## 3 Numerical Examples
### Crack in a Drying Wall  
In this example, we design a problem of masonry wall associated with climate change. The stability issues of the masonry wall are usually caused by the difference of the mechanical and thermal properties between the bricks and their joins, which are mostly made up of mortar. Due to this feature of masonry wall, cracks can be easily triggered and propagated, which can be further accelerated due to a severe change of the environmental condition, that is, the thermal difference between the inside and outside of the wall. In order to show the capabilities of the proposed model, the simulation of a failure of masonry wall under environmental loading is investigated. In this problem, a cross section of two wythes common band of masonry wall is considered in Figure 12. The left figure depicts the whole cross section of a two wythes common band of masonry wall, where a representative part of the wall for modeling is presented in the right figure. A brick is used on the upper part of the wall, while two bricks are placed on its lower part. The bricks' dimensions are presented on Figure 12, and a thin layer of mortar with a thickness of 0.01 m is used between them (dark gray color). The outer surface of the wall is insulated with a 0.01-meter plaster, while no plaster is considered on the inner surface of the wall. For spatial discretization of the model, 2210 bilinear elements are used that their boundaries are matched with material interfaces. The displacements in the y-direction  
Figure 11: Load–displacement for different initial crack lengths (the effect of initial crack length on the proposed model considering two existing initial cracks). Figure 12: The geometry and mesh of the model—the area depicted with dash line (left) is modeled in Problem 2. To the centerline for the right figure (CL) boundary condition is applied. of elements along the axis of symmetry of the model (C.L line) are bounded to zero and no moisture and heat exchanges with the external environment are considered for those elements. The displacement of the node on the bottom-left of the model is restricted in the x-direction to prevent its rigid body motion. The effects of gravity are neglected, and the inner and outer surfaces of the model are exposed to the environment, i.e., they could exchange moisture and heat with the environment. For the moisture exchange with the environment through evaporation or condensation, the following relation is used:  
\[q_{m}=\eta(P_{V}^{env}-P_{v}) \tag{19}\]  
where \(\eta\) is the mass transfer coefficient that is considered to be equal to \(1.4\times 10^{-10}\frac{\text{kg}}{\text{m}^{2}\text{s}}\) for the outdoor environment and \(1.85\times 10^{-11}\frac{\text{kg}}{\text{m}^{2}\text{s}}\) for the indoor environment. \(P_{V}^{env}\) and \(P_{V}\) are environmental and model vapor pressures, respectively [22]. Vapor pressure is obtained from the following equation:  
\[P_{v}=P_{v}^{sat}h \tag{20}\]  
where \(P_{V}^{sat}\) is the saturated vapor pressure per Pascal, which is retrieved from the following [22]:  
\[P_{V}^{sat}=exp\bigg{(}65.8094-\frac{70.66.27}{T}-5.976\ln(T)\bigg{)} \tag{21}\]  
It is assumed that the outdoor environment has a temperature of 40 \({}^{\circ}\)C and relative humidity of 50%. The temperature and relative humidity of indoor environment are taken equal to 10 \({}^{\circ}\)C and 85%, respectively. The model is assumed to be intact initially. The nonlocal stress criterion is checked, along with the loading sequence. If the nonlocal stress passes the tensile strength of the corresponding material, the fracture is embedded into that element. Material properties used in the numerical simulations are given in Table 2. To determine the coefficients of the state surface of the degree of saturation for brick and plaster, the state surfaces' parameters are looked after that have the best fit to proposed surfaces of [22].