- #1
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
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