Eigenvectors for Transverse Isotropy

The $3\times 3$ system (10) can be analyzed fairly easily, and in particular the eigenfunctions and eigenvalues of this system can be obtained in general. However, such general results do not provide much physical insight into the problem we are trying to study, so instead of proceeding in this direction we will now restrict attention to transversely isotropic materials. This case is relevant to many layered earth materials and also industrial systems, and it is convenient because we can immediately eliminate one of the eigenvectors from further consideration. Three mutually orthogonal (but unnormalized) vectors of interest are:

$$\begin{eqnarray} v_1 = \left(\begin{array} {c} 1\\ 1\\ 1\\ \end{array}\right)... ..._3 = \left(\begin{array} {c} 1\\ 1\\ -2\\ \end{array}\right). \end{eqnarray}$$ (12)

Treating these vectors as stresses, the first corresponds to a simple hydrostatic stress, the second to a planar shear stress, and the third to a pure shear stress applied uniaxially along the z-axis (which would also be the symmetry axis for a layered system, but we are not treating such layered systems here). Transverse isotropy of the system under consideration requires: s₁₁ = s₂₂, s₁₃ = s₂₃, and for the poroelastic problem $\beta_1 = \beta_2$. Thus, it is immediately apparent that the planar shear stress v₂ is an eigenvector of the system, and furthermore it results in no contribution from the pore fluid. Therefore, this vector will be of no further interest here, and the system can thereby be reduced to $2\times 2$.

 

• Compliance formulation
  • Case I. $A^*_{33} - A^*_{11}/2 \ne 0$, A^*₁₃ = 0.
  • Case II. A^*₃₃ - A^*₁₁/2 = 0, $A^*_{13} \ne 0$.
  • Case III. $A^*_{33} - A^*_{11}/2 \ne 0$, $A^*_{13} \ne 0$.
• Stiffness formulation
• Effective and undrained shear moduli G_eff and G_u