# -*- 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 ._morph_uniaxial import morph_uniaxial
from .microsphere import affine_force_statevars
[docs]
def morph_representative_directions(F, statevars, p, ε=1e-6):
"""First Piola-Kirchhoff stress tensor of the
`MORPH <https://doi.org/10.1016/s0749-6419(02)00091-8>`_ model formulation [1]_,
implemented by the concept of
`representative directions <https://nbn-resolving.org/urn:nbn:de:bsz:ch1-qucosa-114428>`_
[2]_, [3]_.
Parameters
----------
F : tensortrax.Tensor or jax.Array
Deformation gradient tensor.
statevars : array
Vector of stacked state variables (CTS, λ - 1, SA1, SA2).
p : list of float
A list which contains the 8 material parameters.
ε : float, optional
A small stabilization parameter (default is 1e-6).
Examples
--------
First, choose the desired automatic differentiation backend
.. pyvista-plot::
:context:
>>> import felupe as fem
>>>
>>> # import felupe.constitution.jax as mat
>>> import felupe.constitution.tensortrax as mat
and create the material.
.. pyvista-plot::
:context:
>>> umat = mat.Material(
... mat.models.lagrange.morph_representative_directions,
... p=[0.011, 0.408, 0.421, 6.85, 0.0056, 5.54, 5.84, 0.117],
... nstatevars=84,
... )
>>> ax = umat.plot(
... incompressible=True,
... ux=fem.math.linsteps(
... # [1, 2, 1, 2.75, 1, 3.5, 1, 4.2, 1, 4.8, 1, 4.8, 1],
... [1, 2.75, 1, 2.75],
... num=20,
... ),
... 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] D. Besdo and J. Ihlemann, "A phenomenological constitutive model for
rubberlike materials and its numerical applications", International Journal
of Plasticity, vol. 19, no. 7. Elsevier BV, pp. 1019–1036, Jul. 2003. doi:
`10.1016/s0749-6419(02)00091-8 <https://doi.org/10.1016/s0749-6419(02)00091-8>`_.
.. [2] M. Freund, "Verallgemeinerung eindimensionaler Materialmodelle für die
Finite-Elemente-Methode", Dissertation, Technische Universität Chemnitz,
Chemnitz, 2013.
.. [3] C. Miehe, S. Göktepe and F. Lulei, "A micro-macro approach to rubber-like
materials - Part I: the non-affine micro-sphere model of rubber elasticity",
Journal of the Mechanics and Physics of Solids, vol. 52, no. 11. Elsevier BV, pp.
2617–2660, Nov. 2004. doi:
`10.1016/j.jmps.2004.03.011 <https://doi.org/10.1016/j.jmps.2004.03.011>`_.
See Also
--------
felupe.constitution.tensortrax.models.lagrange.morph : Strain energy function of the
MORPH model formulation.
felupe.constitution.jax.models.lagrange.morph : Strain energy function of the MORPH
model formulation.
"""
def f(λ, statevars, **kwargs):
dψdλ, statevars_new = morph_uniaxial(λ, statevars, **kwargs)
return 5 * dψdλ, statevars_new
return affine_force_statevars(F, statevars, f=f, kwargs={"p": p, "ε": ε})
