Averaging methods

This tutorial explains how to compute the Voigt, Reuss and Hill averages from a given stiffness tensor and a finite or an infinite set of rotations.

The corresponding elastic moduli yet depends on the model, which actually depends on the underlying assumption, namely:

Voigt (i.e. homogeneous strain assumption);
Reuss (i.e. homogeneous stress assumption);
Voigt-Reuss-Hill (or Hill, for short).

The later is just the arithmetic mean between the Voigt and the Reuss stiffnesses.

As an example, let’s consider the stiffness tensor for monoclinic TiNi:

>>> from Elasticipy.FourthOrderTensor import StiffnessTensor
>>> C = StiffnessTensor.fromCrystalSymmetry(symmetry='monoclinic', diad='y', phase_name='TiNi',
...                                        C11=231, C12=127, C13=104,
...                                        C22=240, C23=131, C33=175,
...                                        C44=81, C55=11, C66=85,
...                                        C15=-18, C25=1, C35=-3, C46=3)

Infinite number of orientations (the non-textured case)

If we consider an infinite number of rotations, uniformly distributed on SO3, this necessarily leads to an isotropic behaviour. In this case, the aforementioned averages can be estimated as follows:

>>> C_Voigt = C.Voigt_average()
>>> C_Reuss = C.Reuss_average()
>>> C_Hill = C.Hill_average()

Let’s see how C_Hill looks like:

>>> print(C_Hill)
Stiffness tensor (in Voigt notation):
[[200.92610744 119.59583533 119.59583533   0.           0.
    0.        ]
 [119.59583533 200.92610744 119.59583533   0.           0.
    0.        ]
 [119.59583533 119.59583533 200.92610744   0.           0.
    0.        ]
 [  0.           0.           0.          40.66513606   0.
    0.        ]
 [  0.           0.           0.           0.          40.66513606
    0.        ]
 [  0.           0.           0.           0.           0.
   40.66513606]]
Symmetry: Triclinic

As a comparison, let’s see how the underlying assumption impair the Young moduli:

>>> E_Voigt = C_Voigt.Young_modulus
>>> E_Reuss = C_Reuss.Young_modulus
>>> E_Hill = C_Hill.Young_modulus
>>> print(E_Voigt.mean(), E_Reuss.mean(), E_Hill.mean())
145.66862361382906 76.13802022165396 111.6769053248696

Finite number of orientations

If one wants to consider a finite set of orientations, or consider that these orientations are not uniformly distributed over the SO3 (i.e. if we have a crystallographic texture), the first steps consists in estimating the mean of all the rotated stiffnesses/compliances (depending on the model).

As an example, we assume that the material displays a perfect fiber texture along z. In terms of Bunge-Euler angles, it means that we have Phi=0 for each orientation:

>>> from scipy.spatial.transform import Rotation
>>> import numpy as np
>>> phi1 = np.random.random(10000)*2*np.pi # Random sampling from 0 to 2pi
>>> Phi = phi2 = np.zeros(10000)
>>> Euler_angles = np.array([phi1, Phi, phi2]).T
>>> rotations = Rotation.from_euler('ZXZ', Euler_angles)    # Bunge-Euler angles

Now generate the rotated stiffness tensors:

>>> C_rotated = C * rotations

And compute the Hill averages :

>>> C_Hill = C_rotated.Hill_average()

Let’s see how the Young modulus is distributed over space:

>>> C_Hill.Young_modulus.plot3D() 

It is thus clear that the fiber along the z axis results in a transverse-isotropic behavior in the X-Y plane.