# Identification for Abutment Stress by Drilling Cuttings
## 2 Rationale  
When the coal seam is drilled, its powder discharge is composed of two parts: one is the pulverized coal formed due to drilling broken with the same borehole diameter, the other is the pulverized coal produced with the action of stress changing around the borehole. The former is only related to pore size, while the latter is related to stress state and mechanical properties of the coal seam and stress state of the surrounding rock. In general, in order to explain the relationship between cuttings amount and stress, the Mohr-Coulomb criterion and the Kastner formula based on the ideal elastoplastic assumption are used [20; 21; 22]. Moreover, after strain softening behavior beyond the strength limit is considered, a solution of plane strain of infinite medium with a circular hole based on elastic-plastic assumption is obtained [20]. In an in situ elastic stress zone, a borehole affected by surrounding stress changes, fracture zone, plastic softening zone and elastic zone appear in turn, as shown in Figure 1. The amount of cuttings produced due to elastic displacement of the inner wall is named \(f_{2}\). Another part of coal cuttings produced by the plastic-fractured zone and dilatancy is named \(f_{3}\), which can be approximated as a linear shape in the plastic-fractured zone. Figure 1: Plane strain borehole model in coal seam. \(r_{0}\)—Borehole radius, \(R_{1}\)—Radius of fractured zone, \(R_{2}\)—Radius of plastic zone, I—Fractured zone, II—Plastic zone, III—Elastic zone. The amount of drilling cuttings produced by elastic deformation, \(f_{2}\), is  
\[f_{2}=2\pi r^{2}\rho c_{\rm v}\frac{1+\mu}{E} \tag{1}\]  
where \(\mu\) is the [PERSON]'s ratio of coal, \(E\) is the [PERSON]'s modulus and \(\sigma_{\rm v}\) is surround rock stress. The strain softening constitutive equation of coal is considered as [20]  
\[\sigma=\left\{ \begin{array}{ll}E\epsilon&(\epsilon\leq\epsilon_{c})\\ \sigma_{c}\Big{(}\frac{\epsilon}{\epsilon_{c}}\Big{)}^{m}&(\epsilon\geq \epsilon_{c})\end{array} \right. \tag{2}\]  
where \(m\) is the plastic softening coefficient, \(\sigma_{c}\) is the uniaxial compressive strength (MPa) and \(\epsilon_{c}\) is the strain corresponding to the uniaxial compressive strength. In order to simplify the calculation, the equilibrium equations of plastic and failure zone are the same as follows:  
\[\frac{d\sigma_{\tau}}{dr}+\frac{\sigma_{\tau}-\sigma_{\theta}}{r}=0 \tag{3}\]  
where \(\sigma_{\tau}\) is the radial stress and \(\sigma_{\theta}\) is the tangential stress. The Mohr-Coulomb criterion is considered as yield condition:  
\[\sigma_{\theta}=q\sigma_{\tau}+\sigma_{c} \tag{4}\]  
Considering boundary conditions: when \(r=r_{0}\), \(\sigma_{\tau}=0\); when \(r=R_{2}\), the stress must be continuous at the boundary between the elastic and inelastic regions, the inelastic deformation or plastic-fractured \(R_{2}\) was obtained as follows:  
\[R_{2}=r_{0}\Bigg{[}1+\frac{(2m+\xi-1)(2\sigma_{\rm v}-\sigma_{c})}{\sigma_{ \rm c}^{1-m}\big{[}\sigma_{c}+(\xi-1)\sigma_{\rm v}\big{]}^{m}(\xi+1)}\Bigg{]} ^{\frac{1}{2m+\xi-1}} \tag{5}\]  
where \(\xi=\frac{1+\sin\varphi}{1-\sin\varphi}\), \(\varphi\) is the internal friction angle. If the dilatancy effect is not taken into account, the radial displacement at the junction of the inelastic zone and the elastic zone of borehole, \(u_{R2}\), is  
\[u_{R_{2}}=\frac{1+\mu}{2E}R_{2}\bigg{[}\sigma_{\rm c}+\frac{\xi-1}{\xi+1}(2 \sigma_{\rm v}-\sigma_{c})\bigg{]} \tag{6}\]  
The radial displacement of bore wall is easily obtained from the condition that the volume is constant.