## Constitutive Theory

Some materials are inherently anisotropic due to their microstructure or added reinforcements. Examples of this include fiber reinforced rubbers and thermoplastics, oriented thermoplastics, and biomaterials. The direction-dependent behaviors of these materials can be quite important for their overall mechanical response. If you have not modeled these materials before, the good news is that it is almost as easy to model an anisotropic material as it is to model an isotropic material. The main difference is the amount of experimental testing that is need to extract the response and guide the material model calibration.

ANSYS Mechanical supports the following polynomial-based anisotropic hyperelastic material model energy function:

\[ W = \sum_{i=1}^3 a_i (\overline{I}_1 – 3)^i + \sum_{j=1}^3 b_j (\overline{I}_2 – 3)^j + \sum_{k=2}^6 c_k (\overline{I}_4 – 1)^k\\ + \sum_{l=2}^6 d_l (\overline{I}_5 – 1)^l + \sum_{m=2}^6 e_m (\overline{I}_6 – 1)^m + \sum_{n=2}^6 f_n (\overline{I}_7 – 1)^n + \\ \sum_{o=2}^6 g_o (\overline{I}_8 – 1)^o + \frac{1}{d} (J-1)^2 \]

This equation is very general, in fact, I find it unnecessarily complicated for almost all polymers. I have implemented the following simplified (and more useful) version in MCalibration:

\[ W = \sum_{i=1}^3 a_i (\overline{I}_1 – 3)^i + \sum_{k=2}^6 c_k (\overline{I}_4 – 1)^k + \sum_{m=2}^6 e_m (\overline{I}_6 – 1)^m + \frac{1}{d} (J-1)^2\]

This model consists of a Yeoh isotropic hyperelastic matrix material with independent fibers oriented in two user-defined orientations. The \(c_k\) and \(e_m\) parameters specify the fiber stiffnesses.

## Example

The following screenshot shows the ANSYS anisotropic hyperelastic polynomial material model evaluated in MCalibration.

Here are the final material models in ANSYS ADPL dat-file format:

```
TB, AHYPER, matid, 1, 31, POLY
TBDATA, 1, 1 ! A1
TBDATA, 2, 0 ! A2
TBDATA, 3, 0 ! A3
TBDATA, 4, 0.0 ! B1
TBDATA, 5, 0.0 ! B2
TBDATA, 6, 0.0 ! B3
TBDATA, 7, 1 ! C2
TBDATA, 8, 0 ! C3
TBDATA, 9, 0 ! C4
TBDATA, 10, 0.0 ! C5
TBDATA, 11, 0.0 ! C6
TBDATA, 12, 0.0 ! D2
TBDATA, 13, 0.0 ! D3
TBDATA, 14, 0.0 ! D4
TBDATA, 15, 0.0 ! D5
TBDATA, 16, 0.0 ! D6
TBDATA, 17, 0 ! E2
TBDATA, 18, 0 ! E3
TBDATA, 19, 0 ! E4
TBDATA, 20, 0.0 ! E5
TBDATA, 21, 0.0 ! E6
TBDATA, 22, 0.0 ! F2
TBDATA, 23, 0.0 ! F3
TBDATA, 24, 0.0 ! F4
TBDATA, 25, 0.0 ! F5
TBDATA, 26, 0.0 ! F6
TBDATA, 27, 0.0 ! G2
TBDATA, 28, 0.0 ! G3
TBDATA, 29, 0.0 ! G4
TBDATA, 30, 0.0 ! G5
TBDATA, 31, 0.0 ! G6
TB, AHYPER, matid, 1, 1, PVOL
TBDATA, 1, 0.01 ! D
TB, AHYPER, matid, 1, 3, AVEC
TBDATA, 1, 1
TBDATA, 2, 0
TBDATA, 3, 0
TB, AHYPER, matid, 1, 3, BVEC
TBDATA, 1, 0
TBDATA, 2, 1
TBDATA, 3, 0
MP, DENS, matid, 1e-09
```

You can, of course, also calibrate the ANSYS anisotropic material model to experimental data. Since the material model has been implemented as a built-in feature in MCalibration, the parameter calibration will run extremely fast and will not use ANSYS or any ANSYS license tokens.

The following article shows how you can use the calibrated material model in ANSYS Workbench.