# Simulation of the Fracturing Process of Inclusions Embedded in Rock Matrix under Compression
## 4 Conclusions  
In this study, the RFPA code was applied to conduct a series of numerical simulations to investigate the stress state and fractures evolution in single and multiple inclusions embedded in the rock matrix. The following conclusions can be drawn:  
1. The spacing between two existing fractures directly governs the stress redistribution during fracture filling. With the decreasing of the spacing between the adjacent fractures, the stress is firstly transferred from negative to positive, then transferred from positive to negative. When the fracture spacing reaches the critical value, the tensile stress becomes so small that the magnitude of the stress is lower than the tensile strength of the inclusion. Consequently, there are no more fractures that can be infilled between the adjacent two fractures. Numerical results provide an intuitive way to see the fracturing process and fracture-induced stress re-distribution of inclusions embedded in the rock matrix, which cannot be observed directly in the field.
2. Three types of fractures may occur in either single or multiple inclusions, i.e., consecutive, non-consecutive, and debonding fractures. However, the path of macroscopic fractures may be flexuous with small flaws such as branches because of the heterogeneity of material. Meanwhile, although the effect of heterogeneity on the stress inside inclusion is negligible, the impact of heterogeneity on fracturing mode is not negligible. Fractures typically initiate at local concentrations of tensile stress around flaws. Since flaws produce greater stress concentrations, the location of fracture initiation depends on the distribution of the flaws as well as the magnitude of maximum principal tension stress.
3. The ratio \(E_{i}/E_{r}\) affects the fracturing process of inclusions significantly. It is found that, when \(E_{i}/E_{r}\) ratio is lower than 1.0, the internal fractures even the matrix failure are always the common mode, and few debonding fractures are found. Only a few numbers of consecutive and non-consecutive fractures are formed in some inclusions, even though the compressive load is relatively high. When the \(E_{i}/E_{r}\) ratio is higher than 1.0, mixed fracturing modes are found. With the increasing \(E_{i}/E_{r}\) ratio, a larger number of consecutive and non-consecutive fractures sub-parallel fractures are easily formed with a relatively lower compressive load.
4. The high inclusion content will enhance the disturbing and interaction between inclusions. Therefore, more scattered flaws can be found at the initial loading stage of the model containing multiple inclusions. Nevertheless, the inclusion content has no evident impact on the final fractured mode.  
Figure 17: Numerical results of fracturing behavior with different inclusion content.  
**Author Contributions:** Data curation, C.Y.; formal analysis, N.W.; funding acquisition, C.Y.; investigation, C.Y., P.X. and X.B.; methodology, B.G.; supervision, B.G.; validation, N.W.; writing--original draft, C.Y.; writing--review and editing, B.G. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This study was financially supported by the Inner Mongolia Autonomous Region Natural Science Foundation of China, China (Grant No. 2020 BS05017) and Building Science Institute Open fund of Inner Mongolia University of Science and Technology, China (JYSJ)-2021Q06).  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** The data presented in this study are available on request from the corresponding author.  
**Conflicts of Interest:** The authors declare no conflict of interest.