Source code for felupe.constitution.tensortrax.models.hyperelastic._third_order_deformation
# -*- 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 trace
from tensortrax.math.linalg import det
[docs]
def third_order_deformation(C, C10, C01, C11, C20, C30):
r"""Strain energy function of the isotropic hyperelastic
`Third-Order-Deformation <https://doi.org/10.1002/app.1975.070190723>`_ material
formulation.
Parameters
----------
C : tensortrax.Tensor or jax.Array
Right Cauchy-Green deformation tensor.
C10 : float
Material parameter associated to the linear term of the first invariant.
C01 : float
Material parameter associated to the linear term of the second invariant.
C11 : float
Material parameter associated to the mixed term of the first and second
invariant.
C20 : float
Material parameter associated to the quadratic term of the first invariant.
C30 : float
Material parameter associated to the cubic term of the first invariant.
Notes
-----
The strain energy function is given in Eq. :eq:`psi-tod`
.. math::
:label: psi-tod
\psi &= C_{10} \left(\hat{I}_1 - 3 \right) + C_{01} \left(\hat{I}_2 - 3 \right)
+ C_{11} \left(\hat{I}_1 - 3 \right) \left(\hat{I}_2 - 3 \right)
&+ C_{20} \left(\hat{I}_1 - 3 \right)^2
+ C_{30} \left(\hat{I}_1 - 3 \right)^3
with the first and second main invariant of the distortional part of the right
Cauchy-Green deformation tensor, see Eq. :eq:`invariants-tod`.
.. math::
:label: invariants-tod
\hat{I}_1 &= J^{-2/3} \text{tr}\left( \boldsymbol{C} \right)
\hat{I}_2 &= J^{-4/3} \frac{1}{2} \left(
\text{tr}\left(\boldsymbol{C}\right)^2 -
\text{tr}\left(\boldsymbol{C}^2\right)
\right)
The doubled sum of the material parameters :math:`C_{10}` and :math:`C_{01}` is
equal to the initial shear modulus :math:`\mu` as denoted in Eq.
:eq:`shear-modulus-tod`.
.. math::
:label: shear-modulus-tod
\mu = 2 \left( C_{10} + C_{01} \right)
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:
>>> umat = mat.Hyperelastic(
... mat.models.hyperelastic.third_order_deformation,
... C10=0.5,
... C01=0.1,
... C11=0.01,
... C20=-0.1,
... C30=0.02,
... )
>>> 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()
"""
J3 = det(C) ** (-1 / 3)
I1 = J3 * trace(C)
I2 = (I1**2 - J3**2 * trace(C @ C)) / 2
return (
C10 * (I1 - 3)
+ C01 * (I2 - 3)
+ C11 * (I1 - 3) * (I2 - 3)
+ C20 * (I1 - 3) ** 2
+ C30 * (I1 - 3) ** 3
)