Source code for felupe.constitution.tensortrax.models.hyperelastic._extended_tube
# -*- coding: utf-8 -*-
"""
This file is part of FElupe.
FElupe is free software: you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation, either version 3 of the License, or
(at your option) any later version.
FElupe is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
You should have received a copy of the GNU General Public License
along with FElupe. If not, see <http://www.gnu.org/licenses/>.
"""
from tensortrax.math import log
from tensortrax.math import sum as tsum
from tensortrax.math import trace
from tensortrax.math.linalg import det, eigvalsh
[docs]
def extended_tube(C, Gc, delta, Ge, beta):
r"""Strain energy function of the isotropic hyperelastic
`Extended Tube <https://www.doi.org/10.5254/1.3538822>`_ [1]_ material formulation.
Parameters
----------
C : tensortrax.Tensor or jax.Array
Right Cauchy-Green deformation tensor.
Gc : float
Cross-link contribution to the initial shear modulus.
delta : float
Finite extension parameter of the polymer strands.
Ge : float
Constraint contribution to the initial shear modulus.
beta : float
Global rearrangements of cross-links upon deformation (release of topological
constraints).
Notes
-----
The strain energy function is given in Eq. :eq:`psi-et`
.. math::
:label: psi-et
\psi = \frac{G_c}{2} \left[ \frac{\left( 1 - \delta^2 \right)
\left( \hat{I}_1 - 3 \right)}{1 - \delta^2 \left( \hat{I}_1 - 3 \right)} +
\ln \left( 1 - \delta^2 \left( \hat{I}_1 - 3 \right) \right) \right] +
\frac{2 G_e}{\beta^2} \left( \hat{\lambda}_1^{-\beta} +
\hat{\lambda}_2^{-\beta} + \hat{\lambda}_3^{-\beta} - 3 \right)
with the first main invariant of the distortional part of the right
Cauchy-Green deformation tensor as given in Eq. :eq:`invariants-et`
.. math::
:label: invariants-et
\hat{I}_1 = J^{-2/3} \text{tr}\left( \boldsymbol{C} \right)
and the principal stretches, obtained from the distortional part of the right
Cauchy-Green deformation tensor, see Eq. :eq:`stretches-et`.
.. math::
:label: stretches-et
\lambda^2_\alpha &= \text{eigvals}\left( \boldsymbol{C} \right)
\hat{\lambda}_\alpha &= J^{-1/3} \lambda_\alpha
The initial shear modulus results from the sum of the cross-link and the constraint
contributions to the total initial shear modulus as denoted in Eq.
:eq:`shear-modulus-et`.
.. math::
:label: shear-modulus-et
\mu = G_e + G_c
Examples
--------
First, choose the desired automatic differentiation backend
.. pyvista-plot::
:context:
>>> # import felupe.constitution.jax as mat
>>> import felupe.constitution.tensortrax as mat
and create the hyperelastic material.
.. pyvista-plot::
:context:
>>> import felupe as fem
>>>
>>> umat = mat.Hyperelastic(
... mat.models.hyperelastic.extended_tube,
... Gc=0.1867,
... Ge=0.2169,
... beta=0.2,
... delta=0.09693,
... )
>>> ax = umat.plot(incompressible=True)
.. pyvista-plot::
:include-source: False
:context:
:force_static:
>>> import pyvista as pv
>>>
>>> fig = ax.get_figure()
>>> chart = pv.ChartMPL(fig)
>>> chart.show()
References
----------
.. [1] M. Kaliske and G. Heinrich, "An Extended Tube-Model for Rubber Elasticity:
Statistical-Mechanical Theory and Finite Element Implementation", Rubber
Chemistry and Technology, vol. 72, no. 4. Rubber Division, ACS, pp. 602–632,
Sep. 01, 1999. doi:
`10.5254/1.3538822 <https://www.doi.org/10.5254/1.3538822>`_.
"""
J3 = det(C) ** (-1 / 3)
D = J3 * trace(C)
wC = J3 * eigvalsh(C)
g = (1 - delta**2) * (D - 3) / (1 - delta**2 * (D - 3))
Wc = Gc / 2 * (g + log(1 - delta**2 * (D - 3)))
We = 2 * Ge / beta**2 * tsum(wC ** (-beta / 2) - 1)
return Wc + We
