adtzlr · adtzlr · May 9, 2024 · May 9, 2024 · May 9, 2024 · May 9, 2024
diff --git a/CHANGELOG.md b/CHANGELOG.md
@@ -4,11 +4,14 @@ All notable changes to this project will be documented in this file. The format
 ## [Unreleased]
 
 ### Added
-- Add the isotropic-hyperelastic `alexander(C1, C2, C2, gamma)` model.
+- Add the isotropic-hyperelastic `alexander(C1, C2, C2, gamma, k)` model.
 
 ### Changed
 - Recfactor the `constitution` module.
 
+### Fixed
+- Fix plotting the keyword-arguments of a constitutive material `ConstitutiveMaterial.plot(show_kwargs=True)`. For list-based material parameters of length 1, the brackets aren't shown now. E.g., this affects optimized material parameters.
+
 ## [8.5.1] - 2024-05-08
 
 ### Fixed

diff --git a/src/felupe/constitution/_view.py b/src/felupe/constitution/_view.py
@@ -95,7 +95,9 @@ def plot(self, ax=None, show_title=True, show_kwargs=True, **kwargs):
                 p = []
                 for key, value in self.umat.kwargs.items():
                     if hasattr(value, "__len__"):
-                        val = "[" + " ".join([f"{v:.3g}" for v in value]) + "]"
+                        val = " ".join([f"{v:.3g}" for v in value])
+                        if len(value) > 1:
+                            val = f"[{val}]"
                     else:
                         val = f"{value:.3g}"
                     p.append(f"{key}={val}")

diff --git a/src/felupe/constitution/hyperelasticity/models/__init__.py b/src/felupe/constitution/hyperelasticity/models/__init__.py
@@ -27,7 +27,7 @@
 ]
 
 # default (stable) material parameters
-alexander.kwargs = dict(C1=0, C2=0, C3=0, gamma=100)
+alexander.kwargs = dict(C1=0, C2=0, C3=0, gamma=100, k=0)
 arruda_boyce.kwargs = dict(C1=0, limit=100)
 extended_tube.kwargs = dict(Gc=0, Ge=0, beta=1, delta=0)
 mooney_rivlin.kwargs = dict(C10=0, C01=0)

diff --git a/src/felupe/constitution/hyperelasticity/models/_alexander.py b/src/felupe/constitution/hyperelasticity/models/_alexander.py
@@ -15,11 +15,13 @@
 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 log, trace
+import numpy as np
+from tensortrax import Tensor, Δ, Δδ, f, δ
+from tensortrax.math import exp, log, trace
 from tensortrax.math.linalg import det
 
 
-def alexander(C, C1, C2, C3, gamma):
+def alexander(C, C1, C2, C3, gamma, k):
     r"""Strain energy function of the isotropic hyperelastic
     `Alexander <https://doi.org/10.1016/0020-7225(68)90006-2>`_ material
     formulation [1]_.
@@ -29,23 +31,34 @@ def alexander(C, C1, C2, C3, gamma):
     C : tensortrax.Tensor
         Right Cauchy-Green deformation tensor.
     C1 : float
-        First material parameter associated to the first invariant.
+        Scale factor for the first invariant term.
     C2 : float
-        Second material parameter associated to the second invariant.
+        Scale factor for the second invariant term.
     C3 : float
-        Third material parameter associated to the second invariant.
+        Scale factor for the logarithmic second invariant term.
     gamma : float
-        Dimensionless material parameter associated to the second invariant.
+        Offset-normalization parameter for the logarithmic second invariant term.
+    k : float
+        Scale factor for the exponential first invariant term.
 
     Notes
     -----
+    ..  warning::
+        The strain energy function of the Alexander model formulation is not directly
+        implemented. Only its gradient and hessian w.r.t. the right Cauchy-Green
+        deformation tensor are defined. This is because the imaginary error-function
+        :math:`\text{erfi}(x)` is not included in NumPy - this would require SciPy as a
+        dependency.
+
     The strain energy function is given in Eq. :eq:`psi-alexander`
 
     ..  math::
         :label: psi-alexander
 
-        \psi = C_1 \left(\hat{I}_1 - 3 \right)
-             + C_2 \ln{\frac{\hat{I}_2 - 3 + \gamma}{\gamma}}
+        \psi = C_1 \int_{\hat{I}_1} \exp \left(
+                   k \left(\hat{I}_1 - 3 \right)^2
+               \right) \ d\hat{I}_1
+             + C_2 \ln \left(\frac{\hat{I}_2 - 3 + \gamma}{\gamma} \right)
              + C_3 \left(\hat{I}_2 - 3 \right)
 
     with the first and second main invariant of the distortional part of the right
@@ -78,7 +91,7 @@ def alexander(C, C1, C2, C3, gamma):
         >>> import felupe as fem
         >>>
         >>> umat = fem.Hyperelastic(
-        ...    fem.alexander, C1=0.117, C2=0.137, C3=0.00690, gamma=0.735
+        ...    fem.alexander, C1=17, C2=19.85, C3=1, gamma=0.735, k=0.00015
         ... )
         >>> ax = umat.plot(incompressible=True)
 
@@ -104,4 +117,16 @@ def alexander(C, C1, C2, C3, gamma):
     I1 = J3 * trace(C)
     I2 = (I1**2 - J3**2 * trace(C @ C)) / 2
 
-    return C1 * (I1 - 3) + C2 * log(((I2 - 3) + gamma) / gamma) + C3 * (I2 - 3)
+    def W(I1):
+        "A non-evaluated function with defined first- and second derivatives."
+
+        dWdI1 = exp(k * (I1 - 3) ** 2)
+        return Tensor(
+            x=np.full_like(f(I1), fill_value=np.nan),
+            δx=f(dWdI1) * δ(I1),
+            Δx=f(dWdI1) * Δ(I1),
+            Δδx=δ(dWdI1) * Δ(I1) + f(dWdI1) * Δδ(I1),
+            ntrax=I1.ntrax,
+        )
+
+    return C1 * W(I1) + C2 * log(((I2 - 3) + gamma) / gamma) + C3 * (I2 - 3)
diff --git a/tests/test_constitution.py b/tests/test_constitution.py
@@ -332,7 +332,7 @@ def neo_hooke(C, mu=1):
         ),
         (
             fem.constitution.alexander,
-            dict(C1=0.117, C2=0.137, C3=0.00690, gamma=0.735),
+            dict(C1=0.117, C2=0.137, C3=0.00690, gamma=0.735, k=0.00015),
             True,
         ),
     ]:
@@ -550,6 +550,7 @@ def test_optimize():
         fem.ogden,
         fem.third_order_deformation,
         fem.extended_tube,
+        fem.alexander,
     ]:
         umat = fem.Hyperelastic(model)
         umat_new, res = umat.optimize(ux=[stretches, stresses], incompressible=True)