Source code for felupe.constitution._models_hyperelasticity

# -*- 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,
    ddot,
    det,
    dot,
    dya,
    identity,
    inv,
    trace,
    transpose,
)


[docs]class NeoHooke: r"""Nearly-incompressible isotropic hyperelastic Neo-Hooke 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. 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) Arguments --------- mu : float Shear modulus bulk : float Bulk modulus """ def __init__(self, mu=None, bulk=None, parallel=False): self.parallel = parallel self.mu = mu self.bulk = 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, mu=None, bulk=None): """Strain energy density function per unit undeformed volume of the Neo-Hookean material formulation. Arguments --------- x : list of ndarray List with the Deformation gradient ``F`` (3x3) as first item mu : float, optional Shear modulus (default is None) bulk : float, optional Bulk modulus (default is None) """ F = x[0] if mu is None: mu = self.mu if bulk is None: bulk = self.bulk J = det(F) C = dot(transpose(F), F, parallel=self.parallel) 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, mu=None, bulk=None): """Gradient of the strain energy density function per unit undeformed volume of the Neo-Hookean material formulation. Arguments --------- x : list of ndarray List with the Deformation gradient ``F`` (3x3) as first item mu : float, optional Shear modulus (default is None) bulk : float, optional Bulk modulus (default is None) """ F, statevars = x[0], x[-1] if mu is None: mu = self.mu if bulk is None: bulk = self.bulk J = det(F) iFT = transpose(inv(F, J)) # "physical"-deviatoric (not math-deviatoric!) part of P P = mu * (F - ddot(F, F, parallel=self.parallel) / 3 * iFT) * J ** (-2 / 3) if bulk is not None: # "physical"-volumetric (not math-volumetric!) part of P P += bulk * (J - 1) * J * iFT return [P, statevars]
[docs] def hessian(self, x, mu=None, bulk=None): """Hessian of the strain energy density function per unit undeformed volume of the Neo-Hookean material formulation. Arguments --------- x : list of ndarray List with the Deformation gradient ``F`` (3x3) as first item mu : float, optional Shear modulus (default is None) bulk : float, optional Bulk modulus (default is None) """ F = x[0] if mu is None: mu = self.mu if bulk is None: bulk = self.bulk J = det(F) iFT = transpose(inv(F, J)) eye = identity(F) # "physical"-deviatoric (not math-deviatoric!) part of A4 A4 = ( mu * ( cdya_ik(eye, eye, parallel=self.parallel) - 2 / 3 * dya(F, iFT, parallel=self.parallel) - 2 / 3 * dya(iFT, F, parallel=self.parallel) + 2 / 9 * ddot(F, F, parallel=self.parallel) * dya(iFT, iFT, parallel=self.parallel) + 1 / 3 * ddot(F, F, parallel=self.parallel) * cdya_il(iFT, iFT, parallel=self.parallel) ) * J ** (-2 / 3) ) if bulk is not None: p = bulk * (J - 1) q = p + bulk * J # "physical"-volumetric (not math-volumetric!) part of A4 A4 += J * ( q * dya(iFT, iFT, parallel=self.parallel) - p * cdya_il(iFT, iFT, parallel=self.parallel) ) return [A4]