## 4 Cutter Wear Mechanism and Calculation Model Considering Thermal Effect
### Wear Calculation for Cutting Soft Rock  
There is a difference between the deformation characteristics of soft-rock cutting and hard-rock cutting. The cutter head is embedded in the soft rock for cutting, and the soft rock produces elastic-plastic deformation [35; 36]. As shown in Figure 17, the rock undergoes plastic damage around the cutter head. The outer edge and both sides of the cutter head are prone to lateral continuous wear, resulting in the phenomenon of tip wear. The cutting process of soft rock is more complicated, and the cutting model is simplified. The general solution of the circular hole dilatation problem is applied to the plastic deformation process of soft-rock strata to calculate the elastic-plastic stresses generated in loose strata [37; 38]. According to the characteristics of the isotropic cutter, the cutter ring model is simplified to a U-shape. The soft-rock elastic-plastic deformation zone is simplified to a semicircle, as shown in Figure 18.  
The simplified model truly reflects the cutter-rock contact. The depth of the cut increases, and the elastic deformation zone expands radially. The elastic-plastic zone and the U-shaped cutter circle form a set of concentric semicircles. The elastic-plastic deformation zones of soft soil are divided, and the stresses and strains in different zones are analysed. The radius is in the range of \(R_{t}\leq r\leq R_{p}\). The tiny soil unit model in the elastic deformation region is analysed to solve the contact stress. The stress state of the tiny unit is shown in Figure 19.  
Figure 19: Analysis of unit stress state in plastic region. Figure 17: Cutter wear and rock damage for cutting soft rock. Based on [PERSON]'s law and friction theory, the wear prediction equation can be expressed as:  
\[Q=K\frac{F_{s}L_{s}}{H} \tag{24}\]  
where \(Q\) is the volumetric wear of the cutter ring per unit sliding distance, \(F_{s}\) is the external contact load, \(L_{s}\) is the sliding cutting distance, \(H\) is the hardness of the cutter alloy surface material, and \(K\) is the dimensionless material wear coefficient. Deformation caused by the outward expansion of a circular hole in an ideal elastoplastic soil is studied. The internal pressure \(p\) is uniformly distributed around the circular hole, and the radius increases from \(R_{o}\) to \(R_{l}\). The soft-rock plastic deformation zone has an inner diameter of \(R_{l}\), an outer diameter of \(R_{P}\) and a radial displacement of \(u_{P}\) at the outer boundary. The deformation of the rock and soil bodies satisfies the Mohr-Coulomb yield criterion, and a general solution to the cylindrical hole dilatation problem is obtained. Circular hole expansion pressure gradually increases, the change in stress is independent of the thickness and ring direction, and \(\sigma_{\varphi}=0,\sigma_{s}=0\). The equilibrium of stresses in the plane is considered under axisymmetric condition, and the differential equation is given as:  
\[\frac{d\sigma_{r}}{dr}+\frac{\sigma_{r}-\sigma_{\theta}}{r}=0 \tag{25}\]  
In the cylindrical coordinate system, according to the theory of elastic mechanics, the eigenstructure equation of the elastic phase is the generalised Hooke's law. The stress is expressed in terms of displacement:  
\[ \begin{array}{l}\varepsilon_{r}=\frac{1+\mu}{2}\big{[}(1-\mu)\sigma_{r}-\mu \sigma_{\varphi}\big{]}\ \ \ \ \ \ (a)\\ \varepsilon_{\varphi}=\frac{1+\mu}{2}\big{[}(1-\mu)\sigma_{\varphi}-\mu \sigma_{r}\big{]}\ \ \ \ \ \ (b)\\ \end{array}  \tag{26}\]  
The unit radial displacement and the geometric equation are expressed as:  
\[ \begin{array}{l}\varepsilon_{r}=-\frac{du_{P}}{dr}\ \ \ \ \ (a)\\ \varepsilon_{\varphi}=-\frac{u_{P}}{r}\ \ \ \ \ (b)\\ \end{array}  \tag{27}\]  
where \(u_{P}\) is the outer boundary radial displacement of the plastic deformation zone.