Source code for felupe.constitution.jax._updated_lagrange
# -*- 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/>.
"""
from functools import wraps
import jax
import jax.numpy as jnp
[docs]
def updated_lagrange(material):
r"""Decorate a Cauchy-stress Updated-Lagrange material formulation as a first Piola-
Kirchoff stress function.
Notes
-----
The equilibrium equations for statics are given in Eq. :eq:`equilibrium_statics`.
.. math::
:label: equilibrium_statics
\operatorname{div} \boldsymbol{\sigma} + \boldsymbol{b} = \boldsymbol{0}
The weak form of the equilibrium equations for statics is given in Eq.
:eq:`equilibrium_statics_weak`.
.. math::
:label: equilibrium_statics_weak
\int_v \operatorname{div} \boldsymbol{\sigma} \cdot \delta \boldsymbol{u} \ \mathrm{d}v
+ \int_v \boldsymbol{b} \cdot \delta \boldsymbol{u} \ \mathrm{d}v &= 0
- \int_v \boldsymbol{\sigma} :
\frac{\partial \delta \boldsymbol{u}}{\partial \boldsymbol{x}} \ \mathrm{d}v
+ \int_{\partial v} \left( \boldsymbol{\sigma} \cdot \boldsymbol{n} \right )
\cdot \delta \boldsymbol{u} \ \mathrm{d}a
+ \int_v \boldsymbol{b} \cdot \delta \boldsymbol{u} \ \mathrm{d}v &= 0
This leads to the virtual work of internal forces, see Eq.
:eq:`virtual_work_internal`.
.. math::
:label: virtual_work_internal
\delta W_{\text{int}} = -\int_v
\boldsymbol{\sigma} :
\frac{\partial \delta \boldsymbol{u}}{\partial \boldsymbol{x}} \ \mathrm{d}v
The variation of the total potential energy of internal forces is given in Eq.
:eq:`updated_lagrange_variation`.
.. math::
:label: updated_lagrange_variation
\delta \Pi &= \int_v
\boldsymbol{\sigma} :
\frac{\partial \delta \boldsymbol{u}}{\partial \boldsymbol{x}} \ \mathrm{d}v
\delta \Pi &= \int_V
\boldsymbol{\sigma} : \delta \boldsymbol{F} \boldsymbol{F}^{-1} \ J \mathrm{d}V
\delta \Pi &= \int_V
\boldsymbol{P} : \delta \boldsymbol{F} \ \mathrm{d}V
Finally, the first Piola-Kirchhoff stress tensor is given by Eq.
:eq:`first_piola_kirchhoff_stress`.
.. math::
:label: first_piola_kirchhoff_stress
\boldsymbol{P} = J \boldsymbol{\sigma} \boldsymbol{F}^{-T}
Examples
--------
>>> import felupe as fem
>>> import felupe.constitution.jax as mat
>>> import jax.numpy as jnp
>>>
>>> @fem.updated_lagrange
>>> def neo_hooke_updated_lagrange(F, mu=1):
>>> J = jnp.linalg.det(F)
>>> b = F @ F.T
>>> dev = lambda b: b - jnp.trace(b) / 3 * jnp.eye(3)
>>> τ = mu * dev(J**(-2/3) * b)
>>> return τ / J
>>>
>>> umat = mat.Material(neo_hooke_updated_lagrange, mu=1)
See Also
--------
felupe.constitution.jax.Hyperelastic : A hyperelastic material definition with a
given function for the strain energy density function per unit undeformed volume
with Automatic Differentiation provided by jax.
felupe.constitution.jax.Material : A material definition with a given function for
the partial derivative of the strain energy function w.r.t. the deformation
gradient tensor with Automatic Differentiation provided by jax.
"""
@wraps(material)
def first_piola_kirchhoff_stress(F, *args, **kwargs):
# evaluate the Cauchy stress
res = material(F, *args, **kwargs)
# check if the material formulation returns state variables and extract
# the Cauchy stress tensor
if isinstance(res, jax.Array):
σ = res
statevars_new = None
else:
σ, statevars_new = res
# first Piola-Kirchhoff stress tensor
J = jnp.linalg.det(F)
P = J * σ @ jnp.linalg.inv(F).T
if statevars_new is None:
return P
else:
return P, statevars_new
return first_piola_kirchhoff_stress
