## 4 Numerical Simulation
### Model Configuration  
A numerical model was developed to simulate uniaxial compression on sandstone-like specimens with different LRs and HDRs. The purpose is to observe crack distribution and changes in the damage process of the specimen, and then reveal the cause and mechanism of the SE and LR effect. The Particle Flow Code (PFC) is based on a discrete cell  
Figure 8: Mechanical parameter trends with same HDR at different LRs: (**a**) UCS; (**b**) axial peak strain.  
approach, which treats rock-like materials as an aggregate of rigid particles. This model overcomes the limitations of traditional continuous media analysis methods in the study of rock-like inhomogeneous materials and allows for the spatial and temporal evolution of crack incubation, extension and penetration to be accurately reproduced from a fine-scale perspective [49; 50; 51].  
Rocks are composed of a collection of mineral particles, and the parallel bonding model (PBM) is mostly used in numerical simulations of rock materials. The PBM is capable of transferring both forces and moments, which is more realistic. PBM can be seen as consisting of linear spring set and a parallel bonded spring set, as shown in Figure 9. And in the model, a rock can be interpreted as a combination of discs of finite thickness, which are interconnected by bonding bonds that possess finite stiffness at both the contact points between the discs and in the bonded area. When the applied local stress (i.e., tension, shear or rotational moment) exceeds the specified bond strength, these bonds break to form a microcrack, and the code of destruction is subject to the Mohr-Coulomb theory.  
The PFC\({}^{\text{2D}}\) 6.0 software is based on [PERSON]'s second law and the law of forces and displacements, which relate the forces and displacements of two microelements in contact with each other [52]. The PFC\({}^{\text{2D}}\) software makes the following basic assumptions in its calculations: (1) the particles are rigid; (2) contact occurs over a small area, i.e., point contact; (3) the contact is flexible and a certain amount of \"overlap\" is allowed at the contact; (4) the amount of \"overlap\" is related to the contact force; (5) the contact has a special joint strength; (6) the particle units are represented as discs.  
The PFC\({}^{\text{2D}}\) software explains the damage fracture mechanism of materials from a fine-grained mechanical point of view by recording the cracks that develop during the fracture damage of bonding bonds through the embedded fish language. This paper adopts the recommendation of [PERSON] et al. [53; 54] for the calibration of the fine view  
Figure 9: Parallel bonding model (PBM) and failure criterion.  
parameters, assuming that the particles and the adhesive bonds have common deformation properties as Formula (5), with the maximum to minimum particle size ratio maintained at a constant value of 1.66. To mitigate the influence of end-face friction on the experimental results, we set the contact surface between the loading plate and the specimen as frictionless prior to the numerical simulation experiment.  
\[\left\{\begin{array}{l}Ec=\overline{Ec}\\ kn/ks=\overline{kn}/\overline{ks}\end{array}\right. \tag{5}\]