# Fracture Evolution and Mechanical Properties of Mortar Containing Different Inclusions
## 4 Discussion  
The influence of the inclusion on the strength of the sample mainly depends on its influence on the stress distribution around the hole. The stress solution in the area around the rigid inclusion (\(E=\infty\)) in an infinite field has been provided in the literature [17], which indicated that a rigid backfilled inclusion changes the stress concentration around  
Figure 13: Maximum principal strain vs. time curves of samples. the inclusion and thus changes the rock's stability and failure behavior. However, in practical engineering, the inclusion cannot be regarded as rigid material in most cases, where deformation modules of the inclusion are various and may have different influences on the surrounding materials from case to case. To further determine the influence of inclusion elasticity on the stress distribution around the inclusion, the stress distribution characteristics around the hole-containing incursion with varying elastic moduli are discussed. Figure 14 demonstrates the stress curves of the top and side corners of a unit circular hole in an infinite plate subject to a vertical far field uniaxial, compressive stress \(\sigma_{0}\) with different \(k_{e}\) (\(k_{e}\) refers to the ratio of the elastic modulus of the inclusion to that of the external material). Figure 14 presents the stress curves for both numerical and analytical solutions in the absence of an inclusion. The numerical solution is obtained through finite element numerical methods, while the analytical solution is derived using complex variable function methods [33; 34]. Figure 14,c depicts scenarios where the elastic modulus of the inclusion is less than (\(k_{e}<1\)) and greater than (\(k_{e}>1\)) the elastic modulus of the external material, respectively. \(\sigma_{\theta}\) and \(\sigma_{r}\) refer to the hoop stress and radial stress within the plane, respectively. It can be seen from Figure 14 that for the plate containing a single circular hole, a strong compressive stress concentration can be observed in the middle point of the hole sides (\(\sigma_{\theta}=3\sigma_{0}\)), while tensile stress concentration was formed in the top and bottom of the hole (\(\sigma_{\theta}=-\sigma_{0}\)). Hence, the bottom and top corners experienced a tensile crack, while the side corners of the opening experienced splitting damage due to the compression stress concentration [35; 36; 37]. However, the existence of the inclusion will change the stress distribution around the inclusion and thus change its failure behavior. As shown in Figure 14, when \(k_{e}\) was less than \(1.0\), the stress concentration in both the side and top points decreased with the increase in \(k_{e}\), indicating the support from the inclusion is improving. When \(k_{e}\) was greater than \(1.0\) (Figure 14), \(\sigma_{\theta}\) in the top point was transferred to compressive stress and increased slightly with the increase in \(k_{e}\). Meanwhile, as for the middle point in the side, \(\sigma_{\theta}\) in the form of compressive stress decreased, but \(\sigma_{r}\) tended to have tensile stress under a low level near the middle side point along the horizontal line. Compared with Figure 8, it can be seen that for the samples with a strong inclusion, an initial strain concentration appeared around the inclusion sides but failed to form macro-cracks. According to the maximum principal strain versus time curves and maximum principal strain contours, the crack propagation process can be divided into four stages with an increase in the loading time as shown in Figure 15.  
Figure 14: Stress distribution around an opening or inclusion. Stage 1--Quiet stage: Strain concentration has not yet appeared, and the principal strain remains at a very low level. Stage 2--Formation Stage of the Stress Concentration Area: The principal strain suddenly increases. Strain concentration begins to manifest at the location predisposed to a crack formation.