Double dot product vs double inner product Anything involving tensors has 47 different names and notations, and I am having trouble getting any consistency out of it.
This document (http://www.polymerprocessing.com/notes/root92a.pdf) clearly ascribes to the colon symbol (as "double dot product"):
$\mathbf{T}:\mathbf{U}=T_{ij} U_{ji}$
while this document (http://www.foamcfd.org/Nabla/guides/ProgrammersGuidese3.html) clearly ascribes to the colon symbol (as "double inner product"):
$\mathbf{T}:\mathbf{U}=T_{ij} U_{ij}$
Same symbol, two different definitions.  To make matters worse, my textbook has:
$\mathbf{\epsilon}:\mathbf{T}$
where $\epsilon$ is the Levi-Civita symbol $\epsilon_{ijk}$ so who knows what that expression is supposed to represent.
Sorry for the rant/crankiness, but it's late, and I'm trying to study for a test which is apparently full of contradictions.  Any help is greatly appreciated.

I know this might not serve your question as it is very late, but I myself am struggling with this as part of a continuum mechanics graduate course. The way I want to think about this is to compare it to a 'single dot product.' For example:
\begin{align}
\textbf{A} \cdot \textbf{B} &= A_{ij}B_{kl} (e_i \otimes e_j) \cdot (e_k \otimes e_l)\\
&= A_{ij} B_{kl} (e_j \cdot e_k) (e_i \otimes e_l) \\
&= A_{ij} B_{kl} \delta_{jk} (e_i \otimes e_l) \\
&= A_{ij} B_{jl} (e_i \otimes e_l)
\end{align}
Where the dot product occurs between the basis vectors closest to the dot product operator, i.e. $e_j \cdot e_k$. So now $\mathbf{A} : \mathbf{B}$ would be as following:
\begin{align}
\textbf{A} : \textbf{B} &= A_{ij}B_{kl} (e_i \otimes e_j):(e_k \otimes e_l)\\
&= A_{ij} B_{kl} (e_j \cdot e_k) (e_i \cdot e_l) \\
&= A_{ij} B_{kl} \delta_{jk} \delta_{il} \\
&= A_{ij} B_{jl} \delta_{il}\\
&= A_{ij} B_{ji} 
\end{align}
But I found that a few textbooks give the following result:
$$ \textbf{A}:\textbf{B} = A_{ij}B_{ij}$$
But based on the operation carried out before, this is actually the result of $$\textbf{A}:\textbf{B}^t$$ because
\begin{align}
\textbf{A} : \textbf{B}^t &= A_{ij}B_{kl} (e_i \otimes e_j):(e_l \otimes e_k)\\
&= A_{ij} B_{kl} (e_j \cdot e_l) (e_j \cdot e_k) \\
&= A_{ij} B_{kl} \delta_{jl} \delta_{ik} \\
&= A_{ij} B_{il} \delta_{jl}\\
&= A_{ij} B_{ij} 
\end{align}
But I finally found why this is not the case!
The definition of tensor contraction is not the way the operation above was carried out, rather it is as following:
\begin{align}
\textbf{A} : \textbf{B}^t &= \textbf{tr}(\textbf{AB}^t)\\
&= \textbf{tr}(\textbf{BA}^t)\\
&= \textbf{tr}(\textbf{A}^t\textbf{B})\\
&= \textbf{tr}(\textbf{B}^t\textbf{A}) = \textbf{A} : \textbf{B}^t\\
\end{align}
and if you do the exercise, you'll find that:
$$\textbf{A}:\textbf{B} = A_{ij} B_{ij} $$
