A Fractal Study on Random Distribution of Recycled Concrete and Its Influence on Failure Characteristics

Abstract

In order to quantitatively describe the influence of aggregate distribution on crack development and peak stress of recycled aggregate concrete, a multifractal spectrum theory was proposed to quantitatively characterize aggregate distribution in specimens. A mesomechanical model of reclaimed aggregate concrete mixed with natural aggregate and artificial aggregate was constructed. Numerical simulation tests were conducted on the uniaxial compression mechanical behavior of 25 groups of sample models with the same proportion and different aggregate distribution forms. Based on the box dimension theory, the multiple fractal spectrum method was used to quantitatively characterize the aggregate distribution form, and the key factors affecting cracks were explored based on the gray correlation degree. The research results show that the aggregate distribution in recycled aggregate concrete has multifractal characteristics. The multifractal spectrum was used to effectively characterize the aggregate distribution pattern, which can enlarge local details and provide new ideas for the quantitative analysis of the damage mode of recycled concrete. Secondly, by establishing a statistical model of the correlation between the multifractal spectrum width of the aggregate distribution pattern and the crack distribution box dimension, it was found that there was a positive correlation between the two, that is, the greater the multifractal spectrum width of the aggregate distribution pattern, the greater the crack box dimension, and the more complex the crack distribution. The complexity of aggregate distribution is closely related to the irregularity and complexity of mesoscopic failure crack propagation in recycled concrete specimens. In addition, gray correlation theory was applied to analyze the key factors affecting the formation of cracks in the specimens. The results showed that aggregate distribution had a first-order correlation with crack formation, and changes in aggregate distribution were an important factor affecting the performance of recycled concrete. Secondly, the poor mechanical properties of NAITZ led to obvious material damage, while NCA and MZ had a significant impact on the skeleton effect in the stress–strain process due to their large areas. This study deepens people’s understanding of the damage characteristics and cracking failure modes of recycled concrete. The study verifies the feasibility of the application of recycled aggregates and provides a valuable reference for engineering practice.

1. Introduction

Recycled aggregate concrete, by effectively utilizing waste building materials and industrial by-products, can not only promote the sustainable development of the construction industry but also provide innovative solutions to solve the global resource shortage and environmental pollution problems and help build a green and sustainable building environment [1]. Therefore, it has become a hot topic in the construction industry in recent years, and many domestic and foreign scholars have conducted extensive research in the field of recycled concrete.

Recycled concrete is widely used, and its macroscopic mechanical properties are closely related to its internal fine and microscopic structure. The non-uniformity of its internal structure is the root cause of the macroscopic failure characteristics of recycled concrete. As the skeleton of a concrete structure, the shape and spatial position of coarse aggregate will have an important impact on the macroscopic failure path of concrete [2,3,4,5]. In order to better describe the failure characteristics of recycled concrete, it is of great scientific significance to study the shape and spatial distribution of aggregate. At present, many domestic and foreign scholars have conducted relevant studies on the characteristics and fractal characteristics of recycled aggregate. Unger et al. [6] showed that the spatial distribution form of aggregates is the most important factor influencing the random non-uniformity of concrete materials. Wang et al. [7], Zhou Jinghai et al. [8,9], and Xiao Jianzhuang et al. [10,11] found that different spatial distributions of aggregates would lead to different crack propagation routes of concrete specimens under external load due to the tip effect generated by aggregates, which also resulted in significant differences in the compressive strength of multiple groups of specimens. In order to evaluate the impact of aggregate distribution on the stress intensity factor, Alih et al. [12] adopted the extended two-dimensional numerical simulation method to conduct a numerical study on the cracking behavior of asphalt mixture of semi-circular bending (SCB) specimens with angular edge cracks generated by the generation and filling algorithm. By comparing the crack models of specimens with different aggregate distribution patterns, it was found that under the same crack length, the difference between the K_{eff (Hety)} of heterogeneous specimens and the K_{eff (Hom)} of homogeneous specimens is obvious (in the range of 20 and 90%). In addition, they also reached the same conclusion in two other articles using different models and theoretical techniques [13,14]. Du Xiuli et al. [15] analyzed multiple groups of concrete stress–strain curves by establishing probabilistic and statistical methods and pointed out that the compressive strength and softening curves of concrete under different aggregate distribution forms all follow the two-parameter Weibull distribution, and the dispersion of stress value in the softening stage is greater than that at the peak point.

Although the above research work has improved people’s understanding of the differences in mechanical properties and failure patterns caused by the heterogeneity of concrete interiors, further research is needed in the quantitative characterization of aggregate distribution patterns. In recent years, multifractal theory and method [16,17,18,19,20,21] have gradually emerged, which is an effective method to study the irregular problem of aggregates and can well quantify and characterize the problem of uneven distribution of aggregates. Based on fractal theory, Guo Lixia et al. [22] established a numerical model considering aggregate grading and shape characteristics. Through a simulated uniaxial compression test, it was found that the greater the fractal dimension of aggregate grading, the greater the peak stress of recycled concrete. Ren Qingwen et al. [23] simulated 36 groups of randomly distributed concrete aggregate models by using the Monte Carlo method and quantitatively characterized the distribution form of aggregate by using the multifractal spectrum width. Their study found that, compared with the box dimension, the multifractal spectrum could better reveal the local details of the randomness of aggregate distribution. Yang et al. [17,24] optimized the calculation of fractal dimension through a binarization matrix, established a numerical aggregate model of random fractal concrete, and studied the influence of coarse aggregate content and fractal dimension on the macroscopic mechanical properties of concrete.

Although the above literature mostly uses methods such as fractal dimension to analyze the impact of aggregate distribution on concrete crack development and mechanical properties, most of them focus on the quantitative analysis of simple concrete three-phase materials in a single fractal dimension. Compared with the above studies, this study established a more complex multi-phase material model of recycled aggregate concrete, which is a typical five-phase random aggregate model, consisting of a natural coarse aggregate phase (NCA) and a natural coarse aggregate interface. The model is composed of five materials: the transition zone (NAITZ), the cement mortar phase (CM), the artificial silt-encrusted coarse aggregate phase (CAWS), and the encrusted aggregate interface transition zone (SAITZ). Based on the fractal theory and multifractal spectrum theory, a quantitative study was conducted on the distribution of aggregates and the formation of cracks. At the same time, with the help of the gray correlation theory, the mechanism of relative crack formation of different materials and the peak stress of concrete was systematically analyzed to provide new ideas for engineering applications.

2. Theory and Methods

2.1. Box Dimension Theory

CAWS and NCA have obvious fractal characteristics, and exploring them is conducive to revealing the regularity behind these irregular shapes. Wang [25] verified that aggregates in concrete cross-sections have fractal characteristics. Mad [26] recommends the following fractal formula for aggregate grading:

$$P(X)={\displaystyle \frac{X^{3-D_{\mathrm{g}}}-X_{\min }^{3-D_g}}{X_{\max }^{3-D_g}-X_{\min }^{3-D_g}}}\times 100\%$$

(1)

where (X) is the passing rate of the corresponding aggregate sieve hole, %; $D_g$ is the fractal dimension of the current aggregate gradation; $X_{\max }$ is the largest screen size studied; and $X_{\min }$ is the minimum sieve size.

If $X_{max}$ is much greater than $X_{\min }$, we can simplify the above equation to

$$P(X)={({\displaystyle \frac{X}{X_{\max }}})}^{3-D_g}\times 100\%$$

(2)

Here, the fractal dimension of the formula $D_g$ is selected as the aggregate characteristic model considering gradation, determined as follows:

$$D_{\mathrm{g}}=3-{\displaystyle \frac{\ln P(X)}{\ln X-\ln X_{\max }}}$$

(3)

The fractal dimension can quantitatively describe the shape of the aggregate. For recycled aggregate concrete, CAWS pairs are regular, while NCA pairs are irregular polygons. The contrast is very obvious, so the fractal dimension can be used to characterize its characteristics.

Set the concrete specimen size as 1*1, set $F$ as the non-empty real subset, and cover all aggregates with box size $\delta$ ($\delta$ < 1). At least $N_{\delta }$ boxes are required. Then, the calculation method of box dimension is as follows [27]:

$$D_{\mathrm{x}}=\underset{\delta \to 0}{\lim }{\displaystyle \frac{\ln N_{\delta }(F)}{\ln (1/\delta )}}$$

(4)

where $\delta$ is the continuous distribution and decreases at the rate of 1/2^k, and $N_{\delta }$ is the number of boxes corresponding to this scale, that is, covering all minimum number of boxes for aggregate.

2.2. Multifractal Spectrum Theory

The box dimension can be used to describe the gradation and particle size distribution of aggregates and can only encompass the number of pixels inside the box, thus reflecting the overall fractal characteristics. The multifractal spectrum, also known as the multifractal measure, can be used to represent the set of different scale indices on the same box dimension [28], which has the effect of magnifying details.

Let $R^{\mathrm{d}}$ be the metric space in dimension d, and F be a subset of $R^{\mathrm{d}}$ and a support set of the measure $\mu$. If a partition rule is selected, the resulting fractal set ($F,\mu$) can be represented as a union of the subsets that should be fractal; then, ($F,\mu$) is called a multifractal.

The research object is divided into N small regions: ${\epsilon }_i$ is the linearity of the i-th calculation region, that is, the maximum length width when measured from all directions, and $P_i$ is the measure of the small region. The measure in this paper is the probability according to the probability distribution law ${\displaystyle {\displaystyle \sum }_{i=1}^N}{P}_i=1$. Select ${\alpha }_i$ as the scale index; then, there are

$$P_i\propto {\epsilon }_i^{a_i}\hspace{1em}(i=1,2,\cdots ,N)$$

(5)

where ${\alpha }_i$ is called the Holder index for short, which controls the singularity of the probability density and therefore also becomes the singularity index. If the linearity approaches 0, that is, the interval is small, then the above formula can be simplified as follows:

$${\alpha }_i=\underset{\epsilon \to 0}{\lim }{\displaystyle \frac{\lg P_i}{\lg {\epsilon }_i}}$$

(6)

where ${\alpha }_i$ is the probability size of the selected i-th region, and if the measure on the research object is uniform, then ${\alpha }_i$ is a constant. After obtaining ${\alpha }_i$, the research object can be divided into a series of subsets, with the number of units $N_{\alpha }\left(\epsilon \right)$, which is only related to the $\epsilon$ size; then, the original formula can be rewritten as follows:

$$N_{\alpha }\left(\epsilon \right)\propto {\epsilon }^{-f\left(\alpha \right)},\epsilon \to 0$$

(7)

$$f(\alpha )=-{\lim }_{\epsilon \to 0}{\displaystyle \frac{\lg N_{\alpha }(\epsilon )}{\lg \epsilon }}$$

(8)

where $f(\alpha )$ is called a multifractal spectrum, which represents the fractal characteristics of a series of subsets corresponding to different A-values in complex fractals. $f(\alpha )$ is a continuous spectrum, and for objects satisfying multifractal, $f(\alpha )$ shows the unimodal phenomenon.

In the scale-free interval, it is pointed out that the characteristic function $\tau (q)$ of fractal behavior can be obtained using a simple transformation as follows:

$$\alpha (q)={\displaystyle \frac{d(\tau (q))}{dq}}$$

(9)

$$f(\alpha )=\alpha q-\tau (q)$$

(10)

These formulas are the theoretical core of the multifractal spectrum, and the correlation between variables can be used to describe the internal structure of multifractals. The internal correlation of multifractals reveals the influence of aggregate characteristics on the mechanical properties of recycled aggregate concrete. Therefore, the multifractal theory can be used to study the crack distribution characteristics.

2.3. Gray Relational Analysis

Gray relational analysis (GRA) is an important part of gray system theory [29,30] and is used as the basis to measure the degree of correlation among factors according to the similar development trend or degree of dissimilarity among factors. The essence of correlation is the difference in geometric shapes of curves. The more similar the shapes are, the closer the changes are, and the greater the degree of correlation is.

GRA was used to analyze the influence of various phase materials on crack development, and the internal failure mechanism of recycled aggregate concrete was explored. The steps of gray relational degree calculation are as follows: (1) establishing feature sequences and other sequences; (2) dimensionless processing of parameters; (3) determining difference sequence, maximum difference, and minimum difference sequence; (4) calculating correlation degree coefficient; and (5) calculating Deng’s gray correlation degree.

3. Numerical Simulation Method

3.1. Development of the Five-Phase Random Aggregate Model

According to the physical experiment, the total size of the model and the proportion of coarse aggregate were selected, and the reclaimed aggregate concrete was regarded as a composite model composed of five phases: the natural coarse aggregate phase, the natural aggregate interface transition zone, the mortar phase, the shell coarse aggregate phase, and the shell aggregate interface transition zone. CAWS is nearly spherical, the model is replaced by a circle, NCA is polyhedral, and the aggregate particles are assumed to be polygonal. The placement of aggregates requires the setting of random variables. According to the Monte Carlo method [31,32], random variables in any interval can be obtained based on the transformation of random variables in the interval [0, 1] as follows:

$$f(x)=\left\{\begin{array}{c}1,x\in [0,1]\\ 0,x\notin [0,1]\end{array}\right.$$

(11)

For example, the uniformly distributed random variable Y over any interval [a,] can be obtained by $Y=\mathrm{a}+(b-a)X$. Thus, random variables that satisfy uniform distribution in each interval can be generated, which can be realized based on the Rand function in ANSYS 2017.

The central location of the aggregate is randomly generated in the model, and the intrusion judgment criterion of the aggregate adopts the area discrimination method. First, a random circle with a fixed particle size range of 5~10 needs to be generated, and its particle size and center coordinates are recorded. Then, a second random circle is generated, and the distance between the centers of the two circles is determined. The center and radius of the newly generated circle are recorded as the base of the random polygonal aggregate; in turn, the second circle is regenerated, and the determination is repeated until the number of circles is equal to the initial set number. After all round aggregates are generated, random angle generation and aggregate volume rate control are carried out, and random polygonal aggregates are generated from round aggregates, and the polygonal aggregates are expanded outward by the fixed center point of the circle in equal proportion to generate the interface transition zone. The other parts in the test block are mortar, so the five-phase model is generated.

The nonlinear failure model is adopted in the constitutive model, and the first strength theory is adopted in the failure theory. The uniaxial compression simulation test adopts the displacement loading method, which is the same as the loading method used in the laboratory test, and the analysis type is static analysis. An automatic time step is used to control the load substep and kill the element whose tensile stress exceeds the tensile strength of each phase material to simulate the failure process. The random aggregate model is shown in Figure 1.

3.2. Determination of Mesoscopic Parameters

In this study, the maximum aggregate particle size was 15 mm, and the thickness of the interfacial transition zone was 150 microns, which basically meets the requirements of the literature [4]. In the mesoscopic study of reclaimed soil mixture materials, the mesoscopic parameters of materials in the interfacial transition zone often play a crucial role. The relevant inversion method based on BBD (Box–Behnken design) was adopted [33], and the mesoscopic parameters of each material are listed in

Table 1. Values of each microphase material.

	Modulus of Elasticity (MPa)	Poisson’s Ratio	Tensile Strength (MPa)
NCA	3500	0.20	4.50
NAITZ	1450	0.20	2.00
CM	2330	0.22	2.75
CAWS	5500	0.20	3.29
SAITZ	3275	0.20	2.19

.

The shell of the silt-regenerated aggregate is cement, so the interfacial transition zone of the silt-regenerated aggregate is closer to the mortar material than that of the natural aggregate, so the tensile strength of SAITZ is better than that of NAITZ. The mechanical properties of recycled aggregate are obviously inferior to those of natural aggregate, but the overall difference is not significant. Considering the different shapes of aggregates and the differences in mechanical properties of each phase material, the development of cracks will also be significantly different from that of traditional concrete specimens. The mechanical properties of the material accord with the characteristics of composite soft base materials, and the mechanical properties of the mortar also determine the macroscopic strength of the specimen to a large extent. The cracks bypass the stronger aggregate and pass through the interface and mortar.

According to the physical experiment conditions and experimental results, the lower boundary of the model was fixed, the uniform load was applied to the upper boundary of the specimen, displacement loading control was adopted, and both sides of the model were free boundaries, as shown in Figure 2a. The nonlinear solver in ANSYS was selected to solve the problem. It stopped when the strain reached 0.093 and was divided into 40 load steps. The ratio of the short and short sides of the element was not greater than 3, and the mesh of the interface was required to be finely divided. The mesh division results are shown in Figure 2b.

3.3. Numerical Experiment of Uniaxial Compression Based on Meso-Concrete Model

The uneven distribution of coarse aggregate in slurry and the inconsistent mechanical properties and durability of hardened concrete members also affect the appearance quality of concrete. Meanwhile, the rheological properties of concrete also caused inconsistent aggregate distribution in the same batch of specimens. In order to fully study the influence of aggregate distribution, a total of 25 groups of recycled concrete specimens with different aggregate distributions were established. Through the finite element simulation of uniaxial compression, the peak stress and crack failure zone of concrete were obtained, which are shown in Figure 3.

The results of the mesoscopic numerical simulation of recycled concrete uniaxial compression are shown in Figure 4.

At the same time, the peak stress values of the 25 groups of model specimens were obtained, as shown in Figure 5.

Through uniaxial compression simulation process analysis, it was found that cracks usually started in the interfacial transition zone and then gradually extended to mortar. In the process of crack propagation, when the shell aggregate with low strength is encountered, crack penetration easily occurs. When encountering natural aggregates with higher strength and a more regular shape, cracks around aggregates often occur. For some aggregates, due to their irregular shape, they are easy to break at the edge at the tip, and there are fewer cracks through them. Finally, an X-shaped failure zone was formed in the middle part of the specimen. With the change in aggregate distribution, the initiation, propagation, and penetration of cracks were also different. Because crack propagation is often carried out with minimal energy consumption, the interfacial transition zone between aggregate and mortar, especially the interfacial transition zone between natural aggregate and mortar, is often the weakest place, forming a chain effect and resulting in increased damage.

4. Influence of Aggregate Distribution on Mechanical Behavior of Concrete

4.1. Fractal Study of Aggregate Distribution

4.1.1. Box Dimension

The influence of aggregate distribution on mechanical properties was determined using the box dimension, and the shape characteristics of CAWS and NCA were studied using the fractal dimension. Then, a square box with a side length of l was used to cover the whole image, and a small box without aggregate was used as a non-counting box. The box with aggregate counted as 1 and the other as 0. The number of all boxes $s(L)$ was summed up. If the selected box size was not used, it corresponded to different $s(L)$, and thus different coordinates $(L,(s(L))$ were obtained. By taking L each time as half of the value of L last time, that is, $L_{i+1}=L_i*0.5$, the $\ln (s(L))-\ln (1/L)$ relation curve was drawn. If the relation curve was a straight line, the aggregate had fractal characteristics and its slope would be the fractal dimension, which is the principle of box dimension.

Because the aggregate volume rates of the 25 groups of specimens and the ratios of recycled aggregate and natural aggregate were completely consistent, the shapes of different aggregates could not be completely consistent due to the randomness of the test. However, on the whole, the irregular distribution of aggregates made the mechanical properties of different groups of concrete similar in the linear elastic stage, but there were different failure characteristics once cracking occurred.

The box dimension of the 25 groups of specimens was calculated according to relevant formulas, Shapiro–Wilk normality test was performed, and the cumulative probability distribution histogram was drawn to visually display the distribution of the data(as shown in Figure 6).

$D_{min}=1.833,D_{max}=1.843$, the calculated Shapiro–Wilk eigenvalue W = 0.960, and p = 0.418 >> 0.01 rejected the null hypothesis, that is, the fractal dimension of the 25 groups of specimens is normal, indicating that the calculation difference in the fractal dimension is affected by randomness. The calculation difference in the fractal dimension is 0.01, so it cannot be directly considered that the difference in the mechanical properties of the specimens is caused by the difference in the fractal dimension of the specimens. In other words, it is difficult to use the box dimension method for research, so it is necessary to introduce the multifractal dimension on the basis of a single fractal to further enhance the properties and characteristics of aggregate.

4.1.2. Multifractal Study of Aggregates

Multifractal spectrum theory is a tool for describing the distribution of data in complex systems. This theory holds that the behavior of many phenomena is multi-scale and cannot be described by a single fractal dimension but requires the use of multiple fractal dimensions. Multifractal spectrum theory is like a ruler that can be used to conduct multifractal research on aggregates. For the calculation of the multifractal spectrum, we first need to determine the probability measure. With the help of ANSYS software 2017, the aggregate distribution image was redivided into more detailed units and used as the aggregate distribution pixel map, that is, the unit with aggregate was assigned a value of 1, and the unit without aggregate was assigned a value of 0. If a certain part of the aggregate was inside the unit and the aggregate area exceeded half of the unit area, the value was assigned a value of 1. Otherwise, it was 0. In this study, the box dimension method was used to calculate the probability distribution. The probability measure of each small square is the ratio of the number of characteristic points of the small square to the total number of pixel points of the aggregate.

$$P_i(\epsilon )={\displaystyle \frac{N_i(\epsilon )}{N}}$$

(12)

where $\epsilon$ is the box size, $N$ is the total number of aggregate pixels in the aggregate distribution pixel map, and $N_i(\epsilon )$ is the number of aggregate pixels in the box of the i-th size $\epsilon$.

Based on the abovementioned formulas, taking the first group of specimens as an example, its $D_{\mathrm{q}}-q$ was calculated to determine whether it conformed to the multifractal characteristics(as shown in Figure 7).

It can be intuitively seen from the Figure 7 that the multifractal dimension $D_{\mathrm{q}}$ decreases monotonically with the increase in $\mathrm{q}$. On the whole, $D_{\mathrm{q}}$ is a decreasing function of $\mathrm{q}$ and conforms to the properties of multiple fractal spectra; that is, it is considered that the aggregate distribution has the characteristics of multifractal spectra. At the same time, the larger the value is, with $D_{\mathrm{qmax}}=3.69$, the more it can reflect the uneven distribution of aggregates, and the more obvious the fractal characteristics.

The multifractal spectrum of each specimen was calculated, and the relationship of the multifractal spectrum $f(\alpha )$ − $\alpha$ was explored(as shown in Figure 8).

It can be seen from the Figure 8 that the curves of each group are not the same but have a similar trend of image change; that is, $f(\alpha )$ is a unimodal convex function of α, which is an increasing function on the side of $q$ > 0 and a decreasing function on the side of $q$ < 0. The maximum value is also the maximum $fmax=D\left(0\right)$, that is, the number of capacity bits, which is obtained at $q$ = 0 and conforms to the multifractal characteristics. Therefore, the distribution of aggregate can be characterized by spectral width $\Delta \alpha ={\alpha }_{\max }-{\alpha }_{\min }$, which can realize the magnification details that cannot be realized by a single fractal, and the role of aggregate distribution can be quantitatively studied.

4.2. Influence of Aggregate Distribution on Peak Stress

The initiation and development of cracks in the failure process of concrete materials are affected by the distribution of aggregate, and the crack evolution is random. Due to the complex characteristics of external stress conditions, the stress inside the material is time-sensitive. When cracks develop in the aggregate, cracks will occur through or around the aggregate due to the strength and shape factors of the aggregate, resulting in a chain effect affecting stress development. At the same time, aggregate is the main load-bearing unit, and the section strength of the dense distribution of aggregate will be significantly increased, but it will also lead to sparse aggregate in other parts, forming a weak structure, which is unfavorable to the overall stress. Therefore, a uniform distribution of aggregate leads to uniform stress development and finally a larger peak stress.

The relationship between spectral width $\Delta \alpha$ and peak stress σ was plotted (as shown in Figure 9).

The multifractal spectral width is a measure of distribution inhomogeneity, and the larger the spectral width, the more obvious the difference in the graph. According to the existing literature [23], when the spectral width difference is larger, the overall spectral width is larger, and the peak stress is smaller. In this study, which builds upon the existing literature, the maximum fractal spectral width of each group was 1.942, the minimum value was 1.973, and the overall difference was not large, indicating that the overall distribution of aggregates was relatively uniform. Therefore, the peak stress was relatively stable in the range of 39–47 MPa. When the overall difference in the spectral width is not large, the larger the spectral width, the greater the peak stress.

4.3. Influence of Aggregate Distribution on Crack Morphology

Taking model 1 as an example, the crack distribution diagram during the failure of the model was plotted using the peak stress, and the crack distribution characteristics were the most intuitive manifestation of the failure characteristics of the aggregate. The fractal dimension was used to study the crack distribution characteristics, and finally, it was reduced to the calculation of $\lg N(L)-\lg (1/L)$ curve to determine whether the curve met the linear relationship(as shown in Figure 10).

It was found that the fractal dimension satisfied the linear relationship and the fitting degree was close to 1, indicating that the crack has fractal characteristics and the distribution pattern of the crack can be studied by the box dimension (as shown in Figure 11).

Formula (4) in Section 2 (Theory and Method) was used to calculate the box dimensions of the 25 groups of specimens, and the results are shown in

Table 2. The box dimensions of the 25 specimens.

Class Number	D	Class Number	D	Class Number	D
Group 1	1.842	Group 10	1.835	Group 19	1.838
Group 2	1.839	Group 11	1.836	Group 20	1.836
Group 3	1.838	Group 12	1.840	Group 21	1.838
Group 4	1.836	Group 13	1.833	Group 22	1.836
Group 5	1.842	Group 14	1.838	Group 23	1.842
Group 6	1.838	Group 15	1.840	Group 24	1.836
Group 7	1.841	Group 16	1.836	Group 25	1.841
Group 8	1.835	Group 17	1.843
Group 9	1.837	Group 18	1.838

below.

Due to the different aggregate distributions of the 25 groups of specimens, the initiation, expansion, and penetration paths of cracks were also different, resulting in different final fractal characteristics. The larger the box dimension was, the more complex the image, and the more pronounced the positive correlation between the spectrum width and the box dimension of cracks, that is, the crack distribution increased with the increase in the aggregate distribution spectrum width, and the more complex the aggregate distribution, the more complex the crack distribution.

When the spectrum width is larger, it indicates that the more irregular aggregate distribution makes the crack propagation path more complicated under the same stress condition, and the complex path prolongs the cracking process, resulting in a certain mechanical bite between the cracks. The peak stress is increased to a certain extent [34].

4.4. Influence of Aggregate Distribution on the Damage Area of Each Phase Material at the Peak Value

Compared with ordinary concrete, the composition of reclaimed meso-phase five-phase material is more complex. In order to further explore the influence of the fractal spectrum width of aggregate on the area factors of the failure element of each phase material, the gray correlation method was adopted for the study. Gray relational analysis is a statistical method used to study the relationship between systems, especially in situations with incomplete information or small samples. The gray relational analysis is like a “comparison tool” that evaluates the degree of association by calculating the similarities between different variables. The area of the failure element of each phase material was considered the characteristic subsequence, and the fractal spectrum width of the aggregate and the dimension of the crack box were considered the parent sequence (as shown in

Table 3. Feature sequence and the area of the failure unit of each material.

		Fractal Dimension	Δα	The Failure Unit Area of Each Phase Material mm2
Group				NCA	CM	NAITZ	CAWS	SAITZ
1		1.446	1.9503	93	225	19.75	21	5.55
2		1.512	1.9422	72	173	31.33	5	4.04
3		1.463	1.9583	58	132	27.90	1	2.31
4		1.508	1.9583	56	138	29.80	3	2.02
5		1.506	1.9515	45	107	22.95	4	4.61
6		1.422	1.9538	37	89	25.08	4	2.91
7		1.400	1.9465	37	76	26.87	1	1.30
8		1.523	1.9510	68	166	31.38	6	1.96
9		1.525	1.9635	59	112	26.10	2	1.22
10		1.442	1.9585	55	125	23.84	1	3.45
11		1.500	1.9626	59	155	29.17	2	4.30
12		1.482	1.9424	57	112	26.26	1	2.23
13		1.467	1.9602	65	149	29.24	3	2.28
14		1.428	1.9658	57	125	28.10	1	1.59
15		1.512	1.9427	53	188	29.44	6	4.09
16		1.419	1.9629	42	90	30.87	1	0.80
17		1.471	1.9651	51	130	26.06	5	1.81
18		1.488	1.9569	67	168	24.75	19	5.63
19		1.398	1.9730	43	102	28.16	3	2.07
20		1.461	1.9542	61	156	31.79	1	0.53
21		1.443	1.9603	66	164	29.31	3	2.40
22		1.486	1.9595	41	121	29.97	3	2.18
23		1.526	1.9575	55	156	21.26	22	6.30
24		1.471	1.9654	42	118	30.38	5	3.35
25		1.410	1.9634	31	81	25.12	3	1.81

Note: Due to the large volume of calculation data and space limitations, the above calculation results only show part of the data.

).

Sequence [1] is the characteristic sequence of the system, and the remaining sequence and its gray relation analysis were calculated as follows:

Step 1: Calculate the initial value of the sequence and the infinite tempering treatment and calculate the difference sequence calculation table (as shown in

Table 4. Calculation table of difference sequences.

0	0	0	0	0	0
0.271	0.277	0.541	0.808	0.318	0.05
0.388	0.425	0.401	0.964	0.596	0.008
… …	… …	… …	… …	… …	… …
0.565	0.493	0.521	0.779	0.413	0.009
0.641	0.615	0.297	0.832	0.649	0.032

).

Step 2: Calculate the correlation coefficient.

Maximum range: 0.8985;

Minimum range: 0.0000.

Step 3: Calculate the correlation coefficient (as shown in

Table 5. Correlation coefficient calculation table.

1	1	1	1	1	1
0.643	0.638	0.475	0.377	0.606	0.907
0.557	0.535	0.549	0.336	0.451	0.985
… …	… …	… …	… …	… …	… …
0.464	0.498	0.484	0.385	0.542	0.981
0.432	0.443	0.622	0.37	0.43	0.939

).

Step 4: Perform the gray relation analysis (as shown in

Table 6. Gray relation analysis.

The Failure Unit Area of Each Phase Material mm2					Δα
NCA	CM	NAITZ	CAWS	SAITZ
0.549	0.556	0.594	0.427	0.531	0.949

).

The collated results are presented in Figure 12.

The analysis revealed that the overall gray correlation value was 0.4275–0.5937, belonging to the order of spectral width correlation value as follows: Δα > NAITZ > CM > NCA > SAITZ > CAWS. The six subsequences were highly correlated with their parent sequences, belonging to first-order correlation and fourth-order correlation [35], which is of research significance. The analysis showed that there were three factors affecting crack formation: the area proportion of each phase material, the aggregate distribution, and the mechanical properties of each phase material. The distribution spectrum width of aggregate has a first-order correlation with the formation of cracks, and it can be inferred that the fractal spectrum width directly affects the formation of cracks. Combined with Figure 12, it can be concluded that the larger the fractal spectrum is, the more complex the cracks are. In the development of stress, the killing of NAITZ material is more closely related to the stress, which is due to the short-plate effect. The failure of the material will cause the stress of the specimen to decay rapidly, so it shows a higher gray correlation degree. NCA and CM are close to each other because the mortar material occupies a relatively large area, has a major role in the force, and has good mechanical properties, so the correlation degree is higher. The correlation degree of shell aggregate is relatively low, indicating that the short-plate effect of recycled aggregate is not obvious. Subsequent quantitative analysis verified the feasibility of the application of recycled aggregate.

Therefore, the distribution of aggregate has an important effect on the failure characteristics of concrete. This study highlights the influence law of aggregate distribution and the fracture morphology of concrete and provides a scientific basis for the research of concrete cracking control.

5. Conclusions

In this study, a micromechanical model of recycled aggregate concrete with different aggregate distribution patterns was established by means of numerical simulation. Based on fractal theory, the multifractal spectrum width and fractal dimension of cracks of specimens under different aggregate distribution patterns were calculated, and the relationship among aggregate distribution patterns, peak stress, and failure modes was established. Finally, the gray correlation theory was introduced to obtain the key factors affecting the crack formation of specimens. The conclusions are as follows:

1. The multifractal spectrum can well characterize the aggregate distribution pattern and amplify the local details, providing new ideas for the quantitative study of the failure pattern of recycled aggregate concrete.

2. Using multifractal spectrum as a quantitative characterization method of aggregate distribution, a statistical model was established between the multifractal spectrum width of the aggregate distribution pattern and the box dimension of crack distribution. It was found that the multifractal spectrum width of the aggregate distribution pattern was positively correlated with the box dimension D of meso-failure cracks of recycled aggregate concrete specimens, indicating that the more complex the aggregate distribution was, the more irregular the crack growth was, and the more complex the crack distribution was.

3. The introduction of gray correlation theory shows that the most critical factor affecting the crack formation of specimens is aggregate distribution, which is a first-order correlation, followed by significant material damage caused by poor mechanical properties of NAITZ, and then the skeleton factor of NCA and MZ in the stress–strain process due to their large area, and the short-plate effect of recycled aggregate is not obvious. The feasibility of the application of recycled aggregate was verified, thus providing a reference for engineering applications.

It should be noted that this study only considers the influence of aggregate distribution on the mechanical properties and failure forms of recycled concrete. In the future, it is necessary to further explore the micro-inducements and internal driving mechanisms of the influence of other factors such as aggregate strength characteristics, aggregate shape, grading, etc., on the macroscopic mechanical properties of recycled concrete to provide a theoretical reference for engineering applications. At the same time, we only conducted research based on numerical simulation methods. Although this method is highly flexible and suitable for complex boundary conditions, it also has certain limitations, such as being highly dependent on the setting of grids, boundary conditions, and initial conditions. Inaccurate condition settings may lead to significant deviations in simulation results. Therefore, when using the finite element method, reasonably setting model parameters, selecting correct boundary conditions, and grid division are key to ensuring its accuracy and efficiency so as to improve the effectiveness and reliability of numerical simulation in practical applications.