What is the the bulk modulus formula for anisotropic material?

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Discussion Overview

The discussion revolves around the derivation of the bulk modulus formula for anisotropic materials, contrasting it with the established understanding for isotropic materials. Participants explore definitions, assumptions, and the application of Hooke's law in this context.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant expresses difficulty in deriving the bulk modulus for anisotropic materials, starting with the definition of mean stress.
  • There is a question regarding whether mean stress for anisotropic materials is defined as the average of the normal stresses or if it differs from that.
  • The participant suggests that under mean stress, shear stresses will be zero, similar to isotropic cases.
  • Another participant references a PDF that discusses the effective bulk modulus, indicating that it aligns with the conventional definition of bulk modulus when subjected to hydrostatic stress.
  • A later reply claims that the initial derivation represents the lower bound of the bulk modulus known as the Reuss effective bulk modulus, asserting that the assumptions and derivation are correct.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the derivation of the bulk modulus for anisotropic materials. There are competing views regarding the definitions and implications of the effective bulk modulus.

Contextual Notes

The discussion includes assumptions about the nature of stress and strain in anisotropic materials, as well as the implications of using different definitions of bulk modulus. There are unresolved mathematical steps and dependencies on specific definitions that are not fully clarified.

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

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