https://github.com/adtzlr/hyperelastic
Constitutive hyperelastic material formulations for FElupe.
https://github.com/adtzlr/hyperelastic
automatic-differentiation computational-mechanics constitution fea fem finite-elements-analysis hyperelasticity python pytorch scientific-computing solid-mechanics-codes torch
Last synced: 3 months ago
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Constitutive hyperelastic material formulations for FElupe.
- Host: GitHub
- URL: https://github.com/adtzlr/hyperelastic
- Owner: adtzlr
- License: gpl-3.0
- Created: 2023-06-21T19:47:42.000Z (about 2 years ago)
- Default Branch: main
- Last Pushed: 2024-05-01T20:34:05.000Z (about 1 year ago)
- Last Synced: 2024-10-18T17:03:54.715Z (8 months ago)
- Topics: automatic-differentiation, computational-mechanics, constitution, fea, fem, finite-elements-analysis, hyperelasticity, python, pytorch, scientific-computing, solid-mechanics-codes, torch
- Language: Python
- Homepage: https://hyperelastic.readthedocs.io/
- Size: 566 KB
- Stars: 7
- Watchers: 3
- Forks: 0
- Open Issues: 0
-
Metadata Files:
- Readme: README.md
- License: LICENSE
- Citation: CITATION.cff
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README
Constitutive hyperelastic material formulations for FElupe.
[](https://pypi.python.org/pypi/hyperelastic/) [](https://www.gnu.org/licenses/gpl-3.0) -0c674a) [](https://codecov.io/gh/adtzlr/hyperelastic) [](https://zenodo.org/badge/latestdoi/656860854)  [](https://hyperelastic.readthedocs.io/en/latest/?badge=latest) [](https://hyperelastic.readthedocs.io/_/downloads/en/latest/pdf/)
This package provides the essential building blocks for constitutive hyperelastic material formulations. This includes material behaviour-independent spaces and frameworks as well as material behaviour-dependent model formulations.
**Spaces** ([`hyperelastic.spaces`](https://github.com/adtzlr/hyperelastic/tree/main/src/hyperelastic/spaces)) are full or partial deformations on which a given material formulation should be projected to, e.g. to the distortional (part of the deformation) space. Generalized *Total-Lagrange* **Frameworks** ([`hyperelastic.frameworks`](https://github.com/adtzlr/hyperelastic/tree/main/src/hyperelastic/frameworks)) for isotropic hyperelastic material formulations based on the invariants of the right Cauchy-Green deformation tensor and the principal stretches enable a clean coding of isotropic material formulations.
The [`hyperelastic.math`](https://github.com/adtzlr/hyperelastic/tree/main/src/hyperelastic/math)-module provides helpers in reduced vector ([Voigt](https://en.wikipedia.org/wiki/Voigt_notation)) storage for symmetric three-dimensional second-order tensors along with a matrix storage for (at least minor) symmetric three-dimensional fourth-order tensors. Shear terms are not doubled for strain-like tensors, instead all math operations take care of the reduced vector storage.
$$ \boldsymbol{C} = \begin{bmatrix} C_{11} & C_{22} & C_{33} & C_{12} & C_{23} & C_{13} \end{bmatrix}^T $$
# Installation
Install Python, fire up 🔥 a terminal and run 🏃
```shell
pip install hyperelastic
```
# Usage
Material model formulations have to be created as classes with methods for the evaluation of the `gradient` (stress) and the `hessian` (elasticity) of the strain energy function. It depends on the framework which derivatives have to be defined, e.g. the derivatives w.r.t. the invariants of the right Cauchy-Green deformation tensor or w.r.t. the principal stretches. An instance of a **Framework** has to be *finalized* by the application on a **Space**.
- [Deformation (Full) Space](https://github.com/adtzlr/hyperelastic/blob/main/src/hyperelastic/spaces/_deformation.py)
- [Distortional Space](https://github.com/adtzlr/hyperelastic/blob/main/src/hyperelastic/spaces/_distortional.py)
- [Dilatational Space](https://github.com/adtzlr/hyperelastic/blob/main/src/hyperelastic/spaces/_dilatational.py)
> ⓘ **Note**
> Define your own material model formulation with manual, automatic or symbolic differentiation with the help of your favourite package, e.g. [PyTorch](https://pytorch.org/), [JAX](https://jax.readthedocs.io/en/latest/), [Tensorflow](https://www.tensorflow.org/), [TensorTRAX](https://github.com/adtzlr/tensortrax), [SymPy](https://www.sympy.org/en/index.html), etc.
First, let's import hyperelastic (and its math module).
```python
import hyperelastic as hel
import hyperelastic.math as hm
```
## Invariant-based material formulations
A minimal template for an invariant-based material formulation applied on the distortional space:
```python
class MyInvariantsModel:
def gradient(self, I1, I2, I3, statevars):
"""The gradient as the partial derivative of the strain energy function w.r.t.
the invariants of the right Cauchy-Green deformation tensor."""
return dWdI1, dWdI2, dWdI3, statevars
def hessian(self, I1, I2, I3, statevars_old):
"""The hessian as the second partial derivatives of the strain energy function
w.r.t. the invariants of the right Cauchy-Green deformation tensor."""
return d2WdI1I1, d2WdI2I2, d2WdI3I3, d2WdI1I2, d2WdI2I3, d2WdI1I3
model = MyInvariantsModel()
framework = hel.InvariantsFramework(model)
umat = hel.DistortionalSpace(framework)
```
### Available isotropic hyperelastic invariant-based material formulations
The typical polynomial-based material formulations ([Neo-Hooke](https://en.wikipedia.org/wiki/Neo-Hookean_solid), [Mooney-Rivlin](https://en.wikipedia.org/wiki/Mooney%E2%80%93Rivlin_solid), [Yeoh](https://en.wikipedia.org/wiki/Yeoh_hyperelastic_model)) are all available as submodels of the third order deformation material formulation.
- [Third-Order-Deformation (James-Green-Simpson)](https://onlinelibrary.wiley.com/doi/abs/10.1002/app.1975.070190723) ([code](https://github.com/adtzlr/hyperelastic/blob/main/src/hyperelastic/models/invariants/_third_order_deformation.py))
- [TorchModel](https://pytorch.org/docs/stable/autograd.html) ([code](https://github.com/adtzlr/hyperelastic/blob/main/src/hyperelastic/models/invariants/_torch.py))
## Principal stretch-based material formulations
A minimal template for a principal stretch-based material formulation applied on the distortional space:
```python
class MyStretchesModel:
def gradient(self, λ, statevars):
"""The gradient as the partial derivative of the strain energy function w.r.t.
the principal stretches."""
return [dWdλ1, dWdλ2, dWdλ3], statevars
def hessian(self, λ, statevars_old):
"""The hessian as the second partial derivatives of the strain energy function
w.r.t. the principal stretches."""
return d2Wdλ1dλ1, d2Wdλ2dλ2, d2Wdλ3dλ3, d2Wdλ1dλ2, d2Wdλ2dλ3, d2Wdλ1dλ3
model = MyStretchesModel()
framework = hel.StretchesFramework(model)
umat = hel.DistortionalSpace(framework)
```
### Available isotropic hyperelastic stretch-based material formulations
- [Ogden](https://en.wikipedia.org/wiki/Ogden_(hyperelastic_model)) ([code](https://github.com/adtzlr/hyperelastic/blob/main/src/hyperelastic/models/stretches/_ogden.py))
## Lab
In the [`Lab`](https://github.com/adtzlr/hyperelastic/tree/main/src/hyperelastic/lab), [`Simulations`](https://github.com/adtzlr/hyperelastic/tree/main/src/hyperelastic/lab/_simulation.py) on homogeneous load cases provide a visualization of the material response behaviour.
```python
import numpy as np
import hyperelastic
stretch = np.linspace(0.7, 2.5, 181)
parameters = {"C10": 0.3, "C01": 0.2}
def material(C10, C01):
tod = hyperelastic.models.invariants.ThirdOrderDeformation(C10=C10, C01=C01)
framework = hyperelastic.InvariantsFramework(tod)
return hyperelastic.DeformationSpace(framework)
ux = hyperelastic.lab.Simulation(
loadcase=hyperelastic.lab.Uniaxial(label="uniaxial"),
stretch=np.linspace(0.7, 2.5),
material=material,
labels=parameters.keys(),
parameters=parameters.values(),
)
ps = hyperelastic.lab.Simulation(
loadcase=hyperelastic.lab.Planar(label="planar"),
stretch=np.linspace(1.0, 2.5),
material=material,
labels=parameters.keys(),
parameters=parameters.values(),
)
bx = hyperelastic.lab.Simulation(
loadcase=hyperelastic.lab.Biaxial(label="biaxial"),
stretch=np.linspace(1.0, 1.75),
material=material,
labels=parameters.keys(),
parameters=parameters.values(),
)
fig, ax = ux.plot_stress_stretch(lw=2)
fig, ax = ps.plot_stress_stretch(ax=ax, lw=2)
fig, ax = bx.plot_stress_stretch(ax=ax, lw=2)
ax.legend()
ax.set_title(rf"Mooney-Rivlin (C10={parameters['C10']}, C01={parameters['C01']})")
```
# License
Hyperelastic - Constitutive hyperelastic material formulations for FElupe (C) 2024 Andreas Dutzler, Graz (Austria).
This program 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.
This program 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 this program. If not, see .