Temperature-dependent mechanical properties of ZrC and HfC from first principles

In order to gain insight into the effect of elevated temperature on mechanical performance of ZrC and HfC, the temperature-dependent elastic constants of these two carbides have been systematically studied using two different methods (first-principles qusi-harmonic approximation and first-principles qusi-static approximation). Isoentropic C$_{11}$ gradually decrease and the isoentropic C$_{12}$ slightly increases for ZrC and HfC under temperature. While the isoentropic C$_{44}$ of both ZrC and HfC is insensitive to the temperature. In both ZrC and HfC cases, the decline of temperature-dependent C$_{11}$ calculated from QHA is more pronounced than from QSA. Temperature effects on modulus of elasticity, poisson's ratio, elastic anisotropy, hardness and fracture toughness are further explored and discussed. The results indicate that degradation of B, G and E estimated from QHA is more serious than these from QSA at high temperature. From room temperature to 2500 K, the theoretical decreasing slope (from QHA) of B, G, and E is 0.38 GPa$\cdot$K$^{-1}$, 0.30 GPa$\cdot$K$^{-1}$, and 0.32 GPa$\cdot$K$^{-1}$ for ZrC and 0.33 GPa$\cdot$K$^{-1}$, 0.29 GPa$\cdot$K$^{-1}$, and 0.30 GPa$\cdot$K$^{-1}$ for HfC.