The unsaturated porous media model described in the preceding section were developed as part of θ-STOCK, a powerful computational package for thermo-hydro-mechanical analysis of porous media [
9]. In this section, two numerical problems are designed to show the robustness of the developed model associated with climate change. In the first example (Problem 1), the isothermal propagation of two cracks in an unsaturated porous plate is investigated. The effects of some factors such as porosity, intrinsic permeability, and initial crack length on crack propagation are examined, followed by a comparison with single crack propagation in that plate. In the second example (Problem 2), the more practical problem is simulated by considering crack propagation in a masonry wall due to drying shrinkage. Due to the climatic loadings and the difference between material properties, the plaster on the outer layer of a peripheral wall dries and bends in a hole created between the bricks. Lastly, a parametric study is carried out on the outside temperature and mass transfer coefficient to determine their effects on the obtained results.
3.1. Edge Discontinuities in Unsaturated Porous Media
This problem is designed to investigate the isothermal behavior of an unsaturated porous plate for verification purposes of the proposed model. The problem is solved in a single edge crack by [
20] and two discontinuities are considered in this study. A square plate in the plane strain conditions is given, where two existing cracks with an initial length of 0.05 m are aligned, as in
Figure 3. The top and bottom surfaces of the domain are stretched out with the constant displacement rate of
. All boundaries are assumed to be impermeable to the moisture.
In order to detect the presence of discontinuities via the XFEM, the level-set method is implemented. To define the geometry of each crack independently, three separate functions are required; one represents the crack body, and the other two show crack tips. For a large number of fractures, the calculation of Level Set functions for each crack requires high computational cost. To resolve this problem, the Level Set functions are only calculated for a narrow band of nodes around each crack [
25]. Moreover, the leakage flux is calculated for each crack based on Equation (9), and then is added to general equilibrium equations. The material properties are presented in
Table 1.
Due to the geometrical symmetry of the problem, two tensile cracks are initiated and propagate toward each other along the horizontal direction. Since the stress significantly varies around the crack tip, the nonlocal stress may better capture the onset of crack propagation. For stress regularization around a specified radius in the neighborhood of the crack tip, the averaged stress values are calculated instead of the real crack tip stresses. When the value of this nonlocal stress exceeds the material’s cohesive strength, the crack starts to propagate. The length of the crack extension is determined based on the stress value at the Gauss points on the potential crack trajectory. If this stress on both Gauss points of an element surpasses the material’s tensile strength, the crack bisects that element. In this manner, it is possible that the crack can pass several elements in one step. The crack tip may exist inside an element. To satisfy the zero openings for crack tip displacement, however, it must be located on the edge of the element where the connected nodes are not enriched.
The normal gradient of the suction presented in
Figure 4d indicates moisture leakage flux. The normal gradient of suction has opposite values on the two sides of discontinuities, and the difference between normal gradients of suction on two sides of fractures activates the leakage flux term. As a consequence of moisture leakage flux, moisture is absorbed to the discontinuity, and it ends in the change of water content around the cracks. Thus, as depicted in
Figure 4c, the degree of saturation decreases around the cracks.
To better analyze the distribution of the cohesive traction along the crack, the normal stress and crack opening at different times of crack propagation are presented in
Figure 5a,b, respectively. As the crack propagates, traction-separation force is applied to the crack body. It is clear that the stress value at the crack tip reaches the cohesive strength of the material where the crack has zero openings. The value of cohesive traction decreases from tensile strength for zero openings to zero for initial notches.
To minimize the crack opening and inhibit crack propagation, as is shown in
Figure 5c, the suction increases around the fracture. As the crack opening increases at a specific point, the exerted cohesive force on the crack body decreases. At that point, suction plays a more active role as a preventative force against the crack opening. Still, since the intensity of suction is much lower than the cohesive force, its influence on the reduction in the crack aperture at that point is less that the cohesive force influence. As depicted in
Figure 5d, the degree of saturation is also significantly affected by the presence of the cracks and varies intensely along the crack up to the maximum value of one at the crack tip where the crack is closed and there is no leakage flux.
To better understand the impact that the presence of two cracks has on the unsaturated porous plate behavior, a comparison is made with the case when a single crack propagates from the left edge of the same plate. The distributions of various variables for the case of single crack propagation are presented in
Figure 6 at different times.
As expected, the distributions of cohesive traction imposed on the crack body have the same upper limit in the two cases of one and two cracks. However, since the crack mouth opening in the final steps of single crack case analysis is more than its double crack counterpart (see
Figure 5b and
Figure 6b), and as cohesive force depends directly on the crack opening, cohesive force at the crack mouth of single crack case analysis becomes less than its double crack counterpart at the final steps (
Figure 5a and
Figure 6a). To compensate for a less cohesive force in a single crack case, the crack mouth suction increases more in this case than its double crack equivalent to prevent further crack mouth opening. This behavior accentuates the preventative role of suction to inhibit crack propagation. In addition, this comparison depicts the ability of the proposed model in its successful demonstration of natural force for preventing crack to propagate.
To compare the load carrying capacity of the plate in the presence of two cracks with the case of single crack, load–displacement curves for the two cases are depicted in
Figure 7. Since in the case of double cracks, the load is transferred via the suction instead of the solid matrix in a longer width along with the plate, two graphs are different from the beginning. The higher load carrying capacity is observed in the case of a single crack, the more displacement is observed to reach the failure of the plate.
Sensitivity analysis is further performed to investigate the effect of permeability and initial crack length on the propagation of two cracks in the unsaturated porous plate. A series of reference permeability values and initial crack lengths are adopted to examine their influences on the response of the model. When the reference permeability is high, much fewer suction gradients are required for water exchange that leads to lower suction around the fracture (
Figure 8a), which results in higher normal displacement (
Figure 9c). With lower permeability, on the other hand, the water exchange needs a higher suction gradient that affects a smaller zone around the crack (
Figure 8). This larger suction gradient generates higher suction, reducing the crack opening displacement (
Figure 9c). The load carrying capacity curves of the plate for different intrinsic permeabilities that are shown in
Figure 10 are supportive of the previous explanation. As intrinsic permeability decreases, suction increases around the crack, and the ultimate bearing capacity of the plate rises. To investigate the effect of initial crack length, as the initial crack length increases, load carrying capacity of plate decreases (
Figure 11) since the initial stiffness of plate decreases.
3.2. 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 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:
where
is the mass transfer coefficient that is considered to be equal to
for the outdoor environment and
for the indoor environment.
and
are environmental and model vapor pressures, respectively [
22]. Vapor pressure is obtained from the following equation:
where
is the saturated vapor pressure per Pascal, which is retrieved from the following [
22]:
It is assumed that the outdoor environment has a temperature of 40 °C and relative humidity of 50%. The temperature and relative humidity of indoor environment are taken equal to 10 °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]. If
and
for brick and
and
for plaster are considered (Equation (5)), a good agreement is found with recommended surfaces of [
22]
Figure 13.
At the first step of the simulation, the moisture reduces quickly in the surface layer of the plaster. To make up for the moisture reduction in the surface layer, moisture flows from bricks toward the plaster. Since the air entry pressure is much more in the plaster than in the bricks, as is shown in
Figure 14c, the degree of saturation reduces more in the bricks compared to the plaster at their interface. As the cavity behind the plaster hinders the absorption of moisture by the plaster, the shrinkage of the plaster will not be uniform, and the plaster bulges to the cavity. The maximum stress that develops in the inner surface of plaster in place of the cavity causes nucleation of fracture in the plaster.
With the continuation of drying, the crack progresses and bisects the plaster in the initial steps of simulation. The increase in temperature along the crack over time (
Figure 15a) leads to the drying of plaster in the neighborhood of the crack, followed by the increase in suction around the crack (
Figure 15b) that prevents excessive changes of opening along the crack due to the drying (
Figure 15c). In other words, as a result of the increasing temperature, the degree of saturation decreases (
Figure 14c) with increasing suction (
Figure 14b) that prevents the crack from further propagation. This process limits an excessive change of opening along the crack (
Figure 15c).
The deformation and distribution of suction, saturation, and wall temperature two hours after the beginning of the analysis are shown in
Figure 14. As the crack opens, suction increases around the crack (
Figure 14b) to prevent further fracture opening. The growth of the suction causes absorption of the moisture to the crack and reduction in the degree of saturation in the proximity of the crack (
Figure 14c).
The effects of the outside temperature and mass transfer coefficient are investigated on the obtained results. In addition to the exterior temperature of 40 °C, two temperatures of 30 and 50 °C are considered as outside temperatures as well. By increasing the temperature of the outdoor environment and as the time passes, the temperature increases at the crack mouth (
Figure 16a). The rise in the temperature at the crack mouth induces more drying and, hence, more suction at the crack mouth (
Figure 16b), but as it is evident in
Figure 16c, the increased suction at the crack mouth is insufficient to reduce the crack’s mouth opening displacement (CMOD) as outside temperature rises. In order to study the response of the model to the outside mass transfer coefficient, two values of
and
are considered in addition to the present mass transfer coefficient for the outdoor environment. As the mass transfer coefficient increases for the outdoor environment, surface drying of the plaster happens faster, and suction increases more at the mouth of the crack (
Figure 17a). The increase in the suction due to the rise of the mass transfer coefficient reaches the extent that leads to the decline of CMOD (
Figure 17b), but as drying of the plaster happens faster, the analyses terminate prematurely for higher mass transfer coefficients.