# How should I calculate fourth order tensor times second order tensor?

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.

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 $$\max\{r,\,s\}>2$$.