# -*- 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, ddot, det, dot, dya, identity, inv, trace, transpose
from .._base import ConstitutiveMaterial
[docs]
class NeoHooke(ConstitutiveMaterial):
r"""Nearly-incompressible 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 isochoric part of the right Cauchy-Green
deformation tensor.
Parameters
----------
mu : float or None, optional
Shear modulus (default is None)
bulk : float or None, optional
Bulk modulus (default is None)
Notes
-----
.. note::
At least one of the two material parameters must not be None.
In a nearly-incompressible constitutive framework the strain energy
density is an additive composition of an isochoric and a volumetric
part. While the isochoric part is defined on the distortional part of
the deformation gradient, the volumetric part of the strain
energy function is defined on the determinant of the deformation
gradient.
.. math::
\psi &= \hat{\psi}(\hat{\boldsymbol{C}}) + U(J)
\hat\psi(\hat{\boldsymbol{C}}) &= \frac{\mu}{2}
(\text{tr}(\hat{\boldsymbol{C}}) - 3)
with
.. math::
J &= \text{det}(\boldsymbol{F})
\hat{\boldsymbol{F}} &= J^{-1/3} \boldsymbol{F}
\hat{\boldsymbol{C}} &= \hat{\boldsymbol{F}}^T \hat{\boldsymbol{F}}
The volumetric part of the strain energy density function is a function
the volume ratio.
.. math::
U(J) = \frac{K}{2} (J - 1)^2
The first Piola-Kirchhoff stress tensor is evaluated as the gradient
of the strain energy density function. The hessian of the strain
energy density function enables the corresponding elasticity tensor.
.. math::
\boldsymbol{P} &= \frac{\partial \psi}{\partial \boldsymbol{F}}
\mathbb{A} &= \frac{\partial^2 \psi}{\partial \boldsymbol{F}\ \partial
\boldsymbol{F}}
A chain rule application leads to the following expression for the stress tensor.
It is formulated as a sum of the **physical**-deviatoric (not the mathematical
deviator!) and the physical-hydrostatic stress tensors.
.. math::
\boldsymbol{P} &= \boldsymbol{P}' + \boldsymbol{P}_U
\boldsymbol{P}' &= \frac{\partial \hat{\psi}}{\partial \hat{\boldsymbol{F}}} :
\frac{\partial \hat{\boldsymbol{F}}}{\partial \boldsymbol{F}} =
\bar{\boldsymbol{P}} - \frac{1}{3} (\bar{\boldsymbol{P}} : \boldsymbol{F})
\boldsymbol{F}^{-T}
\boldsymbol{P}_U &= \frac{\partial U(J)}{\partial J} \frac{\partial J}{\partial
\boldsymbol{F}} = U'(J) J \boldsymbol{F}^{-T}
with
.. math::
\frac{\partial \hat{\boldsymbol{F}}}{\partial \boldsymbol{F}} &= J^{-1/3} \left(
\boldsymbol{I} \overset{ik}{\otimes} \boldsymbol{I} - \frac{1}{3} \boldsymbol{F}
\otimes \boldsymbol{F}^{-T} \right)
\frac{\partial J}{\partial \boldsymbol{F}} &= J \boldsymbol{F}^{-T}
\bar{\boldsymbol{P}} &= J^{-1/3} \frac{\partial \hat{\psi}}{\partial
\hat{\boldsymbol{F}}}
With the above partial derivatives the first Piola-Kirchhoff stress
tensor of the Neo-Hookean material model takes the following form.
.. math::
\boldsymbol{P} = \mu J^{-2/3} \left( \boldsymbol{F} - \frac{1}{3} (
\boldsymbol{F} : \boldsymbol{F}) \boldsymbol{F}^{-T} \right) + K (J - 1) J
\boldsymbol{F}^{-T}
Again, a chain rule application leads to an expression for the elasticity tensor.
.. math::
\mathbb{A} &= \mathbb{A}' + \mathbb{A}_{U}
\mathbb{A}' &= \bar{\mathbb{A}} - \frac{1}{3} \left( (\bar{\mathbb{A}} :
\boldsymbol{F}) \otimes \boldsymbol{F}^{-T} + \boldsymbol{F}^{-T} \otimes
(\boldsymbol{F} : \bar{\mathbb{A}}) \right ) + \frac{1}{9} (\boldsymbol{F} :
\bar{\mathbb{A}} : \boldsymbol{F}) \boldsymbol{F}^{-T} \otimes
\boldsymbol{F}^{-T}
\mathbb{A}_{U} &= (U''(J) J + U'(J)) J \boldsymbol{F}^{-T} \otimes
\boldsymbol{F}^{-T} - U'(J) J \boldsymbol{F}^{-T} \overset{il}{\otimes}
\boldsymbol{F}^{-T}
with
.. math::
\bar{\mathbb{A}} = J^{-1/3} \frac{\partial^2 \hat\psi}{\partial
\hat{\boldsymbol{F}}\ \partial \hat{\boldsymbol{F}}} J^{-1/3}
With the above partial derivatives the (physical-deviatoric and
-hydrostatic) parts of the elasticity tensor associated
to the first Piola-Kirchhoff stress tensor of the Neo-Hookean
material model takes the following form.
.. math::
\mathbb{A} &= \mathbb{A}' + \mathbb{A}_{U}
\mathbb{A}' &= J^{-2/3} \left(\boldsymbol{I} \overset{ik}{\otimes} \boldsymbol{I}
- \frac{1}{3} \left( \boldsymbol{F} \otimes \boldsymbol{F}^{-T} +
\boldsymbol{F}^{-T} \otimes \boldsymbol{F} \right ) + \frac{1}{9} (\boldsymbol{F}
: \boldsymbol{F}) \boldsymbol{F}^{-T} \otimes \boldsymbol{F}^{-T} \right)
\mathbb{A}_{U} &= K J \left( (2J - 1) \boldsymbol{F}^{-T} \otimes
\boldsymbol{F}^{-T} - (J - 1) \boldsymbol{F}^{-T} \overset{il}{\otimes}
\boldsymbol{F}^{-T} \right)
Examples
--------
.. pyvista-plot::
:context:
>>> import felupe as fem
>>>
>>> umat = fem.NeoHooke(mu=1.0, bulk=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, bulk=None, parallel=False):
self.parallel = parallel
self.mu = mu
self.bulk = bulk
self.kwargs = {}
if self.mu is not None:
self.kwargs["mu"] = self.mu
if self.bulk is not None:
self.kwargs["bulk"] = self.bulk
# 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
bulk = self.bulk
J = det(F)
C = dot(transpose(F), F, parallel=self.parallel)
W = np.zeros_like(J)
if mu is not None:
W += mu / 2 * (J ** (-2 / 3) * trace(C) - 3)
if bulk is not None:
W += bulk * (J - 1) ** 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
bulk = self.bulk
J = det(F)
iFT = transpose(inv(F, J))
P = out
if P is None:
P = np.zeros_like(F)
if mu is not None:
# "physical"-deviatoric (not math-deviatoric!) part of P
trC = ddot(F, F, parallel=self.parallel)
trC_3 = np.divide(trC, 3, out=trC)
np.multiply(trC_3, iFT, out=P)
np.add(F, -P, out=P)
Jm23 = np.power(J, -2 / 3, out=trC)
np.multiply(P, Jm23, out=P)
np.multiply(P, mu, out=P)
if bulk is not None:
# "physical"-volumetric (not math-volumetric!) part of P
JiFT = np.multiply(J, iFT, out=iFT)
J_1 = np.add(J, -1, out=J)
dUdJ = np.multiply(bulk, J_1, out=J_1)
dUdF = np.multiply(dUdJ, JiFT, out=JiFT)
np.add(P, dUdF, 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
bulk = self.bulk
J = det(F)
iFT = transpose(inv(F, J))
A4 = out
if A4 is None:
A4 = np.zeros((*F.shape[:2], *F.shape[:2], *F.shape[-2:]), dtype=F.dtype)
else:
np.multiply(A4, 0, out=A4)
trC = None
A4b = None
A4c = None
if mu is not None:
# "physical"-deviatoric (not math-deviatoric!) part of A4
eye = identity(F)
trC = ddot(F, F, parallel=self.parallel, out=trC)
np.add(A4, cdya_ik(eye, eye), out=A4)
A4b = dya(F, iFT, parallel=self.parallel, out=A4b)
np.multiply(-2 / 3, A4b, out=A4b)
np.add(A4, A4b, out=A4)
np.add(A4, np.transpose(A4b, [2, 3, 0, 1, 4, 5]), out=A4)
A4b = dya(iFT, iFT, out=A4b)
trC_3 = np.divide(trC, 3, out=trC)
A4c = np.multiply(A4b, trC_3, out=A4c)
np.add(A4, np.transpose(A4c, [0, 3, 2, 1, 4, 5]), out=A4)
np.multiply(A4c, 2 / 3, out=A4c)
np.add(A4, A4c, out=A4)
np.multiply(mu, A4, out=A4)
Jm23 = np.power(J, -2 / 3, out=trC)
np.multiply(Jm23, A4, out=A4)
if bulk is not None:
# "physical"-volumetric (not math-volumetric!) part of A4
if A4b is None:
A4b = dya(iFT, iFT, out=A4b)
J_1 = np.add(J, -1, out=trC)
p = np.multiply(bulk, J_1, out=J_1)
pJ = np.multiply(p, J, out=p)
J2 = np.power(J, 2, out=J)
bulk_J2 = np.multiply(J2, bulk, out=J2)
qJ = np.add(pJ, bulk_J2, out=bulk_J2)
A4c = np.multiply(qJ, A4b, out=A4c)
np.add(A4, A4c, out=A4)
np.multiply(-pJ, np.transpose(A4b, [0, 3, 2, 1, 4, 5]), out=A4c)
np.add(A4, A4c, out=A4)
return [A4]
