#### Abstract

To investigate the fracture process and failure mechanism of concrete subjected to uniaxial compressive loading, a new finite element method—the base force element method (BFEM)—was adopted in the modeling of numerical simulation. At mesoscale, concrete is considered as a three-phase heterogeneous material composed of aggregate particles, cement mortar, and the interfacial transition zones between the two phases. A two-dimensional random convex aggregate model was established using the principle of the area equivalence method. A multistage linear damage constitutive model that can describe nonlinear behavior of concrete under mechanical stress was proposed. The mechanical properties of concrete mesoscopic components are determined. The numerical simulation results indicate that the base force element method can be applied to predict the failure pattern of concrete under compressive loading, which have a good accordance with the available experiment data. The stress contour plots were given and used to analyze the failure mechanism of concrete. The effects of specimen size on the strength of concrete material were studied. It is found that compressive strength of concrete decreases as the specimen size increases. In addition, the influences of aggregate distribution, coarse aggregate content, and end friction on concrete performance are explored.

#### 1. Introduction

The usage of the concrete is increasing with the development of modern civil engineering. As the most widely used construction materials, the researches on the macromechanical properties and the strength for concrete materials of concrete material has important theoretical significance for the reliability in designing concrete structures, especially for the complex structures. It is found that the failure behavior of concrete is complex even under normal loading conditions. In macrolevel, it is hard and incomprehensive to describe the mechanical properties and failure mechanism of concrete material due to the limitations of the laboratory experiment, time-consumption, and environmental factors of the test.

Wittmann et al. [1] Schlangen and van Mier [2] are those of the early scholars to study the mechanical properties of concrete from the microscopic level using numerical analysis methods. The use of numerical analysis methods to study concrete is also known as “numerical concrete.” Until now, advanced mesoscopic mechanics analysis theory and computer technology efforts contribute to explore the numerical simulation of damage process and the influence on material damage in concrete with the finite element method. In mesoscopic structure, concrete is a complex heterogeneous composite material, whose fracture pattern and the macroscopic mechanical nonlinear behavior can be attributed to the properties and mix proportion of internal mesoscopic composition, such as aggregate particles distribution, aggregate particles size, aggregate particles shape, cement mortar properties, interfacial properties, and microcrack [3]. That is to say, the obviously random characteristics of macroscopic mechanical properties were reflected entirely by the global strength and stress-strain full curve which are mainly caused by its internal mesoscopic heterogeneity and randomness of concrete materials.

There is an obvious size effect phenomenon that the larger dimension structures more likely fail at relatively smaller loading. This phenomenon in quasi-brittle structures is related to a transition from ductile behavior of small specimens to a totally brittle response of large ones [4, 5]. Also, the study of size effect has been observed in [6–8]. The behavior of size effect has been widely accepted which is contributed by the effect of internal materials, and some were based on the principles of fracture mechanics [9–11]. Fracture mechanics was first employed in the study of fracture mechanism in concrete materials [12], and a large amount of classical cracking models was established, such as two parameter fracture model [13], virtual fracture model [14], and equivalent fracture model. Nevertheless, without take into account the existence of microcracks that is the main reason for formation and propagation of cracks and also eventual failure of concrete [9, 15], the mechanical properties and fracture mechanism of concrete was difficult to reasonably predict using fracture mechanics theory.

Therefore, the microscopic damage mechanics theory was proposed to make up the limitation of the fracture mechanics concept which neglected the process before macrocracks appeared (i.e., microcracks and microdefects). Accordingly, numerous researches have focused on simulating the damage process of concrete considering its mesoscopic composition characteristics. Typically, the lattice model [16], the micromechanical model [17], the random aggregate model [18], and the mesoelement equivalent model [19]. Varieties of numerical models of mesoscopic mechanics were developed from two dimension to three dimension which made significant progress in ensuring the accuracy of numerical simulation [20, 21].

The concept of damage variable was proposed because of the rise of damage mechanics theory and was combined with continuum mechanics in the effort of [22] to describe the continuous change of material damage, especially the strain softening characteristic. As one of the essential issues of continuous damage mechanics, the damage constitutive relation of material has been studied, which includes elastic damage, plastic damage, and elastoplastic damage. In the previous work, concrete is the quasi-brittle material that had no obvious irreversible deformation after external stress. The investigations [23–25] have indicated that the failure behavior of concrete subjected to uniaxial loading was able to be described by the elastic damage constitutive model considering the influence of damage on stiffness because of the brittle characteristics.

So far, the technology of concrete aggregate placement is the most concerned in numerical simulation on the mesostructure. In order to simplify the calculation, a single aggregate model was proposed [26] to analyze the damage of concrete under uniaxial compression loading. It was found that aggregate-mortar interface is the weakest link and main reason of inelasticity. Buyukozturk [27] developed the nine circular aggregates model to study the strength and failure mechanism of concrete with the loadings of uniaxial and biaxial compressions. Considering the influence of aggregate distribution on the complex performance of concrete, several efforts have been made in consistent with the actual concrete in the statistical sense. Bazant et al. [28] used the random circle particle model that can describe realistically the localization and propagation of cracks which agreed with experiments. In addition, they [29, 30] had published a number of papers in recent years on comparisons between Bazant size effect model (SEM) and Hu-Duan boundary effect model (BEM) for quasi-brittle fracture of concrete. Hu et al. [31, 32] introduced the maximum aggregate size into BEM, established the corresponding prediction equations, and analyzed and compared the essential difference and application function of SEM and BEM. Based on this, they explained the internal mechanism of structural damage size effect.

However, few researches have focused on that the aggregate shape may also affect the properties of concrete. Wittmann et al. [1] presented the random distributed geometry of the natural aggregates model including round and polygon to investigate the effective properties, e.g., the diffusion coefficient and the modulus of elasticity of concrete material. The methods for generating polygon or polyhedron aggregate can be concluded as two types: one was generated new vertices and surfaces from the surroundings of a convex matrix until the requirement was satisfied [18, 33, 34] and the other was using a Delaunay triangulation which divided the available space into separate areas.

In recent years, a new type of finite element method, the base force element method (BFEM), has been developed by Peng and Liu [37] based on the concept of the base forces by Gao [38]. This method takes the base force vector (first-order tensor) as an unknown basic quantity, and it has many advantages. For example, it has concise and integral explicit finite element formulation without numerical integration. Moreover, the programming is simple, and the calculation accuracy is high. Furthermore, the base force element method (BFEM) on potential energy principle was used to analyze recycled aggregate concrete on mesoscale [39].

In this paper, the base force element method (BFEM) is applied to investigate the failure pattern of common concrete with polygon aggregate subjected to uniaxial compressive loading in two dimensions. In mesostructure, concrete is assumed as a three-phase composite material taking its heterogeneous into account. To generate random convex polygonal aggregates on the basis of circle aggregates, correspondingly, according to the Monte Carlo principle, the mechanical properties of mesostructure for concrete were assumed to obey the Weibull distribution. An elastic damage constitutive model is adopted with the nonlinear behavior analysis of concrete material. The good relation with available test data confirms the accuracy and feasibility of the present mesoscopic simulation approach. A series of concrete specimens with different sizes, different aggregate distributions, different aggregate contents, and different end frictions are designed to analyze failure mechanism and the mechanical behavior of concrete from mesolevel.

#### 2. Theory of Base Force Element

The concept of base forces was proposed by Gao [38]. The base forces are used to replace various stress tensors for the description of the stress state at a point. These base forces can be directly obtained from the strain energy. By means of the base forces, the equilibrium equation, boundary condition, and elastic law are written in very simple forms. For large deformation problems, the derivation of basic formulae was simplified.

The base force element method (BFEM) on complementary energy principle uses the base forces as fundamental variables to establish control equations of the novel finite element method. In the literature [38], Gao gave an new idea of deriving compliance matrix of an arbitrary polyhedron element. Peng and Liu [37] gave an explicit expression of the compliance matrix and derived governing equations of the BFEM on complimentary energy principle using the Lagrange multiplier method. The new finite element method based on the concept of base forces was called as the base force element method by Peng and Liu [37].

The BFEM on potential energy principle uses the displacement gradients which are the conjugate variables of base forces to establish control equations of the new finite element method. Peng and Liu [37] derived governing equations of the BFEM on potential energy principle using Gao’s thought for a triangular element as shown in Figure 1. The stiffness matrix of a base force element [39] can be obtained aswhere is Young’s modulus of an element, is Poisson’s ratio, is the area of the element, is the unit tensor, expressed as , and , and can be calculated fromwhere is the basic vector and is the conjugate vector of ; and are the length of edges and of the element; and and denote the external normal of edges and , respectively.

For the plane stress problem, it is necessary to replace by and by in equation (1).

#### 3. Convex Polygon Aggregate

Concrete is an inhomogeneous composite material that the macroperformance is strongly related to its mesocomposition. Generally speaking, concrete was regarded as a three-phase composite material consisting of coarse aggregates, cement mortar, and the interface transition zones (ITZs) between aggregates and cement mortar at mesolevel. As known, internal structural characteristics will change causing the different properties of concrete when applied loading. As the main reason of this change, its internal material heterogeneity has the influence on damage behavior of concrete. Therefore, the aggregate shapes should be noted to ensure that the characteristic of mesostructure material is accurately represented.

The aggregate type of concrete determines the shape of aggregate particles, generally speaking, including two types; gravel aggregates have a round shape, while crushed rock aggregates has an angular shape [18, 19]. A convex polygon aggregate model was created in this paper to simulate real aggregates. The concrete material containing convex aggregate is considered to have three-phase media, as shown in Figure 2. Compared with the work of [39], the effect of aggregate type is explored.

##### 3.1. Polygon-Based Framework

The method is to generate particles with a randomly shaped polygon-based framework which is divided into triangles, quadrilaterals, and pentagons according to the classification of aggregate size on the boundary of the round coarse aggregates at first. The adjacent vertexes of the polygon-based framework are controlled by the following equation:where is the minimum distance between adjacent vertexes of the base frame. is the radius of the aggregate. is the number of vertexes in the polygon-based framework. The area of the newly generated convex aggregate must reach the original aggregate area. The new convex polygon aggregate is illustrated in Figure 3.

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##### 3.2. Convex Polygon Generation

In Figure 4, the vertices of any convex polygon can be ordered in counterclockwise as ; correspondingly, the coordinates are . A new vertex in the outer semicircle is inserted so that the diameter is the longer side of the polygon. In order to improve the generation efficiency of the convex polygons, the slope of the longest side and the angle of the *X*-axis should be first calculated, as shown in Figure 4. Therefore, the coordinates of the new insertion point can be evaluated by equation (4) [18] in the following forms:where and are, respectively, the random numbers from 0 to 1. Herein, the new insertion point cannot exceed the scope of the specimen, namely, and that ranges between to and to , where and are the width and height of the specimen, respectively.

After the new vertex of the polygon is obtained, it is necessary to determine whether the two new edges, and , formed by the new point and the longest side, are longer than the minimum limit length , which is designed to ensure not to affect the division precision of the element, as expressed in the following equation:

Also, taking the efficiency of the extension into consideration, the growth of the newly formed polygon should be greater than the minimum , as in the following form:where and are the area of polygons before and after the new point is inserted, respectively.

Furthermore, the question is whether polygons can be achieved the requirement of convex polygons. According to the work in [40], the area criterion has been widely used to discriminate the convexity of polygon.

(two conditions of the positions in point are shown in Figures 5(a) and 5(b)) is a point in the plane, and its coordinate is . Besides, the area of the triangle can be calculated easily by the following equation:

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It is to be noted that when point is inside the polygon, ; when point is on the polygon boundary, at least ; when point is outside the polygon, at least . Besides, the internal area of the polygon is . In a word, the definition of point *P* is the inner point of the convex polygon that satisfies the following equations:

For these three concisions, the area of polygon should be the sum of area of every new formed polygon; that is,

As mentioned above, point is a newly inserted vertex on the outside of the edge , which formed two triangles and with adjacent sides and , respectively, shown in Figure 6. According to the area criterion when both the areas of the two triangles mentioned in equation (7) are positive, the newly formed is the convex polygon.

In addition, to prevent the occurrence of the aggregates invasion in Figure 7, the insertion point should be checked. For the two cases of with different aggregate invasion, the point P can be inserted according to equations (7) and (10). The similar work was done in [18].

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When the new insertion point meets all the above criteria, the insertion point is confirmed. And the extension ends until the area of generated polygonal is larger than the corresponding circle. Figure 8 presents the flowchart for the extension of the polygon-based framework.

#### 4. Simulation Procedure

It has been known that the random aggregate model can simulate the internal structure of concrete. The heterogeneity of mesostructure will affect the mechanical properties of the material. To simulate more accurately, the random aggregate model was thus adopted. Besides, concrete is treated as a three-phase composite material consisting of aggregates, cement mortar, and the interfacial transition zones (ITZs) between the former two phases in the present model.

##### 4.1. Mesoscale Model of Concrete

According to the experimental work, the particle size of fine aggregates is less than 5 mm which is uniformly distributed in the mortar matrix as part of the mortar phase. Namely, the mortar matrix is viewed as the homogenous phase in mesoscopic structure simplified for the purpose of convenience.

A typical “take-and-place” method was used to ensure that the coarse aggregates are placed randomly in the concrete specimen, which is similar with the work in [43]. Also, the coarse aggregates with large size assumed to be circle are divided into four levels ranging from 5 mm to 20 mm. Based on Fuller’s grading curve, a simplified formula derived by Walraven J. C. is applied to calculate the amount of aggregate particles with different sizes in the two-dimensional plane. Moreover, as mentioned above, the circle aggregates are converted into convex aggregates. It should be noted that the take process and place process are conducted at the same time. Hence, newly generated aggregates are placed immediately one by one into the concrete specimen assuring that there is no overlapping with aggregates placed before and other components [18].

The divided regular triangular finite element mesh is projected directly onto the random aggregate model of concrete, as shown in Figure 9. It is found that dividing the size of the mesh element less than 1/4 of minimum aggregate particle size would make the macroscopic mechanical properties of concrete more stable [44]. Accordingly, choose the mesh size of 1 mm considering that the particle size of the minimum aggregate is 5 mm in this model.

The element type is determined by the position of the element node, and the processes are as follows. When all the nodes of an element are in the area of coarse aggregate, it attributes to coarse aggregate. When all the nodes of an element are in the zone of cement mortar, it belongs to cement mortar. When some nodes of an element fall on the aggregate and some of the nodes are in the cement mortar, the element is judged as ITZ. At mesostructure, as depicted in Figure 9, blue, gray, and green regions denote aggregate, cement mortar, and ITZ, respectively. Then, according to the various types of elements, the corresponding elements mechanical attributes are assigned.

To discover and analyze the fracture process and failure mechanism of concrete material subject to uniaxial compression, the load test with a square sample sized by 150 mm × 150 mm as shown in Figure 9 was simulated. In the test, the center at the bottom of the specimen overlaps with the center of the pressure plate with the testing machine, and the load is applied uniformly, as illustrated in Figure 10. Using the two-dimensional computation model of Figure 9 for simplicity and computational efficiency, the bottom surface boundary is fixed in the *y*-axis and the intermediate point is constrained in the *x* and *y* directions to prevent movements of the rigid body; besides the left, right, and top edges of the model are free in all directions. Additionally, in this study, without taking the end friction of the experimental device into account, displacements are applied to the stage loading as the step of 0.0045 mm/s in the *y* direction at the top boundary of the model. Furthermore, in the present calculation model, the maximum principal stress criterion is adopted.

##### 4.2. Damage Constitutive

Because of the inhomogeneity of concrete internal structure, after subjected to the static loads, the damage and fracture processes within concrete are more complicated. With the characteristics of quasi-brittleness, the nonlinear behavior of concrete stress-strain full curve in the macroscopic structure is caused by the existence and development of microdefects, which is named as damage evolution in the mesoscale. It was found that the elastic damage mechanics model can effectively describe the mechanical properties of concrete composite [23–25]. Hence, mechanical properties of each component in concrete are assumed to be elastic damage in the present model. The phenomenon of stiffness degeneration is the macroscopic performance for the reason of nonlinear behavior of concrete. To predict the deformation and destruction of concrete structures, the damage factor is proposed to describe the state of material damage and mechanical effects.

For the constitutive relation behavior, aggregates, cement mortar, and ITZs are treated as isotropic material, while the material parameters are assigned to corresponding elements. In Figure 11, the softening behavior occurred until the peak strain corresponding to the compression strength is reached. The elastic damage constitutive relation of concrete can be easily obtained, according to Lemaitre strain equivalence principle [45], assuming that the deformation state of any damaged material can be expressed through the constitutive equation of the original nondestructive material depicted in Figure 12.

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Therefore, the stress-strain relationship and effective elastic modulus of damage material can be written aswhere and mean stress and strain of material, respectively, and and are the elastic moduli of nondestructive and damaged material, respectively. The damage factor is a constant, ranging from 0 to 1, according to the elastic damage constitutive model shown in Figure 11, compressive damage factor and tensile damage factor can be determined by equations (13) and (14), respectively. It can be noted from equations (13) and (14) that *d* is at the nondestructive state at 0 and *d* is completely destroyed at 1.

For compressive damage, a multistage linear damage constitutive model was proposed as follows:where , , , and are yield compressive strain, peak compressive strain, residual compressive strain, and ultimate compressive strain, respectively, corresponding to , , and . And yield compressive strain coefficient , residual compressive strain coefficient , and ultimate compressive strain coefficient of aggregates, cement mortar, and ITZs are listed in Table 1. Furthermore, in the value of the yield compressive stress coefficient , residual and ultimate compressive stress coefficient depends on the different components as listed in Table 1.

For tensile damage,where , , and mean peak tensile strain, residual tensile strain, and ultimate tensile strain, respectively. The residual tensile strain coefficient is and ultimate tensile strain coefficient is which are summarized in Table 1. In addition, the relation between peak tensile stress and residual tensile stress can be expressed as , where residual tensile stress coefficient values vary with different components for concrete. Besides, ultimate tensile stress is selected as zero herein the study.

Figure 11 plots the elastic damage constitutive relationship for mesoscopic composition of concrete. It can be noticed that the material properties (i.e., aggregates, cement mortar, and ITZs) are supposed to be linear until the peak stress is reached. Furthermore, the damage and deformation of the concrete material increase continuously with the increase of loading, while its stiffness, i.e., the elastic modulus decreased continuously, indicating the development of the softening phase. When the deformation reaches the residual strain, the concrete material is considered to be completely destroyed with stable bearing capacity.

##### 4.3. Material Parameters and Weibull Distribution Model

For the three-phase medium, including aggregates, cement mortar, and ITZs, the parameters used initially in mesoscopic analysis models are summarized in Table 1, refer [40, 46–49]. It should be noted that the compressive strength and tensile strength of three media within concrete are available in Table 1.

As it is known that concrete is a material with randomness and discreteness, mechanical properties of its internal structure correspond to a certain distribution in statistical sense that can be handled with the probability method. In the past, numerous experimental researches [50–52] have indicated that the mechanical parameters of concrete obey the Weibull distribution, including elastic modulus and strength. According to the principle of the Weibull distribution, several attempts [53, 54] have been devoted to the study of fracture behavior for concrete. Therefore, in order to describe the heterogeneity of concrete materials more reasonably, the numerical simulation of this paper not only considers the random distribution of each phase material but also introduces the method of probability statistics. The mechanical properties of each microelement are assumed to satisfy the Weibull distribution function, considering the heterogeneity of constituent materials within concrete, and it can be defined aswhere is the variable that satisfies this distribution and means the uniformity of the material that determines the shape of the Weibull distribution density function. As listed in Table 1, the uniformity for three phases (i.e., aggregates, cement mortar, and ITZs) of concrete was selected as 15, 6, and 3, respectively. The change can be found in Figure 13 that the intensity of elements tends to be uniform as the value of increases. Besides, the parameter is related to the average value of the variable .

#### 5. Simulation Results and Analyses

The base force element method was extended on the previous work of [39] to model the damage process of the convex aggregate concrete by using the programming language FORTRAN in the numerical simulations.

In this study, the focus is on the investigation of fracture process and failure mechanism, and three specimens of concrete are established for the size of 150 mm × 150 mm. Figure 14 shows the comparison of the obtained experiment data of [55] (refer the work of Jin et al.) and simulation results. The good agreement illustrates that the base force element is able to apply to concrete modeling and has great feasibility and accuracy.

##### 5.1. Failure Pattern

When subjected to external static loading, the crack position and failure pattern of concrete vary with the change of stress level, leading to an alteration of internal mechanical behavior for concrete [49]. Figure 15 demonstrates the whole fracture process from crack to destruction of concrete specimens under uniaxial compressive loading based on the present mesostructure method. What is more, the stress contour plots of two-dimensional concrete specimens are shown in Figure 16. In addition, the corresponding macroscopic failure relationship curve between stress and strain of concrete material is illustrated in Figure 14.

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It can be seen from the stress contour plots in Figure 16 that the inhomogeneity of stress distribution, indicating the difference of mechanical properties between units, leads to the nonlinear behavior of concrete materials under external stress. In Figure 15, black means mesoscopic elements that achieve its compressive strength have entered the stage of failure under compressive loading. During the loading process, damage initially develops slowly with an elastic response at the low loading level. For the reason that aggregates are surrounded by mortar and ITZs, its strength does not work; in other words, the maximum principal stress in mortar is the largest (Figures 15 and 16). As the loading increases, the behavior of deformation accumulation and stress concentration causes the microcracks to appear at the ITZs, which is the weakest area in concrete with many microdefects. The damage to the specimens is getting faster, observing that the stress-strain curve shows a slightly convex curve in Figure 14. The damage is mainly in the area where the aggregate is concentrated. When the loading reaches the peak stress in Figure 14, the concrete specimen is almost totally in the damage stage. As the new damage zones constantly expand, destruction concentrates in the most damaged regions, resulting several local cracks. However, the cracks gradually extend into the mortar and go around when the aggregate particles with relatively high strength is encountered until the cracks penetrate a section of the concrete specimen. Finally, more than one macroscopic cracks are formed that is parallel to the loading direction, causing the fracture damage of the concrete specimens and losing the bearing capacity, similar to the failure consequences of the test.

It can be observed from Figure 14 that initially the concrete internal material is stable, in which the whole specimen is mainly under elastic deformation state, and the stress-strain curve shows the characteristics of linear elasticity. Following this, the existence of microcracks reduced the effective sectional area of concrete; as a result, the stress continues to increase as strain increasing, while the elastic modulus decreases. After the stress peak strength, the concrete material enters the softening stage and the stress decreases with the increase of strain, presenting a nonlinear performance.

##### 5.2. Effect of Aggregates Distribution

The difference in aggregate distribution will change the internal stress field under external loading. A group of the random aggregate model with three different aggregate distributions is established in the simulation as presented in Figure 17.

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It can be observed that concrete with different aggregate distributions has distinct fracture paths in Figure 18. However, the corresponding stress-strain curve segment geometry overlaps in Figure 19, indicating that different aggregate distribution has little effect on the elastic modulus and strength of concrete in mesoscale. After the peak stress, the descending segments are significantly different mainly due to the damage process as shown in Figure 19.

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##### 5.3. Size Effect

The strength of concrete is measured by a cube specimen. As the size effect is widely recognized in the earlier efforts, it has practical significance to reveal the different size of concrete. A group of square concrete samples with the size of 100 mm × 100 mm, 150 mm × 150 mm, and 300 mm × 300 mm has been simulated in this study as showed in Figure 20.

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Figure 21 shows the fracture damage of concrete specimens with different sizes, it can be found that the specimen size has slight effect on the failure pattern and fracture appears as tensile failure along the vertical direction.

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Compared with the stress-strain curve presented in Figure 22 to study the influence of specimen size on strength, it can be noted that compressive peak strength of concrete decreases with the increase in the specimen size. Furthermore, the softening curve of the posterior peak segment is obviously different for varying sizes. The residual strength and ultimate strain decrease with the increase in the specimen size, indicating the brittleness of concrete specimen increases. This is because of the characteristics of nonhomogeneity in internal structure of concrete, including microcracks and microdefects. As the size of concrete specimen increases, the probability of defects increases and the dispersion of material performance increases. Concrete material is regarded with the characteristic of quasi-brittleness, and the damage is mainly caused by localization defects, which make the compressive strength reduction in the larger concrete specimen. And the results are consistent with the observation in [56].

##### 5.4. Effect of Coarse Aggregate Content

As known, there is an optimum amount of coarse aggregate in concrete. In practical engineering, increasing the volume fraction of coarse aggregate can save the cost of concrete materials. Moreover, since the coarse aggregate has a higher strength than the other two mediums (i.e., cement mortar and ITZs), the strength of the concrete specimen increases with the increase of coarse aggregate volume fraction. However, too much coarse aggregate will reduce its adhesion to cement mortar, and correspondingly, the concrete integrity will deteriorate. To discover the effect of coarse aggregate content on the mechanical properties of concrete materials, the samples of concrete with seven different kinds of coarse aggregate content are employed. The volume fraction of coarse aggregate contains 25.61%, 27.92%, 30.27%, 32.60%, 34.93%, 37.25%, and 40.00% in the computations.

Figure 23 plots the corresponding macroscopic compressive stress-strain curve for concrete specimens with different coarse aggregate volume fractions. It can be seen from Figures 23 and 24 that the elastic modulus of the initial elastic stage increases as the volume of coarse aggregate increases, indicating the stiffness of the concrete material increases. While with the increase of coarse aggregate content, the yield state is shorter and the peak strain decreases correspondingly. What is more, the peak stress is obviously improved, with the increasing coarse aggregate content; that is to say, the compressive strength of concrete specimen is improved. After the peak point, the bigger the volume content of coarse aggregate is, the faster the curve decreases, and the effect of softening will be obvious. Also one can know that the damage ductility of concrete specimen is better. To sum up, the impact of coarse aggregate content on mechanical properties of concrete specimens cannot be ignored.

##### 5.5. Effect of End Friction

The constraint type has a great impact on the macroperformance of concrete which is determined by the boundary conditions of specimens. This is because that the change of end friction will take place for different boundary constraints. Therefore, to explore the end effect, two types of boundary conditions are utilized in the numerical simulation. As a comparison of the presented boundary condition, the bottom boundary of the concrete specimen is completely fixed in all the directions (i.e., *x*, *y*, and *xy* directions).

Figures 25 and 15, respectively, plot the damage pattern of nonfriction and friction (i.e., coefficient of friction and ) at the end of the concrete specimen. From the comparison of Figure 15, it can be noted that the end friction has a significant impact on the failure pattern of concrete material in uniaxial compression. In Figure 15, when , the crack first appears at the position of the loading end of the specimen. With the increase of loading, the cracks extend along the oblique direction from four corners at the end of the concrete specimen. The width of cracks after peak stress is wider, and the damage becomes more serious. Finally, the specimens lost its stability to be destroyed. This phenomenon is because that the friction at the end limits the transverse deformation of the two edges of the specimen while end limit is reduced in the middle of the specimen. A few oblique cracks are formed that is caused by maximum tensile stress. In one word, when the end friction exists, the concrete specimen is broken in the form of shear, and oblique cracks transfer from quadrangle to center, eventually resulting in a conical shape. As shown in Figure 25, for , it is mainly expressed as compressive failure forming the vertical cracks through the upper and lower surfaces in which the damage characteristic is obviously distinct from the boundary condition for end friction (i.e., ).

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The uniaxial compressive stress-strain curves for two types of end constraints are shown in Figure 26. One can find that for specimens of the same shape, the rising segment of the curve is almost identical. However, the softening slope after the peak of the stress-strain curve is steeper without considering the end friction. Also, it is found that the peak stress increases as the end friction increases. Generally speaking, end friction can effectively enhance the load-bearing capacity and improve the ductility capacity of concrete specimens.

#### 6. Conclusions

The base force element method, a mesoscopic analysis method, is adopted for investigating the damage process and failure pattern of concrete subjected to uniaxial compression loading, taking the heterogeneity of its internal composition into account. In mesostructure, concrete is assumed to be a three-phase composite material including aggregate particles, cement mortar, and ITZs.

Based on the area criterion method, the aggregates within concrete are considered as convex polygon and a random aggregate model is established. Considering the nonlinear behavior of concrete after stress, an elastic damage constitutive relationship is employed in the model to describe the material parameters since the concrete material has the characteristics of heterogeneity and each medium owns its independent mechanical properties obeying the Weibull probability distribution. In comparison with the obtained test data, the feasibility of the numerical simulation method is confirmed. What is more, the detailed failure mechanism of concrete under uniaxial compression loading is analyzed. In addition, the influence of aggregates distribution, mesh sensitivity, prism specimen size, and end friction on strength and damage process is investigated. Some conclusions of this simulation results are obtained as follows:(1)The numerical simulation results are in good agreement with the obtained experimental data which indicates the feasibility and accuracy of base force element method to model concrete in uniaxial compression.(2)It is found that the multistage linear damage constitutive model proposed in this paper can better simulate the experiment than the linear elastic damage constitutive model.(3)Numerical examples show that the convex polygonal aggregate can simulate the mesostructure of concrete well.(4)The microcracks occur in the weakest zones as ITZs and then go around the aggregate and extend along the cement mortar forming several vertical cracks. Moreover, the failure pattern can be described as four stages, involving appearance of microcracks, propagation of cracks, the increasing of cracks, and fracture damage.(5)Aggregate distribution form only has an effect on the generation of crack location and failure path; however, it does not affect the macroscopic elastic modulus and strength of concrete material.(6)Size effect had significant effect on the macroscopic mechanical properties of the concrete material because the concrete material has the characteristics of the mesoscopic heterogeneity that larger size specimens have more internal defects after stress. Therefore, with the increase of the size of the concrete specimen, the uniaxial compressive strength decreases, while the brittleness of the concrete specimen becomes stronger after the damage.(7)The uniaxial compressive strength of concrete specimens increased significantly with the increase of coarse aggregate content, but the brittleness of the softening segment is stronger. Therefore, it is of great practical significance to choose the content of coarse aggregate reasonably.(8)The end friction can constrain the transverse deformation of the concrete specimen, which decreases with the increase of the distance from the end boundary. The end friction constraint causes the change of failure pattern for the concrete. Additionally, the existence of this effect can obviously improve the ductility capacity and uniaxial compression strength of the concrete square specimen.

#### Data Availability

The data used to support the findings of this study are included within the article.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

This work was supported by the National Science Foundation of China (10972015 and 11172015), Beijing Natural Science Foundation (8162008), and Pre-Exploration Project of Key Laboratory of Urban Security and Disaster Engineering, Ministry of Education, Beijing University of Technology (USDE201404).