[en] Zirconium alloys have been extensively used as fuel cladding materials in nuclear light water reactors (LWRs) due to their favorable mechanical properties, corrosion resistance, low thermal neutron capture cross section and criticality. Because the main functions of these structural materials are to maintain the integrity of UO₂ fuel rods and prevent the release of fission products into the biosphere, a detailed knowledge of their structural stability is of critical importance to ensure the safe operation of nuclear reactors, and the safety of storage, transportation or disposition of used nuclear fuel assemblies. While the Zr-H phase diagram has been known since the 1950's and has remained largely unchanged, the structures, formation mechanisms, and stability of Zr-hydride phases in pure α-Zr or Zr alloys matrices have been subjects of much debate over the last few decades. The formation of hydride phases in the Zr matrix has been linked in previous studies to the degradation of mechanical properties of Zr/Zr-alloys materials. Therefore, an accurate knowledge of the interplay between the structures, stability and mechanical properties of Zr/Zr-alloys and Zr hydrides is of paramount significance to ensure materials reliability during their in- service lifetime. Despite few first-principles studies have investigated the mechanical stability of Zr hydrides polymorphs proposed so far, no ab initio studies have been carried out, to the best of our knowledge, on the mechanical properties of modern nuclear-grade zirconium alloys such as Zircaloy-4 (Zry-4; 1.2-1.7% Sn), Zirconium low oxidation alloy (ZIRLO; 0.7-1.0% Sn and ∼1.0% Nb), or M5 alloy (0.8-1.2% Nb). Therefore, it is important to carry out a series of first-principles calculations systematically to investigate the structure-property relationship of zirconium hydrides and zirconium alloys, that can help explain differences in elastic properties between high-purity zirconium and nuclear-grade zirconium alloys. (authors)