Orthotropic elastic constitutive law for solid elements at constant temperature.

This law is used for mechanical analysis of orthotropic elasticity undergoing large strains.

There are 9 independent parameters : $E_1$, $E_2$, $E_3$, $\nu_{12}$, $\nu_{13}$, $\nu_{23}$, $G_{12}$, $G_{13}$, $G_{23}$. The elastic compliance tensor is : \[D^e_{ijkl} = \begin{bmatrix} \frac{1}{E_1} & \frac{-\nu_{21}}{E_2} & \frac{-\nu_{31}}{E_3} & & & \\ \frac{-\nu_{12}}{E_1} & \frac{1}{E_2} & \frac{-\nu_{32}}{E_3} & & & \\ \frac{-\nu_{13}}{E_1} & \frac{-\nu_{23}}{E_2} & \frac{1}{E_3} & & & \\ & & & \frac{1}{2G_{12}} & & \\ & & & & \frac{1}{2G_{13}} & \\ & & & & & \frac{1}{2G_{23}}\\ \end{bmatrix}\]

The compliance elastic tensor is symmetric (Love, 1944), thus the equalities blow must be satisfied : \[\frac{\nu_{21}}{E_2} = \frac{\nu_{12}}{E_1} \quad , \quad \frac{\nu_{31}}{E_3} = \frac{\nu_{13}}{E_1} \quad , \quad \frac{\nu_{23}}{E_2} = \frac{\nu_{32}}{E_3} \]

Moreover, the strain energy function must be positive (i.e. the quadratic form is said positive definite) : \[\mathbf{U} = \frac{1}{2} \varepsilon_{ij}\varepsilon_{kl}C_{ijkl} > 0\]

Thus, the condition below must be satisfied : \[1 - \nu_{12}\nu_{21} > 0 \quad ; \quad 1 - \nu_{13}\nu_{31} > 0 \quad ; \quad 1 - \nu_{23}\nu_{32} > 0\]\[1-\nu_{12}\nu_{23}\nu_{31}-\nu_{21}\nu_{13}\nu_{32} - \nu_{12}\nu_{21} - \nu_{13}\nu_{31} - \nu_{23}\nu_{32} > 0 \]\[E_1 > 0 \quad ; \quad E_2 > 0 \quad ; \quad E_3 > 0\]\[G_1 > 0 \quad ; \quad G_2 > 0 \quad ; \quad G_3 > 0\]

By inverting the matrix, the elastic tensor is then : \[C^e_{ijkl} = \begin{bmatrix} \frac{1-\nu_{23}\nu_{32}}{E_2E_3det} & \frac{\nu_{21}+\nu_{31}\nu_{23}}{E_2E_3det} & \frac{\nu_{21}\nu_{32}+\nu_{31}}{E_2E_3det} & & & \\ \frac{\nu_{12}+\nu_{13}\nu_{32}}{E_1E_3det} & \frac{1-\nu_{13}\nu_{31}}{E_1E_3det} & \frac{\nu_{32}+\nu_{31}\nu_{12}}{E_1E_3det} & & & \\ \frac{\nu_{13}+\nu_{23}\nu_{12}}{E_1E_2det} & \frac{\nu_{23}+\nu_{21}\nu_{13}}{E_1E_2det} & \frac{1-\nu_{21}\nu_{12}}{E_1E_2det} & & & \\ & & & 2G_{12} & & \\ & & & & 2G_{13} & \\ & & & & & 2G_{23} \\ \end{bmatrix}\] with $det=\dfrac{1-\nu_{31}\nu_{13}-\nu_{21}\nu_{12}-\nu_{32}\nu_{23}-2\nu_{31}\nu_{12}\nu_{23}}{E_1E_2E_3}$

There are 5 independent parameters : ${E_{\parallel}}$, ${E_{\perp}}$, ${\nu_{\parallel\parallel}}$, ${\nu_{\parallel\perp}}$, ${G_{\parallel,\perp}}$.

From orthotropic elasticity let us consider ($e_1$,$e_2$) as the isotropic plane (bedding plane for sedimentary rock) and $e_3$ the normal to this plane. The subscripts ${\parallel}$ and $\perp$ indicates, respectively, the direction parallel to bedding and perpendicular to bedding. \[{E_1=E_2=E_{\parallel}}\quad , \quad {E_3=E_{\perp}}\]

The elastic compliance tensor becomes : \[D^e_{ijkl} = \begin{bmatrix} \frac{1}{E_{\parallel}} & \frac{-\nu_{\parallel\perp}}{E{\parallel}} & \frac{-\nu_{\perp\parallel}}{E_{\perp}} & & & \\ \frac{-\nu_{\parallel\parallel}}{E_{\parallel}} & \frac{1}{E_{\parallel}} & \frac{-\nu_{\perp\parallel}}{E_{\perp}} & & & \\ \frac{-\nu_{\parallel\perp}}{E_{\parallel}} & \frac{-\nu_{\parallel\perp}}{E_{\parallel}} & \frac{1}{E_{\perp}} & & & \\ & & & \frac{1}{2G_{\parallel\parallel}} & & \\ & & & & \frac{1}{2G_{\parallel\perp}} & \\ & & & & & \frac{1}{2G_{\parallel\perp}}\\ \end{bmatrix}\]

The compliance elastic tensor is symmetric (Love, 1944), thus the equalities blow must be satisfied : \[\frac{\nu_{\perp\parallel}}{E_{\perp}} = \frac{\nu_{\parallel\perp}}{E_{\parallel}}\]

In the isotropic plane, the shear modulus is obtained as follow : \[G_{\parallel\parallel} = \frac{E_{\parallel}}{2(1+\nu_{\parallel\parallel})}\]

Because of the symmetry of the stress and strain tensors : \[G_{\parallel\perp}=G_{\perp\parallel}\]

The Hooke’s law is defined in the orthotropic axes for orthotropic elasticity. As a result, a change of the reference system is needed to obtain the stress in the global axes. In the purpose of estimating the stresses in the global axes, a relation taking into account this change in the reference system is proposed. This relation is (Cescotto, 1995) : \[\sigma_{ij} = R_{ik}R_{jl}\sigma'_{kl}\] where $R_{ij}$ is a component of the matrix of rotation, $\sigma_{ij}$ the stress in the orthotropic axes and $\sigma'_{ij}$ the stresses in the current configuration. 

Generally, the matrix of rotation is characterized by the Euler’s angles. The positive direction of rotation is counter-clockwise.

The (X,Y,Z) space represents the current configuration (or global configuration) while the ($\underline{e_1}$,$\underline{e_2}$,$\underline{e_3}$) space represents the orthotropic configuration. To define the rotation, let consider that the cartesian system are equal. The rotation is decomposed in 3 steps and the definition of the angles will be:

• The angle $\alpha$ defines a rotation around the axis Z, thus the axis $\underline{e_2}$ and $\underline{e_1}$ rotates from an angle of amplitude $\alpha$ in the (X,Y) plane.
• The angle $\phi$ defines a rotation around the axis $\underline{e_2}$.
• The angle $\theta$ defines a rotation around the axis $\underline{e_1}$.
  

The matrix which defines the rotation may be written : \[R = \begin{bmatrix} \cos\alpha\cos\phi & \sin\alpha\cos\phi & -\sin\phi \\ -\sin\alpha\cos\theta+\sin\theta\sin\phi\cos\alpha & \cos\alpha\cos\theta+\sin\theta\sin\phi\sin\alpha & \sin\theta\cos\phi \\ \sin\theta\sin\alpha +\cos\alpha\sin\phi\cos\theta & \sin\phi\sin\alpha\cos\theta-\sin\theta\cos\alpha & \cos\phi\cos\theta \end{bmatrix}\]

Prepro : LORTHO.F
Lagamine: ORTHO3D.F

6 for 3D analysis

The stresses are the components of CAUCHY stress tensor in global (X,Y,Z) coordinates.

For the 3-D state:

0 (see ELA3D)