# Study on Mechanical Behavior of a New Three-Dimensional Sensor Under Rock Rheology
## 2 Three-Dimensional borehole Full Stress Monitoring Theoretical Analysis
### Pressure Film Stress Analysis  
When the rheological stress of the rock and soil around the borehole recovers to a certain stage, that is, the borehole shrinks to a certain extent, the surrounding rock pressure is transmitted to the full stress monitor. At this time, the stress of the pressure film is shown in Figure 5. The pressure film can be regarded as a thin-walled spherical shell under internal and external pressure [33], where \(P\) is the oil pressure in the closed oil chamber and the monitoring pressure of the pressure sensor. \(\sigma\) is the pressure of the hole wall to the pressure film, that is, the measured pressure, \(r_{0}\) and \(R_{0}\) are the initial inner diameter and outer diameter of the pressure film, respectively, \(\phi\) is the starting angle of the inner diameter, \(\varphi\) is the starting angle of the outer diameter, and \(\delta\) is the wall thickness.  
Figure 4: The working principle of the full stress detector.  
It can be seen from Figure 5 that the pressure film is completely symmetrical in the \(x\)-axis direction, so only the equilibrium equation needs to be established for the \(y\)-axis direction, and the pressure film is applied with a certain prestress, that is, the film thickness radius increases \(\Delta r\) after the initial internal pressure is given, and the equilibrium condition is:  
\[\int_{\phi_{o}}^{\pi-\phi_{o}}P(r_{m0}+\Delta r)sin\theta d\theta+2F=\int_{\phi _{l}}^{\pi-\phi_{l}}\sigma_{r}(R_{m0}+\Delta r)sin\theta d\theta \tag{23}\]  
In the rectangular co-ordinate system, the distance between the center \(O\) of the outer diameter and the center \(O\prime\) of the inner diameter is \(t\), and the film thickness \(\Delta\delta\) can be obtained by solving:  
\[\Delta\delta=\left\{\begin{array}{cc}R_{m}-\frac{t}{zsin\theta},\ \phi_{o}\leq \theta<\pi-\phi_{o}\\ R_{m}-\sqrt{R_{m}^{2}cos^{2}\theta+r_{m}^{2}sin^{2}\theta},\ \varphi_{l}\leq \theta<\phi_{o},\pi-\varphi_{l}\leq\theta\leq\pi-\phi_{o}\\ \delta,\ 0\leq\theta\leq\varphi_{l},\pi-\varphi_{l}\leq\theta\leq\pi\end{array}\right. \tag{24}\]  
It is known that the pressure film is an elastic material, the elastic modulus is \(E_{M}\) and the strain is \(\varepsilon_{m}\). According to the circumferential strain formula, the following can be obtained:  
\[F=E_{M}\varepsilon_{m}\Delta\delta=\left\{\begin{array}{cc}E_{M}\varepsilon _{m}\big{(}R_{m}-\frac{t}{2 sin\theta}\big{)},\ \phi_{o}\leq\theta<\pi-\phi_{o}\\ E_{M}\varepsilon_{m}\Big{(}R_{m}-\sqrt{R_{m}^{2}cos^{2}\theta+r_{m}^{2}sin^{2} \theta}\Big{)},\ \varphi_{l}\leq\theta<\phi_{o},\pi-\varphi_{o}\leq\theta\leq\pi-\phi_{o}\\ E_{M}\varepsilon_{m}\delta_{p}=E_{M}\delta\varphi_{p}\frac{\Delta r}{r_{0}},\ 0\leq \theta\leq\varphi_{l},\pi-\varphi_{l}\leq\theta\leq\pi\end{array}\right. \tag{25}\]  
From the knowledge of elastic mechanics, the change value under the uniform vertical force \(\sigma\) in the inner diameter range is:  
\[\Delta r=\frac{1+\mu_{Y}}{E_{Y}}(r_{m0}+\Delta\delta)\sigma_{r} \tag{26}\]  
Among them, \(E_{Y}\) is the elastic modulus of the rock mass of the hole wall, and \(\mu_{Y}\) is the Poisson's ratio of the rock mass of the contact hole wall. Then, the simultaneous Equations (23)-(26) can be obtained:  
\[P_{L}=\frac{\sigma_{r}\Big{(}R_{m0}+\Big{(}\frac{1+\mu_{Y}}{E_{Y}}(r_{m0}+ \Delta\delta)\sigma_{r}\Big{)}\Big{)}(cos\varphi_{i}-cos(\pi-\varphi_{i}))-2E _{M}\varepsilon_{m}\big{(}R_{m}-\frac{t}{2 sin\theta}\big{)}}{\Big{(}r_{m0}+ \Big{(}\frac{1+\mu_{Y}}{E_{Y}}(r_{m0}+\Delta\delta)\sigma_{r}\Big{)}\Big{)}( cos\phi_{o}-cos(\pi-\phi_{o}))} \tag{27}\]  
It can be obtained from Equation (27) that there is a certain functional relationship between the internal pressure \(P_{L}\) and the external load \(\sigma_{r}\), so the stress change of the sur  
Figure 5: Force analysis of pressure membrane.  
rounding rock can be described by the pressure film deformation of the three-dimensional borehole total stress meter.