The Effects of Stochastic Circular Pores on Splitting Tensile Behavior of Concrete Based on the Multifractal Theory

Abstract

Concrete naturally contains a large number of circular-like stochastic pores which weaken the tensile strength of concrete and change the crack propagation path. This study investigates the influences of the size distribution and the spatial distribution of stochastic pores on the fracture behavior of concrete based on the splitting tensile test. The mesoscale model of concrete containing coarse aggregate, mortar, interface transition zone (ITZ), and circular pores is established to simulate the crack initiation, propagation, and coalescence of concrete. Concrete samples with a single hole are prepared to verify the effectiveness of the numerical simulation method. Numerical tests are conducted on numerous mesoscale concrete samples with various porosities, pore size distributions, and pore spatial distributions. The numerical simulation results indicate that the tensile strength decreases with the increase of pore size at the same porosity. Based on multifractal theory, a quantitative indicator to describe the spatial distribution uniformity of concrete stochastic pores is proposed. There is a positive correlation between the spatial distribution uniformity of stochastic pores and the tensile strength. The stochastic circular pores can have a profound effect on the concrete’s fracture pattern, which results in three typical macro-crack patterns in the numerical simulation of the splitting tensile test. The presented results deepen the understanding of the influence of stochastic circular pores on the tensile mechanical properties of concrete and provide a reference for the design of concrete structures.

1. Introduction

Concrete is the most widely used building material in the world. The mechanical behavior of concrete is difficult to accurately capture in experiments, while the analytical methods become very complicated due to the nonlinearity and randomness of concrete [1]. The numerical methods seem to become more effective to describe mechanical response, which makes it possible to explicitly study the influences of the mesostructure of concrete on mechanical behavior. At the mesoscale level, concrete is treated as a heterogeneous material consisting of coarse aggregate, mortar, interface transition zone (ITZ), and pore.

At first, concrete heterogeneity is considered as three phases, namely coarse aggregate, mortar, and ITZ [2,3]. The geometry [4,5,6], volume ratio [7,8], and size distribution [9,10] of coarse aggregates affect the mechanical behavior of concrete. The geometry of coarse aggregate was considered to be circular, elliptical, or polygonal, which was found to be sensitive to the tensile strength of concrete [6]. Wu et al. [8] pointed out the compressive strength decreases with increasing volume ratio of coarse aggregate when the maximum size of coarse aggregate is 10 mm. The ITZ parameter affects the stress-strain behavior of concrete, and the compressive strength increases with increasing ITZ strength and fracture energy [11]. The tensile strength of concrete increases when the ratio of ITZ to mortar strength increases [12].

Concrete naturally contains a large number of pores [13,14,15]. A series of technologies, computed tomography, mercury intrusion method, and ultrasonic scattering techniques, are widely used to measure the pore distribution of concrete. Scholars employed CT to construct the mesoscale model of concrete and found the lognormal distribution [16,17] is suitable for the pore size distribution of concrete and the spatial distribution of pores can be described by a uniform distribution [17]. Guo et al. [18] proposed that the range of small pores and large pores is 0~1 mm and greater than 1 mm respectively by ultrasonic scattering techniques and pointed out that the normal distribution can describe the small pores and the lognormal distribution can characterize the large pores.

With the deepening of research, we consider the heterogeneity of concrete from three phases to four phases, i.e., coarse aggregate, mortar, ITZ, and pore. There are two main methods to establish the mesoscale model of concrete containing pores, namely, image-based modeling and parametric modeling. The image-based modeling method uses computed tomography (CT) to obtain the mesoscale model of concrete containing pores [19,20]. The distribution of pores obtained by the image-based modeling method is realistic, but it is expensive to acquire hundreds of the mesoscale models using this method, and it is difficult to quantitatively describe the distribution of pores. The other method is that parametric modeling can control the key parameters, such as geometry, size and space distribution of pores, and porosity in the concrete mesoscale model. This method has been widely used in the mesoscale modeling of concrete [6,21,22].

The influence of pores on the mechanical behavior of concrete cannot be underestimated [23]. The porosity has a severe adverse influence on concrete, which cannot be neglected in the mesoscale fracture behavior of concrete [6]. The tensile strength of concrete decreases as the porosity increases under different specimen sizes due to the size effect of concrete [22]. Zhang Y et al. employed the cohesive element model to study the effect of the pore on the tensile and compressive behavior of concrete and found both tensile strength and compressive strength decrease with the increase of porosity [24]. For a given porosity, the reduction of pore size increases the static compressive strength of foam concrete under uniaxial compression, and the pore size is linearly related to the compressive strength in foam concrete with a uniform pore size [25]. It can be seen that porosity is the main factor for scholars to study the influence of the pores on the mechanical behavior of concrete. The size distribution and spatial distribution may affect the mechanical behavior of concrete. Yet, these factors have not been taken into account. Quantitative description of the spatial distribution of pores is the key to study the influence of the spatial distribution of pores on mechanical property. The multifractal theory has been used to investigate the uniformity of pore spatial distribution in cement paste [26,27]. The spectral width is generally introduced to describe the uniformity of different materials [28,29,30], which is an effective means to quantitatively characterize the pore spatial distribution uniformity of concrete. Thus, it is interesting to study the influence of the uniformity of pore spatial distribution on the mechanical behavior of concrete by the multifractal theory.

Concrete is a quasi-brittle material whose tensile strength is much lower than its compressive strength. The pore may weaken the tensile mechanical property and influence the safety, serviceability, and durability of concrete structures. Thus, it is significantly meaningful to study the effects of pore on tensile behavior. In this paper, a mesoscale model of concrete containing four phases of coarse aggregate, mortar, ITZ, and pore is established. The effect of stochastic pores on the tensile mechanical behavior of concrete is investigated by the splitting tensile test. Firstly, the concrete specimens with a hole are prepared, and the splitting tensile test is carried out to verify the numerical simulation method in this paper. Secondly, three crack patterns are compared by combining the stress-displacement curves and crack patterns. Thirdly, the effect of three pore size distributions on the tensile strength is discussed. Finally, an indicter based on the multifractal theory is proposed to investigate the uniformity of pore spatial distribution. The relationship between the indictor and the tensile strength is discussed in detail.

2. Numerical Simulation Methods

2.1. Cohesive Element Model

The cohesive element model is widely used to simulate crack initiation, propagation, and coalescence of concrete in numerical simulation [21,22], which is employed to simulate the process of crack propagation in this study. The cohesive element theory related to the constitutive response has been well documented in the ABAQUS user manual. Thus, this section merely points out the key model configurations. The Equations (1)–(7) are derived from the ABAQUS user manual.

The crack begins when the stresses satisfy certain damage initiation criteria. This criterion can be represented as

$${\left(\frac{\langle t_n\rangle }{t_n^o}\right)}^2+{\left(\frac{\langle t_s\rangle }{t_s^o}\right)}^2=1$$

(1)

where t_n and t_s represent the normal and shear tractions, respectively. $t_n^o$ and $t_s^o$ denote the peak normal and shear tractions, respectively. The Macaulay brackets $\langle \rangle$ are used to signify that a pure compressive stress does not initiate damage.

$$\langle t_n\rangle =\{\begin{array}{ll}{t}_n,\hfill & t_n\ge 0 \left(Tensile\right)\hfill \\ 0,\hfill & t_n<0 (Compression)\hfill \end{array}$$

(2)

The material stiffness begins to degrade once the corresponding damage initiation criterion is reached. The equivalent displacement and a scalar damage variable $D$ are introduced to describe the damage evolution under coupled normal and shear tractions.

$$\{\begin{array}{l}{\delta }_m^{\max }=\sqrt{{\langle {\delta }_n^{\max }\rangle }^2+{\left({\delta }_s^{\max }\right)}^2}\hfill \\ {\delta }_m^o=\sqrt{{\langle {\delta }_n^o\rangle }^2+{\left({\delta }_s^o\right)}^2}\hfill \\ {\delta }_m^f=\sqrt{{\langle {\delta }_n^f\rangle }^2+{\left({\delta }_s^f\right)}^2}\hfill \end{array}$$

(3)

$$D=\frac{{\delta }_m^f\left({\delta }_m^{\max }-{\delta }_m^o\right)}{{\delta }_m^{\max }\left({\delta }_m^f-{\delta }_m^o\right)}$$

(4)

where ${\delta }_m^{\max }$ denotes the maximum value of the effective displacement during the loading history. ${\delta }_m^o$ and ${\delta }_m^f$ refer to the effective displacement at damage initiation and complete failure, respectively. ${\delta }_n^{\max }$ and ${\delta }_s^{\max }$ represent the maximum value of normal and shear displacement during the loading history, respectively. ${\delta }_n^o$ and ${\delta }_s^o$ denote the normal and shear displacement at damage initiation, respectively. ${\delta }_n^f$ and ${\delta }_s^f$ represent the normal and shear displacement at complete failure, respectively. These corresponding parameters are shown in Figure 1. The range of the scalar damage variable $D$ from 0 (represents no damage) to 1 (represents complete failure). The displacement at complete failure can be obtained by the fracture energy and the peak tractions.

$$\{\begin{array}{c}{G}_{\mathrm{I}}={\displaystyle {\int }_0^{{\delta }_n^f}t\left(\delta \right)d\delta =\frac{1}{2}}{t}_n^o{\delta }_n^f\\ G_{\mathrm{II}}={\displaystyle {\int }_0^{{\delta }_s^f}t\left(\delta \right)d\delta =\frac{1}{2}}{t}_s^o{\delta }_s^f\end{array}$$

(5)

where G_I and G_II denote the mode I fracture energy and the mode II fracture energy, respectively.

The traction state during damage evolution is as follows.

$$t_n=\{\begin{array}{cc}\left(1-D\right)k_n{\delta }_n,\hfill & {\delta }_n\ge 0\hfill \\ k_n{\delta }_n,\hfill & otherwise (no damage to compressive stiffness)\hfill \end{array}$$

(6)

$$t_s=\left(1-D\right)k_s{\delta }_s$$

(7)

where k_n and k_s represent the normal and shear initial stiffness, respectively. δ_n and δ_s denote the normal and shear displacement without damage, respectively.

2.2. Size Distribution and Generation of Coarse Aggregate

The size distribution of coarse aggregate in concrete satisfies the Fuller gradation curve [31].

$$P\left(d\right)=100{\left(\frac{d}{d_{\max }}\right)}^n$$

(8)

where $P\left(d\right)$ is cumulative percentage of coarse aggregate passing through the sieve size. d_max refers to maximum diameter of coarse aggregate. The range of constant $n$ from 0.45 to 0.70. $n=0.5$ in this study [5].

Therefore, the amount of coarse aggregate in the gradation segment can be calculated according to Equation (8).

$$A_{agg}\left[d_i,d_{i+1}\right]=\frac{P\left(d_{i+1}\right)-P\left(d_i\right)}{P\left(d_{\max }\right)-P\left(d_{\min }\right)}\times A_{agg}$$

(9)

where A_agg[d_i, d_i+1] denotes the amount of coarse aggregate in the gradation segment [d_i, d_i+1]. d_max and d_min represent maximum and minimum of coarse aggregate, respectively. A_agg refers to the total amount of coarse aggregate.

In the numerical simulation, the geometry of coarse aggregate from circle to ellipse, and finally develops to the polygon. Its geometry is more and more consistent with the crushed coarse aggregate [32]. Thus, this study selects the polygon to represent the coarse aggregate in concrete. The steps for generating the coarse aggregate are as follows.

Step 1: Determine the diameter range of coarse aggregate and calculate the amount of coarse aggregate according to Equation (9). Step 2: Generate an auxiliary circle and judge whether the auxiliary circle intersects with the generated coarse aggregates. Step 3: Four points are randomly generated in the auxiliary circle to form a quadrilateral. Step 4: Generate polygonal with any number of sides from the quadrilateral. Determine the longest side of the quadrilateral, make the vertical line of the longest side, intersect two points with the auxiliary circle, determine the nearest point to the longest side, and obtain the fifth point, to generate a pentagon. In this way, polygonal with any number of sides can be generated. Step 5: Determine whether the amount of coarse aggregate meets the requirements.

2.3. Size Distribution and Generation of Stochastic Circular Pores

There are a large number of pores in concrete. The size of these pores can be described by the lognormal distribution [16,17]. The size of these pores greater than 1 mm still can be represented by the lognormal distribution [18] as Equation (10).

$$g\left(x\right)=\frac{1}{\sqrt{2\pi }bx}{e}^{-{\left(\ln x-a\right)}^2/\left(2b^2\right)},x>0$$

(10)

where $x$ denotes the diameter of pore. $a$ and $b$ are parameters.

In the numerical simulation, the geometry of the pore is considered to be circular, whose diameter range is 2~4 mm [5,22]. In this study, the pore range is widened to 1~4 mm. The size of stochastic circular pores is random and is described using Equation (10). The size distribution of stochastic pores may affect fracture behavior, which deepens the understanding of the influence of stochastic circular pores on the tensile mechanical properties of concrete. The spatial distribution of pores obeys uniform distribution in the concrete [17]. Thus, the pores are uniformly distributed in the concrete in this paper.

The porosity range is 1~5% according to reference [33], which is 1%, 2%, 3%, 4% and 5%, respectively. For each porosity, three groups of pore size distribution conforming to Equation (10) are considered, as shown in Figure 2. The ‘S’ in Figure 2a represents the parameters $a=0.18, b=0.40$ and expectation 1.30. The ‘M’ in Figure 2b denotes the parameters $a=0.89, b=0.21$ and expectation 2.49. The ‘L’ in Figure 2c represents the parameters $a=1.32, b=0.20$ and expectation 3.82. The expected values of pore size distribution ‘S’, ‘M’, and ‘L’ are 1.30, 2.49 and 3.82, respectively. Naming rules: for example, P01-S, ‘P01′ represents the porosity is 1%, and ‘S’ indicates the pore obeys the lognormal distribution with parameters $a=0.18, b=0.40$.

The steps for generating stochastic circular pores are as follows. Firstly, to obtain a stochastic circular pore, a stochastic number satisfying Equation (10) is generated in the diameter range of 1~4 mm. Secondly, judge whether the stochastic circular pore intersects with the generated coarse aggregates and pores. Finally, determine whether the porosity meets the requirements (The porosity range is 1~5% in this study).

2.4. Insertion Method of Cohesive Element

The zero-thickness cohesive elements are inserted between solid elements to simulate potential initiation and propagation of crack with the cohesive element model. When the scalar damage variable $D$ is equal to 1, the cohesive element completely failed. The cohesive element is deleted, thus forming a crack. The steps for cohesive element insertion are as follows. Firstly, the mesoscale geometric model is meshed to obtain the solid elements of aggregate and mortar. The triangular elements are utilized for the solid elements because they are flexible in modeling realistic crack propagation paths [6].

Secondly, renumber the nodes of each aggregate and mortar solid elements as shown in Figure 3. Insertion of aggregate cohesive elements (Coh-agg) between aggregate solid elements. Insertion of mortar cohesive elements (Coh-mor) between mortar solid elements. Insertion of ITZ cohesive elements (Coh-ITZ) between aggregate solid elements and mortar solid elements. Finally, output the information of Coh-agg, Coh-mor, Coh-ITZ, and renumbered aggregate and mortar solid elements to the file.

3. Validation of Numerical Simulation

Compare the crack propagation and tensile strength of concrete specimens in numerical simulation and experiment to verify the validity of cohesive element model parameters in numerical simulation.

3.1. Splitting Tensile Test of Single Circular Hole Concrete

The splitting tensile test is widely used to obtain the tensile strength of concrete and rock materials. This method is recommended by The International Society for Rock Mechanics (ISRM) and American Society for Testing Material (ASTM) standards [34,35,36]. In this study, the tensile strength of concrete is obtained using the splitting tensile test, which is calculated as

$${\sigma }_T=\frac{2P}{\pi Dt}$$

(11)

where $P$ denotes the peak load, $D$ represents the diameter of specimen, $t$ is the thickness of specimen.

The raw materials for concrete are as follows: the coarse aggregate is the crushed stone with the particle size range from 5 mm to 10 mm. Its apparent density is 2410 kg/m³, and the bulk density is 1350 kg/m³. The cement is selected from 32.5R ordinary Portland cement. The fine aggregate is river sand with a fineness modulus of 2.8. The mix proportion is shown in

Table 1. The mix proportion for concrete (kg/m³).

Cement	Sand	Coarse Aggregate	Water
454	617	1121	209

.

This study prepares two types of concrete specimens, namely intact cylinder specimen and central hole cylinder specimen, as seen in Figure 4. The diameters of the hole are 10 mm and 16 mm, respectively. In the concrete pouring, a plastic pipe is used to prefabricate a circular hole at the center of concrete specimen. Remove the plastic pipe after initial setting of concrete. Nine specimens are poured for this study. Each group contains 3 specimens.

The experiment device of the splitting tensile test is shown in Figure 5. The material testing machine is carried out with a force loading rate of 0.5 kN/s according to the industry standard [37]. A high-speed camera with a sampling frequency of 500 Hz is used to capture the crack propagation of concrete.

3.2. Validation of the Cohesive Element Model

The geometry and size of the concrete specimen in the numerical simulation are the same as the experiment in Section 3.1, as shown in Figure 4. The concrete specimen is located between the lower and upper platens as illustrated in Figure 6. The lower platen is fixed, and the upper platen moves downward. An average element size of 1 mm and loading time in ABAQUS/Explicit solver of 0.01 s are adopted in the numerical simulation according to the reference [38].

According to reference [39], the properties of the mortar and aggregate solid elements are illustrated in

Table 2. The properties of cohesive elements.

Parameters	Value
	Aggregate	Mortar	ITZ
Solid elements [39]
density, $\rho$ (kg/m3)	2500	2200
Young’s modulus, E (GPa)	47.2	29.2
Poisson’s ratio, $\upsilon$	0.2	0.2
Cohesive elements
$\mathrm{Tensile} \mathrm{strength}, t_n^o$ (MPa)	16	4.2	2.1
$\mathrm{Shear} \mathrm{strength}, t_s^o$ (MPa)	45	36	18
Mode I fracture energy, GI (N/mm)	0.16	0.054	0.027
Mode II fracture energy, GII (N/mm)	0.45	0.36	0.18
Initial normal stiffness, kn (MPa/mm)	6.0 × 105	6.0 × 105	6.0 × 105
Initial shear stiffness, ks (MPa/mm)	2.5 × 105	2.5 × 105	2.5 × 105

. Then, the initial stiffness of the cohesive elements is obtained by the Zhou et al. [40] as follows:

$$\{\begin{array}{l}{k}_n=\frac{aE}{l_e}\hfill \\ k_s=\frac{cG}{l_e}=\frac{cE}{2l_e(1+\upsilon )}\hfill \end{array}$$

(12)

where $G$, $E$ and $\upsilon$ denote shear modulus, Young’s modulus and Poisson’s ratio, respectively. l_e represents the average size of the solid elements. $c$ is a constant greater than 1. When $c$ greater than 10, the effect of the cohesive elements on the stiffness is negligible [39].

The properties of ITZ cohesive elements are half of mortar cohesive elements [5,38]. Then, compare the crack propagation and tensile strength of concrete specimens in numerical simulation and experiment to determine the properties of cohesive elements in Table 2.

Next, verify the validity of the properties of cohesive elements.

Figure 7a–i show the crack propagation of the contact specimen, pore diameter 10mm specimen and pore diameter 16mm specimen, respectively. As shown in Figure 7, for the intact concrete specimen, the cracks appear near the center of the specimen (The red arrow in Figure 7b), and they propagate toward both ends of the specimen as the load increases. For central hole concrete specimen, the cracks start near the hole (The red arrows in Figure 7e,h), and they propagate toward both ends of the specimen as the load increases. It can be seen from Figure 7 that the crack path in numerical simulation is consistent with the experiment.

The tensile strength of concrete is calculated by Equation (11), as shown in

Table 3. The error of Tensile strength between the experiment and numerical simulation.

Specimen Types	Mean Tensile Strength of Experiment (MPa)	Tensile Strength of Numerical Simulation (MPa)	Error (%)
Intact	4.00	4.09	2.25
10 mm-hole	3.25	3.10	−4.61
16 mm-hole	2.48	2.58	4.03

. The error between the experiment and numerical simulation of intact specimen, central 10 mm-hole specimen and central 16 mm-hole specimen is 2.25%, −4.61% and 4.03%, respectively. The maximum error of tensile strength is less than 5%, which indicates that the error between the experiment and numerical simulation is small.

The comparison between numerical simulation and the experiment is carried out from two aspects of crack path and tensile strength, which verifies the validity of the cohesive element model in Table 2.

3.3. Element Size (Mesh Density)

The fine mesh can reduce the dependence of numerical simulation results on the size of elements, but this increases the computational cost. Therefore, a mesh convergence study is performed to find a balance between mesh dependence and computational efficiency. This study conducts three groups of average element size (i.e., L = 0.8 mm, L = 1.0 mm, and L = 1.2 mm). The stress-displacement curves for three groups of element size are shown in Figure 8. The stress is calculated according to the Equation (11). The stress-displacement curves of 0.8 mm and 1.0 mm element size are in good agreement. It can be observed that the mesh dependence of the stress-displacement curves is ignored for the models when the average element size is less than or equal to 1.0 mm.

Consequently, considering the balance between mesh dependence and computational efficiency, the average element size is selected to be 1.0 mm in the following simulations. For the models with pores, the average size element of the pore is 0.5 mm due to the size of the pore is small. The element size of the pore is shown in Figure 3.

4. Simulation Results and Discussion

This study employs the Monte Carlo method to investigate the effect of stochastic circular pores on splitting tensile behavior of concrete. Thus, a large number of Monte Carlo samples are prepared. The geometry and spatial distribution of coarse aggregate remain the same, only changing the porosity, spatial distribution and size distribution of pores.

4.1. Number of Monte Carlo Samples

The Monte Carlo method is a method to solve mathematical and physical problems by repeated tests and is widely used to study the effect of mesoscale structure on the mechanical behavior of concrete. This study investigates the influence of stochastic circular pores on the splitting tensile behavior of concrete using the Monte Carlo method.

The Monte Carlo method requires numerous repeated tests. Therefore, it is important to obtain the number of repeated tests. Taking P01-L as an example, the number and size of circular pores remain the same and only change the spatial distribution of the circular pores. The number of numerical concrete samples containing the circular pores is 60. The stress-displacement curves of 60 samples are shown in Figure 9a. The peak stress in each stress-displacement curve in Figure 9a is the tensile strength of the sample. The mean tensile strength of the first n tests is shown in Figure 9b. The mean tensile strength of the first 30 samples is the same as that of the first 60 samples, which is 3.85 MPa. Figure 9b shows the error between the mean tensile strength of the 60 samples and the 60th sample. From the 30th sample, the error range is between −0.23% and 0.12%, which is small.

Thus, in the following numerical simulation, 30 Monte Carlo samples are used for each type of pore size. In total, 450 Monte Carlo samples are generated and analyzed in this investigation.

4.2. Crack Patterns

Three typical macro-crack patterns are observed from the Monte Carlo simulations under splitting tensile test: Type I cracking that appears at the center of the sample illustrated in Figure 10a; Type II cracking that appears at the eccentricity of the sample showed in Figure 10a; and Type III cracking that occurs at the loading end of the sample illustrated in Figure 10a. Figure 10b shows the crack propagation process. Figure 10b demonstrates the load-displacement curves of the three typical macro-crack patterns. Combine the load-displacement curve and crack propagation to find the relationship between the load and the crack propagation. ‘A’ represents the unloaded state; ‘B’ represents the peak load; ‘C’ represents the next frame of the peak load; ‘D’ represents the final damage. To observe the crack propagation process of the sample, the deformation magnification factor is 30.

Type I cracking: The deformation of the sample is small when the load is far from the peak value. When the load is close to the peak value, the deformation of the sample increases rapidly and mainly appears near the center of the sample. In the next frame after the peak load, the cracks occur at the center (see the black arrow in Figure 10a) of the sample; subsequently, the cracks propagate symmetrically from the sample center to the loading end. The damage of the sample is symmetrical.

Type II cracking: The deformation of the sample is small when the load is far from the peak value. When the load is close to the peak value, the deformation of the sample increases rapidly and mainly appears near the center of the sample. In the next frame after the peak load, the cracks occur at the eccentricity (see the black arrow in Figure 10a) of the sample. Subsequently, the cracks propagate from the eccentricity of the sample to the loading end.

Type III cracking: When the load reaches its peak value, the deformation of the sample center is less than the Type I cracking and the Type II cracking. In the next frame after the peak load, the cracks occur at the loading end (see the black arrow in Figure 10a) owing to the stress concentration caused by the pore. Subsequently, the cracks propagate from the loading end to the other loading end.

The pore changes the position of crack initiation thus affecting the crack pattern in the splitting tensile test. There are three types of crack initiation positions, i.e., center, eccentric, and loading end, resulting in three typical macro-crack patterns.

The splitting tensile test requires that the crack initiation appears near the center of the sample according to the principle of the splitting tensile test [41,42]. Type I cracking and Type II cracking meet the requirement. However, Type III cracking does not meet the requirement. This study obtains the tensile strength by the splitting tensile test. Thus, when discussing the influence of stochastic circular pores on tensile strength, the sample with Type III cracking is not considered.

Figure 11 shows the typical crack path of the concrete specimens with different porosity and pore size distribution. There are different crack propagation paths of specimens with different porosity and pore size distribution, which indicates that the stochastic circular pores change the crack propagation path.

4.3. Effect of Stochastic Pores Size Distribution on Tensile Strength

Table 4. Effect of stochastic pore size distribution on tensile strength of concrete.

Porosity	Tensile Strength (MPa)
	S	M	L
1%	3.90	3.87	3.85
2%	3.74	3.70	3.68
3%	3.59	3.55	3.51
4%	3.49	3.43	3.37
5%	3.32	3.29	3.27

Note: ‘S’ represents the expectation of 1.30 for pore size distribution. ‘M’ denotes the expectation of 2.49 for pore size distribution. ‘L’ indicates the expectation of 3.82 for pore size distribution.

shows the effect of stochastic pore size distribution on tensile strength. In order to facilitate observation, the tensile strengths in Table 4 are plotted as a bar graph in Figure 12. It can be clearly seen from Figure 12 that stochastic pore size distribution affects tensile strength of concrete. With the same porosity, the tensile strength decreases with the increase of pore size. The reasons for this phenomenon are that under static load, cracks start from the weakest zone and propagate along the weakest path. With the same porosity, the larger the pore size in the sample, the weaker the sample, resulting in a lower bearing capacity of the sample.

The tensile strength in Table 4 is linearly fitted, as shown in Figure 13. For pore size distribution ‘S’, ‘M’, and ‘L’, the relationship between tensile strength $\sigma$ and porosity $p$ is $\sigma =4.031-0.141p$, $\sigma =3.997-0.143p$, and $\sigma =3.977-0.147p$, respectively. The slopes of pore size distribution ‘S’, ‘M’ and ‘L’ are −0.141, −0.143 and −0.147. This means that the tensile strength decreases linearly with the increase of porosity. Among the three distributions, the pore size distribution ‘S’ has the least effect on tensile strength, while the pore size distribution ‘L’ has the most influence on tensile strength. The absolute value of slope increases with the increase of pore size. The larger the pore size, the faster the tensile strength decreases.

4.4. Effect the Spatial Distribution Uniformity of Stochastic Pores on Tensile Strength

4.4.1. An Indicator for Spatial Distribution Uniformity of Stochastic Pores

The spatial distribution uniformity of stochastic pores may affect the tensile strength of concrete. This study proposes an indicator to describe the spatial distribution uniformity of stochastic pores based on the multifractal theory. Since the 1980s, the multifractal theory has gradually matured and has been widely used to investigate crack distribution inhomogeneity [28] and aggregate distribution inhomogeneity [29] of concrete. The multifractal spectrum width $\Delta \alpha$ = α_max − α_min is employed to quantitatively described spatial distribution uniformity of material. The lower multifractal spectrum width means greater uniformity of spatial distribution [29]. Next, we introduce how to obtain the multifractal spectrum width.

Since the research object is the pore, the concrete is regarded as two phases. One phase is the pore, and the other phase is the aggregate, mortar and ITZ. After black-and-white binarization of pore distribution image, a square box with length $\delta$ is used to cover the image. The number of pixels that pore in the ith box is denoted as S_i(δ). The box length changes with the value of 1/2^k$\left(k=0,1,2,\cdots \right)$. The total number of the box is represented by $N\left(\delta \right)$. The probability measure can be calculated as Equation (13).

$$P_i\left(\delta \right)=\frac{S_i\left(\delta \right)}{{\displaystyle {\sum }_{i=1}^{N\left(\delta \right)}{S}_i\left(\delta \right)}}$$

(13)

The scaling exponent $\tau \left(q\right)$ can be obtained by the following equation [29]:

$$\tau \left(q\right)=\underset{\delta \to 0}{\lim }\frac{{\displaystyle {\sum }_{i=1}^{N\left(\delta \right)}{P}_i{\left(\delta \right)}^q}}{{\log }_{10}\delta }$$

(14)

$q$ is the weight factor, which represents the specific weight of the probability measure P_i(δ). A nonlinear relationship between $\tau \left(q\right)$ and $q$, which means multifractal of this object. While a linear relationship indicates single fractal.

Finally, the $\mathrm{H}\ddot{\mathrm{o}}\mathrm{l}\mathrm{d}\mathrm{e}\mathrm{r}$ exponent $\alpha \left(q\right)$ and the Hausdorff dimension $f\left(\alpha \right)$ can be calculated via a Legendre transform as follows [26]:

$$\{\begin{array}{c}\alpha \left(q\right)=\frac{d\left(\tau \left(q\right)\right)}{dq}\\ f\left(\alpha \right)=\alpha q-\tau \left(q\right)\end{array}$$

(15)

The multifractal spectrum of pore spatial distribution image can be determined by the Legendre transform as shown in Figure 14. Then, the indicator $\Delta \alpha$= α_max − α_min is proposed to describe spatial distribution uniformity of stochastic pores based on the multifractal spectrum. Finally, the relationship between tensile strength of concrete and spatial distribution uniformity of stochastic pores can be analyzed according to the indictor.

4.4.2. Effect of the Spatial Distribution Uniformity

From the splitting tensile test principle and experimental phenomena, it can be seen that crack initiation appears near the central area of the sample. The pore located in the central area changes the stress field of the sample and further affects the tensile strength. Therefore, the spatial distribution of pores located in the central area may influence the tensile strength of concrete.

Table 5. The slopes of the 15 groups.

Number	W = 0.1	W = 0.2	W = 0.3	W = 0.4	W = 0.6	W = 0.8	W = 1.0
P01-S	−0.073	−0.024	−0.035	−0.025	0.014	0.027	−0.022
P01-M	−0.135	−0.072	−0.013	−0.100	−0.112	−0.082	−0.105
P01-L	−0.087	−0.080	−0.065	0.018	−0.018	0.020	0.008
P02-S	−0.088	0.047	0.097	0.100	0.023	0.065	0.061
P02-M	−0.037	−0.042	−0.050	−0.033	−0.011	−0.009	0.009
P02-L	−0.071	0.079	0.068	0.088	0.081	0.098	0.116
P03-S	−0.015	0.087	0.038	−0.059	−0.046	−0.044	0.006
P03-M	−0.087	−0.064	−0.191	−0.085	−0.077	−0.023	−0.059
P03-L	−0.020	−0.023	−0.009	−0.067	0.018	−0.035	−0.003
P04-S	−0.102	0.055	−0.021	0.026	0.014	−0.035	0.025
P04-M	−0.047	0.006	0.059	0.048	0.019	0.002	−0.004
P04-L	−0.052	−0.099	0.011	0.021	0.063	0.027	0.013
P05-S	−0.013	0.129	−0.041	0.004	−0.023	0.014	0.031
P05-M	−0.013	−0.030	−0.015	−0.003	−0.035	−0.063	−0.069
P05-L	−0.029	−0.035	−0.019	−0.071	0.062	0.035	0.015

is gradually expanded to determine the range of the central area. The center line of the grey area in Figure 15 is perpendicular to the loading plate and coincides with the center line of the sample. ‘W’ represents the ratio of grey area width to sample diameter. This study considers seven areas, namely the ratio of ‘W’ is 0.1, 0.2, 0.3, 0.4, 0.6, 0.8, and 1.0, respectively. Taking P02-M as an example, a total of 30 samples are simulated. The multifractal spectrum width of each sample is obtained according to the proposed seven areas. Then, scatter diagrams between $\Delta \alpha$ and tensile strength can be obtained for the seven areas as shown in Figure 16. Referring to the reference [29], linear fit to the scatter in Figure 16 is performed to find the relationship between $\Delta \alpha$ in the seven areas and tensile strength.

Figure 16 shows the variation of tensile strength with $\Delta \alpha$ according to the slope of the fitting curve. The effect of pore spatial distribution in the seven areas on tensile strength is different. If the slope is negative, the tensile strength decreases with the decrease of the uniformity of spatial distribution. On the contrary, the tensile strength increases with the decrease of the uniformity of spatial distribution. This study considers 15 groups in Section 2.3, i.e., five porosities and three groups of pore size distribution for each porosity. To find the relationship between $\Delta \alpha$ and tensile strength under different porosities and pore size distribution, linear fit to the scatter of tensile strength with $\Delta \alpha$ in the 15 groups is performed. Table 5 shows the slopes of the 15 groups.

Table 5 shows that the slopes of 15 groups are negative in the areas ‘W = 0.1’, indicating that tensile strength decreases with the decrease of uniformity of pore spatial distribution under different porosities and pore size distribution. A part of the slopes of 15 groups are negative in the other 6 areas, which means the influence of uniformity of pore spatial distribution in these areas on tensile strength is not obvious. Thus, the area ‘W = 0.1’ is an effective area where the effect of uniformity of pore spatial distribution on tensile strength. According to the principle of splitting tensile test [41,42], cracks occur at the center of the sample and propagate towards the loading end. Thus, the stress field near the middle of the sample is the key to failure. The pore is located in the middle of the sample, which significantly changes the stress field of the middle of the sample and affects the fracture behavior of sample. This study finds the range of the middle of the sample, namely ‘W = 0.1’. In this area ‘W = 0.1’, the concentration of a large number of pores may enhance the stress concentration under loading conditions. Thus, in order to improve the tensile strength of concrete, it is profound to guarantee the uniformity of spatial distribution of pores in the area ‘W = 0.1’.

5. Conclusions

In this study, the numerical simulation results are compared with concrete samples containing a hole to verify the validation of the numerical simulation method. The Monte Carlo method is used to place stochastic circular pores in concrete. The influence of stochastic circular pores on the tensile mechanical behavior of concrete is studied from two aspects, namely, pore size distribution and pore spatial distribution. The effect of pore size distribution on tensile strength is studied from the statistical point of view. The relationship between pore spatial distribution and tensile strength is discussed by using the multifractal theory. Conclusions are as follows:

• The multifractal spectrum width can effectively characterize the spatial distribution of stochastic circular pores, which provides an indicator for quantitative investigation of the uniform of pore spatial distribution. This paper obtains the area ‘W = 0.1’, which is an effective area to study the relationship between tensile strength and the multifractal spectrum width in the splitting tensile test. The tensile strength in this area is positively correlated with the uniformity of pore spatial distribution.
• Within the porosity range of 1~5 %, for pore size distribution ‘S’, ‘M’, and ‘L’, the relationship between tensile strength and porosity is σ = 4.031 − 0.141p, σ = 3.997 − 0.143p, and σ = 3.977 − 0.147p, respectively. The tensile strength of concrete decreases linearly with the increase of porosity. With the same porosity, the tensile strength decreases with the increase of pore size. The larger the pore size, the faster the tensile strength decreases.
• The pores change the position of crack initiation and crack path. There are three types of crack initiation positions in the splitting tensile test, i.e., center, eccentric, and loading end, resulting in three typical macro-crack patterns. The cracks occur at the loading end owing to the stress concentration caused by the circular pore.
• It is reasonable to have 30 Monte Carlo samples for splitting tensile test because the error range is between −0.23% and 0.12%.