$$ \newcommand{\C}{{\mathbb{{C}}}} \newcommand{\R}{{\mathbb{{R}}}} \newcommand{\Q}{{\mathbb{{Q}}}} \newcommand{\Z}{{\mathbb{{Z}}}} \newcommand{\N}{{\mathbb{{N}}}} \newcommand{\uu}[1]{{\boldsymbol{{#1}}}} \newcommand{\uuuu}[1]{{\symbb{{#1}}}} \newcommand{\uv}[1]{{\underline{{#1}}}} \newcommand{\ve}[1]{{\uv{{e}}_{{#1}}}} \newcommand{\x}{{\uv{{x}}}} \newcommand{\n}{{\uv{{n}}}} \newcommand{\eps}{{\uu{{\varepsilon}}}} \newcommand{\E}{{\uu{{E}}}} \newcommand{\sig}{{\uu{{\sigma}}}} \newcommand{\Sig}{{\uu{{\Sigma}}}} \newcommand{\cod}{{\uv{{\symscr{b}}}}} \newcommand{\trans}[1]{{{}^{t}{#1}}} \newcommand{\sotimes}{{\stackrel{s}{\otimes}}} \newcommand{\sboxtimes}{\stackrel{s}{\boxtimes}} \newcommand{\norm}[1]{{\lVert{{#1}}\rVert}} \newcommand{\ud}{{\,\mathrm{d}}} \newcommand{\mat}{\mathsf} \DeclareMathOperator{\arcosh}{arcosh} \DeclareMathOperator{\divz}{div} \DeclareMathOperator{\divu}{\uv{div}} \DeclareMathOperator{\hess}{hess} \DeclareMathOperator{\gradu}{\uv{grad}} \DeclareMathOperator{\graduu}{\uu{grad}} \DeclareMathOperator{\Mat}{Mat} \DeclareMathOperator{\tr}{tr} \DeclareMathOperator{\ISO}{ISO} \newcommand{\jump}[1]{[\hspace*{-.15em}[\hspace*{.1em}{#1}% \hspace*{.1em}]\hspace*{-.15em}]} $$

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 =
 [[0.68926093 0.38749989 0.07905665]
 [0.38749989 0.73063341 0.79515912]
 [0.07905665 0.79515912 0.25769377]]
KM(ε) =
 [0.68926093 0.73063341 0.25769377 1.12452481 0.11180298 0.5480076 ]

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 =
 [[3.39924889 2.61561741 1.68906414 1.6082464  2.37004987 1.96310658]
 [2.61561741 4.93763362 2.45114019 2.03665861 2.86213159 2.91525212]
 [1.68906414 2.45114019 2.71242505 1.23348354 2.05770511 1.61612909]
 [1.6082464  2.03665861 1.23348354 2.33178588 1.52670798 1.52328004]
 [2.37004987 2.86213159 2.05770511 1.52670798 3.80606145 1.85934243]
 [1.96310658 2.91525212 1.61612909 1.52328004 1.85934243 3.46582955]]
c =
 [[[[3.39924889 1.38812598 1.67587834]
   [1.38812598 2.61561741 1.13720193]
   [1.67587834 1.13720193 1.68906414]]

  [[1.38812598 1.73291477 0.92967122]
   [1.73291477 2.06139455 0.76164002]
   [0.92967122 0.76164002 1.14277584]]

  [[1.67587834 0.92967122 1.90303072]
   [0.92967122 2.02383266 0.76335399]
   [1.90303072 0.76335399 1.45501724]]]


 [[[1.38812598 1.73291477 0.92967122]
   [1.73291477 2.06139455 0.76164002]
   [0.92967122 0.76164002 1.14277584]]

  [[2.61561741 2.06139455 2.02383266]
   [2.06139455 4.93763362 1.44013512]
   [2.02383266 1.44013512 2.45114019]]

  [[1.13720193 0.76164002 0.76335399]
   [0.76164002 1.44013512 1.16589294]
   [0.76335399 1.16589294 0.87220458]]]


 [[[1.67587834 0.92967122 1.90303072]
   [0.92967122 2.02383266 0.76335399]
   [1.90303072 0.76335399 1.45501724]]

  [[1.13720193 0.76164002 0.76335399]
   [0.76164002 1.44013512 1.16589294]
   [0.76335399 1.16589294 0.87220458]]

  [[1.68906414 1.14277584 1.45501724]
   [1.14277584 2.45114019 0.87220458]
   [1.45501724 0.87220458 2.71242505]]]]

$\,$

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