# -*- 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 dya
[docs]class OgdenRoxburgh:
r"""A Pseudo-Elastic material formulation for an isotropic treatment of the
load-history dependent Mullins-softening of rubber-like materials.
.. math::
\eta(W, W_{max}) &= 1 - \frac{1}{r} erf\left( \frac{W_{max} - W}
{m + \beta~W_{max}} \right)
\boldsymbol{P} &= \eta \frac{\partial \psi}{\partial \boldsymbol{F}}
\mathbb{A} &= \frac{\partial^2 \psi}{\partial \boldsymbol{F} \partial
\boldsymbol{F}} + \frac{\partial \eta}{\partial \psi} \frac{\partial \psi}
{\partial \boldsymbol{F}} \otimes \frac{\partial \psi}{\partial \boldsymbol{F}}
"""
def __init__(self, material, r, m, beta):
# isotropic hyperelastic material formulation
self.material = material
# ogden-roxburgh material parameters
self.r = r
self.m = m
self.beta = beta
# initial variables for calling
# ``self.gradient(self.x)`` and ``self.hessian(self.x)``
self.x = [np.eye(3), np.zeros(1)]
[docs] def gradient(self, x):
# unpack variables into deformation gradient and state variables
F, statevars = x[0], x[-1]
# material parameters alias
r, m, beta = self.r, self.m, self.beta
# isotropic material formulation: evaluate
# * the strain energy function and
# * the first Piola-Kirchhoff stress tensor
W = self.material.function([F, statevars])[0]
P = self.material.gradient([F, statevars])[0]
# get the maximum load-history strain energy function
Wmax = np.maximum(W, statevars[0])
z = (Wmax - W) / (m + beta * Wmax)
# softening function
eta = 1 - np.tanh(z) / r
# update the state variables
statevars_new = statevars.copy()
statevars_new[0] = Wmax
return [eta * P, statevars_new]
[docs] def hessian(self, x):
# unpack variables into deformation gradient and state variables
F, statevars = x[0], x[-1]
# material parameters alias
r, m, beta = self.r, self.m, self.beta
# isotropic material formulation: evaluate
# * the strain energy function and
# * the first Piola-Kirchhoff stress tensor as well as
# * the according fourth-order elasticity tensor
W = self.material.function([F, statevars])[0]
P = self.material.gradient([F, statevars])[0]
A = self.material.hessian([F, statevars])[0]
# get the maximum load-history strain energy function
Wmax = np.maximum(W, statevars[0])
z = (Wmax - W) / (m + beta * Wmax)
# softening function
eta = 1 - np.tanh(z) / r
# derivative of softening function
detadz = (-1 / r) * 1 / np.cosh(z) ** 2
dzdW = -1 / (m + beta * Wmax)
detadW = detadz * dzdW
# set non-softened derivative to zero
detadW[np.isclose(eta, 1)] = 0
return [eta * A + detadW * dya(P, P)]