# Numerical Modeling of Temperature Effect on Tensile Strength of Granitic Rock
## 4 Discussion  
The aspects of the modeling approach and the results are discussed here. The simulation results are first compared to experimental observations. To this end, the predicted normalized tensile strengths are plotted as a function of temperature in Figure 9, along with the experimental data collected in [11]. The weakening effect predicted with the present approach is within the experimental deviation for several granites, except the a slight overprediction at 500 \({}^{\circ}\)C, which could be easily remeded by fine tuning the thermal expansion coefficient (15). It should be noted that the curve, \(f_{\sigma_{1}/\sigma_{0}}=0.9912(1-4.100/2483.30)^{1/4.10}\), in Figure 9 representing the average of the experimental results for several granites [11] is averaged over a wide range of test temperatures--hence it does not cross the deviation bars at the middle. Moreover, only relevant deviation bars, i.e., those at 300 \({}^{\circ}\)C and 500 \({}^{\circ}\)C, are shown in Figure 9.  
Then some aspects of the numerical approach are discussed. The present approach falls within the class of continuum models, thus having all the advantages, over the discontinuum approach, of this class, such as the maturity of the method (FEM), the ease of calibration and the physical meaning of the model parameters, and, most importantly,  
Figure 8: Simulation results for uniaxial tension test (heat treated rock, 500 \({}^{\circ}\)C): (**a**) cracks with NumRock1; (**b**) failure modes represented by crack opening magnitude; (**c**) average stress–strain curves. the computational effectiveness. Moreover, the poor crack description of the continuum approach (FEM) is improved by the embedded discontinuity technology which is capable of representing cracking in rocks. A staggered explicit approach was employed in simulating the slow heating of rock samples up to a uniform target temperature. As this process, which takes several hours in a lab, is quasi-static in nature, extreme mass scaling using a million-fold density for the mechanical part of the problem, resulting in a 1000-fold critical time step, could be used. This aspect combined with volume heating of the sample resulted in a simulation method capable of performing this kind of simulation in practical CPU times, even with fully explicit time marching. Moreover, the fully explicit method evades the convergence problems related to the Newton-Raphson iteration involved in the implicit methods, which, however, are unconditionally stable in time. The thermal weakening effect of elevated uniform temperatures in rock samples is caused by heterogeneity, which becomes more pronounced in granites due to the deviant behavior of quartz. In the present approach, only the tensile strength heterogeneity of rock-forming minerals was included, while homogenized values for the elastic and thermal properties were used. All heterogeneity effects were reduced to accounting only for the temperature dependence of the quartz thermal expansion using a linear fit representing the net effect between quartz and the rest of the granite-forming minerals (feldspars and micas). This simplified approach has the advantage that it uses the easily measurable mechanical and thermal properties of a rock sample, instead of the far more effort needing properties of constituent minerals. Anyways, this approach appeared to work very well, as the weakening effect of thermal cracking on the tensile strength of granite was predicted with a surprisingly good accuracy. It should be emphasized that the weakening effect was predicted in a non-circular way, i.e., without using the experimental data on the temperature dependence of the tensile strength of granite as a model input. However, the present approach ignores most of the textural aspects of rock as a polycrystalline material. In particular, the grain boundaries were not accounted for. Where their inclusion is deemed crucial, more detailed models, such as the DEM mentioned in the Introduction, should be applied.