*5.3. Water Permeability*

Wagner et al. [96] made water permeability of strain-hardening cement-based composites dependent on quantitative parameters of the cracking patterns. The samples that were pre-cracked as a result of the uniaxial tension test were tested. To better describe the relationship between the cracking pattern and water permeability a new parameter was introduced, which was called the hydraulic crack pattern parameter *RWP*. It was the product of chosen class width, the crack density and weighted crack width value. There was a strong correlation between *RWP* and the water flow rate. In the course of the analyses, an equation binding the crack width with water permeability was also formulated for which the Hagen–Poiseuille equation was the starting point.

Lepech and Li [108] studied the effect of cracking degree of high performance fiber reinforced cementitious composites on water permeability. They limited the characteristics of the cracking pattern to determining the maximum crack width and to counting the number of cracks in the analyzed section. They found that with the increase of crack width, the value of the coefficient of permeability also increases. Moreover, normalized permeability per single crack was calculated. It was observed that this parameter also increases with the increase in crack opening width, although the number of cracks in the analyzed section is higher. The study also indicated a beneficial effect of PVA fibers in the context of reducing crack opening width and thus increasing water-tightness of the cement composite.

In the studies carried out by Aldea et al. [109], the relationships between cracks and water permeability of a normal strength concrete were observed. A controlled splitting tensile test was used to introduce cracks into the concrete structure and then the material was tested for water permeability. The results obtained indicated that water permeability increased significantly as the opening width of the cracks increases. Moreover, high repeatability of results for the same cracking levels was obtained. It was also found that the relationship between crack length and water flow is not linear, which was found for the relationship between the crack area and water flow.

An extensive research program was carried out by Torrijos et al. [112]. Two series of ordinary concretes were tested. Within the framework of the conducted works, the influence of the characteristics of the cracking patterns, expressed in the form of crack width and the crack density, was determined on a number of parameters related to permeability, i.e., the water absorption, water penetration, capillary absorption, water permeability. The cracking pattern was created by subjecting the samples to various degradation processes, including: low humidity drying, load at 150 ◦C and 500 ◦C and alkali-silica reaction. The highest crack density was found for concretes subjected to 500 ◦C load, cracks mainly occurred in the interface and to a lesser extent in the cement matrix. In case of samples subjected to drying in low humidity it was found that the velocity and capacity of capillary absorption increased with the crack density. This increase occurred to a certain maximum and then decreased, although the crack width and the crack density continued to increase. For concretes exposed to the alkali-silica reaction a high permeability was initially noticed, followed by a sharp drop after the first few hours. Rehydration of the gel at the crack sites was given as the reason, which resulted in sealing of the structure. The degree of internal destruction of the samples was evidenced by measuring the ultrasonic wave flow rate. A direct relationship between the pulse velocity and the crack density was found.

## *5.4. Mechanical Strength*

Fahridzadeh et al. [105] proposed a new approach to assess the degree of degradation of reinforced concrete structures. They proposed to use the fractal dimension to describe the morphology of the cracking pattern on the surface of reinforced concrete. The effect was to develop the damage index, which was calculated according to the equation:

$$DI = \frac{D\_i - D\_l}{2 - D\_l} \tag{4}$$

where:

*DI*—the damage index; 0 ≤ *DI* ≤ 1,

*Di*—the fractal dimension of the current status of the cracking pattern,

*Dl*—the fractal dimension of the cracking pattern computed during the first inspection.

The first inspection shall be considered to be the moment when the appearance of cracks on the surface of the component under test is observed as a result of the applied load. The tests carried out on the reversed cycled loaded reinforced concrete walls indicated high practical suitability of *DI*. It was found that the value of both the fractal dimension and *DI* increases with the next load cycle. It was also shown that *DI* can be used to estimate the remnant lateral stiffness of the wall.

The continuation of the above studies is presented in [115,117]. It was noted that in the literature there are practically no works that would correlate the morphology of the cracking patterns with the structural integrity of the studied material, in this case concrete. The multifractal analysis was used in the study and the considerations were carried out on synthetic crack patterns as well as on two real reinforced concrete walls subjected to cyclic mechanical loading. It was found that the development of cracks structure through an increase in their opening width, length, density, is accompanied by a correlated increase of the multifractal parameters. A rapid increase of these parameters was also observed when the concrete walls tested showed severe stiffness loss.

The subject of Ebrahimkhanlou et al. [118] was the analysis of the cracking patterns on the surface of pre-stressed concrete beams. The analysis was focused on the fractal and multifractal characteristics of cracks patterns, which were formed by a mechanical force. Once again, the high usefulness of the fractal dimension for quantitative description of the degree of development of the cracking patterns was confirmed. Moreover, it was proved that fractal analysis can be used to identify different cracking mechanisms.

Fooladi et al. [119] conducted research to determine the effect of the aggregate grading on the multifractal characteristics of the cracking patterns of concrete, with different compressive strengths. The study was carried out on two series of concretes for which the aggregate grading curve assumed the well-graded and gap-graded characteristics respectively. In the well-graded aggregate samples, it was observed that the value of the fractal dimension reached a saturation level, for which the increase in load did not cause any further change in the value of the fractal dimension. For samples with the gap-graded aggregate a continuous increase in the value of the fractal dimension was observed along with an increase in the value of mechanical load. In the study, the singularity spectrum analysis was also carried out, which, combined with the analysis of the fractal dimension, allowed to conclude that the development of cracks in both series of samples is characterized by multifractal character, but with different intensity. The study concluded that the multifractal approach allows to explain local irregularities of the cracking pattern.
