Study on Macroscopic and Mesoscopic Mechanical Behavior of CSG based on Inversion of Mesoscopic Material Parameters

Abstract Cement Sand and Gravel (CSG) is a low-cost, environment-friendly composite material mixed of unscreened aggregate, cement, fly ash and water, and its properties differ from ordinary concrete due to different aggregate characteristics. In order to investigate the effect of aggregate characteristics on the mechanical behavior of CSG, this paper used numerical simulation method to divide the CSG into aggregate unit, cement mortar unit and interface unit at the mesoscopic level and randomly generate aggregate, then used laboratory uniaxial compression test results to inverse the said mesoscopic component parameters, and finally verified the rationality of mesoscopic numerical simulation. Based on the inversed parameters, the numerical simulation test of different aggregate grading was carried out and analyzed. The results showed that: (1) From the perspective of macroscopic mechanical properties, as the sand ratio increased, the aggregate occupancy and the peak stress decreased; under the same aggregate occupancy (the same sand ratio), the stress peak became higher with the improvement of aggregate grading (aggregates of small particle size increased); (2) At the mesoscopic level, the crack of CSG usually appeared on the interface and around the aggregate; the smaller the sand ratio was, the higher the aggregate occupancy was, the more obvious the stress concentration was, and the earlier the cracking of the test piece was, but there were many aggregates, so the eventual failure time was delayed. These research results can provide theoretical basis for engineering design and construction.


Introduction
CSG is made of a few cementing materials (cement and fly ash) and a little water, mixed with the raw sand gravels in the natural river course. It features unscreened aggregates and less cementing materials (usually less than 100 kg/m 3 ) [1,2], which makes it different from the conventional concrete. Therefore, it is necessary to make a study on the CSG characteristics, thus providing theoretical basis for engineering design.
Most previous researches used the experimental method, however, with the development of measurement technology and computer technology, the numerical simulation technology has been more used in the research of material characteristics.Wang Yongsheng [3] et al. conducted numerical simulation test research on indirect tensile strength of cement cold recycling mixtures by using particle flow code of two-dimension, Through laboratory test, inversion analysis reached the value of microscopic mechanical parameters, and verified the reliability of analysis simulation results. Li Chaohong et al. [4] made a mesoscopic numerical simulation of concrete damage and fracture process, and the compressive strength of the simulated cubic concrete was close to that measured in the test; Wang Jiang et al. [5] took into account the random distribution and shapes of aggregate, and concluded the damage distribution characteristics and macro stress-strain relationship of concrete under uniaxial tension through numerical analysis; Dang Nana et al. [6] [7][8][9][10]. Many studies stay focused on the testing of CSG as a new material, while this paper investigates the effect of CSG aggregate on macroscopic and mesoscopic mechanical behavior of a dam toe by means of numerical simulation, based on the CSG's unscreened aggregate feature.

CSG characteristics
Because of the unscreened aggregate, the CSG is different from the conventional concrete. Figure 1 showed the grading and Fuller curves of 12 groups of sand gravels on the site of the CSG dam at Shoukoubao, Shanxi, from which, we could see that the grading of each group of sand gravels had a large deviation from the Fuller curve, indicating poor sand gravel grading at the project site. Calculated based on the grading of the said 12 groups of CSG, the average sand ratio was 41.8%.
During the mix design of CSG, a sand ratio selection test is generally adopted to obtain a proper range of sand ratios, then a test on whether the consumption and strength of cementing materials meet the design requirements is carried out, and finally an appropriate mix proportion of cementing material consumption and sand ratio is proposed. It can be seen that the mix design of CSG requires a large number of tests, and the tests consume a lot of manpower and material resources. however, if the

Random aggregate model
CSG is composed of unscreened aggregate, cementing material and water. From a mesoscopic perspective, CSG can be regarded as a three-phase composite material consisting of natural sand gravel aggregate, mortar matrix, and interface between the mortar matrix and the aggregate. When a random aggregate model is used for finite element numerical analysis, the finite element model is divided into aggregate unit, cement mortar unit and interface unit at a mesoscopic level.

Random aggregate generation
The aggregate particles are randomly distributed on the cross section of the test piece with the help of the pseudo random numbers generated by the Monte Carlo method [11][12][13][14]. In computer simulation, the fundamental random variables were those uniformly distributed within the interval [0,1]. The probability density function with X was: A sampling sequence [x 1 , . . . , xn] of the random variable X can be generated in the computer. The random variables uniformly distributed within the interval [0,1] were the fundamental random variables, and based on them, the random variables in other distribution forms can also be obtained. For example, a random variable x ′ uniformly distributed within the interval [a,b] can be obtained by Similarly, the random numbers of random variables in other distribution forms can also be obtained by transformation of random numbers of random variables uniformly distributed in the interval [0,1].
Monte Carlo method was used to randomly determine the location of aggregates of various particle sizes in the test piece. As to the number of aggregate particles, the method of Walraven et al. [15] was used to convert the three-dimensional aggregate grading curve into the probability of occurrence of aggregates of any diameter in a two-dimensional plane. The probability Pc(D < D 0 ) of aggregates with diameter D < D 0 at any point in the two-dimensional plane was In formula (2), D 0 was the mesh diameter; Dm was the maximum aggregate particle size; P k was the percentage of the aggregate volume to the total volume of the test piece, generally 75%. The probability distribution curve was obtained according to the formula (1), based on which, the number of aggregates of different particle size on the cross section of the test piece was obtained. The aggregates were substantially convex and can be simplified into irregular polygons. The specific steps were as follows: 1. Determine the required number of aggregates of different particle size according to the aggregate grading curve; 2. Generate random circles of the aggregates and random circles of the cementing belt boundaries; 3. Divide quadrants in the circles inside and outside, determine the number of corner points in each quadrant and generate the coordinates of the corner points; 4. Connect the corner points to generate a polygon.

Constitutive model
From a mesoscopic perspective, CSG consists of two materials and one interface namely aggregate, cement mortar of different sand ratios and interface between the aggregate and the cement mortar. Therefore, the random polygon aggregate model included three units: sand gravel aggregate unit, mortar unit, and interface unit. To simplify the calculation, the constitutive relation and failure criterion of the mesoscopic components generally selected a simple form: the constitutive model adopted the linear elastic model [16][17][18][19][20][21][22][23][24][25], the constitutive equation is Where E the modulus of elasticity. And the failure criterion adopted the theory of maximum tensile stress, that is, when the maximum tensile stress exceeds the tensile strength, the material cracks [26,27].

Mesoscopic material parameters
Since the constitutive model of each constituent material used linear elastic model, the material parameters included tensile strength, elastic modulus and Poisson's ratio. The aggregate of CSG is mostly pebbles or gravels with the particle size of less than 150 mm. As to the mesoscopic parameters of the aggregates [28][29][30], the existing reports mostly took macroscopic values, and it was reported [31] that: the rock parameters were affected by the size, the elastic modulus decreased with the increase in size, the strength increased with the increase in size, and the Poisson's ratio remained essentially unchanged. The CSG aggregate was small in particle size and diverse in shape. Due to the limitation of testing methods, it was hard to obtain the actual parameters of the aggregate particles. This paper used inverse analysis to obtain the actual parameters of aggregate particles and the elastic modulus of interface materials and mortar matrix. The failure criterion of mesoscopic components adopted the maximum tensile stress criterion, with reference to the pertinent literature [32][33][34][35], and the values were shown in Table 1.

Inverse analysis
Contrary to the normal analysis, the inverse analysis is to deduce the mesoscopic parameters by inversion based on the laboratory macroscopic test results. The parametric inversion optimization used genetic algorithm and was realized through the data transmission between MATLAB and finite element software. The realization process was as follows: (1) Generate the initial parameter population through MATLAB, namely the parameters required for mesoscopic numerical simulation; (2) Read the generated data file into the finite element software for uniaxial compression test and simulation analysis, and output the nu-merical simulation result; (3) MATLAB read the numerical simulation results file, and compared it with the laboratory uniaxial compression test result to judge whether the iteration was stopped. If the accuracy met requirements, the algorithm ended. If failed, a new population was re-selected and cross-generated. Repeat (2)-(3) until an optimal solution was obtained. Figure 4 showed a photo of a damaged cubic CSG after laboratory uniaxial compression. The size was 100mm*100mm*100mm, and the aggregate was of standard second grade, as shown in Table 2. "d" expressed the particle size and "w" the particle composition. The mix proportion of the CSG was shown in Table 4.
The laboratory uniaxial compression test used a universal testing machine and a displacement control method. The displacement was controlled with different load steps, which corresponded to different stresses, and thus a stressstrain curve was obtained according to the laboratory measurements. In the numerical simulation analysis, a screened second-grade wet aggregate was adopted, with the boundary conditions of full constraints at the bottom and no constraints at the top, which were detailed in the lower right corner of Figure 2; when the genetic algorithm was used for the inverse analysis, the crossover probability was taken as 70%, and the mutation probability 10%. The objective function was: Where: σ j k was the calculated value of the stress corresponding to the k-th load step, σ m k was the experimental value of the stress corresponding to the k-th load step, and N was the total number of load steps, controlled by the total displacement.
After 16 iterative solutions, the elastic modulus of the aggregate in identified CSG by inversion was finalized as E G = 500Mpa, the elastic modulus of the mortar matrix Em = 30Mpa, and the elastic modulus of the interface E i = 20Mpa. The numerical simulation results and test results were shown in Figure 2.
Through the analysis of parametric inversion results, the stress peak in the laboratory test was slightly lower than that in the numerical simulation due to the test condi-

Compression test simulation
We carried out a numerical simulation test on 100*100*100 CSG of different aggregate grades to investigate the effect of grading. In the calculation, the mesoscopic material parameters were assumed unchanged, see Section 2.3, and the aggregates of 30 mm or more were removed in order to compare with the laboratory test results of the screened wet test block. The sand ratios selected were 18.5% (Fuller standard), 34.0%, 43.1% and 46.0% respectively. According to the random aggregate generated and its interface in 2.1 , the material numbers of aggregate, interface and mortar are determined based on the surface number, the corresponding material numbers are assigned to the corresponding material parameters in ANSYS, after grid division, and the established random aggregate models were shown in Figure 3 (a, b, c and d). If the grading of aggregates on the site was poor, the large aggregates could be crushed to improve the grading; therefore, 34.0% and 43.1% were added in the simulation test as control groups.
In the control groups, the number of aggregates of small particle size was increased and the number of aggregates   Figure 3 (e, f).

Results analysis 4.2.1 Analysis of macroscopic mechanical behavior
The uniaxial compression simulation test used displacement loading to obtain a stress-strain curve. Figure 4(a) showed the stress-strain curves at different sand ratios; Figure 4(b) showed the stress-strain curves at different grades with the same sand ratio.
In Figure 4(a), with the increase of the sand ratio, the aggregate occupancy and the peak stress decreased, indicating an inverse relation between the sand ratio and the peak stress, which was consistent with the conclusion in the literature [36]; as the sand ratio increased, the slope of tangent of the stress-strain curve decreased, that was, the elastic modulus became smaller. It was because that with the increase of sand ratio, the aggregate occupancy decreased, and the mechanical properties of the CSG were affected more by the mortar matrix and less by the aggregate; thus the strength and the elastic modulus were both reduced.
In Figure 4(b), under the same aggregate occupancy (the same sand ratio), as the aggregate grading was improved (the aggregates of small particle size increased),  the stress peak became higher, and vice versa. It was because that the elastic modulus of the aggregate and the cement mortar was different, leading to inconsistent deformation, and if the grading was worse, the stress concentration was more obvious, resulting in lower strength of CSG overall. This result can provide guidance for the onsite construction. Figure 5 showed the damaged test blocks of 20% sand ratio and standard second grade in the numerical simulation and the laboratory test. Through comparison between Figure 5(a) and (b), the white part in (a) was the crack after damaged, mostly located in the transition zone between the mortar matrix and the aggregate and a few in the mortar matrix and around the aggregate, which was consistent with (b) in terms of the cracking site and the cracking rule. It was because that the aggregate and the mortar matrix were different in characteristics, resulting in inconsistent deformation (see (a), the displacement was uneven along the aggregate distribution), the stress was concentrated near the aggregate, and the strength at the interface of transition zone between the mortar matrix and the aggregate was the lowest due to the micro-cracks; therefore, the micro-cracks were easier to appear on the transition zone [37], and as the load increased, the energy would release from the weakest point and then extend to the mortar matrix, and the crack bypassed the aggregate to expand along the interface.

Analysis of macroscopic and mesoscopic damage mechanism
After the test block was loaded, the elasticity deformed first, and with the pressure increase, some of the mesoscopic components began to rupture. Table 4 showed the initial failure load step and the eventual failure load step at different sand ratios. It could be seen from Table 4 that: under the same mix proportion, the smaller the sand ratio of the CSG was, the higher the aggregate occupancy was, causing more obvious stress concentration. The low strength of the mortar matrix resulted in earlier cracking of the test piece; the high strength of the aggregate led to strong bearing capacity of the test piece and later eventual failure. On the contrary, the higher the sand ratio was, the lower the aggregate occupancy was, resulting in later cracking and earlier eventual failure.

Conclusion
By analyzing the aggregate characteristics of CSG, establishing two-dimensional random polygon aggregate models and calling of MATLAB and calculation software, this paper realized the inverse analysis of the mesoscopic component parameters, and the results were consistent with the test results. According to the typical characteristics of aggregates on the project site, this paper analyzed the ef-fects of aggregate grading on the macroscopic and mesoscopic properties of CSG through mesoscopic numerical simulation, and came to the following conclusions: 1. With the increase of the sand ratio, the aggregate occupancy and the peak stress decreased, indicating an inverse relation between the sand ratio and the peak stress; as the sand ratio increased, the slope of tangent of the stress-strain curve decreased, that was, the elastic modulus became smaller. Under the same aggregate occupancy (the same sand ratio), as the aggregate grading was improved (the aggregates of small particle size increased), the stress peak became higher, and vice versa. This result can provide guidance for the on-site construction. 2. The cracks of CSG mostly appeared at the transition zone between the aggregate and the mortar matrix and around the aggregate. The smaller the sand ratio was, the higher the aggregate occupancy was, the more obvious the stress concentration was, and the earlier the cracking of the test piece was, but there were many aggregates, so the eventual failure time was delayed; the higher the sand ratio was, the lower the aggregate occupancy was, resulting in later cracking and earlier eventual failure.