# -*- 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 ...math import cdya_ik, cdya_il, det, dot, dya, identity, inv, trace, transpose
from .._base import ConstitutiveMaterial
[docs]
class NeoHookeCompressible(ConstitutiveMaterial):
r"""Compressible isotropic hyperelastic Neo-Hookean material formulation. The strain
energy density function of the Neo-Hookean material formulation is a linear function
of the trace of the right Cauchy-Green deformation tensor.
Parameters
----------
mu : float or None, optional
Shear modulus (second Lamé constant). Default is None.
lmbda : float or None, optional
First Lamé constant (default is None)
Notes
-----
.. math::
\psi &= \psi(\boldsymbol{C})
\psi(\boldsymbol{C}) &= \frac{\mu}{2} \text{tr}(\boldsymbol{C})
- \mu \ln(J) + \frac{\lambda}{2} \ln(J)^2
with
.. math::
J = \text{det}(\boldsymbol{F})
The first Piola-Kirchhoff stress tensor is evaluated as the gradient
of the strain energy density function.
.. math::
\boldsymbol{P} &= \frac{\partial \psi}{\partial \boldsymbol{F}}
\boldsymbol{P} &= \mu \left( \boldsymbol{F} - \boldsymbol{F}^{-T} \right)
+ \lambda \ln(J) \boldsymbol{F}^{-T}
The hessian of the strain energy density function enables the corresponding
elasticity tensor.
.. math::
\mathbb{A} &= \frac{\partial^2 \psi}{\partial \boldsymbol{F}\ \partial
\boldsymbol{F}}
\mathbb{A} &= \mu \boldsymbol{I} \overset{ik}{\otimes} \boldsymbol{I}
+ \left(\mu - \lambda \ln(J) \right)
\boldsymbol{F}^{-T} \overset{il}{\otimes} \boldsymbol{F}^{-T}
+ \lambda \boldsymbol{F}^{-T} {\otimes} \boldsymbol{F}^{-T}
Examples
--------
.. pyvista-plot::
:context:
>>> import felupe as fem
>>>
>>> umat = fem.NeoHookeCompressible(mu=1.0, lmbda=2.0)
>>> ax = umat.plot()
.. pyvista-plot::
:include-source: False
:context:
:force_static:
>>> import pyvista as pv
>>>
>>> fig = ax.get_figure()
>>> chart = pv.ChartMPL(fig)
>>> chart.show()
"""
def __init__(self, mu=None, lmbda=None, parallel=False):
self.parallel = parallel
self.mu = mu
self.lmbda = lmbda
self.kwargs = {"mu": self.mu}
if self.lmbda is not None:
self.kwargs["lmbda"] = self.lmbda
# aliases for function, gradient and hessian
self.energy = self.function
self.stress = self.gradient
self.elasticity = self.hessian
# initial variables for calling
# ``self.gradient(self.x)`` and ``self.hessian(self.x)``
self.x = [np.eye(3), np.zeros(0)]
[docs]
def function(self, x):
"""Strain energy density function per unit undeformed volume of the Neo-Hookean
material formulation.
Parameters
----------
x : list of ndarray
List with the Deformation gradient ``F`` (3x3) as first item
"""
F = x[0]
mu = self.mu
lmbda = self.lmbda
lnJ = np.log(det(F))
C = dot(transpose(F), F, parallel=self.parallel)
W = mu * (trace(C) / 2 - lnJ)
if lmbda is not None:
W += lmbda * lnJ**2 / 2
return [W]
[docs]
def gradient(self, x, out=None):
"""Gradient of the strain energy density function per unit undeformed volume of
the Neo-Hookean material formulation.
Parameters
----------
x : list of ndarray
List with the Deformation gradient ``F`` (3x3) as first item
out : ndarray or None, optional
A location into which the result is stored (default is None).
"""
F, statevars = x[0], x[-1]
mu = self.mu
lmbda = self.lmbda
J = det(F)
iFT = transpose(inv(F, J))
lnJ = np.log(J, out=J)
P = np.multiply(mu, F, out=out)
if lmbda is None:
Pb = np.multiply(iFT, -mu, out=iFT)
else:
lmbda_lnJ = np.multiply(lmbda, lnJ, out=lnJ)
Pb = np.multiply(iFT, -mu + lmbda_lnJ, out=iFT)
np.add(P, Pb, out=P)
return [P, statevars]
[docs]
def hessian(self, x, out=None):
"""Hessian of the strain energy density function per unit undeformed volume of
the Neo-Hookean material formulation.
Parameters
----------
x : list of ndarray
List with the Deformation gradient ``F`` (3x3) as first item
out : ndarray or None, optional
A location into which the result is stored (default is None).
"""
F = x[0]
mu = self.mu
lmbda = self.lmbda
J = det(F)
iFT = transpose(inv(F, J))
lnJ = np.log(J, out=J)
eye = identity(F)
iFTiFT = cdya_il(iFT, iFT, out=out)
A4a = cdya_ik(eye, eye)
np.multiply(mu, A4a, out=A4a)
if lmbda is not None:
lmbda_lnJ = np.multiply(lmbda, lnJ, out=lnJ)
A4b = np.multiply(mu - lmbda_lnJ, iFTiFT, out=iFTiFT)
iFT_iFT = dya(iFT, iFT)
A4c = np.multiply(lmbda, iFT_iFT, out=iFT_iFT)
np.add(A4a, A4b, out=A4b)
A4 = np.add(A4b, A4c, out=A4c)
else:
A4b = np.multiply(mu, iFTiFT, out=iFTiFT)
A4 = np.add(A4a, A4b, out=A4b)
return [A4]
