# A Discrete Model of Rock-Soil Medium Response in the Determination of Horizontal Strain Values
## 2 Theoretical Model of Deformation Accumulation  
A rock that is subjected to a load accumulates the generated stress (energy), which is released discretely and recorded as an elementary, discrete deformation increase once it exceeds a certain value that is characteristic for the given rock type. This cyclical \"accumulation-release of energy\" process characterises a significant part of the deformation process. It is caused by the structure of the rock material and its internal \"resistance\". This process is related not only to the behaviour of rock samples during strength tests that are performed in a laboratory but also to the deformation of the rock-soil medium that is caused by underground mining operations. Online extensometric measurementsof horizontal deformations that were carried out in mining areas and many years of experience of the authors in the field of measurements and analysis of both continuous and discontinuous deformations clearly confirmed such a course of the horizontal strain accumulation (Figure 1). The assumption that this discrete course of the measured deformation is caused by the internal resistance of the measuring device was shown to be false. The performed tests of the measuring device showed that this deformation picture was dominated by the reaction of the rock-soil medium. To estimate the uncertainty of such a deformation process, let us suppose that a single elementary deformation increment \(\Delta\varepsilon\) is a random variable with a normal distribution with a defined expected (average) value \(h\) and a standard deviation \(\sigma_{h}\). Furthermore, let us assume that this applies to the entire deformation process, i.e., to both tensile and compressive deformations (Equation (3)):  
\[|\Delta\varepsilon|\,\sim N(h,\sigma_{h}) \tag{3}\]  
For the schematic course of deformation accumulation over time presented in Figure 2, the standard deviation of any single value \(\varepsilon(t)\) can be determined according to Equation (4):  
\[\sigma_{\varepsilon(t)}=\sigma_{h}\cdot\sqrt{n(t)} \tag{4}\]  
Figure 1: The accumulation of horizontal strain that was recorded from 29 March 1991 to 14 June 1991 using three extensometers installed at Station 2 Fahrradweg at Kapellen (own study). Figure 2: A schematic representation of deformation accumulation over time (own study). where  
\(n(t)\)--the number of elementary discrete reactions up to time \(t\). The average number of elementary discrete responses can be estimated using Equation (5):  
\[n(t)=\frac{\sum_{0}^{t}\lvert\Delta\varepsilon\rvert}{h} \tag{5}\]  
where  
\(\sum\limits_{0}^{t}\lvert\Delta\varepsilon\rvert\)--cumulative value of the deformation increases at time \(t\). For the first section of the curve \(\sum\limits_{0}^{t}\lvert\Delta\varepsilon\rvert\) (Figure 2), we therefore obtain Equation (6):  
\[\sigma_{\varepsilon(t)}=\frac{\sigma_{h}}{\sqrt{h}}\cdot\sqrt{\varepsilon(t) }=c\cdot\sqrt{\varepsilon(t)} \tag{6}\]  
where  
\(c\)--a certain constant that is dependent on the type and properties of the deformed rock-soil medium. It can be seen from the above considerations that for the first phase of the influence of progressing mining operations, i.e., the phase with increasing tensile strain, the standard deviation is proportional to the square root of the predicted or measured strain. The variation coefficient of the strain therefore assumes the form of Equation (7):  
\[M_{\varepsilon(t)}=\frac{\sigma_{\varepsilon(t)}}{\varepsilon(t)}=\frac{c}{ \sqrt{\varepsilon(t)}} \tag{7}\]  
The solutions presented above are illustrated by the following example in which the profile of horizontal deformation that is caused by mining operations of a single longwall (Figure 3) corresponds to the full characteristics of the deformation profile in the direction of the mining operation, which includes the phases of tensile and compressive deformation. Let us assume that the analysed object is located at the point \(P\) (Figure 3).