$$ \newcommand{\C}{{\mathbb{{C}}}} \newcommand{\R}{{\mathbb{{R}}}} \newcommand{\Q}{{\mathbb{{Q}}}} \newcommand{\Z}{{\mathbb{{Z}}}} \newcommand{\N}{{\mathbb{{N}}}} \newcommand{\uu}[1]{{\boldsymbol{{#1}}}} \newcommand{\uuuu}[1]{{\symbb{{#1}}}} \newcommand{\uv}[1]{{\underline{{#1}}}} \newcommand{\ve}[1]{{\uv{{e}}_{{#1}}}} \newcommand{\x}{{\uv{{x}}}} \newcommand{\n}{{\uv{{n}}}} \newcommand{\eps}{{\uu{{\varepsilon}}}} \newcommand{\E}{{\uu{{E}}}} \newcommand{\sig}{{\uu{{\sigma}}}} \newcommand{\Sig}{{\uu{{\Sigma}}}} \newcommand{\cod}{{\uv{{\symscr{b}}}}} % \newcommand{\trans}[1]{{{}^{t}{#1}}} \newcommand{\trans}[1]{{{#1}{}^\intercal}} \newcommand{\sotimes}{{\stackrel{s}{\otimes}}} \newcommand{\sboxtimes}{\stackrel{s}{\boxtimes}} \newcommand{\norm}[1]{{\lVert{{#1}}\rVert}} \newcommand{\ud}{{\,\mathrm{d}}} \newcommand{\mat}{\mathsf} \DeclareMathOperator{\arcosh}{arcosh} \DeclareMathOperator{\divz}{div} \DeclareMathOperator{\divu}{\uv{div}} \DeclareMathOperator{\hess}{hess} \DeclareMathOperator{\gradu}{\uv{grad}} \DeclareMathOperator{\graduu}{\uu{grad}} \DeclareMathOperator{\Mat}{Mat} \DeclareMathOperator{\tr}{tr} \DeclareMathOperator{\ISO}{ISO} \newcommand{\volt}[1]{{#1}^{-1\circ}} \newcommand{\dcirc}{\overset{\circ}{:}} \newcommand{\jump}[1]{\mathopen{[\![}\,#1\,\mathclose{]\!]}} $$

1 Kelvin-Mandel notation

Objectives

Before introducing the specific objects of echoes devoted to tensor calculations in isotropic or anisotropic contexts, this tutorial aims at providing the syntax allowing to represent second-order or fourth-order tensors under the form of matrices in the Kelvin-Mandel notation as detailed in Section A.2.

Imports

import numpy as np
from echoes import *
import math, random

np.set_printoptions(precision=8, suppress=True)
# to display only 8 significant digits of array components

A symmetric $3×3$ second-order matrix can be transformed in a vector of $\R^6$ by the function KM consistently with (A.14). The inverse is done by invKM.

α = np.random.rand(3, 3) ; ε = (α+α.T)/2
print("ε =\n",ε)
print("KM(ε) =\n",KM(ε))
assert np.allclose(invKM(KM(ε)), ε), "error"

ε =
 [[0.69709813 0.53506694 0.32442103]
 [0.53506694 0.94504071 0.77793264]
 [0.32442103 0.77793264 0.3123916 ]]
KM(ε) =
 [0.69709813 0.94504071 0.3123916  1.10016289 0.45880062 0.75669892]

Given a $3×3×3×3$ array c (of type numpy.ndarray) satisfying major and minor symmetries (see Section A.2), the corresponding $6×6$ matrix C obtained by Kelvin-Mandel transform is calculated by C = KM(c). Conversely, if C is a positive definite matrix, c is calculated by c = invKM(C).

A = np.random.rand(6,6)
C = A.T.dot(A) + np.eye(6) # generation of an arbitrary positive definite matrix
c = invKM(C)
print("C =\n",C)
print("c =\n",c)
assert np.allclose(KM(c), C), "error: KM(c) should be equal to C"

C =
 [[2.8344474  1.34244392 1.68007537 1.99254191 1.60339856 1.25473437]
 [1.34244392 2.80416731 1.9010566  1.6950774  1.97903622 0.90911807]
 [1.68007537 1.9010566  3.18344589 2.10720072 2.31111208 1.38175589]
 [1.99254191 1.6950774  2.10720072 3.43130947 2.12506884 1.71139854]
 [1.60339856 1.97903622 2.31111208 2.12506884 3.68201779 1.60227351]
 [1.25473437 0.90911807 1.38175589 1.71139854 1.60227351 2.64520247]]
c =
 [[[[2.8344474  0.88723118 1.133774  ]
   [0.88723118 1.34244392 1.4089399 ]
   [1.133774   1.4089399  1.68007537]]

  [[0.88723118 1.32260124 0.80113676]
   [1.32260124 0.64284356 0.85569927]
   [0.80113676 0.85569927 0.97704896]]

  [[1.133774   0.80113676 1.8410089 ]
   [0.80113676 1.39938993 1.06253442]
   [1.8410089  1.06253442 1.63420302]]]


 [[[0.88723118 1.32260124 0.80113676]
   [1.32260124 0.64284356 0.85569927]
   [0.80113676 0.85569927 0.97704896]]

  [[1.34244392 0.64284356 1.39938993]
   [0.64284356 2.80416731 1.19860072]
   [1.39938993 1.19860072 1.9010566 ]]

  [[1.4089399  0.85569927 1.06253442]
   [0.85569927 1.19860072 1.71565474]
   [1.06253442 1.71565474 1.49001592]]]


 [[[1.133774   0.80113676 1.8410089 ]
   [0.80113676 1.39938993 1.06253442]
   [1.8410089  1.06253442 1.63420302]]

  [[1.4089399  0.85569927 1.06253442]
   [0.85569927 1.19860072 1.71565474]
   [1.06253442 1.71565474 1.49001592]]

  [[1.68007537 0.97704896 1.63420302]
   [0.97704896 1.9010566  1.49001592]
   [1.63420302 1.49001592 3.18344589]]]]

$\,$

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