This program calculates hardness and fracture toughness from elastic tensor as described in the following article:

E. Mazhnik and A. R. Oganov. A model of hardness and fracture toughness of solids. J. Appl. Phys. 126, 125109 (2019).

Please, write the symmetric 6x6 elastic tensor for the crystal structure in the Voigt notation:

Volume per atom, Å^{3}: (only for fracture toughness)
Please, be aware that for metals, intermetallics and alloys the values of hardness are not accurate and the fracture toughness
should include the enhancement factor, which depends on the chemical bond and is not implemented by the current program.

Calculated properties

Averaging scheme

Bulk modulus, GPa

Young's modulus, GPa

Shear modulus, GPa

Poisson's ratio

Voigt

Reuss

Hill

Vickers Hardness, GPa

Fracture toughness, MPa m^{1/2}

Calculation using the average values

If you don't have a full elastic tensor, but only the average values, you can use the following
form where you need to specify only 2 of 4 values – the rest are calculated using isotropic approximation.