Source code for felupe.constitution._models_pseudo_elasticity

# -*- coding: utf-8 -*-
"""
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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)]