I am currently working on a subject that involves a lot of 4th order tensors computations including double dot product and inverse of fourth order tensors.
First the definitions so that we are on the same page. What I call the double dot product is :
$$ (A:B)_{ijkl} = A_{ijmn}B_{mnkl} $$
and for the double dot product between a fourth order tensor and a second order tensor :
$$ (A:s)_{ij} = A_{ijkl}s_{kl}$$
Using the convention of sommation over repeating indices.
What I call the identity of the fourth order tensors is the only tensor such that :
$$ A:I = I:A = A $$
it is defined by $ I = \delta_{ik}\delta_{jl} e_{i} \otimes e_{j} \otimes e_{k} \otimes e_{l} $.
What I call the inverse of a fourth order tensor is the inverse with respect to the double dot product, that is, the inverse of $A$ is the only tensor $B$ such that $AB = BA = I$.
The double dot product is easy to compute if you don't think about the efficiency of the code, just create an array and loop over the four indices. Computing the inverse is something else. Every tensor I use has the minor symmetries $A_{ijkl} = A_{jikl} = A_{ijlk}$ so I thought I would use the Mandel representation for second order and fourth order tensors mentioned on Wikipedia. The fourth order tensor can be put into a $6 \times6$ matrix with the following components :
$$ [C] =
\begin{bmatrix}
C_{1111} & C_{1122} & C_{1133} & \sqrt{2}C_{1123} & \sqrt{2}C_{1131} & \sqrt{2}C_{1112}\\
C_{2211} & C_{2222} & C_{2233} & \sqrt{2}C_{2223} & \sqrt{2}C_{2231} & \sqrt{2}C_{2212}\\
C_{3311} & C_{3322} & C_{3333} & \sqrt{2}C_{3323} & \sqrt{2}C_{3331} & \sqrt{2}C_{3312}\\
\sqrt{2}C_{2311} & \sqrt{2}C_{2322} & \sqrt{2}C_{2333} & 2C_{2323} & 2C_{2331} & 2C_{2312}\\
\sqrt{2}C_{3111} & \sqrt{2}C_{3122} & \sqrt{2}C_{3133} & 2C_{3123} & 2C_{3131} & 2C_{3112}\\
\sqrt{2}C_{1211} & \sqrt{2}C_{1222} & \sqrt{2}C_{1233} & 2C_{1223} &2C_{1231} & 2C_{1212}
\end{bmatrix}
$$
$C$ is a fourth order tensor with minor symmetries and $[C]$ is its Mandel representation. The reason why Mandel's representation exists according to different sources is such that the matrix-matrix and matrix-vector usual products coincide with the fourth order tensors double dot product and the inverse in each respective space (fourth order tensors and $6\times 6$ matrices) coincides as well, that is
$$
[A:B] = [A] \cdot [B] \qquad \qquad (1)
$$
and
$$
[A^{-1}] = [A]^{-1} \qquad \qquad (2)
$$
where $ \cdot $ is the usual matrix-matrix product. But it doesn't work or at least there must be something I don't understand. If I put the identity 4th order tensor defined above into Mandel's notation, I get the following matrix :
$$ [I] =
\begin{bmatrix}
1&0&0&0&0&0\\
0&1&0&0&0&0\\
0&0&1&0&0&0\\
0&0&0&2&0&0\\
0&0&0&0&2&0\\
0&0&0&0&0&2
\end{bmatrix}
$$
which is obviously different from the identity of $6 \times 6$ matrices so if I compute $[C] \cdot [I]$ using the usual matrix-matrix product I won't get the same $[C]$.
I also wrote a little script to check relations (1) and (2) but wasn't able to find this result with random $4^{th}$ order tensors possessing minor symmetries.
What am I missing here ?
Thanks a lot for your help and the discussions to come 🙂
Best Answer
I'll answer my own question since I was able to find the solution to my problem with the help of one commentator. The definition of the identity tensor $I = \delta_{ik}\delta_{jl} e_{i} \otimes e_{j} \otimes e_{k} \otimes e_{l}$ is correct but it does not lead to a tensor with minor symmetries.
My mistake was in using this definition of the identity and applying the Mandel transformation to it. The Mandel transformation preserves the double dot product and the inverse if and only if the transformed tensors have the minor symmetries. As suggested in the review by Helnwein mentioned by @user3658307 whom I thank for his help, the definition one should use for the identity tensor in the case of minor symmetries is :
$$ I = \frac{1}{2}( \delta_{ik}\delta_{jl} + \delta_{il}\delta_{jk})e_{i} \otimes e_{j} \otimes e_{k} \otimes e_{l} $$
Which has the minor symmetries and can thus be put into the Mandel representation that yields :
$$ [I] = \begin{bmatrix} 1 & 0&0&0&0&0 \\ 0&1&0&0&0&0\\0&0&1&0&0&0\\0&0&0&1&0&0\\0&0&0&0&1&0\\0&0&0&0&0&1\end{bmatrix} $$
If you are interested in a more rigorous presentation of this subject (matrix representation of tensors), I highly recommend reading the aforementioned review. In particular, one should be really careful of the covariance and contravariance of the tensors he or she is handling as in some particular cases, their matrix representation can differ even if $A^{ij}_{kl} = A_{ij}^{kl}$.
Note on the inverse :
My original goal was to find an easy way to inverse fourth order tensors with minor symmetries using usual inversion algorithms for matrices. It is not always possible in the general case since the matrix representation of a general fourth order tensor possessing only minor symmetries is not always invertible in the space of $6\times 6$ matrices. However it is known that given a random $n \times n$ matrix with real coordinates, the 'probability' of it being invertible is $1$ in the sense of the Lebesgue measure. In linear elasticity or physics in general, one should thus be able to compute an inverse in every case.