## 3 Force Model and Relative Motion Analysis of Cutter  
### Rock-Breaking Force Pattern of the Cutter  
Most of the research on cutter wear focuses on wear factor analysis, as well as force modelling. The soft and hard rock conditions determine the cutter force state. The external vertical force and rolling force make the cutter rotate and cut and create contact with the rock body at the bottom and two sides. Different stratigraphic transition interfaces will intensify the impact and wear, and the rock-breaking force pattern and the cutter motion directly affect the wear condition of the cutter ring [30].  
Comprehensive prediction theory of rock-breaking force is based on the semi-theoretical and semi-empirical theory Colorado School of Mines (CSM) force model. According to the rock-breaking theory and wear prediction model proposed by scholars at home and abroad, thermal stress and soft-hard composite stratum conditions are considered. We explore the cutter wear mechanism and prediction model under the action of temperature effect [31].  
The force state and motion mode of the cutter breaking process are analysed, as shown in Figure 11. The cutter is rotated in the direction of the rotation axis under the external loads \(F_{r}\) and \(F_{n}\) and cuts the rock. The penetration of the cutter ring is \(p\), the material is uniform, and the friction of the cutter shaft and body is ignored. The cutter ring model is simplified as a thin, hollow disc model, and the cutter is subjected to vertical force and rolling force, whose combined force \(F_{s}\) is directed towards the centre of the disc. Based on the CSM theoretical rock-breaking force model, the mathematical relationship between cutting parameters is established. The cutter combined force \(F_{z}\) is:  
\[F_{z}=C\frac{g\
u Ir_{e}}{1+\gamma}F_{p} \tag{1}\]  
Figure 10: Relationship between abrasive index and abrasion degree for rocks of different strengths.  
The contact angle \(\varphi\) of cutter-rock is expressed as:  
\[\varphi=\arccos\left(\frac{r_{e}-p}{r_{e}}\right) \tag{2}\]  
The distribution pressure \(F_{P}\) in the crushing zone at the edge of the cutter is:  
\[F_{P}=\left[\frac{S\sigma_{c}^{2}\sigma_{t}}{\varphi\sqrt{UI_{r}}}\right]\ ^{\frac{1}{3}} \tag{3}\]  
where \(F_{P}\) is the distributed pressure in the cutter-rock contact crushing zone.  
The vertical load \(F_{n}\) and tangential load \(F_{r}\) applied to the disc cutter are:  
\[F_{n}=F_{z}cos\left(\frac{\varphi}{2}\right)=cos\left(\frac{ \varphi}{2}\right)\frac{C\varphi UL_{r}}{1+\gamma}\left[\frac{S\sigma_{c}^{2} \sigma_{t}}{\varphi\sqrt{UI_{r}}}\right]\ ^{\frac{1}{3}} \tag{4}\] \[F_{r}=F_{z}sin\left(\frac{\varphi}{2}\right)=sin\left(\frac{ \varphi}{2}\right)\frac{C\varphi UL_{r}}{1+\gamma}\left[\frac{S\sigma_{c}^{2} \sigma_{t}}{\varphi\sqrt{UI_{r}}}\right]\ ^{\frac{1}{3}}\]  
where \(F_{n}\) is the vertical force, \(F_{r}\) is the rolling force, \(F_{z}\) is the combined force, \(r_{e}\) is the outer radius of the cutter, \(\varphi\) is the contact angle between the cutter and rock; \(U\) is the width of the cutter edge; \(\sigma_{c}\) is the compressive strength of the rock; \(\sigma_{t}\) is the tensile strength of the rock; \(\gamma\) is the coefficient of distribution of the cutter edge pressure, which takes a value of 0.1; \(S\) is the distance between cutters, where \(S=75\) mm; and \(C\) is a dimensionless coefficient, where \(C=2.12\).