FLAC3D Theory and Background • Constitutive Models

Orthotropic Elastic Model

The orthotropic model accounts for three orthogonal planes of elastic symmetry. Principal coordinate axes of elasticity, labeled 1’,2’,3’, are defined in the directions normal to those planes.

The incremental strain-stress relations in the local axes have the form

(1)\[\begin{split}\begin{Bmatrix} \Delta {\epsilon'}_{11} \\ \Delta {\epsilon'}_{22} \\ \Delta {\epsilon'}_{33} \\ 2\Delta {\epsilon'}_{12} \\ 2\Delta {\epsilon'}_{13} \\ 2\Delta {\epsilon'}_{23} \end{Bmatrix} = \begin{bmatrix} {{1}\over{E_1}} & -{{\nu_{12}}\over{E_2}} & -{{\nu_{13}}\over{E_3}} & & & \\ -{{\nu_{21}}\over{E_1}} & {{1}\over{E_2}} & -{{\nu_{23}}\over{E_3}} & & & \\ -{{\nu_{31}}\over{E_1}} & -{{\nu_{32}}\over{E_2}} & {{1}\over{E_3}} & & & \\ & & & {{1}\over{G_{12}}} & & \\ & & & & {{1}\over{G_{13}}} & \\ & & & & & {{1}\over{G_{23}}} \end{bmatrix} \begin{Bmatrix} \Delta {\sigma'}_{11} \\ \Delta {\sigma'}_{22} \\ \Delta {\sigma'}_{33} \\ \Delta {\sigma'}_{12} \\ \Delta {\sigma'}_{13} \\ \Delta {\sigma'}_{23} \end{Bmatrix}\end{split}\]

The model involves nine independent elastic constants:

\(E_1; E_2; E_3\)	Young’s moduli in the directions of the local axes
\(G_{23}; G_{13}; G_{12}\)	shear moduli in planes parallel to the local coordinate planes
\(\nu_{12}; \nu_{13}; \nu_{23}\)	Poisson’s ratio where \(\nu_{ij}\) characterizes lateral contraction in local direction \(i'\) caused by tensile stress in local direction \(j'\)

By virtue of the symmetry of the strain-stress matrix, we have

(2)\[{\nu_{21} \over {E_1}} = {\nu_{12} \over {E_2}}\]

(3)\[{\nu_{31} \over {E_1}}= {\nu_{13} \over {E_3}}\]

(4)\[{\nu_{32} \over {E_2}} = {\nu_{23} \over {E_3}}\]

In addition to those nine properties, the user prescribes the orientation of the local axes by giving the dip and dip direction of the (1’,2’) plane, and the rotation angle between the 1’ axis and the dip-direction vector (defined in positive sense from the dip direction vector). Default values for all properties are zero.

In the FLAC3D implementation of this model, the local stiffness matrix \([K']\) is found by inversion of the symmetric matrix in Equation (1). Using \(\Delta [\sigma']\) and \(\Delta [\epsilon']\) to represent the incremental stress and strain vectors present in the right and left members of Equation (1):

(5)\[\Delta [\sigma'] = [K'] \Delta [\epsilon']\]

In the global axes, the incremental stress-strain relations are

(6)\[\Delta [\sigma] = [K] \Delta [\epsilon]\]

In FLAC3D, the global stiffness matrix \([K]\) is calculated by applying a transformation of the form

(7)\[[K] = [Q]^T[K'][Q]\]

where \([Q]\) is a suitable 6 × 6 matrix involving direction cosines of local axes in global axes (Q is derived from the relations \({\sigma'}_{ij} = c_{ik} \sigma_{kl} c_{jl}\), where \(c_{ij}\) is direction cosine \(j\) of local axis \(i\)).

In particular, if the local axes are obtained from the global axes by positive rotation through an angle \(\theta\) about the common 3 ≡ 3’ axis, we have

(8)\[\begin{split}Q = \begin{bmatrix} {\cos^2 \theta} & {\sin^2 \theta} & & -\sin \theta \cos \theta & & \\ {\sin^2 \theta} & {\cos^2 \theta} & & +\sin \theta \cos \theta & & \\ & & 1 & & & \\ 2 \sin \theta \cos \theta & -2 \sin \theta \cos \theta & & {\cos^2 \theta} - {\sin^2 \theta} & & \\ & & & & \cos \theta & -\sin \theta \\ & & & & \sin \theta & \cos \theta \end{bmatrix}\end{split}\]

The matrix for rotation about the 1 ≡ 1’ or 2 ≡ 2’ axis may be obtained by cyclic permutation of indices.

orthotropic Model Properties

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

orthotropic

dip f

dip angle [degrees] of the plane defined by axes 1’-2’

dip-direction f

dip direction [degrees] of the plane defined by axes 1’-2’

normal v

normal direction of the planes of symmetry, (\(n_x,n_y,n_z\))

normal-x f

\(x\)-component of unit normal to plane defined by axes 2’-3’, \(n_x\)

normal-y f

\(y\)-component of unit normal to plane defined by axes 1’-3’, \(n_y\)

normal-z f

\(z\)-component of unit normal to plane defined by axes 1’-2’, \(n_z\)

poisson-12 f

Poisson’s ratio characterizing lateral contraction in direction 1’ when tension is applied in direction 2’, \({\nu}_{12}\)

poisson-13 f

Poisson’s ratio characterizing lateral contraction in direction 1’ when tension is applied in direction 3’, \({\nu}_{13}\)

poisson-23 f

Poisson’s ratio characterizing lateral contraction in direction 2’ when tension is applied in direction 3’, \({\nu}_{23}\)

rotation f

rotation [degrees] angle of axes 1’ and 2’ around 3’ in the plane defined by axes 1’-2’, or the angle between axis 1’ and the steepest direction in the plane defined by axes 1’-2’.

shear-12 f

shear modulus in planes parallel to axes 1’-2’, \(G_{12}\)

shear-13 f

shear modulus in planes parallel to axes 1’-3’, \(G_{13}\)

shear-23 f

shear modulus in planes parallel to axes 2’-3’, \(G_{23}\)

young-1 f

Young’s modulus in direction 1’, \(E_{1}\)

young-2 f

Young’s modulus in direction 2’, \(E_{2}\)

young-3 f

Young’s modulus in direction 3’, \(E_{3}\)