Double dot contraction of 4 order tensors

[Paul_S]

Hello,

could anybody help me please how to calculate the following expression:

P : C : Pt

where Pt is transpose(P) and P, C are tensors of order four.

I know all the components of P, C, Pt but do not know how to calculate the double dot contraction operation between these tensors. In literature I have found contraction operation between tensors of order 2 and 2 or of order 4 and 2, but have not found any description of this operation between tensors of order 4 and 4.
Any advice or literature reference should be helpfull for me - thank you for that.

Paul

[Jorgen]

Hello Paul,

By dot contraction between two 4th order tensors I assume you mean an operation that results in another 4th order tensor.

This is the definition of such an opertion in index notation:

[A : B]_ijkl = A_ijmn B_mnkl

In this equation m and n are dummy indices that should be summed over following Einsteins summation convention.

- Jorgen

[Paul_S]

Hello Jorgen,

thank you for your reply. The definition in index notation is exactly what I needed - and it really works.

Paul