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In this work, an ab-initio calculation is used to investigate the elastic constants and some other mechanical and thermal parameters of tetragonal Cu2ZnSnS4 (CZTS) quaternary semiconducting bulk material in Kesterite (KS) and Stannite (ST) phases. The Quantum Espresso code within the Ultra Soft pseudo potentials (USPP) and the local density approximation (LDA) approach were used in the calculation. Firstly,, studies are started with the prediction of the elastic stiffness constants Cij and the normal and shear anisotropy factors. Then some other mechanical moduli, especially the isotropic bulk modulus B, the shear modulus G, the Young modulus E, the Poisson’s ratio ν, and the Pugh’s criteria (G/B) are delivered. The analysis of the mechanical stability criteria at equilibrium shows that our elastic stiffness constants Cij of CZTS material obey all the stability conditions. Additionally, some other parameters of the CZTS semiconductor, especially: the Vickers hardness HV , the sound velocity, the Debye temperature θD and the melting temperature Tm were also calculated. The obtained values of the elastic constants Cij and other mechanical and thermal parameters agree well with experimental and other theoretical results of the literature. The Debye temperature θD of the KS phase was found at around 332.7 K, and that of the stannite phase was found equal to 329.1 K, respectively.