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Rock Physics

Elasticity

Basics of Elasticity for the Geophysicist in a Hurry.

Hooke’s law

Strain ε (dimensionless : ε=uu, where u is a distance). Stress σ (Pa, equivalent to N.m2) Modulus of elasticity or Young’s Modulus E (Pa, equivalent to N.m2) σ=E.ε

Tensors

σxy=(σ11σ12σ13 σ21σ22σ33 σ31σ32σ33 ),εxy=(ε11ε12ε13 ε21ε22ε33 ε31ε32ε33 )

Symmetry implies σij=σji and εij=εji which leaves 6 unique terms in both tensors.

Cauchy Stress Tensor

Cauchy Stress Tensor. By Sanpaz (Own work) CC BY-SA 3.0 or GFDL], via Wikimedia Commons

Elastic material constants

Young’s Modulus , (PaN.m2kg.m1.s2).

E=σ11ε11

As a function of seismic velocities and density: E=ρV2s(3V2p4V2s)V2pV2s As a fonction of P and S Impedances: Eρ=I2s(3I2p4I2s)I2pI2s

Poisson’s Ratio (dimensionless : ratio of strains).

ν=ε33ε11

As a function of seismic velocities and impedances ν=12V2pV2s2V2pV2s1=12I2pI2s2I2pI2s1

Shear Modulus μ , (PaN.m2kg.m1.s2).

μ=12.σ13ε13

Bulk Modulus K , (PaN.m2kg.m1.s2).

K=σ00ε00=σ11+σ22+σ33ε11+ε22+ε33

Seismic velocities as a function of Lame parameters

P-Wave velocity (m.s1)

Vp=K+43μρ=λ+2μρ

S-Wave velocity (m.s1)

Vs=μρ

Lame parameters (PaN.m2kg.m1.s2):

μ=V2sρ λ=ρ(V2p2V2s)

Stiffness pmatrix - Voigt notation

In the anisotropic case, the relation between stress and strain is a 6 by 6 tensor. (σ1 σ2 σ3 σ4 σ5 σ6 )=(c11c12c13c14c15c16 c21c22c23c24c25c26 c31c32c33c34c35c36 c41c42c43c44c45c46 c51c52c53c54c55c56 c61c62c63c64c65c66 ).(ε1 ε2 ε3 ε4 ε5 ε6 )

where (σ1 σ2 σ3 σ4 σ5 σ6 )=(σ11 σ22 σ33 σ23 σ13 σ12 ),and,(ε1 ε2 ε3 ε4 ε5 ε6 )=(ε11 ε22 ε33 ε23 ε13 ε12 )

Stiffness matrix - Isotropic Material

In the isotropic case, the relation between stress and strain depends only on 2 parameters : c11 and c44. The isotropic solid has no preferred direction regarding elastic properties. C=(c11c112c44c112c44000 c112c44c11c112c44000 c112c44c112c44c11000 000c4400 0000c440 00000c44 ) C=(λ+2μλλ000 λλ+2μλ000 λλλ+2μ000 000μ00 0000μ0 00000μ )

Stiffness matrix - VTI Material

Material with a vertical axis of symmetry (layer cake), 5 parameters in the stiffness matrix. C=(c11c112c66c13000 c112c66c11c13000 c13c13c33000 000c4400 0000c440 00000c66 ) VPH=c11ρ,VPV=c33ρ,VSH=c44ρ,VSV=c66ρ;

Stiffness matrix - HTI Material

Material with an horizontal axis of symmetry, this is the transposed of the VTI case, 5 parameters. C=(c11c13c13000 c13c33c332c44000 c13c332c44c33000 000c4400 0000c550 00000c55 ) VPH=c11ρ,VPV=c33ρ,VSH=c44ρ,VSV=c55ρ;

Stiffness matrix - Orthorhombic Material

Material with three mutually orthogonal axis of symmetry : 9 parameters in the stiffness matrix. C=(c11c12c13000 c12c22c23000 c13c23c33000 000c4400 0000c550 00000c66 ) VPH1=c11ρ,VPH2=c22ρ,VPV=c33ρ; VSH1=c44ρ,VSH2=c55ρ,VSV=c66ρ;