# Relationship between bulk modulus and elastic constant

1. May 31, 2007

### only1892

I am getting the bulk modulus from fitting the equation of states and calculating the elastic constant. It seems that there is some relationship between bulk modulus B and elastic constant Cij, B=Cij/9,where i,j run from 1 to 3.
But my result is far away from this, say B=50 from fitting EOS and B=300 from elastic constant. Is this resonable or B=Cij/9 cann't be applied to some material(structure)?

Thanks a lot!

2. Jul 25, 2007

### Modey3

The relationship between the bulk modulus and the elastic constants is:

B = 1/3 (C11 + 2*C12) for polycrystalline, isotropic cubic crystals

If the material has lower symmetry like a orthorhombic material it would have more independent elastic constants. Remember the concept of the bulk modulus was developed for polycrystalline material only. Not that single crystals don't resist volumetric distortion - of course they do. This resistance is going to be system specific and to give further details I need more info.

Regards

Modey3

3. Nov 9, 2007

### handsomecat

To add, if the system is not cubic, there are two possibilities for the polycrystalline Bulk Modulus from the elastic constants.

Voight Average:

9B = ( c11 + c22 + c33 ) + 2*(c12 + c13 + c23)

Reuss Average

1/B = ( s11 + s22 + s33 ) + 2*(s12 + s13 + s23);

Details in the paper

"The Elastic Behaviour of a Crystalline Aggregate", by R.Hill.

Proceedings of the Physical Society of London, Section A. Vol 65, No 5. May 1952,pp349-354.