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

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 =
 [[0.35473279 0.66569612 0.37851704]
 [0.66569612 0.49881953 0.34467884]
 [0.37851704 0.34467884 0.91026962]]
KM(ε) =
 [0.35473279 0.49881953 0.91026962 0.48744949 0.53530393 0.94143648]

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 =
 [[3.43130579 1.02195245 1.50621265 2.46912517 2.82552258 2.01149492]
 [1.02195245 2.06658272 0.53164441 0.71906412 1.37138385 1.14551359]
 [1.50621265 0.53164441 2.24862943 1.51201039 1.66657135 1.17429339]
 [2.46912517 0.71906412 1.51201039 3.87954039 2.73599011 1.91678353]
 [2.82552258 1.37138385 1.66657135 2.73599011 4.52891943 2.49894942]
 [2.01149492 1.14551359 1.17429339 1.91678353 2.49894942 2.882882  ]]
c =
 [[[[3.43130579 1.4223417  1.99794618]
   [1.4223417  1.02195245 1.74593515]
   [1.99794618 1.74593515 1.50621265]]

  [[1.4223417  1.441441   1.24947471]
   [1.441441   0.81000043 0.95839176]
   [1.24947471 0.95839176 0.83035082]]

  [[1.99794618 1.24947471 2.26445972]
   [1.24947471 0.96971482 1.36799506]
   [2.26445972 1.36799506 1.1784439 ]]]


 [[[1.4223417  1.441441   1.24947471]
   [1.441441   0.81000043 0.95839176]
   [1.24947471 0.95839176 0.83035082]]

  [[1.02195245 0.81000043 0.96971482]
   [0.81000043 2.06658272 0.50845512]
   [0.96971482 0.50845512 0.53164441]]

  [[1.74593515 0.95839176 1.36799506]
   [0.95839176 0.50845512 1.93977019]
   [1.36799506 1.93977019 1.0691528 ]]]


 [[[1.99794618 1.24947471 2.26445972]
   [1.24947471 0.96971482 1.36799506]
   [2.26445972 1.36799506 1.1784439 ]]

  [[1.74593515 0.95839176 1.36799506]
   [0.95839176 0.50845512 1.93977019]
   [1.36799506 1.93977019 1.0691528 ]]

  [[1.50621265 0.83035082 1.1784439 ]
   [0.83035082 0.53164441 1.0691528 ]
   [1.1784439  1.0691528  2.24862943]]]]

$\,$

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