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

3  Special tensors

Important Objectives

This tutorial presents the matrix representation of isotropic tensors of second and fourth orders as well as Walpole tensors useful for transverse isotropy. It strongly relies on conventions of tensor algebra introduced in Appendix A especially in terms of products and contractions.

Tip Imports

import numpy as np
from echoes import *
import math

np.set_printoptions(precision=6, suppress=True)
# to display only 6 significant digits of array components

Second-order identity

The second-order identity \(\uu{1}=\delta_{ij}\ve{i}\otimes\ve{j}\) is given in Kelvin-Mandel notation (i.e. vector of \(\R^6\)) by the constant vector Id2

δ = Id2
print("𝟏 (Kelvin-Mandel notation) =\n",δ)
print("𝟏 =\n",invKM(δ))

𝟏 (Kelvin-Mandel notation) =
 [1. 1. 1. 0. 0. 0.]
𝟏 =
 [[1. 0. 0.]
 [0. 1. 0.]
 [0. 0. 1.]]

Fourth-order isotropic tensors

As detailed in Section A.4, the fourth-order identity tensor is

\[ \uuuu{I}=\uu{1}\sboxtimes\uu{1}= \frac{\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}}{2}\,\ve{i}\otimes\ve{j}\otimes\ve{k}\otimes\ve{l} \]

and the projectors

\[ \uuuu{J}=\frac{1}{3}\uu{1}\otimes\uu{1} \quad \textrm{and} \quad \uuuu{K}=\uuuu{I}-\uuuu{J}=\uu{1}\sboxtimes\uu{1}-\frac{1}{3}\uu{1}\otimes\uu{1} \]

which are provided in echoes by Id4, J4 and K4

for T in [Id4, J4, K4]:
    print(T)

[[1. 0. 0. 0. 0. 0.]
 [0. 1. 0. 0. 0. 0.]
 [0. 0. 1. 0. 0. 0.]
 [0. 0. 0. 1. 0. 0.]
 [0. 0. 0. 0. 1. 0.]
 [0. 0. 0. 0. 0. 1.]]
[[0.333333 0.333333 0.333333 0.       0.       0.      ]
 [0.333333 0.333333 0.333333 0.       0.       0.      ]
 [0.333333 0.333333 0.333333 0.       0.       0.      ]
 [0.       0.       0.       0.       0.       0.      ]
 [0.       0.       0.       0.       0.       0.      ]
 [0.       0.       0.       0.       0.       0.      ]]
[[ 0.666667 -0.333333 -0.333333  0.        0.        0.      ]
 [-0.333333  0.666667 -0.333333  0.        0.        0.      ]
 [-0.333333 -0.333333  0.666667  0.        0.        0.      ]
 [ 0.        0.        0.        1.        0.        0.      ]
 [ 0.        0.        0.        0.        1.        0.      ]
 [ 0.        0.        0.        0.        0.        1.      ]]

Walpole bases

The Walpole bases are useful to decompose transversely isotropic fourth-order tensors. They are presented in Section A.5. The Kelvin-Mandel representation of the \(i^\textrm{th}\) Walpole tensor oriented along an axis \(\n\) is constructed by W(i,n=e₃) (\(i \in \{0,..,5\}\) and the normal is by default oriented along the third axis). The symmetrized version is provided by WS(i,n=e₃) (\(i \in \{0,..,4\}\)).

WarningWarning

Note again the shift in indices between the Python convention starting at 0 and the tensors presented in Section A.5.

for i in range(6):
    print("𝕎"+str(i+1)+" =\n",W(i))

for i in range(5):
    print("𝕎ˢ"+str(i+1)+" =\n",WS(i))

𝕎1 =
 [[0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0.]
 [0. 0. 1. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0.]]
𝕎2 =
 [[0.5 0.5 0.  0.  0.  0. ]
 [0.5 0.5 0.  0.  0.  0. ]
 [0.  0.  0.  0.  0.  0. ]
 [0.  0.  0.  0.  0.  0. ]
 [0.  0.  0.  0.  0.  0. ]
 [0.  0.  0.  0.  0.  0. ]]
𝕎3 =
 [[0.       0.       0.       0.       0.       0.      ]
 [0.       0.       0.       0.       0.       0.      ]
 [0.707107 0.707107 0.       0.       0.       0.      ]
 [0.       0.       0.       0.       0.       0.      ]
 [0.       0.       0.       0.       0.       0.      ]
 [0.       0.       0.       0.       0.       0.      ]]
𝕎4 =
 [[0.       0.       0.707107 0.       0.       0.      ]
 [0.       0.       0.707107 0.       0.       0.      ]
 [0.       0.       0.       0.       0.       0.      ]
 [0.       0.       0.       0.       0.       0.      ]
 [0.       0.       0.       0.       0.       0.      ]
 [0.       0.       0.       0.       0.       0.      ]]
𝕎5 =
 [[ 0.5 -0.5  0.   0.   0.   0. ]
 [-0.5  0.5  0.   0.   0.   0. ]
 [ 0.   0.   0.   0.   0.   0. ]
 [ 0.   0.   0.   0.   0.   0. ]
 [ 0.   0.   0.   0.   0.   0. ]
 [ 0.   0.   0.   0.   0.   1. ]]
𝕎6 =
 [[0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 1. 0. 0.]
 [0. 0. 0. 0. 1. 0.]
 [0. 0. 0. 0. 0. 0.]]
𝕎ˢ1 =
 [[0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0.]
 [0. 0. 1. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0.]]
𝕎ˢ2 =
 [[0.5 0.5 0.  0.  0.  0. ]
 [0.5 0.5 0.  0.  0.  0. ]
 [0.  0.  0.  0.  0.  0. ]
 [0.  0.  0.  0.  0.  0. ]
 [0.  0.  0.  0.  0.  0. ]
 [0.  0.  0.  0.  0.  0. ]]
𝕎ˢ3 =
 [[0.       0.       0.707107 0.       0.       0.      ]
 [0.       0.       0.707107 0.       0.       0.      ]
 [0.707107 0.707107 0.       0.       0.       0.      ]
 [0.       0.       0.       0.       0.       0.      ]
 [0.       0.       0.       0.       0.       0.      ]
 [0.       0.       0.       0.       0.       0.      ]]
𝕎ˢ4 =
 [[ 0.5 -0.5  0.   0.   0.   0. ]
 [-0.5  0.5  0.   0.   0.   0. ]
 [ 0.   0.   0.   0.   0.   0. ]
 [ 0.   0.   0.   0.   0.   0. ]
 [ 0.   0.   0.   0.   0.   0. ]
 [ 0.   0.   0.   0.   0.   1. ]]
𝕎ˢ5 =
 [[0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 0. 0. 0.]
 [0. 0. 0. 1. 0. 0.]
 [0. 0. 0. 0. 1. 0.]
 [0. 0. 0. 0. 0. 0.]]

\(\,\)