2.6 Navier's equation

2.6.1 Derivation

2.6.2 Analytical solutions

2.6.3 Numerical solutions

2.6.4 Example with FreeFEM++

2.6.5 Exercise 2: Vertical transverse isotropic elastic properties

[To be developed] VTI compliance matrix (3 - vertical direction perpendicular to bedding):

$\begin{displaymath}%compliance matrix \left[ \begin{array}{c} \varepsilon_{11} ... ...ma_{13} \cfrac{}{}\\ \sigma_{12} \cfrac{}{} \end{array}\right]\end{displaymath}$

(2.1)

where and $G_h = \cfrac{E_h}{2(1+\nu_h)}$ and $G_v$ is not related to the other parameters.

In terms of stiffness coefficients:

$\displaystyle E_h = \frac{(C_{11}-C_{12}) \left[ C_{33}(C_{11}+C_{12})-2\: C_{13}^2 \right]}{C_{11}C_{33}–C_{13}^2}$

$\displaystyle E_v = C_{33} - \frac{2\:C_{13}^2}{C_{11}+C_{12}}$

$\displaystyle \nu_h = \frac{C_{12}C_{33}-C_{13}^2}{C_{11}C_{33}-C_{13}^2}$

$\displaystyle \nu_v = \frac{C_{13}}{C_{11}+C_{12}}$

$\displaystyle G_v = C_{44}$

$\displaystyle G_h = C_{66} = \frac{C_{11}-C_{12}}{2}$

VTI stiffness matrix (3 - vertical direction perpendicular to bedding):

$\begin{displaymath}%compliance matrix \left[ \begin{array}{c} \sigma_{11} \\ \... ...\ 2 \varepsilon_{13} \\ 2 \varepsilon_{12} \end{array}\right]\end{displaymath}$

(2.2)

or in terms of Young moduli and Poisson ratios

$\displaystyle C_{11} = \left[ \frac{1}{(1-\nu_h) E_v - 2 \nu_v^2 E_h} \right] \left( \frac{E_h E_v - \nu_v^2 E_h^2}{1+\nu_h} \right)$

$\displaystyle C_{33} = \left[ \frac{1}{(1-\nu_h) E_v - 2 \nu_v^2 E_h} \right] (E_v^2 - \nu_h E_v^2)$

$\displaystyle C_{12} = \left[ \frac{1}{(1-\nu_h) E_v - 2 \nu_v^2 E_h} \right] \left( \frac{\nu_v^2 E_h^2 + \nu_h E_h E_v}{1+\nu_h} \right)$

$\displaystyle C_{13} = \left[ \frac{1}{(1-\nu_h) E_v - 2 \nu_v^2 E_h} \right] (\nu_v E_h E_v)$

$\displaystyle C_{66} = \frac{C_{11}-C_{12}}{2} = G_h = \cfrac{E_h}{2(1+\nu_h)}$

The parameter $C_{44}$ is independent of all other parameters.

2.6.6 Exercise 3: Displacement Field and Strain Field

[To be developed]