Fractal Feature of Manufactured Sand Ultra-High-Performance Concrete (UHPC) Based on MIP

Abstract

To alleviate environmental pressures, manufactured sand (MS) are increasingly being used in the production of ultra-high-performance concrete (UHPC) due to their consistent supply and environmental benefits. However, manufactured sand properties are critically influenced by processing and production techniques, resulting in substantial variations in fundamental characteristics that directly impact UHPC matrix pore structure and ultimately compromise performance. Traditional testing methods inadequately characterize UHPC’s pore structure, necessitating multifractal theory implementation to enhance pore structural interpretation capabilities. In this study, UHPC specimens were fabricated with five types of MS exhibiting distinct properties and at varying water to binder (w/b) ratios. The flowability, mechanical strength, and pore structure of the specimens were systematically evaluated. Additionally, multifractal analysis was conducted on each specimen group using mercury intrusion porosimetry (MIP) data to characterize pore complexity. SM-type sands have a more uniform distribution of pores of different scales, better pore structure and matrix homogeneity due to their finer particles, moderate stone powder content, and better cleanliness. Both excessively high and low stone powder content, as well as low cleanliness, will lead to pore aggregation and poor closure, degrading the pore structure.

1. Introduction

Ultra-high-performance concrete (UHPC) has attracted much attention in modern civil engineering and construction materials science as a high-performance composite material due to its excellent tensile strength, flexural properties, and enhanced toughness [1,2,3]. However, the widespread use of conventional UHPC is also associated with huge energy consumption [4,5,6]. Usually, UHPC consists of cementitious materials, aggregates, admixtures, and water to form a matrix; the cementitious materials and water ratio is usually 0.16–0.2, so the water to binder (w/b) ratio has become one of the key elements affecting the performance of UHPC [7]. Traditional UHPC usually uses quartz sand (QS) and river sand (RS) as aggregates, but in recent years, the massive mining of QS has exacerbated crystalline silica dust pollution and caused significant damage to the surrounding ecological environment and the health of local residents; on the other hand, the massive mining of RS has almost depleted sand and gravel resources [8,9,10]. Therefore, there is a need to find a more environmentally friendly, economical, and consistent-quality aggregate to replace QS and RS as the aggregate used in UHPC.

Manufactured sand (MS) are man-made aggregates obtained from natural rocks through a series of processes such as crushing, grinding, and screening [11]. MS has the advantages of stable yield, lower cost, and environmental friendliness compared to RS and QS [12]. However, the preparation process of MS inherently incorporates varying amounts of stone powder. Additionally, since MS is mined from mountain rocks, impurities such as clay become intermixed. Notably, Wang et al. [13] clearly established that MS characteristics significantly impact concrete performance. As a result, UHPC made from MS is prone to a number of problems such as poor aggregate formation, poor fresh concrete microstructure, poor workability, and degradation of mechanical properties [14,15,16,17]. There are many researchers who have studied, among other things, the performance of UHPC made from MS. Li et al. [9] investigated the effect of MS replacing QS on the properties of UHPC and concluded that a moderate amount of stone powder in MS has a positive effect on strength improvement, while excessive stone powder deteriorates the pore structure. Shen et al. [18] investigated the effects of MS particle morphology and stone powder on concrete properties and found that stone powder and clay content in MS had a greater impact on concrete properties than particle morphology. Ma et al. [10] also found that an increase in MS doping resulted in an increase in micropore volume, porosity, and mesopore number. Therefore, the use of manufactured sand (MS) in UHPC preparation is a method with both advantages and disadvantages. Due to factors such as stone powder content in MS and inherent sand properties, the performance variability of MS-prepared UHPC is significantly greater than that of quartz sand (QS) and river sand (RS)-based UHPC, necessitating tailored analysis for MS with varying properties. It is worth noting that, among the effects of MS on UHPC, the effects of MS on the densest packing structure and pore space of the UHPC matrix are more important, but very few studies have discussed the pore space distribution, the degree of concentration of pore space, the mechanism of the effects of MS on pore space, and the quantitative relationship between pore space structural characteristics and micromechanical properties of UHPC according to the different types of MS.

In recent years, interdisciplinary research has developed an important mathematical tool, fractal analysis and multifractal theory, which provides innovative perspectives on complex natural phenomena by quantifying the self-similarity characteristics of the system and the heterogeneous distribution law [19,20,21]. In the field of building materials science, this mathematical framework—grounded in non-integer dimensional analysis—provides novel methodologies for decoding the microstructural topology of cementitious systems [20,22]. Existing studies have primarily utilized both single fractal analysis and multifractal analysis. The former reveals the scaling properties of irregular material behavior by identifying complexity in datasets, patterns, or signals through individual fractal dimensions, while the latter employs spatial variability in probabilistic measures to achieve fine-grained descriptions of multiscale pore network features using continuous spectral functions, such as α-chiral spectra and Dq-generalized spectra [23,24]. In terms of characterizing the internal structure of matrix pores, a number of studies have suggested that fractal and multifractal theories have great potential for application [25,26,27,28,29,30,31]. Han et al. [32] analyzed the pore structure of fly ash–cement composites using the Menger sponge model for fractal characterization. Their findings demonstrated that fractal analysis provides a more intuitive understanding of the relationship between the material’s compressive strength and its pore system. Zhu et al. [33] investigated the effect of cellulose nanocrystals (CNCs) on the pore structure and drying shrinkage of UHPC using multifractal analysis. Their results demonstrated that multifractal indices effectively characterize the evolution of the material’s pore structure. Furthermore, they concluded that CNCs improve pore homogeneity and significantly reduce drying shrinkage. Sun et al. [34] used diatomite for cement pastes and used nitrogen adsorption to improve the piezoelectric mercury data and revealed the pore structure evolution mechanism of diatomite cement mortar for freeze–thaw cycles by the multiple fractal analysis method. Therefore, based on fractal theory, this study seeks to analyze the impact of MS on UHPC pore structure through non-integer dimensional metrics, quantify and compare concrete pore complexity, and ultimately establish correlations between microscale pore architecture and macroscale mechanical performance.

In this study, five types of MS with distinct properties, commonly available in the market, were selected, and UHPC specimens with varying binder to sand ratios were prepared. The flowability, expansion time, compressive strength, flexural strength, and pore distribution of UHPC specimens were evaluated under different MS types. To further elucidate the influence of MS on the pore structure of the UHPC matrix, multifractal theory was applied to analyze the pore characteristics of MS-based UHPC. The pore structures of different MS types under varying w/b ratios were systematically characterized using multifractal parameters. Finally, this study fills the gap in how variations in MS types affect the fractal characteristics of UHPC pore structures and establishes a relationship between microscopic pore structure and macroscopic mechanical strength using multifractal analysis.

2. Materials and Methods

2.1. Raw Materials

UHPC specimens were prepared from each of the five MSused in this study. The cementitious materials used included: Huaxin P.O. 52.5 ordinary Portland cement (OPC) (Kunming, China); semi-dense silica fume (SF) from Fangke New Material Technology Co. (Xi’an, China), SiO₂ content of 96.54%; class II fly ash (FA) from Xuanwei Power Plant (Xuanwei, China), 28-day activity index of 81%; polycarboxylic acid high-performance water-reducing agent from Sobute New Materials Co., solids content of 50%; copper-coated steel fibers, flat-straight type, length: 13 mm. The five MS used were: M1 and M2 from Wang Shiheng Sandpit Yangbi County (Dali, China); M3 from Yuzhou Stone Plant Yunlong County (Dali, China); SM1 from the Sand and Gravel Plant of Fumin County (Kunming, China); SM2 from Hongyan Zhanfeng Sandpit Midu County (Dali, China). The details of the various treatments of the sands produced are shown in

Table 1. Different treatments for MS.

Type	Sand	Disposal Method
M-type	M1	After production, undergoes multistage washing, dewatering, and moisture control, indoor storage, and sun drying prior to use.
	M2	After production, undergoes multistage washing, stockpiled in the open, used directly.
	M3	After production, stored indoors without washing, used directly.
SM-type	SM1	After production, stored indoors without washing, used directly.
	SM2	After production, stored indoors without washing, used directly.

. Microscope photographs are shown in Figure A1 of Appendix A and basic properties are shown in

Table 2. Physical properties of MS.

Type	M1	M2	M3	SM1	SM2
Loose bulk density (kg/m3)	1403.2	1436.6	1491.4	1470.1	1462.5
Packed bulk density (kg/m3)	1532.5	1580.705	1541.2	1658.5	1672.6
Apparent density (kg/m3)	2687.8	2696.9	2708.4	2724.1	2716.2
Porosity (%)	40.5	46.9	38.8	40.9	40.3
Crushing value (%)	14.3	16.5	15.9	14.9	15.2
Stone powder (%)	3.2	9.7	6.8	4.7	7.8
Methylene blue value	1.00	2.00	0.25	1.25	0.75

. Photographs of binding materials are shown in Figure A2. The XRF results are shown in

Table 3. The chemical composition of used cement (in wt%).

Composition	OPC	SF	FA	M1	M2	M3	SM1	SM2
F	-	-	-	-	0.27	-	-	-
CaO	66.47	0.85	14.19	83.42	75.10	93.03	85.51	98.10
SiO2	16.55	95.12	41.53	7.91	12.24	0.29	6.34	0.35
Fe2O3	4.39	0.09	6.33	1.68	2.11	0.23	1.33	0.28
Na2O	0.20	0.17	2.3607	0.08	0.09	0.06	0.02	-
MgO	2.03	0.67	3.79	1.9	3.05	6.08	3.23	0.83
Al2O3	4.34	0.31	18.88	3.56	5.49	0.22	2.52	0.17
P2O5	16.55	0.20	1.06	0.02	0.02	0.01	0.01	0.02
SO3	3.64	0.21	3.91	0.08	0.08	0.02	0.10	0.03
Cl	0.03	0.02	0.05	0.01	0.01	-	0.021	0.02
K2O	1.04	0.63	2.28	0.66	0.97	0.01	0.44	0.01
TiO2	0.64	-	1.20	0.35	0.25	-	0.18	-
LOI	0.45	1.73	4.43	0.35	0.25	0.06	0.29	0.20

, the particle size distribution curves are shown in Figure 1, and the XRD results are shown in Figure 2.

2.2. Preparation Process of UHPC

The UHPC mix design in this study was guided by the densest packing theory, with ratios optimized via the modified Andreasen and Andersen model (MAA) [35,36]. Five types of MS were utilized to prepare UHPC specimens at w/b ratios of 0.17 and 0.18. Specimens were labeled as M1, M2, M3, SM1, SM2 (where M1 refers to UHPC prepared with M1 sand, etc.). Detailed mix proportions are provided in

Table 4. Mix proportions of UHPC (kg/m³).

Group	w/b	b/s	OPC	SF	FA	Sand	Water	SP	Fiber
M1	0.18	1.2	840	180	180	1000	216	24	195
M2
M3
SM1
SM2
M1	0.17	1.2	840	180	180	1000	196	24	195
M2
M3
SM1
SM2

.

The preparation process is as follows: (1) weigh the raw materials, add them into the single-horizontal-axis mixing pot, and stir for 200 s to ensure uniform dispersion; (2) mix the weighed water and water-reducing agent thoroughly; (3) add the mixed liquid into the mixing pot and stir for 400 s; (4) use a 10-mesh sieve funnel to uniformly add steel fibers into the mixing pot while stirring, and continue mixing for 200 s; (5) finally, mix for an additional 200 s to achieve full homogeneity. The specimen preparation workflow is illustrated in Figure 3.

The mixed slurry was poured into six 100 mm × 100 mm × 100 mm cubic molds and three 100 mm × 100 mm × 400 mm prism molds for compressive and flexural strength testing. Excess slurry was reserved for fluidity and expansion time measurements. Immediately after molding, the specimens were covered with polyethylene film to minimize water-evaporation-induced cracks and pores. They were then cured in a standard curing chamber (20 °C, 99% humidity) for 2 days to set. After hardening, specimens underwent steam curing in a controlled chamber at 60 °C for 3 days. Finally, they were stored in a laboratory environment (23 °C, 50% humidity) for 7 days prior to testing. Figure A3 shows experimental images related to the sample preparation process.

2.3. Flowability and Expansion Time

Flowability and expansion time were tested in accordance with GB/T 50080-2016 [37]. The flow test utilized a 1500 mm × 1500 mm × 5 mm steel flow table and a conical flow bucket with the following dimensions: top diameter d = 100 mm, bottom diameter D = 200 mm, and height h = 300 mm. Fresh UHPC slurry was poured into the bucket, which was then vertically lifted to allow free flow. Timing commenced immediately upon lifting. The expansion time (T500) was recorded as the duration for the slurry to spread to a 500 mm diameter. After 90 s, two perpendicular spread diameters were measured and averaged to determine the final flowability.

2.4. Mechanical Properties

Mechanical strength testing was performed in compliance with GB/T 17671-2021 [38]. For compressive strength, a preload of 500 N was applied, followed by continuous loading at 2.5 kN/s until failure; results were averaged from six specimens. For flexural strength, a preload of 100 N was applied, with loading maintained at 333 N/s; results were averaged from three specimens. The testing equipment used is WANCE’s microcomputer-controlled oil–electric hybrid servo pressure testing machine HCT206E from Shenzhen, China, manufactured by WANCE Equipment Co., Ltd.. Its maximum load is 2000 kN and the loading frame is a high-rigidity four-column hydraulic loading frame model, the load cell sensitivity is 2.3 mV/V. Images of the compression and flexural testing processes are shown in Figure A4.

2.5. MIP

In this study, pore structure analysis of all UHPC specimens was performed using a Micromeritics AutoPore V 9620 (USA) instrument. The specimens used for MIP testing are fragments taken from the center of the sample (average length: 3–7 mm). For each group, 20–25 fragments were initially collected, followed by removal of protruding steel fibers to ensure structural integrity. Final results were averaged across 10–15 validated fragments per group to ensure representative sampling.

2.6. Multifractal Theory

Fractal theory fundamentally quantifies the complexity of pore surfaces through the fractal dimension, a mathematical parameter that characterizes irregularity and self-similarity [26,30,39]. Cementitious matrices inherently exhibit multiscale chaotic pore networks, where fractal analysis serves as a robust tool to quantify and compare structural complexity [40,41].

Multifractal theory, an extension of classical fractal principles, quantifies non-uniform scale invariance in complex systems and structures. In concrete materials science, it serves as a powerful tool to analyze multiscale heterogeneous architectures and track the evolution of structural complexity [42,43]. By integrating MIP data, multifractal analysis addresses key challenges in characterizing pore structure irregularities, such as connectivity anomalies and size distribution hierarchies, and uncovers the hierarchical multiscale organization of pores within the matrix [44]. This study employed multifractal theory to characterize pore structure evolution in UHPC samples incorporating distinct MS, utilizing singularity spectrum analysis. Building on prior methodologies [34,44,45,46], the multifractal analysis procedure included the following steps (the analysis flowchart is shown in Figure A5):

To conduct multifractal analysis, the sample aperture range I (Equation (1)) must first be discretized. The interval I is partitioned into uniform subintervals using the partitioning scale ε, defined by Equation (2). The resulting subintervals are mathematically expressed via Equation (3).

$$I=[a,b]$$

(1)

$$\epsilon =L\times 2^{-k}$$

(2)

$$N(\epsilon )=2^k$$

(3)

where L is the length of the aperture region I, k is an integer from 0 to 6.

In each of these subintervals, the probability measure is defined as in Equation (4).

$$pi(\epsilon )={\displaystyle \frac{v_i}{{\sum }_{i=1}^{N(\epsilon )}{v}_i}}$$

(4)

where v_i is the relative pore volume of each subinterval.

After that, the partition function χ(q,ε) is established as in Equation (5), and the partition function is obtained based on the pore volume statistics. After that, the set of data points for ln χ(q,ε) and lnε can be calculated from Equation (4).

$$\chi (q,\epsilon )={\displaystyle {\sum }_{i=1}^{N(\epsilon )}p{i}^q(\epsilon )}={\epsilon }^{\tau (q)}$$

(5)

where q is the matrix order, and in this study the range of q is taken as [−5, 5]. τ(q) is the quality index.

According to the partition function of Equation (5), if a power relation between χ(q,ε) and ε exists for the analyzed object, then it indicates that the object is characterized by multiple fractals.

τ(q) in Equation (5) can be defined as the linear regression parameter of χ(q,ε) and ε in the double natural logarithmic coordinate system, i.e., the slope of the ln χ(q,ε)~lnε curve, as shown in Equation (6).

$$\tau (q)={\displaystyle \frac{\ln \chi (q,\epsilon )}{\ln \epsilon }},(\epsilon \to 0)$$

(6)

Finally, the definition of the generalized fractal dimension is then constructed from Equation (7). Multiple fractal properties are revealed as the value of q varies over the range.

$$D_q={\displaystyle \frac{{\tau }_q}{q-1}}={\displaystyle \frac{\ln \chi (q,\epsilon )}{(q-1)\ln \epsilon }},(\epsilon \to 0)$$

(7)

And the mapping relation of the multifractal singular spectrum f(α)~α function corresponding to a specific q value can be finally expressed by Equations (8) and (9) through Legendre transformation.

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

(8)

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

(9)

2.7. SEM

In order to obtain clear SEM images and energy dispersive spectroscopy (EDS) test results, the surface of the sample blocks was sprayed with gold to improve electrical conductivity.

3. Results and Discussion

3.1. Flowability and Expansion Time

The flowability and expansion time of UHPC specimens prepared using different types of sand are shown in Figure 4.

It can be seen that the w/b ratio correlates well with flowability and expansion time, but the effect of different types of manufactured sand on flowability is significantly different. In terms of flow performance, M1 exhibited the highest UHPC slurry flowability at 715 mm, followed by M3 with intermediate flowability, while M2 showed the lowest flow at 565 mm. This disparity is attributed to differences in stone powder content and clay/silt impurity levels among the three MS. The M1 sand, subjected to washing, exhibits low stone powder content, high particle cleanliness, and a low methylene blue (MB) value (reflecting minimal clay content within the sand [47]), which results in high flowability of UHPC slurry. In contrast, M2 sand’s open-air post-mining storage exposed it to rainwater infiltration, introducing clay-like impurities and significant dust into the sand matrix, which resulted in an elevated MB value. Clay-like impurities interfere with adsorption–dispersion mechanisms on cementitious particle surfaces [48], diminishing the water reducer’s effectiveness and consequently yielding the poorest flow performance. M3 was unwashed and stored indoors, and the sand particles were clean, but the stone powder content was higher than that of M1, so the flowability was somewhat affected. SM1 and SM2 exhibited flowability comparable to M3 across varying w/b ratios, despite differences from the M-type sand groups. This similarity arises because SM1/SM2 sands were sieved to reduce stone powder content, resulting in lower MB values and improved flowability relative to M2. However, the sieving process increased fine particle content in SM1/SM2, elevating water demand [49], which balanced their flow performance to levels akin to M3.

The expansion time performance of SM-type sands is more stable than that of M-type sands, with minimal fluctuations under different w/b ratios. Among these, M1 exhibits the best performance, indicating that, after washing, most stone powder is removed. This allows water’s lubricating effect to be fully expressed, reducing the viscosity of UHPC slurry [50] and resulting in the shortest T500. M2 sand, containing harmful clay and excessive dust, adsorbed a significant amount of free water, weakening the water-reducing agent’s effectiveness and resulting in a 71.0% shorter expansion time compared to M1 at w/b = 0.18. The UHPC slurries produced by M3, SM1, and SM2 showed more moderate expansion time performance.

3.2. Mechanical Properties

Figure 5 illustrates the compressive and flexural strengths of UHPC prepared with different MS at varying w/b ratios. As shown, both compressive and flexural strengths increased proportionally with decreasing w/b ratios. Notably, all specimen groups exhibited identical compressive strength trends under both w/b = 0.18 and w/b = 0.17 conditions.

As shown in Figure 5a, the SM-type of MS specimens are the group with the highest average compressive strength under different w/b ratio conditions. SM1 and SM2 reached 157.5 MPa and 155.2 MPa, respectively, at w/b= 0.18 and 164.2 MPa and 160.0 MPa at w/b = 0.17, which were much higher than those of the M-type sands. This is attributed to the SM-type MS, which contains moderate stone powder content and enhanced fine particle distribution. These properties optimize the filling effect and nucleation effect, resulting in a denser interfacial transition zone [12,51]. The lower average strength of SM2 specimens compared to SM1 is attributed to excessive fine particles and stone powder in SM2 sand, which alters concrete grading and induces a grading distribution offset. This reduces the proportion of large-grained aggregates, diminishing the aggregate skeleton effect and compromising mechanical performance [9]. M2 had the lowest average strengths of only 138.9 MPa (w/b = 0.18) and 148.6 MPa (w/b = 0.17). On the one hand, the high MB value of M2 weakens its strength performance [52]; on the other hand, XRF and XRD results of M2 (Table 3 and Figure 2) indicate the presence of fluoride ions (F⁻) and calcium fluoride (CaF₂). During the early stages of cement hydration, a film of semi-permeable hydration products forms on the surface of cement particles [53,54]. This film reacts with soluble F⁻ in the alkaline cement paste, precipitating to create an insoluble CaF₂ barrier layer on the surface of clinker minerals [55], thereby inhibiting further hydration of the cement.

Figure 5b presents the flexural strength of each test group. SM1 demonstrates the highest performance (26.0 MPa at w/b = 0.18 and 27.9 MPa at w/b = 0.17), followed by SM2, while that of M2 remains the lowest (20.4 MPa at w/b = 0.18 and 21.1 MPa at w/b = 0.17). Notably, the average flexural strength trends align closely with those observed for compressive strength across varying w/b ratios.

The mechanical properties of UHPC are closely linked to the pore structure within its matrix. The MS evaluated in this study exhibit significant variations in composition, which directly influence the pore distribution and morphology in the UHPC matrix; these effects will be analyzed in detail in subsequent sections.

3.3. Pore Diameter Distribution

Figure 6 shows the cumulative pore distribution curves and differential curves of UHPC produced from different types of sands at different w/b ratios. As shown in the figure, the pores are divided into five types [56]: gel pores (≤ 0 nm), capillary pores (10 nm–50 nm), mesopores (50 nm–100 nm), macropores (100 nm–10 μm), and macrovoids (≥10 μm).

As can be seen in Figure 6a,b, the cumulative pore volume curves are significantly different depending on the type of sand used for the same w/b ratio. At w/b = 0.18, SM2 exhibits the lowest cumulative pore volume in the large pore size range (>50 nm). In contrast, at w/b = 0.17, both SM1 and SM2 show reduced cumulative pore volumes across all pore size ranges. It is demonstrated that the use of SM sands reduces the total pore volume. Specifically, the threshold pore sizes of M1, M2, and M3 are in the range of 10–20 nm, while SM1 exhibits a smaller threshold range of 5–10 nm, and SM2 falls within 10–20 nm. This improvement is primarily attributed to the finer particle size distribution and moderate stone powder content in SM sands, which enhance the filling effect within the cementitious matrix [57]. It is noteworthy that the cumulative pore volume of M2 in the 10 nm–10 μm range is significantly higher than those of the other groups. This is attributed to M2’s excessive stone powder content and the presence of clay and deleterious impurities in the sand fraction, which disrupt the matrix particle gradation and thereby increase porosity.

Figure 6c,d display the pore size distribution curves for each group. It is evident that the pores in the matrix of all groups are predominantly nanoscale, with primary pore sizes concentrated below 10 nm. This observation confirms that mix proportions optimized via the MAA model achieve the densest particle packing, effectively minimizing matrix porosity.

3.4. Pore Volume Distribution

Figure 7a,b present the cumulative pore volume across pore size ranges in UHPC fabricated with different MS under varying w/b ratios. Figure 7c,d illustrate the pore size distribution percentages across distinct pore size ranges for these samples.

According to Figure 7a,b, the pore volume of each sample group exhibits a strong correlation with the w/b ratio, with total pore volume decreasing as the w/b ratio decreases. This trend aligns closely with the mechanical property improvements observed in Section 3.2. Notably, under all tested w/b ratios, samples fabricated with SM-type sands exhibit consistently lower total pore volumes compared to those prepared with M-type sands. Specifically, at a 0.18 w/b ratio, SM1 and SM2 exhibit 5.5% and 12.6% lower pore volumes, respectively, compared to M3 (the group with the lowest pore volume). At the 0.17 w/b ratio, these reductions increase to 23.8% for SM1 and 12.8% for SM2.

Based on pore size, researchers have classified the harmful effects of pores on concrete [9]. According to the intervals used to classify pores in this study, we consider gel pores (≤10 nm) as harmless pores, capillary pores (10 nm–50 nm) as slightly harmful pores, and pores >50 nm as harmful pores. According to Figure 7c,d, it can be seen that the sum of harmless and slightly harmful pores of SM-type sand specimens is always higher than that of M-type sand specimens. Specifically, M1 totaled 39% and 43% in 0.18 and 0.17 w/b ratio conditions, respectively, M2 44% and 45%, M3 49% for both, SM1 49% and 50%, and SM2 52% for both. Therefore, it can be noted that the matrix is denser using SM sand specimens. In addition, as the w/b ratio decreased, all groups except M2 showed pore refinement, i.e., the harmful pores were reduced and transformed into slightly harmful and harmless pores [58]. Studies indicate that the strength failure of concrete matrices is governed by pore size distribution [6,58]. Specifically, compressive strength degradation correlates strongly with pore size, as small harmful pores (capillary pores) act as start sites for microcracks. These microcracks propagate and interconnect with larger macropores, forming a network that accelerates structural failure under load [59]. This also explains one of the reasons for the differences in mechanical performance of the different types of manufactured sand in Section 3.2.

3.5. Multifractal Analysis

Figure 8 displays scaled plots of the partition function characterizing the pore structure of UHPC samples prepared with different MS under varying w/b ratios. A distinct linear relationship is established between ln χ(q,ε) and lnε for all specimen groups across varying w/b ratios, demonstrating that the pore structure exhibits pronounced multifractal characteristics [60]. Furthermore, the pore distributions of all specimen groups exhibit self-similarity across scales. Specifically, the scaling behavior of all q values for each group demonstrates strong linearity (R² > 0.9) [34], confirming robust multifractal characteristics.

3.5.1. Generalized Fractal Spectrum Analysis

Figure 9a,b present the generalized fractal spectra of UHPC samples at varying w/b ratios. The corresponding quantitative generalized fractal analytical parameters for each group are detailed in

Table 5. Generalized analytical parameters for UHPC of different MS.

Group	D0	D1	H	D−5–D0	D0–D5	ΔD
0.18
M1	1	0.8371	0.8245	0.1807	0.5443	0.7250
M2	1	0.7975	0.8024	0.2371	0.5675	0.8046
M3	1	0.8353	0.8430	0.2046	0.4452	0.6498
SM1	1	0.8906	0.8846	0.1691	0.4116	0.5807
SM2	1	0.8607	0.8659	0.2179	0.3712	0.5891
0.17
M1	1	0.8815	0.8779	0.1811	0.4291	0.6103
M2	1	0.8474	0.8474	0.2162	0.4868	0.7030
M3	1	0.8593	0.8681	0.1879	0.3845	0.5724
SM1	1	0.9010	0.8876	0.1043	0.4305	0.5349
SM2	1	0.8804	0.8862	0.1832	0.3440	0.5372

. The specific meanings of the generalized analysis parameters are given below: D₀ (the capacity dimension or box-counting dimension) depends exclusively on the topological support of a fractal set and is independent of its measurement distribution. The condition D₀ = 1.000 holds if and only if the support set of the studied object is a one-dimensional continuum [61]. This implies that D₀ is unaffected by the quantity or density of the object within each measurement box; instead, it requires only the binary presence of data in each box to satisfy D₀ = 1.000 [39]. D₁ (the information dimension) quantifies the non-uniformity of the measure distribution across the fractal system. Specifically, it reflects the uniformity of pore distribution within a defined pore size range. When D₁ approaches D₀, it signifies a more uniform spatial distribution of pores. Conversely, lower D₁ values indicate marked local clustering of pores, highlighting stronger aggregation in specific regions [62]. H (the Hurst exponent) is derived from the correlation dimension (D₂) using the relationship defined in Equation (10).

$$H={\displaystyle \frac{(D_2+1)}{2}}$$

(10)

The H exponent characterizes the spatial correlation of localized pores across distinct pore size ranges. High or low H exponents reflect reduced or enhanced pore connectivity, respectively, within the matrix [45]. ΔD (the difference between D₋₅ and D₅) represents the width of the generalized fractal spectrum, reflecting the heterogeneity strength of the pore system. Larger ΔD values indicate a more complex pore structure and a heightened heterogeneity in pore distribution across multiple scales [34].

Figure 9 shows that the curves of the positive and negative regions of the q value behave differently. The positive and negative values of q distinguish between pores of different sizes. q < 0 pertains to small pores below 100 nm, corresponding to D₋₅–D₀ values in Table 5. q > 0 aligns the generalized fractal spectra with large pores above 100 nm, matching D₀–D₅ values [61]. q > 0 yields a more dispersed curve, indicating pore structure differences are predominantly governed by large pores. However, this also demonstrates that the stone powder’s filling effect within the system reduces small and medium-sized pores and exerts a diminished influence on the matrix. Examining the fractal parameters across groups, SM-type specimens exhibit higher D₁ values than M-type specimens at varying w/b ratios, indicating more homogeneous pore distribution in SM specimens versus pore aggregation zones in M. The H index further confirms SM-type specimens consistently surpass M counterparts, demonstrating SM’s greater proportion of closed pores and structurally superior pore network relative to M-type specimens.

Figure 10 displays variations in the fractal dimension spectral width (ΔD) across samples. All sample groups exhibit reduced ΔD values upon reducing the w/b ratio, signifying that pores formed by excess water in the matrix diminish when lowering this ratio. This void space becomes occupied by hydration products or stone powder, resulting in increased matrix density. Comparative analysis of intergroup ΔD values reveals SM-type specimens maintain consistently lower levels than M-type specimen counterparts, demonstrating weaker pore structure heterogeneity in SM-type sands.

3.5.2. Multifractal Singular Spectrum Analysis

Figure 11 shows the multiple fractal singularity spectra of the samples at different w/b ratios. The specific parameters of each group of samples are given in

Table 6. Multifractal singular parameters for UHPC of different MS.

Group	αmax	αmin	Δα	Δf(α)
0.18
M1	1.2600	0.3660	0.8939	−0.7773
M2	1.3244	0.3506	0.9737	−0.7774
M3	1.2885	0.4947	0.7938	−0.5309
SM1	1.2413	0.4814	0.7198	−0.7546
SM2	1.2972	0.5797	0.7176	−0.4386
0.17
M1	1.2643	0.4639	0.8005	−0.7295
M2	1.3188	0.4199	0.8989	−0.6563
M3	1.2609	0.5508	0.7101	−0.5314
SM1	1.1461	0.4587	0.6875	−0.8797
SM2	1.2520	0.5877	0.6883	−0.5244

. Δα represents the multifractal singularity spectral width, defined as the difference between maximum (α_max) and minimum (α_min) singularity intensities, reflecting pore size distribution uniformity [30]. A smaller Δα value indicates a simpler pore structure and more homogeneous pore arrangement. This study additionally employed Δf(α) to quantify differences between the system’s most complex and simplest regions [61].

As shown in Figure 11a,b, all groups’ multifractal singularity spectral curves exhibit bell-shaped profiles with left–right asymmetric configurations [63], which further confirms that the specimens’ pore structures possess significant multifractal characteristics. The heterogeneity spectral curves of UHPC specimens fabricated with different MS exhibit notable variations. As evidenced by the curves in Figure 10, all groups display slightly right-skewed spectral peaks, trailing patterns in high α value regions (α > 1.0), and Δf(α) < 0 universally observed, signifying slightly right-skewed curves across all sample groups [63].

Table 6 reveals SM-type samples exhibit lower Δα values than M-type counterparts across w/b ratios, indicating SM-type sands possess more refined pore structures with greater distribution uniformity. M-type sands exhibit identical Δα progression (M3 < M1 < M2), revealing pronounced multiscale pore heterogeneity in M2 where pore/defect distributions occur across multiple scales. M1 maintains elevated multiscale heterogeneity attributable to its lower stone powder content, leaving unfilled pores within the matrix. Analyzing Δα and Δf(α) synergistically, M1 and M2 specimens exhibit elevated Δα and |Δf(α)| values at w/b = 0.18, signifying pervasive presence of multiscale pores within these matrices. This suggests pore structures dominated by larger, structurally simpler pores, exhibiting greater fractal complexity compared to matrices dominated by finer pores. M3 and SM2-type specimens exhibit low Δα and |Δf(α)| values, contrasting with other M-type sand specimens. SM1 demonstrates low Δα paired with high Δf(α), indicating its dense matrix features coexisting macropores: small pores appear primarily as homogeneously distributed nanoscale pores, while large pores exist sparsely as isolated features. This stems from SM1’s lower stone powder content compared to SM2, resulting in reduced large-pore filling capacity.

Figure 12 illustrates w/b ratio variations on Δα across sample groups. All groups exhibit reduced pore heterogeneity following w/b ratio reduction, reflecting enhanced multiscale pore homogeneity. Comparative analysis of multifractal singularity spectral curves reveals a diminished right-skewed curve deviation w/b ratio decrease, signifying weakened large-pore dominance and increased matrix compaction.

3.6. SEM

Figure 13 shows the SEM and EDS results for five specimens under the w/b = 0.17 condition.

As can be seen, in terms of pore distribution uniformity, M2 clearly exhibits distinct pore aggregation areas, whereas the other specimens are relatively more uniform. This corresponds well with the results for parameter D₁. Additionally, pores in M1 and M2 coexist at multiple scales, while M3, SM1, and SM2 exhibit a denser matrix. This also corresponds with the analysis presented in Section 3.5.2.

Figure 13c,d,e also shows some pores filled with particles. Based on the EDS test results, it can be inferred that these particles consist of stone powder. All samples demonstrate the filling effect of stone powder on pores. However, M1 and M2 exhibit poorer filling effects, whereas M3, SM1, and SM2 show more pronounced filling effects.

3.7. Correlation Analysis

Figure 14 illustrates correlations between compressive/flexural strength and porosity, average pore size, ΔD, and Δα independently. Conventional porosity parameters exhibit weak correlations with specimen strength. However, analysis of strength versus multifractal parameters reveals strong correlations, with distinct relationships established between mechanical properties and multifractal characteristics. To further validate these findings, t-tests and F-tests were employed to assess the statistical significance of regression functions and parameters, respectively, with results detailed in

Table 7. The regression parameter and significance test results.

VS.	Parameter	Value	Standard Error	t Value	Prob > |t|	R2
P-C	a	184.150	14.531	12.670	1.41529 × 10−6	0.39
	b	−4.930	2.203	0.056	0.05562
P-F	a	35.501	4.37849	8.108	3.96445 × 10−5	0.48
	b	−1.786	0.66378	−2.690	0.0275
M-C	a	160.576	3.7048	43.343	8.85623 × 10−11	0.47
	b	−0.352	0.1313	−2.681	0.02787
M-F	a	26.539	1.249	21.241	2.53681 × 10−8	0.44
	b	−0.110	0.044	−2.489	0.03757
ΔD-C	a	201.258	7.210	27.910	3.65 × 10−9	0.86
	b	−77.212	11.332	−6.810	0.000132
ΔD-F	a	38.780	3.16512	12.250	3.5 × 10−7	0.74
	b	−23.70	4.97439	−4.760	0.0013
Δα-C	a	203.957	8.978	22.717	1.494 × 10−8	0.81
	b	−65.990	11.302	−5.839	3.87775 × 10−4
Δα-F	a	39.240	3.867	10.147	7.60932 × 10−6	0.66
	b	−19.544	4.868	−4.015	0.00387

and

Table 8. The regression function and significance test results.

VS.	Functions	F Value	Prob > F
P-C	y = 184.15 − 4.93x	5.00774	0.05562
P-F	y = 35.50 − 1.79x	7.23555	0.0275
M-C	y = 160.58 − 0.35x	7.18898	0.02787
M-F	y = 26.54 − 0.11x	6.19612	0.03757
ΔD-C	y = 201.26 − 77.212x	47.63779	1.24276 × 10−4
ΔD-F	y = 38.78 − 23.70x	22.69123	0.00142
Δα-C	y = 203.96 − 65.99x	34.08884	3.87775 × 10−4
Δα-F	y = 39.24 − 19.54x	16.11618	0.00387

. t-test results confirm statistically significant relationships between multifractal parameters and strength metrics. The analysis reveals strong correlations between both ΔD (R² = 0.86) and Δα (R² = 0.81) with compressive strength, though their associations with flexural strength prove less pronounced [56]. This discrepancy arises primarily from steel fibers’ reinforcement mechanism in UHPC, where their incorporation enhances ductility and energy absorption capacity, thereby mitigating pore-related impacts on flexural performance [6,64].

Collectively, multifractal analysis proves particularly effective for UHPC pore structure characterization. Leveraging multifractal analysis’s capacity to amplify, identify, and quantify pore heterogeneity, it successfully bridges microstructural features with macroscopic engineering properties.

4. Further Discussion

Based on the conclusions of various previous studies and the above analysis, it is possible to provide some discussion on the factors and mechanisms by which the properties of MS affect the performance of UHPC.

The gradation of MS primarily influences UHPC’s pore structure. Grading curves must remain within specific thresholds, as excessive coarse or fine sand alters pore characteristics. Oversized particles reduce packing efficiency, while excessive fines disrupt optimal particle distribution, reducing pore distribution uniformity, promoting pore clustering, and enhancing interconnectivity.

Secondly, the MB value of manufactured sand significantly influences UHPC’s pore structure, with storage conditions strongly affecting sand cleanliness. Open-air storage introduces detrimental impurities (dusty clay, contaminants) that elevate MB values, reflecting reduced sand purity and elevated stone powder content. This triple-effect mechanism compromises UHPC performance: (1) impurities disrupt UHPC’s dense particle packing, degrading pore structure through clustered voids; (2) reactive contaminants interfere with cement hydration, impairing strength development; (3) clay minerals adsorb polycarboxylate superplasticizers, diminishing workability.

Third, stone powder content requires strict control. While manufactured sand inherently contain stone powder from production processes, multifractal analysis reveals no specimen group exhibits small-pore dominance. This confirms appropriate stone powder quantities effectively fill UHPC’s pore structure, enhance pore distribution uniformity, and refine pore morphology. However, excess stone powder displaces original sand particles, compromising the aggregate skeleton’s functionality and ultimately degrading UHPC performance.

5. Conclusions

This study investigated the effects of five common manufactured sand types on UHPC properties. UHPC specimens were evaluated for flowability, mechanical performance, and MIP across varying w/b ratios. Multifractal analysis of MIP data quantified pore structure characteristics in UHPC containing these sands, revealing how sand properties govern matrix porosity. The methodology further established microstructure–macrostructure relationships through multifractal parameters, linking pore architecture to mechanical behavior. Key conclusions are as follows:

(1)

UHPC prepared with sieved MS exhibits superior flowability and T500 performance than using traditional MS. Sand gradation, stone powder content, and cleanliness collectively influence flowability. Excessively high fine particle content, elevated stone powder content, or low cleanliness all impair both flowability and T500 performance.

(2)

The UHPC prepared with sieved MS has better mechanical properties than traditional MS. The addition of an appropriate amount of stone powder can have a filling effect, making the matrix denser. MS with a finer particle size distribution exhibits superior performance. However, excessively fine particle distributions adversely affect mechanical properties. Low sand cleanliness indicates excessive clay and harmful substances, which can adversely affect hydration and strength development.

(3)

Regarding pore size and pore volume distributions, stone powder reduces cumulative pore volume and refines pore structure. And finer-graded sands exhibit lower cumulative pore volume and finer pore structures.

(4)

Generalized dimensions analysis reveals that MS-UHPC pore structures are primarily composed of larger voids, while smaller pores maintain relative compactness, with decreasing w/b ratios promoting further pore densification in all sample groups. Singularity spectrum analysis confirms medium-sized pore dominance across specimens, evidenced by a consistent slight right skew in spectral distributions.

(5)

ΔD analysis demonstrates SM-type sands exhibit greater pore uniformity and closure across scales. Δα values confirm SM-type specimens possess finer, more uniformly distributed pores. The correlative analysis of Δα and Δf(α) precisely characterizes pore size distribution patterns, matrix density, and structural complexity in manufactured sand UHPC systems.

(6)

SEM-EDS analysis revealed alignment between pore distribution patterns and multifractal analysis conclusions, while also demonstrating the filler effect of stone powder particles within pore structures. The correlation between multifractal parameters and mechanical properties reveals a significant linear relationship.