large_deformation/hyperelastic.pyΒΆ

Description

Nearly incompressible Mooney-Rivlin hyperelastic material model.

Large deformation is described using the total Lagrangian formulation. Models of this kind can be used to model e.g. rubber or some biological materials.

Find \ul{u} such that:

\intl{\Omega\suz}{} \left( \ull{S}\eff(\ul{u})
+ K(J-1)\; J \ull{C}^{-1} \right) : \delta \ull{E}(\ul{v}) \difd{V}
= 0
\;, \quad \forall \ul{v} \;,

where

\ull{F} deformation gradient F_{ij} = \pdiff{x_i}{X_j}
J \det(F)
\ull{C} right Cauchy-Green deformation tensor C = F^T F
\ull{E}(\ul{u}) Green strain tensor E_{ij} = \frac{1}{2}(\pdiff{u_i}{X_j} +
\pdiff{u_j}{X_i} + \pdiff{u_m}{X_i}\pdiff{u_m}{X_j})
\ull{S}\eff(\ul{u}) effective second Piola-Kirchhoff stress tensor

The effective stress \ull{S}\eff(\ul{u}) is given by:

\ull{S}\eff(\ul{u}) = \mu J^{-\frac{2}{3}}(\ull{I}
- \frac{1}{3}\tr(\ull{C}) \ull{C}^{-1})
+ \kappa J^{-\frac{4}{3}} (\tr(\ull{C}\ull{I} - \ull{C}
- \frac{2}{6}((\tr{\ull{C}})^2 - \tr{(\ull{C}^2)})\ull{C}^{-1})
\;.

../../_images/large_deformation-hyperelastic.png

source code

# -*- coding: utf-8 -*-
r"""
Nearly incompressible Mooney-Rivlin hyperelastic material model.

Large deformation is described using the total Lagrangian formulation.
Models of this kind can be used to model e.g. rubber or some biological
materials.

Find :math:`\ul{u}` such that:

.. math::
    \intl{\Omega\suz}{} \left( \ull{S}\eff(\ul{u})
    + K(J-1)\; J \ull{C}^{-1} \right) : \delta \ull{E}(\ul{v}) \difd{V}
    = 0
    \;, \quad \forall \ul{v} \;,

where

.. list-table::
   :widths: 20 80

   * - :math:`\ull{F}`
     - deformation gradient :math:`F_{ij} = \pdiff{x_i}{X_j}`
   * - :math:`J`
     - :math:`\det(F)`
   * - :math:`\ull{C}`
     -  right Cauchy-Green deformation tensor :math:`C = F^T F`
   * - :math:`\ull{E}(\ul{u})`
     - Green strain tensor :math:`E_{ij} = \frac{1}{2}(\pdiff{u_i}{X_j} +
       \pdiff{u_j}{X_i} + \pdiff{u_m}{X_i}\pdiff{u_m}{X_j})`
   * - :math:`\ull{S}\eff(\ul{u})`
     - effective second Piola-Kirchhoff stress tensor

The effective stress :math:`\ull{S}\eff(\ul{u})` is given by:

.. math::
    \ull{S}\eff(\ul{u}) = \mu J^{-\frac{2}{3}}(\ull{I}
    - \frac{1}{3}\tr(\ull{C}) \ull{C}^{-1})
    + \kappa J^{-\frac{4}{3}} (\tr(\ull{C}\ull{I} - \ull{C}
    - \frac{2}{6}((\tr{\ull{C}})^2 - \tr{(\ull{C}^2)})\ull{C}^{-1})
    \;.
"""
from __future__ import print_function
from __future__ import absolute_import
import numpy as nm

from sfepy import data_dir

filename_mesh = data_dir + '/meshes/3d/cylinder.mesh'

options = {
    'nls' : 'newton',
    'ls' : 'ls',
    'ts' : 'ts',
    'save_times' : 'all',
    'post_process_hook' : 'stress_strain',
}


field_1 = {
    'name' : 'displacement',
    'dtype' : nm.float64,
    'shape' : 3,
    'region' : 'Omega',
    'approx_order' : 1,
}

material_1 = {
    'name' : 'solid',
    'values' : {
        'K'  : 1e3, # bulk modulus
        'mu' : 20e0, # shear modulus of neoHookean term
        'kappa' : 10e0, # shear modulus of Mooney-Rivlin term
    }
}

variables = {
    'u' : ('unknown field', 'displacement', 0),
    'v' : ('test field', 'displacement', 'u'),
}

regions = {
    'Omega' : 'all',
    'Left' : ('vertices in (x < 0.001)', 'facet'),
    'Right' : ('vertices in (x > 0.099)', 'facet'),
}

##
# Dirichlet BC + related functions.
ebcs = {
    'l' : ('Left', {'u.all' : 0.0}),
    'r' : ('Right', {'u.0' : 0.0, 'u.[1,2]' : 'rotate_yz'}),
}

centre = nm.array( [0, 0], dtype = nm.float64 )

def rotate_yz(ts, coor, **kwargs):
    from sfepy.linalg import rotation_matrix2d

    vec = coor[:,1:3] - centre

    angle = 10.0 * ts.step
    print('angle:', angle)

    mtx = rotation_matrix2d( angle )
    vec_rotated = nm.dot( vec, mtx )

    displacement = vec_rotated - vec

    return displacement

functions = {
    'rotate_yz' : (rotate_yz,),
}

def stress_strain( out, problem, state, extend = False ):
    from sfepy.base.base import Struct, debug

    ev = problem.evaluate
    strain = ev('dw_tl_he_neohook.i.Omega( solid.mu, v, u )',
                mode='el_avg', term_mode='strain')
    out['green_strain'] = Struct(name='output_data',
                                 mode='cell', data=strain, dofs=None)

    stress = ev('dw_tl_he_neohook.i.Omega( solid.mu, v, u )',
                mode='el_avg', term_mode='stress')
    out['neohook_stress'] = Struct(name='output_data',
                                   mode='cell', data=stress, dofs=None)

    stress = ev('dw_tl_he_mooney_rivlin.i.Omega( solid.kappa, v, u )',
                mode='el_avg', term_mode='stress')
    out['mooney_rivlin_stress'] = Struct(name='output_data',
                                         mode='cell', data=stress, dofs=None)

    stress = ev('dw_tl_bulk_penalty.i.Omega( solid.K, v, u )',
                mode='el_avg', term_mode= 'stress')
    out['bulk_stress'] = Struct(name='output_data',
                                mode='cell', data=stress, dofs=None)

    return out

##
# Balance of forces.
integral_1 = {
    'name' : 'i',
    'order' : 1,
}
equations = {
    'balance' : """dw_tl_he_neohook.i.Omega( solid.mu, v, u )
                 + dw_tl_he_mooney_rivlin.i.Omega( solid.kappa, v, u )
                 + dw_tl_bulk_penalty.i.Omega( solid.K, v, u )
                 = 0""",
}

##
# Solvers etc.
solver_0 = {
    'name' : 'ls',
    'kind' : 'ls.scipy_direct',
}

solver_1 = {
    'name' : 'newton',
    'kind' : 'nls.newton',

    'i_max'      : 5,
    'eps_a'      : 1e-10,
    'eps_r'      : 1.0,
    'macheps'    : 1e-16,
    'lin_red'    : 1e-2, # Linear system error < (eps_a * lin_red).
    'ls_red'     : 0.1,
    'ls_red_warp': 0.001,
    'ls_on'      : 1.1,
    'ls_min'     : 1e-5,
    'check'      : 0,
    'delta'      : 1e-6,
}

solver_2 = {
    'name' : 'ts',
    'kind' : 'ts.simple',

    't0'    : 0,
    't1'    : 1,
    'dt'    : None,
    'n_step' : 11, # has precedence over dt!
    'verbose' : 1,
}