Mesoscopic Fracture Model of Coarse Aggregate Interlocking Concrete

e skeleton formed by coarse aggregate has an important inuence on the macroscopic mechanical properties of concrete, especially the fracture properties. Based on mesomechanics and fracture mechanics, this paper conducts theoretical simulations and experimental studies on the mechanical and fracture properties of coarse aggregate interlocking concrete through the theory of mesomechanics homogenization as well as compressive strength tests, exural strength tests, axial compressive strength tests, elastic modulus tests, and fracture toughness tests. e results show that when the coarse aggregate is within a certain volume increase, the fracture energy and ultimate strength of concrete are signicantly improved. At the same time, the proposed mesomechanics calculation model has high accuracy for calculating the fracture characteristics of concrete when the coarse aggregate increment is less than 20%.


Introduction
e mechanical strength and fracture toughness of concrete are material parameters that characterize the crack resistance and failure criteria of the material and are extremely important for the safe design of concrete members [1][2][3]. Concrete can be considered as a two-phase composite material containing mortar and aggregate [4][5][6], and its crack tips will form irregular intermittent cracks or crack extension zones during fracture. In recent years, relevant studies have shown that material property parameters such as fracture toughness and fracture energy of concrete are a ected by signi cant size e ects, and to obtain material parameters independent of the size of the concrete material, su ciently large specimens must be cast [7][8][9]. In the concrete single-sided incised beam (SENB) specimen proposed by the International Federation for Research in Structures and Materials (RILEM) [5], the fracture parameters of concrete independent of size will not be obtained directly from this test because the concrete specimen size relative to the maximum particle size of the aggregate is not su ciently large (W/dmax 5∼30). For this reason, revealing the e ect of size on the fracture performance of concrete has become a hot spot for domestic and international research.
Cement paste forms a large component in conventional concrete, while the coarse aggregate content is very small. Coarse aggregates are the strength framework of concrete and are usually the strongest, most durable, and volumetrically stable structural units of concrete materials [10,11]. As a composite material, the mechanical properties of the aggregate improve with increasing aggregate volume fraction as long as the aggregate is well integrated with the cement paste [12][13][14]. However, the mechanical behavior of concrete is not an absolute property and it is in uenced by many factors. Traditional concrete has some problems in its mechanical design [15][16][17]: (1) e large amount of cement and cementitious materials leads to signi cant cracks in the concrete at an early stage, which in turn a ects its mechanical and durability properties at a later stage. (2) e low-or high-volume content of coarse aggregates not only fails to give full play to the strength contribution of aggregates but also causes adverse effects such as increased concrete cost, waste of resources, and increased environmental load. (3) In the process of concrete fracture, the initial crack tip between the aggregate and the cementitious material fails the cementitious occlusion leading to crack expansion. erefore, the content of coarse aggregates in concrete will directly affect the mechanical properties of concrete. Hillerborg [18] proposed earlier that the development of the fracture process zone (FPZ) of concrete is essential in the development of aggregate bond bite cracking and proposed a virtual cracking model, which was widely used and became the basis for the study of quasi-brittle fracture of concrete. e influence of aggregate content on fracture energy and fracture process zone was analyzed; Zollinger et al. [19] showed through tests that the FPZ will increase with the increase in average aggregate size during concrete fracture. Currently, many studies have been conducted to construct high-performance, high-volume stability aggregate interlocking concrete by increasing the amount of coarse aggregates in order to greatly optimize the comprehensive performance and application of concrete materials [20,21].
Among them, cast-in-place aggregate concrete, a newly proposed and promising coarse aggregate interlocking concrete, is considered as a new component to improve the service life of concrete by maximizing the material performance and reducing the incidence of diseases [22,23]. Compared with conventional concrete, aggregate interlocking concrete has some significant advantages in terms of mechanical, durability, and stability properties [24][25][26][27]. In addition, the establishment of fracture models for aggregate interlocking concrete is basically based on three-point loading tests, and equally important mechanical tests such as compressive strength and modulus of elasticity and their mechanical mechanisms, especially fracture theory, have not been systematically studied. Fracture or cracking characteristics are the most required engineering and material properties of concrete, which are necessary signs to ensure the stability of concrete. Also, based on the influence of coarse aggregates, the internal nanostructure and hydration products of aggregate interlocking concrete are redistributed. Traditional mechanical models are not comprehensive in characterizing its fracture mechanism, and a multiscale model is needed for theoretical analysis.
Based on these backgrounds, the main objective of this paper is to investigate the fracture mechanism and theoretical calculation model of aggregate interlocking concrete. Based on the integration of fine-scale mechanics and finescale structural changes of concrete, a fine-scale transport model based on description, theoretical calculation, and numerical analysis is established and determined and its reliability is verified by macroscopic mechanical tests to provide a theoretical supplement and support for the mechanical mechanism of aggregate interlocking concrete.

Theoretical Analysis
e theory of fracture and damage of traditional concrete materials is established based on the energy law, crack development law, and other factors. However, it does not consider the changes in mechanical response caused by the increase in coarse aggregate and the interlocking shape. Considering the influence of coarse aggregate on the microstructure of concrete interface, a new fracture mechanism of aggregate interlocking concrete based on the theory of mesomechanics homogenization (RVE model) was established.

2.1.
e RVE Model of Coarse Aggregate Interlocking Concrete. Suppose there is an elastic solid with a volume of V in the concrete matrix at the mesoscale and an inclusion subdomain in the inclusion and disturbance caused by coarse aggregate in elastic solids. e base domain and the inclusion domain have different elastic tensors and flexibility tensors. e inhomogeneity caused by the existence of pore structure inclusions due to the interaction of coarse aggregates will produce stress and strain field disturbances in a uniform solid. erefore, the stress and strain equations of an elastic solid with pore structure inclusions can be expressed as where σ(X, x) is the stress of a cement-based elastic solid with pore structure inclusions, ε(X, x) is the strain of a cement-based elastic solid with pore structure inclusions, σ 0 (X) is the stress of the cement-based elastic solid, ε 0 (X) is the strain of the cement-based elastic solid, σ d (x) is the disturbance stress of the pore structure with inhomogeneous bodies, and ε d (x) is the disturbance strain of the pore structure with inhomogeneous bodies.
rough the superposition of the characteristic strain field and the actual strain field, the relationship between the equivalent uniform solid stress field and the original nonuniform stress field can be obtained: x, X ⊂ M, where C is the elasticity tensor of nonuniform solids, ε * (x) is the characteristic strain for simulating material mismatch, defect, pore structure, or inhomogeneity mismatch, M is the characteristic domain at the mesoscale of solids, and Ω is the inclusion subdomains in solids. erefore, the characteristic stress field of the pore structure in the hardened concrete due to the interaction of the coarse aggregate can be obtained: 2 Advances in Materials Science and Engineering where C I is the elastic tensor of the inclusion field.
In an infinite space, the induced displacement field caused by the characteristic strain field in the hole structure is determined as where is the Green transformation formula, and R 3 is the unlimited space domain.
For the coarse aggregate lock concrete, the elastic strain field can be expressed as

e Elastic-Plastic Constitutive Relationship of Aggregate
Interlocking Concrete. According to Hooke's law, the relationship between elastic strain and Cauchy stress is e theoretical solution of the elastic-plastic constitutive relationship of coarse aggregate lock-type concrete is determined as where [C FAC ] ijkl is the effective modulus of the concrete in the elastic stage, [C FAC ′ ] ijkl is the effective modulus of the concrete in the plastic stage, [ε FAC ] kl and [ε FAC ′ ] kl is the strain of concrete in the elastic-plastic stage, and [σ FAC ] ij is the loading stress. e effective modulus of concrete in the elastic stage is deduced based on the theory of generalized self-consistent method, which is the mechanical behaviour influenced by the interlocking of coarse aggregate and the mortar between the interface: where C pq FAC is the effective modulus of concrete, C pq f(V Coarse aggregate interlocking ) is the modulus of coarse aggregate under the interlocking action, C pq ITZ is the modulus of ITZ, ϑ 1 is Poisson's ratio of coarse aggregate under the interlocking action, ϑ 2 is Poisson's ratio of ITZ, f(V Coarse aggregate interlocking ) is the density distribution of coarse aggregate, and f(ϕ pore ) is the distribution characterization function of pore structure.

e Fracture Toughness Equation of Aggregate Interlocking Concrete.
e fracture process of materials is predicted by the yield function during the plastic development.
Advances in Materials Science and Engineering erefore, the fracture yield equation of concrete is established based on the Gurson model as where Φ( ij , σ e , f * (ϕ pore )) is the yield function of concrete, σ ij ′ (x) is the macroscopic stress, σ e is the equivalent yield stress, eq is the Mises yield stress, m is the hydrostatic stress, f * (ϕ pore ) is the porosity distribution function of the ITZ during the yielding process, f c (ϕ pore ) is the porosity distribution function when pore confluence begins, f fra (ϕ pore ) is the critical porosity distribution function when the material fractures, D ′ is the damage factor under load, ΔR is the pore size change value of ITZ, and R N is the pore size characteristic value of ITZ. Based on the fracture criterion, the fracture toughness equation of the coarse aggregate lock concrete is determined as where G FAC is the fracture toughness of coarse aggregate interlocking concrete.

Raw Materials and Mixing
Proportions. Raw materials including local ordinary Portland cement of grade P.O 42.5 N from Guangzhou Yuebao Cement Co., local river sand with a fineness modulus of 2.3, and water were used throughout the study. Local limestone was used as a coarse aggregate, with a size ranging from 5 mm to 25 mm. Tables 1 and 2 show the mechanical and chemical properties of ordinary Portland cement. e physical properties of the river sand and coarse aggregates are presented in Table 3. In this experiment, the water-tocement ratio (w/c) was 0.45, and the mixing proportion of concrete is shown in Table 4 (coarse aggregates increase by 5%, 10%, 15%, 20%, and 25% according to the volume percentage of the basic aggregate).

Test Method for Strength of the Coarse Aggregate
Interlocking Concrete. 150 mm × 150 mm × 150 mm (144 pieces) specimens were used for the compressive strength test, 100 mm × 100 mm × 400 mm (108 pieces) were used for the flexural strength test, and 150 mm × 150 mm × 300 mm (108 pieces) were used for the axial compressive strength test for 28 days according to GB/T50081-2019 [29].

Test of Elastic Modulus of the Coarse Aggregate
Interlocking Concrete. e secant modulus under 30% axial compressive strength stress is taken as the elastic modulus value according to GB/T50081-2019 [29]. e size of the experimentally produced specimen is 150 mm × 150 mm × 300 mm (108 pieces).

Test of Fracture Toughness of the Coarse Aggregate
Interlocking Concrete. A notched three-point bending beam was used to determine the fracture toughness of coarse aggregate interlocking concrete. For crack generation, the specimen is made to produce cracks by cutting on the side of the specimen. e slit width is controlled at 3 mm ± 1 mm, the slit length is controlled at 80 mm ± 2 mm, and the angle between the joint surface and the specimen is 90°± 0.5°a ccording to GB/T50081-2019 [29]. e size of the experimentally produced specimen is 100 mm × 100 mm × 400 mm (108 pieces).

Mercury Intrusion Porosimetry (MIP) Analysis.
Mercury intrusion porosimetry (MIP) is the most common method used to study the pore properties of cement-based materials.
is method is relatively straightforward and generally yields reproducible pore size distributions. e significant characteristic parameters, such as the total pore diameter, total porosity, and theoretical pore diameter, can be deduced from these pore size distributions. With increasing pressure, mercury is pressed into the pore structures of concrete samples. After the mercury fills the samples, the relationship between the intrusion pressure and the pore radius can be obtained. e fine particles of concrete (5 groups and 6 solid samples in each group) were subjected to mercury intrusion test analysis.

Experimental Values of Mechanical Characteristics of
Coarse Aggregate Interlocking Concrete. Figures 1 and 2 show the experimental results of compressive and flexural strength of coarse aggregate interlocking concrete. e results show the following: (1) e macroscopic mechanical strength increases as the volume rate of the coarse aggregate increases. (2) When the volume fraction of the external coarse aggregate is 0%-20%, the strength increases with the increase in the aggregate replacement rate. When the volume rate of coarse aggregate is 15%∼20%, the compressive and flexural strengths reach the maximum values, which are increased by 40% and 36%, respectively. (3) However, when  the volume rate of the coarse aggregate is 25%, the macromechanical properties are significantly decreased. is is because too much coarse aggregate in the concrete makes compaction difficult, which reduces the mechanical strength. However, due to the interlocking effect between the aggregates, the final mechanical strength is still higher than the reference value. In addition, the comparative analysis of the interface morphology subjected to compression fracture is shown in Figure 3. e results show the following: (1) e increase in the volume ratio of the coarse aggregate promotes the formation of the spatial distribution of the mutual embedment and occlusion of aggregates during the concrete hydration process. (2) e increase in coarse aggregate also reduces the distribution probability of through cracks under load. (3) When the volume ratio of coarse aggregate is 20%, there is no obvious crack in the core part of the specimen during crushing failure. Figure 4 shows the elastic modulus of the coarse aggregate interlocking concrete. 5%, 10%, 15%, and 20% of the aggregate volume increased by 3.3%, 6.6%, 13%, and 19.8% compared with the elastic modulus of the benchmark group, respectively. Subsequently, when the volume of the aggregate is 25%, the elastic modulus began to decrease. e variation and evolution are similar as the abovementioned results of strength characteristics.

Experimental Values of Fracture Characteristics of Coarse
Aggregate Interlocking Concrete. Figures 5 and 6 show the results of initial fracture load, ultimate fracture load, and initial fracture toughness during fracture experiment of the coarse aggregate interlocking concrete. Compared with the benchmark concrete, with the increase in aggregate volume, the cracking load shows an increasing trend. However, when the aggregate volume increases by 20%, the crack initiation load begins to show a downward trend. When the increase in the volume of the coarse aggregate is 0%-20%, the reason for the phenomenon of crack initiation and ultimate load enhancement is that more aggregate makes the material inside the concrete fill and compact, which improves the overall compactness of the material. When the increment reaches 20%-25%, the workability of concrete will be affected by the excessive coarse aggregate, which will reduce the cohesion of concrete and generate more internal cracks. is leads to a reduction in the initial and ultimate loads during the concrete fracture process.
According to the classical calculation equations in fracture mechanics, the test fracture energy under different coarse aggregate volume increments of the coarse aggregate interlocking concrete can be obtained and is shown in Figure 7.
e results show the following: (1) When the volume increase in the coarse aggregate is 0%-20%, the fracture energy shows an increasing trend. For every 5% increase in volume, the fracture energy increases by 12.3%, 17.6%, and 19.4%, respectively. In the fracture process, a certain friction force is formed between the coarse aggregates, which indirectly increases the bite force between the coarse aggregate and the cementitious material, and the fracture energy is improved. (2) However, when the volume increase fraction of the coarse aggregate reaches 25%, the fracture energy shows a downward trend (compared to 20% aggregate volume fraction, it decreased to 25.3%). When the volume increase of the coarse aggregate reaches 25%, the bite force between the materials begins to decrease due to the decrease in the compactness, which causes the evolution of the fracture energy to show a downward trend. pore size distribution curves based on MIP testing data are shown in Figure 8. Characteristic pore parameters such as porosity and critical pore size are shown in Figure 9. We can obtain the following: (1) With the increase in coarse aggregate volume, the porosity and critical pore size of concrete gradually decrease. (2) e appropriate increase in the volume of the coarse aggregate not only plays the role of the original strength skeleton but also improves the internal compactness of the concrete and reduces the internal pore density, which improves the ultimate strength and fracture resistance of the concrete. (3) However, the pore density increases when the volume increase of coarse aggregate is 20%∼25% based on the influence of workability and insufficient wrapping of cementitious materials, in which the ultimate strength and fracture energy are significantly reduced. In summary, the coarse aggregate has an important relationship with the construction of pore structure of concrete. e change of porosity and critical pore size significantly affects the fracture toughness and ultimate strength, which provides relevant calculation parameters for the theoretical simulation calculation in this paper.

Comparison of the Fracture Characteristics Calculated by the Mesoscopic Model and Experimental Results.
is section reports the results of the fracture characteristics based on the mesoscopic model (equations (8)- (10)). e comparison results and errors are shown in Figures 10 and 11. Comparing the calculation results with the experimental results, we can find the following: (1) For the simulation of compressive strength and elastic modulus, the simulation results are higher than the experimental results. However, as the volume of coarse aggregate increases, the calculation error gradually stabilizes (3∼4%). e accuracy of the calculation results of higher volume increase of the coarse aggregate (>20%) is not ideal. is may be attributed to the fact that when the amount of coarse aggregates exceeds 20%, the distribution of aggregates inside the cement matrix is intricate and complex, which is prone to produce uneven forces on the aggregates, thus causing the calculation results to be affected and producing large fluctuations and errors.
(2) For the simulation results of fracture energy, the simulation results have the same laws as the experimental results and the calculation error is small (the increase of coarse aggregate is less than 20%, and the error is 2%). e simulation results can well reflect the fracture energy development characteristics of the concrete fracture process. However, when the volume increase of coarse aggregate is large (>20%), the simulation results of fracture energy have very large errors. How to improve the accuracy of related calculation results in the future is the research content that the model still needs to consider in the future. In summary, the mesoscale multimechanical model established in this paper has certain theoretical basis and calculation accuracy.

Conclusions
A mesoscopic model, based on the micromechanics RVE model and fracture mechanics mechanism, was validated and applied to the experimental test for calculating the influence of coarse aggregate on fracture characteristics of the concrete. e following conclusions of this study can be drawn: (1) Experiments and theoretical studies are conducted on the ultimate strength and fracture characteristics of concrete. e increase in coarse aggregate can significantly improve the macroscopic strength and fracture resistance of concrete within a certain range. However, as the volume increases, due to the influence of too much coarse aggregate on the working performance, the wrapping performance of the cementitious material, the pore size, and the macromechanical properties decrease (still higher than the reference value).
(2) Based on the new model, some mesoscopic mechanical models were established for describing the mechanism of coupling relationship between pore size, mechanical strength, and fracture characteristics. e proposed model was proved to be effective for evaluating mechanical response of the coarse aggregate interlocking concrete.

Data Availability
e research data used to support the findings of this study are included within the article.

Disclosure
e Key Laboratory of Transport Industry of Road Structure and Material and Foshan University is the co-first unit. Ben Li and Ying Yu are considered as the co-first authors.