====== Elastic constants ======

===== Equations =====
[[https://sci-hub.ru/10.1016/j.cpc.2021.108068|MechElastic: A Python library for analysis of mechanical and elastic
properties of bulk and 2D materials]]

==== Hexagonal ====

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).

==== Orthorhombic ====

[[https://link.springer.com/article/10.1007/s10870-008-9370-6|Young’s Modulus Surface and Poisson’s Ratio Curve for Orthorhombic Crystals]]

==== Monoclinic and Triclinic ====
[[http://journals.cambridge.org/abstract_S1727719100000691|The Stationary Values of Young's Modulus For Monoclinic and Triclinic Materials]]

===== Theory =====
{{:theory:pasted:20230316-141329.png}}
{{:theory:pasted:20230316-143120.png}}
===== Python modules =====
  * [[https://github.com/materialsproject/pymatgen/blob/v2023.3.10/pymatgen/analysis/elasticity/elastic.py#L247-L252|Pymatgen]]
  * [[https://pymatgen.org/pymatgen.analysis.elasticity.elastic.html|ASE]]