FLAC3D Theory and Background • Constitutive Models

Anisotropic (Transversely) Elastic Model

The transversely isotropic model takes a plane of isotropy into consideration. Let the axis of rotational symmetry, normal to the plane of isotropy, correspond to the local \(3'\) axis. This axis is a principal direction of elasticity. Also, any two perpendicular directions \(1',2'\), which are principal directions of elasticity, can be selected in the isotropic plane. With this convention, the transversely isotropic model may be considered as a particular case of the orthotropic model for which

(1)\[E_1 = E_2\]

(2)\[G_{13} = G_{23}\]

(3)\[\nu_{13} = \nu_{23}\]

(4)\[G_{12} = {{E_1}\over{2(1 + \nu_{12})}}\]

If, for clarity, we write

\(E = E_1= E_2\)	Young’s moduli in the plane of isotropy
\(E'= E_3\)	Young’s moduli in the direction normal to the plane of isotropy
\(\nu = \nu_{12}\)	Poisson’s ratio characterizing lateral contraction in the plane of isotropy when tension is applied in this plane
\(\nu' = \nu_{13} = \nu_{23}\)	Poisson’s ratio characterizing lateral contraction in the plane of isotropy when tension is applied in the direction normal to it
\(G = G_{12}\)	shear modulus for the plane of isotropy
\(G' = G_{13} = G_{23}\)	shear modulus for any plane normal to the plane of isotropy

The strain-stress relations in the local axes take the form

(5)\[\begin{split}\begin{Bmatrix} \Delta {\epsilon'}_{11} \\ \Delta {\epsilon'}_{22} \\ \Delta {\epsilon'}_{33} \\ 2\Delta {\epsilon'}_{12} \\ 2\Delta {\epsilon'}_{13} \\ 2\Delta {\epsilon'}_{23} \end{Bmatrix} = \begin{bmatrix} {{1}\over{E}} & -{{\nu}\over{E}} & -{{\nu'}\over{E'}}& & & \\ -{{\nu}\over{E}} & {{1}\over{E}} & -{{\nu'}\over{E'}}& & & \\ -{{\nu'}\over{E'}}& -{{\nu'}\over{E'}}& {{1}\over{E'}} & & & \\ & & & {{1}\over{G}} & & \\ & & & & {{1}\over{G'}} & \\ & & & & & {{1}\over{G'}} \end{bmatrix} \begin{Bmatrix} \Delta {\sigma'}_{11} \\ \Delta {\sigma'}_{22} \\ \Delta {\sigma'}_{33} \\ \Delta {\sigma'}_{12} \\ \Delta {\sigma'}_{13} \\ \Delta {\sigma'}_{23} \end{Bmatrix}\end{split}\]

The model involves the five independent elastic constants \(E, E', \nu, \nu'\), and \(G'\). The shear modulus, \(G\), is calculated by the code from the relation \(G = E/2 (1 + \nu)\) (See Equation (4)). In addition to those five properties, the user prescribes the orientation of the isotropic plane by giving its dip and dip direction in 3D, or angle in 2D. Default values for all properties are zero.

For symmetry of constitutive matrix, the following relation must be satisfied:

(6)\[\frac{\nu}{E} = \frac{\nu'}{E'}\]

The cross shear modulus, \(G_{xy}\), for anisotropic elasticity must be determined. Lekhnitskii (1981) suggests the following equation, based on laboratory testing of rock:

(7)\[G_{xz} = {{E_x E_z} \over {E_x(1 + 2 \nu_{xz}) + E_z}}\]

assuming the \(xy\)-plane is the plane of isotropy.

The FLAC3D implementation proceeds as described in the context of the orthotropic model.

Examples

Borehole Behavior in Elastic Anisotropic Materials

anisotropic Model Properties

Use the following keywords with the zone property (FLAC3D) or block zone property (3DEC) command to set these properties of the anisotropic (transversely-elastic) model.

anisotropic

dip f (3D ONLY)

dip angle [degrees] of the plane of isotropy

dip-direction f (3D ONLY)

dip direction [degrees] of the plane of isotropy

angle f (2D ONLY)

angle [degrees] of the plane of isotropy, taken counterclockwise from the x-axis

normal v

normal direction of the plane of isotropy, (2D or 3D vector)

normal-x f

\(x\)-component of unit normal to the plane of isotropy, \(n_x\)

normal-y f

\(y\)-component of unit normal to the plane of isotropy, \(n_y\)

normal-z f

\(z\)-component of unit normal to the plane of isotropy, \(n_z\) (3D ONLY)

poisson-normal f

Poisson’s ratio characterizing lateral contraction in the plane of isotropy when tension is applied normal to the plane, \({\nu}'\) = \({\nu}'_{13}\) = \({\nu}'_{23}\)

poisson-plane f

Poisson’s ratio characterizing lateral contraction in the plane of isotropy when tension is applied in the plane, \(\nu\) = \({\nu}_{12}\)

shear-normal f

shear modulus for any plane normal to the plane of isotropy, \(G'\) = \(G'_{13}\) = \(G'_{23}\)

young-plane f

Young’s modulus in the plane of isotropy, \(E\) = \(E_1\) = \(E_2\)

young-normal f

Young’s modulus normal to the plane of isotropy, \(E'\) = \(E_3\)