How to Simplify a Double Dot Product of Tensors?

[nikphd]

Hello,

Been a long time lurker, but first time poster. I hope I can be very thorough and descriptive. So, I have been battling with a double inner product of a 2nd order tensor with a 4th order one. So my question is:

How do we expand (using tensor properties) a double dot product of the basis vectors to a simpler one?

$$(e_ie_je_ke_l):(e_me_n)=?$$

and

$$(e_ie_j):(e_ke_le_me_n)=?$$

Thanks a lot!

 

[Geometry_dude]

Are you missing the definitions or what exactly is the problem?
I haven't seen this being used before.

 

[nikphd]

What exactly haven't you seen been used before? The double dot product of a tensor of n=4 with one of n=2? You mean you have only seen it being used for tensors of equal order?

The properties I am referring to, is actually expanding the double dot product to two single dot products!

 

[Geometry_dude]

The "double dot product". What I am saying is that you will rarely find this being used in modern differential geometry, for the plain reason that we have tensor products, contractions, etc. . I believe that this is the reason why you are not getting any responses. Moreover, this is a linear algebra question, not a geometry one.

Either way, if you provide the definitions in terms of said concepts and show how far you got, I am sure people will be able to help you.

 

[nikphd]

Double dot is used extensively in continuum mechanics, even in 2014! For example, 4th order tensors represent orientation of rigid fibers in a 3D space, and 2nd order tensor is the velocity gradient of a flow field.
So should I move my question to the linear algebra section?
My basic question is this actually: "Is the following statement correct? D:uuuu=(D:uu)uu, meaning can i represent a 4th order tensor as a dyad of two 2nd order tensors?And if yes which are the requirements? Symmetry?" I thought it was a pretty straightforward question!

 

[Bteranto]

The product contracts the order of the 4th order tensor to a 2nd order tensor. i.e.
Aijkl ei ej ek el : Bmn em en = Aijkl Bmn ei ej dkm dln = Aijkl Bkl ei ej