Correlation Analysis of Macro–Micro Parameters of Sandstone Based on PFC^3D

Abstract

To address the issue of the low compression–tension ratio in the traditional parallel bond model (PBM), this study proposes an improved PBM incorporating a random distribution strategy of strong–weak contact groups. An L₂₇(3¹²) orthogonal experimental design was employed to construct 27 sets of numerical simulation schemes. Combined with Pearson correlation coefficient analysis and multivariate regression, the influence of twelve microscopic parameters on seven of the macroscopic mechanical properties of sandstone was systematically investigated, including elastic modulus (E), Poisson’s ratio (v), uniaxial compressive strength (σ_c), internal friction angle (φ), cohesion (c), crack damage stress ratio (σ_cd/σ_c), and compressive–tensile strength ratio (σ_c/σ_t). Based on these analyses, a quantitative relationship model between the macro and micro parameters was established and validated through numerical simulation and experimental comparison. The proposed method provides a theoretical foundation for the mechanical modeling of sandstone and the inversion of microscopic parameters.

1. Introduction

Rock is an aggregate composed of various minerals, whose complex mechanical properties and damage evolution mechanisms hold significant theoretical and engineering value for disaster prevention and mitigation in geotechnical engineering. In rock mechanics research, PFC^3D 6.0 (Particle Flow Code) software has attracted considerable attention due to its ability to simulate particle interactions, bonding failure, and crack propagation, among other microscopic mechanical mechanisms. The parallel bond model is a core model within PFC, widely used for simulating the macro–microscopic mechanical behavior of rock materials. The PBM comprises linear and parallel bond components, which can simultaneously transmit forces and moments to resist relative displacement and rotation between particles. When the interaction force between particles exceeds the bond strength, the model reverts to a linear model. This characteristic closely mirrors the processes of compaction, elastic deformation, bond failure in the plastic stage, and eventual rupture of rock under initial loading, thereby justifying the use of the Parallel Bond Model for simulation studies.

In recent years, scholars both domestically and internationally have conducted extensive research on rocks using PFC (Particle Flow Code) software. For example, Wang, X. et al. (2025) [1] constructed a PFC^2D model based on real granite, accurately representing mineral features and overcoming the limitations of traditional random parameter simulations. Their work revealed the effects of mineral particle size and fractures on the dynamic mechanical behavior of granite. Xiao Fukun et al. [2] proposed a method for calibrating macroscopic and microscopic parameters by combining shear tests and particle column collapse tests, and they verified its feasibility in collapsed-body research through phase analysis and weight analysis. Zhang Jiafan et al. [3] conducted triaxial tests on red sandstone and PFC numerical simulations, finding that the rock strength decreases as fracture connectivity increases and that higher connectivity leads to more complex crack propagation during failure. Zhang Jie et al. [4] analyzed the shear strength characteristics and failure mechanisms of rock under constant normal and disturbed loads at different fracture angles using direct shear tests on fractured sandstone and PFC^2D simulations, revealing the influence of disturbed loads on crack propagation and energy dissipation. Xu, Z et al. (2022) [5] proposed a parameter calibration system based on the Parallel Bond Model and Smooth-Joint Model, applicable to the mesoscopic study of transversely isotropic rocks. Wu Luyuan et al. [6] used laboratory experiments combined with numerical simulations and regression analysis to reveal the impact of mesoscopic parameters on the macroscopic mechanical properties of sandstone, and they established a regression relationship. Xu Chaohua et al. [7] used the PB+CCD method to construct a quantitative relationship model for the macroscopic and microscopic mechanical parameters of rocks, providing a reference for discrete element simulation parameter calibration.

Although significant progress has been made in rock mechanics research based on PFC, most studies have still primarily used the traditional parallel bond model (PBM) for simulations. However, the PBM generally suffers from a low compressive–tensile strength ratio (usually between five and eight), which makes it difficult to accurately represent the true mechanical characteristics of rock materials [8]. To address this issue, researchers have proposed various improvements. For instance, Zhang Baoyu et al. [9] simulated rocks using the FJM model, finding that it effectively improves the compressive–tensile strength ratio and mimics the actual mechanical properties of rock materials. Huang Da et al. [10] also pointed out that the FJM model has significant advantages in simulating the mechanical properties of rocks under triaxial stress states. Sun Hao et al. [11] established a rigid block model using Voronoi tessellation and simulated sandstone using the SBM model, discovering that this model more accurately represents the mechanical properties of sandstone, including a higher compressive–tensile strength ratio. Liu Zhe et al. [12] introduced viscoelastic constitutive relations into PFC simulations, derived the force–displacement equation for the viscoelastic model, and developed related contact failure criteria, making the contact force response and bonding strength between particles more correlated, thus improving simulation accuracy.

While the methods mentioned above have made some progress in improving model performance, most research has relied on the PBM model. Therefore, it is of great significance to study how to effectively improve the compressive–tensile strength ratio of the PBM model. Additionally, existing research has relied on trial-and-error methods or empirical formulas during the parameter calibration process, which are inefficient and cannot meet the demands for high-efficiency and accurate simulations. Based on this, this paper takes sandstone as the research subject, uses an improved parallel bond model, and conducts 12 orthogonal experiments on mesoscopic parameters to analyze their influence on macroscopic mechanical properties. This study reveals the quantitative relationships between macro and micro parameters as well as verifies the feasibility of the model and parameters through practical examples.

2. Methodology

2.1. Improvement of the Parallel Bond Model

To address the issue of the low compressive–tensile strength ratio in simulating sandstone using the traditional parallel bond model, a strategy involving the random distribution of strong–weak contact groups was introduced, as illustrated in Figure 1. The strong contact group simulates the stronger cementation characteristics between particles. Conversely, the weak contact group is derived from the strong contact group by applying scaling coefficients (modulus ratio R_E, stiffness ratio R_k, strength ratio R_σ), thereby simulating the weaker cementation characteristics between particles. This enhanced method more accurately reproduces the heterogeneity within rock, thereby improving the model’s capability to simulate the actual mechanical behavior of rock. The parameter settings for each contact group in the model are presented in

Table 1. Parameters of different contact groups.

Model Microscopic Parameters	Strong Contact Group	Weak Contact Group
Effective bond modulus	${\overline{E}}_c$	${\overline{E}}_c$ × RE
Bond stiffness ratio	k	k × Rk
Tensile strength	${\overline{\sigma }}_t$	${\overline{\sigma }}_t$× Rσ
Cohesion	$\overline{c}$	$\overline{c}$× Rσ
Effective bond modulus	${\overline{E}}_c$	${\overline{E}}_c$ × RE

.

2.2. Mesoscopic Parameter Settings and Physical Significance

The microscopic parameters in the parallel bond model include the effective contact modulus of the particles E_c, minimum particle size R_min, friction coefficient μ, particle stiffness ratio k_n/k_s, particle density ρ, particle size ratio R_max/R_min, effective bond modulus ${\overline{E}}_c$, bond stiffness ratio ${\overline{k}}_n/{\overline{k}}_s$, tensile strength ${\overline{\sigma }}_t$, cohesion $\overline{c}$, and friction angle $\overline{\varphi }$. Based on research by Jiang Yue et al. [13] and Huang Yisheng et al. [14], the following values for these microscopic parameters were set: R_min = 0.8 mm, ρ = 2600 kg/m³, R_max/R_min = 1.66, with the particle stiffness ratio consistent with the bond stiffness ratio.

The improved parallel bond model comprises 12 microscopic parameters and 7 macroscopic parameters (see

Table 2. Macro- and microscopic parameters of the improved linear parallel bond model.

Microscopic Parameters	Macroscopic Parameters
Effective bond modulus ${\overline{E}}_c$	Elastic modulus E
Particle effective modulus/effective bond modulus Ec$/{\overline{E}}_c$	Poisson’s ratio ν
Bond stiffness ratio k	Uniaxial compressive strength σc
Tensile strength ${\overline{\sigma }}_t$	Uniaxial compressive strength/uniaxial tensile strength σc/σt
Cohesion/tensile/strength $\overline{c}$ /${\overline{\sigma }}_t$	Crack damage stress σcd
Friction coefficient μ	Internal friction angle φ
Friction angle $\overline{\varphi }$	Cohesion c
Moment distribution coefficient $\overline{\beta }$
Packing ratio Rf
Strength ratio Rσ
Modulus ratio RE
Stiffness ratio Rk

). Among these, the physical interpretation of certain parameters is as follows: R_f denotes the packing ratio, which indicates the proportion of weak contact particle interactions relative to the total number of contacts; $\overline{\beta }$ represents the moment distribution coefficient, describing the ratio of the distribution between the normal and tangential moments; the crack damage stress σ_cd signifies the stress value at the inflection point of volumetric strain during the compressive dilation process of rock. When stress surpasses σ_cd, even if the external stress remains constant, the microcracks within the rock progressively increase, resulting in failure.

2.3. Orthogonal Experimental Design

Orthogonal experimental design is a method that optimizes experimental schemes to effectively quantify the impact of multiple factors on the target response. In this study, the improved parallel bond model (PBM) involved 12 microscopic parameters. Conducting a full-factor test would have resulted in an excessive number of trials, so an orthogonal experimental design was employed to reduce the number of experiments while ensuring the validity of the results. For the 12 microscopic parameters, three levels were selected per parameter for the experimental design. The specific factors and levels are presented in

Table 3. Levels of each factor.

Number	${\overline{\mathit{E}}}_{\mathit{c}}$ (GPa)	Ec/ ${\overline{\mathit{E}}}_{\mathit{c}}$	k	${\overline{\mathit{\sigma }}}_{\mathit{t}}$ (MPa)	$\overline{\mathit{c}}/{\overline{\mathit{\sigma }}}_{\mathit{t}}$	μ	$\overline{\mathit{\varphi }}$	$\overline{\mathit{\beta }}$	Rf	Rσ	RE	Rk
1	75	0.2	2	50	1.5	0.5	30	0.3	0.3	0.05	0.1	1
2	125	0.3	3	100	2.0	0.7	40	0.5	0.5	0.1	0.2	2
3	157	0.4	4	150	2.5	0.9	50	0.7	0.7	0.15	0.3	3

. Based on these factor levels, an L₂₇ (3¹²) orthogonal array design was utilized to generate 27 sets of experimental schemes, detailed in

Table 4. Orthogonal experimental design.

Number	${\overline{\mathit{E}}}_{\mathit{c}}$ (GPa)	Ec/ ${\overline{\mathit{E}}}_{\mathit{c}}$	k	${\overline{\mathit{\sigma }}}_{\mathit{t}}$ (MPa)	$\overline{\mathit{c}}/{\overline{\mathit{\sigma }}}_{\mathit{t}}$	μ	$\overline{\mathit{\varphi }}$	$\overline{\mathit{\beta }}$	Rf	Rσ	RE	Rk
1	75	0.2	2	50	1.5	0.5	30	0.3	0.3	0.05	0.1	1
2	75	0.2	2	50	2	0.7	40	0.5	0.5	0.1	0.2	2
3	75	0.2	2	50	2.5	0.9	50	0.7	0.7	0.15	0.3	3
4	75	0.3	3	100	1.5	0.5	30	0.5	0.5	0.15	0.3	3
5	75	0.3	3	100	2	0.7	40	0.7	0.7	0.05	0.1	1
6	75	0.3	3	100	2.5	0.9	50	0.3	0.3	0.1	0.2	2
7	75	0.4	4	150	1.5	0.5	30	0.7	0.7	0.1	0.2	2
8	75	0.4	4	150	2	0.7	40	0.3	0.3	0.15	0.3	3
9	75	0.4	4	150	2.5	0.9	50	0.5	0.5	0.05	0.1	1
10	125	0.2	3	150	1.5	0.7	50	0.3	0.5	0.05	0.2	3
11	125	0.2	3	150	2	0.9	30	0.5	0.7	0.1	0.3	1
12	125	0.2	3	150	2.5	0.5	40	0.7	0.3	0.15	0.1	2
13	125	0.3	4	50	1.5	0.7	50	0.5	0.7	0.15	0.1	2
14	125	0.3	4	50	2	0.9	30	0.7	0.3	0.05	0.2	3
15	125	0.3	4	50	2.5	0.5	40	0.3	0.5	0.1	0.3	1
16	125	0.4	2	100	1.5	0.7	50	0.7	0.3	0.1	0.3	1
17	125	0.4	2	100	2	0.9	30	0.3	0.5	0.15	0.1	2
18	125	0.4	2	100	2.5	0.5	40	0.5	0.7	0.05	0.2	3
19	175	0.2	4	100	1.5	0.9	40	0.3	0.7	0.05	0.3	2
20	175	0.2	4	100	2	0.5	50	0.5	0.3	0.1	0.1	3
21	175	0.2	4	100	2.5	0.7	30	0.7	0.5	0.15	0.2	1
22	175	0.3	2	150	1.5	0.9	40	0.5	0.3	0.15	0.2	1
23	175	0.3	2	150	2	0.5	50	0.7	0.5	0.05	0.3	2
24	175	0.3	2	150	2.5	0.7	30	0.3	0.7	0.1	0.1	3
25	175	0.4	3	50	1.5	0.9	40	0.7	0.5	0.1	0.1	3
26	175	0.4	3	50	2	0.5	50	0.3	0.7	0.15	0.2	1
27	175	0.4	3	50	2.5	0.7	30	0.5	0.3	0.05	0.3	2

.

2.4. Establishment of the PFC^3D

Using PFC^3D 6.0 software, a three-dimensional discrete element numerical model of sandstone particles was created. This model featured a standard cylindrical specimen with a diameter of 50 mm and a height of 100 mm. The particle size ranged from 0.8 to 1.3 mm, and the model had a porosity of 0.35, consisting of 30,776 particles. Rigid loading plates were installed at both the top and bottom of the model to apply loads and record the stress–strain curve, as illustrated in Figure 2.

According to the orthogonal experimental scheme, the inter-particle contact characteristic parameters were adjusted to sequentially conduct uniaxial compression and uniaxial tension numerical simulations, with the corresponding stress–strain curves recorded. Based on the simulation data, the macroscopic mechanical parameters of the sandstone specimen were determined as follows: σ_t and σ_c were taken as the peak stress values from the stress–strain curves in the tensile and compression tests, respectively, as shown in Figure 3a,b. The value of E was determined by the slope of the stress–strain curve in the elastic stage.

$$E={\displaystyle \frac{{\sigma }_1-{\sigma }_2}{{\epsilon }_1-{\epsilon }_2}}$$

(1)

In this context, σ₁, σ₂, ε₁, and ε₂ denote the stress and strain values at the two points during the elastic stage. v is the ratio of lateral strain to axial strain under uniaxial compression conditions.

$$\nu =-{\displaystyle \frac{{\epsilon }_1}{{\epsilon }_2}}$$

(2)

In the equation, ε₁ and ε₂ represent the lateral and axial strains of the specimen, respectively. The internal friction angle φ and cohesion c are determined by fitting the Mohr–Coulomb criterion to the triaxial compression test data.

3. Results and Discussion

3.1. Orthogonal Experiment Results

Based on the orthogonal experimental design, 27 sets of numerical simulation analyses were conducted with the improved PBM. Through simulations of uniaxial compression, uniaxial tension, and triaxial compression, seven macroscopic parameters corresponding to each curve were extracted. These parameters included the elastic modulus (E), Poisson’s ratio (v), uniaxial compressive strength (σ_t), internal friction angle (φ), cohesion (c), ratio of crack damage stress to uniaxial compressive strength (σ_cd/σ_c), and compressive–tensile strength ratio (σ_c/σ_t). The results of the numerical simulation are presented in

Table 5. Results of orthogonal numerical experiments.

Number	E (GPa)	ν	σc (MPa)	φ (°)	c (MPa)	σcd/σc	σc/σt
1	26.209	0.174	43.029	37.787	5.782	0.464	7.759
2	24.737	0.188	42.573	38.764	5.482	0.546	8.900
3	23.265	0.202	42.118	39.741	5.183	0.627	10.041
4	23.911	0.211	70.244	37.125	8.413	0.464	7.591
5	21.534	0.237	16.725	34.858	4.571	0.546	9.685
6	25.798	0.182	113.573	44.287	9.364	0.294	10.818
7	20.708	0.260	44.396	33.218	7.501	0.465	8.376
8	24.972	0.205	141.244	42.647	12.294	0.212	9.509
9	22.595	0.231	87.725	40.380	8.453	0.294	11.603
10	27.209	0.219	49.118	41.004	5.363	0.494	14.314
11	26.721	0.210	65.563	39.574	7.312	0.441	10.313
12	28.494	0.176	87.352	38.876	9.682	0.315	5.773
13	24.064	0.236	13.204	36.737	3.433	0.626	11.628
14	28.670	0.197	45.680	39.365	5.583	0.445	7.748
15	28.336	0.197	66.102	41.237	6.908	0.450	10.169
16	32.396	0.156	35.697	40.655	4.531	0.540	8.811
17	29.663	0.178	112.414	43.864	8.957	0.255	7.564
18	27.347	0.197	45.970	38.237	5.638	0.439	10.530
19	29.731	0.227	22.151	41.117	3.081	0.591	14.977
20	31.504	0.193	43.940	40.419	5.452	0.464	10.438
21	31.016	0.185	60.385	38.989	7.400	0.412	6.436
22	35.983	0.148	83.914	44.663	7.116	0.366	8.175
23	33.668	0.168	17.470	39.035	3.797	0.549	11.141
24	30.935	0.190	94.187	42.244	8.223	0.265	9.894
25	31.765	0.190	15.068	39.152	2.635	0.540	9.131
26	31.431	0.190	35.490	41.024	3.959	0.546	11.552
27	36.037	0.152	67.966	43.651	6.110	0.365	7.672

.

The results indicate that the configuration of different microscopic parameters significantly impacts the macroscopic performance of the sandstone model. The elastic modulus (E) ranges from 21.534 GPa to 36.037 GPa, the uniaxial compressive strength (σ_c) varies from 13.204 MPa to 141.244 MPa, the internal friction angle (φ) spans from 33.218°to 44.663°, and the cohesion (c) ranges from 2.635 MPa to 12.294 MPa. The compressive–tensile strength ratio (σ_c/σ_t) fluctuates between 5.773 and 14.977, demonstrating that the improved PBM model effectively simulates the compressive–tensile strength ratio characteristics of sandstone, thereby resolving the issue of the low compressive–tensile strength ratio in the traditional PBM model.

3.2. Macro–Mesoscopic Parameter Correlation Analysis

To quantify the linear relationship between microscopic parameters and macroscopic parameters, the Pearson correlation coefficient was utilized for analysis, as illustrated in Equation (3) and the corresponding heatmap in Figure 4.

$$r={\displaystyle \frac{{\displaystyle \sum _{i=1}^n(x_i-\overline{x}})(y_i-\overline{y})}{\sqrt{{\displaystyle \sum _{i=1}^n{(x_i-\overline{x})}^2}}\sqrt{{\displaystyle \sum _{i=1}^n{(y_i-\overline{y})}^2}}}}$$

(3)

where x_i and y_i represent the values for the i group of microscopic parameters and macroscopic parameters, respectively; $\overline{x}$ and $\overline{y}$ denote the mean values of the microscopic parameters and macroscopic parameters, respectively; n signifies the sample size. The correlation coefficient, r, ranges from −1 to 1, with |r| ≥ 0.7 indicating a strong correlation, 0.5 ≤ |r| < 0.7 indicating a moderate correlation, 0.3 ≤ |r| < 0.5 indicating a weak correlation, and |r| < 0.3 indicating essentially no correlation.

As shown in the figure, the effects of the various microscopic parameters on the macroscopic parameters are as follows:

(1) The bonding effective modulus (r = 0.876) exerts the most significant influence on the elastic modulus E, followed by the packing ratio R_f (r = −0.416). The elastic modulus E increases with the rise in the bonding effective modulus but decreases as R_f increases. This indicates that variations in the bonding effective modulus directly affect the sample’s stiffness, whereas R_f impacts stiffness by modifying the ratio of strong to weak inter-particle contacts. The other microscopic parameters with an |r| less than 0.3 have a negligible effect on E and are omitted from further discussion.

(2) The impact of various microscopic parameters on Poisson’s ratio v, from greatest to least, is as follows: filler ratio R_f (r = 0.612), tangential stiffness ratio k (r = 0.549), and effective bond modulus (r = −0.484). Poisson’s ratio v increases with increases in R_f and k but decreases with the increase in the effective bond modulus. This indicates that changes in R_f directly affect the sample’s deformation capacity. An increase in k and the effective bond modulus enhances the inter-particle bonding stiffness, which suppresses relative particle movement, thereby limiting deformation and resulting in a decrease or increase in Poisson’s ratio.

(3) The uniaxial compressive strength σ_c is primarily influenced by the moment distribution coefficient $\overline{\beta }$ (r = −0.477), the packing ratio R_f (r = −0.434), the tensile strength ${\overline{\sigma }}_t$(r = 0.414), $\overline{c}/{\overline{\sigma }}_t$ (r = 0.386), and the strength ratio R_σ (r = 0.342). The negative correlation between $\overline{\beta }$ and R_f indicates that an increase in either leads to a decrease in σ_c. $\overline{\beta }$ affects the overall stability of the specimen through uneven moment distribution, while R_f reduces the compressive performance by increasing the proportion of weak contact. Changes in ${\overline{\sigma }}_t$, $\overline{c}/{\overline{\sigma }}_t$, and R_σ impact the inter-particle bond strength and homogeneity, thereby influencing the specimen’s compressive performance.

(4) The distribution coefficient $\overline{\beta }$ (r = −0.597) exerts the most significant influence on the internal friction angle φ, followed by the packing ratio R_f (r = −0.495), the friction coefficient μ (r = 0.408), and the effective bond modulus ${\overline{E}}_c$ (r = 0.342). φ increases with the rise in μ but decreases with the increase in $\overline{\beta }$ and R_f. This indicates that variations in μ directly impact inter-particle friction, whereas changes in $\overline{\beta }$ and R_f affect the effective inter-particle contact, thereby influencing friction.

(5) The factors influencing cohesion c, ranked from most to least significant, are tensile strength ${\overline{\sigma }}_t$ (r = 0.502), effective bond modulus ${\overline{E}}_c$ (r = −0.415), cohesion/tensile strength $\overline{c}/{\overline{\sigma }}_t$ (r = 0.386), packing ratio R_f (r = −0.368), strength ratio R_σ (r = 0.364), and friction angle $\overline{\varphi }$ (r = −0.341). R_σ is positively correlated with c, indicating that an increase in these factors enhances inter-particle bonding, thereby increasing c. Conversely, R_f and $\overline{\varphi }$ are negatively correlated with c, suggesting that a decrease in the effective inter-particle contact reduces c. Generally, an increase in bond strength enhances inter-particle bonding, but excessively strong bonding restricts particle movement, thereby increasing the specimen’s deformation resistance and leading to a decrease in c.

(6) The critical deviatoric stress ratio is primarily influenced by the tensile strength ${\overline{\sigma }}_t$(r = −0.488), packing ratio R_f (r = 0.453), the ratio of cohesion to tensile strength $\overline{c}/{\overline{\sigma }}_t$ (r = −0.439), the moment distribution coefficient $\overline{\beta }$ (r = 0.365), and the friction angle $\overline{\varphi }$ (r = 0.361). This ratio decreases as ${\overline{\sigma }}_t$ and $\overline{c}/{\overline{\sigma }}_t$ increase and rises with the increase in R_f, $\overline{\beta }$, and $\overline{\varphi }$. This suggests that variations in ${\overline{\sigma }}_t$ and $\overline{c}/{\overline{\sigma }}_t$ directly impact the expansion of microcracks, whereas R_f, $\overline{\beta }$, and $\overline{\varphi }$ limit crack propagation by decreasing particle contact and enhancing friction, thereby elevating the ratio.

(7) The friction angle $\overline{\varphi }$ (r = 0.569) exerts the most significant influence on σ_c/σ_t, followed by the moment distribution coefficient $\overline{\beta }$ (r = −0.486), the strength ratio R_σ (r = −0.435), and the packing ratio R_f (r = 0.417). The positive correlation between $\overline{\varphi }$ and R_f indicates that increasing both parameters elevates the compression–tension ratio. $\overline{\varphi }$ impacts this ratio by modifying the inter-particle friction force, whereas R_f affects it by enhancing the material’s deformation capacity. Conversely, the negative correlation between $\overline{\beta }$ and R_σ suggests that their increase leads to a reduction in the compression–tension ratio. An increase in $\overline{\beta }$ causes local weakening in the sample, while a rise in R_σ results in a more uniform strength distribution, thereby decreasing the compression–tension ratio.

3.3. Macro–Mesoscopic Parameter Regression Modeling

Based on the results of the aforementioned Pearson correlation coefficient analysis, to further investigate the quantitative relationship between the microscopic parameters and macroscopic parameters, the microscopic parameters with an |r| ≥ 0.3 were identified as the primary factors influencing the macroscopic parameters and are presented in

Table 6. Main influencing factors corresponding to each macroscopic parameter.

Macroscopic Parameters	Main Influencing Factors
Elastic modulus	${\overline{E}}_c$ , Rf
Poisson’s ratio	Rf, k, ${\overline{E}}_c$
Compressive strength	$\overline{\beta }$ , Rf, ${\overline{\sigma }}_t$$,\overline{c}/{\overline{\sigma }}_t$ , Rσ
Internal friction angle	$\overline{\beta }$ , Rf, μ${\overline{E}}_c$
Cohesion	${\overline{\sigma }}_t$$,{\overline{E}}_c$$,\overline{c}/{\overline{\sigma }}_t$ , Rf, Rσ$,\overline{\varphi }$
σcd/σc	${\overline{\sigma }}_t$ , Rf, $\overline{c}/{\overline{\sigma }}_t$$,\overline{\beta },\overline{\varphi }$
Compression–tension ratio	$\overline{\varphi },\overline{\beta }$ , Rσ, Rf

.

We established a multiple linear regression model using the main influencing factors from Table 6 as the independent variables and the macroscopic parameters as the dependent variables. The regression analysis was conducted using SPSS Statistics 26.0 software, with significant variables selected through the stepwise regression method. The linear regression model was fitted based on the ordinary least squares method (OLS), resulting in the regression equations for each macroscopic parameters as follows:

$$E=0.087{\overline{E}}_c-9.5351R_f+21.9868$$

(4)

$$\nu =-0.0003{\overline{E}}_c+0.0183k+0.1011R_f+0.1252$$

(5)

$${\sigma }_c=0.3330{\overline{\sigma }}_t+32.0619\overline{c}/{\overline{\sigma }}_t-86.7826\overline{\beta }-78.4979R_f+278.3672R_{\sigma }+15.275$$

(6)

$$\varphi =0.0239{\overline{E}}_c+6.9954\mu -8.7006\overline{\beta }-7.1117R_f+39.9752$$

(7)

$$c=-0.0214{\overline{E}}_c+0.0274{\overline{\sigma }}_t+2.1228\overline{c}/{\overline{\sigma }}_t-0.0875\overline{\varphi }-4.7258R_f+20.0655R_{\sigma }+5.9245$$

(8)

$${\sigma }_{cd}/{\sigma }_c=-0.0013{\overline{\sigma }}_t-0.1209\overline{c}/{\overline{\sigma }}_t+0.0048\overline{\varphi }+0.2414\overline{\beta }+0.3008R_f+0.3587$$

(9)

$${\sigma }_c/{\sigma }_t=0.150\overline{\varphi }-5.3925\overline{\beta }+5.6370R_f-19.0706R_{\sigma }+5.4365$$

(10)

Table 7. Statistical test results of each macro parameter.

Macroscopic Parameters	R2	Variance Inflation Factor	Maximum Condition Number	Jarque–Bera Test	Durbin–Watson Test
E	0.9	108.5	924	0.658	0.983
ν	0.891	62.44	970	0.958	2.272
σc	0.819	19.0	2660	0.736	1.747
φ	0.793	21.08	1180	0.397	1.825
c	0.903	31.16	4180	0.824	2.168
σcd/σc	0.843	22.59	1080	0.892	1.506
σc/σt	0.855	32.31	1010	0.417	0.971

presents the statistical test results of the regression models for each macroscopic parameters, showing the following:

(1) The R² values of all models are greater than 0.79, with E (R² = 0.9) and c (R² = 0.903) demonstrating the best fit and relatively strong fitting effects; φ (R² = 0.793), though slightly lower, still has good explanatory capability.

(2) The multicollinearity test indicates that the maximum condition number reaches 4180 (cohesion c model), but the variance inflation factors (VIFs) for all models are below five, suggesting that although there is some degree of correlation among the independent variables, it remains within an acceptable range.

(3) The normality of the residuals is confirmed by the Jarque–Bera test (p > 0.05), with the models for Poisson’s ratio v (p = 0.958) and the critical deviatoric stress ratio σ_cd/σ_c (p = 0.892) demonstrating optimal residual normality.

(4) The autocorrelation tests show that except for models E (DW = 0.983) and σ_c/σ_t (DW = 0.971), which exhibit slight autocorrelation, the DW values for the other models range between 1.5 and 2.3, satisfying the independence assumption.

(5) It is noteworthy that although some models have a high condition number, all regression coefficients have significant p-values, and the variance inflation factors are all less than five, indicating that the model results are reliable. The regression model, established based on the selection of microscopic parameters with an |r| ≥ 0.3, can adequately explain the changes in the macroscopic parameters.

3.4. Analysis and Discussion of Results

The correlation and regression analysis results indicate significant differences in the influence of different mesoscopic parameters on the macroscopic mechanical properties. Among them, the effective bond modulus ${\overline{E}}_c$ and elastic modulus E exhibit the strongest positive correlation (r = 0.876), which suggests that the bond stiffness between particles is a key factor in determining the overall deformation stiffness of the material. The negative correlation between the filler ratio R_f and E reflects that an increase in the proportion of weak contacts weakens the overall continuity and stiffness of the structure, thereby reducing its deformation resistance.

Regarding Poisson’s ratio ν, the positive correlation between the filler ratio R_f and the bond normal stiffness ratio k indicates that a higher proportion of weak contacts and stronger normal resistance help suppress particle sliding, leading to more significant lateral deformation. In contrast, a larger ${\overline{E}}_c$ limits particle micro-motion, resulting in a decrease in Poisson’s ratio.

The uniaxial compressive strength σ_c is influenced by multiple factors, with the negative correlation between the moment distribution coefficient $\overline{\beta }$ and R_f being particularly prominent. This suggests that unreasonable moment transfer and a high proportion of weak contacts may reduce the overall load-bearing capacity between particles. The positive correlation between the tensile strength σ_t and strength ratio R_σ further confirms that high strength and a uniform structure contribute to improving compressive performance.

The strong positive correlation (r = 0.569) between the compressive–tensile strength ratio σ_c/σ_t and the friction angle φ reveals the decisive role of frictional resistance in determining the failure mode of the structure. An increase in frictional force suppresses crack propagation, enhancing the overall compressive strength of the structure. The negative correlation between the moment distribution coefficient $\overline{\beta }$ and R_σ also suggests that mechanical heterogeneity within the structure can reduce the compressive–tensile strength ratio.

This study established logical relationships between mesoscopic parameters and macroscopic responses through orthogonal design and quantitative analysis methods, providing theoretical support for parameter inversion in rock particle flow modeling. The statistical test results of the regression models indicate a high fitting accuracy, with R² values exceeding 0.79, meeting the needs for engineering prediction and analysis. Compared with traditional empirical calibration methods, the process proposed in this paper is systematic and repeatable, providing a more reliable parameter basis for numerical rock simulations.

4. Instance Verification

4.1. Laboratory Test

To verify the accuracy and applicability of the improved PBM, a Longhu coal mine sandstone rock sample was selected for compression and tensile strength tests. The testing machine used was a TYJ-2000KN microcomputer-controlled electro-hydraulic servo rock triaxial pressure machine from the Key Laboratory of Ground Pressure Control and Gas Treatment for Deep Mining of Heilongjiang Coal Mines. The experimental equipment and the physical diagram of the rock sample are shown in Figure 5.

Uniaxial compression, tensile, and shear tests were performed on the rock sample, resulting in the corresponding stress–strain curve (see Figure 6). By analyzing and fitting the curve data, the mechanical parameters of the rock sample were further calculated, with specific data presented in

Table 8. Test results of mechanical parameters of sandstone.

Macroscopic Parameters	E (GPa)	ν	σc (MPa)	φ (°)	c(MPa)	σcd/σc	σc/σt
Experimental values	32.2	0.25	54.5	43	3.92	0.53	12.1

.

4.2. Numerical Simulation

To ensure that the compression–tension ratio in the simulation aligned more closely with the actual rock conditions, a compression–tension ratio σ_c/σ_t of 12.1, measured in the laboratory, was directly used as the basis for calibrating the microscopic parameters. The specific calibration steps for the microscopic parameters were as follows:

(1) The results of the aforementioned correlation analysis clearly show that E_c/${\overline{E}}_c$, R_E, μ, k, and R_k have almost no impact on the macroscopic mechanical response values. Based on the value of σ_c/σ_t and using the 13th set of data from the orthogonal numerical test results in Table 5 as a reference, the preliminary values were set as E_c/${\overline{E}}_c$ = 0.3, R_E = 0.1, μ = 0.7, k = 4, R_k = 2, and $\overline{\beta }$ = 0.5, R_f = 0.7.

(2) By substituting the known experimental values (see Table 8) and the microscopic parameters values into Equations (5)–(10), the values of other microscopic parameters are obtained as ${\overline{E}}_c$= 194, k = 6, ${\overline{\sigma }}_t$ = 111, $\overline{c}/{\overline{\sigma }}_t$ = 2, μ = 1.0, $\overline{\varphi }$ = 50, R_σ = 0.11.

(3) We conducted preliminary PFC simulations using known microscopic parameters. We obtain the corresponding simulation values through uniaxial compression, uniaxial tension, and triaxial compression tests. We compared these values with the experimental values and analyzed the trend in the stress–strain curve. We further adjusted the microscopic parameters to minimize errors. The final values of the microscopic parameters are shown in

Table 9. Microscopic parameter values of sandstone.

Microscopic Parameters	${\overline{\mathit{E}}}_{\mathit{c}}$ (GPa)	Ec/ ${\overline{\mathit{E}}}_{\mathit{c}}$	k	${\overline{\mathit{\sigma }}}_{\mathit{t}}$	$\overline{\mathit{c}}/{\overline{\mathit{\sigma }}}_{\mathit{t}}$	μ	$\overline{\mathit{\varphi }}$	$\overline{\mathit{\beta }}$	Rf	Rσ	RE	Rk
Value	200	0.3	4.0	115	2	1.0	60	0.5	0.7	0.2	0.2	2

.

4.3. Verification of Results

Based on the values of the aforementioned microscopic parameters, uniaxial compression, tensile, and triaxial compression numerical simulations were conducted with the numerical model. The obtained simulation results were compared and verified against the laboratory test results, as follows: Figure 7 shows the failure mode of the rock sample under uniaxial compression. It can be observed that the crack propagation in the uniaxial compression simulation was generally consistent with the splitting pattern of the experimental rock sample. The stress–strain curve (Figure 8) obtained from the numerical simulation and the laboratory test aligned well for the elastic stage, peak strength, and residual strength, verifying the rationality of the microscopic parameters. In conclusion, after applying the mesoscopic parameters obtained through regression model inversion to PFC numerical simulations, the resulting stress–strain curves closely matched the laboratory test results for three critical mechanical properties: elastic modulus, peak strength, and residual strength. The error between the simulation values and experimental values was controlled within 5% (see

Table 10. Comparison of macroscopic parameter values of sandstone.

Macroscopic Parameters	E (GPa)	ν	σc (MPa)	φ (°)	c(MPa)	σcd/σc	σc/σt
Experimental values	32.2	0.25	54.5	43	3.92	0.53	12.1
Simulated values	33.4	0.24	54.6	43.7	3.89	0.53	12.7
Percentage error (%)	3.7	4	0.2	1.6	0.7	0	4.9

), and the compressive–tensile strength ratio was significantly improved from the traditional PBM range of 5–8 to 12.7, which is close to the measured value of 12.1. The simulated failure modes also aligned with the experimental observations, showing that cracks propagated along the shear plane and exhibit tensile failure characteristics. This fully verifies the reliability and applicability of the mesoscopic parameter regression model established in this study for mechanical modeling of sandstone. This method not only enhances the scientific accuracy of parameter calibration but also provides a transferable technical pathway for the numerical simulation of similar rock materials in the future.

5. Conclusions

This study addressed the issue of the low compressive–tensile strength ratio in the traditional parallel bond model (PBM) when simulating rock materials. The PBM model was improved by introducing a strong–weak contact group parameter configuration strategy to enhance its adaptability to the mechanical behavior of sandstone. Based on this improvement, orthogonal experimental design and regression analysis methods were used to systematically study the effects of 12 mesoscopic parameters on seven macroscopic mechanical properties. A quantitative relationship model between the macro and micro parameters was established, and the model’s applicability and accuracy were verified through laboratory experiments.

The main contributions and value of this research are reflected in the following aspects: (1) The introduction of strong–weak contact groups optimized the particle contact parameters, thereby improving the compressive–tensile strength ratio and providing a reference for improving the PBM model; (2) Through orthogonal design and correlation analysis, we identified the mesoscopic parameters that dominantly influence macroscopic responses, thereby establishing a foundation for model parameter selection and optimization; (3) a multiple regression model was constructed to enable the inverse prediction of macroscopic performance and mesoscopic parameter calibration, improving modeling efficiency and reliability to a certain extent, with practical engineering application value.

The verification results showed that the model’s predictions for key parameters, such as the elastic modulus, compressive strength, and Poisson’s ratio, exhibited errors controlled within 5% compared to the laboratory test results. The simulated compressive–tensile strength ratio was close to the measured value, and the failure mode was also consistent. The research findings provide a reference for future parameter inversion, discrete element modeling, and numerical simulation studies of similar rock materials.