disciplines of Material Sciences and Building Physics. In this context, the improvement of hygro-thermal models and the development of new numerical tools have allowed for more complex and detailed hygro-thermal simulations. Nonetheless, the link between the heat and moisture fields and solid mechanics has not been exploited yet. On one hand, experimental studies on structural elements are normally conducted under controlled, standard laboratory conditions. On the other hand, numerical simulations usually neglect the influence of hygro-thermal loads and assume ideal conditions. The study of HTM effects has received more attention in concrete structures, e.g., curing of concrete for application in dams [ 19 , 20 ]. For masonry, some research has been devoted to the mechanical behavior of walls under extreme scenarios such as high temperatures during fire, e.g., [ 21 , 22 ]. However, the influence of temperature and moisture actions caused by normal environmental conditions are usually disregarded and so are the hygrothermally induced stresses. To date, few studies have approached the mechanical response of masonry as affected by this type of environmental conditions, e.g., [23–25]. Ramézani and Jeong [ 24 ] developed a one-way coupled HTM model to analyze the behavior of a limestone masonry wall subjected to environmental loads. The authors created a 2D finite element model and performed a series of non-isothermal, non-isohygral, steadystate analyses while isolating and combining the different physics. From a comparison between the mechanical, thermo-mechanical, and HTM results, they were able to identify the contribution of each field to the overall response. Moreover, it was shown that the distribution of stresses in the wall was geometry-dependent, and the highest tensile values were concentrated in the vertical head joints and the corners of the masonry units. Transient analyses were also performed to analyze the evolution of the moisture and temperature distributions from the initial condition to the steady-state equilibrium. The authors did not perform transient HTM analyses for variable environmental conditions and this type of analysis is still missing in the literature. Some authors have dealt with independent thermo-mechanical or hygro-mechanical problems in masonry. For the former, most research is devoted to the mechanical response of structural elements under high-temperature conditions. Among the hygro-mechanical Appl. Sci. 2023,13, 4514 3 of 35 studies, Castellazzi et al. [ 16 ] developed a multi-scale numerical framework for the analysis of masonry walls subjected to water absorption, e.g., rising damp. In particular, the authors employed a two-dimensional discrete coupled hygro-mechanical model to assess the influence of the moisture field on the structural response as well as to incorporate the effect of mechanical degradation on the moisture transport process. In line with the hygro-mechanical studies, research on salt transport and crystallization in porous materials has received more attention in recent years, e.g., [ 25 – 31 ]. The proposed models have been proved able to predict the transport of salts and subsequent crystallization under different environmental scenarios. An all-encompassing analysis considering the additive effects of thermal dilation, hygric swelling, and crystallization pressure is still missing. On the other hand, scientific works on fully coupled HTM problems in masonry are notably scarce. Regarding this aspect, further investigation is needed to determine the influence of moisture on the mechanical performance of the material, including linear and non-linear behavior as well as damage. Similarly, more research is necessary to quantify and reproduce the reverse coupling, that is, the mechanical effects (stress variation, creep, volumetric changes, damage, cracking) on moisture transport. The studies developed by Mertens [ 32 ] and Carmeliet [ 33 ] are worth mentioning, and they focused on the non-linear hysteretic behavior of quasi-brittle porous materials. Different finite element computational strategies have been developed for the mechanical analysis of masonry: namely, micro- and macro-modeling approaches [ 34 ]. These strategies can be further extended to hygro-thermal analyses as presented in Figure 1. In the case of micro-modeling, which is the focus of this work, the constituent materials (brick units and mortar) and brick–mortar interfaces are modeled separately. On the other hand, macro-modeling approaches assume a homogenized continuum material. Appl. Sci. 2023, 13, 4514 3 of 38 studies, Castellazzi et al. [16] developed a multi-scale numerical framework for the analysis of masonry walls subjected to water absorption, e.g., rising damp. In particular, the authors employed a two-dimensional discrete coupled hygro-mechanical model to assess the influence of the moisture field on the structural response as well as to incorporate the effect of mechanical degradation on the moisture transport process. In line with the hygromechanical studies, research on salt transport and crystallization in porous materials has received more attention in recent years, e.g., [25–31]. The proposed models have been proved able to predict the transport of salts and subsequent crystallization under different environmental scenarios. An all-encompassing analysis considering the additive effects of thermal dilation, hygric swelling, and crystallization pressure is still missing. On the other hand, scientific works on fully coupled HTM problems in masonry are notably scarce. Regarding this aspect, further investigation is needed to determine the influence of moisture on the mechanical performance of the material, including linear and non-linear behavior as well as damage. Similarly, more research is necessary to quantify and reproduce the reverse coupling, that is, the mechanical effects (stress variation, creep, volumetric changes, damage, cracking) on moisture transport. The studies developed by Mertens [32] and Carmeliet [33] are worth mentioning, and they focused on the non-linear hysteretic behavior of quasi-brittle porous materials. Different finite element computational strategies have been developed for the mechanical analysis of masonry: namely, micro- and macro-modeling approaches [34]. These strategies can be further extended to hygro-thermal analyses as presented in Figure 1. In the case of micro-modeling, which is the focus of this work, the constituent materials (brick units and mortar) and brick–mortar interfaces are modeled separately. On the other hand, macro-modeling approaches assume a homogenized continuum material. Figure 1. Modeling strategies for masonry. Adapted from [35], following [34,36]. This paper presents a hygro-thermo-mechanical (HTM) model to simulate the impact of hygro-thermal loads on the structural response of masonry walls. Moreover, this paper provides a new insight into the HTM behavior of masonry when subjected to normal environmental actions, considering both constant and variable temperature and relative humidity conditions. These aspects are usually disregarded in experimental and simulation works, and therefore, there is an evident lack of scientific knowledge. This paper is structured as follows. First, a fully coupled heat and moisture transport model is proposed and hygro-thermal simulations are performed using a 2D finite element model of a threewythe brick wall. The results of the hygro-thermal studies are evaluated in terms of temperature and relative humidity profiles across the thickness of the wall as well as for Figure 1. Modeling strategies for masonry. Adapted from [35], following [34,36]. This paper presents a hygro-thermo-mechanical (HTM) model to simulate the impact of hygro-thermal loads on the structural response of masonry walls. Moreover, this paper provides a new insight into the HTM behavior of masonry when subjected to normal environmental actions, considering both constant and variable temperature and relative humidity conditions. These aspects are usually disregarded in experimental and simulation works, and therefore, there is an evident lack of scientific knowledge. This paper is structured as follows. First, a fully coupled heat and moisture transport model is proposed and hygro-thermal simulations are performed using a 2D finite element model of a threewythe brick wall. The results of the hygro-thermal studies are evaluated in terms of temperature and relative humidity profiles across the thickness of the wall as well as for Appl. Sci. 2023,13, 4514 4 of 35 different points on the external and internal surfaces. The evolution of temperature and relative humidity obtained by transient analyses show that heat transfer is significantly faster than moisture transport. Subsequently, the analyses were extended to incorporate mechanical effects by the use of a one-way HTM model. For all the analyses, two different mortars were considered for the wall: namely, natural hydraulic lime and cement mortar. It is proven that the wall with cement mortar develops higher stresses when compared to the one with lime mortar, due to the dissimilar hygro-thermal behavior between the constituent materials. Moreover, the results show that the HTM response of masonry walls is a complex 2D phenomenon. Finally, the environmental loads change the natural stress distribution caused by gravitational loads alone and induce localized concentrations of compressive and tensile stresses. 2. Hygro-Thermal Model This section describes the numerical model adopted for hygro-thermal simulations. The moisture transport model was validated by the authors in [ 37 ]. In the present paper, this model is extended to incorporate the thermal field. Thus, the theoretical bases and main assumptions of the extended model are briefly presented here. Masonry is described as a porous medium made up of a matrix of interconnected pores inside a solid skeleton. The solid phase is assumed continuous, homogeneous, inert (no chemical reactions), isotropic, and non-deformable. Moreover, pores are assumed cylindrical, with a homogeneous, isotropic distribution inside the material. The void space defined by the pore system is filled with a liquid and a gaseous phase in different proportions. The liquid phase is pure and incompressible, whereas the air phase is an ideal mixture of dry air and water vapor at atmospheric pressure. For simplicity, advective air flow is not implemented in the model and pressure gradients are considered negligible. Additionally, a local instantaneous thermodynamic equilibrium is assumed between the liquid and gaseous phases. Consequently, the contributions of liquid water and water vapor may be combined to define a global moisture content. Nonetheless, the contribution of water vapor to the total moisture content is negligible. Knudsen flow is not considered, and the effect of gravity is neglected since capillary forces are dominant. The volume fraction of (moist) air inside the porous space does not contribute to the overall thermal conductivity and specific heat capacity of the material. The formation of ice is out of the scope of the present work; thus, the temperature remains above the freezing point of water. Heat flux induced by moisture transport (Dufour effect) as well as moisture flux induced by temperature gradients (Soret effect) are considered negligible. The coupling between heat and moisture fields will be further discussed in the following sections. The effect of imperfect contact interfaces in multi-layered cases is considered for moisture transport by means of hydraulic resistances [ 9 ]. For heat transfer, the impact of interfaces between adjacent materials is disregarded so that the temperature profiles are continuous between layers. 2.1. Governing Equations The heat transfer model considers conductive heat transfer according to Fourier’s Law and accounts for the heat of vaporization: ρC∂T ∂t=∇·(λ∇T)+Lv∇·δa µ∇(ϕpv,sat), (1) where ρC [J/(m 3· K)] is the volumetric heat capacity at constant pressure, T [K] is the temperature, t [s] is time, λ [W/(m · K)] is the thermal conductivity, Lv [J/kg] is the latent heat of evaporation, δa [kg/(m · s · Pa)] is the water vapor permeability of still air, µ [–] is the water vapor resistance of the material, ϕ [–] is the relative humidity, and pv,sat [Pa] is the vapor saturation pressure. The second term on the right-hand side of Equation (1) represents the latent heat of vaporization associated with liquid/gas phase change. Appl. Sci. 2023,13, 4514 5 of 35 On the other hand, the multi-phase moisture transport model stems from Fick’s laws of diffusion and considers a combination of liquid transport by capillarity and vapor diffusion: ∂w ∂ϕ ∂ϕ ∂t=∇·∂w ∂ϕ Dw∇ϕ+∇·δa µ∇(ϕpv,sat), (2) where w [kg/m 3 ] is the moisture content, and Dw [m 2 /s] is the liquid water diffusivity. The modified version of the exponential expression proposed by Künzel [ 38 ] is used to describe the liquid water diffusivity: Dw=γ·Aw wcap 2 ·103·(w wcap −1), (3) where γ [–] is a diffusivity factor depending on the material and transport process (wetting/drying), Aw [kg/(m 2· s 0.5 )] is the capillary absorption coefficient, and wcap [kg/m 3 ] is the capillary moisture content. 2.2. Interface Modeling The brick–mortar interfaces are assumed to have an imperfect contact for moisture transport. Therefore, the moisture flow across the interface, gIF [kg/(m 2· s)], is modeled through the introduction of an interface permeability, KIF [s/m], or an interface resistance RIF [m/s], such as [9]: gIF =KIF ∂pc ∂x=1 1 KIF ∂pc ∂x=∆pc 1 KIF ∆x=∆pc RIF , (4) where pc [Pa] is the capillary pressure, which can be related to the relative humidity using Kelvin’s equation: pc=ρw·Rv·T·ln ϕ, (5) with ρw [kg/m 3 ] being the density of water, and Rv [J/(kg · K)] being the universal gas constant for water vapor. Thus, assuming the applicability of Kelvin’s equation, the moisture flow across the interface can be defined as: gIF =KIF ∂pc ∂ϕ ∂ϕ ∂x=KIF ρw·Rv·T ϕ ∂ϕ ∂x=ρw·Rv·T ϕ ∆ϕ 1 KIF ∆x=ρw·Rv·T ϕ ∆ϕ RIF . (6) 2.3. Hygro-Thermal Coupling Simultaneous heat and mass transfer in porous media constitutes a highly coupled problem: that is, temperature balance and thermal properties vary according to moisture content, while the moisture-related properties and moisture equilibrium are dependent on temperature. 2.3.1. Moisture Dependency of Thermal Parameters It is known that the heat capacity of a porous material is directly related to its moisture content. In particular, if the material is wet, the effect of moisture on heat capacity can be considered using: ρC=ρbulkCp+w·Cw, (7) where ρbulk [kg/m 3 ] is the bulk density of the material, Cp [J/(kg · K)] is the specific heat capacity of the dry material, and Cw [J/(kg · K)] is the specific heat capacity of liquid water. The specific heat capacity of water has a weak thermal dependence for the range of temperatures of interest in the present study [ 39 ]. Therefore, it is taken as a constant, Cw=4182 J/(kg·K), which corresponds to the specific heat capacity of water at 20 ◦C. Appl. Sci. 2023,13, 4514 6 of 35 Additionally, the moisture influence on the thermal conductivity of a wet porous material can be determined through the following expression: λ=λ0(1+b·w/ρbulk)=λ01+b·wg, (8) where λ0 [W/(m · K)] is the thermal conductivity of the solid matrix, and b [–] is a materialdependent thermal conductivity supplement. The supplement b can be described as the fractional increase (sometimes expressed as a percentage) of the thermal conductivity per mass-percent moisture content, i.e., moisture mass per unit mass of dry material × 100. Thermal conductivity has a minor dependence on temperature for normal environmental conditions, and therefore, its thermal dependency is disregarded [40]. For practical applications, the thermal conductivity of porous materials is often given as a linear relationship based on experimental values determined for different moisture content conditions. For instance, an expression of the form λ=A+B·w has been extensively used in the literature to describe the thermal conductivity of brick [ 41 – 43 ]. In the previous expression, the intercept λ(w=0) stands for the thermal conductivity of the dry material, λ0 . In turn, the slope of the curve represents the increase in thermal conductivity per moisture content, and according to Equation (8), it must be equal to λ0·b/ρbulk. 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◦ 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, φo [–], and bulk density, ρbulk [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, wsat [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 ◦ C and 50 ◦ C [ 46 ]. As a result, wsat [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, wcap [kg/m 3 ], which is considered constant. However, it must be noted that a widely accepted theory to describe the relationship between wcap and temperature is still lacking in the literature. Apart from the saturation state boundaries, wcap and wsat , 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, Castellazzi et al. [49] proposed the following expression: w(ϕ)=wcap·ψ(T)−1 ψ(T)−ϕ·ϕ, (9) where ψ(T) [–] is a fitting parameter dependent on temperature. It is noted that ψ is inversely proportional to the temperature. The correlation between ψ 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. Appl. Sci. 2023,13, 4514 7 of 35 The influence of temperature on vapor diffusion is partly considered through the water vapor permeability of still air, δa[kg/(m·s·Pa)], defined empirically as [50]: δa=2.31·10−5 Rv·T p0 pT 273.151.81 , (10) where p0= 101,325 Pa is the standard atmospheric pressure, and p [Pa] is the ambient barometric pressure. Additionally, the saturation vapor pressure, pv,sat [Pa], is a non-linear function of temperature defined empirically as [51,52]: pv,sat =610.7[Pa]·107.5(T−273.15 T−35.85 ). (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, Aw [kg/(m 2· s 0.5 )], or sorptivity, S=Aw/ρw [m/s 0.5 ], e.g., [ 44 , 53 – 56 ]. The reported results support the validity of the Lucas–Washburn law, which describes the water penetration depth x [m] after a period of time t[s] as [57]: x=sσrcosθ 2η·√t, (12) where σ [N/m] is the surface tension of water, r [m] is the equivalent hydraulic capillary radius, θ is the contact angle, and η [Pa · s] is the viscosity of water. This expression states that liquid water flow is influenced by a series of factors. In order to understand how the overall liquid flow is affected by variations of temperature, it is convenient to study the impact of temperature on each of those factors separately. First, assuming that the porous structure undergoes negligible changes due to thermal (or hygric) expansion, it is reasonable to assume a constant r . Moreover, temperature has only a slight impact on the contact angle θ, so its influence can be disregarded. Therefore, the remaining temperature-dependent parameters are the surface tension σ and the viscosity η . Comparing the definition of the capillary absorption coefficient with the Lucas–Washburn equation, it is possible to establish a relation of proportionality as suggested by Gummerson et al. [53]: Aw∝pσ/η. (13) Considering the previous relation, Feng and Janssen [ 44 ] fitted tabulated values for σ and η (temperatures between 0 ◦ C and 50 ◦ C) and established a linear model to describe the relationship between Awand T: Aw=kpσ/η=k[0.095(T−273.15)+6.566], (14) where k [kg/m 2.5 ] is a micro-structure-dependent but temperature-independent factor. As an example, these authors studied the absorption behavior of different building materials under several temperature conditions and determined k= 0.0059 for brick ( Aw= 0.052 at 22.5 ◦ C). For cement mortars, Hanumanthu and Sarkar [ 56 ] found k= 0.0019–0.0023 (Aw=0.016–0.020 at 20 ◦C), whereas for NHL mortar, the data presented by Karagiannis et al. [ 58 ] were reinterpreted by Feng and Janssen [ 55 ], who established k= 0.0125 (Aw=0.106 at 20 ◦C). 2.4. Initial and Boundary Conditions The hygro-thermal model is completed by introducing initial and boundary conditions. The Dirichlet boundary condition yields: T=− T, (15) Appl. Sci. 2023,13, 4514 8 of 35 ϕ=− ϕ, (16) where − T [K] and − ϕ [–] are the prescribed temperature and relative humidity at the boundary, respectively. Conversely, the Neumann boundary condition is defined as a flux derived from a temperature or vapor pressure difference: q=hT(Text −T), (17) g=hm(pv,ext −pv), (18) where q [W/m 2 ] is the convective heat flux, hT [W/(m 2· K)] is the heat transfer coefficient, Text [K] is the temperature of the environment, T [K] is the temperature at the boundary, g [kg/(m 2· s)] is the convective moisture flux, hm [s/m] is the convective mass transfer coefficient, and pv,ext [Pa] and pv [Pa] are the partial vapor pressures defined for the environment and at the boundary, respectively. In the present work, hT accounts for the combined effects of convection and long-wave radiation exchanges with the environment. Thus, this lumped convective/radiative heat transfer coefficient is defined as hT=hT,c+hT,r, (19) where hT,c [W/(m 2· K)] is the convective heat transfer coefficient, and hT,r [W/(m 2· K)] is the radiative heat transfer coefficient. The heat transfer coefficient is dependent on the local air flow conditions, the temperature, the geometry, and the orientation of the studied element, which makes an accurate estimation very complex. For building applications, a simplified approach is usually adopted, and the heat transfer coefficient is assumed to be constant [38]. 3. Hygro-Thermal Simulations and Discussion The model described in the previous section is used here to simulate the hygrothermal behavior of a brick masonry wall. First, the main features of the numerical model are described: geometry, material properties, and boundary conditions. Then, two sets of analyses are performed: namely, steady-state and time-dependent analyses. Finally, the results of the simulations are presented in terms of temperature and relative humidity profiles, and the obtained trends are discussed. The analyses described in the following paragraphs are based on the finite element method and were performed using the software COMSOL Multiphysics [59]. 3.1. Description of the Numerical Model A common type of load-bearing masonry wall was selected as a case study to perform the hygro-thermal simulations. In particular, the chosen wall is made up of bricks with dimensions 205 mm × 95 mm × 50 mm, and mortar joints 15 mm thick. The resulting structure is a 315 mm thick (three-wythe brick) masonry wall with a total height of 2650 mm. The geometrical configuration of the wall is depicted in Figure 2. The described geometry was used to prepare a two-dimensional (2D) model. The material properties used for the hygro-thermal simulations are presented in Table 1. Two types of mortar were considered for the analyses: namely, natural hydraulic lime mortar (LM) and cement mortar (CM). Moreover, the brick–mortar interfaces were assumed to have perfect contact for heat transfer, whereas imperfect contact was considered for moisture transport. Thus, a hydraulic resistance RIF = 2.0 × 10 9 m/s was assigned to the brick–mortar interfaces. The wall is assumed to be located in Guimarães (Portugal), which has an annual average temperature ϑ= 13.5 ◦ C and an annual average relative humidity RH = 74% [ 60 ]. Additionally, the wall is considered to have initial conditions equal to these yearly average Appl. Sci. 2023,13, 4514 9 of 35 values; thus, ϑ= 13.5 ◦ C and ϕ0= 0.74. One face of the wall is exposed to the exterior, whereas the opposite face is in contact with a conditioned interior space with constant temperature and relative humidity: namely, 20 ◦ C and 50% RH. It must be noted that the presented configuration is rather unusual for a building envelope in contact with a modern habitable interior space since it lacks any type of dedicated moisture barrier or thermal insulation. Nonetheless, this configuration is reasonable for certain historical walls and expedient for the analyses at hand, in which thermal and moisture gradients are of interest. In the case of a masonry wall with good insulation on the interior side, adiabatic boundary conditions could be assumed for the internal face. Appl. Sci. 2023, 13, 4514 9 of 38 modern habitable interior space since it lacks any type of dedicated moisture barrier or thermal insulation. Nonetheless, this configuration is reasonable for certain historical walls and expedient for the analyses at hand, in which thermal and moisture gradients are of interest. In the case of a masonry wall with good insulation on the interior side, adiabatic boundary conditions could be assumed for the internal face. (a) (b) (c) Figure 2. Schematic geometrical configuration of the brick masonry wall: (a) transversal cross-sec- tion; (b) vertical cross-section; (c) bond arrangement and geometry of the brick units. Dimensions in mm. The hygro-thermal simulations were performed through both steady-state (SS) and time-dependent (TD) analyses. The SS analysis was used to evaluate the final hygro-ther- mal equilibrium attained by the wall considering constant environmental conditions as explained in the previous paragraph. Moreover, two sets of time-dependent analyses were defined. In particular, TD1 was performed to evaluate the evolution of temperature and relative humidity profiles across the wall from the initial state to hygro-thermal equilibrium with the environment. Therefore, the final stage of TD1 is analogous to the results obtained with the stationary analysis SS. Additionally, the transient study TD2 considered the wall already in equilibrium with the environment (the initial conditions were taken from the results of SS or the final stage of TD1) and then imposed variable external conditions. In particular, the external conditions for TD2 were assumed to vary with time following a sinusoidal law, given by: 𝜗(𝑡)=13.5+10·cos(2𝜋𝑡−𝜋/2), (20) 𝜑(𝑡)=0.74+0.12·cos(2𝜋𝑡+𝜋/2), (21) for the external temperature and relative humidity, respectively, with 𝑡 expressed in days. The proposed thermal variation corresponds to the average maximum temperature excursion that can be experienced in a single day during mid-season periods, i.e., ∆𝜗≈ 20 °C. Additionally, average values of the seasonal relative humidity were selected to determine the range of daily moisture variation, and the corresponding sinusoidal function was set to follow an opposite trend to the one defined for the temperature. The initial and boundary conditions used for the hygro-thermal simulations of the masonry wall are summarized in Table 2. Neumann boundary conditions were imposed Figure 2. Schematic geometrical configuration of the brick masonry wall: ( a ) transversal cross-section; ( b ) vertical cross-section; ( c ) bond arrangement and geometry of the brick units. Dimensions in mm. The hygro-thermal simulations were performed through both steady-state (SS) and time-dependent (TD) analyses. The SS analysis was used to evaluate the final hygrothermal equilibrium attained by the wall considering constant environmental conditions as explained in the previous paragraph. Moreover, two sets of time-dependent analyses were defined. In particular, TD1 was performed to evaluate the evolution of temperature and relative humidity profiles across the wall from the initial state to hygro-thermal equilibrium with the environment. Therefore, the final stage of TD1 is analogous to the results obtained with the stationary analysis SS. Additionally, the transient study TD2 considered the wall already in equilibrium with the environment (the initial conditions were taken from the results of SS or the final stage of TD1) and then imposed variable external conditions. In particular, the external conditions for TD2 were assumed to vary with time following a sinusoidal law, given by: ϑ(t)=13.5 +10·cos(2πt−π/2), (20) ϕ(t)=0.74 +0.12·cos (2πt+π/2), (21) for the external temperature and relative humidity, respectively, with t expressed in days. The proposed thermal variation corresponds to the average maximum temperature excursion that can be experienced in a single day during mid-season periods, i.e., ∆ϑ≈ 20 ◦ C. Additionally, average values of the seasonal relative humidity were selected to determine Appl. Sci. 2023,13, 4514 16 of 35 As was pointed out for the SS analysis, the presence of mortar joints can be distinguished by slope changes in the temperature and relative humidity profiles at the location of the brick–mortar interfaces. It is noted that the TD1 results for the CM case show the location of the vertical head joints more evidently. Once again, this can be explained by the more dissimilar properties between bricks and cement mortar in comparison with lime mortar. Besides the previous analyses, a parametric study was performed in order to evaluate the influence of the hydraulic resistance on the evolution of relative humidity. In particular, different values of the interface resistance, RIF , were adopted, but the results did not show significant differences. Thus, it was concluded that the impact of the interface resistance was not significant for the moisture conditions considered in this problem. It must be pointed out, however, that the hydraulic resistance may become non-negligible for higher moisture levels. Considering the results of the time-dependent analysis TD1 for both LM and CM cases, only slight differences are noticed between the three studied sections of the wall. Therefore, a macro-modeling strategy with equivalent masonry properties could be used as well to simulate the observed temperature and moisture content evolution. Notwithstanding the loss of detail, a simplified 1D model could be considered as well with a minor impact on the overall accuracy. 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 7and 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 Appl. Sci. 2023,13, 4514 17 of 35 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 ϕ=pv/pv,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. Appl. Sci. 2023, 13, 4514 17 of 38 Besides the previous analyses, a parametric study was performed in order to evaluate the influence of the hydraulic resistance on the evolution of relative humidity. In particular, different values of the interface resistance, 𝑅, were adopted, but the results did not show significant differences. Thus, it was concluded that the impact of the interface resistance was not significant for the moisture conditions considered in this problem. It must be pointed out, however, that the hydraulic resistance may become non-negligible for higher moisture levels. Considering the results of the time-dependent analysis TD1 for both LM and CM cases, only slight differences are noticed between the three studied sections of the wall. Therefore, a macro-modeling strategy with equivalent masonry properties could be used as well to simulate the observed temperature and moisture content evolution. Notwithstanding the loss of detail, a simplified 1D model could be considered as well with a minor impact on the overall accuracy. 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. Figure 7. Time-dependent analysis with variable external boundary conditions (TD2). Temperature evolution for different points of the studied three-wythe brick masonry wall: ( a ) joints with natural hydraulic lime mortar (LM); (b) joints with cement mortar (CM). 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. Again, the results obtained with the new RIF values did not show significant differences. Therefore, it was concluded that the impact of the hydraulic resistance is not significant for the boundary conditions considered in this problem. Nonetheless, it must be recalled that the influence of the interface may become non-negligible for higher moisture levels. Appl. Sci. 2023,13, 4514 18 of 35 Appl. Sci. 2023, 13, 4514 19 of 38 Figure 8. Time-dependent analysis with variable external boundary conditions (TD2). Evolution of the relative humidity for different points of the studied three-wythe brick masonry wall: (a) joints with natural hydraulic lime mortar (LM); (b) joints with cement mortar (CM). 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. Again, the results obtained with the new 𝑅 values did not show significant differences. Therefore, it was concluded that the impact of the hydraulic resistance is not significant for the boundary conditions considered in this problem. Nonetheless, it must be recalled that the influence of the interface may become non-negligible for higher moisture levels. Independently of the studied case, the response of the wall stabilizes (dynamic equilibrium) after a couple of cycles. It is noted that external/internal phase delay entails temperature and relative humidity gradients higher than the ones expected if the shift was not considered, and maximum and minimum values were occurring simultaneously at one and the other sides of the wall. In particular, the maximum thermal gradient obtained for the average temperatures between the two faces is ∆𝑇= 13.0 °C for the LM case and ∆𝑇= 12.4 °C for the CM case. In turn, the maximum moisture gradient obtained for the average relative humidity values between the two faces is ∆𝜑= 0.18 for the LMJ case and ∆𝜑= 0.17 for the CM case. Considering these aspects, a time-dependent analysis with variable boundary conditions may produce more severe scenarios and is, therefore, recommended for a better approximation of real environmental actions. Considering the results described in the previous paragraphs, it is evident that the hygro-thermal behavior of brick masonry walls depends, to a great extent, on the type of mortar. When considering the wall with natural hydraulic lime mortar (LM), the temperature and relative humidity trends showed comparable values for the groups of studied points, independent of their location on brick or mortar joints. Therefore, this type of Figure 8. Time-dependent analysis with variable external boundary conditions (TD2). Evolution of the relative humidity for different points of the studied three-wythe brick masonry wall: ( a ) joints with natural hydraulic lime mortar (LM); (b) joints with cement mortar (CM). Independently of the studied case, the response of the wall stabilizes (dynamic equilibrium) after a couple of cycles. It is noted that external/internal phase delay entails temperature and relative humidity gradients higher than the ones expected if the shift was not considered, and maximum and minimum values were occurring simultaneously at one and the other sides of the wall. In particular, the maximum thermal gradient obtained for the average temperatures between the two faces is ∆T= 13.0 ◦ C for the LM case and ∆T=12.4 ◦C for the CM case. In turn, the maximum moisture gradient obtained for the average relative humidity values between the two faces is ∆ϕ= 0.18 for the LMJ case and ∆ϕ= 0.17 for the CM case. Considering these aspects, a time-dependent analysis with variable boundary conditions may produce more severe scenarios and is, therefore, recommended for a better approximation of real environmental actions. Considering the results described in the previous paragraphs, it is evident that the hygro-thermal behavior of brick masonry walls depends, to a great extent, on the type of mortar. When considering the wall with natural hydraulic lime mortar (LM), the temperature and relative humidity trends showed comparable values for the groups of studied points, independent of their location on brick or mortar joints. Therefore, this type of hygro-thermal problem could be simulated for the LM case by means of a macromodeling approach with an equivalent masonry material or a simplified 1D model with hardly any loss of information. Conversely, the wall with cement mortar (CM) showed some differences in temperature and relative humidity depending on the material where Appl. Sci. 2023,13, 4514 19 of 35 the studied point was located. Thus, for the CM case, the macro-modeling approach or the reduction to a 1D problem would entail a certain loss of detail and accuracy. 4. Hygro-Thermo-Mechanical Model This section presents the numerical model employed for hygro-thermo-mechanical analysis. The main hypotheses and simplifications previously explained for the hygrothermal models (see Section 2) are valid here as well. With respect to the structural mechanics field, a series of new assumptions must be introduced. In general, the environmental loads induced by normal exposure conditions are not expected to cause damage (at least not in the short term). Thus, the materials are likely to behave within the linear elastic range and material non-linearity is disregarded. Moreover, materials and structural elements are considered to be in an undamaged initial state. Finally, infinitesimal strains and small deformations and rotations are assumed. 4.1. 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]: ε=εelastic +εplastic +εthermal +εswelling (22) The mechanical strains εelastic and εplastic 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: εthermal =αT∆T=αTT−Tre f (23) where αT [1/K] is the coefficient of thermal expansion of the material, and Tre f [K] is the reference temperature at which the thermally induced strain is zero. An analogous relation may be written for the moisture-induced strain: εswelling =αh∆wg=αhwg−wg,re f (24) where αh [–] is the moisture swelling coefficient of the material, wg [kg/kg] is the gravimetric moisture content, and wg,re f [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 αh is initially given in terms of gravimetric moisture content (dimensionless), it must be divided by the density of the material, ρbulk [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, Carmeliet et al. [ 66 ] studied the so-called mechano-sorptive behavior of porous Appl. Sci. 2023,13, 4514 20 of 35 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 ]. Nonetheless, considering the lack of specific research dedicated to the moisture-dependent mechanical behavior of masonry materials, these aspects are neglected in the present study and further works on this topic are needed. Under the assumption of linear elasticity and the described one-way coupled problem, the temperature- and moisture-induced strains are directly proportional to the results of the hygro-thermal problem by means of the thermal and hygric expansion coefficients. Appl. Sci. 2023, 13, 4514 21 of 38 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, Carmeliet 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]. Nonetheless, considering the lack of specific research dedicated to the moisture-dependent mechanical behavior of masonry materials, these aspects are neglected in the present study and further works on this topic are needed. Under the assumption of linear elasticity and the described one-way coupled problem, the temperature- and moisture-induced strains are directly proportional to the results of the hygro-thermal problem by means of the thermal and hygric expansion coefficients. Figure 9. Schematic diagram of the proposed one-way coupled hygro-thermo-mechanical model. 4.2. Initial and Boundary Conditions The equilibrium state of a porous building material is intimately related to the surrounding environmental conditions. Specifically, temperature and moisture content increments cause material dilation, whereas shrinkage takes place when temperature and relative humidity decrease. Moreover, these material responses give rise to the appearance and evolution of internal stresses. It is worth noting that the stresses induced by thermal or hygric volumetric variations are necessarily associated with a deformation constraint. In other words, if deformations are not restrained, the material will experience a free dimensional change and internal stresses will not appear. Nonetheless, constraints always exist in real applications, usually linked to the external boundary conditions or the presence of interfaces. Therefore, internal stresses caused by temperature and moisture content variations are expected. In this context, the initial conditions of the system are Figure 9. Schematic diagram of the proposed one-way coupled hygro-thermo-mechanical model. 4.2. Initial and Boundary Conditions The equilibrium state of a porous building material is intimately related to the surrounding environmental conditions. Specifically, temperature and moisture content increments cause material dilation, whereas shrinkage takes place when temperature and relative humidity decrease. Moreover, these material responses give rise to the appearance and evolution of internal stresses. It is worth noting that the stresses induced by thermal or hygric volumetric variations are necessarily associated with a deformation constraint. In other words, if deformations are not restrained, the material will experience a free dimensional change and internal stresses will not appear. Nonetheless, constraints always exist in real applications, usually linked to the external boundary conditions or the presence of interfaces. Therefore, internal stresses caused by temperature and moisture content variations are expected. In this context, the initial conditions of the system are usually assumed as the initial equilibrium state. Since they are associated with no internal deformations, the initial conditions are commonly known as free-strain or zero-strain conditions. Appl. Sci. 2023,13, 4514 21 of 35 5. Hygro-Thermo-Mechanical Simulations and Discussion This section demonstrates the application of the hygro-thermo-mechanical (HTM) model using the previously introduced brick masonry wall as a case study. First, the description of the model is recalled, and the new parameters and boundary conditions needed for the mechanical component are defined. Then, the results of the HTM analyses are presented and discussed. 5.1. Description of the Numerical Model and Boundary Conditions The same three-wythe brick masonry wall presented for the hygro-thermal simulations was used for the HTM analyses (see description of the model in Section 3.1). Similarly, the physical, thermal, and hygric material properties presented in Table 1were employed for the model definition. The newly added properties needed for the mechanical field are introduced in Table 3. The numerical simulations were performed with a 2D plane strain finite element model using the software COMSOL Multiphysics [59]. Two different scenarios were studied: namely, a gravitational load case and a combined hygro-thermo-mechanical case. For the gravitational case, body forces were considered, including a uniformly distributed load, L= 12.5 kN/m, applied on top of the wall to simulate the weight of a simply supported roof. The HTM scenario was built upon the gravitational case so that the body forces were included. Moreover, for the HTM case, the temperature and moisture conditions defined previously for the time-dependent hygro-thermal analysis TD2 (see Section 3.1) were imposed on the wall. It is recalled that the TD2 analysis considers an initial step for the wall to reach the steady-state equilibrium with the environment. In other words, the wall is already in a deformed configuration (initial conditions to steady-state equilibrium) before the application of the variable boundary conditions. Table 3. Summary of mechanical, thermo-mechanical, and hygro-mechanical properties used for the hygro-thermo-mechanical simulations. Material Property Symbol Value/Expression Units Source Brick Young’s modulus E5580 MPa Ghiassi et al. [67] Poisson’s ratio ν0.20 – Compressive strength fc19.80 MPa Tensile strength ft2.00 MPa Coefficient of thermal expansion αT2.56 ×10−61/◦CRamirez et al. [18] Coefficient of hygric expansion αh 1.40 × 10 −4/wg0.72 – Lime mortar (LM) Young’s modulus E3330 MPa Penas et al. [68] Poisson’s ratio ν0.20 – Compressive strength fc1.10 MPa Tensile strength fb0.40 MPa Coefficient of thermal expansion αT3.04 ×10−61/◦CRamirez et al. [18] Coefficient of hygric expansion αh 3.00 × 10 −4/wg0.62 – Cement mortar (CM) Young’s modulus E2370 MPa D’Altri et al. [35] Poisson’s ratio ν0.20 – Compressive strength fc8.20 MPa Tensile strength ft1.50 MPa Coefficient of thermal expansion αT7.38 ×10−61/◦CRamirez et al. [18] Coefficient of hygric expansion αh 1.32 × 10 −4/wg0.78 – Regarding the mechanical boundary conditions, the bottom of the wall is assumed to be fixed: that is, all degrees of freedom are constrained. Conversely, the uppermost edge is not restrained, and all degrees of freedom remain free. The wall is assumed to be in a nondeformed configuration before the application of the gravitational and environmental loads. Moreover, the initial temperature and relative humidity are ϑ0= 13.5 ◦ C and ϕ0=0.74, respectively. For these conditions, the temperature- and moisture-induced strains are zero. Appl. Sci. 2023,13, 4514 22 of 35 Assuming the applicability of Saint Venant’s principle, the following discussion on the distribution of stresses focuses on a section of the mid-portion of the wall (height around 1.20 m). In other words, the area selected for the analysis of stresses is reasonably distant from any applied load or boundary condition that could directly influence the local stress/strain distribution. 5.2. Hygro-Thermo-Mechanical Behavior of a Brick Masonry Wall The results of the HTM model presented in the previous section are examined in the following paragraphs. In particular, the behavior of the wall is studied in terms of minimum and maximum principal stresses, σ3 and σ1 , according to the type of material and case scenario. The general sign convention is followed: thus, negative stresses indicate compression, and positive stresses denote tension. For the HTM analyses, the moment with maximum induced deformation (maximum horizontal displacement of the top of the wall) was chosen for the study. The selected deformed state is shown in Figures 10 and 11 for the LM and the CM cases, respectively, together with the history of displacements for the top of the wall and the imposed boundary conditions. Here, u and v are the displacements in the X and Y directions, respectively. It is recalled that the stress discussion is focused solely on a selected mid-height section of the wall. Appl. Sci. 2023, 13, 4514 23 of 38 Regarding the mechanical boundary conditions, the bottom of the wall is assumed to be fixed: that is, all degrees of freedom are constrained. Conversely, the uppermost edge is not restrained, and all degrees of freedom remain free. The wall is assumed to be in a non-deformed configuration before the application of the gravitational and environmental loads. Moreover, the initial temperature and relative humidity are 𝜗= 13.5 °C and 𝜑= 0.74, respectively. For these conditions, the temperature- and moisture-induced strains are zero. Assuming the applicability of Saint Venant’s principle, the following discussion on the distribution of stresses focuses on a section of the mid-portion of the wall (height around 1.20 m). In other words, the area selected for the analysis of stresses is reasonably distant from any applied load or boundary condition that could directly influence the local stress/strain distribution. 5.2. Hygro-Thermo-Mechanical Behavior of a Brick Masonry Wall The results of the HTM model presented in the previous section are examined in the following paragraphs. In particular, the behavior of the wall is studied in terms of minimum and maximum principal stresses, 𝜎 and 𝜎, according to the type of material and case scenario. The general sign convention is followed: thus, negative stresses indicate compression, and positive stresses denote tension. For the HTM analyses, the moment with maximum induced deformation (maximum horizontal displacement of the top of the wall) was chosen for the study. The selected deformed state is shown in Figures 10 and 11 for the LM and the CM cases, respectively, together with the history of displacements for the top of the wall and the imposed boundary conditions. Here, 𝑢 and 𝑣 are the displacements in the X and Y directions, respectively. It is recalled that the stress discussion is focused solely on a selected mid-height section of the wall. (a) (b) Figure 10. Hygro-thermo-mechanical analysis of the three-wythe brick masonry wall with natural hydraulic lime mortar: ( a ) imposed boundary conditions and displacement history measured at the top of the wall; (b) deformed shape selected for the study (maximum horizontal displacement). When considering the displacement history for the LM case (see Figure 10), it is possible to observe the initial deformed configuration of the wall before the application of the variable external boundary conditions. In particular, the wall presents a horizontal displacement to- Appl. Sci. 2023,13, 4514 23 of 35 wards the interior caused by moisture shrinkage of the internal face (ϕint =0.50 <ϕ0=0.74), which prevails over thermal dilation (ϑint =20 ◦C>ϑ0=13.5 ◦C). This horizontal displacement is approximately equal on the exterior and interior sides of the wall. On the other hand, the vertical displacement is rather small and remains always below 0.1 mm during the duration of the analysis. This vertical displacement is different on the exterior and interior sides of the wall due to rotation of the section. When the variable boundary conditions are implemented, the wall oscillates following the sinusoidal trend of the external temperature and relative humidity. In fact, the maximum horizontal displacement occurs with some delay after the external temperature peak (overall thermal expansion with greater dilation of the exterior face, i.e., ϑext >ϑint >ϑ0 ) and after the minimum value of the external relative humidity (general shrinkage of the wall still more prominent on the internal face, i.e., ϕint <ϕext <ϕ0 ). The displacements obtained for the wall with cement mortar (see Figure 11) follow the same trends described above for the LM case. However, the CM case shows larger deformations. This was expected considering the higher thermal expansion and hygric expansion coefficients of cement with respect to the lime mortar. Appl. Sci. 2023, 13, 4514 24 of 38 Figure 10. Hygro-thermo-mechanical analysis of the three-wythe brick masonry wall with natural hydraulic lime mortar: (a) imposed boundary conditions and displacement history measured at the top of the wall; (b) deformed shape selected for the study (maximum horizontal displacement). When considering the displacement history for the LM case (see Figure 10), it is possible to observe the initial deformed configuration of the wall before the application of the variable external boundary conditions. In particular, the wall presents a horizontal displacement towards the interior caused by moisture shrinkage of the internal face (𝜑= 0.50 <𝜑= 0.74), which prevails over thermal dilation (𝜗= 20 °C >𝜗= 13.5 °C). This horizontal displacement is approximately equal on the exterior and interior sides of the wall. On the other hand, the vertical displacement is rather small and remains always below 0.1 mm during the duration of the analysis. This vertical displacement is different on the exterior and interior sides of the wall due to rotation of the section. When the variable boundary conditions are implemented, the wall oscillates following the sinusoidal trend of the external temperature and relative humidity. In fact, the maximum horizontal displacement occurs with some delay after the external temperature peak (overall thermal expansion with greater dilation of the exterior face, i.e., 𝜗>𝜗>𝜗) and after the minimum value of the external relative humidity (general shrinkage of the wall still more prominent on the internal face, i.e., 𝜑<𝜑<𝜑). The displacements obtained for the wall with cement mortar (see Figure 11) follow the same trends described above for the LM case. However, the CM case shows larger deformations. This was expected considering the higher thermal expansion and hygric expansion coefficients of cement with respect to the lime mortar. (a) (b) Figure 11. Hygro-thermo-mechanical analysis of the three-wythe brick masonry wall with cement mortar: (a) imposed boundary conditions and displacement history measured at the top of the wall; (b) deformed shape selected for the study (maximum horizontal displacement). Figure 11. Hygro-thermo-mechanical analysis of the three-wythe brick masonry wall with cement mortar: ( a ) imposed boundary conditions and displacement history measured at the top of the wall; (b) deformed shape selected for the study (maximum horizontal displacement). The stress state obtained for the LM case under the self-weight condition (body force plus roof weight) is shown in Figure 12. As expected, the distribution of minimum principal stresses (see Figure 12a,b) reveals a uniform level of compression in bricks and bed mortar joints, around − 0.09 MPa for the selected area (note that compression increases with wall depth). Conversely, a 2D effect can be seen in the vertical head joints, which experience a lower compression level, around − 0.06 MPa. A more detailed analysis of the stress Appl. Sci. 2023,13, 4514 24 of 35 distribution in bricks brings to light two aspects: (a) the compression stress is slightly lower, around − 0.08 MPa, in the mid-section of the header course bricks, that is, the portion of the brick aligned with the vertical head joints; (b) higher compression values, ca. −0.10 MPa, are found on the vertical edges of the bricks adjacent to mortar head joints. Therefore, the brick–mortar interaction is associated with 2D phenomena that affect the distribution of stresses within the masonry wall. The maximum principal stresses obtained for the gravitational case reveal expected trends as well (see Figure 12c,d). On one hand, the positive component of the principal stress for bricks is practically zero, whereas it presents low positive values for the vertical head joints, around 5.0 × 10 −3 MPa. Even though the obtained values are not significant, this observation is consistent with the fact that head joints typically act as tension planes within the wall geometry. On the other hand, the results for the horizontal joints show small negative values for σ1 , around − 8.0 × 10 −3 MPa, which indicates that the bed mortar joints experience biaxial compression. Moreover, the maximum principal stress in the bed joints drops to zero in the regions near the external and internal faces of the wall. The distribution of stresses for the CM wall under gravitational loads is shown in Figure 13. It is noted that the obtained results are analogous to the ones of the LM case. In this sense, the distribution of minimum principal stresses for bricks and mortar bed joints (see Figure 13a,b) presents the same average compression, i.e., − 0.09 MPa. Likewise, the areas of the bricks aligned with vertical joints show lower stresses, ca. − 0.08 MPa. Some differences between the two cases can be pointed out as well. For instance, the compression values on the vertical edges of the bricks adjacent to mortar head joints are slightly higher, around −0.11 MPa. Moreover, the vertical joints show less compressive stresses, ca. −0.05 MPa. The distribution of maximum principal stresses for the wall with cement mortar (see Figure 13c,d) shows some differences with respect to the LM case as well. The positive component of the principal stress for bricks is still negligible for most of the wall, but higher values appear close to the brick corners, between 5.0 × 10 −3 and 6.0 × 10 −3 MPa. Furthermore, higher σ1 values can be found in the vertical head joints, ca. 7.0 × 10 −3 MPa. Although the obtained tensile stresses are low, they may contribute to crack propagation when cracks are already present. On the other hand, the horizontal joints show lower negative values, around − 0.01 MPa; thus, they undergo a biaxial compression state. Finally, the maximum principal stress in bed joints falls to zero in the regions near the faces of the wall. In conclusion, for the two studied models, the outcomes of the gravitational case reveal a complex 2D stress distribution derived from the multi-layered nature of the material. Moreover, the stress distribution results for the LM and CM walls are analogous, although slightly higher stress states are obtained for the wall with cement mortar. Regarding the HTM analysis, the stress distribution results obtained for the LM case are shown in Figure 14. It is recalled that the instant with maximum horizontal displacement is selected for the analysis (see Figure 10). At that moment, the imposed boundary conditions lead to shrinkage of the internal face of the wall. It is noted that the corresponding deformed configuration is intimately linked to the stress distribution. As expected, the distribution of stresses is not uniform across the wall thickness. Considering the minimum principal stresses (see Figure 14a,b), lower compression values are found in the middle portion of the wall, around − 0.07 MPa. Conversely, higher stresses appear near the external and internal faces, ca. − 0.12 MPa. The compressive stresses in bed mortar joints follow a similar trend with values around − 0.13 MPa near the wall boundaries. On the other hand, the results of σ3 in head mortar joints show lower compression levels, with values around − 0.05 MPa and − 0.04 MPa for the vertical joints located closer to the external and internal surfaces, respectively. Finally, the bricks show the same trends observed for the gravitational case: namely, lower compressive stresses in the areas aligned with the vertical joints and higher concentration of stresses near the vertical edges adjacent to mortar head joints. Appl. Sci. 2023,13, 4514 25 of 35 Appl. Sci. 2023, 13, 4514 26 of 38 (a) (b) (c) (d) Figure 12. Hygro-thermo-mechanical analysis of the three-wythe brick masonry wall with natural hydraulic lime mortar. Principal stress distribution obtained for the gravitational loads alone: minimum principal stress, 𝜎 [MPa], in (a) bricks and (b) mortar joints; maximum principal stress, 𝜎 [MPa], in (c) bricks and (d) mortar joints. The distribution of stresses for the CM wall under gravitational loads is shown in Figure 13. It is noted that the obtained results are analogous to the ones of the LM case. In Figure 12. Hygro-thermo-mechanical analysis of the three-wythe brick masonry wall with natural hydraulic lime mortar. Principal stress distribution obtained for the gravitational loads alone: minimum principal stress, σ3 [MPa], in ( a ) bricks and ( b ) mortar joints; maximum principal stress, σ1[MPa], in (c) bricks and (d) mortar joints. Appl. Sci. 2023,13, 4514 32 of 35 the implementation of mechanical boundary conditions as well as the initial or free-strain conditions. A careful definition of initial and boundary conditions is of special importance for mechanical analyses since the appearance and distribution of stresses largely depend on these parameters. In particular, when considering the non-linear behavior of the coefficient of hygric expansion, the definition of the initial moisture conditions (and, therefore, the associated free-strain state) greatly affects the stress evolution. The three-wythe brick masonry model defined for the hygro-thermal analysis was further used to study the internal distribution of stresses of the wall subjected to temperature and moisture gradients. In particular, the temperature and moisture results from the time-dependent analysis TD2 were used as input for the mechanical study. The instant of maximum displacement was chosen, and the corresponding stress distribution induced by the hygro-thermal loads was compared with the self-weight scenario. The analysis focused on a section of the mid-portion of the wall in order to avoid disturbances caused by the proximity of loads or boundary conditions. It was proven that temperature and moisture variations change the natural stress distribution of the wall subjected to gravitational loads alone. Moreover, the brick wall with cement mortar underwent higher stress levels in comparison with the wall with natural hydraulic lime mortar. This was caused by the higher thermal expansion and hygric expansion coefficients of cement in comparison with lime mortar. Although the induced tensile stresses were low, they may contribute to crack propagation of existing cracks. Similarly, the hygro-thermally induced deformations may add up to previous damage and magnify existing problems such as tilting in structures with moisture deterioration at the base or soil permanent deformation. Finally, it is noted that the hygro-thermo-mechanical behavior of the brick wall is a 2D phenomenon derived from the multi-layered nature of masonry. The numerical approach presented in this paper can be used to evaluate the hygrothermal performance of masonry walls as well as to complement the structural assessment when considering other static or dynamic loads. The methodology can be further extended to study the compatibility between masonry units and mortars. Taking into consideration the everyday nature and relevance of hygro-thermo-mechanical problems, the need for further investigation on this topic is evident. Similarly, experimental studies aimed at acquiring data for validation of the corresponding models are needed. Future works should extend the hygro-thermo-mechanical model to incorporate non-linear material behavior, with different constitutive laws for tension and compression and the capacity to reproduce creep. Likewise, improvements could be achieved by extending the simulations to 3D models, which are more suitable to reproduce complex hygro-thermal phenomena such as thermal bridges in envelope walls. Moreover, the study may be expanded to incorporate different heat and moisture regimes according to specific climatic zones. Finally, considering the cyclic nature of hygro-thermal phenomena, the study of material fatigue is strongly recommended. Author Contributions: Conceptualization, R.R., B.G., P.P. and P.B.L.; methodology, R.R., B.G., P.P. and P.B.L.; software, R.R.; validation, R.R.; formal analysis, R.R.; investigation, R.R.; writing—original draft preparation, R.R.; writing—review and editing, B.G., P.P. and P.B.L.; visualization, R.R.; supervision, B.G., P.P. and P.B.L.; project administration, P.B.L.; funding acquisition, P.B.L. All authors have read and agreed to the published version of the manuscript. Funding: This work was partly financed by FCT/MCTES through national funds (PIDDAC) under the R&D Unit Institute for Sustainability and Innovation in Structural Engineering (ISISE), under reference UIDB/04029/2020, and under the Associate Laboratory Advanced Production and Intelligent Systems ARISE under reference LA/P/0112/2020. The first author acknowledges the financial support provided by FCT—Foundation for Science and Technology, under grant agreement SFRH/BD/117114/2016. The APC was funded by MDPI’s Invited Paper Program. Institutional Review Board Statement: Not applicable. Data Availability Statement: Not applicable. Conflicts of Interest: The authors declare no conflict of interest. Appl. Sci. 2023,13, 4514 33 of 35 References 1. Ghiassi, B.; Lourenço, P.B. (Eds.) Long-Term Performance and Durability of Masonry Structures: Degradation Mechanisms, Health Monitoring and Service Life Design, 1st ed.; Woodhead Publishing: Cambridge, UK, 2019; ISBN 9780081021101. 2. Brooks, J.J. 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