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Let's say I have two second-order tensors ${\mathbf{S}} = {S_{ij}}{{\mathbf{e}}_i} \otimes {{\mathbf{e}}_j}$ and ${\mathbf{T}} = {T_{ij}}{{\mathbf{e}}_i} \otimes {{\mathbf{e}}_j}$ . Then, I know ${\mathbf{ST}} = \left( {{S_{ij}}{{\mathbf{e}}_i} \otimes {{\mathbf{e}}_j}} \right)\left( {{T_{kl}}{{\mathbf{e}}_k} \otimes {{\mathbf{e}}_l}} \right) = {S_{ij}}{T_{kl}}{\delta _{jk}}{{\mathbf{e}}_i} \otimes {{\mathbf{e}}_l} = {S_{ij}}{T_{jl}}{{\mathbf{e}}_i} \otimes {{\mathbf{e}}_l}$ .

Using the following property

$$\begin{gathered} \left( {\left( {{\mathbf{a}} \otimes {\mathbf{b}}} \right)\left( {{\mathbf{c}} \otimes {\mathbf{d}}} \right)} \right){\mathbf{v}} = \left( {{\mathbf{a}} \otimes {\mathbf{b}}} \right)\left( {\left( {{\mathbf{c}} \otimes {\mathbf{d}}} \right){\mathbf{v}}} \right) \hfill \\ = \left( {{\mathbf{a}} \otimes {\mathbf{b}}} \right){\mathbf{c}}\left( {{\mathbf{d}} \cdot {\mathbf{v}}} \right) = {\mathbf{a}}\left( {{\mathbf{b}} \cdot {\mathbf{c}}} \right)\left( {{\mathbf{d}} \cdot {\mathbf{v}}} \right) = \left( {{\mathbf{b}} \cdot {\mathbf{c}}} \right){\mathbf{a}}\left( {{\mathbf{d}} \cdot {\mathbf{v}}} \right) = \left( {{\mathbf{b}} \cdot {\mathbf{c}}} \right)\left( {{\mathbf{a}} \otimes {\mathbf{d}}} \right){\mathbf{v}} \hfill \\ \end{gathered} $$

Then, how do i calculate forth order tensor times second order tensor like

$\underline{\underline {\mathbf{A}}} = {A_{ijkl}}{{\mathbf{e}}_i} \otimes {{\mathbf{e}}_j} \otimes {{\mathbf{e}}_k} \otimes {{\mathbf{e}}_l}$

and

${\mathbf{B}} = {B_{ij}}{{\mathbf{e}}_i} \otimes {{\mathbf{e}}_j}$

$\underline{\underline {\mathbf{A}}} {\mathbf{B}} = {A_{ijkl}}\left( {{{\mathbf{e}}_i} \otimes {{\mathbf{e}}_j} \otimes {{\mathbf{e}}_k} \otimes {{\mathbf{e}}_l}} \right){B_{pq}}\left( {{{\mathbf{e}}_p} \otimes {{\mathbf{e}}_q}} \right) = ??$

By the way, what do I call the operation between ${\mathbf{S}}$ and ${\mathbf{T}}$ ? just multiplication? Usually operator has name in continuum mechacnis like 'dot product', 'double dot product' and so on. But, I have no idea how to call it when they omit a operator like this case.

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1 Answer 1

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The equivalent of what you did with $S,\,T$ is to cancel two inner $\mathbf{e}$s viz.$$A_{ijkl}B_{pq}\delta_{lp}\mathbf{e}_i\otimes\mathbf{e}_j\otimes\mathbf{e}_k\otimes\mathbf{e}_q=A_{ijkl}B_{lq}\mathbf{e}_i\otimes\mathbf{e}_j\otimes\mathbf{e}_k\otimes\mathbf{e}_q.$$It's hard to have terminology for every way to multiply tensors. In general, a rank $r$ tensor and a rank $s$ tensor can be multiplied with the contraction of anywhere from $0$ to $\min\{r,\,s\}$ index pairs, and when several pairs are cancelled there will be many ways to choose them. For example, your original calculation took $r=s=2$ and contracted one pair of indices, but there's a special name for a way to contract two, and it generalizes to cases with $\max\{r,\,s\}>2$.

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