Using first-principles calculations based on density functional theory, the elastic constants and some of the related physical quantities, such as the bulk, shear, and Young’s moduli, Poisson’s ratio, anisotropic factor, acoustic velocity, minimum thermal conductivity, and Debye temperature, are reported in this paper for the hexagonal intermetallic compound Ti ${}_3$ Al. The obtained results are well consistent with the available experimental and theoretical data. The effect of pressure on all studied parameters was investigated. By the mechanical stability criteria under isotropic pressure, it is predicted that the compound is mechanically unstable at pressures above 71.4 GPa. Its ductility, anisotropy, and Debye temperature are enhanced with pressure.

A theoretical study of the structural, elastic, and mechanical properties of B19 TiAl intermetallic compound with orthorhombic structure has been carried out by using ab initio density functional theory calculations based on the projector augmented wave and the generalized gradient approximation. The optimized structure parameters are found to be in agreement with the available experimental data. The orthorhombic B19 structure of TiAl is found to be mechanically stable at ground state in terms of formation energy and single crystal elastic constants. Additionally, the polycrystalline bulk, shear and Young’s moduli, Poisson’s ratio, Cauchy pressure, and anisotropy factors are obtained from the single crystal elastic constants. At ground state, the B19 TiAl is found to not only have intrinsic brittleness in terms of Pugh’s ratio, Poisson’s ratio, and Cauchy pressure, but also exhibit elastic anisotropy in terms of elastic anisotropy factors and orientation dependence of Young’s modulus.

The pressure dependence of the lattice and elastic constants of the orthorhombic YBa ${}_2$ Cu ${}_3$ O ${}_7$ are firstly investigated using the first principles calculations based on the density functional theory. The calculated lattice parameters at 0 GPa are in agreement with the available experimental data. By the elastic stability criteria under isotropic pressure, it is predicted that YBa ${}_2$ Cu ${}_3$ O ${}_7$ with and orthorhombic structure is mechanically stable under pressure up to 100 GPa. On the basis of the elastic constants, Pugh’s modulus ratio, Poisson’s ratio, elastic anisotropy, Debye temperature, and the minimum thermal conductivity of YBa ${}_2$ Cu ${}_3$ O ${}_7$ under pressure up to 100 GPa are further investigated. It is found that its ductility, Debye temperature, and minimum thermal conductivity increase with pressure.