# Simulation of Rock Failure Cone Development Using a Modified Load-Transferring Anchor Design
## 2 Materials and Methods
### Model Geometry, Finite Element Mesh, and Boundary Conditions  
An axisymmetric, two-dimensional (2D) model was adopted to represent the interaction between the anchor and the rock mass, as illustrated in Figure 8. Based on previous experimental observations [50], the extent of the failure zone at the rock's free surface was assumed to reach approximately \(3h_{\text{ref}}\), where \(h_{\text{ref}}\) denotes the effective embedment depth of the anchor. Using this assumption, the angle of the potential breakout cone was estimated to be \(\alpha=20^{\circ}\), corresponding to the trajectory of the hypothetical fracture surface [33]. The selection of \(\alpha=20^{\circ}\) was not arbitrary but was based on an integrative approach combining field evidence and results of prior numerical simulations. This angle was used as an auxiliary parameter to define the local mesh refinement zone in anticipation of the likely crack path. According to existing standards, a cone angle of approximately \(35^{\circ}\) is assumed for concrete breakout [39]. However, in the case of natural rocks, our field investigations demonstrated failure cone angles ranging from 13 to \(17^{\circ}\) in gray sandstone to approximately \(25^{\circ}\) in porphyry [50; 64]. Additionally, 3D FEM simulations conducted in earlier studies showed values between \(27^{\circ}\) and \(29^{\circ}\), depending on the lithology and anchor geometry [46]. In this study, the value \(\alpha=20^{\circ}\) was selected as a representative average based on these findings. It provided satisfactory agreement between expected crack propagation zones and numerical predictions, especially in defining regions of high mesh density near the failure trajectory. Figure 8: Axisymmetric model of the detachment anchor acting on the rock mass: \(\alpha\)—angle of the breakout cone; \(\beta\)—undercut angle of the rock (anchor head geometry); \(\Delta\gamma\)—imposed displacement of the anchor components along the \(\gamma\)-axis of the adopted coordinate system. This estimated failure geometry was explicitly considered in the model to ensure increased finite element mesh refinement in the region of expected crack propagation. By concentrating the mesh in this critical zone, the accuracy of stress distribution analysis and fracture simulation was significantly improved, allowing for detailed observation of damage evolution under the applied load. The interaction between the tested anchor and the rock mass exhibits an analogy to a linear screw-driven mechanism. The rotational motion of the drive screw around its own axis is transformed into a forced linear displacement of the conical undercutting head along the axis of symmetry of the model (i.e., the screw's rotational axis). This motion results in a gradual increase in the parameter \(\Delta\)Y, which represents the distance between the tip of the drive screw and the base of the conical cutting head, as illustrated in Figure 8.  
Based on the results of field investigations [50], the analyzed domain of the rock mass in the model was defined as a rectangular area with the dimensions \(R=500\) mm (horizontal) and \(H=300\) mm (vertical), as shown in Figure 9. The borehole for the anchor is located along the axis of rotation of the model and was modeled with dimensions corresponding to those presented in the referenced figure. Using the finite element library available in the ABAQUS software (ABAQUS, Version 2022, Dassault Systemes Simulia Corp., Providence, RI, USA), the simulation of kinematic loading in the analysis of anchor-rock interaction was implemented through the application of a \"connector displacement\" element, as illustrated in Figure 10. This element type enables the imposition of prescribed relative motion--specifically, axial displacement--between designated reference points, effectively replicating the linear movement of the anchor head induced by the rotational actuation of the drive screw.