Elastic properties of Nb-based alloys by using the density functional theory

A first-principles density functional approach is used to study the electronic and the elastic properties of Nb ₁₅ X (X = Ti, Zr, Hf, V, Ta, Cr, Mo, and W) alloys. The elastic constants c ₁₁ and c ₁₂, the shear modulus C′, and the elastic modulus E ₁₀₀ are found to exhibit similar tendencies, each as a function of valence electron number per atom (EPA), while c ₄₄ seems unclear. Both c ₁₁ and c ₁₂ of Nb ₁₅X alloys increase monotonically with the increase of EPA. The C′ and E ₁₀₀ also show similar tendencies. The elastic constants (except c ₄₄) increase slightly when alloying with neighbours of a higher d-transition series. Our results are supported by the bonding density distribution. When solute atoms change from Ti(Zr, Hf) to V(Ta) then to Cr(Mo, W), the bonding electron density between the central solute atom and its first neighbouring Nb atoms is increased and becomes more anisotropic, which indicates the strong interaction and thus enhances the elastic properties of Nb - Cr(Mo, W) alloys. Under uniaxial 100 tensile loading, alloyed elements with less (more) valence electrons decrease (increase) the ideal tensile strength.