## 4 Investigation of Influence of Model Size and Joint Dip Angle on Rock Brittleness
### Analysis of Brittleness Characteristics  
This study quantitatively analyzed the influence of joint dip angle and model size on the brittleness features of the specimens using Equation (15). Figure 8a shows changes in the brittleness index of the specimens concerning the model size. The brittleness index displays a monotonic increasing pattern as the model size increases when the joint dip angle is constant at 0\({}^{\circ}\). As the dip angle of the joints increases to 30\({}^{\circ}\), the specimen gradually shifts from plasticity to brittleness with the growth in model size. Similarly, as for the joint dip angle of 60\({}^{\circ}\), the brittleness of the specimen maintains an upward tendency with the model size. When the joint dip angle reaches 90\({}^{\circ}\), the brittleness index rises from 3.74 to 4.45 with the increasing model size. Among these four joint dip angles, the least significant impact of model size on the specimen brittleness is found in the case where the joint dip angle is 60\({}^{\circ}\).  
Figure 8: Effects of model size (**a**) and joint dip angle (**b**) on brittleness characteristics of jointed specimens.  
The impact of joint dip angle on the rock brittleness properties was quantitatively discussed using Equation (15), as illustrated in Figure 8b. For a model size of 40 mm, the brittleness index initially drops and then increases with the increasing joint dip angle. For a model size of 80 mm, there is no clear correlation between the specimen brittleness and the joint dip angle, which could result from the inherent discreteness of the specimen material. The brittleness of the jointed specimen with a model size of 120 mm initially declines and then grows with the increasing joint dip angle, while it decreases for a model size of 160 mm. For a model size of 200 mm, the brittleness index initially decreases and then increases as the joint dip angle grows. Among the mentioned sizes, the effect of the joint dip angle on the brittleness of the specimens is relatively insignificant for a model size of 160 mm. Furthermore, the brittleness index of the specimens has been observed to decrease at joint dip angles of 30\({}^{\circ}\) or 60\({}^{\circ}\), which may be attributed to the impact of the initial damage on the jointed rock mass at these angles, as demonstrated by [PERSON] et al. [23].  
Figures 9 and 10 illustrate the relationships between the two fitting parameters (m and F) and the model size (L) and joint dip angle (\(\alpha\)). The parameter \(m\) generally increases with the model size, while it shows a trend of first increasing and then declining as the joint dip angle grows. Previous research has demonstrated that the parameter \(m\) refers to a shape parameter related to the shape of the post-peak stress-strain curve [15]. The larger the value of m, the more brittle the material. This result aligns with the overall results of the proposed index and offers further confirmation supporting the proposed index's soundness. In contrast, the parameter \(F\) tends to decrease as the model size increases and there is no clear relationship with the joint dip angle.  
As depicted in Figure 11, the brittleness distribution of jointed rock masses was plotted using an interpolation method for various combinations of model sizes (ranging from 40 mm to 200 mm) and joint dip angles (ranging from 0\({}^{\circ}\) to 90\({}^{\circ}\)). It can be observed that the rock brittleness index increases with the increase in model size, and the brittleness index shows an approximate V-shaped variation with the increase in joint dip angle. In fact, the brittleness index of rock mass is very complex, which is not only related to the mechanical properties of the rock itself, but also to the geometric distribution of joints in the rock mass and the size of the model. Additionally, the results show that large-sized specimens with horizontal or vertical joints are more prone to brittle damage.  
Figure 9: The relationship between the fitting parameter (m) and model size of the jointed specimens.