What is the the bulk modulus formula for anisotropic material?

[cylee]

I can understand the derivation of bulk modulus (K) for isotropic material. However I have difficulty to do the same for anisotropic material.

to start with we have the definition:
mean_stress = K * (strain_xx+strain_yy+strain_zz)

My question is for anisotropic material:
Is mean_stress = (stress_xx+stress_yy+stress_zz) / 3 or something else?

when the material is subjected to mean_stress (or hydrostatic pressure if you would like), the shear stresses will be zero, the same as the isotropic case, correct?

Then how do we derive the bulk modulus formula for anisotropic material using hooke's law (compliance) coefficients?

Thanks!

By the way, here is my guess. Please feel free to correct it.

mean_stress = K * (volumetric_strain) (By definition)

mean_stress = K * (strain_xx+strain_yy+strain_zz)

mean_stress = K * [(S11+S21+S31)*stress_xx + (S12+S22+S32)*stress_yy + (S13+S23+S33)*stress_zz + (S14+S24+S34)*stress_xy + (S15+S25+S35)*stress_xz + (S16+S26+S36)*stress_yz] (From hooke's law)

mean_stress = K * (S11+S21+S31+S12+S22+S32+S13+S23+S33) * mean_stress (subjected to mean_stress)

K = 1/sum(Sij) for i,j=1:3

 

[AlephZero]

See section 2.8 of http://www.colorado.edu/engineering/CAS/Felippa.d/FelippaHome.d/Publications.d/Report.CU-CAS-02-09.pdf

 

[cylee]

The pdf suggests the use of effective bulk modulus. But as far as bulk modulus is concerned, it is the ratio between mean normal stress and volumetric strain, subjected to hydrostatic stress (which is the mean normal stress). This statement is the same as writing w=[1 1 1 0 0 0] for the effective bulk modulus for anisotropic material, which again degenerates to the conventional bulk modulus definition.
Sorry, I can't see the point of your attached pdf. In specific, I am still wondering whether my derivation of bulk modulus for anisotropic material is correct or not.

 

[cylee]

CONFIRMED FOUNDING HERE

Long story short: My derivation represents the lower bound of the bulk modulus called Reuss effective bulk modulus. My assumption and derivation are correct. Thank you for all your input I very much appreciate it.