# 16.10. Elastic Three-Dimensional Anisotropic Model

    BEGIN PARAMETERS FOR MODEL ELASTIC_3D_ANISOTROPIC
      #
      # Elastic constants
      #
      YOUNGS MODULUS = <real>
      POISSONS RATIO = <real>
      SHEAR MODULUS  = <real>
      BULK MODULUS   = <real>
      LAMBDA         = <real>
      TWO MU         = <real>
      #
      # Material coordinates system definition
      #
      COORDINATE SYSTEM              = <string>  coordinate_system_name
      DIRECTION FOR ROTATION         = <real>    1|2|3
      ALPHA                          = <real>    (degrees)
      SECOND DIRECTION FOR ROTATION  = <real>    1|2|3
      SECOND ALPHA                   = <real>    (degrees)
      #
      # Required parameters
      #
      STIFFNESS MATRIX 11 = <real>
      STIFFNESS MATRIX 22 = <real>
      STIFFNESS MATRIX 33 = <real>
      STIFFNESS MATRIX 12 = <real>
      STIFFNESS MATRIX 13 = <real>
      STIFFNESS MATRIX 23 = <real>
      STIFFNESS MATRIX 44 = <real>
      STIFFNESS MATRIX 55 = <real>
      STIFFNESS MATRIX 66 = <real>
      STIFFNESS MATRIX 45 = <real>
      STIFFNESS MATRIX 46 = <real>
      STIFFNESS MATRIX 56 = <real>
      STIFFNESS MATRIX 14 = <real>
      STIFFNESS MATRIX 15 = <real>
      STIFFNESS MATRIX 16 = <real>
      STIFFNESS MATRIX 24 = <real>
      STIFFNESS MATRIX 25 = <real>
      STIFFNESS MATRIX 26 = <real>
      STIFFNESS MATRIX 34 = <real>
      STIFFNESS MATRIX 35 = <real>
      STIFFNESS MATRIX 36 = <real>
      #
      # Thermal strain functions
      #
      THERMAL STRAIN 11 FUNCTION = <real>
      THERMAL STRAIN 22 FUNCTION = <real>
      THERMAL STRAIN 33 FUNCTION = <real>
      THERMAL STRAIN 12 FUNCTION = <real>
      THERMAL STRAIN 23 FUNCTION = <real>
      THERMAL STRAIN 13 FUNCTION = <real>
      #
      END [PARAMETERS FOR MODEL ELASTIC_3D_ANISOTROPIC]

The ELASTIC 3D ANISOTROPIC model is an extension of the ELASTIC model which allows for full anisotropy in both the material stiffness and thermal expansion. Each stiffness component is labeled with \(i\) and \(j\) indices which correspond to the components of stress and strain vectors in contracted notation,

\[\begin{split}\begin{bmatrix} \sigma_{11} \\ \sigma_{22} \\\sigma_{33} \\\sigma_{12} \\\sigma_{23} \\\sigma_{13} \end{bmatrix} = \begin{bmatrix} C_{11} & C_{12} & C_{13} & C_{14} & C_{15} & C_{16} \\ C_{12} & C_{22} & C_{23} & C_{24} & C_{25} & C_{26} \\ C_{13} & C_{23} & C_{33} & C_{34} & C_{35} & C_{36} \\ C_{14} & C_{24} & C_{34} & C_{44} & C_{45} & C_{46} \\ C_{15} & C_{25} & C_{35} & C_{45} & C_{55} & C_{56} \\ C_{16} & C_{26} & C_{36} & C_{46} & C_{56} & C_{66} \end{bmatrix} \begin{bmatrix} \epsilon_{11}^{\mathrm{mech}} \\ \epsilon_{22}^{\mathrm{mech}} \\\epsilon_{33}^{\mathrm{mech}} \\\epsilon_{12}^{\mathrm{mech}} \\\epsilon_{23}^{\mathrm{mech}} \\\epsilon_{13}^{\mathrm{mech}} \end{bmatrix}\,,\end{split}\]

where the stress and strain components are with respect to principal material directions. The thermal strains are defined in a similar manner,

\[\epsilon = \epsilon^{\mathrm{mech}} + \epsilon^{\mathrm{th}}, \epsilon^{\mathrm{th}} = [\epsilon_{11}^{\mathrm{th}}(\theta)\, \epsilon_{22}^{\mathrm{th}}(\theta)\, \epsilon_{33}^{\mathrm{th}}(\theta)\, \epsilon_{12}^{\mathrm{th}}(\theta)\, \epsilon_{23}^{\mathrm{th}}(\theta)\, \epsilon_{13}^{\mathrm{th}}(\theta)]^\mathrm{T}\,.\]

In a finite strain situation, the anisotropic model is formulated in a hypoelastic manner with a constitutive equation of

\[\dot{\sigma}_{ij} = C_{ijkl}\left( D_{kl} - D_{kl}^{\mathrm{th}}\right)\,,\]

where \(D_{kl}\) and \(D_{kl}^\mathrm{th}\) are the total and thermal strain rates, respectively, and the components of the fourth order stiffness tensor \(C_{ijkl}\) are related to the contracted notation by

\[\begin{split}\left[\mathbb{C}\right] = \begin{bmatrix} C_{1111} & C_{1122} & C_{1133} & C_{1112} & C_{1123} & C_{1113} \\ C_{1122} & C_{2222} & C_{2233} & C_{2212} & C_{2223} & C_{2213} \\ C_{1133} & C_{2233} & C_{3333} & C_{3312} & C_{3323} & C_{3313} \\ C_{1112} & C_{2212} & C_{3312} & C_{1212} & C_{1223} & C_{1213} \\ C_{1123} & C_{2223} & C_{3323} & C_{1223} & C_{2323} & C_{2313} \\ C_{1113} & C_{2213} & C_{3313} & C_{1213} & C_{2313} & C_{1313} \end{bmatrix}\,.\end{split}\]