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$\def\dblContInOut{{\scriptscriptstyle\bullet\circ}}\def\dblContOutIn{{\scriptscriptstyle\circ\bullet}}$

Acknowledgment

The notation of tensors and operators is largly adopted from O. Kintzel & Y. Başar (2006). One notable exception is the notation change from $\tnsr A \times \tnsr B$ to $\tnsr A \odot \tnsr B$ for one particular second order tensor product (see below).

Notation scheme

Scalars $x$, vectors $\vctr v$, second order tensors $\tnsr A$, fourth order tensors $\tnsrfour B$, and basis vectors $\vctr g$.

Tensor products

\begin{eqnarray*} \tnsrfour C &= \tnsr A \otimes \tnsr B &= A^{ij} B^{kl}\,\vctr g_i\otimes\vctr g_j\otimes\vctr g_k\otimes\vctr g_l \\ \tnsrfour C &= \tnsr A \odot \tnsr B &= A^{ij} B^{kl}\,\vctr g_i\otimes\vctr g_k\otimes\vctr g_l\otimes\vctr g_j \\ & &= A^{il} B^{jk}\,\vctr g_i\otimes\vctr g_j\otimes\vctr g_k\otimes\vctr g_l \\ \tnsrfour C &= \tnsr A \boxtimes \tnsr B &= A^{ij} B^{kl}\,\vctr g_i\otimes\vctr g_k\otimes\vctr g_j\otimes\vctr g_l \\ & &= A^{ik} B^{jl}\,\vctr g_i\otimes\vctr g_j\otimes\vctr g_k\otimes\vctr g_l \end{eqnarray*}

Double contraction

\begin{eqnarray*} \tnsrfour C &= \tnsrfour A :\tnsrfour B &= A^{ijkl} B_{klmn}\,\vctr g_i\otimes\vctr g_j\otimes\vctr g^m\otimes\vctr g^n \\ \tnsr C &= \tnsrfour A \dblContInOut \tnsr B &= A^{ijkl} B_{jk}\,\vctr g_i\otimes\vctr g_l \\ \tnsr C &= \tnsr A \dblContInOut \tnsrfour B &= A_{il} B^{ijkl}\,\vctr g_j\otimes\vctr g_k \\ \tnsrfour C &= \tnsrfour A \dblContInOut \tnsrfour B &= A^{ijkl} B_{jmnk}\,\vctr g_i\otimes\vctr g^m\otimes\vctr g^n\otimes\vctr g_l \\ \tnsr C &= \tnsrfour A \dblContOutIn \tnsr B &= A^{ijkl} B_{il}\,\vctr g_j\otimes\vctr g_k \\ \tnsr C &= \tnsr A \dblContOutIn \tnsrfour B &= A_{jk} B^{ijkl}\,\vctr g_i\otimes\vctr g_l \\ \tnsrfour C &= \tnsrfour A \dblContOutIn \tnsrfour B &= A^{ijkl} B_{miln}\,\vctr g^m\otimes\vctr g_j\otimes\vctr g_k\otimes\vctr g^n \end{eqnarray*}

Fourth order identity tensors

\begin{align*} \tnsrfour I &= \tnsr I\otimes\tnsr I = \vctr g_i\otimes\vctr g^i\otimes\vctr g_j\otimes\vctr g^j \\ \tnsrfour I^\text R &= \tnsr I\boxtimes\tnsr I = \vctr g_i\otimes\vctr g^j\otimes\vctr g^i\otimes\vctr g_j \\ \tnsrfour I^\text L &= \tnsr I\odot\tnsr I = \vctr g_i\otimes\vctr g_j\otimes\vctr g^j\otimes\vctr g^i \end{align*}

Tensor derivatives

\begin{align*} \tnsr A,_{\scriptscriptstyle\tnsr B} &= \frac{\partial A_{ij}}{\partial B_{kl}}\,\vctr g^i\otimes\vctr g_k\otimes\vctr g_l\otimes\vctr g^j \end{align*} \begin{align*} \tnsr A,_{\scriptscriptstyle\tnsr B} &= \tnsr A,_{\scriptscriptstyle\tnsr C}\dblContInOut\tnsr C,_{\scriptscriptstyle\tnsr B} \\ (\tnsr A \tnsr B),_{\scriptscriptstyle\tnsr C} &= \tnsr A,_{\scriptscriptstyle\tnsr C}\tnsr B + \tnsr A\tnsr B,_{\scriptscriptstyle\tnsr C} \\ (\tnsr A : \tnsr B),_{\scriptscriptstyle\tnsr C} &= \tnsr A,_{\scriptscriptstyle\tnsr C}\dblContOutIn\tnsr B + \tnsr A\dblContInOut\tnsr B,_{\scriptscriptstyle\tnsr C} \\ (f \tnsr A),_{\scriptscriptstyle\tnsr C} &= \tnsr A\otimes f,_{\scriptscriptstyle\tnsr C} + f\tnsr A,_{\scriptscriptstyle\tnsr C} \end{align*}

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References

[1]

O. Kintzel & Y. Başar
Fourth-order tensors – tensor differentiation with applications to continuum mechanics. Part I: Classical tensor analysis
Z. angew. Math. Mech. 86 (2006) 291–311