# Numerical Modeling of Temperature Effect on Tensile Strength of Granitic Rock
## 3 Numerical Simulations
### Simulations of Uniaxial Tension Test of Intact and Heat Treated Rock  
First, a uniaxial tension test is carried out on intact rock. The loading is applied as a constant velocity boundary condition at the upper edge of the specimen. The velocity is set to \(5\) mm/s, which means a strain rate of \(0.1\) s\({}^{-1}\) with the present sample size. The results for the numerical rocks in Figure 3 are shown in Figure 6.  
Figure 4: Simulation results for heat treatment (300 \({}^{\circ}\)C, NumRock1): (**a**) crack opening magnitude; (**b**) stress-like softening variable; (**c**) temperature field; (**d**) thermally induced cracks; (**e**) temperature evolution in time; (**f**) polar histogram showing crack orientation angles (2186 cracks). Figure 5: Simulation results for heat treatment (500 \({}^{\circ}\)C, NumRock1): (**a**) crack opening magnitude; (**b**) stress-like softening variable; (**c**) temperature field; (**d**) thermally induced cracks (3098 cracks). All the numerical rock samples show the experimental transversal splitting failure mode with differing details. Cracks, 565 in total, appear all over the sample (Figure 6b). Only some of them open to form the final failure plane. Moreover, the crack orientation is mostly horizontal (Figure 6f). The predicted tensile strengths vary from 9.1 to 9.8 MPa. A slight pre-peak bent can be observed in the average stress-strain curves when the stress is 7 MPa. This corresponds to the failures (microcrack events) of biotite grains. After the peak stress, very brittle failure of the sample is attested in each case. Next, a uniaxial tension test is carried out on the heat-treated samples. In order to mimic the experiments, these simulations are performed on the cooled down specimens. Thus, no thermal stresses exist in the sample. Moreover, it is assumed that the cracks are closed, meaning that their opening vectors are nullified but their orientations and residual strengths are included in the initial state here. Thereby, the residual tensile strength for an element with a crack is calculated by \(\sigma_{\text{tb}}+\textit{q}_{\text{d}}\) where \(\sigma_{\text{tb}}\) is the intact tensile strength of the mineral and \(\textit{q}_{\text{d}}\) is the stress-like softening variable at the end of the heat treatment simulations. The simulation results for the target temperature of 300 \({}^{\circ}\)C is shown in Figure 7.  
Figure 6: Simulation results for uniaxial tension test (intact rock): (**a**) crack opening magnitude with NumRock1; (**b**) cracks with NumRock1; (**c**) crack opening magnitude with NumRock2; (**d**) crack opening magnitude with NumRock3; (**e**) average stress–strain curves; (**f**) polar histogram showing crack orientation angles (565 cracks) with NumRock1. The weakening effect of the thermal cracks induced by heating to 300 \({}^{\circ}\)C is not very strong. Indeed, the tensile strength of the heated samples is about 90% that of the intact one (Figure 7c). Moreover, the failure modes differ only in details from the intact ones (compare Figure 7d to Figure 6. As to the cracks plotted for the numerical rock 1 in Figure 7a,b, many new cracks (green color) have initiated in elements already having an unfavorably oriented thermally induced crack (blue color). Between the thermal cracks, a considerable number of new horizontal cracks, due to the tensile loading, have initiated. Next, the simulations were re-performed continuing the heating up to 500 \({}^{\circ}\)C. The results are plotted in Figure 8.  
When the heating is continued to 500 \({}^{\circ}\)C, the weakening effect is substantial, as seen in the average stress-strain curves in Figure 8c. Moreover, the pre-peak nonlinear part of the curves is much more pronounced than those at 300 \({}^{\circ}\)C. This is due to the opening of the thermally induced cracks, which have reduced tensile strengths, during uniaxial tension loading. Furthermore, the post-peak softening part of the average responses is much more ductile due to dissipation occurring through the opening of a considerable number of cracks.