  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.
