.. _models-hyperelastic:

*******************
Hyperelastic Models
*******************

Hyperelastic materials are in many ways easier to understand than hypoelastic materials, and are often considered more thermodynamically consistent. On the other hand, it may be difficult to consistently extend a small deformation model to the finite deformation regime in a hyperelastic framework. Regardless of the pluses and minuses of the two formulations, hyperelastic models are in LAMÉ and will be reviewed here.

Hyperelastic models generally assume a scalar valued strain energy density that is a function of invariants of the deformation through the deformation gradient, :math:`F_{ij}`. Using the principle of material frame indifference, the strain energy density is written as a function of the symmetric right Cauchy-Green tensor, :math:`C_{ij} = F_{ki}F_{kj}`

.. math::

   W = W \left( C_{ij} \right)

The stress, in particular the second Piola-Kirchhoff stress, is found by taking the derivative of :math:`W` with respect to :math:`C_{ij}`. This relation comes from the stress-power relations. From the second Piola-Kirchhoff stress, we can find the Cauchy stress

.. math::

   S_{ij} = 2 \frac{\partial W}{\partial C_{ij}}
   \;\;\; ; \;\;\;
   \sigma_{ij} = \frac{1}{J} F_{ik}S_{kl}F_{jl}

Hyperelastic models are generally of two types. The most common are written in terms of the three invariants of :math:`C_{ij}`: :math:`I_{1}`, :math:`I_{2}`, and :math:`I_{3}`

.. math::

   I_{1} = \mbox{tr}{\bf C} = C_{ii}
   \;\;\; ; \;\;\;
   I_{2} = \frac{1}{2} \left( C_{ii}C_{jj} - C_{ij}C_{ij} \right)
   \;\;\; ; \;\;\;
   I_{3} = \mbox{det}{\bf C}

The second Piola-Kirchhoff stress is then

.. math::

   S_{ij} = 2 \left(
   \frac{\partial W}{\partial I_{1}}\frac{\partial I_{1}}{\partial C_{ij}} +
   \frac{\partial W}{\partial I_{2}}\frac{\partial I_{2}}{\partial C_{ij}} +
   \frac{\partial W}{\partial I_{3}}\frac{\partial I_{3}}{\partial C_{ij}}
   \right)

Evaluating this expression requires the derivatives of the invariants with respect to the components :math:`C_{ij}`

.. math::

   \frac{\partial I_{1}}{\partial C_{ij}} = \delta_{ij}
   \;\;\; ; \;\;\;
   \frac{\partial I_{2}}{\partial C_{ij}} = I_{1} \delta_{ij} - C_{ij}
   \;\;\; ; \;\;\;
   \frac{\partial I_{3}}{\partial C_{ij}} = I_{3} C_{ij}^{-1}

Using this in the expression for the second Piola-Kirchhoff stress, and converting it to the Cauchy stress, we have

.. math::

   \sigma_{ij} = \frac{2}{J} \left\{ \frac{\partial W}{\partial I_{3}} \delta_{ij} + \left( \frac{\partial W}{\partial I_{1}} + I_{1} \frac{\partial W}{\partial I_{2}} \right) B_{ij} - \frac{\partial W}{\partial I_{2}} B_{ij}^{2} \right\}

The majority of hyperelastic models calculate the stress in this manner.

Some hyperelastic models, however, have their strain energy densities written in terms of the principal stretches [:footcite:`ogden`]. When this is the case the calculation of the stress is more complex. The right stretch can be written as

.. math::

   {\bf U} = \sum_{i=1}^{3} \lambda_{i} \bar{\bf e}_{i} \otimes \bar{\bf e}_{i}

where :math:`\lambda_{i}` are the principal stretches, or eigenvalues, and :math:`\bar{\bf e}_{i}` are the principal directions, or eigenvectors.
The strain energy density is :math:`W(\lambda_{i})`.
We calculate the stress components of the second Piola-Kirchhoff stress, :math:`\bar{S}_{ij}`, with respect to the principal directions

.. math::

   {\bf S} = \bar{S}_{ij} \bar{\bf e}_{i} \otimes \bar{\bf e}_{j}

This is done by calculating :math:`\partial W/\partial {\bf C}` in the following manner

.. math::
   :label: eq-hypervar

   \delta W = \frac{\partial W}{\partial\lambda_{i}} \delta \lambda_{i} = \frac{\partial W}{\partial {\bf C}}: \delta{\bf C}

Writing the right Cauchy-Green tensor with respect to the principal directions we have

.. math::

   {\bf C} = \sum_{i=1}^{3} \lambda_{i}^{2} \, \bar{\bf e}_{i} \otimes \bar{\bf e}_{i}
   \;\;\; ; \;\;\;
   \delta {\bf C} = \sum_{i=1}^{3} 2 \lambda_{i} \delta\lambda_{i} \bar{\bf e}_{i} \otimes \bar{\bf e}_{i}
   + \lambda_{i}^{2} \delta\bar{\omega}_{ij} \left( \bar{\bf e}_{i} \otimes \bar{\bf e}_{j} + \bar{\bf e}_{j} \otimes \bar{\bf e}_{i}\right)

Equating terms on both sides of :eq:`eq-hypervar` we get

.. math::

   \bar{S}_{11} = \frac{1}{\lambda_{1}} \frac{\partial W}{\partial\lambda_{1}}
   \;\;\; ; \;\;\;
   \bar{S}_{22} = \frac{1}{\lambda_{2}} \frac{\partial W}{\partial\lambda_{2}}
   \;\;\; ; \;\;\;
   \bar{S}_{33} = \frac{1}{\lambda_{3}} \frac{\partial W}{\partial\lambda_{3}}
   \;\;\; ; \;\;\;
   \bar{S}_{ij} = 0 \;\;\; \text{otherwise}

These calculations can also be checked by writing the invariants in terms of the principal stretches. For a hyperelastic model written in terms of the invariants the results should be the same.

The differences between hypoelastic and hyperelastic models should not matter for the analyst. For the constitutive modeler, however, the benefits and drawbacks of the two formulations must be considered.

.. raw::
   html

   <hr>

.. footbibliography::
