## 2 Parallel-Bond Model and Its Micro-Parameters  
PFC3D simulates the mechanical behavior of continuous medium materials by defining the contact model to bond mutually independent particle units. Several studies have addressed the calibration of micro-parameters for various contact models in PFC and also analyzed the relationships between these micro-parameters and macro-properties [18; 38; 39; 40; 41; 42]. The following parameters in PBM have been shown to exert major effects on the macro-properties [22; 26; 28; 37; 38]: \(R_{\rm max}/R_{\rm min}\) (\(R_{\rm min}\) is minimum particle radius in the entire population of particles, \(R_{\rm max}\) is particle radius in the entire population of particles), \(R_{\rm min}\), \(L/d\) (\(L\) is the smallest characteristic model length, such as the diameter of a cylindrical specimen, or the smallest width of a parallelepiped specimen, \(d\) is the median particle diameter), \(k^{*}\) (bond normal-to-shear stiffness ratio, \(k^{*}=k_{n}/k_{s}\), \(k_{n}\) is normal stiffness, \(k_{s}\) is shear stiffness), \(E_{c}\) (effective modulus, \(E_{c}=k^{*}L_{R}/A\), \(L_{R}\) is the sum of particle radius on both sides of the bond, \(A=2r\) (2D) or \(A=\pi n^{2}\) (3D), \(r\) is the minimum radius of particles on both sides of the bond), \(\mu\) (grain contact friction coefficient), \(\vec{k}^{*}\) (bond normal-to-shear stiffness ratio, \(\vec{k}=\vec{k}_{n}/\vec{k}_{s}\), \(\vec{k}_{n}\) is bond normal stiffness, \(\vec{k}_{s}\) is bond shear stiffness), \(\vec{E}_{c}\) (bond effective modulus, \(\vec{E}_{c}=\vec{k}\,L_{R}\)), \(\vec{\sigma}_{n}\) (bond normal stress), \(\vec{\tau}_{n}\) (bond shear stress), and \(\lambda\) (bond width multiplier). \(\vec{R}=\lambda r\), \(\vec{R}\) is the bond radius. By setting \(\lambda\), the bond radius is enlarged or reduced. Among these parameters, \(\vec{k}^{*}\), \(\vec{\sigma}_{n}\), and \(\vec{\tau}_{n}\) have significant effects on model macro-properties. The \(\vec{k}^{*}\) value has a substantial effect on the model [PERSON]'s ratio and [PERSON]'s modulus [17; 19; 22]. With increasing \(\vec{k}^{*}\), the [PERSON]'s ratio increases, while [PERSON]'s modulus decreases [19; 22; 19; 22]. In contrast, the existing studies on the effect of \(\vec{k}^{*}\) on uniaxial compressive strength (UCS) are insufficient. Most researchers have not considered this effect or only considered it on the condition that other micro-parameters are unchanged. They have not considered coupling effects between \(\vec{k}\) and other micro-parameters on UCS, as well as the model deformation and failure characteristics [18; 19; 20; 21; 22; 38; 43]. Table 1 shows the results of the relationship between \(\vec{k}\) and UCS in different researchers' studies. Previous studies have shown that UCS of the model is also highly dependent on \(\overline{\sigma}_{n}\) and \(\overline{\tau}_{n}\), both of which have a positive effect [18; 19]. [PERSON] [21] reported that when \(\overline{\sigma}_{n}/\overline{\tau}_{n}>0.5\), the strength of the model was mainly affected by \(\overline{\tau}_{n}\), and when \(\overline{\sigma}_{n}/\overline{\tau}_{n}<0.5\), it was mainly affected by \(\overline{\sigma}_{n}\). However, changes in \(\overline{\sigma}_{n}\) and \(\overline{\tau}_{n}\) generally do not affect [PERSON]'s modulus or [PERSON]'s ratio. Earlier studies have primarily investigated the effect of a particular single micro-parameter on the macro-properties of the model, whereas the combined effects of multiple micro-parameters on the macro-properties of the model have not been taken into account. Additionally, some studies [18; 19; 20; 21; 23; 24; 25; 26; 27] set \(k^{\star}=\vec{k}\) as a default to facilitate analyses, even though these values have been shown to have considerable effects on the model macro-properties under different conditions [7; 22]. In summary, the effects of \(\vec{k}\), \(\overline{\sigma}_{n}\), and \(\overline{\tau}_{n}\) in PFC3D on model macro-properties require a more thorough investigation.