# -*- 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 ..assembly import IntegralForm
from ..constitution import AreaChange
from ..math import dot
from ._helpers import Assemble, Results
[docs]
class SolidBodyCauchyStress:
r"""A Cauchy stress boundary on a solid body.
Parameters
----------
field : FieldContainer
A field container with fields created on a boundary region.
stress : ndarray of shape (3, 3, ...) or None, optional
The prescribed Cauchy stress components :math:`\sigma_{ij}` (default is None).
If None, all Cauchy stress components are set to zero.
Notes
-----
.. math::
\delta W_{ext} = \int_{\partial V}
\delta \boldsymbol{u} \cdot \boldsymbol{\sigma}\ d\boldsymbol{a} =
\int_{\partial V}
\delta \boldsymbol{u} \cdot \boldsymbol{\sigma} J \boldsymbol{F}^{-T}\
d\boldsymbol{A}
Examples
--------
.. pyvista-plot::
:force_static:
>>> import numpy as np
>>> import felupe as fem
>>>
>>> mesh = fem.Rectangle(n=6)
>>> region = fem.RegionQuad(mesh)
>>> field = fem.FieldContainer([fem.FieldAxisymmetric(region, dim=2)])
>>>
>>> boundaries = {"fixed": fem.Boundary(field[0], fx=0)}
>>> solid = fem.SolidBody(umat=fem.NeoHooke(mu=1, bulk=2), field=field)
>>>
>>> mask = np.logical_and(mesh.x == 1, mesh.y > 0.5)
>>> region_stress = fem.RegionQuadBoundary(
... mesh=mesh,
... mask=mask, # select a subset of faces on the surface
... ensure_3d=True, # True for axisymmetric/plane strain
... )
>>> field_boundary = fem.FieldContainer([fem.FieldAxisymmetric(region_stress, dim=2)])
>>> stress = fem.SolidBodyCauchyStress(field=field_boundary)
>>>
>>> table = (
... fem.math.linsteps([0, 1], num=5, axis=1, axes=9)
... + fem.math.linsteps([0, 1], num=5, axis=3, axes=9)
... ).reshape(-1, 3, 3)
>>>
>>> step = fem.Step(
... items=[solid, stress], ramp={stress: 1.0 * table}, boundaries=boundaries
... )
>>> job = fem.Job(steps=[step]).evaluate()
>>> solid.plot("Principal Values of Cauchy Stress").show()
"""
def __init__(self, field, cauchy_stress=None):
self.field = field
self._normals = self.field.region.normals
self.results = Results()
self.results.kinematics = self._extract(self.field)
self.results.cauchy_stress = np.zeros((3, 3))
if cauchy_stress is not None:
self.results.cauchy_stress = cauchy_stress
self.assemble = Assemble(
vector=self._vector, matrix=self._matrix, multiplier=-1.0
)
self._area_change = AreaChange()
[docs]
def update(self, cauchy_stress):
self.__init__(self.field, cauchy_stress)
def _extract(self, field):
self.field = field
self.results.kinematics = self.field.extract()
return self.results.kinematics
def _vector(self, field=None, parallel=False, resize=None):
if field is not None:
self._update(field)
self.results.kinematics = self._extract(self.field)
fun = self._area_change.function(
self.results.kinematics,
self._normals,
parallel=parallel,
)
fun[0] = dot(self.results.cauchy_stress, fun[0], mode=(2, 2), out=fun[0])
self.results.force = IntegralForm(
fun=fun, v=self.field, dV=self.field.region.dV, grad_v=[False]
).assemble(parallel=parallel)
if resize is not None:
self.results.force.resize(*resize.shape)
return self.results.force
def _matrix(self, field=None, parallel=False, resize=None):
if field is not None:
self._update(field)
self.results.kinematics = self._extract(self.field)
fun = self._area_change.gradient(
self.results.kinematics,
self._normals,
parallel=parallel,
)
fun[0] = dot(self.results.cauchy_stress, fun[0], mode=(2, 4), out=fun[0])
self.results.stiffness = IntegralForm(
fun=fun,
v=self.field,
u=self.field,
dV=self.field.region.dV,
grad_v=[False],
grad_u=[True],
).assemble(parallel=parallel)
if resize is not None:
self.results.stiffness.resize(*resize.shape)
return self.results.stiffness
def _update(self, other_field, field=None):
if field is not None:
self.field = field
self.field[0].values = other_field[0].values
self.results.kinematics = self._extract(self.field)
return self.field
