Elasticity

Hooke’s law

Strain \(\varepsilon\) (dimensionless : \(\varepsilon = \frac{\partial u}{u}\), where \(u\) is a distance). Stress \(\sigma\) (\(Pa\), equivalent to \(N.m^{-2}\)) Modulus of elasticity or Young’s Modulus \(E\) (\(Pa\), equivalent to \(N.m^{-2}\)) \[\sigma = E.\varepsilon\]

Tensors

\[\sigma_{xy} = \begin{pmatrix} \sigma_{11}&\sigma_{12}&\sigma_{13}& \\\ \sigma_{21}&\sigma_{22}&\sigma_{33}& \\\ \sigma_{31}&\sigma_{32}&\sigma_{33}& \\\ \end{pmatrix} , \varepsilon_{xy} = \begin{pmatrix} \varepsilon_{11}&\varepsilon_{12}&\varepsilon_{13}& \\\ \varepsilon_{21}&\varepsilon_{22}&\varepsilon_{33}& \\\ \varepsilon_{31}&\varepsilon_{32}&\varepsilon_{33}& \\\ \end{pmatrix}\]

Symmetry implies \(\sigma_{ij}=\sigma_{ji}\) and \(\varepsilon_{ij}=\varepsilon_{ji}\) which leaves 6 unique terms in both tensors.

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

Elastic material constants

Young’s Modulus , (\(Pa \Leftrightarrow N.m^{-2} \Leftrightarrow kg.m^{-1}.s^{-2}\)).

\[E = \frac{\sigma_{11}}{\varepsilon_{11}}\]

As a function of seismic velocities and density: \(E = \rho \frac{V_s^2(3V_p^2-4V_s^2)}{V_p^2 - V_s^2}\) As a fonction of P and S Impedances: \(E\rho = \frac{I_s^2(3I_p^2-4I_s^2)}{I_p^2 - I_s^2}\)

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

\[\nu = \frac{\varepsilon_{33}}{\varepsilon_{11}}\]

As a function of seismic velocities and impedances \(\nu = \frac{1}{2} \frac{\frac{V_p^2}{V_s^2} - 2 } {\frac{V_p^2}{V_s^2} - 1 } = \frac{1}{2} \frac{\frac{I_p^2}{I_s^2} - 2 } {\frac{I_p^2}{I_s^2} - 1 }\)

Shear Modulus \(\mu\) , (\(Pa \Leftrightarrow N.m^{-2} \Leftrightarrow kg.m^{-1}.s^{-2}\)).

\[\mu = \frac{1}{2} . \frac{\sigma_{13}}{\varepsilon_{13}}\]

Bulk Modulus \(K\) , (\(Pa \Leftrightarrow N.m^{-2} \Leftrightarrow kg.m^{-1}.s^{-2}\)).

\[K = \frac{\sigma_{00}}{\varepsilon_{00}} = \frac{\sigma_{11}+\sigma_{22}+\sigma_{33}}{\varepsilon_{11}+\varepsilon_{22}+\varepsilon_{33}}\]

Seismic velocities as a function of Lame parameters

P-Wave velocity (\(m.s^{-1}\))

\[V_{p}=\sqrt{\frac{K+\frac{4}{3}\mu}{\rho}} = \sqrt{\frac{\lambda+2\mu}{\rho}}\]

S-Wave velocity (\(m.s^{-1}\))

\[V_{s}=\sqrt{\frac{\mu}{\rho}}\]

Lame parameters (\(Pa \Leftrightarrow N.m^{-2} \Leftrightarrow kg.m^{-1}.s^{-2}\)):

\[\mu = V_s^2 \rho\] \[\lambda = \rho (V_p^2 - 2V_s^2)\]

Stiffness pmatrix - Voigt notation

In the anisotropic case, the relation between stress and strain is a 6 by 6 tensor. \[\begin{pmatrix} \sigma_{1}\\\ \sigma_{2}\\\ \sigma_{3}\\\ \sigma_{4}\\\ \sigma_{5}\\\ \sigma_{6}\\\ \end{pmatrix} = \begin{pmatrix} c_{11}&c_{12}&c_{13}&c_{14}&c_{15}&c_{16}\\\ c_{21}&c_{22}&c_{23}&c_{24}&c_{25}&c_{26}\\\ c_{31}&c_{32}&c_{33}&c_{34}&c_{35}&c_{36}\\\ c_{41}&c_{42}&c_{43}&c_{44}&c_{45}&c_{46}\\\ c_{51}&c_{52}&c_{53}&c_{54}&c_{55}&c_{56}\\\ c_{61}&c_{62}&c_{63}&c_{64}&c_{65}&c_{66}\\\ \end{pmatrix} . \begin{pmatrix} \varepsilon_{1}\\\ \varepsilon_{2}\\\ \varepsilon_{3}\\\ \varepsilon_{4}\\\ \varepsilon_{5}\\\ \varepsilon_{6}\\\ \end{pmatrix}\]

where \[\begin{pmatrix} \sigma_{1}\\\ \sigma_{2}\\\ \sigma_{3}\\\ \sigma_{4}\\\ \sigma_{5}\\\ \sigma_{6}\\\ \end{pmatrix} = \begin{pmatrix} \sigma_{11}\\\ \sigma_{22}\\\ \sigma_{33}\\\ \sigma_{23}\\\ \sigma_{13}\\\ \sigma_{12}\\\ \end{pmatrix} , and, \begin{pmatrix} \varepsilon_{1}\\\ \varepsilon_{2}\\\ \varepsilon_{3}\\\ \varepsilon_{4}\\\ \varepsilon_{5}\\\ \varepsilon_{6}\\\ \end{pmatrix} = \begin{pmatrix} \varepsilon_{11}\\\ \varepsilon_{22}\\\ \varepsilon_{33}\\\ \varepsilon_{23}\\\ \varepsilon_{13}\\\ \varepsilon_{12}\\\ \end{pmatrix}\]

Stiffness matrix - Isotropic Material

In the isotropic case, the relation between stress and strain depends only on 2 parameters : \(c_{11}\) and \(c_{44}\). The isotropic solid has no preferred direction regarding elastic properties. \[C= \begin{pmatrix} c_{11}&c_{11}-2c_{44}&c_{11}-2c_{44}&0&0&0\\\ c_{11}-2c_{44}&c_{11}&c_{11}-2c_{44}&0&0&0\\\ c_{11}-2c_{44}&c_{11}-2c_{44}&c_{11}&0&0&0\\\ 0&0&0&c_{44}&0&0\\\ 0&0&0&0&c_{44}&0\\\ 0&0&0&0&0&c_{44}\\\ \end{pmatrix}\] \[C= \begin{pmatrix} \lambda + 2\mu & \lambda & \lambda&0&0&0\\\ \lambda & \lambda + 2\mu & \lambda&0&0&0\\\ \lambda & \lambda & \lambda + 2\mu &0&0&0\\\ 0&0&0&\mu&0&0\\\ 0&0&0&0&\mu&0\\\ 0&0&0&0&0&\mu\\\ \end{pmatrix}\]

Stiffness matrix - VTI Material

Material with a vertical axis of symmetry (layer cake), 5 parameters in the stiffness matrix. \[C= \begin{pmatrix} c_{11}&c_{11}-2c_{66}&c_{13}&0&0&0\\\ c_{11}-2c_{66}&c_{11}&c_{13}&0&0&0\\\ c_{13}&c_{13}&c_{33}&0&0&0\\\ 0&0&0&c_{44}&0&0\\\ 0&0&0&0&c_{44}&0\\\ 0&0&0&0&0&c_{66}\\\ \end{pmatrix}\] \[V_{PH} = \sqrt{\frac{c_{11}}{\rho}},\quad V_{PV} = \sqrt{\frac{c_{33}}{\rho}},\quad V_{SH} = \sqrt{\frac{c_{44}}{\rho}},\quad V_{SV} = \sqrt{\frac{c_{66}}{\rho}};\]

Stiffness matrix - HTI Material

Material with an horizontal axis of symmetry, this is the transposed of the VTI case, 5 parameters. \[C= \begin{pmatrix} c_{11}&c_{13}&c_{13}&0&0&0\\\ c_{13}&c_{33}&c_{33}-2c_{44}&0&0&0\\\ c_{13}&c_{33}-2c_{44}&c_{33}&0&0&0\\\ 0&0&0&c_{44}&0&0\\\ 0&0&0&0&c_{55}&0\\\ 0&0&0&0&0&c_{55}\\\ \end{pmatrix}\] \[V_{PH} = \sqrt{\frac{c_{11}}{\rho}},\quad V_{PV} = \sqrt{\frac{c_{33}}{\rho}},\quad V_{SH} = \sqrt{\frac{c_{44}}{\rho}},\quad V_{SV} = \sqrt{\frac{c_{55}}{\rho}};\]

Stiffness matrix - Orthorhombic Material

Material with three mutually orthogonal axis of symmetry : 9 parameters in the stiffness matrix. \[C= \begin{pmatrix} c_{11}&c_{12}&c_{13}&0&0&0\\\ c_{12}&c_{22}&c_{23}&0&0&0\\\ c_{13}&c_{23}&c_{33}&0&0&0\\\ 0&0&0&c_{44}&0&0\\\ 0&0&0&0&c_{55}&0\\\ 0&0&0&0&0&c_{66}\\\ \end{pmatrix}\] \[\begin{aligned} V_{PH1} = \sqrt{\frac{c_{11}}{\rho}},\quad V_{PH2} = \sqrt{\frac{c_{22}}{\rho}},\quad V_{PV} = \sqrt{\frac{c_{33}}{\rho}}; \end{aligned}\] \[\begin{aligned} V_{SH1} = \sqrt{\frac{c_{44}}{\rho}},\quad V_{SH2} = \sqrt{\frac{c_{55}}{\rho}},\quad V_{SV} = \sqrt{\frac{c_{66}}{\rho}}; \end{aligned}\]