Source code for felupe.constitution.tensortrax.models.hyperelastic._alexander
# -*- 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/>.
"""
import numpy as np
from tensortrax import Tensor, Δ, Δδ, f, δ
from tensortrax.math import exp, log, trace
from tensortrax.math.linalg import det
[docs]
def alexander(C, C1, C2, C3, gamma, k):
r"""Strain energy function of the isotropic hyperelastic
`Alexander <https://doi.org/10.1016/0020-7225(68)90006-2>`_ material
formulation [1]_.
Parameters
----------
C : tensortrax.Tensor
Right Cauchy-Green deformation tensor.
C1 : float
Scale factor for the first invariant term.
C2 : float
Scale factor for the second invariant term.
C3 : float
Scale factor for the logarithmic second invariant term.
gamma : float
Offset-normalization parameter for the logarithmic second invariant term.
k : float
Scale factor for the exponential first invariant term.
Notes
-----
.. warning::
The strain energy function of the Alexander model formulation is not directly
implemented. Only its gradient and hessian w.r.t. the right Cauchy-Green
deformation tensor are defined. This is because the imaginary error-function
:math:`\text{erfi}(x)` is not included in NumPy - this would require SciPy as a
dependency.
The strain energy function is given in Eq. :eq:`psi-alexander`
.. math::
:label: psi-alexander
\psi = C_1 \int_{\hat{I}_1} \exp \left(
k \left(\hat{I}_1 - 3 \right)^2
\right) \ d\hat{I}_1
+ C_2 \ln \left(\frac{\hat{I}_2 - 3 + \gamma}{\gamma} \right)
+ C_3 \left(\hat{I}_2 - 3 \right)
with the first and second main invariant of the distortional part of the right
Cauchy-Green deformation tensor, see Eq. :eq:`invariants-alexander`.
.. math::
:label: invariants-alexander
\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 initial shear modulus :math:`\mu` is given in Eq.
:eq:`shear-modulus-alexander`.
.. math::
:label: shear-modulus-alexander
\mu = 2 \left( C_1 + \frac{C_2}{\gamma} + C_3 \right)
Examples
--------
.. pyvista-plot::
:context:
>>> import felupe as fem
>>>
>>> umat = fem.Hyperelastic(
... fem.alexander, C1=17, C2=19.85, C3=1, gamma=0.735, k=0.00015
... )
>>> ux = fem.math.linsteps([0.6, 5], num=50)
>>> ps = fem.math.linsteps([1, 5], num=50)
>>> bx = fem.math.linsteps([1, 3], num=50)
>>>
>>> ax = umat.plot(ux=ux, ps=ps, bx=bx, 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] H. Alexander, "A constitutive relation for rubber-like materials",
International Journal of Engineering Science, vol. 6, no. 9. Elsevier BV, pp.
549–563, Sep. 1968. doi:
`10.1016/0020-7225(68)90006-2 <https://doi.org/10.1016/0020-7225(68)90006-2>`_.
"""
J3 = det(C) ** (-1 / 3)
I1 = J3 * trace(C)
I2 = (I1**2 - J3**2 * trace(C @ C)) / 2
def W(I1):
"A non-evaluated function with defined first- and second derivatives."
dWdI1 = exp(k * (I1 - 3) ** 2)
return Tensor(
x=np.full_like(f(I1), fill_value=np.nan),
δx=f(dWdI1) * δ(I1),
Δx=f(dWdI1) * Δ(I1),
Δδx=δ(dWdI1) * Δ(I1) + f(dWdI1) * Δδ(I1),
ntrax=I1.ntrax,
)
return C1 * W(I1) + C2 * log(((I2 - 3) + gamma) / gamma) + C3 * (I2 - 3)
