The bulk modulus, shear modulus, Young’s modulus, and Poisson’s ratio were estimated according to Hooke’s law and the Voigt-Reuss-Hill (VRH) model [25 – 26]. For hexagonal polycrystalline crystal:

• B=[2(C11+C12)+4C13+C33]/9,
• G=(C11+C12+2C33−4C13+12C44+12C66)/30,
• E=9BG/(3B+G),
• ν=(3B−2G)/2(3B+G),

The Vickers hardness (HV) was calculated according to the empirical formula [19]:

• HV= 2(K^2G)0.585−3,
• K=G/B.

The values of universal anisotropy factor (A^U) and anisotropy factor of shear modulus (A^G), which are associated with plastic deformation, have been calculated in agreement with [17, 27] and shown below:

* A^U=5(GV/GR)+(BV/BR)−6≥0
* A^G=(GV−GR)/(GV+GR),

where BV (BR) and GV (GR) are bulk and shear moduli in Voigt [28] (Reuss [29]) approximation, respectively

• 24. D. Sholl, J. Steckel. Density functional theory: a practical introduction. Wiley (2011).
• 25. G. Sin’ko. Physical Review B. 77 (10), 104118 (2008). Crossref
• 26. D. Chung, W. Buessem. Journal of Applied Physics. 38, 2535 (1967). Crossref
• 27. S. Ranganathan , M. Ostoja-Starzewski. Physical Review Letters. 101 (5), 055504 (2008). Crossref
• 28. W. Voigt. Lehrbuch der kristallphysik. (1928) 962 p.
• 29. A. Reuss, Z. Angew. Math. Mech. 9, 49 (1929).

Young’s Modulus Surface and Poisson’s Ratio Curve for Orthorhombic Crystals

The Stationary Values of Young's Modulus For Monoclinic and Triclinic Materials

• Pymatgen
• ASE