Source code for felupe.constitution.tensortrax.models.hyperelastic._finite_strain_viscoelastic
# -*- 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, inv
from tensortrax.math.special import from_triu_1d, triu_1d
[docs]
def finite_strain_viscoelastic(C, Cin, mu, eta, dtime):
r"""Multiplicative
`finite strain viscoelastic <https://doi.org/10.1016/j.cma.2013.07.004>`_ [1]_
material formulation.
Notes
-----
The material formulation is built upon the multiplicative decomposition of the
distortional part of the deformation gradient tensor into an elastic and an
inelastic part, see Eq. :eq:`multiplicative-split`.
.. math::
:label: multiplicative-split
\hat{\boldsymbol{F}} &= \boldsymbol{F}_e \boldsymbol{F}_i
\ (= \hat{\boldsymbol{F}_e} \boldsymbol{F}_i)
\boldsymbol{C}_e &= \boldsymbol{F}_e^T \boldsymbol{F}_e
\boldsymbol{C}_i &= \boldsymbol{F}_i^T \boldsymbol{F}_i
\text{tr}\left( \boldsymbol{C}_e \right) &= \text{tr}\left(
\hat{\boldsymbol{C}} \boldsymbol{C}_i^{-1}
\right)
The components of the inelastic right Cauchy-Green deformation tensor are used as
state variables with the evolution equation and its explicit update formula as given
in Eq. :eq:`evolution` [1]_. The elastic part of the multiplicative decomposition of
the deformation gradient tensor is also enforced to be an unimodular tensor which
leads to the constraint :math:`\det(\boldsymbol{F_i})=1`. Hence, the inelastic right
Cauchy-Green deformation tensor must be an unimodular tensor
:math:`\det(\boldsymbol{C_i})=1`.
.. math::
:label: evolution
\dot{\boldsymbol{C}}_i &= \frac{\mu}{\eta}\ \hat{\boldsymbol{C}}
\boldsymbol{C}_i &= \hat{\overline{\boldsymbol{C}_{i,n}
+ \frac{\Delta t \mu}{\eta} \hat{\boldsymbol{C}}}}
The distortional part of the strain energy density per unit undeformed volume is
assumed to be of a Neo-Hookean form, see Eq. :eq:`nh-w`.
.. math::
:label: nh-w
\hat{\psi} = \frac{\mu}{2} \left( \text{tr}\left(
\hat{\boldsymbol{C}} \boldsymbol{C}_i^{-1}
\right) - 3 \right)
Examples
--------
.. pyvista-plot::
:context:
>>> import felupe as fem
>>>
>>> umat = fem.Hyperelastic(
... fem.finite_strain_viscoelastic, mu=1.0, eta=1.0, dtime=1.0, nstatevars=6
... )
>>> ax = umat.plot(
... incompressible=True,
... ux=fem.math.linsteps([1, 1.5, 1, 2, 1], num=[5, 5, 10, 10]),
... ps=None,
... bx=None,
... )
.. pyvista-plot::
:include-source: False
:context:
:force_static:
>>> import pyvista as pv
>>>
>>> fig = ax.get_figure()
>>> chart = pv.ChartMPL(fig)
>>> chart.show()
References
----------
.. [1] A. V. Shutov, R. Landgraf, and J. Ihlemann, "An explicit solution for
implicit time stepping in multiplicative finite strain viscoelasticity",
Computer Methods in Applied Mechanics and Engineering, vol. 265. Elsevier BV,
pp. 213–225, Oct. 2013. doi:
`10.1016/j.cma.2013.07.004 <https://doi.org/10.1016/j.cma.2013.07.004>`_.
"""
J3 = det(C) ** (-1 / 3)
# update of state variables by evolution equation
Ci = from_triu_1d(Cin[:6], like=C) + (mu / eta * dtime) * (J3 * C)
Ci = det(Ci) ** (-1 / 3) * Ci
# first invariant of elastic part of right Cauchy-Green deformation tensor
I1 = J3 * trace(C @ inv(Ci))
# strain energy function and state variable
return mu / 2 * (I1 - 3), triu_1d(Ci)
