# Simulation of the Fracturing Process of Inclusions Embedded in Rock Matrix under Compression
## 2 Fracturing Behaviors of Single Inclusion Embedded in Rock Matrix
### Failure Analysis of Single Inclusion under Uniaxial Compression  
In this section, a numerical model similar to Figure 2 is employed to investigate the fracturing behavior in a single inclusion subjected to uniaxial compression. The homogeneity index \(m\) of inclusion is 3.0, representing the assumption that the inclusion is a heterogeneous material. The Poisson ratio of the inclusion is 0.25 and the Poisson ratio of the matrix is 0.45. All other mechanical parameters are the same as those listed in Table 1. The fracture-induced AE energy accumulation under compression is shown in Figure 5.  
Figure 6 shows the sequence of fracture infilling during the failure of an inclusion modeled in Figure 2. Figure 6a shows the acoustic emission (AE) distribution with red and white circles indicating tensile and compressive failure of the elements at the current loading step, while the black circles are used to indicate the damaged elements at previous loading steps. The AE events correlate well with the number of failed elements. The failure process is accompanied by a rapid increase in the AE event count [32; 33; 34]. Figure 6b shows the evolution of the fracture and maximum principal stress field. The grey scale indicates the relative magnitude of the stress within the elements. The elements with a lighter shade of grey have relatively higher stresses. The dark elements in Figure 6b represent the nucleated flaw. Fractures form through the connection of flaws. According to the sequence of the fractures in Figure 6, the failure process of inclusion can be divided into three stages. In the first stage, the initial flaws are obviously initiated at the interface between the inclusion and matrix. In step 49, the macroscopic fracture starts at the interface, and several isolated fractures initiate randomly at the location inside the inclusion where the local tensile stress reaches its local tensile strength depending on the high heterogeneity of the inclusion. The AE events increase slightly because only a small number of elements with lower strength meet the failure criterion. In the second stage, from step 50 to step 80, the isolated fractures at the interface soon coalesce and propagate to form multiple sub-parallel fractures. There are sudden and rapid increases in the number of failure events. Each sharp increase in the number of AE events corresponds to the formation of a new macroscopic fracture. Almost all the fractures show tortuous paths which depend on the distribution of the heterogeneity in the material. As new fractures are continuously infilled between the earlier formed fractures, the spacing between adjacent fractures decreases gradually. At the last stage, as shown in step 100, the spacing between adjacent fractures is so closely that no more new fractures are infilled. In this stage, the AE events generally remain small and stable until new fractures nucleate and propagate. Figure 5: Fracture-event counts and AE-released energy accumulation versus remote compressive loading. consecutive fracture. The macroscopic consecutive fracture completely cut through the inclusion. The non-consecutive fractures include the fractures initiated from the interface and the fractures initiated inside the inclusion. These non-consecutive fractures did not completely pass through the inclusion body. Parts of the non-consecutive fractures can develop into consecutive fractures. The debonding fracture is the cracks formed around the interface. Although the interface between the inclusion and matrix is assumed to be perfectly bonded, the delamination effect may occur for the obvious change of the local stress near the inclusion boundary [23].