## 5 Discussion
### Mechanisms Underpinning the Effects of Micro-Parameters on Macro-Properties  
The failure envelope for the PBM of PFC3D is shown in Figure 14. PBM determines the bond fracture between particles, mainly by the following method [37]. First, the tensile strength limit was determined. If the tensile strength limit was exceeded (\(\overline{\sigma}>\overline{\sigma}_{n}\)), the bond underwent tensile fracture. If the bond did not undergo tensile fracture, the shear strength limit was determined. The shear strength was expressed as:  
\[\overline{\tau}_{c}=\overline{\tau}_{n}-\sigma\tan\overline{\varphi} \tag{3}\]  
\[\sigma=\overline{F}_{n}/\overline{A} \tag{4}\]  
where \(\overline{F}_{n}\) is the normal component of the parallel-bond contact force, and \(\overline{A}=\pi R^{2}\) is the cross-sectional area of the contact bond between particles. If the shear strength limit was exceeded (\(\overline{\tau}>\overline{\tau}_{c}\)), then the bond underwent shear fracture. Combined with Table 4, the results showed that under the conditions of different \(\overline{\sigma}_{n}/\overline{\tau}_{n}\), the effects of \(\overline{k}\) on the macro-properties could be divided into three types. \(\overline{\sigma}_{n}/\overline{\tau}_{n}<0.8\): In this case, the strength of specimens was mainly restricted by \(\overline{\sigma}_{n}\), and the inter-particle bond failure was dominated by tensile fracture. Figure 14: Failure envelope for the parallel-bond. By selecting the contact force chain, which is an option in the PFC software, the stress state of the bond between particles could be observed. Additionally, Figure 15 shows a growing number of contact force chains under tension within the specimens. Owing to the relatively low \(\overline{\sigma}_{n}\), the bond within the specimens first underwent tensile fracture because the external load exceeded \(\overline{\sigma}_{n}\), and the sample strength decreased continuously. \(0.8\leq\overline{\sigma}_{n}/\overline{\tau}_{n}<4\): When \(\overline{k}\)  was relatively low, the sample strength was mainly restricted  
As shown in Figure 2, the sample strength increased with \(\overline{\sigma}_{n}/\overline{\tau}_{n}\). However, Figure 16 shows that for \(\overline{\sigma}_{n}/\overline{\tau}_{n}>10\), the sample strength ceased to increase with increasing \(\overline{\sigma}_{n}/\overline{\tau}_{n}\). Furthermore, Figure 16 shows that an increase in \(\overline{\sigma}_{n}\) or \(\overline{\tau}_{n}\) resulted in higher strength. In the research scheme of this paper, increasing \(\overline{\sigma}_{n}/\overline{\tau}_{n}\) referred to maintaining a constant \(\overline{\tau}_{n}\) while increasing \(\overline{\sigma}_{n}\). When \(\overline{\sigma}_{n}/\overline{\tau}_{n}\) reached a certain value (that we would call N), the \(\overline{\sigma}_{n}\) was relatively high, and the bond did not undergo tensile fracture. The shear strength limit was then determined. Because \(\overline{\tau}_{n}\) did not increase, the sample strength was restricted by shear strength and, thus, ceased to increase. As demonstrated in Figures 14 and 17, the value of \(n\) was related to \(\overline{k}^{*}\) and the internal friction angle (\(\overline{\phi}\)). The N decreased with increasing \(\overline{\phi}\) and increased with increasing \(\overline{k}^{*}\) (Figure 18). This indicated that the range of the effect of \(\overline{\sigma}_{n}/\overline{\tau}_{n}\) (\(0<\overline{\sigma}_{n}/\overline{\tau}_{n}<\) N) was affected by a combination of \(\overline{\phi}\) and \(\overline{k}^{*}\). According to Figure 19, the parallel-bond model degenerated into a linear contact model after bond failure [10; 37]. In this case, the overall sample stiffness decreased rapidly, whereas the KK increased; that is, the \(k^{*}\) of the linear contact portion increased. There was a larger decrease in the overall sample stiffness, while Young's modulus decreased, and the Poisson's ratio increased. The deformation parameter of the specimens deteriorated, and their strength declined.