## 4 Cutter Wear Mechanism and Calculation Model Considering Thermal Effect
### Calculation of Radial Thermal Stresses on the Cutter Ring  
The wear prediction model is established, and the radial thermal stress of the cutter ring is calculated by considering the soft and hard composite strata with thermal effect conditions. According to the thermal stress theory of elastomer, the cutter ring is simplified to a two-dimensional model without considering the thickness change, \(\sigma_{z}=0\) and the stress-displacement in the radial and annular directions is obtained as:  
\[ \begin{array}{l}\sigma_{t}=\frac{E}{(1+\mu)(1-\mu)}\frac{du}{dr}+\frac{E\mu} {(1+\mu)(1-\mu)}\frac{u}{r}-\frac{E\alpha}{1-\mu}\Delta t\\ \sigma_{\varphi}=\frac{E}{(1+\mu)(1-\mu)}\frac{u}{r}+\frac{E\mu}{(1+\mu)(1- \mu)}\frac{du}{dr}-\frac{E\alpha}{1-\mu}\Delta t\end{array} \Bigg{\}} \tag{7}\]  
According to the force balance equation of the cutter ring:  
\[\left. \begin{array}{l}\frac{dr_{1}}{dt}+\frac{\sigma_{t}-\sigma_{\varphi}}{ t}=0\hskip 28.452756 pt(a)\\ \frac{dr_{2}}{dt2}=0\hskip 28.452756 pt(b)\end{array} \right\} \tag{8}\]  
The cutter radius is known for the determined parameters \(r_{i},r_{e}\) The radial integral of the disc cutter ring is calculated to obtain the thermal stress constants, denoted as \(C_{1},C_{2}\). Combined with the theory of heat transfer, calculations were carried out using the calculus  
Figure 14: Analysis of abrasive wear calculation model. The volume of abrasion when a single abrasive grain moves a unit distance is \(ah_{p}\). The distance travelled by an abrasive grain is \(dl\), and the volume of material removed is:  
\[dV=h_{p}^{2}\tan\beta dl \tag{12}\]  
Defining the wear volume produced per unit displacement as the wear degree \(dV/dl\), the wear volume \(Q\) produced by motion of \(n\) abrasive grains is:  
\[Q=n\frac{dV}{dl} \tag{13}\]  
Associating Equations (11)-(13) yields the volumetric wear:  
\[Q=\frac{F_{p}}{\pi\sigma_{s}tan\beta} \tag{14}\]  
The sliding distance is \(L_{s}\), and the volumetric wear of all conical particles on the cutter surface \(\Delta V\) is:  
\[\Delta V=n\frac{dV}{dl}L_{s}=\frac{F_{p}L_{s}}{\pi\sigma_{s}tan\beta} \tag{15}\]  
The cutter rotates around the axis to cut, and the amount of wear lost on the outer edge surface is equal to the amount of wear removed by the plasticity of the abrasive grains. The cutting distance is the relative sliding distance. The cross-section of the wear model is shown in Figure 16a,b. According to the wear volume balance, the wear volume is equal to the total volume of surface particles lost due to friction. The expression for the predicted volumetric wear \(\Delta V\) of the cutter is:  
\[\Delta V=(U\frac{L_{m}}{p}+\Delta htan\theta)2\pi R \tag{16}\]  
where \(p\) is the depth of penetration, \(L_{m}\) is the advancement distance of the cutter, \(U\) is the width of the cutting edge, \(\Delta h\) is the depth of wear, \(\theta\) is the angle of the cutting edge on one side of the cutter head, and \(R\) is the mounting radius of the cutter. Figure 16: Wear analysis model for cutter rings. (**a**) Wear model of the outer edge of the cutter; (**b**) volumetric wear analysis of the cutter ring surface. The cutter ring analysis model is shown in blue and the amount of changing wear is shown in red. The volumetric wear of the cutter ring increases with the cutting distance, and the wear depth \(\Delta h\) is expressed as:  
\[\Delta h=(\frac{\Delta V}{2\pi Rtan\theta}-\frac{UL_{m}}{ptan\theta}) \tag{17}\]  
The friction force is mainly the crushing pressure at the bottom of the cutter. Substituting the hard wear amount and external load expression (3) into Equation (17) gives us:  
\[\Delta h=\frac{L_{s}*10^{2}}{2\pi^{2}R\sigma_{s}tan\theta tan\beta}\bigg{[} \frac{S\sigma_{c}{}^{2}\sigma_{t}}{\sigma\sqrt{Ur_{e}}}\bigg{]}^{\frac{1}{3}}- \frac{UL_{m}}{ptan\theta} \tag{18}\]  
A purely rolling cutter does not do work and has no wear. Frontal wear of the cutter is mainly generated by relative sliding friction.