T. Qin, Z. Wang, Y. Wang, F. Besenbacher, M. Otyepka, and M. Dong, Recent Progress in Emerging Two-Dimensional Transition Metal Carbides References, Nano-Micro Lett. 13, 143 (2021); https://doi.org/10.1007/s40820-021-00710-7.

P. Prysyazhnyuk, O. Ivanov, O. Matvienkiv, S. Marynenko, O. Korol, and I. Koval, Impact and abrasion wear resistance of the hardfacings based on high-manganese steel reinforced with multicomponent carbides of Ti-Nb-Mo-V-C system, Procedia Struct. Integr. 36, 130 (2022); https://doi.org/10.1016/j.prostr.2022.01.014.

S. T. A. Shihab, P. Prysyazhnyuk, R. Andrusyshyn, L. Lutsak, O. Ivanov, and I. Tsap, Forming the structure and the properties of electric arc coatings based on high manganese steel alloyed with titanium and niobium carbides, East.-Eur. J. Enterp. Technol. 1, 38 (2020); https://doi.org/10.15587/1729-4061.2020.194164.

M. Mhadhbi, Titanium Carbide: Synthesis, Properties and Applications, Brilliant Eng. 2, 1 (2020); https://doi.org/10.36937/ben.2021.002.001.

L. Trinh et al., J. Am., Selective laser sintering and spark plasma sintering of (Zr,Nb,Ta,Ti,W)C compositionally complex carbide ceramics, Ceram. Soc. 107, 7175 (2024); https://doi.org/10.1111/jace.20019.

W. Wei, X. Ying, W. Xu, H. Zhiquan, and L. Shengxin, Effect of V content on microstructures and properties of TiC cermet fusion welding interface, China Weld. 33, 40 (2024); https://doi.org/10.12073/j.cw.20231212019.

M. Chen et al., Effect of VC addition on the microstructure and properties of TiC steel-bonded carbides fabricated by two-step sintering, Int. J. Refract. Met. Hard Mater. 108, 105948 (2022); https://doi.org/10.1016/j.ijrmhm.2022.105948.

R. Picha, A. Kroupa, and P. Broz, Phase Diagram of the Ti-V-C System at 1000 and 1200° C, Arch. Metall. Mater. No 2, 139 (2001).

M. Chen, X. Zhang, X. Xiao, and H. Zhao, Effect of VC additions on the microstructure and mechanical properties of TiC-based cermets, Mater. Res. Express 7, 106527 (2020); https://doi.org/10.1088/2053-1591/abc2a1.

Q. Wang, Q. Li, H. Ding, and F. Tian, Elastic properties of solid-solution refractory metal carbides with vacancy from virtual crystal approximation and supercell method, Comput. Condens. Matter 32, e00721 (2022); https://doi.org/10.1016/j.cocom.2022.e00721.

P. Prysyazhnyuk et al., Analysis of the effects of alloying with Si and Cr on the properties of manganese austenite based on AB INITIO modelling, East.-Eur. J. Enterp. Technol. 6, 28 (2020); https://doi.org/10.15587/1729-4061.2020.217281.

J. Kim, First-principles investigation of the elastic properties and phase stability of (Ti1-xNix)C1-y ternary metastable carbides, J. Alloys Compd. 853, 157349 (2021); https://doi.org/10.1016/j.jallcom.2020.157349.

A. van de Walle, M. Asta, and G. Ceder, The alloy theoretic automated toolkit: A user guide, Calphad 26, 539 (2002); https://doi.org/10.1016/s0364-5916(02)80006-2.

J. Hafner and G. Kresse, The Vienna AB-Initio Simulation Program VASP: An Efficient and Versatile Tool for Studying the Structural, Dynamic, and Electronic Properties of Materials, in Properties of Complex Inorganic Solids (Springer US, 1997) pp. 69–82; https://doi.org/10.1007/978-1-4615-5943-6_10.

J. W. Furness, A. D. Kaplan, J. Ning, J. P. Perdew, and J. Sun, Accurate and Numerically Efficient r2SCAN Meta-Generalized Gradient Approximation, J. Phys. Chem. Lett. 11, 8208 (2020); https://doi.org/10.1021/acs.jpclett.0c02405.

G. Kresse and D. Joubert, From ultrasoft pseudopotentials to the projector augmented-wave method, Phys. Rev. B 59, 1758 (1999); https://doi.org/10.1103/physrevb.59.1758.

H. J. Monkhorst and J. D. Pack, Special points for Brillouin-zone integrations, Phys. Rev. B 13, 5188 (1976); https://doi.org/10.1103/PhysRevB.13.5188.

V. Wang, N. Xu, J.-C. Liu, G. Tang, and W.-T. Geng, VASPKIT: A user-friendly interface facilitating high-throughput computing and analysis using VASP code, Comput. Phys. Commun. 267, 108033 (2021); https://doi.org/10.1016/j.cpc.2021.108033.

R. Hill, The Elastic Behaviour of a Crystalline Aggregate, Proc. Phys. Soc. A 65, 349 (1952); https://doi.org/10.1088/0370-1298/65/5/307.

D. M. Teter, Computational alchemy: The search for new superhard materials, MRS Bull. 23, 22 (1998); https://doi.org/10.1557/S0883769400031420.

X.-Q. Chen, H. Niu, D. Li, and Y. Li, Modeling hardness of polycrystalline materials and bulk metallic glasses, Intermetallics 19, 1275 (2011); https://doi.org/10.1016/j.intermet.2011.03.026.

Y. Tian, B. Xu, and Z. Zhao, Microscopic theory of hardness and design of novel superhard crystals, Int. J. Refract. Met. Hard Mater. 33, 93 (2012); https://doi.org/10.1016/j.ijrmhm.2012.02.021.

N. Miao, B. Sa, J. Zhou, and Z. Sun, Theoretical investigation on the transition-metal borides with Ta3B4-type structure: A class of hard and refractory materials, Comput. Mater. Sci. 50, 1559 (2011); https://doi.org/10.1016/j.commatsci.2010.12.015.

E. Mazhnik and A. R. Oganov, A model of hardness and fracture toughness of solids, J. Appl. Phys. 126, 125109 (2019); https://doi.org/10.1063/1.5113622.

P. Prysyazhnyuk and D. Di Tommaso, The thermodynamic and mechanical properties of Earth-abundant metal ternary boride Mo2(Fe,Mn)B2 solid solutions for impact- and wear-resistant alloys, Mater. Adv. 4, 3822 (2023); https://doi.org/10.1039/D3MA00313B.

G. te Velde and E. J. Baerends, Precise density-functional method for periodic structures, Phys. Rev. B 44, 7888 (1991); https://doi.org/10.1103/PhysRevB.44.7888.

J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized Gradient Approximation Made Simple, Phys. Rev. Lett. 77, 3865 (1996); https://doi.org/10.1103/PhysRevLett.77.3865.

X. Zhang, J.D. Comins, A.G. Every and P.R. Stoddart, Surface Brillouin scattering studies on vanadium carbide, Int. J. Refract. Met. Hard Mater. 16, 303 (1998); https://doi.org/10.1016/S0263-4368(98)00046-8.

S. F. Pugh, XCII. Relations between the elastic moduli and the plastic properties of polycrystalline pure metals, Philos. Mag. 45, 823 (1954); https://doi.org/10.1080/14786440808520496.