# Numerical Parametric Studies on the Stress Distribution in Rocks around Underground Silo
## 3 Theoretical Background  
### Material Model Description  
The Generalized Hoek and Brown Model [18; 19; 20] was applied to this numerical analysis. This model represents the constitutive relationship of rocks or soils. The failure surface is described in its generalized form by the following equation:  
\[\mathrm{F}(\mathrm{p},\ \mathrm{q},\ \mathrm{\theta})=\mathrm{q}-\left\{\left( \alpha+\mathrm{\beta}\mathrm{p}\right)^{\mathrm{n}}+\kappa\right\}\ \mathrm{R}(\mathrm{\theta})=0 \tag{1}\]  
where the stress invariants (\(\mathrm{p},\mathrm{q}\), and \(\mathrm{\theta}\)) are defined in Appendix A along with the expression for \(\mathrm{R}(\mathrm{\theta})\).  
The function \(\mathrm{R}(\mathrm{\theta})\) denotes the shape of the yield surface projected onto the octahedral plane. The effect of the change of parameter \(\mathrm{k}\) on the shape of the yield surface is shown in Figures 6 and 7. The value \(\mathrm{k}\) denotes the ratio of the shear strength in the triaxial extension state to the shear strength in the triaxial compression state at the same mean pressure. It can vary from 0.5 to 1.0 and is a measure of the influence of the intermediate principal stress on the yield surface. When it is unity, \(\mathrm{R}(\mathrm{\theta})\) is circular, representing a von Mises or Drucker-Prager failure model. When it is less than unity, \(\mathrm{R}(\mathrm{\theta})\) is a smooth cornered approximation to the Mohr-Coulomb failure envelope [21; 22; 23]. The parameter \(\mathrm{n}\) in Equation (1) determines the shape of the yield surface in the p-q plane.  
When \(\mathrm{n}=0\), the strength envelope decreases to the von Mises or Tresca yield surface, and the shear strength shows a constant value for mean pressure. When \(\mathrm{n}=1/2\), the strength envelope represents a Hoek and Brown failure surface [22]. This nonlinear failure model means a multidimensional generalization of the original one-dimensional axial symmetric Hoek and Brown model, based on extensive laboratory experiments and field data [24]. When \(\mathrm{n}=1\), the strength envelope in the p-q plane is representative of the Drucker-Prager or Mohr-Coulomb failure surface and shear strength is linearly proportional to the mean pressure.  
Figure 6: Triaxial compression and extension modes in an octahedral plane.  
The parameters \(\alpha\), \(\beta\), and \(\kappa\) of Equation (1) define the failure envelope in the p-q plane as values determined in laboratory tests. Recommended relationships for determining these parameters for various types of materials are listed in Table 2. The empirical material parameters for n = 1/2 are tabulated in Table A1 in Appendix A for several different rock types, as descriptors of rock quality. A detailed description of rock quality is presented in Table A2 in Appendix A.  
The model includes the classical von Mises, Drucker-Prager, and Mohr-Coulomb failure equations, as well as the empirically based Hoek and Brown failure equation [25, 26]. One of its useful features is that the model can use empirical data as a basis for modelling the strength of an in situ rock mass when the in situ strength data are not available. The three-dimensional elasto-plastic matrix for the Generalized Hoek and Brown Model is presented in Appendix A. The model assumes an elastic state below the failure surface, mutually dependent volumetric and deviatoric behavior once it reaches the failure surface, and a perfectly plastic state along the failure surface.