# Numerical Modeling of Temperature Effect on Tensile Strength of Granitic Rock
## 1 Introduction  
Rocks often face high temperature conditions and thermal shocks in geotechnical engineering applications, such as harvesting deep geothermal energy [1], nuclear waste disposal [2], and thermal drilling [3]. Quartz-bearing rocks are particularly susceptible to temperature effects in their material properties and, consequently, in their response under thermal loading due to the \(\alpha\)-\(\beta\) transition of quartz at its Curie point (\(\sim\)573 \({}^{\circ}\)C) [4]. Naturally, the temperature effects on rock mechanical properties have been extensively studied, experimentally [1, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14] and numerically [11, 12, 13, 14, 15, 16, 17]. The temperature effects in rocks manifest as a degradation of mechanical properties ([PERSON]'s modulus and strength), while thermal properties show mixed behavior, i.e., some increase (thermal expansion coefficient, specific heat) while others decrease (thermal conductance) [7, 11]. The degradation of mechanical properties, especially that of the tensile strength, is the most important aspect from a geotechnical engineering point of view. The main mechanism behind the deterioration of mechanical properties, i.e., damage, is thermal cracking due the pronounced heterogeneity of a rock material [13]. More precisely, a mismatch in the elastic properties leads to thermal stresses at mineral grain boundaries, which causes intra- and intergranular cracking, even under a uniform temperature field. With quartz-bearing rocks, such as granite, this heterogeneity becomes even more pronounced as the thermal expansion coefficient of quartz increases nonlinearly upon approaching the\(\alpha\)-\(\beta\) transition of Quartz at its Curie point while the other granite forming minerals, i.e., felspars and micas, behave linearly as a function of temperature [7]. Numerical modeling of temperature effects in rocks in an important task in geotechnical engineering. It enables to gain insights into the phenomena in a manner impossible to achieve using laboratory experiments or in situ testing due to physical or economic reasons. Predicting temperature effects in rock through numerical modeling is challenging due to thermal cracking. Modeling cracks involves numerical description of displacement discontinuities. There are basically two approaches in computational mechanics to model cracks: the continuum approach (mostly the finite element method, or FEM) and the discontinuity approach, based on particle or discrete element methods (DEM). For general reviews on numerical methods in rock mechanics, the reader is referred to [18; 19]. Here, it suffices to say generally that the discontinuum approach is naturally superior to the continuum approach in fracture modeling (see [20] for an example). However, the critical shortcoming of particle methods is the computational labor required to keep track and update the particle contact configurations and neighbors. As to the continuum approach, the classical FEM can only model fracture in the smeared sense, i.e., as localized deformation, by damage, and/or plasticity models. Notwithstanding, continuum models have the advantage of computational efficiency and relative simplicity in terms of calibration of material and model parameters. For this reason, FEM has been enhanced to better describe discontinuities. The enriched FEM methods include embedded discontinuity FEM [21] and extended FEM [22]. Embedded discontinuity FEM is adopted in the present study as it allows to recast the problem of solving the crack-opening vector into a format similar to plasticity models [23], which is a considerable implementational advantage over the extended FEM. In the present work, the detrimental effect of thermally induced cracking on the tensile strength of granitic rock is numerically studied. A staggered explicit solution method is developed to solve the underlying thermo-mechanical problem. As mentioned above, rock fracture is modeled with embedded discontinuity finite elements.