Add the Blatz-Ko hyperelastic foam model (#901)

docs/felupe/constitution/autodiff/jax.rst

@@ -47,6 +47,7 @@ These material model formulations are defined by a strain energy density functio
 
 .. autosummary::
 
+   blatz_ko
    miehe_goektepe_lulei
    mooney_rivlin
    storakers
@@ -90,6 +91,8 @@ formulations in :class:`~felupe.constitution.jax.Material`.
 
 .. autofunction:: felupe.constitution.jax.updated_lagrange
 
+.. autofunction:: felupe.constitution.jax.models.hyperelastic.blatz_ko
+
 .. autofunction:: felupe.constitution.jax.models.hyperelastic.miehe_goektepe_lulei
 
 .. autofunction:: felupe.constitution.jax.models.hyperelastic.mooney_rivlin

docs/felupe/constitution/autodiff/tensortrax.rst

@@ -32,6 +32,7 @@ These material model formulations are defined by a strain energy density functio
    alexander
    anssari_benam_bucchi
    arruda_boyce
+   blatz_ko
    extended_tube
    finite_strain_viscoelastic
    lopez_pamies
@@ -96,6 +97,8 @@ and :func:`~felupe.constitution.tensortrax.updated_lagrange` material formulatio
 
 .. autofunction:: felupe.constitution.tensortrax.models.hyperelastic.arruda_boyce
 
+.. autofunction:: felupe.constitution.tensortrax.models.hyperelastic.blatz_ko
+
 .. autofunction:: felupe.constitution.tensortrax.models.hyperelastic.extended_tube
 
 .. autofunction:: felupe.constitution.tensortrax.models.hyperelastic.finite_strain_viscoelastic

src/felupe/__init__.py

@@ -249,6 +249,7 @@
         alexander,
         anssari_benam_bucchi,
         arruda_boyce,
+        blatz_ko,
         constitutive_material,
         extended_tube,
         finite_strain_viscoelastic,
@@ -278,6 +279,7 @@
         "alexander",
         "anssari_benam_bucchi",
         "arruda_boyce",
+        "blatz_ko",
         "extended_tube",
         "finite_strain_viscoelastic",
         "isochoric_volumetric_split",

src/felupe/constitution/__init__.py

@@ -58,6 +58,7 @@
         alexander,
         anssari_benam_bucchi,
         arruda_boyce,
+        blatz_ko,
         extended_tube,
         finite_strain_viscoelastic,
         lopez_pamies,
@@ -78,6 +79,7 @@
         "alexander",
         "anssari_benam_bucchi",
         "arruda_boyce",
+        "blatz_ko",
         "extended_tube",
         "finite_strain_viscoelastic",
         "isochoric_volumetric_split",

src/felupe/constitution/jax/models/hyperelastic/__init__.py

@@ -1,10 +1,12 @@
+from ._blatz_ko import blatz_ko
 from ._miehe_goektepe_lulei import miehe_goektepe_lulei
 from ._mooney_rivlin import mooney_rivlin
 from ._storakers import storakers
 from ._third_order_deformation import third_order_deformation
 from ._yeoh import yeoh
 
 __all__ = [
+    "blatz_ko",
     "miehe_goektepe_lulei",
     "mooney_rivlin",
     "storakers",
@@ -13,6 +15,7 @@
 ]
 
 # default (stable) material parameters
+blatz_ko.kwargs = dict(mu=1)
 miehe_goektepe_lulei.kwargs = dict(mu=0, N=100, U=0, p=2, q=2)
 mooney_rivlin.kwargs = dict(C10=0, C01=0)
 storakers.kwargs = dict(mu=[0], alpha=[2], beta=[1])

src/felupe/constitution/jax/models/hyperelastic/_blatz_ko.py

@@ -0,0 +1,33 @@
+# -*- 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
+
+from jax.numpy import sqrt, trace
+from jax.numpy.linalg import det
+
+from ....tensortrax.models.hyperelastic import blatz_ko as blatz_ko_docstring
+
+
+@wraps(blatz_ko_docstring)
+def blatz_ko(C, mu):
+
+    I1 = trace(C)
+    I2 = (I1**2 - trace(C @ C)) / 2
+    I3 = det(C)
+
+    return mu * (I2 / I3 + 2 * sqrt(I3) - 5)

src/felupe/constitution/tensortrax/models/hyperelastic/__init__.py

@@ -12,6 +12,7 @@
 from ._alexander import alexander
 from ._anssari_benam_bucchi import anssari_benam_bucchi
 from ._arruda_boyce import arruda_boyce
+from ._blatz_ko import blatz_ko
 from ._extended_tube import extended_tube
 from ._finite_strain_viscoelastic import finite_strain_viscoelastic
 from ._lopez_pamies import lopez_pamies
@@ -31,6 +32,7 @@
     "alexander",
     "anssari_benam_bucchi",
     "arruda_boyce",
+    "blatz_ko",
     "extended_tube",
     "finite_strain_viscoelastic",
     "lopez_pamies",
@@ -51,6 +53,7 @@
 alexander.kwargs = dict(C1=0, C2=0, C3=0, gamma=100, k=0)
 anssari_benam_bucchi.kwargs = dict(mu=0, N=100)
 arruda_boyce.kwargs = dict(C1=0, limit=100)
+blatz_ko.kwargs = dict(mu=0)
 extended_tube.kwargs = dict(Gc=0, Ge=0, beta=1, delta=0)
 lopez_pamies.kwargs = dict(mu=[0, 0], alpha=[1, 4])
 miehe_goektepe_lulei.kwargs = dict(mu=0, N=100, U=0, p=2, q=2)

src/felupe/constitution/tensortrax/models/hyperelastic/_blatz_ko.py

@@ -0,0 +1,94 @@
+# -*- 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 tensortrax.math import sqrt, trace
+from tensortrax.math.linalg import det
+
+
+def blatz_ko(C, mu):
+    r"""Strain energy function of the Blatz-Ko isotropic hyperelastic
+    `foam <https://doi.org/10.1122/1.548937>`_ material formulation [1]_.
+
+    Parameters
+    ----------
+    C : tensortrax.Tensor or jax.Array
+        Right Cauchy-Green deformation tensor.
+    mu : float
+        The shear modulus.
+
+    Notes
+    -----
+    The Poisson ratio of the Blatz-Ko model formulation is :math:`\nu = 0.25`. The
+    strain energy function is given in Eq. :eq:`psi-blatz-ko`
+
+    ..  math::
+        :label: psi-blatz-ko
+
+        \psi = \frac{\mu}{2} \left(\frac{I_2}{I_3} + 2 \sqrt{I_3} - 5 \right)
+
+    The shear modulus :math:`\mu` is related to young's modulus as denoted in Eq.
+    :eq:`shear-modulus-blatz-ko`.
+
+    ..  math::
+        :label: shear-modulus-blatz-ko
+
+        \mu = \frac{2 E}{5}
+
+    Examples
+    --------
+    First, choose the desired automatic differentiation backend
+
+    ..  pyvista-plot::
+        :context:
+
+        >>> # import felupe.constitution.jax as mat
+        >>> import felupe.constitution.tensortrax as mat
+
+    and create the hyperelastic material.
+
+    ..  pyvista-plot::
+        :context:
+
+        >>> import felupe as fem
+        >>>
+        >>> umat = mat.Hyperelastic(mat.models.hyperelastic.blatz_ko, mu=1.0)
+        >>> ax = umat.plot()
+
+    ..  pyvista-plot::
+        :include-source: False
+        :context:
+        :force_static:
+
+        >>> import pyvista as pv
+        >>>
+        >>> fig = ax.get_figure()
+        >>> chart = pv.ChartMPL(fig)
+        >>> chart.show()
+
+    References
+    ----------
+    .. [1] P. J. Blatz and W. L. Ko, "Application of Finite Elastic Theory to the
+       Deformation of Rubbery Materials", Transactions of the Society of Rheology, vol.
+       6, no. 1. Society of Rheology, pp. 223–252, Mar. 01, 1962. doi: 10.1122/1.548937.
+    """
+
+    I1 = trace(C)
+    I2 = (I1**2 - trace(C @ C)) / 2
+    I3 = det(C)
+
+    return mu * (I2 / I3 + 2 * sqrt(I3) - 5)

tests/test_constitution.py

@@ -386,6 +386,7 @@ def neo_hooke(C, mu=1):
             ),
             False,
         ),
+        (fem.blatz_ko, dict(mu=1.0), False),
     ]:
         umat = fem.Hyperelastic(model, **kwargs)
 

tests/test_constitution_jax.py

@@ -68,6 +68,7 @@ def test_hyperelastic_jax():
         md.third_order_deformation,
         md.miehe_goektepe_lulei,
         md.storakers,
+        md.blatz_ko,
     ]:
         umat = mat.Hyperelastic(W, **W.kwargs)
         solid = fem.SolidBody(umat=umat, field=field)