Add Phase speeds as vis for stiffness tensor

continuum_mechanics/visualization.py

@@ -9,6 +9,7 @@
 """
 import numpy as np
 import matplotlib.pyplot as plt
+from mpl_toolkits.mplot3d import Axes3D
 from scipy.linalg import eigvalsh
 from continuum_mechanics.tensor import christ_stiff
 
@@ -287,12 +288,12 @@ def plot_surf(R, phi, theta, title="Wave speed", **kwargs):
 
     # Phi is the azimut angle
     # theta is the cenital angle
-    V1, V2, V3, phi_vec, theta_vec = christofel_eig(C) 
+    V1, V2, V3, phi_vec, theta_vec = christofel_eig(C, 100, 100) 
     V1 = np.sqrt(V1*1e9/rho)
     V2 = np.sqrt(V2*1e9/rho)
     V3 = np.sqrt(V3*1e9/rho)
 
-    plot_surf(V1, phi_vec, theta_vec, title="qS1 speed")
-    plot_surf(V2, phi_vec, theta_vec, title="qS2 speed")
-    plot_surf(V3, phi_vec, theta_vec, title="qP speed")
+    plot_surf(V1, phi_vec, theta_vec)
+    plot_surf(V2, phi_vec, theta_vec)
+    plot_surf(V3, phi_vec, theta_vec)
     plt.show()

docs/usage.rst

@@ -322,3 +322,71 @@ Now, let us visualize the tensor
 .. image:: img/mohr3d_2.png
   :width: 600px
   :align: center
+
+
+Elasticity tensor visualization
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+Let us consider β-brass that is a cubic material and
+has the following material properties in Voigt notation:
+
+.. math::
+    C_{11} = 52\text{ GPa},\quad
+    C_{12} = 27.5\text{ GPa},\quad
+    C_{44} = 173\text{ GPa}.
+
+.. code:: python
+
+    C11 = 52e9
+    C12 = 27.5e9
+    C44 = 173e9
+    rho = 7600
+    C = np.zeros((6, 6))
+    C[0:3, 0:3] = np.array([[C11, C12, C12],
+                            [C12, C11, C12],
+                            [C12, C12, C11]])
+    C[3:6, 3:6] = np.diag([C44, C44, C44])
+
+One way to visualize a stiffness tensor is to use the
+Christofel equation
+
+.. math::
+
+    \det(\Gamma_{ij} - v_p^2\delta_{ij}) = 0\, ,
+
+where :math:`\Gamma_{ij}` is the Christofel stiffness and depends
+on the material properties (:math:`c_{ijkl}`) and unit vectors
+(:math`n_i`):
+
+.. math:
+
+    \Gamma_{ij} = c_{iklj}n_k n_l\, .
+
+This provides the eigenvalues that represent the phase
+speed for three propagation modes in the material.
+
+
+.. code:: python
+
+    V1, V2, V3, phi_vec, theta_vec = christofel_eig(C, 100, 100) 
+    V1 = np.sqrt(V1/rho)
+    V2 = np.sqrt(V2/rho)
+    V3 = np.sqrt(V3/rho)
+
+Phase speed for the first quasi-transverse mode.
+
+.. image:: img/qS1.png
+  :width: 600px
+  :align: center
+
+Phase speed for the second quasi-transverse mode.
+
+.. image:: img/qS2.png
+  :width: 600px
+  :align: center
+
+Phase speed for the quasi-longitudinal mode.
+
+.. image:: img/qP.png
+  :width: 600px
+  :align: center

setup.py

@@ -11,7 +11,10 @@
 with open('HISTORY.rst') as history_file:
     history = history_file.read()
 
-requirements = [ ]
+requirements = ['sympy>=1.3',
+                'matplotlib>=3',
+                'numpy',
+                'scipy']
 
 setup_requirements = ['pytest-runner', ]