## 2 Theoretical Analysis
### Normal-Tangential Contact
#### 2.2.1 High-Normal Force Level  
In the case of the high-normal force level, the maximum static frictional force on the contact surface between the two spheres is larger than the tangential force required for any one sphere crushing, that is \(\mu F_{ncr}>F_{tcr}\), and the breakage occurs without relative sliding between the two particles. \(F_{ncr}\) and \(F_{tcr}\) are the normal force and the tangential force at crushing, respectively, and \(\mu\) is the surface sliding friction coefficient of the material. In the stages of local compaction and elastic deformation, only the normal force does work, while the tangential force does not. As a result, the mechanism of these two stages is similar to that under normal contact conditions, and only the integral crushing of the particle is to be discussed next. The solutions of stresses within the half-space along the line normal to and through the center of the contact area and the displacement at the center of the contact area are of interest as the area of the contact zone is much smaller than the particle size and they lend themselves to simple mathematical manipulation. The normal stress components at a point in the half-space (see Figure 2), under the action of a concentrated force \(P\), are as follows [49]:  
\[\sigma_{\rho}\ =\ \frac{P}{2\pi^{2}}\left[-\frac{3\rho^{2}z}{r^{3}}+\frac{(1 \ -\ 2\
u)r}{r+z}\right], \tag{6}\]  
\[\sigma_{\varphi}\ =\ \frac{(1\ -\ 2\
u)P}{2\pi r^{2}}\left(\frac{z}{r}-\frac{r }{r+z}\right), \tag{7}\]  
\[\sigma_{z}\ =\ -\frac{3 pz^{3}}{2\pi r^{5}}. \tag{8}\]  
The cylindrical coordinate system shown in Figure 3a is established with the central point of the compacted area as the origin, and the \(z\)-axis is in the same direction as the resultant force \(F\) between the two spheres. The stress components at the point \(M\) on the \(z\)-axis inside the sphere can be obtained by superimposing Equations (6)-(8) on the contact  
Figure 2: Cylindrical coordinates \(\rho\)–\(\varphi\)–\(z\) and orthogonal normal stresses \(\sigma_{\rho}\), \(\sigma_{\varphi}\) and \(\sigma_{z}\). \(\rho\) and \(z\) are the horizontal and vertical distances from the action point of the force \(P\) to the location of interest, respectively. \(\varphi\) is the horizontal rotation angle of the location of interest relative to the \(y\)-axis. surface. Figure 3b shows an inclined plane subjected to two vertical concentrated loads \(P\) symmetrical to the origin \(O\), in which \(\theta\) is very small, and the normal stress component \(\sigma_{z}\) of the point \(M\) on the \(z\)-axis is as follows:  
\[\sigma_{z} = -\frac{3P(z+\rho\tan\theta)^{3}}{2\pi\Big{[}\rho^{2}{+}(z+\rho\tan \theta)^{2}\Big{]}^{5/2}}-\frac{3P(z\ -\ \rho\tan\theta)^{3}}{2\pi\Big{[}\rho^{2}{+}(z\ -\ \rho\tan\theta)^{2}\Big{]}^{5/2}} \tag{9}\] \[\approx-\frac{6P^{2}}{2\pi\big{(}\rho^{2}{+}z^{2}\big{)}^{5}},\]  
which are close to the calculation results in Figure 3c. The same conclusions are obtained for \(\sigma_{\rho}\) and \(\sigma_{\varphi}\). Consequently, the stress components at the point \(M\) in Figure 3a are obtained through equivalenting the boundary condition as a uniform load \(q\) acting on a horizontal circular area with radius \(a\), as follows:  
\[\sigma_{z}{=}\ q\Bigg{[}1\ -\frac{z^{3}}{(a^{2}{+}z^{2})^{3/2}}\Bigg{]}, \tag{10}\]  
\[\sigma_{\rho}{=}\ \sigma_{\varphi}\ =\ \frac{q}{2}\Bigg{[}(1\ -\ 2\
u){+}\frac{z^{3}}{(a^{2}{+}z^{2})^{3/2}}-\frac{2(1+\
u)z}{(a^{2}{+}z^{2})^ {1/2}}\Bigg{]}. \tag{11}\]  
In this study, the simple two-parameter multiaxial failure criterion for brittle materials proposed by [PERSON] [50; 51] is applied. The criterion states that a material is at failure when the following applies:  
\[\frac{\psi K}{\sqrt{3}}\sigma_{ii}{+}(1+\psi)^{2}s_{ij}s_{ij}>\frac{K^{2}}{1+ \psi}, \tag{12}\]  
Figure 3: **(a)** The coordinate system with the center of the compacted area as the coordinate origin. \(F_{n}\) and \(F_{t}\) are the normal and tangential forces between the two spheres, respectively. \(F\) is the resultant force of \(F_{n}\) and \(F_{t}\).