Add the hyperelastic foam model (tensortrax)

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

@@ -14,6 +14,7 @@
 from ._arruda_boyce import arruda_boyce
 from ._extended_tube import extended_tube
 from ._finite_strain_viscoelastic import finite_strain_viscoelastic
+from ._foam import foam
 from ._lopez_pamies import lopez_pamies
 from ._miehe_goektepe_lulei import miehe_goektepe_lulei
 from ._mooney_rivlin import mooney_rivlin
@@ -32,6 +33,7 @@
     "arruda_boyce",
     "extended_tube",
     "finite_strain_viscoelastic",
+    "foam",
     "lopez_pamies",
     "miehe_goektepe_lulei",
     "mooney_rivlin",
@@ -50,6 +52,7 @@
 anssari_benam_bucchi.kwargs = dict(mu=0, N=100)
 arruda_boyce.kwargs = dict(C1=0, limit=100)
 extended_tube.kwargs = dict(Gc=0, Ge=0, beta=1, delta=0)
+foam.kwargs = dict(mu=[0, 0], alpha=[2, -2], beta=[1, 1])
 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)
 mooney_rivlin.kwargs = dict(C10=0, C01=0)

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

@@ -0,0 +1,122 @@
+# -*- 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 sum as tsum
+from tensortrax.math.linalg import det, eigvalsh
+
+
+def foam(C, mu, alpha, beta):
+    r"""Strain energy function of the isotropic hyperelastic
+    `Foam <https://doi.org/10.1016/0022-5096(86)90033-5>`_ material formulation [1]_.
+
+    Parameters
+    ----------
+    C : tensortrax.Tensor
+        Right Cauchy-Green deformation tensor.
+    mu : list of float
+        List of moduli.
+    alpha : list of float
+        List of stretch exponents.
+    beta : list of float
+        List of coefficients for the degree of compressibility.
+
+    Notes
+    -----
+    The strain energy function is given in Eq. :eq:`psi-foam`
+
+    ..  math::
+        :label: psi-ogden
+
+        \psi = \sum_i \frac{2 \mu_i}{\alpha^2_i} \left[
+            \hat{\lambda}_1^{\alpha_i} +
+            \hat{\lambda}_2^{\alpha_i} +
+            \hat{\lambda}_3^{\alpha_i} - 3
+            + \frac{1}{\beta_i} \left( J^{-\alpha \beta} - 1 \right)
+        \right]
+
+    The sum of the moduli :math:`\mu_i` is equal to the initial shear modulus
+    :math:`\mu`, see Eq. :eq:`shear-modulus-foam`.
+
+    ..  math::
+        :label: shear-modulus-ogden
+
+        \mu = \sum_i \mu_i
+
+    The initial bulk modulus is given in Eq. :eq:`bulk-modulus-foam`.
+
+    ..  math::
+        :label: bulk-modulus-ogden
+
+        K = \sum_i 2 \mu_i \left( \frac{1}{3} + \beta_i \right)
+
+    Examples
+    --------
+    First, import the automatic differentiation backend
+
+    ..  pyvista-plot::
+        :context:
+
+        >>> import felupe.constitution.tensortrax as mat
+
+    and create the hyperelastic material.
+
+    ..  pyvista-plot::
+        :context:
+
+        >>> import felupe as fem
+        >>>
+        >>> umat = mat.Hyperelastic(
+        ...     mat.models.hyperelastic.foam,
+        ...     mu=[4.5 * (1.85 / 2), -4.5 * (-9.2 / 2)],
+        ...     alpha=[1.85, -9.2],
+        ...     beta=[0.92, 0.92],
+        ... )
+        >>> ax = umat.plot(
+        ...     ux=fem.math.linsteps([1, 2], 20),
+        ...     ps=fem.math.linsteps([1, 2], 20),
+        ...     bx=fem.math.linsteps([1, 1.5], 10),
+        ... )
+
+    ..  pyvista-plot::
+        :include-source: False
+        :context:
+        :force_static:
+
+        >>> import pyvista as pv
+        >>>
+        >>> fig = ax.get_figure()
+        >>> chart = pv.ChartMPL(fig)
+        >>> chart.show()
+
+    References
+    ----------
+    .. [1] B. Storåkers, "On material representation and constitutive branching in
+       finite compressible elasticity", Journal of the Mechanics and Physics of Solids,
+       vol. 34, no. 2. Elsevier BV, pp. 125–145, Jan. 1986. doi:
+       10.1016/0022-5096(86)90033-5.
+    """
+
+    I3 = det(C)
+    λ2 = I3 ** (-1 / 3) * eigvalsh(C)
+
+    return tsum(
+        [
+            2 * μ / α**2 * ((tsum(λ2 ** (α / 2)) - 3) + (I3 ** (-α * β / 2) - 1) / β)
+            for μ, α, β in zip(mu, alpha, beta)
+        ]
+    )