Determination of mechanical parameters for elements in meso-mechanical models of concrete

Abstract

The responses of cement mortar specimens of different dimensions under compression and tension were calculated based on the discrete element method with the modified-rigid-body-spring concrete model, in which the mechanical parameters derived from macro-scale material tests were applied directly to the mortar elements. By comparing the calculated results with those predicted by the Carpinteri and Weibull size effects laws, a series of formulas to convert the macro-scale mechanical parameters of mortar and interface to those at the meso-scale were proposed through a fitting analysis. Based on the proposed formulas, numerical simulation of axial compressive and tensile failure processes of concrete and cement mortar materials, respectively were conducted. The calculated results were a good match with the test results.

Keywords

concrete / meso-mechanical model / discrete element method / size effect / mechanical parameter

Introduction

Concrete is a heterogeneous material that consists of mortar, coarse aggregates and a meso-level interfacial transition zone. It is known that the mechanical behavior and failure modes of concrete at the macro-scale are strongly affected by its meso-structures [1]. Therefore, an approach to research the failure process of concrete in the meso-scale is significant for a further understanding of the failure mechanism, strength and deformation characteristics of concrete at the macro-scale.

When the mechanical properties of concrete are analyzed using meso-scale numerical methods, key points are the establishment of rational meso-mechanical models and an approach to determine the material parameters for elements in the models. The meso-mechanic models of concrete can be divided into two categories according to the assumptions of materials [2]. Finite element method (FEM) based models, such as microplane model [3], lattice model [4] and random particle model [5], belong to the first category, in which the material is assumed to be continuous. Discrete element method (DEM) based models, such as the modified-rigid-body-spring model [6], a beam-particle model [7] and the extended distinct element method [8], are classified in the second category, in which the continuity assumption is not needed.

In FEM-based models, the boundaries between adjacent elements need to meet the requirement of deformation compatibility, making it difficult to simulate the cracking process in concrete. Moreover, a numerical solution may become unstable when the behavior of concrete is highly nonlinear. In DEM-based models, these shortcomings can be easily overcome, because the adjacent elements can be separated in the analysis of the failure process of concrete.

In the application of FEM- and DEM-based models, the mechanical parameters of elements, such as the strength of mortar and the interface elements at a meso-scale, first need to be established. Theoretically, the mechanical parameters of elements should be determined through a meso-scale material test. However, it is difficult to carry out that kind of the test, due to the limitations of the current testing technology. The most applicable way is the conversion of the mechanical parameters obtained from traditional macro-scale material tests. However, the question is how to convert the mechanical parameters of materials from the macro-scale to the meso-scale.

Nagai et al. carried out simulations on failure processes of mortar and concrete with two-dimensional(2D) and three-dimensional (3D) rigid-body-spring-models(RBSMs) and concluded that, due to the different dimensions of the elements, the values of the material property of elements at the meso-scale were different from those of the analyzed object in the macro-scale [9,10]. Accordingly, relationships between Poisson’s ratio v_elem and elastic modulus E_elem of elements in the meso-scale and Poisson’s ratio v and elastic modulus E in the macro-scale were proposed, which were used in the 2D and 3D RBSMs.

Xing et al. [11] presented empirical conversion formulas of mechanical parameters, including the strength and elastic modulus, from the macro-scale to the meso-scale by repeated calculation of the compressive concrete specimens using a beam-particle model.

The existing research results are good examples of the adjustment of elements’ mechanical parameter values in the meso-scale. However, the elements are different in different models. Generally, in DEM-based models, concrete is assumed to be a composite material of mortar, coarse aggregates and their interface. For normal concrete, the coarse aggregates usually do not fail when the concrete is subjected to compression or tension; and, the mechanical behavior of the interfaces is mainly affected by the mechanical behavior of the mortar and surface texture of the aggregates. Therefore, it is important to study the mechanical behavior of mortar at the meso-scale if the concrete material is modeled based on DEM.

In this study, we focus on the mechanical parameters of mortar elements in DEM-based models. First, the pure mortar specimens with different dimensions under compression and tension are calculated using the modified-rigid-body-spring concrete model proposed by the authors [6]. The calculated results are then compared with those predicted with the existing size effect laws. The calculated results were verified through a comparison with experimental results. Finally, a series of formulas for conversion of the mechanical parameters of mortar and interface in the macro-scale to the meso-scale are proposed, according to the calculation results for the mortar specimens and the Weibull size effect law. The proposed formulas are applied in the numerical analysis of concrete materials.

Modified-rigid-body-spring concrete model based on DEM

The authors developed a 2D modified-rigid-body-spring concrete model based on DEM [6], in which concrete is assumed to be a three-phase composite material, with coarse aggregates as the dispersed phase, mortar as the continuous phase, and zero-sized interfaces as the interfacial phase. The coarse aggregates in a concrete section were assumed to be circular. According to the Walraven function [12], the diameter and number of coarse aggregates in a cross section of two dimensional model can be calculated and the distribution probability of an aggregate with a certain diameter can be obtained from the Fuller’s gradation [13]. The probability of the diameter (D) of a circular aggregate particle being smaller than D_ij (where i is the number of the grade and j = 1, 2 for the upper or lower bound of grade i, respectively) can be expressed as:

(1)

where d=D_{ij}/D_{\max };D_{\max } is the maximum size of the aggregate particle; and, P_k is the aggregate volume fraction.

According to the different phases, the element mesh is generated with a Voronoi diagram to reduce the influence of the mesh arrangement on the cracking direction. The generated element mesh for a cross section of concrete specimen is shown in Fig. 1.

Each element has one rotational and two transitional degrees of freedom. The polyhedron elements are interconnected by springs including normal and shear springs, as shown in Fig. 2, where x_i, y_i, θ_i and x_j, y_j, θ_j are the horizontal, vertical and rotational displacements of elements i and j, respectively; k_n,s, k_s,s and Δ_n, Δ_s are the stiffness and deformation of the normal and shear springs, respectively; l is the length of the common boundary of the two elements; and, h_i and h_j are the lengths of the perpendicular line from the center of gravity of elements i and j to the common boundary, respectively. There are also dampers with damping coefficients c_n and c_s along the boundary, which are used to dissipate energy when solving static problems using the dynamic relaxation method.

Two kinds of spring groups are defined to simulate the connection of mortar and the connection between mortar and aggregate. All mortar elements are assumed to be rigid, and the deformations of mortar elements are represented by those of the mortar springs. Aggregate elements are considered to be totally rigid and cannot fail. Tensile or compressive forces are developed by normal springs in the normal direction, and shear forces by shear springs in the tangential direction. Cracking of the concrete can be characterized by the fracture of springs between elements.

The fracture criterion described as Eq. (2) [14], which considers compression, tension and shear failures, is used for the mortar springs. The Mohr-Coulomb failure theory, as expressed in Eq. (3) [15], is used to illustrate meso-level interfacial failure.

(2)

(3)

where τ_m and τ_i are the shear stress of mortar and interface, respectively; σ_m and σ_i are the normal stress; and, f_m represents the compressive strength of the mortar.

Before fracture, the normal and shear springs obey the constitutive models shown in Fig. 3 [6], where w_max is the maximum crack width, which is set to be 0.03 mm based on experimental results [16]; Δ_tmax and Δ_cmax are the deformations of the normal spring corresponding to the maximum tensile force F_tmax and the maximum compressive force F_cmax, respectively; and, Δ_smax is the deformation of the shear spring corresponding to the maximum shear force F_smax, which may change with variation of the normal force.

When the fracture occurs, the relationship between elements changes from connecting to contacting; and, contact actions between elements are represented by contact springs. The constitutive models for normal and shear contact springs are shown in Fig. 4 [6], in which F_n,c and F_s,c are the normal and shear contact forces, respectively; and, Δ_n and Δ_s are the normal and shear contact deformation, respectively. To avoid oversized overlapping between the elements, the threshold value for the contact depth in the normal direction Δ_ncri is set as 0.003 mm, according to the numerical analysis results.

The analysis solution is derived from the dynamic relaxation procedure of the DEM. The central finite difference procedure is adopted to solve the dynamic equilibrium equation in every calculation step. The time interval Δt is chosen based on the size and the strength of the mortar elements. For a common concrete specimen with a minimum distance of 5 mm between sites defined in Fig. 1, Δt is set to be smaller than 2 × 10^-5 s to avoid spring failure in the first time step of calculation and to get a convergent solution. The complex calculation of the stiffness matrix in the finite element solution is avoided by the dynamic relaxation procedure of DEM, and there is no need to address the negative stiffness region when concrete begins to crush. Details of the numerical calculation can be found in reference [6].

Determination of mechanical parameters for elements

Mechanical behavior of mortar specimens with different dimensions

In the numerical model mentioned above, the mortar in the concrete is considered as a homogeneous material. To verify whether the material properties obtained from traditional macro-scale material tests could be directly applied to the mortar elements, numerical simulations on the failure processes of mortar specimens were conducted with cube compressive specimens and cylinder tensile specimens of different sizes D that were defined as the length of a side of the cube specimens or the diameter of the cylinder specimens. In the calculation, the element size d that is the minimum distance between adjacent sites defined in Fig. 1 was set as 2.5 mm. The mechanical parameters for mortar elements were obtained from the macro-scale tests, as shown in Table 1 [15], where the compressive strength, elastic modulus and Poisson’s ratio were acquired from uniaxial compressive tests of cube mortar specimens with a length of 70.7mm. The tensile strength was obtained from tensile tests of cylinder specimens that had a diameter of 50mm and a height of 80 mm.

Four strength grades (M1, M2, M3 and M4) were prepared for mortar with different mixture proportions, as shown in Table 1. The water/cement ratios were 0.83, 0.65, 0.49 and 0.4 (by weight), respectively. The cement used in the specimens was local Portland cement (425R), and medium sand with a maximum diameter of 5mm was utilized in mortar specimens.

The calculation results indicated that the macroscopic mechanical parameters of mortar obtained by simulation differed with the variation of specimen size D, as shown in Fig. 5. However, when specimen size D was greater than 40mm, the compressive and tensile strengths changed very little.

It is not appropriate to determine the mechanical parameters of mortar elements in meso-mechanical models for concrete directly from traditional macro-scale material tests. The characteristic shown in Fig. 5 for mortar is usually called the size effect, which is discussed in detail in Section 3.2.

Size effect analysis of mortar specimens under compression and tension

The commonly used size effect principles for concrete materials are the classical Weibull statistical size effect theory, the Bazant energy release size effect theory, and the Carpinteri multi-fractal scaling size effect theory. Different size effect theories have different application scopes, and the tensile and compressive strengths of concrete obey different size effect laws.

Carpinteri et al. [18] studied the size effects of uniaxially compressed concrete specimens. The analytical expression of the multi-fractal scaling law is:

(4){\sigma }_N=f_{\mathrm{c}}\sqrt{\mathrm{1}+{\displaystyle \frac{l_{\mathrm{c}\mathrm{h}}}{D}}},

where σ_N is the nominal compressive strength of concrete of size D, and f_c and l_ch represent the nominal compressive strength of an infinitely large concrete specimen and an internal material length, respectively. l_ch is proportional to d_max, the maximum size of aggregates in concrete specimens [18]. D is the characteristic size of a structure, which can be arbitrary (e.g., the cross-sectional dimension of the prism, the beam depth, the beam span, diameter of the cylinder) for geometrical comparison of similar structures.

Using lg(σ_N/f_c) as the vertical axis and lg(D/l_ch) as the horizontal axis in a coordinate system, the simulation results of the compressive strength of mortar specimens calculated in the previous section were compared with the test results in Refs. [19-21], as shown in Fig. 6. It can be seen that the variation trend of mortar compressive strength from numerical simulation is almost same as that of concrete compressive strength from the tests, which indicates that calculation results can accurately reflect the size effect of the compressive strength of mortar specimens.

The compressive strengths of mortar specimens with different dimensions obtained by numerical simulation and the Carpinteri size effect law are shown in Fig. 5(a), indicating that the size effect law reflected by the calculation results has a good match with the Carpinteri size effect law. Therefore, the previously mentioned meso-scale numerical simulation model for concrete can be employed as an efficient tool to determine the compressive strength for mortar elements in the meso-scale.

For uniaxial tension specimens, fracture usually occurs at the weakest cross section in the longitudinal direction; and, the whole cross section disconnects as soon as the material cracks. The nature of tension fracture basically meets the applicable conditions of the Weibull size effect law [22]: first, the material fails(or is assumed to fail) upon the initiation of a macro-crack; and, second, the material fails with a tiny fracture process zone induced by insignificant stress redistribution. The Weibull size effect law for the tensile strength of materials can be expressed as:/

(5){\sigma }_t=f_t{\left({\displaystyle \frac{D_{\mathrm{0}}}{D}}\right)}^{\raisebox{1ex}{$n$}\!\left/ \!\raisebox{-1ex}{$m$}\right.},

where D₀ is the reference size, f_t is the tensile strength of D₀-sized material, n represents the dimensions of the objective (n = 2 or 3: in this study, the 2D model was adopted, so the value of n is 2), and m is the Weibull modulus, which is generally taken as 12 [23].

Figure 5(b) shows the results calculated by the previously mentioned numerical model for tensile strengths of cylinder mortar specimens with different dimensions and the corresponding results predicted by Eq. (5), using D₀ as the element size d and where the value of d is 2.5 mm. It can be seen in Fig. 5(b) that, with increases in the dimension of the specimen, the tensile strength calculated by the numerical model tended to be constant; however, the tensile strength predicted by Eq. (5) tended to be zero. Obviously, the tensile strength of mortar could not be zero, even if the size of the specimen was large enough. Therefore, the Weibull size effect law can only be used to predict the variation character of tensile strengths of mortar specimens when the ratio of d to D is not very small. In this study, it is recommended that d/D≥1/20.

Meso-scale compressive strength of mortar elements

The size effect of the compressive strength of materials is connected with its fragility: the higher the fragility, the more significant the size effect [24]. The fragility of concrete is enhanced when the strength grade increases. In this study, the numerical simulation model was employed to analyze the size effect of the compressive strength of mortar, taking into account the influence of the mortar strength grade. According to the Chinese standard for the test method of performance on building mortar, JGJ/T70-2009 [25], the standard specimen for the compressive strength of mortar is a cube with a length of 70.7 mm on each side; and, the strength grade is determined by the testing compressive strengths of standard specimens. Mortar cubes of different sizes and different strength grades (grades 1 to 8) under uniaxial compressive force were numerically analyzed. The calculated compressive strengths of the mortar specimens are shown in Fig. 7.

The compressive strengths of mortar specimens of size D can be obtained by the Carpinteri size effect law illustrated in Fig. 7. Moreover, the relationships of compressive strengths between different sizes (D) can be acquired from Fig. 7. As an example, the regressive curve of the relationship between elements with a size of 2.5 mm and specimens with a size of 70 mm is shown in Fig. 8. More generally, once the compressive strength of a specimen of size D has been obtained through the test, the corresponding compressive strength of an element with size d can be determined by using Eq. (6).

(6)f_{c,d}={\left({\displaystyle \frac{\mathrm{1}}{a}}\cdot f_{c,D}^{}\right)}^{\raisebox{1ex}{$\mathrm{1}$}\!\left/ \!\raisebox{-1ex}{$b$}\right.},

where f_c,D is the compressive strength of a specimen of size D; f_c,d is the compressive strength of a corresponding element of size d; and, a and b are parameters related to specimen size D and element size d.

A set values for a and b were obtained according to the regressive curve from Fig. 8, where the specimen size D was equal to70 mm. Other sets of a and b with different specimen sizes can be obtained in the same way. Therefore, a general equation to calculate the parameters of a and b for arbitrary specimen size D and element size d was obtained through regressive analysis, as shown in Eq. (7):

(7)\{\begin{array}{l}a=\mathrm{0.254}\cdot \sqrt{\mathrm{1}+\mathrm{14.511}{\displaystyle \frac{d}{D}}}\hfill \\ b=\mathrm{0.963}\cdot \sqrt{\mathrm{1}+\mathrm{0.195}{\displaystyle \frac{d}{D}}}\hfill \end{array}\begin{array}{lll}& & \end{array}(d\le \mathrm{5}\mathrm{mm}).

Meso-scale tensile strength of mortar elements

According to the analysis of the tensile strength of the calculated results, the Weibull size effect law can be used to determine the tensile strength of elements in the meso-scale, as shown in Eq. (8):

(8)f_{t,d}=f_{t,D}\cdot {\left({\displaystyle \frac{D}{d}}\right)}^{\raisebox{1ex}{$\mathrm{1}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{6}$}\right.}(D/d\le \mathrm{20}),

where f_t,D represents the tensile strength of specimens of size D, and f_t,d is the corresponding tensile strength of elements of size d.

Meso-scale elastic modulus and Poisson’s ratio of mortar elements

The elastic modulus and Poisson’s ratio of mortar, obtained with the above simulation on compressive specimens, also have size effects, as shown in Fig. 9. Few experimental studies on the size effect of the elastic modulus and Poisson’s ratio are found in the literature, and there are no uniform laws to describe the size effect for these two parameters. In this study, the relationships between the macro-scale and meso-scale elastic modulus and Poisson’s ratio were established based on the numerical simulation results of mortar under uniaxial compression. The regressive equations are:

(9){\nu }_e=-\mathrm{91.46}{\nu }_{}^3+\mathrm{45.79}{\nu }_{}^2-\mathrm{4.51}\nu ,

(10)E_e=E\cdot (-\mathrm{34.13}{\nu }_{}^{\mathrm{3}}-\mathrm{1.18}{\nu }_{}^{\mathrm{2}}+\mathrm{2.18}\nu +\mathrm{1}),

where E_e and ν_e are the elastic modulus and Poisson’s ratio of mortar elements, respectively; and, E and ν are those of the corresponding mortar specimens. These two formulas are suitable for the situation where the size of the element is smaller than 5 mm.

Application of mechanical parameters of elements in DEM-based models

The parameters of mortar acquired from tests, together with those of elements (d = 2.5 mm), which were obtained by the formulas in Eqs. (6) to (10) are listed in Table 1. The mechanical parameters of the interface between the mortar and the coarse aggregates were determined based on the relationship between the interface and the mortar proposed in reference [15]. All of the determined mechanical parameters of elements were applied in DEM-based models described in Section 2, in order to study the compressive and tensile behaviors of mortar and concrete. The numerical simulation results were compared with those of the tests.

Concrete specimens (grades C1-C4) were also provided with the same water/cement ratios and cement/sand ratios as those of the mortar listed in Table 1. The sand/coarse aggregates ratios of concrete were set to 0.70, 0.56, 0.45 and 0.38 for grades C1-C4, respectively. Cobblestones with diameters of 5mm to 20mm were used as coarse aggregates. The mass percentages of aggregates with diameters of 5-10 mm, 10-16 mm and 16-20 mm were 70.7%, 18.7% and 10.6%, respectively.

For the uniaxial compressive test, 3 prism specimens, with a height of 230 mm and a cross section of 70.7 mm × 70.7 mm for mortar and a height of 300 mm and a cross section of 100 mm × 100 mm for concrete, were prepared for each grade of mortar and concrete, respectively. Six cylinder specimens with a height of 100 mm and a diameter of 50 mm were used for the tensile analysis.

The tensile strengths of concrete and mortar were evaluated with a Limpet Pull-Off Tester. The compressive strength of the mortar was tested using the TSY-2000 pressure tester. The compression behavior of concrete was measured with the INSTRON-1345 universal material tester. The 2D meso-mechanical model was applied to simulate uniaxial compression (for specimens of 70.7 mm in width and 230 mm in height for mortar and 100 mm in width and 300 mm in height for concrete) and uniaxial tension (for specimens of 50 mm in width and 100 mm in height).

Behavior of cement mortar

The simulated results of compressive strength (average of three specimens) were compared with experimental results, as listed in Table 2,which indicates that the numerical simulated results had a good match with the experimental results when the strength grade of cement mortar was low(i.e., mortar strengths M1 and M2). However, the numerical results were smaller than the test results when the strength grade was high (i.e., M3 and M4).

The average strength value of six specimens was considered as the material tensile strength of cement mortar, and the tensile test results were also compared with the numerical results, as shown in Table 2. It shows that numerical results are generally larger than the test results, which means that the conversion of tensile strength based on the Weibull size effect is slightly overestimated.

Comparisons between the numerically simulated and experimental stress-strain curves of axial compressive mortar specimens are shown in Fig. 10. The simulated ascending branches of the stress-strain curves were approximately linear, while the experimental ascending branches were approximate quadratic, as shown in Fig. 10. The difference was greater for low-strength cement mortar. Strains at the peak loads of the numerical curves were generally lower than the test values. The reason may be that the numerical model in this paper assumes that the cement mortar was homogeneous, without initial defects; however, test specimens cannot meet this ideal condition. Gels, pores and initial meso-cracks inside the material are often the main causes for plastic deformation.

Behavior of concrete

The failure processes of concrete prism specimens under axial compression were simulated. The comparison between the calculated and test results of axial compressive strength is presented in Table 3. Axial compressive strengths with different strength grades calculated by the model were smaller than the test results, except for grade C1, which had lower test results. This may be attributed to the marble slabs used as the aggregate surface in the test to estimate the interface strength in concrete, which may have led a measured interface bond strength lower than the real strength in concrete [13]. Therefore, when the test results were used as the interface mechanical parameter in the numerical model, lower calculated results were obtained. Moreover, in this paper, mechanical parameters of the mortar specimens were used as the parameters of mortar in concrete specimens, which may also underestimate its real strength.

Numerically calculated stress-strain curves of axial compressive specimens (grades C1-C4) are illustrated in Fig. 11. The calculated peak strains of the concrete specimens were also lower than the test values. Pores and initial defects inside the concrete material were not reflected in the numerical model, which is the reason for a lower plastic strain in the calculation. For a low water-cement radio, high-strength concrete generates less pores when it is cast and may be relatively more homogeneous than low-strength concrete. Therefore, the simulated stress-strain curves of higher strength concrete (C3 and C4) are better than lower strength concrete in this paper.

The tensile strength of concrete was also the average strength of six specimens. The calculated results were compared with results of the tensile test, as shown in Table 3. It shows that the numerically calculated tensile strengths agreed well with those of the test results with β₂ (where β₂ = (simulated results-test results)/ test results × 100%) in the range of-3.3% to 12.6%. Tensile strengths calculated by the model were larger than the test results, except for grade C1.This may be related to the tensile failure mechanism of specimens. Cracks in tensile specimens always generate along a weak plane, and interface strength has a great influence on the bearing capacity of a specimen. Once the interface bond fails, the surrounding cement mortar quickly cracks. Moreover, the initial defects of concrete were not considered in the mechanical model during simulation, which may have led to experimental results lower than simulated results.

Conclusions

Mechanical parameters in the meso-mechanical model of cement mortar were analyzed based on DEM. It was found that the calculated material properties of mortar varied with changes in specimen size, which indicates that the mechanical parameters derived from the traditional material tests cannot be directly used in the meso-mechanical model. Analysis of the numerical results of mortar specimens determined that compressive strength obeyed the Carpinteri size effect law. The size effect of the tensile strength conformed to the Weibull size effect law within a certain size range; and, the relationships between macro- and meso-mechanical parameters were obtained according to the calculated results and size effect laws. Therefore, the material properties of mortar and interface measured with macro-scale tests can be easily converted into those of the elements in the meso-scale.

The combination of the converted mechanical parameters of elements with the modified-rigid-body-spring model proposed by the authors can effectively simulate the compressive and tensile behavior of mortar and concrete under uniaxial load.

References

[1] Ueda T. Prediction of structural performance during service life from microstructure. In: Workshop Proceedings—Microstructure and Durability to Predict Service Life of Concrete Structures. Hokkaido University, Sapporo, Japan, 2004: 39–48

[2] Meguro K, Tagel-Din H. Applied element simulation of RC structures under cyclic loading. Journal of Structural Engineering, 2001, 127(11): 1295–1305

[3] Bažant Z P. Chapter 3: Microplane model for strain controlled inelastic behavior. In: Desai C S, Gallagher R H, eds. Proceedings of Mechanics of Engineering Materials. London: Wiley, 1984, 45–59

[4] Cusatis G, Bažant Z P, Cedolin L. Confinement-shear lattice model for concrete damage in tension and compression: I. Theory. Journal of Engineering Mechanics, 2003, 129(12): 1439–1448

[5] Bažant Z P, Tabbara M R, Kazemi M T, Pijaudier-Cabot G. Random particle models for fracture of aggregate or fiber composites. Journal of Engineering Mechanics, 1990, 116(8): 1686–1709

[6] Gu X L, Hong L, Wang Z L, Lin F. A modified-rigid-body-spring concrete model for prediction of initial defects and aggregates distribution effect on behavior of concrete. Computational Materials Science, 2013, 77: 355–365

[7] Xing J B. Introduction of Beam-Particle Model. Beijing: Earthquake Engineering Press, 1999 (in Chinese)

[8] Hakuno M, Meguro K. Simulation of concrete-frame collapse due to dynamic loading. Journal of Engineering Mechanics, 1993, 119(9): 1709–1722

[9] Nagai K, Sato Y, Ueda T. Mesoscopic simulation of failure of mortar and concrete by 2D RBSM. Journal of Advanced Concrete Technology, 2004, 2(3): 359–374

[10] Nagai K, Sato Y, Ueda T. Three-dimensional simulation of mortar and concrete model failure in meso level by rigid body spring model. Journal of Structural and Engineering, JSCE, 2004, 50A: 167–178

[11] Xing J B, Yu L Q. Study of fracture behavior in particle composites with beam-aggregate model. Journal of Basic Science and Engineering, 1997, 5(2): 193–198 (in Chinese)

[12] Walraven J C, Reinhardt H W. Theory and experiments on the mechanical behavior of cracks in plain and reinforced concrete subjected to shear loading. HERON, 1981, 26(1A): 26–35

[13] Fuller W B, Thompson S E. The laws of proportioning concrete. Transactions of the American Society of Civil Engineers, 1906, LVII(2): 67–143

[14] Wang Z L, Gu X L, Lin F. Experimental study on failure criterion of mortar under combined stresses. Journal Building Materials, 2011, 14(4): 437–442 (in Chinese)

[15] Gu X L, Hong L, Wang Z L, Lin F. Experimental study and application of mechanical properties for the interface between cobblestone aggregate and mortar in concrete. Construction & Building Materials, 2013, 46: 156–166

[16] Gopalaratnam V S, Shah S P. Softening response of plain concrete in direct tension. Journal of the American Concrete Institute, 1985, 82(3): 310–323

[17] Wang Z L. Study and Application of Mesoscopic Mechanical Model of Concrete Based on the Discrete Element Method. Shanghai: Tongji University, 2009 (in Chinese)

[18] Carpinteri A, Ferro G, Monetto I. Scale effects in uniaxially compressed concrete specimens. Magazine of Concrete Research, 1999, 51(3): 217–225

[19] Blanks R F, McNamara C C. Mass concrete tests in large cylinders. Journal of the American Concrete Institute, 1935, 31: 280–303

[20] Xu J S, He X X. Size effect on the strength of a concrete member. Engineering Fracture Mechanics, 1990, 35(4–5): 687–695

[21] Bocca P, Carpinteri A. Size effect in the compressive behavior of concrete. Department of Structural Engineering, Politecnico di Torino, Internal Report, 1989, No. 17

[22] Weibull W. A statistical distribution function of wide applicability. Journal of Applied Mechanics. Transactions of the American Society of Civil Engineers, 1951, 18(3): 293–297

[23] Zech B, Wittmann F H. A complex study on the reliability assessment of the containment of a PWR, Part II. Probabilistic approach to describe the behavior of materials. In: Jaeger T A, Boley B A, eds. Transactions of the 4th International Conference on Structural Mechanics in Reactor Technology. European Communities, Brussels, Belgium, 1977, H(J1/11): 1–14

[24] Bažant Z P, Yu Q. Universal size effect law and effect of crack depth on quasi-brittle structure strength. Journal of Engineering Mechanics, 2009, 135(2): 78–84

[25] Industry Standards of the People’s Republic of China. Standard for Test Method of Performance on Building Mortar JGJ/T70–2009 (in Chinese)

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