# 15.1. Elastic Orthotropic Model

    BEGIN PARAMETERS FOR MODEL ELASTIC_ORTHOTROPIC
        #
        # Elastic constants
        YOUNGS MODULUS = <real>youngs_modulus
        POISSONS RATIO = <real>poissons_ratio
        SHEAR MODULUS = <real>shear_modulus
        BULK MODULUS = <real>bulk_modulus
        LAMBDA = <real>lambda
        TWO MU = <real>two_mu
        #
        # required parameters
        E11 = <real>e11
        E22 = <real>e22
        E33 = <real>e33
        NU12 = <real>nu12
        NU13 = <real>nu13
        NU23 = <real>nu23
        G12 = <real>g12
        G13 = <real>g13
        G23 = <real>g23
        COORDINATE SYSTEM = <string>coordinate_system_name
        #
        # optional parameters
        ANGLE_1_ABSCISSA = <real>angle_1_abscissa
        ANGLE_2_ABSCISSA = <real>angle_2_abscissa
        ANGLE_3_ABSCISSA = <real>angle_3_abscissa
        ROTATION_AXIS_1 = <real>rotation_axis_1
        ROTATION_AXIS_2 = <real>rotation_axis_2
        ROTATION_AXIS_3 = <real>rotation_axis_3
        ANGLE_1_FUNCTION = <string>angle_1_function_name
        ANGLE_2_FUNCTION = <string>angle_2_function_name
        ANGLE_3_FUNCTION = <string>angle_3_function_name
        THERMAL_STRAIN_11_FUNCTION =
          <string>thermal_strain_11_function_name
        THERMAL_STRAIN_22_FUNCTION =
          <string>thermal_strain_22_function_name
        THERMAL_STRAIN_33_FUNCTION =
          <string>thermal_strain_33_function_name
    END [PARAMETERS FOR MODEL ELASTIC_ORTHOTROPIC]

The elastic orthotropic model is a Kirchhoff linear elastic constitutive relation that is useful for modeling polymer matrix composite structures undergoing small strains. Required inputs are

- two of the five general elastic material constants,

- directional properties, and

- the coordinate system.

The following is a brief description of each input.

- See the Sierra/SM User Manual for more information on elastic constants input.

- In each material direction, moduli \(E_{11}\), \(E_{22}\), and \(E_{33}\) are defined with the `E11`, `E22`, and `E33` command lines, Poisson’s ratios \(\nu_{12}\), \(\nu_{13}\), and \(\nu_{23}\) are given by the `NU12`, `NU13`, and `NU23` command lines, and shear moduli \(G_{12}\), \(G_{13}\), and \(G_{23}\) are defined using command lines `G12`, `G13`, and `G23`.

- Principal material direction specification requires a user specified coordinate system given by the `COORDINATE SYSTEM` command line, as detailed in the Sierra/SM User Manual. The material orientation may then be specified using one of the following approaches:

  - Three spatially varying, ordered Euler angle functions are given in terms of global coordinates (X, Y, Z = 1, 2, 3) for rotations about a corresponding local axis:

    - The rotation angle function abscissas \(x_1\), \(x_2\), and \(x_3\), corresponding to the global system (X, Y, Z) = (1, 2, 3), are defined with the `ANGLE_1_ABSCISSA`, `ANGLE_2_ABSCISSA`, and `ANGLE_3_ABSCISSA` command lines, respectively.

    - The axes of rotation \(i\), \(j\), and \(k\) are defined with the `ROTATION_AXIS_1`, `ROTATION_AXIS_2`, and `ROTATION_AXIS_3` command lines, respectively.

    - The rotation angle functions \(\theta_{i}(x_{1})\), \(\theta_{j}(x_{2})\), and \(\theta_{k}(x_{3})\) are defined with the `ANGLE_1_FUNCTION`, `ANGLE_2_FUNCTION`, and `ANGLE_3_FUNCTION` command lines, respectively. Angles are specified in degrees. The Sierra/SM User Manual provides additional details about defining functions.

    The rotation axis and angle are applied successively in order (1, 2, 3), or (X, Y, Z); that is, each sequential Euler angle rotation defines a new coordinate system in which the subsequent rotation axis and angle are defined.

  - Alternatively, the angles and axes may be defined directly at each element integration point using `INITIAL CONDITION` command blocks, as described in the Sierra/SM User Manual. Angles may be specified in degrees using the variables `ANGLE_1`, `ANGLE_2`, and `ANGLE_3`, while axes are given by `AXIS_1`, `AXIS_2`, and `AXIS_3`.

  - A final option is to initialize the rotation tensor to correspond to the local coordinate system defined in the `COORDINATE SYSTEM` command line.

  The resulting material directions may be visualized by requesting the element fields `MATERIAL_DIRECTION_1`, `MATERIAL_DIRECTION_2`, and `MATERIAL_DIRECTION_3` in the results output block.

- Functions to describe normal thermal engineering strains along the principal material directions are given by the `THERMAL_STRAIN_11_FUNCTION`, `THERMAL_STRAIN_22_FUNCTION`, and `THERMAL_STRAIN_33_FUNCTION` command lines. See the Sierra/SM User Manual for guidance on defining functions.

Warning

The `ELASTIC_ORTHOTROPIC` model has not been tested in conjunction with the control stiffness implicit solver block.

Output variables available for this model are listed in Table 15.1.

Table 15.1 State Variables for ELASTIC ORTHOTROPIC Model
Index | Name | Description
--- | --- | ---
1 | `Q_XX` | X component of the material 11 unit vector
2 | `Q_YY` | Y component of the material 22 unit vector
3 | `Q_ZZ` | Z component of the material 33 unit vector
4 | `Q_XY` | Y component of the material 11 unit vector
5 | `Q_YZ` | Z component of the material 22 unit vector
6 | `Q_ZX` | X component of the material 33 unit vector
7 | `Q_YX` | X component of the material 22 unit vector
8 | `Q_ZY` | Y component of the material 33 unit vector
9 | `Q_XZ` | Z component of the material 11 unit vector
10 | `ANGLE_1` | Rotation angle about axis 1 (degrees)
11 | `ANGLE_2` | Rotation angle about axis 2 (degrees)
12 | `ANGLE_3` | Rotation angle about axis 3 (degrees)
13 | `AXIS_1` | First axis of rotation
14 | `AXIS_2` | Second axis of rotation
15 | `AXIS_3` | Third axis of rotation
16 | `TH_STR_XX` | Thermal stretch ratio in material 11 direction
17 | `TH_STR_YY` | Thermal stretch ratio in material 22 direction
18 | `TH_STR_ZZ` | Thermal stretch ratio in material 33 direction
19 | `MAT_STRAIN_XX` | Green Lagrange strain in material 11 direction
20 | `MAT_STRAIN_YY` | Green Lagrange strain in material 22 direction
21 | `MAT_STRAIN_ZZ` | Green Lagrange strain in material 33 direction
22 | `MAT_STRAIN_XY` | Green Lagrange strain in material 12 direction
23 | `MAT_STRAIN_YZ` | Green Lagrange strain in material 23 direction
24 | `MAT_STRAIN_ZX` | Green Lagrange strain in material 31 direction
25 | `MAT_STRESS_XX` | 2nd P-K stress in material 11 direction
26 | `MAT_STRESS_YY` | 2nd P-K stress in material 22 direction
27 | `MAT_STRESS_ZZ` | 2nd P-K stress in material 33 direction
28 | `MAT_STRESS_XY` | 2nd P-K stress in material 12 direction
29 | `MAT_STRESS_YZ` | 2nd P-K stress in material 23 direction
30 | `MAT_STRESS_ZX` | 2nd P-K stress in material 31 direction
31 | `MAT_LOG_STRAIN_XX` | Log (Hencky) strain in material 11 direction
32 | `MAT_LOG_STRAIN_YY` | Log (Hencky) strain in material 22 direction
33 | `MAT_LOG_STRAIN_ZZ` | Log (Hencky) strain in material 33 direction
34 | `MAT_LOG_STRAIN_XY` | Log (Hencky) strain in material 12 direction
35 | `MAT_LOG_STRAIN_YZ` | Log (Hencky) strain in material 23 direction
36 | `MAT_LOG_STRAIN_ZX` | Log (Hencky) strain in material 31 direction
37 | `MAT_CAUCHY_STRESS_XX` | Cauchy stress in material 11 direction
38 | `MAT_CAUCHY_STRESS_YY` | Cauchy stress in material 22 direction
39 | `MAT_CAUCHY_STRESS_ZZ` | Cauchy stress in material 33 direction
40 | `MAT_CAUCHY_STRESS_XY` | Cauchy stress in material 12 direction
41 | `MAT_CAUCHY_STRESS_YZ` | Cauchy stress in material 23 direction
42 | `MAT_CAUCHY_STRESS_ZX` | Cauchy stress in material 31 direction