Tensor Analysis

Let’s look at the tensor notation and voigt notation and tensor roatation. For that, we need to first import Tensor module

import numpy as np
import pandas as pd

from santex.tensor import Tensor

Let’s load some atiffness matrix values for forsterite, enstatite and diopside. The values can be seen in voigt matrix format

cij_forsterite = np.array([[320.5,  68.15,  71.6,     0,     0,     0],
  [ 68.15,  196.5,  76.8,     0,     0,     0],
  [  71.6,   76.8, 233.5,     0,     0,     0],
  [   0,      0,      0,     64,     0,     0],
  [   0,      0,      0,      0,    77,     0],
  [   0,      0,      0,      0,     0,  78.7]])

cij_enstatite = np.array([[  236.90,   79.60,   63.20,    0.00,    0.00,    0.00],
  [    79.60,  180.50,   56.80,    0.00,    0.00,    0.00],
  [    63.20,   56.80,  230.40,    0.00,    0.00,    0.00],
  [     0.00,    0.00,    0.00,   84.30,    0.00,    0.00],
  [     0.00,    0.00,    0.00,    0.00,   79.40,    0.00],
  [     0.00,    0.00,    0.00,    0.00,    0.00,   80.10]])
cij_diopside = np.array([[  228.10,   78.80,   70.20,    0.00,    7.90,    0.00],
  [    78.80,  181.10,   61.10,    0.00,    5.90,    0.00],
  [    70.20,   61.10,  245.40,    0.00,   39.70,    0.00],
  [     0.00,    0.00,    0.00,   78.90,    0.00,    6.40],
  [     7.90,    5.90,   39.70,    0.00,   68.20,    0.00],
  [     0.00,    0.00,    0.00,    6.40,    0.00,   78.10]])

Let’s instantiate the Tensor class

tensor = Tensor()

Let’s convert the 66 voigt matrix to 3333 stiffness tensor for forsterite

cijkl_forsterite = tensor.voigt_to_tensor(cij_forsterite)

cijkl_forsterite

array([[[[320.5 ,   0.  ,   0.  ],
         [  0.  ,  68.15,   0.  ],
         [  0.  ,   0.  ,  71.6 ]],

        [[  0.  ,  78.7 ,   0.  ],
         [ 78.7 ,   0.  ,   0.  ],
         [  0.  ,   0.  ,   0.  ]],

        [[  0.  ,   0.  ,  77.  ],
         [  0.  ,   0.  ,   0.  ],
         [ 77.  ,   0.  ,   0.  ]]],


       [[[  0.  ,  78.7 ,   0.  ],
         [ 78.7 ,   0.  ,   0.  ],
         [  0.  ,   0.  ,   0.  ]],

        [[ 68.15,   0.  ,   0.  ],
         [  0.  , 196.5 ,   0.  ],
         [  0.  ,   0.  ,  76.8 ]],

        [[  0.  ,   0.  ,   0.  ],
         [  0.  ,   0.  ,  64.  ],
         [  0.  ,  64.  ,   0.  ]]],


       [[[  0.  ,   0.  ,  77.  ],
         [  0.  ,   0.  ,   0.  ],
         [ 77.  ,   0.  ,   0.  ]],

        [[  0.  ,   0.  ,   0.  ],
         [  0.  ,   0.  ,  64.  ],
         [  0.  ,  64.  ,   0.  ]],

        [[ 71.6 ,   0.  ,   0.  ],
         [  0.  ,  76.8 ,   0.  ],
         [  0.  ,   0.  , 233.5 ]]]])

Now, let’s convert the 333*3 stiffness tensor to voigt matrix again

cij_forsterite = tensor.tensor_to_voigt(cijkl_forsterite)

cij_forsterite

array([[320.5 ,  68.15,  71.6 ,   0.  ,   0.  ,   0.  ],
       [ 68.15, 196.5 ,  76.8 ,   0.  ,   0.  ,   0.  ],
       [ 71.6 ,  76.8 , 233.5 ,   0.  ,   0.  ,   0.  ],
       [  0.  ,   0.  ,   0.  ,  64.  ,   0.  ,   0.  ],
       [  0.  ,   0.  ,   0.  ,   0.  ,  77.  ,   0.  ],
       [  0.  ,   0.  ,   0.  ,   0.  ,   0.  ,  78.7 ]])

Rotate Tensors

Let’s now rotate the above tensor with alpha = 10 degrees, beta = 20 degrees, and gamma = 30 degrees

alpha = 10
beta = 20
gamma = 30 # in degrees

rotated_forsterite = tensor.rotate_tensor(cijkl_forsterite, alpha, beta, gamma)

voigt_rotated_forsterite = tensor.tensor_to_voigt(rotated_forsterite)

rotated_forsterite

array([[[[254.40669486,  32.19107185,   9.1917399 ],
         [ 32.19107185,  83.60527452,   2.25426383],
         [  9.1917399 ,   2.25426383,  74.25170921]],

        [[ 32.19107185,  93.02059007,   5.79247342],
         [ 93.02059007,  24.85704921,   6.26816834],
         [  5.79247342,   6.26816834,  -1.68082112]],

        [[  9.1917399 ,   5.79247342,  73.39492984],
         [  5.79247342,   4.15506621,   7.02836206],
         [ 73.39492984,   7.02836206,   4.58001959]]],


       [[[ 32.19107185,  93.02059007,   5.79247342],
         [ 93.02059007,  24.85704921,   6.26816834],
         [  5.79247342,   6.26816834,  -1.68082112]],

        [[ 83.60527452,  24.85704921,   4.15506621],
         [ 24.85704921, 229.09177813,   3.16662267],
         [  4.15506621,   3.16662267,  77.9967799 ]],

        [[  2.25426383,   6.26816834,   7.02836206],
         [  6.26816834,   3.16662267,  72.58824371],
         [  7.02836206,  72.58824371,  -5.76818554]]],


       [[[  9.1917399 ,   5.79247342,  73.39492984],
         [  5.79247342,   4.15506621,   7.02836206],
         [ 73.39492984,   7.02836206,   4.58001959]],

        [[  2.25426383,   6.26816834,   7.02836206],
         [  6.26816834,   3.16662267,  72.58824371],
         [  7.02836206,  72.58824371,  -5.76818554]],

        [[ 74.25170921,  -1.68082112,   4.58001959],
         [ -1.68082112,  77.9967799 ,  -5.76818554],
         [  4.58001959,  -5.76818554, 228.39399978]]]])

Let’s now look at the rotated forsterite in voigt form and compare with the original non rotated voigt form

voigt_rotated_forsterite

array([[254.40669486,  83.60527452,  74.25170921,   2.25426383,
          9.1917399 ,  32.19107185],
       [ 83.60527452, 229.09177813,  77.9967799 ,   3.16662267,
          4.15506621,  24.85704921],
       [ 74.25170921,  77.9967799 , 228.39399978,  -5.76818554,
          4.58001959,  -1.68082112],
       [  2.25426383,   3.16662267,  -5.76818554,  72.58824371,
          7.02836206,   6.26816834],
       [  9.1917399 ,   4.15506621,   4.58001959,   7.02836206,
         73.39492984,   5.79247342],
       [ 32.19107185,  24.85704921,  -1.68082112,   6.26816834,
          5.79247342,  93.02059007]])