Anisotropic Hyperelasticity

Overview of Strain Energy Potentials

Similar to isotropic hyperelastic behavior, the anisotopic hyperelastic behavior is described in terms of a "strain energy potential," $U$ , that defines the strain energy stored in the material per unit of reference volume (volume in the initial configuration). Broadly, the strain energy potentials can be functions of strain measures directly or of invariants of strain measures. In addition, potentials that are functions of strain invariants can be made to depend on local fiber directions in the material to account for example, fibers embedded in a matrix.

Strain Energy Potential	Description
Generalized Fung form	Simulates both orthotropic and fully anisotropic hyperelasticity. This model assumes strain energy dependence on strains.
Holzapfel-Gasser-Ogden form	Simulates anisotropic hyperelasticity with strain energy dependence on invariants. This model allows separate modeling of fibers with local directions and the collagenous matrix in which they are embedded.
Strain-based and Invariant-based user defined	User-defined strain energy potential that depends on strain or strain invariants.

Generalized Fung form

The generalized Fung strain energy $U$ per unit reference volume of the material is composed of two contributions: a contribution in terms of modified Green strain, a distortional strain measure and a contribution in terms of volumetric strain. The actual form is:

Here, $Q={\overline{\mathit{\epsilon }}}^G:\mathbf{b}:{\overline{\mathit{\epsilon }}}^G={\overline{\epsilon }}_{ij}^G{b}_{ijkl}{\overline{\epsilon }}_{kl}^G,$ where $\mathbf{b}$ is a symmetric rank 4 tensor of material constants with components $b_{ijkl}$ . In the orthotropic case, 9 of these constants have to be specified while the number of constants is 21 for the anisotropic case. The modified Green strain is defined as ${\overline{\mathit{\epsilon }}}^G=\frac{1}{2}\left(\overline{\mathbf{C}}-\mathbf{I}\right)$ , where $\overline{\mathbf{C}}=J^{-\frac{2}{3}}\mathbf{C}$ . Here, $\mathbf{C}={\mathbf{F}}^T\mathbf{F}$ is the right Cauchy-Green tensor; $J=det\left(\mathbf{F}\right)$ is the total volume ratio; $\mathbf{F}$ is the deformation gradient; $J^{el}=J/\prod _{i=1}^3(1+{ϵ}_{th}^i)$ is the elastic volume ratio with the thermal effects to volume change excluded; ${ϵ}_{th}^i$ are thermal strains in the principal directions ; $c$ and $D$ are phenomenological material constants.

Table 1. Options for Generalized Fung Form

Input Data	Description
$b_{ijkl}$	$b_{ijkl}$ for various subscript values; 9 constants for the orthotropic case and 21 constants for the anisotropic case.
c	Material parameter $c$
D	Material parameter $D$
Use temperature-dependent data	Specify material parameters that depend on temperature. A Temperature field appears in the data table. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.
Number of field variables	Specifies field variable dependent stress-strain data. A new Field field appears in the data table each time the number of field variables is incremented by one. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.
Moduli time scale	Select Instantaneous or Long Term for the application of viscoelastic effects.

Holzapfel-Glasser-Ogden Potential

The Holzapfel-Glasser-Ogden strain energy is based on strain invariants and provides a convenient form to model materials where fibers with specific orientations are embedded in a matrix. A good example is biological tissue in arterial walls with collagen fibers. The specific form of the potential used is:

with

where $U$ is the strain energy per unit of reference volume of the material; $C_{10}$ , $D$ , $k_1$ , $k_2$ , and $\kappa$ are temperature-dependent material parameters; $N$ is the number of families of fibers ( $N\le 3$ ); ${\overline{I}}_1$ is the first deviatoric strain invariant; $J^{e\mathrm{\ell }}$ is the elastic volume ratio with the thermal effects to volume change excluded; and ${\overline{I}}_{4\left(\alpha \alpha \right)}$ are what are known as pseudo-invariants of modified Green strain $\overline{\mathbf{C}}$ and mean fiber directions ${\mathbf{A}}_{\alpha }$ .

Table 2. Options for Holzapfel-Glasser-Ogden Form

Input Data	Description
Number of local directions	$N$
C10	$C_{10}$
D	$D$
k1	$k_1$
k2	$k_2$
Fiber dispersion parameter k	$\kappa$ . If $\kappa =0$ , the fibers are perfectly aligned; if $\kappa =\frac{1}{3}$ , the fiber distribution is isotropic.
Use temperature-dependent data	Specify material parameters that depend on temperature. A Temperature field appears in the data table. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.
Number of field variables	Specifies field variable dependent stress-strain data. A new Field field appears in the data table each time the number of field variables is incremented by one. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.
Moduli time scale	Select Instantaneous or Long Term for the application of viscoelastic effects.

User-defined Form

Alternatively, you can specify your own strain energy density function as a function of strain measure or as a function of strain invariants and local fiber directions.

The user subroutine UANISOHYPER_STRAIN for implicit time integration simulations and the user subroutine VUANISOHYPER_STRAIN for explicit time integration simulations allow for a strain based dependence on modified Green strain, a distortional strain measure and $J^{e\mathrm{\ell }}$ , the elastic volume ratio with the thermal effects to volume change excluded.

The user subroutine UANISOHYPER_INV for implicit time integration simulations and the user subroutine VUANISOHYPER_INV for explicit time integration simulations allow for specifying strain energy density in terms of strain invariants and invariants (combinations of dot product of local fiber directions) associated with fiber directions.

Table 3. Options to Specify User-defined Form

Input Data	Description
Formulation	Specify whether the user subroutine is strain or strain invariant based.
Compressiblity	Applies only to Implicit simulations; Specify purely incompressible (enforce volume preserving) or compressible (allow volume changing) behavior occuring from mechanical forces.
Number of local directions	Number of local fiber directions with a maximum value of 3. This option only appears when strain energy dependence on strain invariants is chosen.
User defined material parameters	User-defined material parameters to calculate strain energy.

Definition of Preferred Material Directions

For the strain-based forms (such as the Fung form and user-defined forms using user subroutines), you must specify local orientation directions for the directions of anisotropy. Currently, local orientation specification is not supported if the orientations have a spatial variation.

For invariant-based forms of the strain energy function (such as the Holzapfel form and user-defined forms using user subroutines), you have to specify both local orientation directions of anisotropy and local fiber direction vectors ${\mathbf{A}}_{\alpha }$ . Currently, spatial distribution of local orientation directions and local fiber directions are not supported.