# Numerical Modeling of Temperature Effect on Tensile Strength of Granitic Rock
## 3 Numerical Simulations
### Material Properties and Model Parameters  
The homogenized mechanical and thermal properties for granite taken from [11] are: [PERSON]'s modulus \(E=37.35\) GPa; [PERSON]'s ratio \(\
u=0.127\); density \(\rho=2650\) kg/m\({}^{3}\); thermal expansion coeff. \(\alpha_{0}=8\)E\(-6\) K\({}^{-1}\); specific heat \(c=820\) J/kgK; and thermal conductivity \(k=2.6\) W/mK.  
To properly predict the failure mode in tension, the tensile strength is assumed to be heterogeneous, i.e., mineral specific. In this respect, the numerical rock consists of quartz (33%), feldspar (59%) and biotite (8%) minerals, with their respective tensile strengths [27] of 14 MPa, 11 MPa and 7 MPa. Moreover, the mode I specific fracture energies, \(G_{\text{lc}}\), are [28] 40 J/m\({}^{2}\) for quartz and felspar, and 28 J/m\({}^{2}\) for biotite. The rock strength heterogeneity is described by random clusters of finite elements assigned with these strength properties. The numerical rock mesostructures (consisting of 4276 elements) thus generated to be used in the simulations are shown in Figure 3. Finally, the shear effect parameter is \(\beta=1\), and the viscosity is set to \(s_{\text{d}}=0.001\) MPa\(\cdot\)s/m.  
Figure 3: Tensile strength distribution in numerical rock samples (quartz = white, feldspar = gray, biotite = black).  
As mentioned in the Introduction, the temperature dependence of the material parameters is reduced to that of the thermal expansion. The tensile strength of granite at elevated temperatures is measured for a specimen and is first heated slowly to the target temperature and then cooled down to room temperature. This heat treatment induces cracks in the sample, leading to degradation of its tensile strength. For heterogeneous brittle materials with inherent flaws (e.g., microcracks), the tensile strength is an emerging property, representing the sample integrity under extensional loading, not a material property per se at the material point level. Therefore, it would beg the question to feed the experimental tensile strength at certain temperature into the material point level constitutive law and then predict that same strength.  
Thereby, the mechanical and thermal properties, except for the thermal expansion coefficient, are assumed temperature independent during heat treatment simulations. The thermal expansion coefficient of quartz depends on temperature, as follows:  
\[\alpha_{q}(\theta)=\alpha_{q0}\bigg{(}1+\frac{0.75}{823-293}\bigg{)}(\theta- 293),[1/K] \tag{15}\]  
where \(\alpha_{q0}\) is the thermal coefficient at room temperature. According to (15), the thermal expansion of quartz depends linearly on a temperature in the range of 20-550 \({}^{\circ}\)C. This is of course not fully realistic (see [7]) but a consequence of the present simplified modeling approach of using homogenous material properties (except the tensile strength) and reducing all the temperature dependent heterogeneity to that of quartz, as mentioned above. However, it will be shown that this approach predicts the thermal weakening effect with a reasonable accuracy.