Owing to complex aspects of cemented sand and gravel (CSG), such as included unscreened aggregates, CSG properties differ from those of ordinary concrete. Fractal theory is introduced to study the effects of aggregate characteristics on CSG properties, quantifying aggregate gradation and shape. Numerical simulation and analyses show that: (1) improved aggregate gradation decreases the gradation fractal dimension and increases the CSG peak stress and elastic modulus; (2) more irregularly shaped aggregates increase the shape fractal dimension and decrease the CSG peak stress and elastic modulus; (3) the relationship quantified between aggregate characteristics and CSG mechanical properties provides a theoretical basis for aggregate allocation in engineering design and construction. Mixing artificial aggregates can improve aggregate gradation but reduces CSG performance. Appropriately blending artificial and onsite aggregates achieves optimal CSG performance; in this study, this is attained using 20% artificial aggregates added under standard gradation.
La presencia de áridos no cribados en arena y grava cementadas (CSG) hacen que sus propiedades difieran de las del hormigón convencional. Se introduce la teoría fractal para estudiar los efectos de las características de los áridos en las propiedades de CSG, cuantificando la gradación y la forma de los áridos. La simulación numérica y el análisis muestran que: (1) la gradación mejorada de los áridos disminuye la dimensión fractal de la gradación y aumenta la tensión máxima y el módulo elástico de CSG; (2) áridos de formas más irregulares aumentan la dimensión fractal de la forma y disminuyen la tensión máxima y el módulo elástico de CSG; (3) la relación cuantificada entre las características de los áridos y las propiedades mecánicas de CSG proporcionan una base teórica para la asignación de los áridos en el diseño de ingeniería y en la construcción. La mezcla de los áridos artificiales puede mejorar la gradación de los áridos, pero reduce el rendimiento de CSG. Combinaciones adecuadas de áridos artificiales y naturales logran un rendimiento óptimo de CSG; en este estudio, esto se logra añadiendo un 20% de áridos artificiales con gradación estándar.
Cemented sand and gravel (CSG) is a cementitious material of a known strength formed by mixing, paving, vibrating, and rolling a small amount of cementing material together with unscreened and unwashed gravels at the project site (
In recent years, an increasing number of studies have focused on the relationship between aggregate shapes and the resulting macroscopic properties of composite materials. Huang (
In terms of numerical simulation, Xiong and Xiao (
Separately, fractal theory was introduced into the study of material structure, opening a new avenue for quantifying the relationship between material complexity and macroscopic properties. Yu (
Based on different gradation characteristics of natural sand gravels at the project site, this study uses numerical simulation to obtain the maximum density gradation attainable by adding artificial aggregates. Fractal theory is applied to characterize aggregate properties and to further explore the effects of aggregate fractal characteristics on CSG macroscopic and mesoscopic mechanical behavior. This provides a theoretical basis for mix proportion design in other engineering applications.
Fractal geometry (
Aggregate gradation refers to the proportional relationship between the numbers of aggregate particles of different sizes. Traditional aggregates are divided into small stones (520 mm), medium stones (2040 mm), large stones (4080 mm) and superlarge stones (80150 mm) according to particle size. Such an irrational gradation indicates poor aggregate density, and reduces the performance of the composite material.
Based on the gradation method, a fractal model was established for aggregate gradation of CSG (
where r represents the size of a sieve pore for grading particles; r_{min} and r_{max} represent the smallest and largest particle sizes, respectively; D_{g} represents the fractal dimension of the aggregate size mass distribution; and P_{0} represents the pass rate at the maximum nominal particle size. Because r_{min} is much smaller than r_{max},
In accordance with the above formula, the fractal dimension D_{g} of the aggregate gradation can be obtained.
The section of an aggregate is rough and complex with obvious fractal characteristics; this can be represented by the box dimension. One of the most widely used fractal dimensions, its mathematical expression is as follows: supposing that is any nonempty bounded subset on R and N_{δ} is the minimum number of boxes of size δ that can cover the set F, the box dimension can be obtained through the following
where δ represents the unit measurement scale in a continuous distribution, i.e., a unit square used in the twodimensional plane;
In the generation of random aggregates, a group of random variables uniformly distributed on the interval [0, 1] were first generated by the Monte Carlo method (
Random variables on any other intervals can be obtained from the transformation of random variables on the interval [0, 1]. For example, the uniformly distributed random variable Y on any interval [a, b] can be obtained by Y = a + (b  a) X. Thus, random variables that meet the uniform distribution on each interval were generated.
The position of aggregates within different particle size ranges in the test pieces was randomly determined using the Monte Carlo method. The number of aggregate particles was obtained from the gradation of concrete and the occupancy of aggregates was determined using the Fuller gradation theory based on the principle of maximum density.
According to the Walaraven Equation (
where
CSG test cubes of 100 mm × 100 mm × 100 mm were used in the experiment. Based on laboratory conditions, standard grade II aggregates were adopted with the mix proportions shown in
Cement (kg/m^{3})  Flash (kg/m^{3})  Sand (kg/m^{3})  Water (kg/m^{3})  Aggregate (kg/m^{3})  Sand rate 

70  20  434  90  1736  0.2 
From a mesoscopic perspective, CSG can be seen as a threephase composite material composed of sand gravel aggregates, a mortar matrix, and interfaces between the mortar matrix and the aggregates. In the twodimensional plane, it was assumed that the natural sand gravel aggregates are round and the artificial sand gravel aggregates polygonal. In this study, by generating aggregate random circles and adhesive random circles with boundaries, the inner and outer circles were divided into quadrants. The number of corner points for each quadrant was determined, and the corner point coordinates were formed. Finally, the corner points were connected to generate polygons (
The constitutive relationship and failure criterion for mesoscopic component materials were selected simply. The constitutive model of each component adopted a linear elastic model while the failure criterion adopted the maximum stress criterion; that is, when the tensile stress in a material exceeds its maximum tensile strength, the material is assumed to crack.
In the finite element calculations, the material parameters of each component included the tensile strength, elastic modulus, and Poisson’s ratio. Since the parameters of mesocomponent materials were difficult to measure, minimizing the difference between stressstrain curves obtained from experiments and numerical simulation was taken as the optimization goal to obtain the parameters by means of inversion (
Meso component  Elastic modulus (MPa)  Poisson’s ratio  Tensile strength (MPa) 

Aggregate  210  0.16  0.5 
Cement mortar  65  0.20  0.5 
Interface  32  0.16  0.4 
Aggregate shape has a remarkable impact on the interface (
CSG aggregates usually come from the project site and have round or nearround shapes. In principle, they are used without screening, giving rise to a complex gradation. The research of Feng (
For projects lacking natural riverbed sand gravels, artificial aggregates can be used to replace natural aggregates. Artificial aggregates are generally shaped as irregular polygons. Their surfaces are relatively rough and irregular with more edges and corners than natural aggregates. Artificial gravel aggregates supplement natural gravel aggregates at dam sites, allowing the aggregate gradation of CSG to be adjusted. In order to mix aggregates in a logical way, the aggregates’ characteristics must be quantified. Owing to the complex gradations and shapes of aggregates, fractal theory was introduced in this work to quantify their characteristics.
The impact of aggregate gradation characteristics on CSG was compared with Fuller gradation, which was taken as the standard gradation (BZ) of aggregates and was combined with cemented sand and gravel site aggregates. The onsite gradations with sand rates of 29.2% and 38.8% were selected as control gradation 1 (DZ1) and control gradation 2 (DZ2), respectively. The cumulative percentage of particles passing the sieve is illustrated in
No.  Sieving particle size (mm)  

40  35  30  25  20  15  10  5  
BZ  100.0  94.0  87.0  79.0  71.0  61.0  50.0  35.0 
DZ1  100.0  93.0  85.5  77.4  68.6  61.3  52.3  39.9 
DZ2  100.0  93.8  87.2  79.9  71.8  65.4  57.4  45.9 
Since the Fuller curve model was highly consistent with the power function curve, the cumulative aggregate distribution curve was transformed into the same form as
According to the above formula and
The fractal dimensions of the three gradations obtained are shown in
No.  Slope K  Fractal dimension 


BZ  0.5049  2.4951  1.000 
DZ1  0.4404  2.5596  0.993 
DZ2  0.3730  2.6270  0.990 
Natural sand gravel aggregates are generally round and artificial aggregates are generally polygonal. In order to study the effect of the mix proportion of different shapes of sand gravel aggregates on the performance of CSG, the fractal characteristics of mixed aggregates were determined as listed in
No.  Aggregate proportion (%)  

Round  Polygonal  
LC1  0  100 
LC2  20  80 
LC3  40  60 
LC4  60  40 
LC5  80  100 
LC6  100  0 
Based on
According to
No.  Fractal dimension 
R^{2} 

LC1  1.7630  0.998 
LC2  1.6102  0.998 
LC3  1.5000  0.997 
LC4  1.3187  1.000 
LC5  1.1477  0.995 
Based on the results obtained, the following discussion examines the effects of aggregate characteristics on CSG mechanical behavior. In particular, the relationships between aggregate gradation and shape fractal dimensions and resulting CSG mechanical behavior are discussed.
The effect of aggregate gradation on the mechanical behavior of CSG was studied based on the three aggregate gradations described in 3.1: the standard gradation (BZ), control gradation 1 (DZ1), and control gradation 2 (DZ2). The random aggregate models in which all aggregates were artificial (LC1) are shown in
As the aggregate gradation approaches the standard gradation, the fractal dimension of the aggregate gradation decreases and the peak stress increases. Simultaneously, as the aggregate gradation improves, the tangent slope of the stressstrain curve increases; that is, the elastic modulus increases. This occurs because, as the aggregate gradation approaches the standard gradation, the aggregate quantity increases, improving the aggregate density; thus, both the strength and elastic modulus increase. This is consistent with experimental results in the literature (
The effect of aggregate shape on the mechanical behavior of CSG was analyzed in accordance with the relationship between fractal characteristics and mechanical characteristic parameters.
Considering the impact of the corner edges, the area of the interfacial transition zone (ITZ) units around the polygonal aggregates is larger than that around the circular aggregates of the same volume. Under the same stress, more ITZ units around the polygonal aggregates will show damage and fracture than around the circular aggregates. This conclusion is consistent with that in the reference works (
Considering the impact of stress concentration, the stress distribution around polygonal aggregates is more concentrated than that around the circular aggregates. Therefore, ITZ units around the polygonal aggregates are more prone to damage and fracture. Furthermore, under the standard gradation, when the aggregate shapes are all round (LC6), i.e., when all the aggregates are natural sand gravel aggregates, the highest peak stress and elastic modulus are achieved. However, since this study focuses on project sites lacking natural sand gravels, this condition is difficult to achieve and is not considered further.
According to
For use in CSG, artificial aggregates are mixed into natural aggregates sourced from riverbeds. Considering the complex resulting aggregate characteristics, aggregate gradation and shape were quantified using fractal dimensions in this research. On one hand, the results obtained show that mixing artificial aggregates standardizes the resulting gradations, and that the closer a gradation is to the standard gradation, the better its CSG mechanical properties. On the other hand, excessive artificial aggregate content may degrade CSG mechanical properties. For project sites lacking natural aggregates, artificial aggregates should be added appropriately to achieve the best performance. When 20% artificial aggregate content was added under the standard gradation considered in this study, the elastic modulus and peak stress reached their maximum values; this scenario was suitable for onsite mixing. The method used in this study to investigate the impact of complex aggregates on CSG mechanical properties through fractal theory and numerical simulation can provide a theoretical reference for other CSG projects.
In view of the complex characteristics of CSG aggregates, the concept of fractal dimensions was introduced to quantify aggregate gradation and shape. A twodimensional random aggregate model of CSG was established, and mechanical properties of CSG under different aggregate gradation and shape fractal dimensions were studied through parameter inversion. The following conclusions were drawn:
The closer the aggregate gradation to the standard gradation, the smaller the fractal dimension of the aggregate gradation; as the proportion of polygonal aggregates increased, the aggregate shape fractal dimension increased.
According to uniaxial compression numerical testing, as the aggregate gradation fractal dimension decreased, both the peak stress and elastic modulus of CSG increased.
According to uniaxial compression numerical testing, as the aggregate shape fractal dimension increased, both the peak stress and elastic modulus of CSG decreased.
For mixing artificial aggregates with natural aggregates from riverbeds, a mix proportion for optimal mechanical properties was obtained; this could provide a theoretical basis for similar projects.
Due to the limited test methods available for this study, the mesoscopic numerical simulation technique in this work did not consider the effect of aggregate shape on interface performance; this topic requires further research in the future.
This research was funded by National Key research and Development Project of China: (2018YFC0406803) physical and numerical model and evolution law of performance of cemented granular material dam, open project of Research Centre on Levee Safety & Disaster Prevention Ministry of Water Resources: (2018008) research on characteristics and optimization of cemented gravel flood control dike, Graduate Education Innovation Program Fund of North China University of Water Resources and Electric Power: (YK202006) Mesodamage mechanism and evolution rule of cement sand and gravel under freezethaw action and Henan Provincial Natural Science Foundation Project: (202300410270) Research on Frost Resistance Durability Behavior and Deterioration Damage Mechanism of Cemented Sand and Gravel.