According to the mechanical stability criterion for hexagonal crystals [35] and the calculated elastic constants of Ti₃AlB₂, Ti₂ZrAlB₂-1 and Ti₂ZrAlB₂-2 listed in Table 2, it can be said that they are all stable in mechanics.

Based on the investigation of formation enthalpy, phonon dispersion curve and elastic constants of Ti₃AlB₂, Ti₂ZrAlB₂-1 and Ti₂ZrAlB₂-2, it can be concluded that these three compounds are stable at normal pressure. However, we still know little about their performance, which needs further exploration. Here, we first studied their bulk modulus B and shear modulus G according to the Voigt-Reuss-Hill approximation [36-37] and listed the results in Table 3.

The B/G ratio is a parameter to evaluate the brittleness of materials proposed by Pugh [38-39]. In present work, the B/G ratio of Ti₃AlB₂, Ti₂ZrAlB₂-1 and Ti₂ZrAlB₂-2 are larger than 1.75, which means they have good ductility. This is an obvious advantage that the MAX materials studied previously does not have. This excellent characteristic shows that the three layered ceramic materials have good toughness and are not fragile. Fragile is the weakness of traditional ceramic materials, none of the three materials in this study has this disadvantage.

The Poisson ratio values, ν=(3B-2G)/2(3B+G), are calculated to evaluate the degree of the covalent bonding [40]. For Ti₃AlB₂, Ti₂ZrAlB₂-1 and Ti₂ZrAlB₂-2, their Poisson ratios are larger than 0.25, which means ionic bonds dominate in these three crystal materials. And the specific bonding characteristics will be discussed in the following study of electronic properties.

Table 3. Calculated bulk Modulus B (GPa), Shear Modulus G (GPa), Young’s Modulus E (GPa), Poisson’s Ratio ν, and B/G of Ti₃AlB₂, Ti₂ZrAlB₂-1 and Ti₂ZrAlB₂-2 at atmospheric pressure.