Calibration method of mesoscopic parameter in sandy cobble soil triaxial test based on PFC3D

Abstract

This paper presents a rapid and effective calibration method of mesoscopic parameters of a three-dimensional particle flow code (PFC3D) model for sandy cobble soil. The method is based on a series of numerical tests and takes into account the significant influence of mesoscopic parameters on macroscopic parameters. First, numerical simulations are conducted, with five implementation steps. Then, the multi-factor analysis of variance method is used to analyze the experimental results, the mesoscopic parameters with significant influence on the macroscopic response are singled out, and their linear relations to macroscopic responses are estimated by multiple linear regression. Finally, the parameter calibration problem is transformed into a multi-objective function optimization problem. Numerical simulation results are in good agreement with laboratory results both qualitatively and quantitatively. The results of this study can provide a basis for the calibration of microscopic parameters for the investigation of sandy cobble soil mechanical behavior.

Graphical abstract

Keywords

sandy cobble soil / particle flow code / orthogonal experiment / regression analysis / multi-objective optimization / calibration

1 Introduction

Sandy cobble soil is mainly composed of cobbles and gravel, a collective term for discrete clastic deposits containing small amounts of sand and clay [1]. In China and other countries, sandy cobble soil with characteristics of low cohesion, large porosity, and strong permeability is widely distributed. It is a mechanically unstable stratum that is unfavorable to tunnel stability [2]. The key to solving the associated problems lies in a comprehensive understanding of the physical and mechanical properties of the strata [3–10], and formulating appropriate engineering countermeasures. Numerical simulation of particle flow code (PFC) has the advantages of well-defined material properties, clear physical concepts, low cost, high efficiency, and good reproducibility. It can easily handle discontinuous media mechanics and is commonly used to simulate the mechanical properties of discrete materials [11,12].

The study of particle discrete element numerical simulation requires calibration of the numerical model based on a macroscopic mechanical response to determine mesoscopic parameters. Potyondy and Cundall [13] established the ratio of maximum and minimum particle size (R_max/R_min) for Lac du Bonnet granite material as 1.66. The ratio can effectively reflect the size and arrangement of mineral grains, and has been widely adopted by subsequent scholars [14,15]. Van [16] and Mier and Jan [17] studied the size of representative volume units and particles, in which the ratio was at least 7 to 8. For material with the ratio of a numerical test model size to particle average radius greater than 30 to 40, Jensen et al. [18] suggested that the number of particles does not affect the numerical simulation results. Yang et al. [19] calibrated red sandstone using two-dimensional particle flow code (PFC2D) through extensive repeated experiments and verified the numerical simulation and indoor experimental results. Xia and Zeng [20] uses the bonded-particle element method embedded in smooth joints to calibrate the macroscopic mechanical properties of transversely isotropic rock. Zhang et al. [21] described a calibration method for the bond model and the smooth joint model. However, most of these studies are qualitative and rely on a “trial-and-error” approach, which can be time-consuming and inefficient.

To overcome the limitations of the “trial-and-error” method, scholars have conducted extensive research to improve the calibration process [15,22–26]. Yoon [22] used the Plackett-Burman test to establish the linear relationship between macroscopic mechanical indexes and mesoscopic parameters. They then used the response surface method to determine the interaction between significant influencing parameters, and formulated the problem as a nonlinear multi-objective mathematical programming problem. Hanley et al. [23] used the Taguchi method to calibrate the mesoscopic parameters and select appropriate design experiments for two- and three-dimensional models, and provided a reasonable and effective calibration procedure. Chehreghani et al. [24] studied the sensitivity of mesoscopic parameters in bonded particle models by response surface methodology and central composite design, and obtained a quantitative expression relating macroscopic and mesoscopic parameters. Li et al. [25] developed a three-dimensional discrete element model using flat-joint and smooth-joint contact models to investigate the effect of anisotropy on the tensile behavior of slate, and proposed a calibration procedure for the discrete element method (DEM) modeling of transversely isotropic rock. Xu et al. [26] proposed a linear parallel bond model to calibrate limestone and verified the effectiveness of the proposed method. Lu et al. [27] proposed a systematic algorithm to simulate gravelly soils using the commercially available DEM program PFC2D. The shear behaviors of binary mixtures composed of irregular coarse particles or discoidal fines have also been investigated. Cui et al. [28] proposed a hybrid extension method that combines the advantages of the ray extension method and plane extension method and explores the influence of important factors on tunneling. Wu et al. [29] and Chen et al. [30] established a three-dimensional concrete mesostructured modeling algorithm based on cohesive zone models [31]. Xu et al. [15] calibrated the linear parallel bonding model and smooth joint model for rocks, establishing the relationship between macroscopic and mesoscopic parameters. A standardized parameter calibration process was obtained by iterative calibration.

A literature review shows that the current research on calibration methods focuses on rock materials. The mesoscopic parameter calibration method of sandy cobble soil mostly adopts the “trial-and-error” method. In the calibration process, there are profound differences between sandy cobble soil and rock material in terms of the constitutive model, in terms of mesoscopic parameters and their value ranges, and in terms of macroscopic indexes and their variation ranges. Therefore, the mesoscopic parameter calibration method of rock materials is not suitable for sandy cobble soil, and it is necessary to study the rapid calibration method for such soil.

Addressing the lack of an efficient calibration method for mesoscopic parameter calibration of sandy cobble soil, this paper presents a rapid and effective calibration process. First, a numerical test scheme is designed by orthogonal test design. Then, the influence of mesoscopic parameters on macroscopic parameters is analyzed by multi-factor analysis of variance. Through regression analysis, the linear relationship between macroscopic parameters and main mesoscopic parameters is established. Finally, the calibration problem is transformed into a multi-objective function problem. The results are compared with physical test results of typical sandy cobble soil.

2 Numerical test scheme

The orthogonal table is generally expressed by L_n (r^m), where L is the symbol of an orthogonal table; n is the number of rows (test times); r is the number of factor levels; m is the number of vertical columns (maximum number of factors that can be arranged).

2.1 Macroscopic parameters indicators

The obtained stress–strain relationship and failure envelope of the PFC model should be consistent with laboratory test results. This is the standard for calibration of triaxial shear mesoscopic parameters of granular flow. Therefore, the selection of macroscopic characteristic indexes should follow [32]: 1) the selected indicators can be easily and readily obtained from the test; 2) the selected index can better describe the shape of the calibrated curve. The index describing the stress–strain curve of sandy cobble soil triaxial test is initially elastic modulus E₀, peak strength σ_f, and residual strength σ_r, as shown in Fig.1; the indexes describing the Mohr–Coulomb strength envelope include cohesion c and internal friction angle φ. The selection of specific indicators should be determined according to the shape of the calibration curve and the focus of the research.

In this paper, four parameters including initial elastic modulus E₀, peak strength σ_f, cohesion c, and internal friction angle φ are selected as macroscopic indexes. Meanwhile, the initial elastic modulus and peak strength of sandy cobble soil are closely related to the confining pressure. The initial elastic modulus and peak strength vary with confining stress level; the peak strength corresponding to a single confining stress level is not representative. Therefore, three initial elastic modulus E₀ and peak strengths σ_f appear in the triaxial test, and a curve needs to be selected to obtain the corresponding relationship between macro and micro parameters. For the consistency of each working condition, without specifying the case, in this paper, the initial elastic modulus E₀ and peak strength σ_f are based on the condition that the confining pressure is 300 kPa.

2.2 Mesoscopic parameter indicators

Selecting the appropriate contact model is crucial to improving calibration efficiency and accuracy. The linear parallel bond model is mostly used to simulate the cementation between grains of rock materials [32–35]. The linear contact bond model is often used to simulate the occlusion and embedment between sandy cobble materials [36–38], and this model is used in this study. The linear contact bond model can directly increase the bonding between particles, without considering the gradation of raw materials to reflect the occlusion between particles. It saves computation and improves efficiency.

Based on the above considerations, this paper selects the contact effective modulus E_c, the normal and shear stiffness ratio k_n/k_s, the friction coefficient between particles μ, the bond shear strength τ_c, the bond tensile strength σ_c, and the particle clump content as the mesoscopic parameters of this numerical analysis. Clump refers to a series of spheres connected rigidly that can effectively increase the occlusion and embedding effect between particles. Tab.1 shows the range of mesoscopic parameters determined by a series of numerical simulation experiments.

Under different combinations of selected factor levels, the range of macroscopic parameters is as follows: initial elastic modulus E₀ range is 10–100 MPa; peak strength σ_f range is 500–2050 kPa; cohesion c range is 0–600 kPa; internal friction angle φ range is 17°–47°. Therefore, at the selected factor level, the macroscopic mechanical index range obtained in this paper covers the macroscopic mechanical index range of most sandy cobble soil [37–42].

2.3 Implementation steps

1) Sample preparation: the numerical test model size is H × L = 600 mm × 300 mm. The particles are in the form of ball and clump. Since the generation of clump particles is very time-consuming, only 2 and 3 spheres are used to obtain two types of clump units in this paper. These spheres are rigidly connected and randomly generated to simulate the different shapes of particles in sandy cobble soil. The clump model approximately simulates the irregularity of sand cobble soil particles, which makes up for the influence of particle gradation to a certain extent. In addition, in the loading process, the energy needed to overcome the midmovement between particles is large, so the strength is relatively high, which is consistent with experimental findings. According to the relevant Refs. [13–18], this paper takes L/R_min (R_min = 10 mm) = 30, R_max/R_min = 1.66, porosity n = 0.32, the particle density is 2700 kg/m³. The particles are evenly distributed in the test vessel using the expansion method; the number of particles generated is around 20000, as shown in Fig.2.

2) Assignment of contact models: this paper adopts the contact bond model. It requires input mesoscopic parameters, first for the linear group: contact effective modulus (E_c), normal and shear stiffness ratio (k_n/k_s), and friction coefficient between particles (μ); and secondly for the contact bonding group: bond tensile strength (σ_c) and bond shear strength (τ_c).

3) Applying confining pressure: in the numerical experiment of this study, three confining pressures of 100, 200, and 300 kPa are applied to each group. The confining pressures of 100, 200, and 300 kPa are mostly used for the triaxial test in laboratory tests [40,42,43]. Some scholars have also selected these three confining pressures for PFC numerical tests [41,44,45]. To ensure the accuracy of the results, the above confining pressures are also selected for calculation in this paper. By monitoring the difference between contact pressure and set pressure in each calculation step, the displacement of the side wall is adjusted to achieve constant confining pressure.

4) Vertical loading: the loading rate is 1.2 mm/s. In the shearing process, the unbalanced force rate is always less than 1 × 10^–4, and the variation of confining pressure does not exceed 0.3%.

5) Termination of loading: when the peak strength occurs, the loading should be terminated at 3%–5% after the peak strain. If there is no peak strength, the test is terminated at 15% of the axial strain.

3 Numerical results and analysis

Due to the number and level of factors, in this paper a L₃₂ (4⁹) orthogonal table is used for numerical experiments. A total of 32 groups were assessed in this numerical simulation experiment, and the parameters used for calculation of each case are shown in Tab.2.

3.1 Multifactor analysis of variance

Analysis of variance can determine whether there are significant differences among multiple groups of mesoscopic parameters. It accurately estimates the importance of various factors in the test results, and is widely used in data analysis of orthogonal tests. Depending on the sources of influencing factors, analysis of variance is divided into intra-group error and inter-group error. The ratio of mean square between groups and within groups of factor A is F_A statistics. F_α is the critical value of the dominant level α. If F_A > F_α, then it is considered that factor A has a significant effect on the test results. The greater F_A results relative to F_α, the more obvious the effect of factor A [46].

The test in this study is designed as a L_n (r^m) orthogonal table. The test result is y_i (i = 1,2,…,n). The basic steps of analysis of variance are as follows.

1) Calculation of the sum of squares of total deviations:

(1)S{S}_{\mathrm{T}}=\sum _{i=1}^n{(y_i-\overline{y})}^2,

(2)\overline{y}=\sum _{i=1}^n{y}_i.

2) The sum of squares of deviations caused by factor j:

(3)S{S}_j=\frac{n}{r}\sum _{i=1}^n{\left(k_i-\overline{y}\right)}^2,

(4)k_i=\frac{r}{n}{K}_i,

where K_i is the sum of the test results with level number i of column j.

3) The sum of variance squares of calculation error:

(5)S{S}_{\mathrm{e}}=\sum S{S}_{\mathrm{B}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{k}\mathrm{s}\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e}}.

4) Computational freedom.

The degrees of freedom of SS_T, SS_j and SS_e are:

(6)\mathrm{d}{f}_{\mathrm{T}}=n-1,

(7)\mathrm{d}{f}_j=r-1,

(8)\mathrm{d}{f}_{\mathrm{e}}=\sum d{f}_{\mathrm{B}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{k}\mathrm{s}\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e}}.

5) Calculation of mean square:

(9)M{S}_j=\frac{S{S}_j}{\mathrm{d}{f}_j},

(10)M{S}_{\mathrm{e}}=\frac{S{S}_{\mathrm{e}}}{\mathrm{d}{f}_{\mathrm{e}}}.

6) Calculation of F value:

(11)F_j=\frac{M{S}_j}{M{S}_{\mathrm{e}}}.

7) Significance test.

For the given significance level α, if F_j > F_α, the j factor is considered to have a significant effect on the test results. The greater F_j results relative to F_α, the more obvious the effect of factor j on the test results.

3.2 Results of multivariate analysis of variance

The results of numerical tests were analyzed by analysis of variance. The analysis results are shown in Tab.3. From Eqs. (1)–(5), the sum of total deviation squares, the sum of deviation squares of various factors, and the sum of error squares can be calculated. From Eqs. (6)–(8), the total freedom, the freedom of each factor, and the freedom of error can be calculated. The significance levels α = 0.05 and α = 0.01 were selected. Look up the table can be obtained, F_0.05 (3,13) = 3.41, F_0.01 (3,13) = 5.74. When F_0.05 < F_j < F_0.01, it is considered that factor j has a significant influence (95%) on the test results, denoted as “*”. When F_j > F_0.01, it is considered that factor j has a very significant effect (99%) on the test results, denoted as “**”. To more intuitively analyze the influence of test factors on test indexes, the results of analysis of variance are plotted as histograms, as shown in Fig.3–Fig.6.

As shown in Fig.3, the factors of E_c, k_n/k_s, clump, and τ_c all have significant impact on E₀. The maximum F value of E_c is 70.50, much larger than F_0.01. The F values of k_n/k_s, τ_c, and clump are 17.88, 9.52, and 4.30, respectively. Relative to E_c, k_n/k_s, clump, and τ_c, other factors can be ignored. The relative sensitivities of the main mesoscopic parameters to E₀ are as follows: E_c > k_n/k_s > clump > τ_c.

As shown in Fig.4, the F values of μ, clump, τ_c, and E_c are greater than F_0.01, which has a very significant influence on σ_f, and so other factors can be ignored. The peak strength σ_f is most sensitive to the change of μ. The relative sensitivities of the main mesoscopic parameters to σ_f are as follows: μ > clump > τ_c > E_c.

As shown in Fig.5, E_c, τ_c, and σ_c have significant effects on cohesion c, and other factors can be ignored. The relative sensitivities of the main mesoscopic parameters to c are as follows: E_c > τ_c > σ_c.

As shown in Fig.6, μ, E_c, and clump have strong effects on the internal friction angle φ. The friction coefficient has the most obvious effect on the internal friction angle. The relative sensitivities of the main mesoscopic parameters to c are as follows: μ > E_c > clump.

3.3 Regression analysis

Based on the results of analysis of variance, the regression analysis is carried out on mesoscopic parameters that influence macroscopic parameters. The correspondences between macroscopic and mesoscopic parameters are as follows:

(12)\begin{array}{rl}{E}_0=& 0.142E_{\mathrm{c}}-5.4k_{\mathrm{n}}/k_{\mathrm{s}}+9.83{\tau }_{\mathrm{c}}\\ & +15.87\mathrm{c}\mathrm{l}\mathrm{u}\mathrm{m}\mathrm{p}+16.41,(R^2=0.78),\end{array}

(13)\begin{array}{rl}{\sigma }_{\mathrm{f}}=& -2.47E_{\mathrm{c}}+985.27\mu +631.81{\tau }_{\mathrm{c}}\\ & +653.08\mathrm{c}\mathrm{l}\mathrm{u}\mathrm{m}\mathrm{p}-5.71,(R^2=0.903),\end{array}

(14)\begin{array}{rl}c=& -1.33E_{\mathrm{c}}+28.08k_{\mathrm{n}}/k_{\mathrm{s}}+163.68{\sigma }_{\mathrm{c}}\\ & +253.86{\tau }_{\mathrm{c}}-66.61,\left(R^2=0.828\right),\end{array}

(15)\phi =0.046E_{\mathrm{c}}+17.82\mu +9.85\mathrm{c}\mathrm{l}\mathrm{u}\mathrm{m}\mathrm{p}+9.84,(R^2=0.798).

As can be seen from Eq. (12), initial elastic modulus increases with the increase of contact effective modulus, bond shear strength, and clump, but decreases with increase of normal shear stiffness ratio. Similarly, from Eq. (13), peak strength is positively correlated with friction coefficient between particles, bond shear strength and clump, and negatively correlated with contact effective modulus. Equation (14) shows that the cohesion increases with the increase of shear stiffness ratio, bond tensile strength, and bond shear strength, and decreases with the increase of contact effective modulus. From Eq. (15), internal friction angle is positively correlated with contact effective modulus, friction coefficient between particles, and clump. As the macroscopic parameters are affected by many mesoscopic parameters, and some mesoscopic parameters are discarded, the macroscopic and mesoscopic parameters are not completely linear. Therefore, the fitted R-squared value is not ideal, but it can generally reflect the relationship between macro and micro parameters and achieve the effect of rapid calibration.

4 Mesoscopic parameter calibration

To make the macroscopic mechanical indexes of numerical tests and laboratory tests as consistent as possible, the multi-objective mathematical programming method is employed to optimize the mesoscopic parameters of numerical simulation experiment. The specific mesoscopic parameter calibration process is shown in Fig.7.

1) Objective function

To make the physical indexes obtained by numerical simulation and laboratory test as close as possible, the objective function expression is adopted, as follows:

(16)min\{\begin{array}{c}\left|E_0-E_0^{\ast }\right|,\\ \left|{\sigma }_{\mathrm{f}}-{\sigma }_{\mathrm{f}}^{\ast }\right|,\\ \left|c-c^{\ast }\right|,\\ \left|\phi -{\phi }^{\ast }\right|,\end{array}

where E₀, σ_f, c, and φ are obtained by regression equations Eqs. (12)–(15); E₀*, σ_f*, c*, and φ* are the macroscopic parameters obtained from the laboratory test.

2) Constraint conditions

The regression equation is obtained at a certain factor level, the range of mesoscopic parameters should be restricted, and the constraint conditions are as follows:

(17)\{\begin{array}{c}20⩽E_{\mathrm{c}}⩽200,\\ 1⩽k_{\mathrm{n}}/k_{\mathrm{s}}⩽4,\\ 0.25⩽\mu ⩽1,\\ 0.25⩽{\sigma }_{\mathrm{c}}⩽1,\\ 0.25⩽{\tau }_{\mathrm{c}}⩽1,\\ 0.25⩽\mathrm{c}\mathrm{l}\mathrm{u}\mathrm{m}\mathrm{p}⩽1.\end{array}

3) Optimization of parameters

The Fgoalattain function in MATLAB is used for the optimization of the multi-objective function. The goal matrix goal = 0 is set to transform the goal programming problem into the minimum solution problem.

5 Example verification

The macroscopic parameters (see Tab.4, laboratory triaxial compression test) in Ref. [42] are selected. The initial mesoscopic parameters (see Tab.5 preliminary calculation) are obtained by Eq. (16) optimization. After inputting initial mesoscopic parameters into PFC3D numerical test model, the initial macroscopic parameters obtained are shown in Tab.4 (preliminary calculation). Compared with the laboratory test results, it is found that the error in peak strength is 2.46%, and the error in internal friction angle is 3.14%. But the errors in initial elastic modulus and cohesion are 47.04% and 28.55% respectively.

According to Eqs. (12)–(15), the contents of E_c, μ, σ_c, and clump are appropriately optimized, see Tab.5 optimization and adjustment. The macroscopic parameters after optimization and adjustment are shown in Tab.4 (optimization and adjustment). After optimization and adjustment, the error of initial modulus is 4.91%, the peak strength error is 1.32%, the error of cohesion is 0.81%, and the internal friction angle error is 0.95%. The errors are all less than 5%, and within the acceptable range.

Fig.8 shows the stress–strain curves of the laboratory triaxial test and numerical simulation test of sandy cobble soil. Compared with the triaxial test, the peak strain (strain at peak strength) of the stress–strain curve obtained by the numerical simulation test is the smaller of the two. With the increase of strain, there will be a certain softening phenomenon. However, the initial elastic modulus, peak strength, and curve shape of numerical test results are consistent with those of triaxial test results, and can better simulate the characteristics of sandy cobble soil.

6 Conclusions

An integrated parameter calibration method for sandy cobble soil is proposed, based on the contact bond model and clump. The main conclusion of this work are as follows.

1) A calibration parameter index system is proposed through regression analysis of the macroscopic mechanical indexes and the main mesoscopic parameters. The significant influence of mesoscopic parameters on the macroscopic mechanical properties of sandy cobble soil is obtained based on the multi-factor analysis of variance.

2) The contact bond model can increase the bond between particles, which can simulate the occlusion between cobble particles. The numerical results are in good agreement with the laboratory test results. The mesoscopic parameter calibration method of sandy cobble soil can be realized more efficiently by combining the contact bond model with the clump, and the result is in good agreement with the laboratory test results.

3) The initial elastic modulus increases with increasing contact modulus, bond shear strength, and clump, and it decreases with increasing normal shear stiffness ratio. The peak strength is positively correlated with friction coefficient, contact shear strength and clump, and it is negatively correlated with the contact modulus. The cohesion increases with increasing normal shear stiffness ratio, bond tensile strength, and bond shear strength, and decreases with increasing contact modulus. The internal friction angle is positively correlated with the contact modulus, friction coefficient, and clump.

4) By slightly adjusting the mesoscopic parameters, the error between the macroscopic parameters of the laboratory test and the numerical simulation is within 5%, and the mesoscopic parameters consistent with the triaxial test are obtained. The results may provide guidance and reference for the rapid calibration of the mesoscopic parameters of sandy cobble soil.

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