## 5 Analysis for SE and LR Causes  
### The Mechanical Model  
The EFE can increase the shear stress (\(\tau\)) at the end face of the specimen, limiting the lateral deformation, resulting in the specimen showing tapered damage. The mode can be simplified as the specimen is broken into 4 parts, where there is friction between the  
Figure 13: Crack extension in uniaxial compression of the specimens with different LRs. blocks (as shown in Figure 14). According to the Mohr-Coulomb theory, we can obtain the following Formula (6). \[\left\{ \begin{array}{l}\sigma_{1}=\sigma-2f\cos\theta\\ \tau=c+\sigma_{1}\tan\varphi\end{array} \right. \tag{6}\]  
where \(\sigma\) is the stress applied by the press; \(\sigma_{1}\) denotes the actual axial stress; \(\tau\) is shear stress at the end face of the specimen; \(f\) is friction generated between the blocks; \(\theta\) is the angle between conical block and vertical direction; \(c\) is the cohesion at the end face of the specimen; \(\varphi\) is the internal friction angle at the end face of the specimen. [PERSON] et al. [56] assumed that the power consumed by end-face friction satisfied the Weibull distribution, which is given by:  
\[\left\{ \begin{array}{l}W_{\tau}=2\int\tau\Delta vds\\ \Delta v=\sqrt{{v_{x}}^{2}+{v_{y}}^{2}}=\mu\frac{v_{0}}{h}\sqrt{x^{2}+y^{2}} \\ r^{2}=x^{2}+y^{2}\end{array} \right. \tag{7}\]  
where \(W_{\tau}\) is the power consumed by end-face friction; \(\Delta v\) is the difference in speed between the test head and the surface of the specimen; \(v_{0}\) is the loading speed of the test head; \(\mu\) is [PERSON]'s ratio of the specimen; \(S\) is the cross-sectional area of the specimen; \(h\) is the height of the specimen; \(r\) is the radius of the specimen. By combining Formulas (6) and (7), the coupling relationship can be expressed as (8). \[W_{\tau}=\frac{2\mu\pi r^{3}(c+(\sigma-2f\cos\theta)\tan\varphi)v_{0}}{h} \tag{8}\]  
In turn, the relationship between \(W_{\tau}\) with the HDR and the LR is obtained as Formula (9). \[W_{\tau}=\mu Sv_{0}(c+\sigma\tan\varphi-2f\cos\theta\tan\varphi)\frac{d}{h} \tag{9}\]  
\[\Delta_{1}=W_{\tau\theta_{1}}-W_{\tau\theta_{2}}<0 \tag{10}\]  
It can be seen from Formula (9) that \(W_{\tau}\) is proportional to the LR and inversely proportional to the HDR. As the LR increases or the HDR decreases, the \(W_{\tau}\) increases accordingly. Meanwhile, when the height of the specimen is between 30 and 50 mm, the angle \(\theta\) between the conical block produced by the damage and the vertical direction  
Figure 14: Schematic diagram of the EFE. (**a**) Mechanical structural modeling of the end face; (**b**) conical damage angle for specimens of different heights. increases significantly (\(\theta_{2}>\theta_{1}\)). Also, \(\theta\in(0,\pi/4)\), \(\cos\theta\) could decrease as \(\theta\) increases. Therefore, \(W_{\tau}\) will increase, i.e., \(\Delta_{1}<0\) (as Formula (10)). The excessive \(W_{\tau}\) can reduce the actual amount of energy transferred inside the specimen (as Formulas (11)-(14)). In other words, under the same value of work done by the testing machine, the greater the \(W_{\tau}\), the less real energy is actually transferred to the specimen, as illustrated in Figure 15. Therefore, a specimen that produces the EFE with the same \(W_{m}\) (work done by the testing machine) has a passive enhancement of its ability to resist external loads. At the same time, the EFE will limit the specimen to undergo lateral deformation, which is equivalent to applying confining pressure to the specimen, thus increasing the load carrying capacity of the specimen. However, the EFE also resulted in more cracks in the specimen. This can be confirmed in experimental results and numerical simulations. The EFE discussed in this article are generated when the LR is too large (the dynamic loading phase) or when the HDR is too small (06-1). Therefore, the EFE is very harmful to engineering stability and safety, some measures should be taken to effectively avoid the generation of the EFE.