## 4 Materials and Methods
### Developing the Model
#### 4.1.2 Material Properties  
As a non-linear analysis of cable bolts was conducted, elastic, plastic, shear damage and fracture properties were assigned to the steel bolt. The steel properties tested were extracted from [PERSON]'s tested data, where the yield and ultimate strength were provided as a force. ABAQUS requires yield and ultimate strength to be inputted as a pressure load, thus yield and ultimate strength were divided by the undeformed cross-sectional  
Figure 3: Geometric part—steel bolt. Figure 2: Geometric part—concrete box. area of the bolt under investigation. The elastic properties of the 63T Sumo Cable Bolt are summarised in Table 4.  
To define a smooth transition from elastic to plastic deformation, the Giuffre-Menegotto-Pinto model was used [31]. The plastic behaviour of steel was defined in terms of stress-strain data, which was calculated in accordance with AS4100:2020. To accurately reflect laboratory data, the engineering stress-strain was converted into true stress-strain because engineering stress-strain fails to account for a changing cross-sectional area due to plastic failure. The following equations were used to convert the plastic stress-strain values from engineering to true values [32]. \[\sigma_{T}=\sigma_{E}(1+\varepsilon_{E}) \tag{1}\]  
\[\varepsilon_{T}=\ln(1+\varepsilon_{E}) \tag{2}\]  
where \(\sigma_{T}\) is the true stress, \(\sigma_{E}\) is the engineering stress, \(\varepsilon_{T}\) is the true strain and \(\varepsilon_{E}\) is the engineering strain. Figure 5 compares the engineering vs. true stress-strain curve of steel with a yield strength of 922 MPa and an ultimate strength of 1031 MPa, that was used to calibrate the static model. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Density** & **Elastic Modulus** & **Yield Strength** & **Ultimate Strength** & **Poisson’s Ratio** \\
**(kg/m\({}^{3}\))** & **(MPa)** & **(MPa)** & **(MPa)** & **Poisson’s Ratio** \\ \hline
7800 & 200,000 & 922 & 1031 & 0.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Elastic steel properties of the 63T Sumo cable bolt. Figure 4: Geometric part—load sphere. Figure 5: True vs. engineering stress–strain relationship. The shear damage and fracture properties that were defined for steel with a yield strength of 922 MPa and an ultimate strength of 1031 MPa are summarised below in Table 5.  
The fracture energy is defined as the energy required per unit area to change a material surface from its initial unloaded state to a state of complete separation [33]. The fracture energy of a material is estimated by calculating the integral sum of a material's stress-strain curve [34]. To calculate the fracture energy of each tested steel yield and ultimate strength, a Python code was developed, which has been recorded in Appendix A.1. The concrete box acts as an alternative to rock, where each concrete block represents a layer of rock. This simplification is justified as rock possesses similar properties to concrete, and thus allows for easily repeatable experimental double shear testing. Table 6 summarises the defined concrete material properties for ABAQUS. The Drucker-Prager and the Concrete Damage Plasticity (CPD) model were both investigated when calibrating the numerical model. The Drucker-Prager model produced marginally more accurate results to the experimental data. However, the computational time was excessive, thus the CDP model was used. The CDP model defines two primary failure mechanisms, compressive crushing and tensile cracking, which are available in the ABAQUS properties library. The compressive linear elastic range was defined as up to 40% of its compressive strength, and the tensile behaviour was modelled as initially elastic followed by softening after cracking occurred. The damage parameter as a function of inelastic strain were defined in accordance with [PERSON] (2015) [35]. For dynamic testing, the load sphere was also assigned steel properties, however the ball is intended not to deform and hence assigned as a rigid body.