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The brittle fracture of rock with an angled crack under combined tensile and compressive loading conditions is studied using linear elastic fracture mechanics (LEFM). The modified maximum tangential stress criterion (MTSC) and the maximum shear stress criterion (MSSC) are used to check crack initiations in the tensile and shear modes, respectively. The effects of the friction coefficient of the crack surfaces and the nonsingular stresses (T stresses) on the crack initiation are studied for the cases of both low and high compressive confining pressure coefficients. The T stresses include those both parallel (Tx) and perpendicular (Ty) to the crack plane. The type of crack initiation under the combined tensile and compressive loading conditions is found to remain tensile dominated when the compressive confining pressure coefficient is small. However, shear crack extension becomes possible with the compressive confining pressure coefficient and friction coefficient increasing if the crack orientation angle is small. Moreover, the high compressive confining pressure and substantial friction are found to increase the possibility of shear crack extension. The theoretical predictions presented in this study move one step forward than the available analytical solutions for the angled crack subjected to general biaxial load and agree well with those from experimental tests.

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It is essential to monitor the ground temperature over large areas to understand and predict the effects of climate change on permafrost due to its rapid warming on the Qinghai-Tibet Plateau (QTP). Land surface temperature (LST) is an important parameter for the energy budget of permafrost environments. Moderate Resolution Imaging Spectroradiometer (MODIS) LST products are especially valuable for detecting permafrost thermal dynamics across the QTP. This study presents a comparison of MODIS-LST values with in situ near-surface air temperature (Ta), and ground surface temperature (GST) obtained from 2014 to 2016 at five sites in Beiluhe basin, a representative permafrost region on the QTP. Furthermore, the performance of the thermal permafrost model forced by MODIS-LSTs was studied. Averaged LSTs are found to strongly correlated with Ta and GST with R2 values being around 0.9. There is a significant warm bias (4.43–4.67 °C) between averaged LST and Ta, and a slight warm bias (0.67–2.66 °C) between averaged LST and GST. This study indicates that averaged MODIS-LST is supposed to be a useful data source for permafrost monitoring. The modeled ground temperatures and active-layer thickness have a good agreement with the measurements, with a difference of less than 1.0 °C and 0.4 m, respectively.

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