# Numerical Modeling of Temperature Effect on Tensile Strength of Granitic Rock
## 2 Numerical Methods
### Rock Fracture Model
On the left, the  
Figure 1: (a) CST element with two intersecting discontinuities (\(\mathbf{n}_{1}\), \(\mathbf{m}_{1}\) : normal and tangent for crack 1, \(\Gamma_{\text{d}1}\); \(\mathbf{n}_{2}\), \(\mathbf{m}_{2}\) : normal and tangent for crack 2, \(\Gamma_{\text{d}2}\); \(N_{i}\): interpolation function at node \(i\)); (b) illustration of the displacement decomposition to regular and enhanced part and the corresponding functions in a 1D case (\(u_{reg}\) : regular displacement; \(u\): total displacement; \(\alpha_{\text{d}}\); displacement jump; \(M_{\Gamma_{\text{d}}}\) : special function restricting the effect of \(\alpha_{\text{d}}\) inside the element). functions involved in the decomposition are plotted. The decomposition consists of the regular nodal displacement, \(u_{reg}=N_{1}u_{1}+N_{2}u_{2}\), and the enhanced, discontinuous part, \(\alpha_{\rm d}\), the effect of which is restricted inside the element by function \(M_{\rm f_{d}}\). In the 1D case, the selection of function \(\varphi\) is readily identifiable as the interpolation function of node 2, i.e., \(N_{2}\). The finite element formulation of the embedded discontinuity kinematics is based on the enhanced assumed strain concept (EAS). The details of the implementation are presented in [23; 24]. Here, the resulting equations for the thermo-mechanical problem are presented as:  
\[\int_{\Omega_{e}}\rho N_{i}N_{j}\ddot{u}_{j}\mathrm{d}\Omega+\int_{\Omega_{e} }\boldsymbol{\sigma}\cdot\
abla N_{i}\mathrm{d}\Omega=\boldsymbol{0},\;i,j=1,\ldots,N_{nodes} \tag{4}\]  
\[\int_{\Omega_{e}}c\rho N_{i}N_{j}\ddot{\theta}_{j}\mathrm{d}\Omega+\int_{ \Omega_{e}}\
abla N_{i}k\
abla N_{j}\theta_{j}\mathrm{d}\Omega-\int_{\Omega_{ e}}Q_{int}N_{j}\mathrm{d}\Omega=0,\;i,j=1,\ldots,N_{nodes} \tag{5}\]  
\[\phi_{\rm d}(\mathbf{f}_{\rm r_{d}i})=0_{i}\mathbf{f}_{\rm r_{d}i}= \boldsymbol{\sigma}\cdot\mathbf{n}_{i},\;\;\;i=1,2\;(\text{for elements with crack(s)}) \tag{6}\]  
\[\boldsymbol{\sigma}=\mathbf{E}:\left(\hat{\boldsymbol{\varepsilon}}-\sum_{i=1 ^{2}}^{2}\bigl{(}\
abla\varphi_{\rm T_{d}i}\otimes\boldsymbol{\alpha}_{\rm d} \bigr{)}^{sym}-\boldsymbol{\epsilon}_{\theta}\right), \tag{7}\]  
where \(\hat{\boldsymbol{\varepsilon}}=\bigl{(}\
abla N_{i}\otimes\boldsymbol{u}_{i}^{ \prime}\bigr{)}^{sym}\) and \(\boldsymbol{\epsilon}_{\theta}=\alpha\Delta\theta\mathbf{I}\) are the thermal strain, with \(\alpha\) being the thermal expansion coefficient, \(\Delta\theta\) the temperature change, and \(\mathbf{I}\) the unit tensor. Moreover, \(\ddot{u}_{j}\) is the acceleration vector, \(N_{nodes}\) is the number of nodes in the mesh, and \(N_{i}\) is the interpolation function of node \(i\). In addition, \(\boldsymbol{\sigma}\) is the stress tensor, \(\dot{\theta}_{j}\) is the rate of change of the temperature vector, \(\theta_{j}\), at node \(j\), \(\rho\) and \(c\) are the density and the specific heat capacity of the material, \(k\) is the conductivity, and \(Q_{int}\) is the internal heat production. Equation (4) is the discretized form of the balance of the linear momentum in the absence of external forces, while Equation (5) is the discretized equation of heat balance. The first equation in Equation (6), with \(\phi_{\rm d}i\) being the loading function to be defined in the next section, defines the elastic zone of stresses. Moreover, the second equation in (6) is the traction balance over the cracks with \(\mathbf{f}_{\rm r_{d}i}\) being the traction for crack \(i\). Finally, Equation (7) defines the constitutive relation for the material, with \(\mathbf{E}\) being the elasticity tensor. It is noted that Equations (4)-(7) have contributions from only the external heat influx, \(Q_{int}\), which simulates heating in an oven. All other heat generation types, such as thermo-elastic (in the bulk material) and thermo-plastic heat generation (at the crack due to opening dissipation), are neglected as insignificant in comparison to the external heat influx [15].